

small (250x250 max)
medium (500x500 max)
Large
Extra Large
large ( > 500x500)
Full Resolution


NEW IDEAS IN HIGGS PHYSICS By STEVEN GABRIEL Bachelor of Science in Physics Georgia State University Atlanta, Georgia, United States 2002 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful¯llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY May, 2010 COPYRIGHT °c By STEVEN GABRIEL May, 2010 NEW IDEAS IN HIGGS PHYSICS Dissertation Approved: S. Nandi Dissertation Advisor K.S. Babu F. Rizatdinova B. Binegar A. Gordon Emslie Dean of the Graduate College iii TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 2 Unfaithful Representations of Finite Groups and Tribimaximal Neu trino Mixing 7 2.1 The Discrete Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Derivation of Representations . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Invariants Under the Discrete Symmetry . . . . . . . . . . . . . . . . 33 2.5 Calculation of the Neutrino Mass Matrix . . . . . . . . . . . . . . . . 38 2.6 Calculation of the Charged Lepton Mass Matrix . . . . . . . . . . . . 42 3 A 6D Higgsless Standard Model 45 3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 E®ective theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Relation to EWPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 Nonoblique corrections and fermion masses . . . . . . . . . . . . . . 55 3.4.1 Improving the calculability . . . . . . . . . . . . . . . . . . . . 57 4 A New Two Higgs Doublet Model 61 4.1 Model and the Formalism . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Phenomenological Implications . . . . . . . . . . . . . . . . . . . . . 63 4.3 Cosmological Implications . . . . . . . . . . . . . . . . . . . . . . . . 68 iv 5 CONCLUSIONS 71 BIBLIOGRAPHY 73 v LIST OF TABLES Table Page 2.1 This table shows the matrices representing the generators in each irrep. of A4, in a certain basis. . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 This table shows the assignments of the fermions and Higgs ¯elds under SU(2)L £ U(1)Y £ [(S4 3 £ Z3 2 ) o A4] . . . . . . . . . . . . . . . . . . . 11 vi LIST OF FIGURES Figure Page 3.1 Symmetry breaking of SU(2)L £ U(1)Y on the rectangle. At one boundary y1 = ¼R1, SU(2)L is broken to U(1)I3 while on the boundary y2 = ¼R2 the subgroup U(1)I3 £ U(1)Y is broken to U(1)Q, which leaves only U(1)Q unbroken on the entire rectangle. Locally, at the ¯xed point (0; 0), SU(2)L£ U(1)Y remains unbroken. The dashed arrows indicate the propagation of the lowest resonances of the gauge bosons. . . . . . . . . . . . . . . . . . 48 3.2 E®ect of the brane kinetic terms L0 on the KK spectrum of the gauge bosons (for the example of W§). Solid lines represent massive excitations, the bottom dotted lines would correspond to the zero modes which have been removed by the BC's. Without the brane terms (a), the lowest KK excitations are of order 1=R ' 1 TeV . After switching on the dominant brane kinetic terms (b), the zero modes are approximately \restored" with a small mass mW ¿ 1=R (dashed line), while the higher KKlevels receive small corrections to their masses (thin solid lines) and decouple below » 1 TeV . 52 3.3 Oneloop diagram for ÁÁ scattering on S1=Z2. The total incoming momen tum is (p; p0 5) and the total outgoing momentum is (p; p5). Generally, it is possible that jp0 5j 6= jp5j, since the orbifold ¯xed points break 5D transla tional invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1 Left panel: Branching ratio for h ! ¾¾ as a function of mh for the value of the parameter, ¸¤ = 0:1. Right panel: Branching ratio for h ! ¾¾ as a function of ¸¤ for mh = 135 GeV . . . . . . . . . . . . . . 67 vii CHAPTER 1 INTRODUCTION The Standard Model (SM) of electroweak interactions [1], based on the gauge symmetry group SU(2)L £ U(1)Y , provides a highly successful description of elec troweak precision tests (EWPT) [2, 3]. One fundamental ingredient of the SM is the Higgs mechanism [4], which accomplishes electroweak symmetry breaking (EWSB) and at high energies unitarizes massive W§ and Z scattering through the presence of the scalar Higgs doublet [5]. Although the mass of the Higgs boson is not predicted by the SM, accurate measurements of the top quark and the W boson mass at the Tevatron, as well as the Z boson mass at LEP, have narrowed the SM Higgs boson mass between 80 and 200 GeV [3]. Failure to observe the SM Higgs boson at LEP2 has also placed a direct lower bound of 114 GeV on its mass [6]. The dominant decay modes of the SM Higgs boson are to bb, WW, ZZ or tt, depending on its mass. Extensions of the SM may avoid constraints on the Higgs mass, and may allow Higgs bosons with masses less than the above limits. The dominant decay modes of the Higgs bosons can also be altered in such extensions, thus transforming the discovery signals for the Higgs bosons at the Large Hadron Collider (LHC). However, there is as yet no direct evidence of the Higgs boson, so that the details of the Higgs sector, if it even exists, remain a mystery. Thus, it is important to explore alternative Higgs sector scenarios. One interesting scenario involves the role of the Higgs sector in neutrino mixing. The existence of neutrino masses is now well established experimentally [7, 8]. At 1¾, 1 the masssquared di®erences and mixing angles are [7]: ¢m2 21 = 7:65(+0:23= ¡ 0:20) £ 10¡5 eV 2; ¢jm2 31j = 2:40(+0:13= ¡ 0:11) £ 10¡3 eV 2; and sin2 µ23 = 0:50(+0:022= ¡ 0:016); sin2 µ12 = 0:341(+0:07= ¡ 0:06); sin2 µ13 < 0:035: These values are in good agreement with a tribimaximal mixing pattern given by the mixing matrix [9, 10] UMNS = 0 BBBB@ q 2 3 p1 3 0 ¡p1 6 p1 3 ¡p1 2 ¡p1 6 p1 3 p1 2 1 CCCCA P (1.1) where P is a diagonal phase matrix. This corresponds to sin2 µ23 = 1=2; sin2 µ12 = 1=3; sin2 µ13 = 0: It has long been known that such a mixing pattern can be obtained using a ¯nite family symmetry [1025] such as A4 [1925]. In these models, A4 is broken to a Z2 subgroup in the neutrino sector by a triplet Higgs, with the VEV structure (0; 1; 0) or some permutation thereof, and to a Z3 subgroup in the charged lepton sector by a triplet Higgs, with the VEV structure (1; 1; 1). However, there is a serious technical problem with this, in that couplings between the Higgs ¯elds responsible for the symmetry breaking will force the VEV's to align, upsetting the desired breaking pattern [2125]. To overcome this problem, one can introduce more complicated symmetries. In Section 2, we consider models where the SM lepton families belong to representations of the ¯nite symmetry which are not faithful (that is, not every member of the group is represented by a distinct transformation). In e®ect, the 2 Higgs sector knows about the full symmetry while the lepton sector does not. We consider a renormalizable nonsupersymmetric gauge theory with an additional ¯nite symmetry that has the semidirect product structure G = (G1 £ G2) o A4, with G1 = S3£S3£S3£S3 and G2 = Z2£Z2£Z2. A symmetry thus structured will contain G1, G2, and G1 £ G2 as invariant subgroups, so that G will have representations corresponding to the homomorphisms G=(G1 £ G2) » A4, G=G1 » G2 o A4, and G=G2 » G1oA4. SM leptons can then be assigned to representations of A4. Neutrino masses are generated by a Higgs ¯eld Á, belonging to a 16dimensional representation of G1 o A4, while chargedlepton masses are generated by a Higgs ¯eld Â, belonging to a 6dimensional representation of G2oA4. The additional symmetries, G1 and G2, prevent quadratic and cubic interactions between Á and Â and allow only a trivial quartic interaction (i.e., the interaction is the product of quadratic invariants) that does not cause an alignment problem. In this way, the alignment problem is addressed without altering the desired properties of the family symmetry, so that neutrino mixing can be explained using only symmetries which are broken spontaneously by the Higgs mechanism. However, no fundamental scalar particle has been observed yet in nature, and as long as there is no direct evidence for the existence of the Higgs boson, the actual mechanism of EWSB remains a mystery. In case the Higgs boson will also not be found at the Tevatron or the LHC, it will therefore be necessary to consider alternative ways to achieve EWSB without a Higgs. We explore this possibility in Section 3. It is well known, that in extra dimensions, gauge symmetries can also be broken by boundary conditions (BC's) on a compact space [27]. Here, a geometric "Higgs" mechanism ensures treelevel unitarity of longitudinal gauge boson scattering through a tower of KaluzaKlein (KK) [28] excitations [29]. The original model for Higgsless EWSB [30] is an SU(2)L£SU(2)R£U(1)B¡L gauge theory compacti¯ed on an interval [0; ¼R] in ¯vedimensional (5D) °at space. At one end of the interval, SU(2)R £ 3 U(1)B¡L is broken to U(1)Y . At the other end, SU(2)L £ SU(2)R is broken to the diagonal subgroup SU(2)D, thereby leaving only U(1)Q of electromagnetism unbroken in the e®ective fourdimensional (4D) theory. Although this model exhibited some similarities with the SM, the ½ parameter deviated from unity by » 10% and the lowest KK excitations of theW§ and Z were too light (» 240 GeV ) to be in agreement with experiment. These problems have later been resolved by considering the setup in warped space [33]. Based on the same gauge group, similar e®ects can be realized in 5D °at space [32], when 4D brane kinetic terms [34{36] dominate the contribution from the bulk. In 5D Higgsless models, a ½ parameter close to unity is achieved at the expense of enlarging the SM gauge group by an additional gauge group SU(2)R, which introduces a gauged custodial symmetry in the bulk. However, it is possible to obtain consistent 6D Higgsless models of EWSB, which are based only on the SM gauge group SU(2)L £ U(1)Y and allow the ½ parameter to be set equal to unity. We consider a Higgsless model for EWSB in six dimensions, which is based only on the SM gauge group SU(2)L £ U(1)Y , where the gauge bosons propagate in the bulk. The model is formulated in °at space with the two extra dimensions compacti¯ed on a rectangle and EWSB is achieved by imposing consistent BC's. The higher KK resonances of W§ and Z decouple below » 1TeV through the presence of a dominant 4D brane induced gauge kinetic term. The ½ parameter is arbitrary and can be set exactly to one by an appropriate choice of the bulk gauge couplings and compacti¯cation scales. Unlike in the 5D theory, the mass scale of the lightest gauge bosons W and Z is solely set by the dimensionful bulk couplings, which (upon compacti¯cation via mixed BC's) are responsible for EWSB.We calculate the treelevel oblique corrections to the S; T; and U parameters and ¯nd that they are in better agreement with data than in proposed 5D warped and °at Higgsless models. In Section 4, we present a model that includes a second Higgs doublet that pro vides an alternate explanation for the tiny masses of the SM neutrinos, as well as 4 possibilities for altering signals for discovery of the Higgs at the LHC. Our proposal is to extend the SM electroweak symmetry to SU(2)L £ U(1) £ Z2 and introduce three SU(2)£U(1) singlet right handed neutrinos, NR, as well as an additional Higgs doublet, Á. While the SM symmetry is spontaneously broken by the VEV of an EW doublet Â at the 100 GeV scale, the discrete symmetry Z2 is spontaneously broken by the tiny VEV of this additional doublet Á at a scale of 10¡2 GeV . Thus in our model, tiny neutrino masses are related to this Z2 breaking scale. We note that although our model has extreme ¯ne tuning, that is no worse than the ¯ne tuning problem in the usual GUT model. Many versions of the two Higgs doublet model have been exten sively studied in the past [37]. The examples include: a) a supersymmetric two Higgs doublet model, b) nonsupersymmetric two Higgs doublet models i) in which both Higgs doublets have vacuum expectation values (VEV's) with one doublet coupling to the up type quarks only, while the other coupling to the down type quarks only, ii) only one doublet coupling to the fermions, and iii) only one doublet having VEV's and coupling to the fermions [38]. What is new in our model is that one doublet couples to all the SM fermions except the neutrinos, and has a VEV which is same as the SM VEV, while the other Higgs doublet couples only to the neutrinos with a tiny VEV » 10¡2 eV . This latter involves the Yukawa coupling of the lefthanded SM neutrinos with a singlet righthanded neutrino, NR. The lefthanded SM neutrinos combine with the singlet righthanded neutrinos to make massive Dirac neutrinos. The neutrino mass is so tiny because of the tiny VEV of the second Higgs doublet, which is responsible for the spontaneous breaking of the discrete symmetry, Z2. Note that in the neutrino sector, our model is very distinct from the seasaw model. Lepton number is strictly conserved, and hence no NRNR mass terms are allowed. Thus the neutrino is a Dirac particle, and there is no neutrinoless double ¯ decay in our model. In the Higgs sector, in addition to the usual massive neutral scalar and pseudoscalar Higgs, and two charged Higgs, our model contains one essentially massless scalar 5 Higgs. We will show that this is still allowed by the current experimental data and can lead to an invisible decay mode of the SMlike Higgs boson, thus complicating the Higgs searches at the Tevatron and the LHC. 6 CHAPTER 2 Unfaithful Representations of Finite Groups and Tribimaximal Neutrino Mixing 2.1 The Discrete Symmetry As described in the Introduction, we consider a renormalizable nonsupersymmetric gauge theory with an additional ¯nite symmetry given by the semidirect product1 G = (G1 £G2)oA4, with G1 = S3 £S3 £S3 £S3 and G2 = Z2 £Z2 £Z2. The group A4 can be described using two generators obeying the relations, X2 = Y 3 = E; XY X = Y 2XY 2; (2.1) where E is the identity. The irreducible representations are one real singlet, 1; two complex singlets, 10 and 100; and one real triplet, 3. Table 1 gives X and Y in each of these representations for a certain choice of basis. The S3 generators, Ai and Bi, and the Z2 generators, Ci, obey A3i = B2 i = E; BiAiB¡1 i = A¡1 i ; C2 i = E; (2.2) and Ci commutes with Ai and Bi. The remaining relations de¯ning the full symmetry are XA1X¡1 = A2; XA2X¡1 = A1; XA3X¡1 = A4; XA4X¡1 = A3; 1The semidirect product, N oH, contains N and H as subgroups and obeys hnh¡1 2 N for all n 2 N and h 2 H [39]. Thus, N is an invariant subgroup. The number of elements in the group, denoted by jN o Hj, is jNjjHj. The semidirect product exists when H has a factor group which is a subgroup of the automorphism group of N. 7 X Y 1 1 1 10 1 ! 100 1 !2 3 0 BBBB@ ¡1 0 0 0 1 0 0 0 ¡1 1 CCCCA 0 BBBB@ 0 0 1 1 0 0 0 1 0 1 CCCCA ! = e2i¼=3 Table 2.1: This table shows the matrices representing the generators in each irrep. of A4, in a certain basis. XB1X¡1 = B2; XB2X¡1 = B1; XB3X¡1 = B4; XB4X¡1 = B3; (2.3) Y A1Y ¡1 = A1; Y A2Y ¡1 = A3; Y A3Y ¡1 = A4; Y A4Y ¡1 = A2; Y B1Y ¡1 = B1; Y B2Y ¡1 = B3; Y B3Y ¡1 = B4; Y B4Y ¡1 = B2; (2.4) XC1X¡1 = C1C2C2; XC2X¡1 = C3; XC3X¡1 = C2; Y C1Y ¡1 = C2; Y C2Y ¡1 = C3; Y C3Y ¡1 = C1: (2.5) It's easy to see that if C1, C2, and C3 are all represented by the identity matrix, then (6) is trivially satis¯ed. So in this case, one need only ¯nd representations that respect Eqs. (2)(5). But this is equivalent to ¯nding representations of G1oA4. The representations of this type that we will be using are a real 16dimensional represen tation, a real 48dimensional representation, and a real 8dimensional representation. These will be referred to hereafter as 16AB, 48AB, and 8AB. The matrices representing the remaining generators in each of these representations can be found in Section 2.3 8 below. Similarly, if A1, A2, A3, A4, B1, B2, B3, and B4 are all represented by the identity matrix, then (4) and (5) are trivially satis¯ed. Finding these representations corresponds to ¯nding representations of G2 o A4. For this type, we will be using a real 6dimensional representation, which we will call 6C. The matrices representing the remaining generators in this representation can also be found in Section 2.3. Fi nally, if the Ai's, Bi's, and Ci's are all represented by the identity matrix, then the only nontrivial relation is (2), corresponding to the representations of A4 given in Table 1. These representations will be used for SM leptons. 2.2 The Model The SM lepton assignments under A4 are eR1 » 1; eR2 » 10; eR3 » 100; (L1;L2;L3) » 3: (2.6) The ¯nite symmetry is broken at a scale M¤, which is large compared to the weak scale, by two Higgs ¯elds, Á and Â. Neutrino Dirac masses are generated by the real Higgs ¯eld Á belonging to 16AB, while charged lepton masses are generated by the real Higgs ¯eld Â belonging to 6C. Symmetryinvariant interactions between Á and Â must consist of products of G1 invariants constructed from Á with G2 invariants constructed from Â. The 16dimensional representation to which Á belongs is (2; 2; 2; 2) with respect to G1 = S3 £ S3 £ S3 £ S3, so that there is only one quadratic G1 invariant that can be constructed with Á, which is invariant under the full symmetry. Thus, there are no cubic invariants involving both Á and Â, and the only quartic invariant containing both is a trivial product of quadratic invariants, which does not generate a VEV alignment problem. Then the potential of Á and Â has the form VÁÂ = a1f1(Á; Á) + a2f2(Â; Â) + b1g1(Á; Á; Á) + b2g2(Â; Â; Â) + c1h1(Á) + c2h2(Á) +c3h3(Â) + c4h4(Â) + c5h5(Â) + c6h6(Â) + c7f1(Á; Á)f2(Â; Â); (2.7) 9 where the functions f1, f2, g1, g2, h1, h2, h3, h4, h5, and h6 are given in Section 2.4 below. The neutrino masses are generated from Á by integrating out multiplets of heavy righthanded neutrinos, with masses at a scale M¤ which is large compared to the EW scale. These multiplets are N » 3, N0 » 48AB, and N00 » 8AB. If the Z2 subgroup of A4 generated by X is left unbroken by the VEV of Á (along with an additional accidental Z2 that is actually part of S4, see [41]), the light neutrino mass matrix is forced to have the form Mº = 0 BBBB@ aº 0 cº 0 bº 0 cº 0 aº 1 CCCCA : (2.8) This matrix is diagonalized by Uº = 1 p 2 0 BBBB@ 1 0 ¡1 0 p 2 0 1 0 1 1 CCCCA Pº; (2.9) where diagonal Pº is a phase matrix. The charged lepton masses are generated from Â by integrating out multiplets of heavy vectorlike fermions, whose masses are also at the high scale M¤, with the same gauge quantum numbers as righthanded charged leptons. These are EL;R » 3 and E0 L;R » 6C. If the Z3 subgroup of A4 generated by Y is left unbroken by the VEV of Â, the light lefthanded charged lepton mass matrix is forced to have the form My eMe = 1 p 3 0 BBBB@ 1 1 1 1 ! !2 1 !2 ! 1 CCCCA 0 BBBB@ ae 0 0 0 be 0 0 0 ce 1 CCCCA 1 p 3 0 BBBB@ 1 1 1 1 !2 ! 1 ! !2 1 CCCCA = UL 0 BBBB@ ae 0 0 0 be 0 0 0 ce 1 CCCCA Uy L (2.10) 10 SU(2)L U(1)Y (S4 3 £ Z3 2 ) o A4 L 2 1/2 3 eR1 1 1 1 eR2 1 1 10 eR3 1 1 100 N 1 0 3 N0 1 0 48AB N00 1 0 8AB EL 1 1 3 ER 1 1 3 E0L 1 1 6C E0R 1 1 6C Á 1 0 16AB Â 1 0 6C H 2 1/2 1 Table 2.2: This table shows the assignments of the fermions and Higgs ¯elds under SU(2)L £ U(1)Y £ [(S4 3 £ Z3 2 ) o A4] Eqs. (10) and (11) then give the desired form (1) for the mixing matrix UMNS = UT L U¤ º . The symmetry assignments of the fermions and Higgs ¯elds in the model are summarized in Table 2. From the matrices given in Section 2.3, it can be seen that the most general VEV structure for Â that leaves the Z3 subgroup of A4 generated by Y unbroken is hÂi = (vÂ1; vÂ2; vÂ1; vÂ2; vÂ1; vÂ2): (2.11) Upon minimizing the potential, one ¯nds that vÂ2 = 0, vÂ1 6= 0 is allowed. Here, C1C2C3 is left unbroken in addition to Y . Since the SM leptons do not transform un der the Ci's, these additional symmetries do not a®ect the light lepton mass matrices. 11 So the desired minimum is hÂi = (vÂ; 0; vÂ; 0; vÂ; 0): (2.12) Since C1C2C3, and Y commute, the subgroup they generate is Z2 £ Z3. Of course, Â also trivially leaves all Ai's and Bi's unbroken. The most general VEV structure for Á that leaves the Z2 subgroup of A4 generated by X unbroken is hÁi = (vÁ1; vÁ2; vÁ2; vÁ3; vÁ4; vÁ5; vÁ6; vÁ7; vÁ4; vÁ6; vÁ5; vÁ7; vÁ9; vÁ9; vÁ9; vÁ10): (2.13) Upon minimizing the potential, we ¯nd that hÁi = (0; 0; 0; 0; vÁ; vÁ; vÁ; vÁ; vÁ; vÁ; vÁ; vÁ; 0; 0; 0; 0) (2.14) is acceptable. In addition to X, this VEV leaves the generators B1, B2, B3B4, and A3A4 unbroken. These form the subgroup D4 £ S3, with D4 generated by B1, B2, and X and with S3 generated by A3A4 and B3B4. Of course, Á also leaves all Ci's unbroken. (To leave the accidental Z2 ½ S4 mentioned above unbroken requires vÁ5 = vÁ6 in (14), which is satis¯ed in (15).) From (8), we ¯nd that vÁ and vÂ in (13) and (15) must be solutions to 2a1 + 3b1vÁ + 2(c1 + c2)v2Á + 6c7v2Â = 0; 2a2 + 3b2vÂ + 4(c3 + c5)v2Â + 16c7v2Á = 0: Neutrino Dirac masses are generated through Lº = ¸(L1N1 + L2N2 + L3N3)eH + mN(N2 1 + N2 2 + N2 3 ) + m0 Nf3(N0;N0) + m00 Nf4(N00;N00) +®1g3(N; Á;N0) + ®2g4(N00; Á;N0) + ¯g5(Á;N0;N0); (2.15) where the functions f3, f4, g3, g4, and g5 are given in Section 2.4. N » 3 is required because the SM Higgs H only breaks EW symmetry, so that it can only cause left handed neutrinos to mix with a triplet. Since 3 £ 16AB = 48AB, Á » 16AB induces 12 mixing between N and N0 » 48AB. N00 » 8AB is needed to remove unwanted acci dental symmetries. Upon integrating out the heavy righthanded neutrinos, the light neutrino mass matrix (9) is obtained (see Section 2.5). The light neutrino masses are found to be m1 = ¯¯¯¯¯ ¸2v2 2 m0 Nm00 N ¡ 4®2 2v2Á + ¯vÁm00 N ¡2®2 1v2Á m00 N + mN ¡ m0 Nm00 N ¡ 4®2 2v2Á + ¯vÁm00 N ¢ ¯¯¯¯¯ ; m2 = ¯¯¯¯¯ ¸2v2 2 m0 Nm00 N ¡ 2®2 2v2Á + ¯vÁm00 N ¡2®2 1v2Á m00 N + mN ¡ m0 Nm00 N ¡ 2®2 2v2Á + ¯vÁm00 N ¢ ¯¯¯¯¯ ; m3 = ¯¯¯¯¯ ¸2v2 2 m0 N + ¯vÁ ¡2®2 1v2Á + mN(m0 N + ¯vÁ) ¯¯¯¯¯ : Charged lepton masses are generated through Le = ·(ER1L1 + ER2L2 + ER3L3)H + mE(ER1EL1 + ER2EL2 + ER3EL3) + m0 Ef2(E 0 R;E0 L) +°1g6(ER;E0 L; Â) + °2g6(EL;E0 R; Â) + ²1eR1f2(E0 L; Â) + ²2g7(eR2;E0 L; Â) + ²3g8(eR3;E0 L; Â) +´1g2(E 0 R;E0 L; Â) + ´2g2(E 0 L;E0 R; Â) + c:c:; (2.16) where the functions g6, g7, and g8 are once again given in Section 2.4. Upon integrating out the heavy fermions, the light charged lepton masssquared matrix (11) is obtained (see Section 2.6). The masses are m2e = 3j·²1°2v2Â vj2 3j²1vÂ(mE + °2vÂ)j2 + jmE(m0 E + ´1vÂ + ´2vÂ) ¡ °1°2v2Â j2 ; m2 ¹ = 3j·²2°2v2Â vj2 3j²2vÂ(mE + !°2vÂ)j2 + jmE(m0 E + !2´1vÂ + !´2vÂ) ¡ °1°2v2Â j2 ; m2¿= 3j·²3°2v2Â vj2 3j²3vÂ(mE + !2°2vÂ)j2 + jmE(m0 E + !´1vÂ + !2´2vÂ) ¡ °1°2v2Â j2 : 13 2.3 Derivation of Representations Let H ½ G, and assume that we understand the representation theory of H. G can be decomposed into cosets of H, G = Xn i=1 siH = Xn i=1 fsihj h 2 Hg; (2.17) where the number n of cosets is equal to the number of elements in G divided by the number of elements in H. The coset decomposition is independent of the choice of the representative si for each coset. Let ° be a kdimensional irreducible representation of H. It induces a representation °" of G given by °"(g)ij = X h2H °(h)±(h; s¡1 i gsj): (2.18) In other words, the ij subblock of °" is °(s¡1 i gsj) when s¡1 i gsj 2 H and is zero otherwise. Note that the dimension of the induced representation is kn. In general, °" is reducible. Up to this point, it was not necessary to assume that H is invariant. Let us now do so, h 2 H =) ghg¡1 2 H; 8g 2 G: Then for each g 2 G, we can de¯ne a new representation °g from ° °g(h) = °(ghg¡1): (2.19) For g 2 H, °g is equivalent to ° (that is, °g is ° in a di®erent basis), °g(h) = °(g)°(h)°¡1(g): If g is outside of H, then °g may be either equivalent or inequivalent to °. The set of all inequivalent irreducible representations that can be obtained from ° by the transformation (20) (including ° itself) is called the orbit O° of the representation °. Note that the true singlet (i.e., °(h) = 1, 8h 2 H) is always in its own orbit. Two 14 representations that belong to the same orbit have equivalent induced representations (19). If g1 and g2 belong to the same coset in (18), then they di®er by a factor belonging to H. So, by an argument similar to that showing °g is equivalent to ° for g 2 H, °g1 and °g2 are equivalent. Then, to identify the orbit, it su±ces to consider how ° transforms under the coset representatives, si. Let H° be the set of all g 2 G such that °g is equivalent to °. Then, not only does H° contain H, but it consists of a whole number of cosets from (18). H° is an invariant subgroup of G and is called the little group of the representation ° (or of the orbit O°). Note that the little group of the true singlet is always the entire group G. If each coset sends ° to an inequivalent representation, then the number of representations in O° is equal to the number n of cosets, and H° = H. In this case, the induced representation °" in (19) is irreducible. Otherwise, it is reducible. As an example, consider the group A4 = (Z2 £ Z2) o Z3. Let X and Z be the Z2 generators and Y be the Z3 generator. Y cyclicly permutes the three Z2 subgroups of Z2 £ Z2: Y XY ¡1 = Z; Y ZY ¡1 = XZ; Y (XZ)Y ¡1 = X: (2.20) Note that Z is not an independent generator (Z = Y XY ¡1). The decomposition into cosets of Z2 £ Z2 is A4 = fE; X; Z; XZg + fY; Y X; Y Z; Y XZg + fY 2; Y 2X; Y 2Z; Y 2XZg:(2.21) We can choose E, Y , and Y 2 = Y ¡1 as coset representatives. Then the induced representation (19) takes the form °"(X) = 0 BBBB@ °(X) 0 0 0 °(Y ¡1XY ) 0 0 0 °(Y XY ¡1) 1 CCCCA ; °"(Z) = 0 BBBB@ °(Z) 0 0 0 °(Y ¡1ZY ) 0 0 0 °(Y ZY ¡1) 1 CCCCA ; 15 °"(Y ) = 0 BBBB@ 0 0 °(E) °(E) 0 0 0 °(E) 0 1 CCCCA : (2.22) There are four onedimensional irreducible representations of Z2 £ Z2: (a) °(X) = 1; °(Z) = 1 (b) °(X) = ¡1; °(Z) = ¡1 (c) °(X) = ¡1; °(Z) = 1 (d) °(X) = 1; °(Z) = ¡1 As always, the true singlet (a) belongs to its own orbit. The induced representation is °"(X) = °"(Z) = 0 BBBB@ 1 0 0 0 1 0 0 0 1 1 CCCCA ; °"(Y ) = 0 BBBB@ 0 0 1 1 0 0 0 1 0 1 CCCCA : These matrices can be diagonalized simultaneously, yielding three onedimensional representations: 1 : X = Z = 1; Y = 1; 10 : X = Z = 1; Y = !; 100 : X = Z = 1; Y = !¤; with ! = exp(2i¼=3). Using (21), we have for (b), °Y (X) = °(Y XY ¡1) = °(Z) = ¡1; °Y (Z) = °(Y ZY ¡1) = °(XZ) = 1; 16 which is (c), and °Y ¡1(X) = °(Y ¡1XY ) = °(XZ) = 1; °Y ¡1(Z) = °(Y ¡1ZY ) = °(X) = ¡1; which is (d). Thus, (b), (c), and (d) make up a single orbit. Since the number of representations in this orbit is equal to the number of cosets in (22), the induced representation is irreducible. From (23), we have 3 : X = 0 BBBB@ ¡1 0 0 0 1 0 0 0 ¡1 1 CCCCA ; Z = 0 BBBB@ ¡1 0 0 0 ¡1 0 0 0 1 1 CCCCA ; Y = 0 BBBB@ 0 0 1 1 0 0 0 1 0 1 CCCCA : For every group G, there exists a maximal invariant subgroup H; that is, there are no proper invariant subgroups that contain H. For this subgroup, the factor group G=H is simple. If G is itself a simple group then the maximal invariant subgroup is the trivial subgroup, fEg. There also exists a maximal invariant subgroup H0 of H. So, we have a chain, Hi ½ Hi¡1 ½ ::: ½ H2 ½ H1 = G; where Hj+1 is an invariant subgroup of Hj , and Hj=Hj+1 is simple. This chain can be continued until the trivial subgroup fEg is reached, but for our purposes it su±ces to stop at the largest subgroup whose representation theory we already know. Then, if we know how to determine the representation theory of a group from that of its maximal invariant subgroup, we can apply this recursively. So, let G be a group, and let H be its maximal invariant subgroup. We will further assume that G=H is a cyclic group. Since the little group of a representation of H must be an invariant subgroup of G containing H, the little group for each representation must be either H or all of G. If the little group is H, the induced representation is irreducible. So we need only concern ourselves with the case where the little group is G. Let us consider another example. The group S4 is equal to A4 o Z2. The A4 generators given above and the Z2 generator, which we will denote by W, obey the 17 relations WXW¡1 = Z; WZW¡1 = X; WYW¡1 = Y ¡1; (2.23) along with (21). The decomposition into cosets of A4 is S4 = A4+WA4. Noting that W¡1 = W, the induced representations have the form °"(X) = 0 B@ °(X) 0 0 °(WXW) 1 CA ; °"(Z) = 0 B@ °(Z) 0 0 °(WZW) 1 CA ; °"(Y ) = 0 B@ °(Y ) 0 0 °(WYW) 1 CA ; °"(W) = 0 B@ 0 °(E) °(E) 0 1 CA : (2.24) For the 1 of A4, °"(X) = °"(Z) = °"(Y ) = 0 B@ 1 0 0 1 1 CA ; °"(W) = 0 B@ 0 1 1 0 1 CA : Upon diagonalization, this yields X = Z = Y = 1; W = 1; X = Z = Y = 1; W = ¡1: Without much di±culty, we see that 10 and 100 make up an orbit, so that the induced representation is irreducible, X = Z = 0 B@ 1 0 0 1 1 CA ; Y = 0 B@ ! 0 0 !¤ 1 CA ; W = 0 B@ 0 1 1 0 1 CA : The triplet 3 of A4, °(X) = 0 BBBB@ ¡1 0 0 0 1 0 0 0 ¡1 1 CCCCA ; °(Z) = 0 BBBB@ ¡1 0 0 0 ¡1 0 0 0 1 1 CCCCA ; °(Y ) = 0 BBBB@ 0 0 1 1 0 0 0 1 0 1 CCCCA ; 18 must belong to its own orbit because there is no other possibility. We have °W(X) = °(WXW) = °(Z) = 0 BBBB@ ¡1 0 0 0 ¡1 0 0 0 1 1 CCCCA ; °W(Z) = °(WZW) = °(X) = 0 BBBB@ ¡1 0 0 0 1 0 0 0 ¡1 1 CCCCA ; °W(Y ) = °(WYW) = °¡1(Y ) = 0 BBBB@ 0 1 0 0 0 1 1 0 0 1 CCCCA : Since this representation must lie in its own orbit, there exists a matrix S such that S°W(X)S¡1 = °(X); S°W(Z)S¡1 = °(Z); S°W(Y )S¡1 = °(Y ): Indeed, by inspection, we see that we can take S = 0 BBBB@ 1 0 0 0 0 1 0 1 0 1 CCCCA : Note that S¡1 = S. The induced representation of S4 can then be written °"(X) = 0 B@ °(X) 0 0 S°(X)S 1 CA ; °"(Z) = 0 B@ °(Z) 0 0 S°(Z)S 1 CA ; °"(Y ) = 0 B@ °(Y ) 0 0 S°(Y )S 1 CA ; °"(W) = 0 B@ 0 I I 0 1 CA : 19 Let S = 1 p 2 0 B@ I I I ¡I 1 CA 0 B@ I 0 0 S 1 CA : Then S°"(X)S¡1 = 0 B@ °(X) 0 0 °(X) 1 CA ; S°"(Z)S¡1 = 0 B@ °(Z) 0 0 °(Z) 1 CA ; S°"(Y )S¡1 = 0 B@ °(Y ) 0 0 °(Y ) 1 CA ; S°"(W)S¡1 = 0 B@ S 0 0 ¡S 1 CA : So there are two irreducible triplets of S4, X = °(X); Z = °(Z); Y = °(Y ); W = S; and X = °(X); Z = °(Z); Y = °(Y ); W = ¡S: Let us now consider the group (Z2 £ Z2 £ Z2) o A4. Let Ci be the Z2 generators. With the A4 generators in (21), they obey XC1X¡1 = C1C2C2; XC2X¡1 = C3; XC3X¡1 = C2; (2.25) ZC1Z¡1 = C3; ZC2Z¡1 = C1C2C3; ZC3Z¡1 = C2; (2.26) Y C1Y ¡1 = C2; Y C2Y ¡1 = C3; Y C3Y ¡1 = C1: (2.27) We have the chain Z3 2 ½ (Z3 2 ) o Z2 ½ (Z3 2 ) o (Z2 £ Z2) ½ (Z3 2 ) o A4 of invariant subgroups. Without much di±culty, we can see that the representations (a) C1 = 1; C2 = ¡1; C3 = ¡1 20 (b) C1 = ¡1; C2 = 1; C3 = 1 (c) C1 = ¡1; C2 = 1; C3 = ¡1 (d) C1 = 1; C2 = ¡1; C3 = 1 (e) C1 = ¡1; C2 = ¡1; C3 = 1 (f) C1 = 1; C2 = 1; C3 = ¡1; lie in the same orbit with respect to (Z3 2 )oA4. Now consider the subgroup (Z3 2 )oZ2, where the last Z2 is generated by X. From (26), under X C1 ! C1C2C3; C2 $ C3; so that (a) $ (a); (b) $ (b); (c) $ (d); (e) $ (f): So (a) and (b) each give two onedimensional representations of (Z4 2 ) o Z2: (a) C1 = 1; C2 = ¡1; C3 = ¡1; X = 1 (a0) C1 = 1; C2 = ¡1; C3 = ¡1; X = ¡1 (b) C1 = ¡1; C2 = 1; C3 = 1; X = 1 (b0) C1 = ¡1; C2 = 1; C3 = 1; X = ¡1; while (c/d) and (e/f) each give twodimensional irreducible representations: (c=d) C2 = M1; C1 = C3 = M2; X = S 21 (e=f) C3 = M1; C1 = C2 = M2; X = S; where M1 ´ 0 B@ 1 0 0 ¡1 1 CA ; M2 ´ 0 B@ ¡1 0 0 1 1 CA ; S ´ 0 B@ 0 1 1 0 1 CA : (2.28) Now add Z to obtain (Z3 2 ) o (Z2 £ Z2). Under Z C1 ! C3; C2 ! C1C2C3; X $ X; so that (a) $ (b); (a0) $ (b0); (c=d) $ (c=d); (e=f) $ (e=f): Now (a/b) and (a0/b0) each give twodimensional irreducible representations: (a=b) C1 = M1; C2 = C3 = M2; X = I; Z = S (a0=b0) C1 = M1; C2 = C3 = M2; X = ¡I; Z = S For (c/d), the induced representation is C2 = 0 B@ M1 0 0 M1 1 CA ; C1 = C3 = 0 B@ M2 0 0 M2 1 CA; X = 0 B@ S 0 0 S 1 CA ; Z = 0 B@ 0 I I 0 1 CA : This can be blockdiagonalized by inspection, (c=d) C2 = M1; C1 = C3 = M2; X = S; Z = I (c0=d0) C2 = M1; C1 = C3 = M2; X = S; Z = ¡I: For (e/f), the induced representation is C1 = C4 = 0 B@ M1 0 0 M2 1 CA ; C2 = C3 = 0 B@ M2 0 0 M1 1 CA ; X = 0 B@ S 0 0 S 1 CA ; Z = 0 B@ 0 I I 0 1 CA : 22 Noting that M2 = SM1S, we can blockdiagonalize this using the same method that was used in the S4 example for the triplet orbit. This gives (e=f) C1 = C4 = M1; C2 = C3 = M2; X = S; Z = S (e0=f0) C1 = C4 = M1; C2 = C3 = M2; X = S; Z = ¡S: Finally, we add Y . With a little e®ort, it can be seen that (a/b), (c/d), and (e/f) lie in one orbit, and (a0/b0), (c0/d0), and (e0/f0) lie in another. Then, the induced repre sentations are irreducible. So, we ¯nally end up with two sixdimensional irreducible representations of (Z3 2 ) o A4: C1 = 0 BBBB@ M1 0 0 0 M2 0 0 0 M2 1 CCCCA ; C2 = 0 BBBB@ M2 0 0 0 M1 0 0 0 M2 1 CCCCA ; C3 = 0 BBBB@ M2 0 0 0 M2 0 0 0 M1 1 CCCCA ; X = 0 BBBB@ I 0 0 0 S 0 0 0 S 1 CCCCA ; Z = 0 BBBB@ S 0 0 0 I 0 0 0 S 1 CCCCA ; Y = 0 BBBB@ 0 0 I I 0 0 0 I 0 1 CCCCA ; and C1 = 0 BBBB@ M1 0 0 0 M2 0 0 0 M2 1 CCCCA ; C2 = 0 BBBB@ M2 0 0 0 M1 0 0 0 M2 1 CCCCA ; C3 = 0 BBBB@ M2 0 0 0 M2 0 0 0 M1 1 CCCCA ; X = 0 BBBB@ ¡I 0 0 0 ¡S 0 0 0 S 1 CCCCA ; Z = 0 BBBB@ S 0 0 0 ¡I 0 0 0 ¡S 1 CCCCA ; Y = 0 BBBB@ 0 0 I I 0 0 0 I 0 1 CCCCA : It can be checked directly that these matrices respect all of the relations (21), (26), (27), and (28). 23 Now consider the group (S3 £ S3 £ S3 £ S3) o A4. Let Ai and Bi be the S4 generators. They obey A3i = B2 i = E; BiAiB¡1 i = A¡1 i : (2.29) The irreducible representations of S3 are two onedimensional representations given by Ai = 1; Bi = 1; Ai = 1; Bi = ¡1; and one twodimensional representation given by Ai = MA ´ 0 B@ ! 0 0 !2 1 CA ; Bi = MB ´ 0 B@ 0 1 1 0 1 CA ; With the A4 generators in (4), Ai and Bi respect the relations XA1X¡1 = A2; XA2X¡1 = A1; XA3X¡1 = A4; XA4X¡1 = A3; XB1X¡1 = B2; XB2X¡1 = B1; XB3X¡1 = B4; XB4X¡1 = B3; (2.30) ZA1Z¡1 = A3; ZA2Z¡1 = A4; ZA3Z¡1 = A1; ZA4Z¡1 = A2; ZB1Z¡1 = B3; ZB2Z¡1 = B4; ZB3Z¡1 = B1; ZB4Z¡1 = B2; (2.31) Y A1Y ¡1 = A1; Y A2Y ¡1 = A3; Y A3Y ¡1 = A4; Y A4Y ¡1 = A2; Y B1Y ¡1 = B1; Y B2Y ¡1 = B3; Y B3Y ¡1 = B4; Y B4Y ¡1 = B2: (2.32) As in the last example, we have a chain of invariant subgroups, S4 3 ½ (S4 3 ) o Z2 ½ (S4 3 ) o (Z2 £ Z2) ½ (S4 3 ) o A4: 24 The representation (2; 2; 2; 2) under S3 £ S3 £ S3 £ S3 lies in its own orbit. This representation can be written in terms of 16dimensional matrices, °(A1) = MA I I I; °(B1) = MB I I I; °(A2) = I MA I I; °(B2) = I MB I I; °(A3) = I I MA I; °(B3) = I I MB I; °(A4) = I I I MA; °(B4) = I I I MB: Now add X to obtain the subgroup (S4 3 ) o Z2. From (14), under X A1 $ A2; A3 $ A4; B1 $ B2; B3 $ B4: Consider how this rearranges the eigenvalues of each of the 16 basis states under the diagonal generators (A1;A2;A3;A4), (1) (!; !; !; !) ¡! (!; !; !; !) » (1) (2) (!2; !; !; !) ¡! (!; !2; !; !) » (3) (3) (!; !2; !; !) ¡! (!2; !; !; !) » (2) (4) (!2; !2; !; !) ¡! (!2; !2; !; !) » (4) (5) (!; !; !2; !) ¡! (!; !; !; !2) » (9) (6) (!2; !; !2; !) ¡! (!; !2; !; !2) » (11) 25 (7) (!; !2; !2; !) ¡! (!2; !; !; !2) » (10) (8) (!2; !2; !2; !) ¡! (!2; !2; !; !2) » (12) (9) (!; !; !; !2) ¡! (!; !; !2; !) » (5) (10) (!2; !; !; !2) ¡! (!; !2; !2; !) » (7) (11) (!; !2; !; !2) ¡! (!2; !; !2; !) » (6) (12) (!2; !2; !; !2) ¡! (!2; !2; !2; !) » (8) (13) (!; !; !2; !2) ¡! (!; !; !2; !2) » (13) (14) (!2; !; !2; !2) ¡! (!; !2; !2; !2) » (15) (15) (!; !2; !2; !2) ¡! (!2; !; !2; !2) » (14) (16) (!2; !2; !2; !2) ¡! (!2; !2; !2; !2) » (16): 26 This yields the permutation matrix PX ´ 0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : Note that P¡1 X = PX. We can now check directly that PX°(B1)P¡1 X = °(B2); PX°(B2)P¡1 X = °(B1); PX°(B3)P¡1 X = °(B4); PX°(B4)P¡1 X = °(B3): So, we obtain two 16dimensional irreducible representations of (S4 3 ) o Z2, (a) Ai = °(Ai); Bi = °(Bi); X = PX (b) Ai = °(Ai); Bi = °(Bi); X = ¡PX: 27 Next add Z to obtain the subgroup (S4 3 ) o (Z2 £ Z2). Proceeding as is the previous step yields PZ ´ 0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : Again note that P¡1 Z = PZ. This gives four 16dimensional irreducible representations of (S4 3 ) o (Z2 £ Z2), (a) Ai = °(Ai); Bi = °(Bi); X = PX; Z = PZ (a0) Ai = °(Ai); Bi = °(Bi); X = PX; Z = ¡PZ (b) Ai = °(Ai); Bi = °(Bi); X = ¡PX; Z = PZ (b0) Ai = °(Ai); Bi = °(Bi); X = ¡PX; Z = ¡PZ: 28 Finally, add Y to obtain (S4 3 )oA4. Then, (a) lies in its own orbit, while (a0), (b), and (b0) lie in another orbit. First, consider the orbit of (a). We ¯nd that the permutation matrix PY ´ 0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA respects the relations PY °(A1)P¡1 Y = °(A1); PY °(A2)P¡1 Y = °(A3); PY °(A3)P¡1 Y = °(A4); PY °(A4)P¡1 Y = °(A2); PY °(B1)P¡1 Y = °(B1); PY °(B2)P¡1 Y = °(B3); PY °(B3)P¡1 Y = °(B4); PY °(B4)P¡1 Y = °(B2); PY PXP¡1 Y = PZ; PY PZP¡1 Y = PXPZ; PY (PXPZ)P¡1 Y = PX: 29 (Note that P¡1 Y = P2 Y .) The induced representation can then be written °"(Ai) = 0 BBBB@ °(Ai) 0 0 0 P2 Y °(Ai)PY 0 0 0 PY °(Ai)P2 Y 1 CCCCA ; °"(Bi) = 0 BBBB@ °(Bi) 0 0 0 P2 Y °(Bi)PY 0 0 0 PY °(Bi)P2 Y 1 CCCCA °"(X) = 0 BBBB@ °(X) 0 0 0 P2 Y °(X)PY 0 0 0 PY °(X)P2 Y 1 CCCCA ; °"(Z) = 0 BBBB@ °(Z) 0 0 0 P2 Y °(Z)PY 0 0 0 PY °(Z)P2 Y 1 CCCCA ; °"(Y ) = 0 BBBB@ 0 0 I I 0 0 0 I 0 1 CCCCA : Let P = 1 p 3 0 BBBB@ I I I I !I !2I I !2I !I 1 CCCCA 0 BBBB@ I 0 0 0 PY 0 0 0 P2 Y 1 CCCCA : Then P°"(Ai)P¡1 = 0 BBBB@ °(Ai) 0 0 0 °(Ai) 0 0 0 °(Ai) 1 CCCCA ; P°"(Bi)P¡1 = 0 BBBB@ °(Bi) 0 0 0 °(Bi) 0 0 0 °(Bi) 1 CCCCA ; P°"(X)P¡1 = 0 BBBB@ °(X) 0 0 0 °(X) 0 0 0 °(X) 1 CCCCA ; P°"(Z)P¡1 = 0 BBBB@ °(Z) 0 0 0 °(Z) 0 0 0 °(Z) 1 CCCCA ; P°"(Y )P¡1 = 0 BBBB@ PY 0 0 0 !PY 0 0 0 !2PY 1 CCCCA : 30 So the result is three 16dimensional irreducible representations of (S4 3 )oA4. For the other orbit, the induced representation is irreducible. It is given by °"(A1) = 0 BBBB@ °(A1) 0 0 0 °(A1) 0 0 0 °(A1) 1 CCCCA ; °"(B1) = 0 BBBB@ °(B1) 0 0 0 °(B1) 0 0 0 °(B1) 1 CCCCA ; °"(A2) = 0 BBBB@ °(A2) 0 0 0 °(A4) 0 0 0 °(A3) 1 CCCCA ; °"(B2) = 0 BBBB@ °(B2) 0 0 0 °(B4) 0 0 0 °(B3) 1 CCCCA ; °"(A3) = 0 BBBB@ °(A3) 0 0 0 °(A2) 0 0 0 °(A4) 1 CCCCA ; °"(B3) = 0 BBBB@ °(B3) 0 0 0 °(B2) 0 0 0 °(B4) 1 CCCCA ; °"(A4) = 0 BBBB@ °(A4) 0 0 0 °(A3) 0 0 0 °(A2) 1 CCCCA ; °"(B4) = 0 BBBB@ °(B4) 0 0 0 °(B3) 0 0 0 °(B2) 1 CCCCA ; °"(X) = 0 BBBB@ PX 0 0 0 ¡PXPZ 0 0 0 ¡PZ 1 CCCCA ; °"(Z) = 0 BBBB@ ¡PZ 0 0 0 PX 0 0 0 ¡PXPZ 1 CCCCA ; Y (48) = 0 BBBB@ 0 0 I I 0 0 0 I 0 1 CCCCA : We can also see that the representations (2; 1; 1; 1), (1; 2; 1; 1), (1; 1; 2; 1), and (1; 1; 1; 2) of S3 £ S3 £ S3 £ S3 make up an orbit. The 8dimensional representation of (S4 3 ) o A4 this orbit gives rise to is given by A1 = diag(!; !2; 1; 1; 1; 1; 1; 1); A2 = diag(1; 1; !; !2; 1; 1; 1; 1); A3 = diag(1; 1; 1; 1; !; !2; 1; 1); A4 = diag(1; 1; 1; 1; 1; 1; !; !2); 31 B1 = 0 BBBBBBBBBBBBBBBBBBBBB@ 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCA ; B2 = 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCA ; B3 = 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCA ; B4 = 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 CCCCCCCCCCCCCCCCCCCCCA ; X = 0 BBBBBBBBBBBBBBBBBBBBB@ 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 CCCCCCCCCCCCCCCCCCCCCA ; Z = 0 BBBBBBBBBBBBBBBBBBBBB@ 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 CCCCCCCCCCCCCCCCCCCCCA 32 Y = 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 CCCCCCCCCCCCCCCCCCCCCA : 2.4 Invariants Under the Discrete Symmetry In this section, we give the symmetry invariants which are used in our model. These can be computed directly from the matrices given in the previous section. 16AB £ 16AB invariant (xi; x0 j » 16AB): f1(xi; x0 j) = x1x0 16 + x2x0 15 + x3x0 14 + x4x0 13 + x5x0 12 + x6x0 11 + x7x0 10 + x8x0 9 +x9x0 8 + x10x0 7 + x11x0 6 + x12x0 5 + x13x0 4 + x14x0 3 + x15x0 2 + x16x0 1 6C £ 6C invariant (wi;w0j » 6C): f2(wi;w0 j) = w1w0 1 + w2w0 2 + w3w0 3 + w4w0 4 + w5w0 5 + w6w0 6: 48AB £ 48AB invariant (yi; y0j » 48AB): f3(yi; y0 j) = y1y0 16 + y2y0 15 + y3y0 14 + y4y0 13 + y5y0 12 + y6y0 11 + y7y0 10 + y8y0 9 33 y9y0 8 + y10y0 7 + y11y0 6 + y12y0 5 + y13y0 4 + y14y0 3 + y15y0 2 + y16y0 1 + y17y0 32 + y18y0 31 +y19y0 30 + y20y0 29 + y21y0 28 + y22y0 27 + y23y0 26 + y24y0 25 + y25y0 24 + y26y0 23 + y27y0 22 + y28y0 21 +y29y0 20 + y30y0 19 + y31y0 18 + y32y0 17 + y33y0 48 + y34y0 47 + y35y0 46 + y36y0 45 + y37y0 44 + y38y0 43 +y39y0 42 + y40y0 41 + y41y0 40 + y42y0 39 + y43y0 38 + y44y0 37 + y45y0 36 + y46y0 35 + y47y0 34 + y48y0 33 8AB £ 8AB invariant (zi; z0j » 8AB): f4(zi; z0 j) = z1z0 2 + z2z0 1 + z3z0 4 + z4z0 3 + z5z0 6 + z6z0 5 + z7z0 8 + z8z0 7 16AB £ 16AB £ 16AB invariant (xi; x0 j ; x00 k » 16AB): g1(xi; x0 j ; x00 k) = x1x0 1x00 1 + x2x0 2x00 2 + x3x0 3x00 3 + x4x0 4x00 4 + x5x0 5x00 5 + x6x0 6x00 6 + x7x0 7x00 7 + x8x0 8x00 8 x9x0 9x00 9 + x10x0 10x00 10 + x11x0 11x00 11 + x12x0 12x00 12 + x13x0 13x00 13 + x14x0 14x00 14 + x15x0 15x00 15 + x16x0 16x00 16 6C £ 6C £ 6C invariant (wi;w0j ;w00 k » 6C): g2(wi;w0 j ;w00 k) = w1w0 3w00 5 + w5w0 1w00 3 + w3w0 5w00 1 + w1w0 4w00 6 + w6w0 1w00 4 + w4w0 6w00 1 +w2w0 3w00 6 + w6w0 2w00 3 + w3w0 6w00 2 + w2w0 4w00 5 + w5w0 2w00 4 + w4w0 5w00 2 3 £ 16AB £ 48AB invariant (ti » 3, xj » 16AB, yk » 48AB): g3(ti; xj ; yk) = t1(x16y33 + x15y34 + x8y35 + x7y36 + x14y37 + x13y38 + x6y39 + x5y40 34 +x12y41 + x11y42 + x4y43 + x3y44 + x10y45 + x9y46 + x2y47 + x1y48) t2(x16y1 + x7y10 + x6y11 + x5y12 + x4y13 + x3y14 + x2y15 + x1y16 +x15y2 + x14y3 + x13y4 + x12y5 + x11y6 + x10y7 + x9y8 + x8y9) t3(x16y17 + x15y18 + x12y19 + x11y20 + x8y21 + x7y22 + x4y23 + x3y24 +x14y25 + x13y26 + x10y27 + x9y28 + x6y29 + x5y30 + x2y31 + x1y32) 8AB £ 16AB £ 48AB invariant (zi » 8AB, xj » 16AB, yk » 48AB): g4(zi; xk; yj ) = z1(x15y1 + x5y11 + x3y13 + x1y15 + x15y17 + x11y19 + x7y21 + x3y23 + x13y25 + x9y27 + x5y29 + x13y3 +x1y31 + x15y33 + x7y35 + x13y37 + x5y39 + x11y41 + x3y43 + x9y45 + x1y47 + x11y5 + x9y7 + x7y9) +z2(x8y10 + x6y12 + x4y14 + x2y16 + x16y18 + x16y2 + x12y20 + x8y22 + x4y24 + x14y26 + x10y28 + x6y30 +x2y32 + x16y34 + x8y36 + x14y38 + x14y4 + x6y40 + x12y42 + x4y44 + x10y46 + x2y48 + x12y6 + x10y8) +z3(x14y1 + x5y10 + x2y13 + x1y14 ¡ x14y17 ¡ x13y18 ¡ x10y19 + x13y2 ¡ x9y20 ¡ x6y21 ¡ x5y22 ¡ x2y23 ¡x1y24 ¡ x14y33 ¡ x13y34 ¡ x6y35 ¡ x5y36 ¡ x10y41 ¡ x9y42 ¡ x2y43 ¡ x1y44 + x10y5 + x9y6 + x6y9) +z4(x8y11 + x7y12 + x4y15 + x3y16 ¡ x16y25 ¡ x15y26 ¡ x12y27 ¡ x11y28 ¡ x8y29 + x16y3 ¡ x7y30 ¡ x4y31 ¡x3y32 ¡ x16y37 ¡ x15y38 ¡ x8y39 + x15y4 ¡ x7y40 ¡ x12y45 ¡ x11y46 ¡ x4y47 ¡ x3y48 + x12y7 + x11y8) +z5(¡x12y1 ¡ x3y10 ¡ x2y11 ¡ x1y12 + x12y17 + x11y18 ¡ x11y2 + x4y21 + x3y22 + x10y25 + x9y26 + x2y29 ¡x10y3 + x1y30 ¡ x12y33 ¡ x11y34 ¡ x4y35 ¡ x3y36 ¡ x10y37 ¡ x9y38 ¡ x2y39 ¡ x9y4 ¡ x1y40 ¡ x4y9) +z6(¡x8y13 ¡ x7y14 ¡ x6y15 ¡ x5y16 + x16y19 + x15y20 + x8y23 + x7y24 + x14y27 + x13y28 + x6y31 + x5y32 ¡x16y41 ¡ x15y42 ¡ x8y43 ¡ x7y44 ¡ x14y45 ¡ x13y46 ¡ x6y47 ¡ x5y48 ¡ x16y5 ¡ x15y6 ¡ x14y7 ¡ x13y8) +z7(¡x8y1 ¡ x8y17 ¡ x7y18 ¡ x4y19 ¡ x7y2 ¡ x3y20 ¡ x6y25 ¡ x5y26 ¡ x2y27 ¡ x1y28 ¡ x6y3 + x8y33 +x7y34 + x6y37 + x5y38 ¡ x5y4 + x4y41 + x3y42 + x2y45 + x1y46 ¡ x4y5 ¡ x3y6 ¡ x2y7 ¡ x1y8) +z8(x15y36 ¡ x14y11 ¡ x13y12 ¡ x12y13 ¡ x11y14 ¡ x10y15 ¡ x9y16 ¡ x16y21 ¡ x15y22 ¡ x12y23 ¡ x11y24 ¡ x14y29 ¡x13y30 ¡ x10y31 ¡ x9y32 + x16y35 ¡ x15y10 + x14y39 + x13y40 + x12y43 + x11y44 + x10y47 + x9y48 ¡ x16y9) 35 16AB £ 48AB £ 48AB invariant (xi » 16AB; yj ; y0k » 48AB): g5(xi; yj ; y0 k) = x1y1y0 1 + x2y2y0 2 + x3y3y0 3 + x4y4y0 4 + x5y5y0 5 + x6y6y0 6 + x7y7y0 7 + x8y8y0 8 +x9y9y0 9 + x10y10y0 10 + x11y11y0 11 + x12y12y0 12 + x13y13y0 13 + x14y14y0 14 + x15y15y0 15 + x16y16y0 16 +x1y17y0 17 + x2y18y0 18 + x3y25y0 25 + x4y26y0 26 + x5y19y0 19 + x6y20y0 20 + x7y27y0 27 + x8y28y0 28 +x9y21y0 21 + x10y22y0 22 + x11y29y0 29 + x12y30y0 30 + x13y23y0 23 + x14y24y0 24 + x15y31y0 31 + x16y32y0 32 +x1y33y0 33 + x2y34y0 34 + x3y37y0 37 + x4y38y0 38 + x5y41y0 41 + x6y42y0 42 + x7y45y0 45 + x8y46y0 46 +x9y35y0 35 + x10y36y0 36 + x11y39y0 39 + x12y40y0 40 + x13y43y0 43 + x14y44y0 44 + x15y47y0 47 + x16y48y0 48 3 £ 6C £ 6C invariant (ti » 3; wj ;w0k » 6C): g6(ti;wj ;w0 k) = t1(w5w0 5 ¡ w6w0 6) + t2(w1w0 1 ¡ w2w0 2) + t3(w3w0 3 ¡ w4w0 4) 10 £ 6C £ 6C invariant (s0 » 10; wi;w0j » 6C): g7(s0;wi;w0 j) = s0(w1w0 1 + w2w0 2 + !2w3w0 3 + !2w4w0 4 + !w5w0 5 + !w6w0 6) 100 £ 6C £ 6C invariant (s00 » 100; wi;w0j » 6C): g8(s00;wi;wj) = s00(w1w0 1 + w2w0 2 + !w3w0 3 + !w4w0 4 + !2w5w0 5 + !2w6w0 6) For our purposes, it su±ces to have the 16AB£16AB£16AB£16AB and 6C£6C£6C£6C 36 invariants for the case where all four ¯elds are the same. 16AB £ 16AB £ 16AB £ 16AB invariants (xi » 16AB): h1(xi) = x21 x2 16 + x22 x2 15 + x23 x2 14 + x24 x2 13 + x25 x2 12 + x26 x2 11 + x27 x2 10 + x28 x29 ; h2(xi) = x1x2x15x16 + x1x3x14x16 + x2x4x13x15 + x3x4x13x14 + x1x5x12x16 + x4x5x12x13 +x2x6x11x15 + x3x6x11x14 + x5x6x11x12 + x2x7x10x15 + x3x7x10x14 + x5x7x10x12 +x1x8x9x16 + x4x8x9x13 + x6x8x9x11 + x7x8x9x10 6C £ 6C £ 6C £ 6C invariants (wi » 6C): h3(wi) = w4 1 + w4 2 + w4 3 + w4 4 + w4 5 + w4 6; h4(wi) = w2 1w2 2 + w2 3w2 4 + w2 5w2 6; h5(wi) = w2 1w2 3 + w2 1w2 4 + w2 1w2 5 + w2 1w2 6 + w2 2w2 3 + w2 2w2 4 + w2 2w2 5 + w2 2w2 6 +w2 3w2 5 + w2 3w2 6 + w2 4w2 5 + w2 4w2 6; h6(wi) = w1w2w3w4 + w1w2w5w6 + w3w4w5w6 37 2.5 Calculation of the Neutrino Mass Matrix In this section, we show how the neutrino mass matrix is computed. From Section 2.4, the term in Eq. (13) that mixes N and N0 is2 g3(N; hÁi;N0) = vÁN1(N0 35 + N0 36 + N0 39 + N0 40 + N0 41 + N0 42 + N0 45 + N0 46) +vÁN2(N0 5 + N0 6 + N0 7 + N0 8 + N0 9 + N0 10 + N0 11 + N0 12) +vÁN3(N0 19 + N0 20 + N0 21 + N0 22 + N0 27 + N0 28 + N0 29 + N0 30): The term that mixes N0 and N00 is g4(N00; hÁi;N0) = vÁN00 1 (N0 5 + N0 7 + N0 9 + N0 11) + vÁN00 2 (N0 6 + N0 8 + N0 10 + N0 12) +vÁN00 3 (N0 5 + N0 6 + N0 9 + N0 10) + vÁN00 4 (N0 7 + N0 8 + N0 11 + N0 12) ¡vÁN00 5 (N0 1 + N0 2 + N0 3 + N0 4) ¡ vÁN00 6 (N0 13 + N0 14 + N0 15 + N0 16) ¡vÁN00 7 (N0 1 + N0 2 + N0 3 + N0 4) ¡ vÁN00 8 (N0 13 + N0 14 + N0 15 + N0 16) +vÁN00 1 (N0 19 + N0 21 + N0 27 + N0 29) + vÁN00 2 (N0 20 + N0 22 + N0 28 + N0 30) ¡vÁN00 3 (N0 19 + N0 20 + N0 21 + N0 22) ¡ vÁN00 4 (N0 27 + N0 28 + N0 29 + N0 30) +vÁN00 5 (N0 17 + N0 18 + N0 25 + N0 26) + vÁN00 6 (N0 23 + N0 29 + N0 31 + N0 32) ¡vÁN00 7 (N0 17 + N0 18 + N0 25 + N0 26) ¡ vÁN00 8 (N0 23 + N0 29 + N0 31 + N0 32) +vÁN00 1 (N0 35 + N0 39 + N0 41 + N0 45) + vÁN00 2 (N0 36 + N0 40 + N0 42 + N0 46) 2Note that, in the 16AB basis used here, Á17¡i = Á¤i , i = 1 ¡ 8. 38 ¡vÁN00 3 (N0 35 + N0 36 + N0 41 + N0 42) ¡ vÁN00 4 (N0 39 + N0 40 + N0 45 + N0 46) +vÁN00 5 (N0 33 + N0 34 + N0 37 + N0 38) + vÁN00 6 (N0 43 + N0 44 + N0 47 + N0 48) ¡vÁN00 7 (N0 33 + N0 34 + N0 37 + N0 38) ¡ vÁN00 8 (N0 43 + N0 44 + N0 47 + N0 48) Since the symmetries B1, B2, B3B4, and A3A4 are unbroken, components of N0 and N00 that transform under these symmetries cannot mix with the light neutrinos. This leaves p1 = N05 + N06 + N07 + N08 + N09 + N0 10 + N0 11 + N0 p 12 8 ; p2 = N0 19 + N0 20 + N0 21 + N0 22 + N0 27 + N0 28 + N0 29 + N0 p 30 8 ; p3 = N0 35 + N0 36 + N0 39 + N0 40 + N0 41 + N0 42 + N0 45 + N0 p 46 8 ; q1 = N00 1 + N00 p 2 2 ; q2 = N00 3 + N00 p 4 2 : We now have g3(N; hÁi;N0) = p 8vÁ(N1p3 + N2p1 + N3p2); g4(N00; hÁi;N0) = 2vÁ(q1p1 + q2p1 + q1p2 ¡ q2p2 + q1p3 ¡ q2p3) + :::; where the ellipses in the second equation refer to terms involving only decoupled components. The mass matrix for (º1; º2; º3;N1;N2;N3; p1; p2; p3; q1; q2) has the form 1 2 Mº = 0 B@ 0 m mT M 1 CA ; with m = 0 BBBB@ 1 2¸v 0 0 0 0 0 0 0 0 1 2¸v 0 0 0 0 0 0 0 0 1 2¸v 0 0 0 0 0 1 CCCCA ; 39 M = 0 BBBBBBBBBBBBBBBBBBBBB@ mN 0 0 0 0 p 2®1vÁ 0 0 0 mN 0 p 2®1vÁ 0 0 0 0 0 0 mN 0 p 2®1vÁ 0 0 0 0 p 2®1vÁ 0 m0 N + ¯vÁ 0 0 ®2vÁ ®2vÁ 0 0 p 2®1vÁ 0 m0 N + ¯vÁ 0 ®2vÁ ¡®2vÁ p 2®1vÁ 0 0 0 0 m0 N + ¯vÁ ®2vÁ ¡®2vÁ 0 0 0 ®2vÁ ®2vÁ ®2vÁ m00 N 0 0 0 0 ®2vÁ ¡®2vÁ ¡®2vÁ 0 m00 N 1 CCCCCCCCCCCCCCCCCCCCCA : Here, m only contains entries at the EW scale, while M contains entries at the higher scale M¤. To order M2W =M2 ¤ , Mº is blockdiagonalized by Uº = 0 B@ I ¡mM¡1 M¡1mT I 1 CA : The light neutrino mass matrix Mº is given by the upperleft block of UºMºUT º , 1 2 Mº = ¡mM¡1mT : Let S = 1 p 2 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 1 0 0 0 0 p 2 0 0 0 0 1 0 0 0 0 0 ¡1 0 0 0 0 0 p 2 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ¡1 0 0 1 0 0 1 0 0 0 0 ¡1 0 0 1 0 0 1 CCCCCCCCCCCCCCCCCCCCCA : 40 Then, S¡1MS = 0 BBBB@ A 0 0 0 B 0 0 0 C 1 CCCCA ; with A = 0 BBBB@ mN p 2®1vÁ 0 p 2®1vÁ m0 N + ¯vÁ 2®2vÁ 0 2®2vÁ m00 N 1 CCCCA ; B = 0 BBBB@ mN p 2®1vÁ 0 p 2®1vÁ m0 N + ¯vÁ p 2®2vÁ 0 p 2®2vÁ m00 N 1 CCCCA ; C = 0 B@ mN ¡ p 2®1vÁ ¡ p 2®1vÁ m0 N + ¯vÁ 1 CA : So, we can write 1 2 Mº = ¡mS 0 BBBB@ A¡1 0 0 0 B¡1 0 0 0 C¡1 1 CCCCA S¡1mT = ¡ ¸2v2 8 0 BBBB@ (A¡1)11 + (C¡1)11 0 (A¡1)11 ¡ (C¡1)11 0 2(B¡1)11 0 (A¡1)11 ¡ (C¡1)11 0 (A¡1)11 + (C¡1)11 1 CCCCA This mass matrix is diagonalized by (10), and the masses are given by m1 = ¯¯¯¯ ¸2v2 2 (A¡1)11 ¯¯¯¯ ; m2 = ¯¯¯¯ ¸2v2 2 (B¡1)11 ¯¯¯¯ ; m3 = ¯¯¯¯ ¸2v2 2 (C¡1)11 ¯¯¯¯ : 41 2.6 Calculation of the Charged Lepton Mass Matrix In this section, we show how the charged lepton mass matrix is computed. From Section 2.4, the terms in Eq. (14) that mix eR1, eR2, and eR3 with E0L are eR1f2(E0 L; hÂi) + c:c: = vÂeR1(E0 L1 + E0 L3 + E0 L5) + c:c:; g7(eR2;E0 L; hÂi) + c:c: = vÂeR2(E0 L1 + !2E0 L3 + !E0 L5) + c:c:; g8(eR3;E0 L; hÂi) + c:c: = vÂeR3(E0 L1 + !E0 L3 + !2E0 L5) + c:c:: The term that mixes ER and E0L is g6(ER;E0 L; hÂi) + c:c: = vÂ(ER1E0 L5 + ER2E0 L1 + ER3E0 L3) + c:c:; with a similar result for the term that mixes EL and E0R . In the basis with (eL1; eL2; eL3;EL1;EL2; EL3;E 0 L1;E 0 L3;E 0 L5) on the left and (eR1; eR2; eR3;ER1;ER2;ER3;E0R 1;E0R3;E0R 5) on the right, the mass matrix has the form Me = 0 B@ 0 M0 m M 1 CA ; with m = 0 BBBBBBBBBBBBBB@ ·v 0 0 0 ·v 0 0 0 ·v 0 0 0 0 0 0 0 0 0 1 CCCCCCCCCCCCCCA ; M0 = 0 BBBB@ 0 0 0 ²1vÂ ²1vÂ ²1vÂ 0 0 0 ²2vÂ !2²2vÂ !²2vÂ 0 0 0 ²3vÂ !²3vÂ !2²3vÂ 1 CCCCA ; 42 and M = 0 BBBBBBBBBBBBBB@ mE 0 0 0 0 °1vÂ 0 mE 0 °1vÂ 0 0 0 0 mE 0 °1vÂ 0 0 °2vÂ 0 m0 E ´2vÂ ´1vÂ 0 0 °2vÂ ´1vÂ m0 E ´2vÂ °2vÂ 0 0 ´2vÂ ´1vÂ m0 E 1 CCCCCCCCCCCCCCA : Here, m only contains entries at the EW scale, while M and M0 contain entries at the higher scale M¤. To order M2W =M2 ¤ , the lefthanded masssquared matrixMy eMe is blockdiagonalized by UL = 0 B@ I myM(MyM +M0yM0)¡1 (MyM +M0yM0)¡1Mym I 1 CA : The upper left entry of ULMy eMeUy L is the light lefthanded masssquared matrix My eMe = mym ¡ myM(MyM +M0yM0)¡1Mym Let S = 1 p 3 0 BBBBBBBBBBBBBB@ 1 0 1 0 1 0 1 0 ! 0 !2 0 1 0 !2 0 ! 0 0 1 0 1 0 1 0 1 0 ! 0 !2 0 1 0 !2 0 ! 1 CCCCCCCCCCCCCCA : Then SyMS and SyM0yM0S are both block diagonal (three 2£2 blocks each). So we have myM(MyM +M0yM0)¡1Mym = myS 0 BBBB@ A 0 0 0 B 0 0 0 C 1 CCCCA Sym: 43 = j·vj2 3 0 BBBB@ A11 + B11 + C11 A11 + !2B11 + !C11 A11 + !B11 + !2C11 A11 + !B11 + !2C11 A11 + B11 + C11 A11 + !2B11 + !C11 A11 + !2B11 + !C11 A11 + !B11 + !2C11 A11 + B11 + C11 1 CCCCA This has the form (11). The masses are given by m2e = j·vj2(1 ¡ A11); m2 ¹ = j·vj2(1 ¡ B11); m2¿ = j·vj2(1 ¡ C11): 44 CHAPTER 3 A 6D Higgsless Standard Model 3.1 The Model Let us consider a 6D SU(2)L £ U(1)Y gauge theory in a °at spacetime back ground, where the two extra spatial dimensions are compacti¯ed on a rectangle1.The coordinates in the 6D space are written as zM = (x¹; ym), where the 6D Lorentz indices are denoted by capital Roman letters M = 0; 1; 2; 3; 5; 6, while the usual 4D Lorentz indices are symbolized by Greek letters ¹ = 0; 1; 2; 3, and the coordinates ym (m = 1; 2) describe the ¯fth and sixth dimension.2 The physical space is thus de¯ned by 0 · y1 · ¼R1 and 0 · y2 · ¼R2, where R1 and R2 are the compacti¯cation radii of a torus T2, which is obtained by identifying the points of the twodimensional plane R2 under the actions T5 : (y1; y2) ! (y1+2¼R1; y2) and T6 : (y1; y2) ! (y1; y2+2¼R2). We denote the SU(2)L and U(1)Y gauge bosons in the bulk respectively by Aa M(zM) (a = 1; 2; 3 is the gauge index) and BM(zM). The action of the gauge ¯elds in our model is given by S = Z d4x Z ¼R1 0 dy1 Z ¼R2 0 dy2 (L6 + ±(y1)±(y2)L0) ; (3.1) where L6 is a 6D bulk gauge kinetic term and L0 is a 4D brane gauge kinetic term localized at (y1; y2) = (0; 0), which read respectively L6 = ¡ M2L 4 FaMNFMNa ¡ M2 Y 4 BMNBMN; L0 = ¡ 1 4g2Fa ¹ºF¹ºa ¡ 1 4g02B¹ºB¹º; (3.2) 1Chiral compacti¯cation on a square has recently been considered in Ref. [42]. 2For the metric we choose a signature (+;¡;¡;¡;¡;¡). 45 with ¯eld strengths FaM N = @MAa N ¡ @NAa M + fabcAb MAc N (fabc is the structure con stant) and BMN = @MBN ¡ @NBM. In Eqs. (3.2), the quantities ML and MY have mass dimension +1, while g and g0 are dimensionless. Since the boundaries of the manifold break translational invariance and are "singled out" with respect to the points in the interior of the rectangle, brane terms like L0 can be produced by quan tum loop e®ects [34, 35] or arise from classical singularities in the limit of vanishing brane thickness [36]. Unlike in ¯ve dimensions (for a discussion of the » ! 1 limit in generalized 5D R» gauges see, e.g., Ref. [43] and also Ref. [30]), we cannot go to a unitary gauge where all ¯elds Aa 5;6 (a = 1; 2; 3) and B5;6 are identically set to zero. Instead, there will remain after dimensional reduction one combination of physical scalar ¯elds in the spectrum3. To make these scalars su±ciently heavier than the LeeQuiggThacker bound of ¼ 2 TeV , we can assume, e.g., a seventh dimension compacti¯ed on S1=Z2 with compacti¯cation radius R3 . R1;R2. By setting Aa 5;6;7 = B5;6;7 = 0 (Aa7 and B7 are the seventh components of the gauge ¯elds) on all boundaries of this manifold, the associated scalars can acquire for compacti¯cation scales R¡1 1 ;R¡1 2 ' 1 ¡ 2 TeV , masses well above 2 TeV . Therefore, at low energies . 2 ¡ 3 TeV , we have a model without any light scalars and will, in what follows, neglect the heavy scalar degrees of freedom. Since the Lagrangian in Eq. (3.2) does not contain any explicit gauge symmetry breaking, we can obtain consistent new BC's on the boundaries by requiring the variation of the action to be zero. Variation of the action in Eq. (3.2) yields after 3We thank H. Murayama and M. Serone for pointing out this fact. 46 partial integration ±S = Z d4x Z ¼R1 y1=0 dy1 Z ¼R2 y2=0 dy2 £ M2L ¡ @MFaM¹ ¡ fabcFbM¹Ac M ¢ ±Aa ¹ +M2 Y @MBM¹±B¹ ¤ + Z d4x Z ¼R2 y2=0 dy2 £ M2L Fa 5¹±Aa¹ +M2 Y B5¹±B¹¤¼R1 y1=0 + Z d4x Z ¼R1 y1=0 dy1 £ M2L Fa 6¹±Aa¹ +M2 Y B6¹±B¹¤¼R2 y2=0 + Z d4x · 1 g2 (@¹Fa¹º ¡ fabcFb¹ºAc ¹)±Ac º + 1 g02 @¹B¹º±Bº ¸ (y1;y2)=(0;0) = 0; (3.3) where we have (as usual) assumed that the gauge ¯elds and their derivatives go to zero for x¹ ! 1. The bulk terms in in the ¯rst line in Eq. (3.3), lead to the familiar bulk equations of motion. Moreover, since the minimization of the action requires the boundary terms to vanish as well, we obtain from the second and third line in Eq. (3.3) a set of consistent BC's for the bulk ¯elds. We break the electroweak symmetry SU(2)L £ U(1)Y ! U(1)Q by imposing on two of the boundaries following BC's: at y1 = ¼R1 : A1 ¹ = 0; A2 ¹ = 0; (3.4a) at y2 = ¼R2 : @y2(M2L A3 ¹ +M2 Y B¹) = 0; A3 ¹ ¡ B¹ = 0: (3.4b) The Dirichlet BC's in Eq. (3.4a) break SU(2)L ! U(1)I3 , where U(1)I3 is the U(1) subgroup associated with the third component of weak isospin I3. The BC's in Eq. (3.4b) break U(1)I3 £ U(1)Y ! U(1)Q, leaving only U(1)Q unbroken on the entire rectangle (see Fig. 3.1). Note, in Eq. (3.4b), that the ¯rst BC involving the derivative with respect to y2 actually follows from the second BC ±A3 ¹ = ±B¹ by minimization of the action. The gauge groups U(1)I3 and U(1)I3 £ U(1)Y remain unbroken at the boundaries y1 = 0 and y2 = 0, respectively. Locally, at the ¯xed point (y1; y2) = (0; 0), SU(2)L £ U(1)Y is unbroken. We can restrict ourselves, for simplicity, to the solutions which are relevant to EWSB, by imposing on the other 47 (0; R2) ( R1; R2) y2 (0;) y1 A3 (yU2()1)I3 U(1)Y!U(1)Q B (y2) A1 ;2(y1) ( R1;0) SU(2)L!U(1)I3 01 23===RRR 4=R L0 mW123===RRR 4=R (a) (b) k p;0 5 p05k50 pk p5k5p;5 k05 k5 Figure 3.1: Symmetry breaking of SU()L £ U(1)Y on the rectangle. At one boundary y1 = ¼R1, SU(2)L is broken to U(1)I3 while on the boundary y2 = ¼R2 the subgroup U(1)I3£U(1)Y is broken to U(1)Q, which leaves only U(1)Q unbroken on the entire rectangle. Locally, at the ¯xed point 0; 0, SU(2)L £ U(1)Y remains unbroken. The dashed arrows indicate the propagation of the lowest resonances of the gauge bosons. two boundaries the following Dirichlet BC's: at y1 = 0 : A1;2 ¹ (zM) = A 1;2 ¹ (x¹); (3.5a) at y2 = : A3 ¹(zM) = A 3 ¹(x¹); B¹(zM) = B¹(x¹); (3.5b) where the bar indicates a boundary ¯eld. The Dirichlet BC's in Eqs.(3.5) require A1;2 ¹ to be independent of y2, while A3 ¹ and B¹ become independent of y1, such that we can generally write A1;2 ¹ = A1;2(x¹; y1), A3 ¹ = A3 ¹(x¹; y2), and B¹ = B¹(x¹; y2). For the transverse4 components of the gauge ¯elds the bulk equations of motion then take the forms (p2+@2 y1)A1;2 ¹ (x¹; y1) = 0; (p2+@2 y2)A3 ¹(x¹; y2) = 0; (p2+@2 y2)B¹(x¹; y2) = 0; (3.6) where p2 = p¹p¹ and p¹ = i@¹ is the momentum in the uncompacti¯ed 4D space. Since we assume all the gauge couplings to be small, we will, in what follows, treat 4Note that @MFaM¹ = p2P¹º(p)Aa¹ + (@2 y1 + @2 y2 )Aaº = 0, where P¹º(p) = g¹º ¡ p¹pº=p2 is the operator projecting onto transverse states. 48 Aa ¹ approximately as a "free" ¯eld (i.e., without self interaction) and drop all cubic and quartic terms in Aa ¹. We assume that the fermions, in the ¯rst approximation, are localized on the brane at (y1; y2) = (0; 0), away from the walls of electroweak symmetry breaking. This choice will avoid any unwanted nonoblique corrections to the electroweak precision parameters. 3.2 E®ective theory The total e®ective 4D Lagrangian in the compacti¯ed theory Ltotal can be written as Ltotal = L0 +Le® , where Le® = R ¼R1 0 dy1 R ¼R2 0 dy2 L6 denotes the contribution from the bulk, which follows from integrating out the extra dimensions. After partial inte gration along the y1 and y2 directions, we obtain for Le® the nonvanishing boundary term Le® = ¡M2L ¼R2 h A 1 ¹@y1A1¹ + A 2 ¹@y1A2¹ i y1=0 ¡¼R1 h M2L A 3 ¹@y2A3¹ +M2 Y B¹@y2B¹ i y2=0 ; (3.7) where we have applied the bulk equations of motion and eliminated the terms from the boundaries at y1 = ¼R1 and y2 = ¼R2 by virtue of the BC's in Eqs. (3.4). Notice, that in arriving at Eq. (3.7) we have rede¯ned the bulk gauge ¯elds as A¹ ! A0 ¹ ´ A¹= p 2 to canonically normalize the kinetic energy terms of the KK modes. In order to determine Ltotal explicitly, we ¯rst solve the equations of motion in Eq. (3.6) and insert the solutions into the expression for Le® in Eq. (3.7). The most general solutions for Eqs. (3.6) can be written as A1;2 ¹ (x¹; y1) = A 1;2 ¹ (x¹) cos(py1) + b1;2 ¹ (x¹) sin(py1); (3.8a) A3 ¹(x¹; y2) = A 3 ¹(x¹) cos(py2) + b3 ¹(x¹) sin(py2); (3.8b) B¹(x¹; y2) = B¹(x¹) cos(py2) + bY ¹ (x¹) sin(py2); (3.8c) 49 where p = p p¹p¹ and we have already applied the BC's in Eq. (3.5). The coe±cients ba ¹(x¹) and bY ¹ (x¹) are then determined from the BC's in Eqs. (3.4). For b1;2 ¹ (x¹), e.g., we ¯nd from the BC's in Eq. (3.4a) that b1;2 ¹ (x¹) = ¡A¹ 1;2 (x¹) cot(p¼R1) and hence one obtains A1;2 ¹ (x¹; y1) = A 1;2 ¹ (x¹) [cos(py1) ¡ cot(p¼R1) sin(py1)] : (3.9a) In a similar way, one arrives after some calculation at the solutions A3 ¹(x¹; y2) = A 3 ¹(x¹) · cos(py2) + M2L tan(p¼R2) ¡M2 Y cot(p¼R2) M2L +M2 Y sin(py2) ¸ + B¹(x¹) M2 Y tan(p¼R2) +M2 Y cot(p¼R2) M2L +M2 Y sin(py2); (3.9b) B¹(x¹; y2) = A 3 ¹(x¹) M2L tan(p¼R2) +M2L cot(p¼R2) M2L +M2 Y sin(py2) + B¹(x¹) · cos(py2) + M2 Y tan(p¼R2) ¡M2L cot(p¼R2) M2L +M2 Y sin(py2) ¸ (3:.9c) Inserting the wavefunctions in Eqs. (3.9) into the e®ective Lagrangian in Eq. (3.7), we can rewrite Le® as Le® = A a ¹§aa(p2)A a¹ + A 3 ¹§3B(p2)B ¹ + B¹§BB(p2)B ¹ ; (3.10) where (aa) = (11); (22), and (33) and the momentumdependent coe±cients § are given by §11(p2) = §22(p2) = ¼R2M2L p cot(p¼R1); §33(p2) = ¡¼R1M2L p M2L tan(p¼R2) ¡M2 Y cot(p¼R2) M2L +M2 Y ; §3B(p2) = ¡2¼R1M2L M2 Y p tan(p¼R2) + cot(p¼R2) M2L +M2 Y ; §BB(p2) = ¡¼R1M2 Y p M2 Y tan(p¼R2) ¡M2L cot(p¼R2) M2L +M2 Y : (3.11) The §'s can be viewed as the electroweak vacuum polarization amplitudes which summarize in the low energy theory the e®ect of the symmetry breaking sector. The 50 presence of these terms leads at tree level to oblique corrections (as opposed to vertex corrections and box diagrams) of the gauge boson propagators and a®ects electroweak precision measurements [44,45]. Since Le® in Eq. (3.7) generates e®ective mass terms for the gauge bosons in the 4D theory5, the KK masses of the W§ bosons are found from the zeros of the inverse propagator as given by the solutions of the equation §11(p2) ¡ p2 2g2 = 0: (3.12) To determine the KK masses of the gauge bosons, we will from now on assume that the brane terms L0 dominate the bulk kinetic terms, i.e., we take 1=g2; 1=g02 À (ML;Y ¼)2R1R2. As a result, we ¯nd for the W§'s the mass spectrum mn = n R1 µ 1 + 2g2M2L R1R2 n2 + : : : ¶ ; n = 1; 2; : : : ; m20 = 2g2M2L R2 R1 + O(g4M4L R2 2) = m2 W; (3.13) where we identify the lightest state with mass m0 with the W§. Observe in Eq. (3.13), that the inclusion of the brane kinetic terms L0 for 1=R1; 1=R2 & O(TeV ) leads to a decoupling of the higher KKmodes with masses mn (n > 0) from the electroweak scale, leaving only the W§ states with a small mass m0 in the lowenergy theory (see Fig. 3.2). Note that a similar e®ect has been found for warped models in Ref. [47]. The calculation of the mass of the Z boson goes along the same lines as for W§, but requires, due to the mixing of A 3 ¹ with B¹ in Eq. (3.10), the diagonalization of the kinetic matrix Mkin = 0 B@ §33(p2) ¡ p2 2g2 1 2§3B(p2) 1 2§3B(p2) §BB(p2) ¡ p2 2g02 1 CA ; (3.14) which has the eigenvalues ¸§(p2) = 1 2 µ §33(p2) ¡ p2 2g2 + §BB(p2) ¡ p2 2g02 ¶ § 1 2 sµ §33(p2) ¡ p2 2g2 ¡ §BB + p2 2g02 ¶2 + §2 3B(p2); (3.15) 5For an e®ective ¯eld theory approach to oblique corrections see, e.g., Ref. [46]. 51 (0;) y1 ( R1;0) 3 01 23===RRR 4=R L0 mW123===RRR 4=R (a) (b) k p;05 p05k50 pk p5k5p;5 k05 k5 Figure 3.2: E®ect of the brane kinetic terms L0 on the KK spectrum of the gauge bosons (for the example of W§). Solid lines represent massive excitations, the bottom dotted lines would correspond to the zero modes which have been removed by the BC's. Without the brane terms (a), the lowest KK excitations are of order 1=R ' 1 TeV . After switching on the dominant brane kinetic terms (b), the zero modes are approximately \restored" with a small mass mW ¿ 1=R (dashed line), while the higher KKlevels receive small corrections to their masses (thin solid lines) and decouple below » 1 TeV . where the KK towers of the ° and Z are given by the solutions of the equations ¸¡(p2) = 0 (for °) and ¸+(p2) = 0 (for Z), respectively. By taking in Eq. (3.15) the limit p2 ! 0, it is easily seen that ¸¡(p2) = 0 has a solution with p2 = 0, which we identify with the massless ° of the SM, corresponding to the unbroken gauge group U(1)Q. The lowest excitation in the tower of solutions to ¸+(p2) = 0 has a masssquared m2 Z = 2(g2 + g02)M2L M2 Y R1 (M2L+M2 Y )R2 + O(g4M4L R2 2); (3.16) which we identify with the Z of the SM. All other KK modes of the ° and Z have masses of order & 1=R2 and thus decouple for 1=R1; 1=R2 & O(TeV ), leaving only a massless ° and a Z with mass mZ in the lowenergy theory. 52 3.3 Relation to EWPT One important constraint on any model for EWSB results from the measurement of the ½ parameter, which is experimentally known to satisfy the relation ½ = 1 to better than 1% [2]. In our model, we ¯nd from Eqs. (3.13) and (3.16) a ¯t of the natural zerothorder SM relation for the ½ parameter in terms of ½ ´ m2 W m2 Z cos2 µW = g2 g2 + g02 M2L +M2 Y M2 Y µ R2 R1 ¶2 1 cos2µW = 1; (3.17) where µW ¼ 28:8± is the Weinberg angle of the SM. For de¯niteness, we will choose in the following the 4D brane couplings g and g0 to satisfy the usual SM relation g2=(g2 + g02) = cos2µW ¼ 0:77. De¯ning ½ = 1 + ¢½, we then obtain from Eq. (3.17) that ¢½ = 0 if the bulk kinetic couplings and compacti¯cation radii satisfy the relation (M2L +M2 Y )=M2 Y = R2 1=R2 2: (3.18) Although we can thus set ¢½ = 0 by appropriately dialing the gauge couplings and the size of the extra dimensions, we observe in Eq. (3.10) that Le® introduces a manifest breaking of custodial symmetry (which transforms the three gauge bosons Aa ¹ among themselves) and will thus contribute to EWPT via oblique corrections to the SM parameters.6 To estimate the e®ect of the oblique corrections in our model let us consider in the 4D e®ective theory a general vacuum polarization tensor ¦¹º AB(p2) between two gauge ¯elds A and B which can (for canonically normalized ¯elds) be expanded as [46] i¦AB ¹º (p2) = igAgB h ¦(0) AB + p2¦(1) AB i g¹º + p¹pº terms; (3.19) where gA and gB are the couplings corresponding to the gauge ¯elds A and B, re spectively. After going in Le® back to canonical normalization by rede¯ning Aa ¹ ! A0 ¹ ´ Aa ¹=g and B¹ ! B0¹ ´ B¹=g0, we identify §aa(p2) ' 1 2[¦(0) aa + p2¦(1) aa ], for 6Note, however, that in the limit p2 ! 0, we have §11 = §33, which restores custodial symmetry. 53 (aa) = (11); (22); (33); (BB), while §3B(p2) ' ¦(0) 3B + p2¦(1) 3B. From Eqs. (3.11) we then obtain the polarization amplitudes ¦(0) 11 = ¦(0) 22 = 2M2L R2 R1 ; ¦(1) 11 = ¦(1) 22 = ¡2 ¼2M2L 3 R1R2; ¦(0) 33 = 2 M2L M2 Y M2L +M2 Y R1 R2 ; ¦(1) 33 = ¡2 ¼2M2L R1R2 M2L +M2 Y (M2L + 1 3 M2 Y ); ¦(0) 3B = ¡2 M2L M2 Y M2L +M2 Y R1 R2 ; ¦(1) 3B = ¡ 4 3 ¼2M2L M2 Y M2L +M2 Y R1R2: (3.20) A wide range of e®ects from new physics on EWPT can be parameterized in the ²1, ²2, and ²3 framework [45], which is related to the S; T, and U formalism of Ref. [44] by ²1 = ®T, ²2 = ¡®U=4 sin2µW, and ²3 = ®S=4 sin2µW. The experimental bounds on the relative shifts with respect to the SM expectations are roughly of the order ²1; ²2; ²3 . 3¢10¡3 [48]. From Eq. (3.20) we then obtain for these parameters explicitly ²1 = g2(¦(0) 11 ¡ ¦(0) 33 )=m2 W = ¡2g2 M2L m2 W R1 R2 ¡ M2 Y =(M2L +M2 Y ) ¡ (R2=R1)2¢ (3;.21a) ²2 = g2(¦(1) 33 ¡ ¦(1) 11 ) = ¡g2 4¼2 3 M4L M2L +M2 Y R1R2; (3.21b) ²3 = ¡g2¦(1) 3B = g2 4¼2 3 M2L M2 Y M2L +M2 Y R1R2; (3.21c) where we have used in the last equation that ¡²3=(gg0) = ¦(1) 3° =sin2µW ¡ ¦(1) 33 = cot µW¦(1) 3B [45]. Note in Eq. (3.21a), that for our choice of parameters we have ²1 = ¢½ = 0. The quantities j²2j and j²3j, on the other hand, are bounded from below by the requirement of having su±ciently many KK modes below the strong coupling (or cuto®) scale of the theory. Using \naive dimensional analysis" (NDA) [49,50], one obtains for the strong coupling scale ¤ of a Ddimensional gauge theory [51] roughly ¤D¡4 ' (4¼)D=2¡(D=2)=g2D, where gD is the bulk gauge coupling. In our 6D model, we would therefore have ¤ ' p 2(4¼)3=2ML;Y which leads for ML;Y ' 102 GeV to a cuto® ¤ ' 6 TeV . Assuming for simplicity ML = MY , it follows from Eq. (3.18) that R2 = R1= p 2, and using Eqs. (3.21b) and (3.21c) we obtain ²3 ' g2 96 p 2¼ (¤R2)2 ' 2:3 £ 10¡3 £ (g¤R2)2; (3.22) 54 while ²2 ' ²3. It is instructive to compare the value for ²3 in our 6D setup as given by Eq. (3.22) with the corresponding result of the 5D model in Ref. [32]. We ¯nd that by going from 5D to 6D, the strong coupling scale of the theory is lowered from » 10 TeV down to » 6 TeV . Despite the lowering of the cuto® scale, however, the parameter ²3 is in the 6D model by » 15% smaller than the corresponding 5D value7. This is due to the fact that in the 6D model the bulk gauge kinetic couplings satisfy ML = MY ' 100 GeV , while they take in 5D the values ML ' MY ' 10 GeV , which is one order of magnitude below the electroweak scale. From Eq. (3.22) we then conclude that one can take for the inverse loop expansion parameter ¤R2 ' 1=g ¼ 1:6 in agreement with EWPT. Like in the 5D case, however, the 6D model seems not to admit a loop expansion parameter in the regime ¤R2 À 1 as required for the model to be calculable. 3.4 Nonoblique corrections and fermion masses In the previous discussion, we have assumed that the fermions are (approximately) localized at (y1; y2) = (0; 0). This would make the fermions exactly massless, since they have no access to the EWSB at y1 = ¼R1 and y2 = ¼R2. In this limiting case, the e®ects on the electroweak precision parameters (²1; ²2; ²3=S; T;U) come from the oblique corrections due to the vector self energies as given by Eq. (3.10). A more realistic case will be to extend the fermion wave functions to the bulk, i.e., to the walls of EWSB, where fermion mass operators of the form CªLªR (C is some appropriate mass parameter) can be written. Thus, although the fermion wave functions will be dominantly localized at (0; 0), the pro¯le of the wavefunctions in the bulk will be such that it will have small contributions from the symmetry breaking walls, giving rise to fermion masses. The hierarchy of fermion masses would then be accommodated by 7Notice that in Ref. [32], the strong coupling scale is de¯ned by 1=¤ = 1=¤L + 1=¤R, while we assume for ML = MY that ¤ = ¤L = ¤Y . 55 some suitable choice of the parameters C [52]. To make the incorporation of heavy fermions in our model explicit, let us introduce the 6D chiral quark ¯elds Qi, Ui, and Di (i = 1; 2; 3 is the generation index), where Qi are the isodoublet quarks, while Ui and Di denote the isosinglet up and down quarks, respectively. For the cancellation of the SU(3)C £SU(2)L£U(1)Y gauge and gravitational anomalies we assume that Qi have positive and Ui;Di have negative SO(1; 5) chiralities [53]. Next, we consider the action of the top quark ¯elds with zero bulk mass, which is given by Sfermion = Z dx4 Z ¼R1 0 dy1 Z ¼R2 0 dy2 i(Q3¡MDMQ3 + U3¡MDMU3) + Z dx4 Z ¼R1 0 dy1 Z ¼R2 0 dy2 K±(y1)±(y2)i[Q3¡¹D¹Q3 + U3¡¹D¹U3] + Z dx4 Z ¼R1 0 dy1 Z ¼R2 0 dy2 C±(y1 ¡ ¼R1)±(y2 ¡ ¼R2)Q3LU3R + h:(c3:.;23) where we have added in the second line 4D brane kinetic terms with a (common) gauge kinetic parameter K = [m]¡2 at (y1; y2) = (0; 0) and in the third line we included a boundary mass term with coe±cient C = [m]¡1, which mixes Q3L and U3R at (y1; y2) = (¼R1; ¼R2). Note, that the addition of the boundary mass term in the last line of Eq. (3.23) is consistent with gauge invariance, since U(1)Q the only gauge group surviving at (y1; y2) = (¼R1; ¼R2). Consider now ¯rst the limit of a vanishing brane kinetic term K ! 0. Like in the 5D case [31], appropriate Dirichlet and Neumann BC's for Q3L;R and U3L;R would give, in the KK tower corresponding to the top quark, a lowest mass eigenstate, which is a Dirac fermion with mass mt of the order mt » C=R2, where we have de¯ned the length scale R » R1 » R2. Next, by analogy with the generation of the W§ and Z masses, switching on a dominant brane kinetic term K=R2 À 1, ensures an approximate localization of Q3L and U3R at (y1; y2) = (0; 0) and leads to mt » C=K [32]. Now, the typical values of non oblique corrections to the SM gauge couplings coming from the bulk are8 » CR=K » 8The factor C becomes obvious when treating the brane ¯elds in Eq. (3.23) as 4D ¯elds, in which 56 mt=(1=R) and keeping these contributions under control, the compacti¯cation scale 1=R must be su±ciently large. Like in 5D models, this generally introduces a possible tension between the 3rd generation quark masses and the coupling of the Z to the bottom quark. Replacing in the above discussion U3L;R with D3L;R and mt by the bottom quark mass mb(mZ) ¼ 3 GeV , we thus estimate for 1=R » 1 TeV a shift of the SM Z ! bLbL coupling by roughly » 0:3%, which is of the order of current experimental uncertainties9. Similarly, we predict in our model the coupling of the Z to the top quark to deviate by » 10% from the SM value, which can be checked in the electroweak production of single top in the Tevatron Run 2. It can also be tested in the tt pair production in a possible future linear collider. 3.4.1 Improving the calculability To improve the calculability of the model, it seems necessary to raise (for given 1=g2D ) the strong coupling scale ¤, which would allow the appearance of more KK modes below the cuto®. In fact, it has recently been argued that the compacti¯cation of a 5D gauge theory on an orbifold S1=Z2 gives a cuto® which is by a factor of 2 larger than the NDA estimate obtained for an uncompacti¯ed space [48]. Let us now demonstrate this e®ect explicitly by repeating the NDA calculation of Ref. [49] on an orbifold following the methods of Refs. [35] and [54]. For this purpose, consider a 5D scalar ¯eld Á(x¹; y) (where we have de¯ned y = y1), propagating in an S1=Z2 orbifold extra dimension. The radius of the 5th dimension is R and periodicity implies y + 2¼R » y. As a consequence, the momentum in the ¯fth dimension is quantized as p5 = n=R for integer n. Under the Z2 action y ! ¡y the scalar transforms as Á(x¹; y) = §Á(x¹;¡y), where the + (¡) sign corresponds to Á being even (odd) under case C = [m]+1 and K = [m]0. 9The LEP/SLC ¯t of ¡b=¡had in Z decay requires the shift of the Z ! bLbL coupling to be . 0:3% [3]. 57 01 (a) (b) k p;05 p05k50 pk p5k5p;5 k05 k5 Figure 3.3: Oneloop diagram for ÁÁ scattering on S1=Z2. The total incoming momentum is (p; p0 5) and the total outgoing momentum is (p; p5). Generally, it is possible that jp0 5j 6= jp5j, since the orbifold ¯xed points break 5D translational invariance. Z2. The scalar propagator on this space is given by [35, 54] D(p; p5; p0 5) = i 2 ½ ±p5;p0 5 § ±¡p5;p0 5 p2 ¡ p25 ¾ ; (3.24) where the additional factor 1=2 takes into account that the physical space is only half of the periodicity. Consider now the oneloop ÁÁ scattering diagram in Fig. 3.3. The total incoming momentum is (p; p0 5) and the total outgoing momentum is (p; p5), which can in general be di®erent, since 5D translation invariance is broken by the orbifold boundaries. Locally, however, momentum is conserved at the vertices. The diagram then reads i§ = 1 4 ¸2 2 1 2¼R X k5;k05 Z d4k (2¼)4 ½ ±k5;k05 § ±¡k5;k05 k2 ¡ k2 5 ¾½ ±(p5¡k5);(p0 5¡k05 ) § ±¡(p5¡k5);(p0 5¡k05 ) (p ¡ k)2 ¡ (p5 ¡ k5)2 ¾ ; (3.25) where ¸ is the quartic coupling and the additional factor 1=4 results from working on S1=Z2. After summing over k05, the integrand can be written as F(k5) = 1 (k2 ¡ k2 5) [(p ¡ k)2 ¡ (p5 ¡ k5)2] © ±p5p0 5 + ±p5;¡p0 5 § ±2k5;(p5+p0 5) § ±2k5;(p5¡p0 5) ª : (3.26) In Eq. (3.26), the ¯rst two terms in the bracket conserve jp0 5j and contribute to the bulk kinetic terms of the scalar. The last two terms, on the other hand, violate jp0 5j conservation and thus lead to a renormalization of the brane couplings [35]. Note 58 that these brane terms lead in Eq. (3.25) to a logarithmic divergence. Applying, on the other hand, to the bulk terms the Poisson resummation identity 1 2¼R X1 m=¡1 F(m=R) = X1 n=¡1 Z 1 ¡1 dk 2¼ e¡2¼ikRnF(k); (3.27) we obtain a sum of momentum space integrals, where the \local" n = 0 term diverges linearly like in 5D uncompacti¯ed space. This term contributes a linear divergence to the diagram such that the scattering amplitude becomes under order one rescalings of the random renormalization point for the external momenta of the order i§ ! ¸2 4 Z d5k (2¼)5 [k2(p ¡ k)2]¡1 ' ¸2 2 ¤ (4¼)5=2¡(5=2) ; (3.28) where ¤ is an ultraviolet cuto®. On S1=Z2, we thus indeed obtain for the strong coupling scale ¤ ' 48¼3¸¡2, which is two times larger than the NDA value obtained in 5D uncompacti¯ed space. This is also in agreement with the de¯nition of ¤ for a 5D gauge theory on an interval given in Ref. [48]. Similarly, when the 5th dimension is compacti¯ed on S1=(Z2£Z02 ) [55], we expect a raising of ¤ by a factor of 4 with respect to the uncompacti¯ed case. Let us brie°y estimate how far this could improve the calculability of our 6D model. To this end, we assume, besides the two extra dimensions compacti¯ed on the rectangle, two additional extra dimensions with radii R3 and R4, each of which has been compacti¯ed on S1=(Z2 £Z02 ). We assume that the gauge bosons are even under the actions of the Z2£Z02 groups. Moreover, we take for the bulk kinetic coe±cients in eight dimensions M4L = M4 Y and set R3 = R4 = R2 = R1= p 2. From the expression analogous to Eq. (3.21c), we then obtain the estimate ²3 ' g2(¼MLR2)4=3 p 2, where the relative factor (¼R2=2)2, arises from integrating over the physical space on each circle, which is only 1=4 of the circumference. With respect to the NDA value ¤4 ' (4¼)4¡(4)M4L in uncompacti¯ed space, the cuto® gets now modi¯ed as ¤4 ! 16 ¢ ¤4, implying that ²3 ' g2 192 p 2 (¤R2=4)4 ' 1:3 £ 10¡3 £ (¤R2=4)4: (3.29) 59 In agreement with EWPT, the loop expansion parameter could therefore assume here a value (¤R2)¡1 ' 0:25, corresponding to the appearance of 4 KK modes per extra dimension below the cuto®. Taking also a possible additional raising of ¤ by a factor of p 2 due to the reduced physical space on the rectangle into account, one could have (¤R2)¡1 ' 0:2 with 5 KK modes per extra dimension below the cuto®. In conclusion, this demonstrates that by going beyond ¯ve dimensions, the calculability of Higgsless models could be improved by factors related to the geometry. 60 CHAPTER 4 A New Two Higgs Doublet Model 4.1 Model and the Formalism Our proposed model is based on the symmetry group SU(3)c £ SU(2)L £ U(1) £ Z2. In addition to the usual SM fermions, we have three EW singlet righthanded neutrinos, NRi; i = 1 ¡ 3, one for each family of fermions. The model has two Higgs doublets, Â and Á. All the SM fermions and the Higgs doublet Â, are even under the discrete symmetry, Z2, while the RH neutrinos and the Higgs doublet Á are odd under Z2. Thus all the SM fermions except the lefthanded neutrinos, couple only to Â. The SM lefthanded neutrinos, together with the righthanded neutrinos, couple only to the Higgs doublet Á. The gauge symmetry SU(2) £ U(1) is broken spontaneously at the EW scale by the VEV of Â, while the discrete symmetry Z2 is broken by a VEV of Á, and we take hÁi » 10¡2 eV . Thus, in our model, the origin of the neutrino masses is due to the spontaneous breaking of the discrete symmetry Z2. The neutrinos are massless in the limit of exact Z2 symmetry. Through their Yukawa interactions with the Higgs ¯eld Á, the neutrinos acquire masses much smaller than those of the quarks and charged leptons due to the tiny VEV of Á. The Yukawa interactions of the Higgs ¯elds with the leptons are LY = ylª l LlRÂ + yºlª l LNR eÁ + h:c:; (4.1) where ª l L = (ºl; l)L is the usual lepton doublet and lR is the charged lepton singlet. The ¯rst term gives rise to the mass of the charged leptons, while the second term gives a tiny neutrino mass. The interactions with the quarks are the same as in the 61 Standard Model with Â playing the role of the SM Higgs doublet. Note that in our model, a SM lefthanded neutrino, ºL combines with a right handed neutrino, NR, to make a massive Dirac neutrino with a mass » 10¡2 eV, the scale of Z2 symmetry breaking. For simplicity, we do not consider CP violation in the Higgs sector. (Note that in this model, spontaneous CP violation would be highly suppressed by the small VEV ratio and could thus be neglected. However, one could still consider explicit CP violation). The most general Higgs potential consistent with the SM £Z2 symmetry is [56] V = ¡¹21 ÂyÂ ¡ ¹22 ÁyÁ + ¸1(ÂyÂ)2 + ¸2(ÁyÁ)2 + ¸3(ÂyÂ)(ÁyÁ) ¡ ¸4jÂyÁj2 ¡ 1 2 ¸5[(ÂyÁ)2 + (ÁyÂ)2]: (4.2) The physical Higgs ¯elds are a charged ¯eld H, two neutral scalar ¯elds h and ¾, and a neutral pseudoscalar ¯eld ½. In the unitary gauge, the two doublets can be written Â = 1 p 2 0 B@ p 2(VÁ=V )H+ h0 + i(VÁ=V )½ + VÂ 1 CA ; Á = 1 p 2 0 B@ ¡ p 2(VÂ=V )H+ ¾0 ¡ i(VÂ=V )½ + VÁ 1 CA ; (4.3) where VÂ = hÂi, VÁ = hÁi, and V 2 = V 2 Â + V 2 Á . The particle masses are m2 W = 1 4 g2V 2; m2 H = 1 2 (¸4 + ¸5)V 2; m2 ½ = ¸5V 2; m2 h;¾ = (¸1V 2 Â + ¸2V 2 Á ) § q (¸1V 2 Â ¡ ¸2V 2 Á )2 + (¸3 ¡ ¸4 ¡ ¸5)2V 2 Â V 2 Á : (4.4) 62 An immediate consequence of the scenario under consideration is a very light scalar ¾ with mass m2 ¾ = 2¸2V 2 Á [1 + O(VÁ=VÂ)]: (4.5) The mass eigenstates h; ¾ are related to the weak eigenstates h0; ¾0 by h0 = ch + s¾; ¾0 = ¡sh + c¾; (4.6) where c and s denotes the cosine and sine of the mixing angles, and are given by c = 1 + O(V 2 Á =V 2 Â ); s = ¡ ¸3 ¡ ¸4 ¡ ¸5 2¸1 (VÁ=VÂ) + O(V 2 Á =V 2 Â ): (4.7) Since VÁ » 10¡2 eV and VÂ » 250 GeV, this mixing is extremely small, and can be neglected. Hence, we see that h behaves essentially like the SM Higgs (except of course in interactions with the neutrinos). The interactions of the neutral Higgs ¯elds with the Z are given by Lgauge = g 2V (cVÁ + sVÂ)(½@¹h ¡ h@¹½)Z¹ + g 2V (sVÁ ¡ cVÂ)(½@¹¾ ¡ ¾@¹½)Z¹ + g2 4 (sVÁ ¡ cVÂ)hZ¹Z¹ + g2 4 (cVÁ + sVÂ)¾Z¹Z¹ + g2 8 (h2 + ¾2 + ½2)Z¹Z¹ (4.8) where g2 = g2 + g02, and VÂ and VÁ are the two VEV's. 4.2 Phenomenological Implications We now consider the phenomenological implications of this model. There are sev eral interesting phenomenological implications which can be tested in the upcoming 63 neutrino experiments and high energy colliders. The light neutrinos in our model are Dirac particles. So neutrinoless double beta decay is not allowed in our model. This is a very distinctive feature of our model for the neutrino masses compared to the traditional seesaw mechanism. In the seesaw model, light neutrinos are Majorana particles, and thus neutrinoless double beta decay is allowed. The current limit on the double beta decay is mee » 0:3 eV . This limit is expected to go down to about mee » 0:01 eV in future experiments [57]. If no neutrinoless double beta decay is observed to that limit, that will cast serious doubts on the seesaw model. In our model, of course, it is not allowed at any level. Next, we consider the implications of our model for high energy colliders. First we consider the production of the light scalar ¾ in e+e¡ collisions. The only possible decay modes of this particle are a diphoton mode, ¾ ! °° which can occur at the oneloop level and, if it has enough mass, a ¾ ! ºº mode. The one loop decay to two photons takes place with quarks, W bosons, or charged Higgs bosons in the loop. The largest contribution to this decay mode is » e8m5 ¾=mq 4. This gives the lifetime of ¾ to be » 1020 years, which is much larger than the age of the universe. Thus ¾ essentially behaves like a stable particle, and its production at the colliders will lead to missing energy in the event. The couplings of ¾ to quarks and charged leptons takes place only through mixing which is highly suppressed (proportional to the ratio VÁ=VÂ). Thus we need only consider its production via its interactions with gauge bosons. The ZZ¾ coupling is also highly suppressed, so that processes such as e+e¡ ! Z¤ ! Z¾ and Z ! Z¤¾ ! ff¾ are negligible. However, no such suppression occurs for the ZZ¾¾ coupling. Consider the Z decay process Z ! Z¤¾¾ ! ff¾¾. A direct calculation yields the width (neglecting the ¾ and fermion masses), 64 ¡(Z ! ff¾¾) = G3 Fm5 Z(g2V + g2A ) 2 p 2(2¼)5 Z mZ=2 0 dE1 Z mZ=2 0 dE2 £ Z 1 ¡1 d(cos µ) E2 1E2 2(3 ¡ cos µ) (2E1E2 ¡ 2E1E2 cos µ ¡ m2 Z)2 + m2 Z¡2 Z ; (4.9) where gV = T3 ¡ 2Qsin2 µW and gA = T3. This gives X f ¡(Z ! ff¾¾) ' 2:5 £ 10¡7 GeV: (4.10) For the 1:7 £ 107 Z's observed at resonance at LEP1 [58], this gives an expectation of only about two such events. Now we consider the production of the heavy Higgs particles in our model. Since the charged Higgs H§ and the pseudoscalar, ½ can be produced along with the light scalar ¾, there will be stricter mass bound on these particles than in a typical two Higgs doublet model. Let us consider the pseudoscalar ½, and assume m½ < mZ. Then the Z can decay via Z ! ¾½. Since ½ couples negligibly to all SM fermions except the neutrinos, here we need only consider its decay to ºº (or ¾¾ if we consider CP violation), so this process contributes to the invisible decay width of the Z. The width for this process is ¡ = GFm3 Z 24 p 2¼ µ 1 ¡ m2 ½ m2 Z ¶3 (4.11) This is less than the experimental uncertainty in the invisible Z width for m½ & 78 GeV . (The experimental value of the invisible Z width is 499:0 § 1:5 MeV [59].) For m½ > mZ, real pseudoscalar ½ can be produced via e+e¡ ! Z¤ ! ½¾. The total cross section for this process is ¾ = G2 Fm4 Z(g2V + g2A )s 24¼ µ 1 s ¡ m2 Z ¶2 µ 1 ¡ m2 ½ s ¶3 : (4.12) 65 For LEP2, p s ' 200 GeV , we ¯nd that less than one event is expected in ' 3000 pb¡1 [6] of data for m½ & 95 GeV . Note that the bound on the ½ mass we obtain is much less than the mass for which the Higgs potential becomes strongly coupled (¸5 · 2 p ¼ which gives m½ · 470 GeV). For m½ > mZ, the Z can still decay invisibly through Z ! ½¤¾ ! ºº¾. The width for this decay is invwidth¡ = GFm2 Zy2 ºl 3 p 2(2¼)3 Z mZ=2 0 dE E3(mZ ¡ 2E) (m2 Z ¡ 2mZE ¡ m2 ½)2 : (4.13) Summing over generations, this gives ¡(m½ = 100 GeV ) ' (0:1 MeV )( 1 3 X l y2 ºl) ¡(m½ = 200 GeV ) ' (4 £ 10¡3 MeV )( 1 3 X l y2 ºl): (4.14) Even if we take 1 3 P y2 ºl » 1, these values are well within the experimental uncertainty in the invisible Z width of 1:5 MeV . Note that if we allow explicit CP violation in the Higgs sector, the invisible decay Z ! ½¾ ! ¾¾¾ will also occur. Our model has very interesting implications for the discovery signals of the Higgs boson at the high energy colliders, such as the Tevatron and LHC. Note that since VÁ is extremely small compared to VÂ, the neutral Higgs boson, h is like the SM Higgs boson so far its decays to fermions and to W and Z bosons are concerned. However, in our model, h has new decay modes, such as h ! ¾¾ which is invisible. This could change the Higgs signal at the colliders dramatically. The width for this invisible decay mode h ! ¾¾ is given by ¡(h ! ¾¾) = (¸3 + ¸4 + ¸5)2V 2 Â 32¼mh : (4.15) 66 100 150 200 250 300 mh HGeVL 0 0.2 0.4 0.6 0.8 Br Hh s sL 0 0.2 0.4 0.6 0.8 1 l* 0 0.2 0.4 0.6 0.8 1 Br Hh s sL Figure 4.1: Left panel: Branching ratio for h ! ¾¾ as a function of mh for the value of the parameter, ¸¤ = 0:1. Right panel: Branching ratio for h ! ¾¾ as a function of ¸¤ for mh = 135 GeV . Using m2 h = 2¸1V 2 Â + O(V 2 Á =V 2 Â ); (4.16) this can be written ¡(h ! ¾¾) = (¸3 + ¸4 + ¸5)2mh 64¼¸1 : (4.17) Depending on the parameters, it is possible for the dominant decay mode of h to be this invisible mode. The branching ratios for the Higgs decay to this invisible mode are shown in Fig. 4 (left panel), for the Higgs mass range from 100 to 300 GeV , for the choice of the value of the parameter, ¸¤ equal to 0:1 where ¸¤ is de¯ned to be equal to (¸3+¸4+¸5)2 ¸1 . The right panel in Fig. 4 shows how this branching ratio depends on this parameter for a Higgs mass of 135 GeV . (The results for the branching ratio is essentially the same for other values of the Higgs mass between 120 and 160 GeV ). We see that for a wide range of this parameter, for the Higgs mass up to about 160 GeV , the invisible decay mode dominates, thus changing the Higgs search strategy at the Tevatron Run 2 and the LHC . The production rate of the neutral scalar Higgs h in our model are essentially the same as in the SM. This implies that the Higgs mass bound from LEP is not signi¯cantly altered . (The L3 collaboration set a bound of 67 mh ¸ 112:3 GeV for an invisibly decaying Higgs with the SM production rate [60]). However, because of the dominance of the invisible decay mode, it will be very di±cult to observe a signal at the LHC in the usual production and decay channels such as qqh ! qqWW, qqh ! qq¿ ¿ , h ! °°, h ! ZZ ! 4l, tth (with h ! bb) and h ! WW ! lºlºl [61]. However, a signal with such an invisible decay mode of the Higgs (as in our model) can be easily observed at the LHC through the weak boson fusion processes, qq ! qqW+W¡ ! qqH and qq ! qqZZ ! qqH [62] if appropriate trigger could be designed for the ATLAS and CMS detector. For example, with only 10 fb¡1 of data at the LHC, such a signal can be observed at the 95 percent CL with an invisible branching ratio of 31 percent or less for a Higgs mass of upto 400 GeV [62]. Thus our model can be easily tested at the LHC for a large region of the Higgs mass. Of course, establishing that this signal is from the Higgs boson production will be very di±cult at the LHC. For the Higgs search at the Tevatron, the usual signal from the Wh production, and the subsequent decays of h to WW¤ or bb will be absent. The most promising mode in our model will be the production of ZH, with Z decaying to l§l§ (l = e; ¹) and the Higgs decaying invisibly. There will be a peak in the missing energy distribution in the ¯nal state with a Z. We urge the Tevatron collaborations to look for such a signal. 4.3 Cosmological Implications Our model has several interesting astrophysical and cosmological implications. Firstly, there is a problem with primordial nucleosynthesis [63]. This occurs because the relatively strong interactions between left and righthanded neutrinos and the light scalar ¾ will keep righthanded neutrinos and ¾ in thermal equilibrium with left handed neutrinos during nucleosynthesis. So, the e®ective number of light degrees of freedom, g¤ = gb + 7 8gf (gb and gf are the numbers of bosonic and fermionic spin degrees of freedom respectively), is 68 g¤ = (g¤)SM + 1 + 7 8 (6) = 17: (4.18) (Equivalently, the e®ective number of neutrinos is Nº = 6 + 4 7 .) This increases the expansion rate of the universe, which is proportional to p g¤. As a result, reactions which interconvert protons and neutrons freeze out of thermal equilibrium at a higher temperature, increasing the ratio of neutrons to protons during nucleosynthesis. This increase alters the abundances of light elements produced in subsequent nucleosyn thesis reactions, most notably, helium4 is greatly overproduced. The mass fraction of helium4 obtained here is ' 0:3 compared to the observed fraction ' 0:25. To solve this problem, our model requires a nonstandard nucleosynthesis scenario. One possibility is a large neutrino degeneracy. It is assumed in standard nucleosynthe sis that the chemical potential of neutrinos ¹º ' 0. However, since relic neutrinos are not observed, this is not required by observation. A large value of ¹º alters the equilibrium ratio of neutrons to protons, n p = e¡¹º=T µ n p ¶ ¹º=0 ; (4.19) leading to an alteration of light element abundances. Our problem can be solved with ¹º » 0:1 MeV . In depth studies have been conducted, where the e®ective number of neutrinos, neutrino degeneracy and the density of baryons are allowed to vary, in order to ¯nd the most general values consistent with BBN and WMAP [65](as well as studies which ¯x Nº = 3, leading to much stronger bounds on neutrino degeneracy [66]). These studies ¯nd upper bounds on Nº from 7:1 to 8:7, depending on how conservative an interpretation of the data is used. Another possible solution could be the existence of massive particle species that decay after nucleosynthesis. Energetic decay products of these particles interact with background nuclei, causing nonthermal nuclear reactions, such as helium4 dissociation, that reset light element 69 abundances [64]. (We also note that in the above analysis, we have taken three right handed neutrinos. For the oscillation experiments, as well as for direct measurements, the lightest neutrino mass can be zero. So, only two righthanded neutrinos are strictly required. This could make the Big Bang nucleosynthesis problem somewhat milder.) There are also bounds on the e®ective number of neutrinos coming from astro physical observations other than light element abundances. For example, data from WMAP and the Sloan Digital Sky Survey (SDSS) power spectrum of luminous red galaxies, give a bound 0:8 < Nº < 7:6 [67]. The authors of [68] claim that data from the SDSS Lyman® forest power spectrum, along with cosmic microwave background, supernova, and galaxy clustering data, seem to require Nº > 3. Additionally, the ºº¾ interaction can a®ect supernova explosion dynamics,and since this interaction can be fairly strong it may bind ºº, giving rise to the possibility of ºº atoms and a new kind of star formation. Also, the spontaneous breaking of the discrete global symmetry Z2 will lead to the formation of cosmological domain walls. These walls will have energy per unit area ´ » V 3 Á , so their e®ect will be small. The resulting temperature anisotropies are ±T T ' G´H¡1 0 » 10¡20; (4.20) where G is Newton's gravitational constant and H0 is the present Hubble parameter. The observed level of CMB temperature anisotropies is 10¡5 [59], so this is not a problem. 70 CHAPTER 5 CONCLUSIONS We have presented several scenarios that alter the Higgs sector from that of the SM. First, we presented a renormalizable nonsupersymmetric model based on the ¯nite symmetry G = (G1 £G2)oA4, with G1 = S3 £S3 £S3 £S3 and G2 = Z2 £Z2 £Z2, with SM leptons assigned to representations of A4. Neutrino masses are generated by a Higgs ¯eld Á belonging to a 16dimensional representation of G1 o A4 while chargedlepton masses are generated by a Higgs ¯eld Â belonging to a 6dimensional representation of G2 oA4. The additional symmetries, G1 and G2, prevent quadratic and cubic interactions between Á and Â and allow only a trivial quartic interaction that does not cause an alignment problem, addressing the alignment problem without altering the desired properties of the family symmetry. In this way, we are able to explain all aspects of neutrino mixing using only symmetries which are spontaneously broken by the Higgs mechanism. Next, we have considered a 6D Higgsless model for EWSB based only on the SM gauge group SU(2)L £ U(1)Y . The model is formulated in °at space with the two extra dimensions compacti¯ed on a rectangle of size » (TeV )¡2. EWSB is achieved by imposing consistent BC's on the edges of the rectangle. The higher KK resonances of W§ and Z decouple below » 1 TeV through the presence of a dominant 4D brane induced gauge kinetic term at the point where SU(2)L £ U(1)Y remains unbroken. The ½ parameter is arbitrary and can be set exactly to unity by appropriately choosing the bulk gauge couplings and compacti¯cation scales. The resulting gauge couplings 71 in the e®ective 4D theory arise essentially from the brane couplings, slightly modi¯ed (at the level of one percent) by the bulk interaction. Thus, the main role played by the bulk interactions is to break the electroweak gauge symmetry. We calculate the treelevel oblique corrections to the S, T, and U parameters and ¯nd them to be consistent with current data. Finally, we have presented a simple extension of the Standard Model supplemented by a discrete symmetry, Z2. We have also added three righthanded neutrinos, one for each family of fermions, and one additional Higgs doublet. While the electroweak symmetry is spontaneously broken at the usual 100 GeV scale, the discrete symmetry, Z2 remains unbroken to a scale of about 10¡2 eV . The spontaneous breaking of this Z2 symmetry by the VEV of the second Higgs doublet generates tiny masses for the neutrinos. The neutral heavy Higgs in our model is very similar to the SM Higgs in its couplings to the gauge bosons and fermions, but it also couples to a very light scalar Higgs present in our model. This light scalar Higgs, ¾, is essentially stable, or decays to ºº. Thus the production of this ¾ at the high energy colliders leads to missing energy. The SMlike Higgs, for a mass up to about 160 GeV dominantly decays to the invisible mode h ! ¾¾. Thus the Higgs signals at high energy hadron colliders are dramatically altered in our model. Our model also has interesting implications for astrophysics and cosmology. 72 BIBLIOGRAPHY [1] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, p.367 of Elementary Particle Theory, ed. N. Svartholm (Almquist and Wiksells, Stockholm, 1969); S.L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2 (1970) 1285. [2] Particle Data Group Collaboration, K. Hagiwara et al., Phys. Rev. D 66 (2002) 010001. [3] The ElectroWeak Working Group, http://lepewwg.web.cern.ch/LEPEWWG/ [4] F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321; P.W. Higgs, Phys. Lett. 12 (1964) 132 and Phys. Rev. Lett. 13 (1964) 508; T.W. Kibble, Phys. Rev. 155 (1967) 1554. [5] C. H. Llewellyn Smith, Phys. Lett. B 46 (1973) 233; D.A. Dicus, V.S. Mathur, Phys. Rev. D 7 (1973) 3111; J.M. Cornwall, D.N. Levin, and G. Tiktopoulos, Phys. Rev. D 10 (1974) 1145; B.W. Lee, C. Quigg, and H.B. Thacker, Phys. Rev. D 16 (1977) 1519; M.J.G. Veltmann, Acta Phys. Polon. B 8 (1977) 475. [6] ALEPH, DELPHI, L3, and OPAL Collaborations, The LEP working group for Higgs boson searches, G. Abbiendi et al., Phys. Lett. B565 (2003) 61. [7] T. Schwetz, M. Tortola, and J.W.F. Valle, New J. Phys. 10, 113011 (2008). [8] G.L. Fogli, E. Lisi, A. Mirizzi, D. Montanino, and P.D. Serpico, Phys. Rev. D74, 093004 (2006). [9] P.F. Harrison, D.H. Perkins and W.G. Scott, Phys. Lett. B458, 79 (1999); Phys. Lett. B530, 167 (2002). 73 [10] Z.Z. Xing, Phys. Lett. B533, 85 (2002); X.G. He and A. Zee, Phys. Lett. B560, 87 (2003); Phys. Rev. D68, 037302 (2003). [11] A. Aranda, C.D. Carone, and R.F. Lebed, Phys. Rev. D62, 016009 (2000). [12] J. Kubo, A. Mondragon, M. Mondragon, and E. RodriguezJauregui, Prog. Theor. Phys. 109, 795 (2003). [13] C. Hagedorn, M. Lindner, and F. Plentinger, Phys. Rev. D74, 025007 (2006). [14] I. de Medeiros Varzielas, S.F. King, and G.G. Ross, Phys. Lett. B648, 201 (2007). [15] M.C. Chen and K.T. Mahanthappa, Phys. Lett. B652, 34 (2007). [16] F. Feruglio, C. Hagedorn, Y. Lin, and L. Merlo, Nucl. Phys. B775, 120 (2007). [17] P. Frampton and T. Kephart, JHEP 0709, 110 (2007). [18] A. Blum, C. Hagedorn, and M. Lindner, Phys. Rev. D77, 076004 (2008). [19] W. Grimus and L. Lavoura, JHEP 0904, 013 (2009). [20] E. Ma and G. Rajasekaran, Phys. Rev. D64, 113012 (2001) [21] E. Ma, Mod. Phys. Lett. A17, 2361 (2002). [22] K.S. Babu, E. Ma, and J.W.F. Valle, Phys. Lett. B552, 207 (2003). [23] G. Altarelli and F. Feruglio, Nucl. Phys. B720, 64 (2005); Nucl. Phys. B741, 215 (2006). [24] K. S. Babu and X.G. He, hepph/0507217. [25] X.G. He, Y.Y. Keum, and R.R. Volkas, JHEP 0604, 039 (2006). [26] E. Ma, Phys. Lett. B671, 366 (2009). 74 [27] J. Scherk and J.H. Schwarz, Phys. Lett. B 82 (1979) 60; Nucl. Phys. B 153 (1979) 61; E. Cremmer, J. Scherk, and J.H. Schwarz, Phys. Lett. B 84 (1979) 83; Y. Hosotani, Phys. Lett. B 126 (1983) 309 (1983); Phys. Lett. B 129 (1983) 193; An nals Phys. 190 (1989) 233; for a recent review see M. Quiros, hepph/0302189. [28] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1921 (1921) 966; O. Klein, Z. Phys. 37 (1926) 895. [29] R. Sekhar Chivukula, D.A. Dicus, and H.J. He, Phys. Lett. B 525 (2002) 175; R.S. Chivukula, D.A. Dicus, H.J. He, and S. Nandi, Phys. Lett. B 562 (2003) 109. [30] C. Csaki, C. Grojean, H. Murayama, L. Pilo, and J. Terning, Phys. Rev. D 69 (2004) 055006. [31] C. Csaki, C. Grojean, J. Hubisz, Y. Shirman, and J. Terning, hepph/0310355. [32] R. Barbieri, A. Pomarol, and R. Rattazzi, Phys. Lett. B 591 (2004) 141. [33] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690; Phys. Rev. Lett. 83 (1999) 3370. [34] E.A. Mirabelli and M.E. Peskin, Phys. Rev. D 58 (1998) 065002. [35] H. Georgi, A.K. Grant, and G. Hailu, Phys. Lett. B 506 (2001) 207. [36] W.D. Goldberger and M.B. Wise, Phys. Rev. D 65 (2002) 025011. [37] See for example, Higgs Hunters Guide, by J.F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, AddisonWesley Publishing, New York, 1990. [38] R. Barbieri, L.J. Hall, and V.S. Rychkov, Phys. Rev. D74 (2006) 015007. [39] S.K. Kim, Group Theoretical Methods: And Applications to Molecules and Crys tals, Cambridge University Press, New York, 1999. 75 [40] C. Hagedorn, M. Lindner, and R.N. Mohapatra, JHEP 0606, 042 (2006). [41] C.S. Lam, Phys. Rev. D79, 073015 (2008); W. Grimus, L. Lavoura, and P.O. Ludl, 0906.2689 [hepph]. [42] B.A. Dobrescu and E. Ponton, JHEP 0403 (2004) 071 (2004). [43] A. Muck, A. Pilaftsis, and R. Ruckl, Phys. Rev. D 65 (2002) 085037. [44] M.E. Peskin and T. Takeuchi, Phys. Rev. D 46 (1992) 381 (1992). [45] G. Altarelli and R. Barbieri, Phys. Lett. B 253 (1991) 161; G. Altarelli, R. Barbieri, and S. Jadach, Nucl. Phys. B 369 (1992) 3; Erratumibid. B 376 (1992) 444. [46] B. Holdom and J. Terning, Phys. Lett. B 247 (1990) 88; M. Golden and L. Randall, Nucl. Phys. B 361 (1991) 3. [47] M. Carena, E. Ponton, T.M.P. Tait, and C.E.M.Wagner, Phys. Rev. D 67 (2003) 096006. [48] R. Barbieri, A. Pomarol, R. Rattazzi, and A. Strumia, hepph/0405040. [49] A. Manohar and H. Georgi, Nucl. Phys. B 234 (1984) 189. [50] H. Georgi and L. Randall, Nucl. Phys. B 276 (1986) 241. [51] Z. Chacko, M.A. Luty, and E. Ponton, JHEP 0007 (2000) 036. [52] K. Agashe, A. Delgado, M.J. May, and R. Sundrum, JHEP 0308 (2003) 050. [53] B.A. Dobrescu and E. Poppitz, Phys. Rev. Lett. 87 (2001) 031801; N. Arkani Hamed, H.C. Cheng, B.A. Dobrescu, and L.J. Hall, Phys. Rev. D 62 (2000) 096006. [54] H.C. Cheng, K.T. Matchev, and M. Schmaltz, Phys. Rev. D 66 (2002) 036005. 76 [55] R. Barbieri, L.J. Hall, and Y. Nomura, Phys. Rev. D 63 (2001) 105007. [56] S. Nandi, Phys. Lett. B202 (1988) 385, Erratumibid, B207 (1988) 520. [57] L. Baudis et al. Phys. Rev. Lett. 83 (1999) 41; IGEX Collaboration, C.E. Aalseth et al., Phys. Rev. D65 (2002) 092007; I. Abd et al., hepex/0404039. [58] ALEPH, DELPHI, L3, and OPAL Collaborations, The LEP EW working group, The SLD EW and heavy °avour groups, Phys. Rept. 427 (2006) 257. [59] W.M. Yao et al., J. Phys. G33 (2006) 1. [60] L3 Collaboration, P. Achard et al., Phys. Lett. B609 (2005) 35. [61] K. Crammer, B. Mellado, W. Quayle, and S. L. Wu, (ATLAS Collaboration), ATLPHYS2004034. [62] O.J.P. Eboli and D. Zepppenfeld, Phys. Lett. B495 (2000) 147. [63] The Early Universe, by E.W. Kolb and M.S. Turner, AddisonWesley Publishing, New York, 1990. [64] See for example, M. Kawasaki, K. Kohri, and T. Moroi, Phys. Rev. D71 (2005) 083502; S. Dimopoulos, R. Esmailzadeh, L.J. Hall, and G.D. Starkman, Astro phys. J. 330 (1988) 545. [65] V. Barger, J.P. Kneller, P. Langacker, D. Marfatia, and G. Steigman, Phys. Lett. B569 (2003) 123; A. Cuoco, F. Iocco, G. Mangano, G. Miele, O. Pisanti, and P.D. Serpico, Int. J. Mod. Phys. A19 (2004) 4431; G. Steigman, Phys. Scripta T121 (2005) 142. [66] P.D. Serpico and G.G. Ra®elt, Phys. Rev. D71 (2005) 127301. [67] K. Ichikawa, M. Kawasaki, and F. Takahashi, arXiv: astroph/0611784v1. 77 [68] U. Seljak, A. Slosar, and P.McDonald, JCAP 0610 (2006) 014. [69] A. Vilenkin, Phys. Rept. 121 (1985) 263. 78 VITA Steven Gabriel Candidate for the Degree of Doctor of Philosophy Dissertation: NEW IDEAS IN HIGGS PHYSICS Major Field: Physics Biographical: Personal Data: Born in Snellville, Georgia, United States on March 2, 1980. Education: Received the B.S. degree from Georgia State University, Atlanta, Georgia, United States, 2002, in Physics Completed the requirements for the degree of Doctor of Philosophy with a major in Physics, Oklahoma State University in May, 2010. Name: Steven Gabriel Date of Degree: May, 2010 Institution: Oklahoma State University Location: Stillwater, Oklahoma Title of Study: NEW IDEAS IN HIGGS PHYSICS Pages in Study: 78 Candidate for the Degree of Doctor of Philosophy Major Field: Physics The Higgs mechanism, which is responsible for electroweak symmetry breaking and unitarization of massive W§ and Z scattering, is a fundamental ingredient of the Standard Model (SM). However, there is as yet no direct evidence of the Higgs boson, so that the details of the Higgs sector, if it even exists, remain a mystery. Here, we explore several scenarios that alter the Higgs sector from that of the SM. The ¯rst uses additional symmetries of the Higgs sector to address certain issues of neutrino mixing, the second uses extra dimensional boundary conditions to avoid the need for a Higgs entirely, and the last uses additional Higgs ¯elds to provide an alternative explanation for tiny neutrino masses. ADVISOR'S APPROVAL:
Click tabs to swap between content that is broken into logical sections.
Rating  
Title  New Ideas in Higgs Physics 
Date  20100501 
Author  Bagriel, Steven Allen 
Keywords  Particle physics 
Document Type  
Full Text Type  Open Access 
Abstract  The Higgs mechanism, which is responsible for electroweak symmetry a fundamental ingredient of the Standard Model (SM). However, there is as yet no direct evidence of the Higgs boson, so that the details of the Higgs sector, if it even exists, remain a mystery. Here, we explore several scenarios that alter the Higgs sector from that of the SM. The first uses additional symmetries of the Higgs sector to address certain issues of neutrino mixing, the second uses extra dimensional boundary conditions to avoid the need for a Higgs entirely, and the last uses additional Higgs fields to provide an alternative explanation for tiny neutrino masses. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  NEW IDEAS IN HIGGS PHYSICS By STEVEN GABRIEL Bachelor of Science in Physics Georgia State University Atlanta, Georgia, United States 2002 Submitted to the Faculty of the Graduate College of Oklahoma State University in partial ful¯llment of the requirements for the Degree of DOCTOR OF PHILOSOPHY May, 2010 COPYRIGHT °c By STEVEN GABRIEL May, 2010 NEW IDEAS IN HIGGS PHYSICS Dissertation Approved: S. Nandi Dissertation Advisor K.S. Babu F. Rizatdinova B. Binegar A. Gordon Emslie Dean of the Graduate College iii TABLE OF CONTENTS Chapter Page 1 INTRODUCTION 1 2 Unfaithful Representations of Finite Groups and Tribimaximal Neu trino Mixing 7 2.1 The Discrete Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.3 Derivation of Representations . . . . . . . . . . . . . . . . . . . . . . 14 2.4 Invariants Under the Discrete Symmetry . . . . . . . . . . . . . . . . 33 2.5 Calculation of the Neutrino Mass Matrix . . . . . . . . . . . . . . . . 38 2.6 Calculation of the Charged Lepton Mass Matrix . . . . . . . . . . . . 42 3 A 6D Higgsless Standard Model 45 3.1 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2 E®ective theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.3 Relation to EWPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 3.4 Nonoblique corrections and fermion masses . . . . . . . . . . . . . . 55 3.4.1 Improving the calculability . . . . . . . . . . . . . . . . . . . . 57 4 A New Two Higgs Doublet Model 61 4.1 Model and the Formalism . . . . . . . . . . . . . . . . . . . . . . . . 61 4.2 Phenomenological Implications . . . . . . . . . . . . . . . . . . . . . 63 4.3 Cosmological Implications . . . . . . . . . . . . . . . . . . . . . . . . 68 iv 5 CONCLUSIONS 71 BIBLIOGRAPHY 73 v LIST OF TABLES Table Page 2.1 This table shows the matrices representing the generators in each irrep. of A4, in a certain basis. . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 This table shows the assignments of the fermions and Higgs ¯elds under SU(2)L £ U(1)Y £ [(S4 3 £ Z3 2 ) o A4] . . . . . . . . . . . . . . . . . . . 11 vi LIST OF FIGURES Figure Page 3.1 Symmetry breaking of SU(2)L £ U(1)Y on the rectangle. At one boundary y1 = ¼R1, SU(2)L is broken to U(1)I3 while on the boundary y2 = ¼R2 the subgroup U(1)I3 £ U(1)Y is broken to U(1)Q, which leaves only U(1)Q unbroken on the entire rectangle. Locally, at the ¯xed point (0; 0), SU(2)L£ U(1)Y remains unbroken. The dashed arrows indicate the propagation of the lowest resonances of the gauge bosons. . . . . . . . . . . . . . . . . . 48 3.2 E®ect of the brane kinetic terms L0 on the KK spectrum of the gauge bosons (for the example of W§). Solid lines represent massive excitations, the bottom dotted lines would correspond to the zero modes which have been removed by the BC's. Without the brane terms (a), the lowest KK excitations are of order 1=R ' 1 TeV . After switching on the dominant brane kinetic terms (b), the zero modes are approximately \restored" with a small mass mW ¿ 1=R (dashed line), while the higher KKlevels receive small corrections to their masses (thin solid lines) and decouple below » 1 TeV . 52 3.3 Oneloop diagram for ÁÁ scattering on S1=Z2. The total incoming momen tum is (p; p0 5) and the total outgoing momentum is (p; p5). Generally, it is possible that jp0 5j 6= jp5j, since the orbifold ¯xed points break 5D transla tional invariance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.1 Left panel: Branching ratio for h ! ¾¾ as a function of mh for the value of the parameter, ¸¤ = 0:1. Right panel: Branching ratio for h ! ¾¾ as a function of ¸¤ for mh = 135 GeV . . . . . . . . . . . . . . 67 vii CHAPTER 1 INTRODUCTION The Standard Model (SM) of electroweak interactions [1], based on the gauge symmetry group SU(2)L £ U(1)Y , provides a highly successful description of elec troweak precision tests (EWPT) [2, 3]. One fundamental ingredient of the SM is the Higgs mechanism [4], which accomplishes electroweak symmetry breaking (EWSB) and at high energies unitarizes massive W§ and Z scattering through the presence of the scalar Higgs doublet [5]. Although the mass of the Higgs boson is not predicted by the SM, accurate measurements of the top quark and the W boson mass at the Tevatron, as well as the Z boson mass at LEP, have narrowed the SM Higgs boson mass between 80 and 200 GeV [3]. Failure to observe the SM Higgs boson at LEP2 has also placed a direct lower bound of 114 GeV on its mass [6]. The dominant decay modes of the SM Higgs boson are to bb, WW, ZZ or tt, depending on its mass. Extensions of the SM may avoid constraints on the Higgs mass, and may allow Higgs bosons with masses less than the above limits. The dominant decay modes of the Higgs bosons can also be altered in such extensions, thus transforming the discovery signals for the Higgs bosons at the Large Hadron Collider (LHC). However, there is as yet no direct evidence of the Higgs boson, so that the details of the Higgs sector, if it even exists, remain a mystery. Thus, it is important to explore alternative Higgs sector scenarios. One interesting scenario involves the role of the Higgs sector in neutrino mixing. The existence of neutrino masses is now well established experimentally [7, 8]. At 1¾, 1 the masssquared di®erences and mixing angles are [7]: ¢m2 21 = 7:65(+0:23= ¡ 0:20) £ 10¡5 eV 2; ¢jm2 31j = 2:40(+0:13= ¡ 0:11) £ 10¡3 eV 2; and sin2 µ23 = 0:50(+0:022= ¡ 0:016); sin2 µ12 = 0:341(+0:07= ¡ 0:06); sin2 µ13 < 0:035: These values are in good agreement with a tribimaximal mixing pattern given by the mixing matrix [9, 10] UMNS = 0 BBBB@ q 2 3 p1 3 0 ¡p1 6 p1 3 ¡p1 2 ¡p1 6 p1 3 p1 2 1 CCCCA P (1.1) where P is a diagonal phase matrix. This corresponds to sin2 µ23 = 1=2; sin2 µ12 = 1=3; sin2 µ13 = 0: It has long been known that such a mixing pattern can be obtained using a ¯nite family symmetry [1025] such as A4 [1925]. In these models, A4 is broken to a Z2 subgroup in the neutrino sector by a triplet Higgs, with the VEV structure (0; 1; 0) or some permutation thereof, and to a Z3 subgroup in the charged lepton sector by a triplet Higgs, with the VEV structure (1; 1; 1). However, there is a serious technical problem with this, in that couplings between the Higgs ¯elds responsible for the symmetry breaking will force the VEV's to align, upsetting the desired breaking pattern [2125]. To overcome this problem, one can introduce more complicated symmetries. In Section 2, we consider models where the SM lepton families belong to representations of the ¯nite symmetry which are not faithful (that is, not every member of the group is represented by a distinct transformation). In e®ect, the 2 Higgs sector knows about the full symmetry while the lepton sector does not. We consider a renormalizable nonsupersymmetric gauge theory with an additional ¯nite symmetry that has the semidirect product structure G = (G1 £ G2) o A4, with G1 = S3£S3£S3£S3 and G2 = Z2£Z2£Z2. A symmetry thus structured will contain G1, G2, and G1 £ G2 as invariant subgroups, so that G will have representations corresponding to the homomorphisms G=(G1 £ G2) » A4, G=G1 » G2 o A4, and G=G2 » G1oA4. SM leptons can then be assigned to representations of A4. Neutrino masses are generated by a Higgs ¯eld Á, belonging to a 16dimensional representation of G1 o A4, while chargedlepton masses are generated by a Higgs ¯eld Â, belonging to a 6dimensional representation of G2oA4. The additional symmetries, G1 and G2, prevent quadratic and cubic interactions between Á and Â and allow only a trivial quartic interaction (i.e., the interaction is the product of quadratic invariants) that does not cause an alignment problem. In this way, the alignment problem is addressed without altering the desired properties of the family symmetry, so that neutrino mixing can be explained using only symmetries which are broken spontaneously by the Higgs mechanism. However, no fundamental scalar particle has been observed yet in nature, and as long as there is no direct evidence for the existence of the Higgs boson, the actual mechanism of EWSB remains a mystery. In case the Higgs boson will also not be found at the Tevatron or the LHC, it will therefore be necessary to consider alternative ways to achieve EWSB without a Higgs. We explore this possibility in Section 3. It is well known, that in extra dimensions, gauge symmetries can also be broken by boundary conditions (BC's) on a compact space [27]. Here, a geometric "Higgs" mechanism ensures treelevel unitarity of longitudinal gauge boson scattering through a tower of KaluzaKlein (KK) [28] excitations [29]. The original model for Higgsless EWSB [30] is an SU(2)L£SU(2)R£U(1)B¡L gauge theory compacti¯ed on an interval [0; ¼R] in ¯vedimensional (5D) °at space. At one end of the interval, SU(2)R £ 3 U(1)B¡L is broken to U(1)Y . At the other end, SU(2)L £ SU(2)R is broken to the diagonal subgroup SU(2)D, thereby leaving only U(1)Q of electromagnetism unbroken in the e®ective fourdimensional (4D) theory. Although this model exhibited some similarities with the SM, the ½ parameter deviated from unity by » 10% and the lowest KK excitations of theW§ and Z were too light (» 240 GeV ) to be in agreement with experiment. These problems have later been resolved by considering the setup in warped space [33]. Based on the same gauge group, similar e®ects can be realized in 5D °at space [32], when 4D brane kinetic terms [34{36] dominate the contribution from the bulk. In 5D Higgsless models, a ½ parameter close to unity is achieved at the expense of enlarging the SM gauge group by an additional gauge group SU(2)R, which introduces a gauged custodial symmetry in the bulk. However, it is possible to obtain consistent 6D Higgsless models of EWSB, which are based only on the SM gauge group SU(2)L £ U(1)Y and allow the ½ parameter to be set equal to unity. We consider a Higgsless model for EWSB in six dimensions, which is based only on the SM gauge group SU(2)L £ U(1)Y , where the gauge bosons propagate in the bulk. The model is formulated in °at space with the two extra dimensions compacti¯ed on a rectangle and EWSB is achieved by imposing consistent BC's. The higher KK resonances of W§ and Z decouple below » 1TeV through the presence of a dominant 4D brane induced gauge kinetic term. The ½ parameter is arbitrary and can be set exactly to one by an appropriate choice of the bulk gauge couplings and compacti¯cation scales. Unlike in the 5D theory, the mass scale of the lightest gauge bosons W and Z is solely set by the dimensionful bulk couplings, which (upon compacti¯cation via mixed BC's) are responsible for EWSB.We calculate the treelevel oblique corrections to the S; T; and U parameters and ¯nd that they are in better agreement with data than in proposed 5D warped and °at Higgsless models. In Section 4, we present a model that includes a second Higgs doublet that pro vides an alternate explanation for the tiny masses of the SM neutrinos, as well as 4 possibilities for altering signals for discovery of the Higgs at the LHC. Our proposal is to extend the SM electroweak symmetry to SU(2)L £ U(1) £ Z2 and introduce three SU(2)£U(1) singlet right handed neutrinos, NR, as well as an additional Higgs doublet, Á. While the SM symmetry is spontaneously broken by the VEV of an EW doublet Â at the 100 GeV scale, the discrete symmetry Z2 is spontaneously broken by the tiny VEV of this additional doublet Á at a scale of 10¡2 GeV . Thus in our model, tiny neutrino masses are related to this Z2 breaking scale. We note that although our model has extreme ¯ne tuning, that is no worse than the ¯ne tuning problem in the usual GUT model. Many versions of the two Higgs doublet model have been exten sively studied in the past [37]. The examples include: a) a supersymmetric two Higgs doublet model, b) nonsupersymmetric two Higgs doublet models i) in which both Higgs doublets have vacuum expectation values (VEV's) with one doublet coupling to the up type quarks only, while the other coupling to the down type quarks only, ii) only one doublet coupling to the fermions, and iii) only one doublet having VEV's and coupling to the fermions [38]. What is new in our model is that one doublet couples to all the SM fermions except the neutrinos, and has a VEV which is same as the SM VEV, while the other Higgs doublet couples only to the neutrinos with a tiny VEV » 10¡2 eV . This latter involves the Yukawa coupling of the lefthanded SM neutrinos with a singlet righthanded neutrino, NR. The lefthanded SM neutrinos combine with the singlet righthanded neutrinos to make massive Dirac neutrinos. The neutrino mass is so tiny because of the tiny VEV of the second Higgs doublet, which is responsible for the spontaneous breaking of the discrete symmetry, Z2. Note that in the neutrino sector, our model is very distinct from the seasaw model. Lepton number is strictly conserved, and hence no NRNR mass terms are allowed. Thus the neutrino is a Dirac particle, and there is no neutrinoless double ¯ decay in our model. In the Higgs sector, in addition to the usual massive neutral scalar and pseudoscalar Higgs, and two charged Higgs, our model contains one essentially massless scalar 5 Higgs. We will show that this is still allowed by the current experimental data and can lead to an invisible decay mode of the SMlike Higgs boson, thus complicating the Higgs searches at the Tevatron and the LHC. 6 CHAPTER 2 Unfaithful Representations of Finite Groups and Tribimaximal Neutrino Mixing 2.1 The Discrete Symmetry As described in the Introduction, we consider a renormalizable nonsupersymmetric gauge theory with an additional ¯nite symmetry given by the semidirect product1 G = (G1 £G2)oA4, with G1 = S3 £S3 £S3 £S3 and G2 = Z2 £Z2 £Z2. The group A4 can be described using two generators obeying the relations, X2 = Y 3 = E; XY X = Y 2XY 2; (2.1) where E is the identity. The irreducible representations are one real singlet, 1; two complex singlets, 10 and 100; and one real triplet, 3. Table 1 gives X and Y in each of these representations for a certain choice of basis. The S3 generators, Ai and Bi, and the Z2 generators, Ci, obey A3i = B2 i = E; BiAiB¡1 i = A¡1 i ; C2 i = E; (2.2) and Ci commutes with Ai and Bi. The remaining relations de¯ning the full symmetry are XA1X¡1 = A2; XA2X¡1 = A1; XA3X¡1 = A4; XA4X¡1 = A3; 1The semidirect product, N oH, contains N and H as subgroups and obeys hnh¡1 2 N for all n 2 N and h 2 H [39]. Thus, N is an invariant subgroup. The number of elements in the group, denoted by jN o Hj, is jNjjHj. The semidirect product exists when H has a factor group which is a subgroup of the automorphism group of N. 7 X Y 1 1 1 10 1 ! 100 1 !2 3 0 BBBB@ ¡1 0 0 0 1 0 0 0 ¡1 1 CCCCA 0 BBBB@ 0 0 1 1 0 0 0 1 0 1 CCCCA ! = e2i¼=3 Table 2.1: This table shows the matrices representing the generators in each irrep. of A4, in a certain basis. XB1X¡1 = B2; XB2X¡1 = B1; XB3X¡1 = B4; XB4X¡1 = B3; (2.3) Y A1Y ¡1 = A1; Y A2Y ¡1 = A3; Y A3Y ¡1 = A4; Y A4Y ¡1 = A2; Y B1Y ¡1 = B1; Y B2Y ¡1 = B3; Y B3Y ¡1 = B4; Y B4Y ¡1 = B2; (2.4) XC1X¡1 = C1C2C2; XC2X¡1 = C3; XC3X¡1 = C2; Y C1Y ¡1 = C2; Y C2Y ¡1 = C3; Y C3Y ¡1 = C1: (2.5) It's easy to see that if C1, C2, and C3 are all represented by the identity matrix, then (6) is trivially satis¯ed. So in this case, one need only ¯nd representations that respect Eqs. (2)(5). But this is equivalent to ¯nding representations of G1oA4. The representations of this type that we will be using are a real 16dimensional represen tation, a real 48dimensional representation, and a real 8dimensional representation. These will be referred to hereafter as 16AB, 48AB, and 8AB. The matrices representing the remaining generators in each of these representations can be found in Section 2.3 8 below. Similarly, if A1, A2, A3, A4, B1, B2, B3, and B4 are all represented by the identity matrix, then (4) and (5) are trivially satis¯ed. Finding these representations corresponds to ¯nding representations of G2 o A4. For this type, we will be using a real 6dimensional representation, which we will call 6C. The matrices representing the remaining generators in this representation can also be found in Section 2.3. Fi nally, if the Ai's, Bi's, and Ci's are all represented by the identity matrix, then the only nontrivial relation is (2), corresponding to the representations of A4 given in Table 1. These representations will be used for SM leptons. 2.2 The Model The SM lepton assignments under A4 are eR1 » 1; eR2 » 10; eR3 » 100; (L1;L2;L3) » 3: (2.6) The ¯nite symmetry is broken at a scale M¤, which is large compared to the weak scale, by two Higgs ¯elds, Á and Â. Neutrino Dirac masses are generated by the real Higgs ¯eld Á belonging to 16AB, while charged lepton masses are generated by the real Higgs ¯eld Â belonging to 6C. Symmetryinvariant interactions between Á and Â must consist of products of G1 invariants constructed from Á with G2 invariants constructed from Â. The 16dimensional representation to which Á belongs is (2; 2; 2; 2) with respect to G1 = S3 £ S3 £ S3 £ S3, so that there is only one quadratic G1 invariant that can be constructed with Á, which is invariant under the full symmetry. Thus, there are no cubic invariants involving both Á and Â, and the only quartic invariant containing both is a trivial product of quadratic invariants, which does not generate a VEV alignment problem. Then the potential of Á and Â has the form VÁÂ = a1f1(Á; Á) + a2f2(Â; Â) + b1g1(Á; Á; Á) + b2g2(Â; Â; Â) + c1h1(Á) + c2h2(Á) +c3h3(Â) + c4h4(Â) + c5h5(Â) + c6h6(Â) + c7f1(Á; Á)f2(Â; Â); (2.7) 9 where the functions f1, f2, g1, g2, h1, h2, h3, h4, h5, and h6 are given in Section 2.4 below. The neutrino masses are generated from Á by integrating out multiplets of heavy righthanded neutrinos, with masses at a scale M¤ which is large compared to the EW scale. These multiplets are N » 3, N0 » 48AB, and N00 » 8AB. If the Z2 subgroup of A4 generated by X is left unbroken by the VEV of Á (along with an additional accidental Z2 that is actually part of S4, see [41]), the light neutrino mass matrix is forced to have the form Mº = 0 BBBB@ aº 0 cº 0 bº 0 cº 0 aº 1 CCCCA : (2.8) This matrix is diagonalized by Uº = 1 p 2 0 BBBB@ 1 0 ¡1 0 p 2 0 1 0 1 1 CCCCA Pº; (2.9) where diagonal Pº is a phase matrix. The charged lepton masses are generated from Â by integrating out multiplets of heavy vectorlike fermions, whose masses are also at the high scale M¤, with the same gauge quantum numbers as righthanded charged leptons. These are EL;R » 3 and E0 L;R » 6C. If the Z3 subgroup of A4 generated by Y is left unbroken by the VEV of Â, the light lefthanded charged lepton mass matrix is forced to have the form My eMe = 1 p 3 0 BBBB@ 1 1 1 1 ! !2 1 !2 ! 1 CCCCA 0 BBBB@ ae 0 0 0 be 0 0 0 ce 1 CCCCA 1 p 3 0 BBBB@ 1 1 1 1 !2 ! 1 ! !2 1 CCCCA = UL 0 BBBB@ ae 0 0 0 be 0 0 0 ce 1 CCCCA Uy L (2.10) 10 SU(2)L U(1)Y (S4 3 £ Z3 2 ) o A4 L 2 1/2 3 eR1 1 1 1 eR2 1 1 10 eR3 1 1 100 N 1 0 3 N0 1 0 48AB N00 1 0 8AB EL 1 1 3 ER 1 1 3 E0L 1 1 6C E0R 1 1 6C Á 1 0 16AB Â 1 0 6C H 2 1/2 1 Table 2.2: This table shows the assignments of the fermions and Higgs ¯elds under SU(2)L £ U(1)Y £ [(S4 3 £ Z3 2 ) o A4] Eqs. (10) and (11) then give the desired form (1) for the mixing matrix UMNS = UT L U¤ º . The symmetry assignments of the fermions and Higgs ¯elds in the model are summarized in Table 2. From the matrices given in Section 2.3, it can be seen that the most general VEV structure for Â that leaves the Z3 subgroup of A4 generated by Y unbroken is hÂi = (vÂ1; vÂ2; vÂ1; vÂ2; vÂ1; vÂ2): (2.11) Upon minimizing the potential, one ¯nds that vÂ2 = 0, vÂ1 6= 0 is allowed. Here, C1C2C3 is left unbroken in addition to Y . Since the SM leptons do not transform un der the Ci's, these additional symmetries do not a®ect the light lepton mass matrices. 11 So the desired minimum is hÂi = (vÂ; 0; vÂ; 0; vÂ; 0): (2.12) Since C1C2C3, and Y commute, the subgroup they generate is Z2 £ Z3. Of course, Â also trivially leaves all Ai's and Bi's unbroken. The most general VEV structure for Á that leaves the Z2 subgroup of A4 generated by X unbroken is hÁi = (vÁ1; vÁ2; vÁ2; vÁ3; vÁ4; vÁ5; vÁ6; vÁ7; vÁ4; vÁ6; vÁ5; vÁ7; vÁ9; vÁ9; vÁ9; vÁ10): (2.13) Upon minimizing the potential, we ¯nd that hÁi = (0; 0; 0; 0; vÁ; vÁ; vÁ; vÁ; vÁ; vÁ; vÁ; vÁ; 0; 0; 0; 0) (2.14) is acceptable. In addition to X, this VEV leaves the generators B1, B2, B3B4, and A3A4 unbroken. These form the subgroup D4 £ S3, with D4 generated by B1, B2, and X and with S3 generated by A3A4 and B3B4. Of course, Á also leaves all Ci's unbroken. (To leave the accidental Z2 ½ S4 mentioned above unbroken requires vÁ5 = vÁ6 in (14), which is satis¯ed in (15).) From (8), we ¯nd that vÁ and vÂ in (13) and (15) must be solutions to 2a1 + 3b1vÁ + 2(c1 + c2)v2Á + 6c7v2Â = 0; 2a2 + 3b2vÂ + 4(c3 + c5)v2Â + 16c7v2Á = 0: Neutrino Dirac masses are generated through Lº = ¸(L1N1 + L2N2 + L3N3)eH + mN(N2 1 + N2 2 + N2 3 ) + m0 Nf3(N0;N0) + m00 Nf4(N00;N00) +®1g3(N; Á;N0) + ®2g4(N00; Á;N0) + ¯g5(Á;N0;N0); (2.15) where the functions f3, f4, g3, g4, and g5 are given in Section 2.4. N » 3 is required because the SM Higgs H only breaks EW symmetry, so that it can only cause left handed neutrinos to mix with a triplet. Since 3 £ 16AB = 48AB, Á » 16AB induces 12 mixing between N and N0 » 48AB. N00 » 8AB is needed to remove unwanted acci dental symmetries. Upon integrating out the heavy righthanded neutrinos, the light neutrino mass matrix (9) is obtained (see Section 2.5). The light neutrino masses are found to be m1 = ¯¯¯¯¯ ¸2v2 2 m0 Nm00 N ¡ 4®2 2v2Á + ¯vÁm00 N ¡2®2 1v2Á m00 N + mN ¡ m0 Nm00 N ¡ 4®2 2v2Á + ¯vÁm00 N ¢ ¯¯¯¯¯ ; m2 = ¯¯¯¯¯ ¸2v2 2 m0 Nm00 N ¡ 2®2 2v2Á + ¯vÁm00 N ¡2®2 1v2Á m00 N + mN ¡ m0 Nm00 N ¡ 2®2 2v2Á + ¯vÁm00 N ¢ ¯¯¯¯¯ ; m3 = ¯¯¯¯¯ ¸2v2 2 m0 N + ¯vÁ ¡2®2 1v2Á + mN(m0 N + ¯vÁ) ¯¯¯¯¯ : Charged lepton masses are generated through Le = ·(ER1L1 + ER2L2 + ER3L3)H + mE(ER1EL1 + ER2EL2 + ER3EL3) + m0 Ef2(E 0 R;E0 L) +°1g6(ER;E0 L; Â) + °2g6(EL;E0 R; Â) + ²1eR1f2(E0 L; Â) + ²2g7(eR2;E0 L; Â) + ²3g8(eR3;E0 L; Â) +´1g2(E 0 R;E0 L; Â) + ´2g2(E 0 L;E0 R; Â) + c:c:; (2.16) where the functions g6, g7, and g8 are once again given in Section 2.4. Upon integrating out the heavy fermions, the light charged lepton masssquared matrix (11) is obtained (see Section 2.6). The masses are m2e = 3j·²1°2v2Â vj2 3j²1vÂ(mE + °2vÂ)j2 + jmE(m0 E + ´1vÂ + ´2vÂ) ¡ °1°2v2Â j2 ; m2 ¹ = 3j·²2°2v2Â vj2 3j²2vÂ(mE + !°2vÂ)j2 + jmE(m0 E + !2´1vÂ + !´2vÂ) ¡ °1°2v2Â j2 ; m2¿= 3j·²3°2v2Â vj2 3j²3vÂ(mE + !2°2vÂ)j2 + jmE(m0 E + !´1vÂ + !2´2vÂ) ¡ °1°2v2Â j2 : 13 2.3 Derivation of Representations Let H ½ G, and assume that we understand the representation theory of H. G can be decomposed into cosets of H, G = Xn i=1 siH = Xn i=1 fsihj h 2 Hg; (2.17) where the number n of cosets is equal to the number of elements in G divided by the number of elements in H. The coset decomposition is independent of the choice of the representative si for each coset. Let ° be a kdimensional irreducible representation of H. It induces a representation °" of G given by °"(g)ij = X h2H °(h)±(h; s¡1 i gsj): (2.18) In other words, the ij subblock of °" is °(s¡1 i gsj) when s¡1 i gsj 2 H and is zero otherwise. Note that the dimension of the induced representation is kn. In general, °" is reducible. Up to this point, it was not necessary to assume that H is invariant. Let us now do so, h 2 H =) ghg¡1 2 H; 8g 2 G: Then for each g 2 G, we can de¯ne a new representation °g from ° °g(h) = °(ghg¡1): (2.19) For g 2 H, °g is equivalent to ° (that is, °g is ° in a di®erent basis), °g(h) = °(g)°(h)°¡1(g): If g is outside of H, then °g may be either equivalent or inequivalent to °. The set of all inequivalent irreducible representations that can be obtained from ° by the transformation (20) (including ° itself) is called the orbit O° of the representation °. Note that the true singlet (i.e., °(h) = 1, 8h 2 H) is always in its own orbit. Two 14 representations that belong to the same orbit have equivalent induced representations (19). If g1 and g2 belong to the same coset in (18), then they di®er by a factor belonging to H. So, by an argument similar to that showing °g is equivalent to ° for g 2 H, °g1 and °g2 are equivalent. Then, to identify the orbit, it su±ces to consider how ° transforms under the coset representatives, si. Let H° be the set of all g 2 G such that °g is equivalent to °. Then, not only does H° contain H, but it consists of a whole number of cosets from (18). H° is an invariant subgroup of G and is called the little group of the representation ° (or of the orbit O°). Note that the little group of the true singlet is always the entire group G. If each coset sends ° to an inequivalent representation, then the number of representations in O° is equal to the number n of cosets, and H° = H. In this case, the induced representation °" in (19) is irreducible. Otherwise, it is reducible. As an example, consider the group A4 = (Z2 £ Z2) o Z3. Let X and Z be the Z2 generators and Y be the Z3 generator. Y cyclicly permutes the three Z2 subgroups of Z2 £ Z2: Y XY ¡1 = Z; Y ZY ¡1 = XZ; Y (XZ)Y ¡1 = X: (2.20) Note that Z is not an independent generator (Z = Y XY ¡1). The decomposition into cosets of Z2 £ Z2 is A4 = fE; X; Z; XZg + fY; Y X; Y Z; Y XZg + fY 2; Y 2X; Y 2Z; Y 2XZg:(2.21) We can choose E, Y , and Y 2 = Y ¡1 as coset representatives. Then the induced representation (19) takes the form °"(X) = 0 BBBB@ °(X) 0 0 0 °(Y ¡1XY ) 0 0 0 °(Y XY ¡1) 1 CCCCA ; °"(Z) = 0 BBBB@ °(Z) 0 0 0 °(Y ¡1ZY ) 0 0 0 °(Y ZY ¡1) 1 CCCCA ; 15 °"(Y ) = 0 BBBB@ 0 0 °(E) °(E) 0 0 0 °(E) 0 1 CCCCA : (2.22) There are four onedimensional irreducible representations of Z2 £ Z2: (a) °(X) = 1; °(Z) = 1 (b) °(X) = ¡1; °(Z) = ¡1 (c) °(X) = ¡1; °(Z) = 1 (d) °(X) = 1; °(Z) = ¡1 As always, the true singlet (a) belongs to its own orbit. The induced representation is °"(X) = °"(Z) = 0 BBBB@ 1 0 0 0 1 0 0 0 1 1 CCCCA ; °"(Y ) = 0 BBBB@ 0 0 1 1 0 0 0 1 0 1 CCCCA : These matrices can be diagonalized simultaneously, yielding three onedimensional representations: 1 : X = Z = 1; Y = 1; 10 : X = Z = 1; Y = !; 100 : X = Z = 1; Y = !¤; with ! = exp(2i¼=3). Using (21), we have for (b), °Y (X) = °(Y XY ¡1) = °(Z) = ¡1; °Y (Z) = °(Y ZY ¡1) = °(XZ) = 1; 16 which is (c), and °Y ¡1(X) = °(Y ¡1XY ) = °(XZ) = 1; °Y ¡1(Z) = °(Y ¡1ZY ) = °(X) = ¡1; which is (d). Thus, (b), (c), and (d) make up a single orbit. Since the number of representations in this orbit is equal to the number of cosets in (22), the induced representation is irreducible. From (23), we have 3 : X = 0 BBBB@ ¡1 0 0 0 1 0 0 0 ¡1 1 CCCCA ; Z = 0 BBBB@ ¡1 0 0 0 ¡1 0 0 0 1 1 CCCCA ; Y = 0 BBBB@ 0 0 1 1 0 0 0 1 0 1 CCCCA : For every group G, there exists a maximal invariant subgroup H; that is, there are no proper invariant subgroups that contain H. For this subgroup, the factor group G=H is simple. If G is itself a simple group then the maximal invariant subgroup is the trivial subgroup, fEg. There also exists a maximal invariant subgroup H0 of H. So, we have a chain, Hi ½ Hi¡1 ½ ::: ½ H2 ½ H1 = G; where Hj+1 is an invariant subgroup of Hj , and Hj=Hj+1 is simple. This chain can be continued until the trivial subgroup fEg is reached, but for our purposes it su±ces to stop at the largest subgroup whose representation theory we already know. Then, if we know how to determine the representation theory of a group from that of its maximal invariant subgroup, we can apply this recursively. So, let G be a group, and let H be its maximal invariant subgroup. We will further assume that G=H is a cyclic group. Since the little group of a representation of H must be an invariant subgroup of G containing H, the little group for each representation must be either H or all of G. If the little group is H, the induced representation is irreducible. So we need only concern ourselves with the case where the little group is G. Let us consider another example. The group S4 is equal to A4 o Z2. The A4 generators given above and the Z2 generator, which we will denote by W, obey the 17 relations WXW¡1 = Z; WZW¡1 = X; WYW¡1 = Y ¡1; (2.23) along with (21). The decomposition into cosets of A4 is S4 = A4+WA4. Noting that W¡1 = W, the induced representations have the form °"(X) = 0 B@ °(X) 0 0 °(WXW) 1 CA ; °"(Z) = 0 B@ °(Z) 0 0 °(WZW) 1 CA ; °"(Y ) = 0 B@ °(Y ) 0 0 °(WYW) 1 CA ; °"(W) = 0 B@ 0 °(E) °(E) 0 1 CA : (2.24) For the 1 of A4, °"(X) = °"(Z) = °"(Y ) = 0 B@ 1 0 0 1 1 CA ; °"(W) = 0 B@ 0 1 1 0 1 CA : Upon diagonalization, this yields X = Z = Y = 1; W = 1; X = Z = Y = 1; W = ¡1: Without much di±culty, we see that 10 and 100 make up an orbit, so that the induced representation is irreducible, X = Z = 0 B@ 1 0 0 1 1 CA ; Y = 0 B@ ! 0 0 !¤ 1 CA ; W = 0 B@ 0 1 1 0 1 CA : The triplet 3 of A4, °(X) = 0 BBBB@ ¡1 0 0 0 1 0 0 0 ¡1 1 CCCCA ; °(Z) = 0 BBBB@ ¡1 0 0 0 ¡1 0 0 0 1 1 CCCCA ; °(Y ) = 0 BBBB@ 0 0 1 1 0 0 0 1 0 1 CCCCA ; 18 must belong to its own orbit because there is no other possibility. We have °W(X) = °(WXW) = °(Z) = 0 BBBB@ ¡1 0 0 0 ¡1 0 0 0 1 1 CCCCA ; °W(Z) = °(WZW) = °(X) = 0 BBBB@ ¡1 0 0 0 1 0 0 0 ¡1 1 CCCCA ; °W(Y ) = °(WYW) = °¡1(Y ) = 0 BBBB@ 0 1 0 0 0 1 1 0 0 1 CCCCA : Since this representation must lie in its own orbit, there exists a matrix S such that S°W(X)S¡1 = °(X); S°W(Z)S¡1 = °(Z); S°W(Y )S¡1 = °(Y ): Indeed, by inspection, we see that we can take S = 0 BBBB@ 1 0 0 0 0 1 0 1 0 1 CCCCA : Note that S¡1 = S. The induced representation of S4 can then be written °"(X) = 0 B@ °(X) 0 0 S°(X)S 1 CA ; °"(Z) = 0 B@ °(Z) 0 0 S°(Z)S 1 CA ; °"(Y ) = 0 B@ °(Y ) 0 0 S°(Y )S 1 CA ; °"(W) = 0 B@ 0 I I 0 1 CA : 19 Let S = 1 p 2 0 B@ I I I ¡I 1 CA 0 B@ I 0 0 S 1 CA : Then S°"(X)S¡1 = 0 B@ °(X) 0 0 °(X) 1 CA ; S°"(Z)S¡1 = 0 B@ °(Z) 0 0 °(Z) 1 CA ; S°"(Y )S¡1 = 0 B@ °(Y ) 0 0 °(Y ) 1 CA ; S°"(W)S¡1 = 0 B@ S 0 0 ¡S 1 CA : So there are two irreducible triplets of S4, X = °(X); Z = °(Z); Y = °(Y ); W = S; and X = °(X); Z = °(Z); Y = °(Y ); W = ¡S: Let us now consider the group (Z2 £ Z2 £ Z2) o A4. Let Ci be the Z2 generators. With the A4 generators in (21), they obey XC1X¡1 = C1C2C2; XC2X¡1 = C3; XC3X¡1 = C2; (2.25) ZC1Z¡1 = C3; ZC2Z¡1 = C1C2C3; ZC3Z¡1 = C2; (2.26) Y C1Y ¡1 = C2; Y C2Y ¡1 = C3; Y C3Y ¡1 = C1: (2.27) We have the chain Z3 2 ½ (Z3 2 ) o Z2 ½ (Z3 2 ) o (Z2 £ Z2) ½ (Z3 2 ) o A4 of invariant subgroups. Without much di±culty, we can see that the representations (a) C1 = 1; C2 = ¡1; C3 = ¡1 20 (b) C1 = ¡1; C2 = 1; C3 = 1 (c) C1 = ¡1; C2 = 1; C3 = ¡1 (d) C1 = 1; C2 = ¡1; C3 = 1 (e) C1 = ¡1; C2 = ¡1; C3 = 1 (f) C1 = 1; C2 = 1; C3 = ¡1; lie in the same orbit with respect to (Z3 2 )oA4. Now consider the subgroup (Z3 2 )oZ2, where the last Z2 is generated by X. From (26), under X C1 ! C1C2C3; C2 $ C3; so that (a) $ (a); (b) $ (b); (c) $ (d); (e) $ (f): So (a) and (b) each give two onedimensional representations of (Z4 2 ) o Z2: (a) C1 = 1; C2 = ¡1; C3 = ¡1; X = 1 (a0) C1 = 1; C2 = ¡1; C3 = ¡1; X = ¡1 (b) C1 = ¡1; C2 = 1; C3 = 1; X = 1 (b0) C1 = ¡1; C2 = 1; C3 = 1; X = ¡1; while (c/d) and (e/f) each give twodimensional irreducible representations: (c=d) C2 = M1; C1 = C3 = M2; X = S 21 (e=f) C3 = M1; C1 = C2 = M2; X = S; where M1 ´ 0 B@ 1 0 0 ¡1 1 CA ; M2 ´ 0 B@ ¡1 0 0 1 1 CA ; S ´ 0 B@ 0 1 1 0 1 CA : (2.28) Now add Z to obtain (Z3 2 ) o (Z2 £ Z2). Under Z C1 ! C3; C2 ! C1C2C3; X $ X; so that (a) $ (b); (a0) $ (b0); (c=d) $ (c=d); (e=f) $ (e=f): Now (a/b) and (a0/b0) each give twodimensional irreducible representations: (a=b) C1 = M1; C2 = C3 = M2; X = I; Z = S (a0=b0) C1 = M1; C2 = C3 = M2; X = ¡I; Z = S For (c/d), the induced representation is C2 = 0 B@ M1 0 0 M1 1 CA ; C1 = C3 = 0 B@ M2 0 0 M2 1 CA; X = 0 B@ S 0 0 S 1 CA ; Z = 0 B@ 0 I I 0 1 CA : This can be blockdiagonalized by inspection, (c=d) C2 = M1; C1 = C3 = M2; X = S; Z = I (c0=d0) C2 = M1; C1 = C3 = M2; X = S; Z = ¡I: For (e/f), the induced representation is C1 = C4 = 0 B@ M1 0 0 M2 1 CA ; C2 = C3 = 0 B@ M2 0 0 M1 1 CA ; X = 0 B@ S 0 0 S 1 CA ; Z = 0 B@ 0 I I 0 1 CA : 22 Noting that M2 = SM1S, we can blockdiagonalize this using the same method that was used in the S4 example for the triplet orbit. This gives (e=f) C1 = C4 = M1; C2 = C3 = M2; X = S; Z = S (e0=f0) C1 = C4 = M1; C2 = C3 = M2; X = S; Z = ¡S: Finally, we add Y . With a little e®ort, it can be seen that (a/b), (c/d), and (e/f) lie in one orbit, and (a0/b0), (c0/d0), and (e0/f0) lie in another. Then, the induced repre sentations are irreducible. So, we ¯nally end up with two sixdimensional irreducible representations of (Z3 2 ) o A4: C1 = 0 BBBB@ M1 0 0 0 M2 0 0 0 M2 1 CCCCA ; C2 = 0 BBBB@ M2 0 0 0 M1 0 0 0 M2 1 CCCCA ; C3 = 0 BBBB@ M2 0 0 0 M2 0 0 0 M1 1 CCCCA ; X = 0 BBBB@ I 0 0 0 S 0 0 0 S 1 CCCCA ; Z = 0 BBBB@ S 0 0 0 I 0 0 0 S 1 CCCCA ; Y = 0 BBBB@ 0 0 I I 0 0 0 I 0 1 CCCCA ; and C1 = 0 BBBB@ M1 0 0 0 M2 0 0 0 M2 1 CCCCA ; C2 = 0 BBBB@ M2 0 0 0 M1 0 0 0 M2 1 CCCCA ; C3 = 0 BBBB@ M2 0 0 0 M2 0 0 0 M1 1 CCCCA ; X = 0 BBBB@ ¡I 0 0 0 ¡S 0 0 0 S 1 CCCCA ; Z = 0 BBBB@ S 0 0 0 ¡I 0 0 0 ¡S 1 CCCCA ; Y = 0 BBBB@ 0 0 I I 0 0 0 I 0 1 CCCCA : It can be checked directly that these matrices respect all of the relations (21), (26), (27), and (28). 23 Now consider the group (S3 £ S3 £ S3 £ S3) o A4. Let Ai and Bi be the S4 generators. They obey A3i = B2 i = E; BiAiB¡1 i = A¡1 i : (2.29) The irreducible representations of S3 are two onedimensional representations given by Ai = 1; Bi = 1; Ai = 1; Bi = ¡1; and one twodimensional representation given by Ai = MA ´ 0 B@ ! 0 0 !2 1 CA ; Bi = MB ´ 0 B@ 0 1 1 0 1 CA ; With the A4 generators in (4), Ai and Bi respect the relations XA1X¡1 = A2; XA2X¡1 = A1; XA3X¡1 = A4; XA4X¡1 = A3; XB1X¡1 = B2; XB2X¡1 = B1; XB3X¡1 = B4; XB4X¡1 = B3; (2.30) ZA1Z¡1 = A3; ZA2Z¡1 = A4; ZA3Z¡1 = A1; ZA4Z¡1 = A2; ZB1Z¡1 = B3; ZB2Z¡1 = B4; ZB3Z¡1 = B1; ZB4Z¡1 = B2; (2.31) Y A1Y ¡1 = A1; Y A2Y ¡1 = A3; Y A3Y ¡1 = A4; Y A4Y ¡1 = A2; Y B1Y ¡1 = B1; Y B2Y ¡1 = B3; Y B3Y ¡1 = B4; Y B4Y ¡1 = B2: (2.32) As in the last example, we have a chain of invariant subgroups, S4 3 ½ (S4 3 ) o Z2 ½ (S4 3 ) o (Z2 £ Z2) ½ (S4 3 ) o A4: 24 The representation (2; 2; 2; 2) under S3 £ S3 £ S3 £ S3 lies in its own orbit. This representation can be written in terms of 16dimensional matrices, °(A1) = MA I I I; °(B1) = MB I I I; °(A2) = I MA I I; °(B2) = I MB I I; °(A3) = I I MA I; °(B3) = I I MB I; °(A4) = I I I MA; °(B4) = I I I MB: Now add X to obtain the subgroup (S4 3 ) o Z2. From (14), under X A1 $ A2; A3 $ A4; B1 $ B2; B3 $ B4: Consider how this rearranges the eigenvalues of each of the 16 basis states under the diagonal generators (A1;A2;A3;A4), (1) (!; !; !; !) ¡! (!; !; !; !) » (1) (2) (!2; !; !; !) ¡! (!; !2; !; !) » (3) (3) (!; !2; !; !) ¡! (!2; !; !; !) » (2) (4) (!2; !2; !; !) ¡! (!2; !2; !; !) » (4) (5) (!; !; !2; !) ¡! (!; !; !; !2) » (9) (6) (!2; !; !2; !) ¡! (!; !2; !; !2) » (11) 25 (7) (!; !2; !2; !) ¡! (!2; !; !; !2) » (10) (8) (!2; !2; !2; !) ¡! (!2; !2; !; !2) » (12) (9) (!; !; !; !2) ¡! (!; !; !2; !) » (5) (10) (!2; !; !; !2) ¡! (!; !2; !2; !) » (7) (11) (!; !2; !; !2) ¡! (!2; !; !2; !) » (6) (12) (!2; !2; !; !2) ¡! (!2; !2; !2; !) » (8) (13) (!; !; !2; !2) ¡! (!; !; !2; !2) » (13) (14) (!2; !; !2; !2) ¡! (!; !2; !2; !2) » (15) (15) (!; !2; !2; !2) ¡! (!2; !; !2; !2) » (14) (16) (!2; !2; !2; !2) ¡! (!2; !2; !2; !2) » (16): 26 This yields the permutation matrix PX ´ 0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : Note that P¡1 X = PX. We can now check directly that PX°(B1)P¡1 X = °(B2); PX°(B2)P¡1 X = °(B1); PX°(B3)P¡1 X = °(B4); PX°(B4)P¡1 X = °(B3): So, we obtain two 16dimensional irreducible representations of (S4 3 ) o Z2, (a) Ai = °(Ai); Bi = °(Bi); X = PX (b) Ai = °(Ai); Bi = °(Bi); X = ¡PX: 27 Next add Z to obtain the subgroup (S4 3 ) o (Z2 £ Z2). Proceeding as is the previous step yields PZ ´ 0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA : Again note that P¡1 Z = PZ. This gives four 16dimensional irreducible representations of (S4 3 ) o (Z2 £ Z2), (a) Ai = °(Ai); Bi = °(Bi); X = PX; Z = PZ (a0) Ai = °(Ai); Bi = °(Bi); X = PX; Z = ¡PZ (b) Ai = °(Ai); Bi = °(Bi); X = ¡PX; Z = PZ (b0) Ai = °(Ai); Bi = °(Bi); X = ¡PX; Z = ¡PZ: 28 Finally, add Y to obtain (S4 3 )oA4. Then, (a) lies in its own orbit, while (a0), (b), and (b0) lie in another orbit. First, consider the orbit of (a). We ¯nd that the permutation matrix PY ´ 0 BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCA respects the relations PY °(A1)P¡1 Y = °(A1); PY °(A2)P¡1 Y = °(A3); PY °(A3)P¡1 Y = °(A4); PY °(A4)P¡1 Y = °(A2); PY °(B1)P¡1 Y = °(B1); PY °(B2)P¡1 Y = °(B3); PY °(B3)P¡1 Y = °(B4); PY °(B4)P¡1 Y = °(B2); PY PXP¡1 Y = PZ; PY PZP¡1 Y = PXPZ; PY (PXPZ)P¡1 Y = PX: 29 (Note that P¡1 Y = P2 Y .) The induced representation can then be written °"(Ai) = 0 BBBB@ °(Ai) 0 0 0 P2 Y °(Ai)PY 0 0 0 PY °(Ai)P2 Y 1 CCCCA ; °"(Bi) = 0 BBBB@ °(Bi) 0 0 0 P2 Y °(Bi)PY 0 0 0 PY °(Bi)P2 Y 1 CCCCA °"(X) = 0 BBBB@ °(X) 0 0 0 P2 Y °(X)PY 0 0 0 PY °(X)P2 Y 1 CCCCA ; °"(Z) = 0 BBBB@ °(Z) 0 0 0 P2 Y °(Z)PY 0 0 0 PY °(Z)P2 Y 1 CCCCA ; °"(Y ) = 0 BBBB@ 0 0 I I 0 0 0 I 0 1 CCCCA : Let P = 1 p 3 0 BBBB@ I I I I !I !2I I !2I !I 1 CCCCA 0 BBBB@ I 0 0 0 PY 0 0 0 P2 Y 1 CCCCA : Then P°"(Ai)P¡1 = 0 BBBB@ °(Ai) 0 0 0 °(Ai) 0 0 0 °(Ai) 1 CCCCA ; P°"(Bi)P¡1 = 0 BBBB@ °(Bi) 0 0 0 °(Bi) 0 0 0 °(Bi) 1 CCCCA ; P°"(X)P¡1 = 0 BBBB@ °(X) 0 0 0 °(X) 0 0 0 °(X) 1 CCCCA ; P°"(Z)P¡1 = 0 BBBB@ °(Z) 0 0 0 °(Z) 0 0 0 °(Z) 1 CCCCA ; P°"(Y )P¡1 = 0 BBBB@ PY 0 0 0 !PY 0 0 0 !2PY 1 CCCCA : 30 So the result is three 16dimensional irreducible representations of (S4 3 )oA4. For the other orbit, the induced representation is irreducible. It is given by °"(A1) = 0 BBBB@ °(A1) 0 0 0 °(A1) 0 0 0 °(A1) 1 CCCCA ; °"(B1) = 0 BBBB@ °(B1) 0 0 0 °(B1) 0 0 0 °(B1) 1 CCCCA ; °"(A2) = 0 BBBB@ °(A2) 0 0 0 °(A4) 0 0 0 °(A3) 1 CCCCA ; °"(B2) = 0 BBBB@ °(B2) 0 0 0 °(B4) 0 0 0 °(B3) 1 CCCCA ; °"(A3) = 0 BBBB@ °(A3) 0 0 0 °(A2) 0 0 0 °(A4) 1 CCCCA ; °"(B3) = 0 BBBB@ °(B3) 0 0 0 °(B2) 0 0 0 °(B4) 1 CCCCA ; °"(A4) = 0 BBBB@ °(A4) 0 0 0 °(A3) 0 0 0 °(A2) 1 CCCCA ; °"(B4) = 0 BBBB@ °(B4) 0 0 0 °(B3) 0 0 0 °(B2) 1 CCCCA ; °"(X) = 0 BBBB@ PX 0 0 0 ¡PXPZ 0 0 0 ¡PZ 1 CCCCA ; °"(Z) = 0 BBBB@ ¡PZ 0 0 0 PX 0 0 0 ¡PXPZ 1 CCCCA ; Y (48) = 0 BBBB@ 0 0 I I 0 0 0 I 0 1 CCCCA : We can also see that the representations (2; 1; 1; 1), (1; 2; 1; 1), (1; 1; 2; 1), and (1; 1; 1; 2) of S3 £ S3 £ S3 £ S3 make up an orbit. The 8dimensional representation of (S4 3 ) o A4 this orbit gives rise to is given by A1 = diag(!; !2; 1; 1; 1; 1; 1; 1); A2 = diag(1; 1; !; !2; 1; 1; 1; 1); A3 = diag(1; 1; 1; 1; !; !2; 1; 1); A4 = diag(1; 1; 1; 1; 1; 1; !; !2); 31 B1 = 0 BBBBBBBBBBBBBBBBBBBBB@ 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCA ; B2 = 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCA ; B3 = 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 CCCCCCCCCCCCCCCCCCCCCA ; B4 = 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 1 CCCCCCCCCCCCCCCCCCCCCA ; X = 0 BBBBBBBBBBBBBBBBBBBBB@ 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 CCCCCCCCCCCCCCCCCCCCCA ; Z = 0 BBBBBBBBBBBBBBBBBBBBB@ 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 CCCCCCCCCCCCCCCCCCCCCA 32 Y = 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 1 CCCCCCCCCCCCCCCCCCCCCA : 2.4 Invariants Under the Discrete Symmetry In this section, we give the symmetry invariants which are used in our model. These can be computed directly from the matrices given in the previous section. 16AB £ 16AB invariant (xi; x0 j » 16AB): f1(xi; x0 j) = x1x0 16 + x2x0 15 + x3x0 14 + x4x0 13 + x5x0 12 + x6x0 11 + x7x0 10 + x8x0 9 +x9x0 8 + x10x0 7 + x11x0 6 + x12x0 5 + x13x0 4 + x14x0 3 + x15x0 2 + x16x0 1 6C £ 6C invariant (wi;w0j » 6C): f2(wi;w0 j) = w1w0 1 + w2w0 2 + w3w0 3 + w4w0 4 + w5w0 5 + w6w0 6: 48AB £ 48AB invariant (yi; y0j » 48AB): f3(yi; y0 j) = y1y0 16 + y2y0 15 + y3y0 14 + y4y0 13 + y5y0 12 + y6y0 11 + y7y0 10 + y8y0 9 33 y9y0 8 + y10y0 7 + y11y0 6 + y12y0 5 + y13y0 4 + y14y0 3 + y15y0 2 + y16y0 1 + y17y0 32 + y18y0 31 +y19y0 30 + y20y0 29 + y21y0 28 + y22y0 27 + y23y0 26 + y24y0 25 + y25y0 24 + y26y0 23 + y27y0 22 + y28y0 21 +y29y0 20 + y30y0 19 + y31y0 18 + y32y0 17 + y33y0 48 + y34y0 47 + y35y0 46 + y36y0 45 + y37y0 44 + y38y0 43 +y39y0 42 + y40y0 41 + y41y0 40 + y42y0 39 + y43y0 38 + y44y0 37 + y45y0 36 + y46y0 35 + y47y0 34 + y48y0 33 8AB £ 8AB invariant (zi; z0j » 8AB): f4(zi; z0 j) = z1z0 2 + z2z0 1 + z3z0 4 + z4z0 3 + z5z0 6 + z6z0 5 + z7z0 8 + z8z0 7 16AB £ 16AB £ 16AB invariant (xi; x0 j ; x00 k » 16AB): g1(xi; x0 j ; x00 k) = x1x0 1x00 1 + x2x0 2x00 2 + x3x0 3x00 3 + x4x0 4x00 4 + x5x0 5x00 5 + x6x0 6x00 6 + x7x0 7x00 7 + x8x0 8x00 8 x9x0 9x00 9 + x10x0 10x00 10 + x11x0 11x00 11 + x12x0 12x00 12 + x13x0 13x00 13 + x14x0 14x00 14 + x15x0 15x00 15 + x16x0 16x00 16 6C £ 6C £ 6C invariant (wi;w0j ;w00 k » 6C): g2(wi;w0 j ;w00 k) = w1w0 3w00 5 + w5w0 1w00 3 + w3w0 5w00 1 + w1w0 4w00 6 + w6w0 1w00 4 + w4w0 6w00 1 +w2w0 3w00 6 + w6w0 2w00 3 + w3w0 6w00 2 + w2w0 4w00 5 + w5w0 2w00 4 + w4w0 5w00 2 3 £ 16AB £ 48AB invariant (ti » 3, xj » 16AB, yk » 48AB): g3(ti; xj ; yk) = t1(x16y33 + x15y34 + x8y35 + x7y36 + x14y37 + x13y38 + x6y39 + x5y40 34 +x12y41 + x11y42 + x4y43 + x3y44 + x10y45 + x9y46 + x2y47 + x1y48) t2(x16y1 + x7y10 + x6y11 + x5y12 + x4y13 + x3y14 + x2y15 + x1y16 +x15y2 + x14y3 + x13y4 + x12y5 + x11y6 + x10y7 + x9y8 + x8y9) t3(x16y17 + x15y18 + x12y19 + x11y20 + x8y21 + x7y22 + x4y23 + x3y24 +x14y25 + x13y26 + x10y27 + x9y28 + x6y29 + x5y30 + x2y31 + x1y32) 8AB £ 16AB £ 48AB invariant (zi » 8AB, xj » 16AB, yk » 48AB): g4(zi; xk; yj ) = z1(x15y1 + x5y11 + x3y13 + x1y15 + x15y17 + x11y19 + x7y21 + x3y23 + x13y25 + x9y27 + x5y29 + x13y3 +x1y31 + x15y33 + x7y35 + x13y37 + x5y39 + x11y41 + x3y43 + x9y45 + x1y47 + x11y5 + x9y7 + x7y9) +z2(x8y10 + x6y12 + x4y14 + x2y16 + x16y18 + x16y2 + x12y20 + x8y22 + x4y24 + x14y26 + x10y28 + x6y30 +x2y32 + x16y34 + x8y36 + x14y38 + x14y4 + x6y40 + x12y42 + x4y44 + x10y46 + x2y48 + x12y6 + x10y8) +z3(x14y1 + x5y10 + x2y13 + x1y14 ¡ x14y17 ¡ x13y18 ¡ x10y19 + x13y2 ¡ x9y20 ¡ x6y21 ¡ x5y22 ¡ x2y23 ¡x1y24 ¡ x14y33 ¡ x13y34 ¡ x6y35 ¡ x5y36 ¡ x10y41 ¡ x9y42 ¡ x2y43 ¡ x1y44 + x10y5 + x9y6 + x6y9) +z4(x8y11 + x7y12 + x4y15 + x3y16 ¡ x16y25 ¡ x15y26 ¡ x12y27 ¡ x11y28 ¡ x8y29 + x16y3 ¡ x7y30 ¡ x4y31 ¡x3y32 ¡ x16y37 ¡ x15y38 ¡ x8y39 + x15y4 ¡ x7y40 ¡ x12y45 ¡ x11y46 ¡ x4y47 ¡ x3y48 + x12y7 + x11y8) +z5(¡x12y1 ¡ x3y10 ¡ x2y11 ¡ x1y12 + x12y17 + x11y18 ¡ x11y2 + x4y21 + x3y22 + x10y25 + x9y26 + x2y29 ¡x10y3 + x1y30 ¡ x12y33 ¡ x11y34 ¡ x4y35 ¡ x3y36 ¡ x10y37 ¡ x9y38 ¡ x2y39 ¡ x9y4 ¡ x1y40 ¡ x4y9) +z6(¡x8y13 ¡ x7y14 ¡ x6y15 ¡ x5y16 + x16y19 + x15y20 + x8y23 + x7y24 + x14y27 + x13y28 + x6y31 + x5y32 ¡x16y41 ¡ x15y42 ¡ x8y43 ¡ x7y44 ¡ x14y45 ¡ x13y46 ¡ x6y47 ¡ x5y48 ¡ x16y5 ¡ x15y6 ¡ x14y7 ¡ x13y8) +z7(¡x8y1 ¡ x8y17 ¡ x7y18 ¡ x4y19 ¡ x7y2 ¡ x3y20 ¡ x6y25 ¡ x5y26 ¡ x2y27 ¡ x1y28 ¡ x6y3 + x8y33 +x7y34 + x6y37 + x5y38 ¡ x5y4 + x4y41 + x3y42 + x2y45 + x1y46 ¡ x4y5 ¡ x3y6 ¡ x2y7 ¡ x1y8) +z8(x15y36 ¡ x14y11 ¡ x13y12 ¡ x12y13 ¡ x11y14 ¡ x10y15 ¡ x9y16 ¡ x16y21 ¡ x15y22 ¡ x12y23 ¡ x11y24 ¡ x14y29 ¡x13y30 ¡ x10y31 ¡ x9y32 + x16y35 ¡ x15y10 + x14y39 + x13y40 + x12y43 + x11y44 + x10y47 + x9y48 ¡ x16y9) 35 16AB £ 48AB £ 48AB invariant (xi » 16AB; yj ; y0k » 48AB): g5(xi; yj ; y0 k) = x1y1y0 1 + x2y2y0 2 + x3y3y0 3 + x4y4y0 4 + x5y5y0 5 + x6y6y0 6 + x7y7y0 7 + x8y8y0 8 +x9y9y0 9 + x10y10y0 10 + x11y11y0 11 + x12y12y0 12 + x13y13y0 13 + x14y14y0 14 + x15y15y0 15 + x16y16y0 16 +x1y17y0 17 + x2y18y0 18 + x3y25y0 25 + x4y26y0 26 + x5y19y0 19 + x6y20y0 20 + x7y27y0 27 + x8y28y0 28 +x9y21y0 21 + x10y22y0 22 + x11y29y0 29 + x12y30y0 30 + x13y23y0 23 + x14y24y0 24 + x15y31y0 31 + x16y32y0 32 +x1y33y0 33 + x2y34y0 34 + x3y37y0 37 + x4y38y0 38 + x5y41y0 41 + x6y42y0 42 + x7y45y0 45 + x8y46y0 46 +x9y35y0 35 + x10y36y0 36 + x11y39y0 39 + x12y40y0 40 + x13y43y0 43 + x14y44y0 44 + x15y47y0 47 + x16y48y0 48 3 £ 6C £ 6C invariant (ti » 3; wj ;w0k » 6C): g6(ti;wj ;w0 k) = t1(w5w0 5 ¡ w6w0 6) + t2(w1w0 1 ¡ w2w0 2) + t3(w3w0 3 ¡ w4w0 4) 10 £ 6C £ 6C invariant (s0 » 10; wi;w0j » 6C): g7(s0;wi;w0 j) = s0(w1w0 1 + w2w0 2 + !2w3w0 3 + !2w4w0 4 + !w5w0 5 + !w6w0 6) 100 £ 6C £ 6C invariant (s00 » 100; wi;w0j » 6C): g8(s00;wi;wj) = s00(w1w0 1 + w2w0 2 + !w3w0 3 + !w4w0 4 + !2w5w0 5 + !2w6w0 6) For our purposes, it su±ces to have the 16AB£16AB£16AB£16AB and 6C£6C£6C£6C 36 invariants for the case where all four ¯elds are the same. 16AB £ 16AB £ 16AB £ 16AB invariants (xi » 16AB): h1(xi) = x21 x2 16 + x22 x2 15 + x23 x2 14 + x24 x2 13 + x25 x2 12 + x26 x2 11 + x27 x2 10 + x28 x29 ; h2(xi) = x1x2x15x16 + x1x3x14x16 + x2x4x13x15 + x3x4x13x14 + x1x5x12x16 + x4x5x12x13 +x2x6x11x15 + x3x6x11x14 + x5x6x11x12 + x2x7x10x15 + x3x7x10x14 + x5x7x10x12 +x1x8x9x16 + x4x8x9x13 + x6x8x9x11 + x7x8x9x10 6C £ 6C £ 6C £ 6C invariants (wi » 6C): h3(wi) = w4 1 + w4 2 + w4 3 + w4 4 + w4 5 + w4 6; h4(wi) = w2 1w2 2 + w2 3w2 4 + w2 5w2 6; h5(wi) = w2 1w2 3 + w2 1w2 4 + w2 1w2 5 + w2 1w2 6 + w2 2w2 3 + w2 2w2 4 + w2 2w2 5 + w2 2w2 6 +w2 3w2 5 + w2 3w2 6 + w2 4w2 5 + w2 4w2 6; h6(wi) = w1w2w3w4 + w1w2w5w6 + w3w4w5w6 37 2.5 Calculation of the Neutrino Mass Matrix In this section, we show how the neutrino mass matrix is computed. From Section 2.4, the term in Eq. (13) that mixes N and N0 is2 g3(N; hÁi;N0) = vÁN1(N0 35 + N0 36 + N0 39 + N0 40 + N0 41 + N0 42 + N0 45 + N0 46) +vÁN2(N0 5 + N0 6 + N0 7 + N0 8 + N0 9 + N0 10 + N0 11 + N0 12) +vÁN3(N0 19 + N0 20 + N0 21 + N0 22 + N0 27 + N0 28 + N0 29 + N0 30): The term that mixes N0 and N00 is g4(N00; hÁi;N0) = vÁN00 1 (N0 5 + N0 7 + N0 9 + N0 11) + vÁN00 2 (N0 6 + N0 8 + N0 10 + N0 12) +vÁN00 3 (N0 5 + N0 6 + N0 9 + N0 10) + vÁN00 4 (N0 7 + N0 8 + N0 11 + N0 12) ¡vÁN00 5 (N0 1 + N0 2 + N0 3 + N0 4) ¡ vÁN00 6 (N0 13 + N0 14 + N0 15 + N0 16) ¡vÁN00 7 (N0 1 + N0 2 + N0 3 + N0 4) ¡ vÁN00 8 (N0 13 + N0 14 + N0 15 + N0 16) +vÁN00 1 (N0 19 + N0 21 + N0 27 + N0 29) + vÁN00 2 (N0 20 + N0 22 + N0 28 + N0 30) ¡vÁN00 3 (N0 19 + N0 20 + N0 21 + N0 22) ¡ vÁN00 4 (N0 27 + N0 28 + N0 29 + N0 30) +vÁN00 5 (N0 17 + N0 18 + N0 25 + N0 26) + vÁN00 6 (N0 23 + N0 29 + N0 31 + N0 32) ¡vÁN00 7 (N0 17 + N0 18 + N0 25 + N0 26) ¡ vÁN00 8 (N0 23 + N0 29 + N0 31 + N0 32) +vÁN00 1 (N0 35 + N0 39 + N0 41 + N0 45) + vÁN00 2 (N0 36 + N0 40 + N0 42 + N0 46) 2Note that, in the 16AB basis used here, Á17¡i = Á¤i , i = 1 ¡ 8. 38 ¡vÁN00 3 (N0 35 + N0 36 + N0 41 + N0 42) ¡ vÁN00 4 (N0 39 + N0 40 + N0 45 + N0 46) +vÁN00 5 (N0 33 + N0 34 + N0 37 + N0 38) + vÁN00 6 (N0 43 + N0 44 + N0 47 + N0 48) ¡vÁN00 7 (N0 33 + N0 34 + N0 37 + N0 38) ¡ vÁN00 8 (N0 43 + N0 44 + N0 47 + N0 48) Since the symmetries B1, B2, B3B4, and A3A4 are unbroken, components of N0 and N00 that transform under these symmetries cannot mix with the light neutrinos. This leaves p1 = N05 + N06 + N07 + N08 + N09 + N0 10 + N0 11 + N0 p 12 8 ; p2 = N0 19 + N0 20 + N0 21 + N0 22 + N0 27 + N0 28 + N0 29 + N0 p 30 8 ; p3 = N0 35 + N0 36 + N0 39 + N0 40 + N0 41 + N0 42 + N0 45 + N0 p 46 8 ; q1 = N00 1 + N00 p 2 2 ; q2 = N00 3 + N00 p 4 2 : We now have g3(N; hÁi;N0) = p 8vÁ(N1p3 + N2p1 + N3p2); g4(N00; hÁi;N0) = 2vÁ(q1p1 + q2p1 + q1p2 ¡ q2p2 + q1p3 ¡ q2p3) + :::; where the ellipses in the second equation refer to terms involving only decoupled components. The mass matrix for (º1; º2; º3;N1;N2;N3; p1; p2; p3; q1; q2) has the form 1 2 Mº = 0 B@ 0 m mT M 1 CA ; with m = 0 BBBB@ 1 2¸v 0 0 0 0 0 0 0 0 1 2¸v 0 0 0 0 0 0 0 0 1 2¸v 0 0 0 0 0 1 CCCCA ; 39 M = 0 BBBBBBBBBBBBBBBBBBBBB@ mN 0 0 0 0 p 2®1vÁ 0 0 0 mN 0 p 2®1vÁ 0 0 0 0 0 0 mN 0 p 2®1vÁ 0 0 0 0 p 2®1vÁ 0 m0 N + ¯vÁ 0 0 ®2vÁ ®2vÁ 0 0 p 2®1vÁ 0 m0 N + ¯vÁ 0 ®2vÁ ¡®2vÁ p 2®1vÁ 0 0 0 0 m0 N + ¯vÁ ®2vÁ ¡®2vÁ 0 0 0 ®2vÁ ®2vÁ ®2vÁ m00 N 0 0 0 0 ®2vÁ ¡®2vÁ ¡®2vÁ 0 m00 N 1 CCCCCCCCCCCCCCCCCCCCCA : Here, m only contains entries at the EW scale, while M contains entries at the higher scale M¤. To order M2W =M2 ¤ , Mº is blockdiagonalized by Uº = 0 B@ I ¡mM¡1 M¡1mT I 1 CA : The light neutrino mass matrix Mº is given by the upperleft block of UºMºUT º , 1 2 Mº = ¡mM¡1mT : Let S = 1 p 2 0 BBBBBBBBBBBBBBBBBBBBB@ 1 0 0 0 0 0 1 0 0 0 0 p 2 0 0 0 0 1 0 0 0 0 0 ¡1 0 0 0 0 0 p 2 0 0 0 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ¡1 0 0 1 0 0 1 0 0 0 0 ¡1 0 0 1 0 0 1 CCCCCCCCCCCCCCCCCCCCCA : 40 Then, S¡1MS = 0 BBBB@ A 0 0 0 B 0 0 0 C 1 CCCCA ; with A = 0 BBBB@ mN p 2®1vÁ 0 p 2®1vÁ m0 N + ¯vÁ 2®2vÁ 0 2®2vÁ m00 N 1 CCCCA ; B = 0 BBBB@ mN p 2®1vÁ 0 p 2®1vÁ m0 N + ¯vÁ p 2®2vÁ 0 p 2®2vÁ m00 N 1 CCCCA ; C = 0 B@ mN ¡ p 2®1vÁ ¡ p 2®1vÁ m0 N + ¯vÁ 1 CA : So, we can write 1 2 Mº = ¡mS 0 BBBB@ A¡1 0 0 0 B¡1 0 0 0 C¡1 1 CCCCA S¡1mT = ¡ ¸2v2 8 0 BBBB@ (A¡1)11 + (C¡1)11 0 (A¡1)11 ¡ (C¡1)11 0 2(B¡1)11 0 (A¡1)11 ¡ (C¡1)11 0 (A¡1)11 + (C¡1)11 1 CCCCA This mass matrix is diagonalized by (10), and the masses are given by m1 = ¯¯¯¯ ¸2v2 2 (A¡1)11 ¯¯¯¯ ; m2 = ¯¯¯¯ ¸2v2 2 (B¡1)11 ¯¯¯¯ ; m3 = ¯¯¯¯ ¸2v2 2 (C¡1)11 ¯¯¯¯ : 41 2.6 Calculation of the Charged Lepton Mass Matrix In this section, we show how the charged lepton mass matrix is computed. From Section 2.4, the terms in Eq. (14) that mix eR1, eR2, and eR3 with E0L are eR1f2(E0 L; hÂi) + c:c: = vÂeR1(E0 L1 + E0 L3 + E0 L5) + c:c:; g7(eR2;E0 L; hÂi) + c:c: = vÂeR2(E0 L1 + !2E0 L3 + !E0 L5) + c:c:; g8(eR3;E0 L; hÂi) + c:c: = vÂeR3(E0 L1 + !E0 L3 + !2E0 L5) + c:c:: The term that mixes ER and E0L is g6(ER;E0 L; hÂi) + c:c: = vÂ(ER1E0 L5 + ER2E0 L1 + ER3E0 L3) + c:c:; with a similar result for the term that mixes EL and E0R . In the basis with (eL1; eL2; eL3;EL1;EL2; EL3;E 0 L1;E 0 L3;E 0 L5) on the left and (eR1; eR2; eR3;ER1;ER2;ER3;E0R 1;E0R3;E0R 5) on the right, the mass matrix has the form Me = 0 B@ 0 M0 m M 1 CA ; with m = 0 BBBBBBBBBBBBBB@ ·v 0 0 0 ·v 0 0 0 ·v 0 0 0 0 0 0 0 0 0 1 CCCCCCCCCCCCCCA ; M0 = 0 BBBB@ 0 0 0 ²1vÂ ²1vÂ ²1vÂ 0 0 0 ²2vÂ !2²2vÂ !²2vÂ 0 0 0 ²3vÂ !²3vÂ !2²3vÂ 1 CCCCA ; 42 and M = 0 BBBBBBBBBBBBBB@ mE 0 0 0 0 °1vÂ 0 mE 0 °1vÂ 0 0 0 0 mE 0 °1vÂ 0 0 °2vÂ 0 m0 E ´2vÂ ´1vÂ 0 0 °2vÂ ´1vÂ m0 E ´2vÂ °2vÂ 0 0 ´2vÂ ´1vÂ m0 E 1 CCCCCCCCCCCCCCA : Here, m only contains entries at the EW scale, while M and M0 contain entries at the higher scale M¤. To order M2W =M2 ¤ , the lefthanded masssquared matrixMy eMe is blockdiagonalized by UL = 0 B@ I myM(MyM +M0yM0)¡1 (MyM +M0yM0)¡1Mym I 1 CA : The upper left entry of ULMy eMeUy L is the light lefthanded masssquared matrix My eMe = mym ¡ myM(MyM +M0yM0)¡1Mym Let S = 1 p 3 0 BBBBBBBBBBBBBB@ 1 0 1 0 1 0 1 0 ! 0 !2 0 1 0 !2 0 ! 0 0 1 0 1 0 1 0 1 0 ! 0 !2 0 1 0 !2 0 ! 1 CCCCCCCCCCCCCCA : Then SyMS and SyM0yM0S are both block diagonal (three 2£2 blocks each). So we have myM(MyM +M0yM0)¡1Mym = myS 0 BBBB@ A 0 0 0 B 0 0 0 C 1 CCCCA Sym: 43 = j·vj2 3 0 BBBB@ A11 + B11 + C11 A11 + !2B11 + !C11 A11 + !B11 + !2C11 A11 + !B11 + !2C11 A11 + B11 + C11 A11 + !2B11 + !C11 A11 + !2B11 + !C11 A11 + !B11 + !2C11 A11 + B11 + C11 1 CCCCA This has the form (11). The masses are given by m2e = j·vj2(1 ¡ A11); m2 ¹ = j·vj2(1 ¡ B11); m2¿ = j·vj2(1 ¡ C11): 44 CHAPTER 3 A 6D Higgsless Standard Model 3.1 The Model Let us consider a 6D SU(2)L £ U(1)Y gauge theory in a °at spacetime back ground, where the two extra spatial dimensions are compacti¯ed on a rectangle1.The coordinates in the 6D space are written as zM = (x¹; ym), where the 6D Lorentz indices are denoted by capital Roman letters M = 0; 1; 2; 3; 5; 6, while the usual 4D Lorentz indices are symbolized by Greek letters ¹ = 0; 1; 2; 3, and the coordinates ym (m = 1; 2) describe the ¯fth and sixth dimension.2 The physical space is thus de¯ned by 0 · y1 · ¼R1 and 0 · y2 · ¼R2, where R1 and R2 are the compacti¯cation radii of a torus T2, which is obtained by identifying the points of the twodimensional plane R2 under the actions T5 : (y1; y2) ! (y1+2¼R1; y2) and T6 : (y1; y2) ! (y1; y2+2¼R2). We denote the SU(2)L and U(1)Y gauge bosons in the bulk respectively by Aa M(zM) (a = 1; 2; 3 is the gauge index) and BM(zM). The action of the gauge ¯elds in our model is given by S = Z d4x Z ¼R1 0 dy1 Z ¼R2 0 dy2 (L6 + ±(y1)±(y2)L0) ; (3.1) where L6 is a 6D bulk gauge kinetic term and L0 is a 4D brane gauge kinetic term localized at (y1; y2) = (0; 0), which read respectively L6 = ¡ M2L 4 FaMNFMNa ¡ M2 Y 4 BMNBMN; L0 = ¡ 1 4g2Fa ¹ºF¹ºa ¡ 1 4g02B¹ºB¹º; (3.2) 1Chiral compacti¯cation on a square has recently been considered in Ref. [42]. 2For the metric we choose a signature (+;¡;¡;¡;¡;¡). 45 with ¯eld strengths FaM N = @MAa N ¡ @NAa M + fabcAb MAc N (fabc is the structure con stant) and BMN = @MBN ¡ @NBM. In Eqs. (3.2), the quantities ML and MY have mass dimension +1, while g and g0 are dimensionless. Since the boundaries of the manifold break translational invariance and are "singled out" with respect to the points in the interior of the rectangle, brane terms like L0 can be produced by quan tum loop e®ects [34, 35] or arise from classical singularities in the limit of vanishing brane thickness [36]. Unlike in ¯ve dimensions (for a discussion of the » ! 1 limit in generalized 5D R» gauges see, e.g., Ref. [43] and also Ref. [30]), we cannot go to a unitary gauge where all ¯elds Aa 5;6 (a = 1; 2; 3) and B5;6 are identically set to zero. Instead, there will remain after dimensional reduction one combination of physical scalar ¯elds in the spectrum3. To make these scalars su±ciently heavier than the LeeQuiggThacker bound of ¼ 2 TeV , we can assume, e.g., a seventh dimension compacti¯ed on S1=Z2 with compacti¯cation radius R3 . R1;R2. By setting Aa 5;6;7 = B5;6;7 = 0 (Aa7 and B7 are the seventh components of the gauge ¯elds) on all boundaries of this manifold, the associated scalars can acquire for compacti¯cation scales R¡1 1 ;R¡1 2 ' 1 ¡ 2 TeV , masses well above 2 TeV . Therefore, at low energies . 2 ¡ 3 TeV , we have a model without any light scalars and will, in what follows, neglect the heavy scalar degrees of freedom. Since the Lagrangian in Eq. (3.2) does not contain any explicit gauge symmetry breaking, we can obtain consistent new BC's on the boundaries by requiring the variation of the action to be zero. Variation of the action in Eq. (3.2) yields after 3We thank H. Murayama and M. Serone for pointing out this fact. 46 partial integration ±S = Z d4x Z ¼R1 y1=0 dy1 Z ¼R2 y2=0 dy2 £ M2L ¡ @MFaM¹ ¡ fabcFbM¹Ac M ¢ ±Aa ¹ +M2 Y @MBM¹±B¹ ¤ + Z d4x Z ¼R2 y2=0 dy2 £ M2L Fa 5¹±Aa¹ +M2 Y B5¹±B¹¤¼R1 y1=0 + Z d4x Z ¼R1 y1=0 dy1 £ M2L Fa 6¹±Aa¹ +M2 Y B6¹±B¹¤¼R2 y2=0 + Z d4x · 1 g2 (@¹Fa¹º ¡ fabcFb¹ºAc ¹)±Ac º + 1 g02 @¹B¹º±Bº ¸ (y1;y2)=(0;0) = 0; (3.3) where we have (as usual) assumed that the gauge ¯elds and their derivatives go to zero for x¹ ! 1. The bulk terms in in the ¯rst line in Eq. (3.3), lead to the familiar bulk equations of motion. Moreover, since the minimization of the action requires the boundary terms to vanish as well, we obtain from the second and third line in Eq. (3.3) a set of consistent BC's for the bulk ¯elds. We break the electroweak symmetry SU(2)L £ U(1)Y ! U(1)Q by imposing on two of the boundaries following BC's: at y1 = ¼R1 : A1 ¹ = 0; A2 ¹ = 0; (3.4a) at y2 = ¼R2 : @y2(M2L A3 ¹ +M2 Y B¹) = 0; A3 ¹ ¡ B¹ = 0: (3.4b) The Dirichlet BC's in Eq. (3.4a) break SU(2)L ! U(1)I3 , where U(1)I3 is the U(1) subgroup associated with the third component of weak isospin I3. The BC's in Eq. (3.4b) break U(1)I3 £ U(1)Y ! U(1)Q, leaving only U(1)Q unbroken on the entire rectangle (see Fig. 3.1). Note, in Eq. (3.4b), that the ¯rst BC involving the derivative with respect to y2 actually follows from the second BC ±A3 ¹ = ±B¹ by minimization of the action. The gauge groups U(1)I3 and U(1)I3 £ U(1)Y remain unbroken at the boundaries y1 = 0 and y2 = 0, respectively. Locally, at the ¯xed point (y1; y2) = (0; 0), SU(2)L £ U(1)Y is unbroken. We can restrict ourselves, for simplicity, to the solutions which are relevant to EWSB, by imposing on the other 47 (0; R2) ( R1; R2) y2 (0;) y1 A3 (yU2()1)I3 U(1)Y!U(1)Q B (y2) A1 ;2(y1) ( R1;0) SU(2)L!U(1)I3 01 23===RRR 4=R L0 mW123===RRR 4=R (a) (b) k p;0 5 p05k50 pk p5k5p;5 k05 k5 Figure 3.1: Symmetry breaking of SU()L £ U(1)Y on the rectangle. At one boundary y1 = ¼R1, SU(2)L is broken to U(1)I3 while on the boundary y2 = ¼R2 the subgroup U(1)I3£U(1)Y is broken to U(1)Q, which leaves only U(1)Q unbroken on the entire rectangle. Locally, at the ¯xed point 0; 0, SU(2)L £ U(1)Y remains unbroken. The dashed arrows indicate the propagation of the lowest resonances of the gauge bosons. two boundaries the following Dirichlet BC's: at y1 = 0 : A1;2 ¹ (zM) = A 1;2 ¹ (x¹); (3.5a) at y2 = : A3 ¹(zM) = A 3 ¹(x¹); B¹(zM) = B¹(x¹); (3.5b) where the bar indicates a boundary ¯eld. The Dirichlet BC's in Eqs.(3.5) require A1;2 ¹ to be independent of y2, while A3 ¹ and B¹ become independent of y1, such that we can generally write A1;2 ¹ = A1;2(x¹; y1), A3 ¹ = A3 ¹(x¹; y2), and B¹ = B¹(x¹; y2). For the transverse4 components of the gauge ¯elds the bulk equations of motion then take the forms (p2+@2 y1)A1;2 ¹ (x¹; y1) = 0; (p2+@2 y2)A3 ¹(x¹; y2) = 0; (p2+@2 y2)B¹(x¹; y2) = 0; (3.6) where p2 = p¹p¹ and p¹ = i@¹ is the momentum in the uncompacti¯ed 4D space. Since we assume all the gauge couplings to be small, we will, in what follows, treat 4Note that @MFaM¹ = p2P¹º(p)Aa¹ + (@2 y1 + @2 y2 )Aaº = 0, where P¹º(p) = g¹º ¡ p¹pº=p2 is the operator projecting onto transverse states. 48 Aa ¹ approximately as a "free" ¯eld (i.e., without self interaction) and drop all cubic and quartic terms in Aa ¹. We assume that the fermions, in the ¯rst approximation, are localized on the brane at (y1; y2) = (0; 0), away from the walls of electroweak symmetry breaking. This choice will avoid any unwanted nonoblique corrections to the electroweak precision parameters. 3.2 E®ective theory The total e®ective 4D Lagrangian in the compacti¯ed theory Ltotal can be written as Ltotal = L0 +Le® , where Le® = R ¼R1 0 dy1 R ¼R2 0 dy2 L6 denotes the contribution from the bulk, which follows from integrating out the extra dimensions. After partial inte gration along the y1 and y2 directions, we obtain for Le® the nonvanishing boundary term Le® = ¡M2L ¼R2 h A 1 ¹@y1A1¹ + A 2 ¹@y1A2¹ i y1=0 ¡¼R1 h M2L A 3 ¹@y2A3¹ +M2 Y B¹@y2B¹ i y2=0 ; (3.7) where we have applied the bulk equations of motion and eliminated the terms from the boundaries at y1 = ¼R1 and y2 = ¼R2 by virtue of the BC's in Eqs. (3.4). Notice, that in arriving at Eq. (3.7) we have rede¯ned the bulk gauge ¯elds as A¹ ! A0 ¹ ´ A¹= p 2 to canonically normalize the kinetic energy terms of the KK modes. In order to determine Ltotal explicitly, we ¯rst solve the equations of motion in Eq. (3.6) and insert the solutions into the expression for Le® in Eq. (3.7). The most general solutions for Eqs. (3.6) can be written as A1;2 ¹ (x¹; y1) = A 1;2 ¹ (x¹) cos(py1) + b1;2 ¹ (x¹) sin(py1); (3.8a) A3 ¹(x¹; y2) = A 3 ¹(x¹) cos(py2) + b3 ¹(x¹) sin(py2); (3.8b) B¹(x¹; y2) = B¹(x¹) cos(py2) + bY ¹ (x¹) sin(py2); (3.8c) 49 where p = p p¹p¹ and we have already applied the BC's in Eq. (3.5). The coe±cients ba ¹(x¹) and bY ¹ (x¹) are then determined from the BC's in Eqs. (3.4). For b1;2 ¹ (x¹), e.g., we ¯nd from the BC's in Eq. (3.4a) that b1;2 ¹ (x¹) = ¡A¹ 1;2 (x¹) cot(p¼R1) and hence one obtains A1;2 ¹ (x¹; y1) = A 1;2 ¹ (x¹) [cos(py1) ¡ cot(p¼R1) sin(py1)] : (3.9a) In a similar way, one arrives after some calculation at the solutions A3 ¹(x¹; y2) = A 3 ¹(x¹) · cos(py2) + M2L tan(p¼R2) ¡M2 Y cot(p¼R2) M2L +M2 Y sin(py2) ¸ + B¹(x¹) M2 Y tan(p¼R2) +M2 Y cot(p¼R2) M2L +M2 Y sin(py2); (3.9b) B¹(x¹; y2) = A 3 ¹(x¹) M2L tan(p¼R2) +M2L cot(p¼R2) M2L +M2 Y sin(py2) + B¹(x¹) · cos(py2) + M2 Y tan(p¼R2) ¡M2L cot(p¼R2) M2L +M2 Y sin(py2) ¸ (3:.9c) Inserting the wavefunctions in Eqs. (3.9) into the e®ective Lagrangian in Eq. (3.7), we can rewrite Le® as Le® = A a ¹§aa(p2)A a¹ + A 3 ¹§3B(p2)B ¹ + B¹§BB(p2)B ¹ ; (3.10) where (aa) = (11); (22), and (33) and the momentumdependent coe±cients § are given by §11(p2) = §22(p2) = ¼R2M2L p cot(p¼R1); §33(p2) = ¡¼R1M2L p M2L tan(p¼R2) ¡M2 Y cot(p¼R2) M2L +M2 Y ; §3B(p2) = ¡2¼R1M2L M2 Y p tan(p¼R2) + cot(p¼R2) M2L +M2 Y ; §BB(p2) = ¡¼R1M2 Y p M2 Y tan(p¼R2) ¡M2L cot(p¼R2) M2L +M2 Y : (3.11) The §'s can be viewed as the electroweak vacuum polarization amplitudes which summarize in the low energy theory the e®ect of the symmetry breaking sector. The 50 presence of these terms leads at tree level to oblique corrections (as opposed to vertex corrections and box diagrams) of the gauge boson propagators and a®ects electroweak precision measurements [44,45]. Since Le® in Eq. (3.7) generates e®ective mass terms for the gauge bosons in the 4D theory5, the KK masses of the W§ bosons are found from the zeros of the inverse propagator as given by the solutions of the equation §11(p2) ¡ p2 2g2 = 0: (3.12) To determine the KK masses of the gauge bosons, we will from now on assume that the brane terms L0 dominate the bulk kinetic terms, i.e., we take 1=g2; 1=g02 À (ML;Y ¼)2R1R2. As a result, we ¯nd for the W§'s the mass spectrum mn = n R1 µ 1 + 2g2M2L R1R2 n2 + : : : ¶ ; n = 1; 2; : : : ; m20 = 2g2M2L R2 R1 + O(g4M4L R2 2) = m2 W; (3.13) where we identify the lightest state with mass m0 with the W§. Observe in Eq. (3.13), that the inclusion of the brane kinetic terms L0 for 1=R1; 1=R2 & O(TeV ) leads to a decoupling of the higher KKmodes with masses mn (n > 0) from the electroweak scale, leaving only the W§ states with a small mass m0 in the lowenergy theory (see Fig. 3.2). Note that a similar e®ect has been found for warped models in Ref. [47]. The calculation of the mass of the Z boson goes along the same lines as for W§, but requires, due to the mixing of A 3 ¹ with B¹ in Eq. (3.10), the diagonalization of the kinetic matrix Mkin = 0 B@ §33(p2) ¡ p2 2g2 1 2§3B(p2) 1 2§3B(p2) §BB(p2) ¡ p2 2g02 1 CA ; (3.14) which has the eigenvalues ¸§(p2) = 1 2 µ §33(p2) ¡ p2 2g2 + §BB(p2) ¡ p2 2g02 ¶ § 1 2 sµ §33(p2) ¡ p2 2g2 ¡ §BB + p2 2g02 ¶2 + §2 3B(p2); (3.15) 5For an e®ective ¯eld theory approach to oblique corrections see, e.g., Ref. [46]. 51 (0;) y1 ( R1;0) 3 01 23===RRR 4=R L0 mW123===RRR 4=R (a) (b) k p;05 p05k50 pk p5k5p;5 k05 k5 Figure 3.2: E®ect of the brane kinetic terms L0 on the KK spectrum of the gauge bosons (for the example of W§). Solid lines represent massive excitations, the bottom dotted lines would correspond to the zero modes which have been removed by the BC's. Without the brane terms (a), the lowest KK excitations are of order 1=R ' 1 TeV . After switching on the dominant brane kinetic terms (b), the zero modes are approximately \restored" with a small mass mW ¿ 1=R (dashed line), while the higher KKlevels receive small corrections to their masses (thin solid lines) and decouple below » 1 TeV . where the KK towers of the ° and Z are given by the solutions of the equations ¸¡(p2) = 0 (for °) and ¸+(p2) = 0 (for Z), respectively. By taking in Eq. (3.15) the limit p2 ! 0, it is easily seen that ¸¡(p2) = 0 has a solution with p2 = 0, which we identify with the massless ° of the SM, corresponding to the unbroken gauge group U(1)Q. The lowest excitation in the tower of solutions to ¸+(p2) = 0 has a masssquared m2 Z = 2(g2 + g02)M2L M2 Y R1 (M2L+M2 Y )R2 + O(g4M4L R2 2); (3.16) which we identify with the Z of the SM. All other KK modes of the ° and Z have masses of order & 1=R2 and thus decouple for 1=R1; 1=R2 & O(TeV ), leaving only a massless ° and a Z with mass mZ in the lowenergy theory. 52 3.3 Relation to EWPT One important constraint on any model for EWSB results from the measurement of the ½ parameter, which is experimentally known to satisfy the relation ½ = 1 to better than 1% [2]. In our model, we ¯nd from Eqs. (3.13) and (3.16) a ¯t of the natural zerothorder SM relation for the ½ parameter in terms of ½ ´ m2 W m2 Z cos2 µW = g2 g2 + g02 M2L +M2 Y M2 Y µ R2 R1 ¶2 1 cos2µW = 1; (3.17) where µW ¼ 28:8± is the Weinberg angle of the SM. For de¯niteness, we will choose in the following the 4D brane couplings g and g0 to satisfy the usual SM relation g2=(g2 + g02) = cos2µW ¼ 0:77. De¯ning ½ = 1 + ¢½, we then obtain from Eq. (3.17) that ¢½ = 0 if the bulk kinetic couplings and compacti¯cation radii satisfy the relation (M2L +M2 Y )=M2 Y = R2 1=R2 2: (3.18) Although we can thus set ¢½ = 0 by appropriately dialing the gauge couplings and the size of the extra dimensions, we observe in Eq. (3.10) that Le® introduces a manifest breaking of custodial symmetry (which transforms the three gauge bosons Aa ¹ among themselves) and will thus contribute to EWPT via oblique corrections to the SM parameters.6 To estimate the e®ect of the oblique corrections in our model let us consider in the 4D e®ective theory a general vacuum polarization tensor ¦¹º AB(p2) between two gauge ¯elds A and B which can (for canonically normalized ¯elds) be expanded as [46] i¦AB ¹º (p2) = igAgB h ¦(0) AB + p2¦(1) AB i g¹º + p¹pº terms; (3.19) where gA and gB are the couplings corresponding to the gauge ¯elds A and B, re spectively. After going in Le® back to canonical normalization by rede¯ning Aa ¹ ! A0 ¹ ´ Aa ¹=g and B¹ ! B0¹ ´ B¹=g0, we identify §aa(p2) ' 1 2[¦(0) aa + p2¦(1) aa ], for 6Note, however, that in the limit p2 ! 0, we have §11 = §33, which restores custodial symmetry. 53 (aa) = (11); (22); (33); (BB), while §3B(p2) ' ¦(0) 3B + p2¦(1) 3B. From Eqs. (3.11) we then obtain the polarization amplitudes ¦(0) 11 = ¦(0) 22 = 2M2L R2 R1 ; ¦(1) 11 = ¦(1) 22 = ¡2 ¼2M2L 3 R1R2; ¦(0) 33 = 2 M2L M2 Y M2L +M2 Y R1 R2 ; ¦(1) 33 = ¡2 ¼2M2L R1R2 M2L +M2 Y (M2L + 1 3 M2 Y ); ¦(0) 3B = ¡2 M2L M2 Y M2L +M2 Y R1 R2 ; ¦(1) 3B = ¡ 4 3 ¼2M2L M2 Y M2L +M2 Y R1R2: (3.20) A wide range of e®ects from new physics on EWPT can be parameterized in the ²1, ²2, and ²3 framework [45], which is related to the S; T, and U formalism of Ref. [44] by ²1 = ®T, ²2 = ¡®U=4 sin2µW, and ²3 = ®S=4 sin2µW. The experimental bounds on the relative shifts with respect to the SM expectations are roughly of the order ²1; ²2; ²3 . 3¢10¡3 [48]. From Eq. (3.20) we then obtain for these parameters explicitly ²1 = g2(¦(0) 11 ¡ ¦(0) 33 )=m2 W = ¡2g2 M2L m2 W R1 R2 ¡ M2 Y =(M2L +M2 Y ) ¡ (R2=R1)2¢ (3;.21a) ²2 = g2(¦(1) 33 ¡ ¦(1) 11 ) = ¡g2 4¼2 3 M4L M2L +M2 Y R1R2; (3.21b) ²3 = ¡g2¦(1) 3B = g2 4¼2 3 M2L M2 Y M2L +M2 Y R1R2; (3.21c) where we have used in the last equation that ¡²3=(gg0) = ¦(1) 3° =sin2µW ¡ ¦(1) 33 = cot µW¦(1) 3B [45]. Note in Eq. (3.21a), that for our choice of parameters we have ²1 = ¢½ = 0. The quantities j²2j and j²3j, on the other hand, are bounded from below by the requirement of having su±ciently many KK modes below the strong coupling (or cuto®) scale of the theory. Using \naive dimensional analysis" (NDA) [49,50], one obtains for the strong coupling scale ¤ of a Ddimensional gauge theory [51] roughly ¤D¡4 ' (4¼)D=2¡(D=2)=g2D, where gD is the bulk gauge coupling. In our 6D model, we would therefore have ¤ ' p 2(4¼)3=2ML;Y which leads for ML;Y ' 102 GeV to a cuto® ¤ ' 6 TeV . Assuming for simplicity ML = MY , it follows from Eq. (3.18) that R2 = R1= p 2, and using Eqs. (3.21b) and (3.21c) we obtain ²3 ' g2 96 p 2¼ (¤R2)2 ' 2:3 £ 10¡3 £ (g¤R2)2; (3.22) 54 while ²2 ' ²3. It is instructive to compare the value for ²3 in our 6D setup as given by Eq. (3.22) with the corresponding result of the 5D model in Ref. [32]. We ¯nd that by going from 5D to 6D, the strong coupling scale of the theory is lowered from » 10 TeV down to » 6 TeV . Despite the lowering of the cuto® scale, however, the parameter ²3 is in the 6D model by » 15% smaller than the corresponding 5D value7. This is due to the fact that in the 6D model the bulk gauge kinetic couplings satisfy ML = MY ' 100 GeV , while they take in 5D the values ML ' MY ' 10 GeV , which is one order of magnitude below the electroweak scale. From Eq. (3.22) we then conclude that one can take for the inverse loop expansion parameter ¤R2 ' 1=g ¼ 1:6 in agreement with EWPT. Like in the 5D case, however, the 6D model seems not to admit a loop expansion parameter in the regime ¤R2 À 1 as required for the model to be calculable. 3.4 Nonoblique corrections and fermion masses In the previous discussion, we have assumed that the fermions are (approximately) localized at (y1; y2) = (0; 0). This would make the fermions exactly massless, since they have no access to the EWSB at y1 = ¼R1 and y2 = ¼R2. In this limiting case, the e®ects on the electroweak precision parameters (²1; ²2; ²3=S; T;U) come from the oblique corrections due to the vector self energies as given by Eq. (3.10). A more realistic case will be to extend the fermion wave functions to the bulk, i.e., to the walls of EWSB, where fermion mass operators of the form CªLªR (C is some appropriate mass parameter) can be written. Thus, although the fermion wave functions will be dominantly localized at (0; 0), the pro¯le of the wavefunctions in the bulk will be such that it will have small contributions from the symmetry breaking walls, giving rise to fermion masses. The hierarchy of fermion masses would then be accommodated by 7Notice that in Ref. [32], the strong coupling scale is de¯ned by 1=¤ = 1=¤L + 1=¤R, while we assume for ML = MY that ¤ = ¤L = ¤Y . 55 some suitable choice of the parameters C [52]. To make the incorporation of heavy fermions in our model explicit, let us introduce the 6D chiral quark ¯elds Qi, Ui, and Di (i = 1; 2; 3 is the generation index), where Qi are the isodoublet quarks, while Ui and Di denote the isosinglet up and down quarks, respectively. For the cancellation of the SU(3)C £SU(2)L£U(1)Y gauge and gravitational anomalies we assume that Qi have positive and Ui;Di have negative SO(1; 5) chiralities [53]. Next, we consider the action of the top quark ¯elds with zero bulk mass, which is given by Sfermion = Z dx4 Z ¼R1 0 dy1 Z ¼R2 0 dy2 i(Q3¡MDMQ3 + U3¡MDMU3) + Z dx4 Z ¼R1 0 dy1 Z ¼R2 0 dy2 K±(y1)±(y2)i[Q3¡¹D¹Q3 + U3¡¹D¹U3] + Z dx4 Z ¼R1 0 dy1 Z ¼R2 0 dy2 C±(y1 ¡ ¼R1)±(y2 ¡ ¼R2)Q3LU3R + h:(c3:.;23) where we have added in the second line 4D brane kinetic terms with a (common) gauge kinetic parameter K = [m]¡2 at (y1; y2) = (0; 0) and in the third line we included a boundary mass term with coe±cient C = [m]¡1, which mixes Q3L and U3R at (y1; y2) = (¼R1; ¼R2). Note, that the addition of the boundary mass term in the last line of Eq. (3.23) is consistent with gauge invariance, since U(1)Q the only gauge group surviving at (y1; y2) = (¼R1; ¼R2). Consider now ¯rst the limit of a vanishing brane kinetic term K ! 0. Like in the 5D case [31], appropriate Dirichlet and Neumann BC's for Q3L;R and U3L;R would give, in the KK tower corresponding to the top quark, a lowest mass eigenstate, which is a Dirac fermion with mass mt of the order mt » C=R2, where we have de¯ned the length scale R » R1 » R2. Next, by analogy with the generation of the W§ and Z masses, switching on a dominant brane kinetic term K=R2 À 1, ensures an approximate localization of Q3L and U3R at (y1; y2) = (0; 0) and leads to mt » C=K [32]. Now, the typical values of non oblique corrections to the SM gauge couplings coming from the bulk are8 » CR=K » 8The factor C becomes obvious when treating the brane ¯elds in Eq. (3.23) as 4D ¯elds, in which 56 mt=(1=R) and keeping these contributions under control, the compacti¯cation scale 1=R must be su±ciently large. Like in 5D models, this generally introduces a possible tension between the 3rd generation quark masses and the coupling of the Z to the bottom quark. Replacing in the above discussion U3L;R with D3L;R and mt by the bottom quark mass mb(mZ) ¼ 3 GeV , we thus estimate for 1=R » 1 TeV a shift of the SM Z ! bLbL coupling by roughly » 0:3%, which is of the order of current experimental uncertainties9. Similarly, we predict in our model the coupling of the Z to the top quark to deviate by » 10% from the SM value, which can be checked in the electroweak production of single top in the Tevatron Run 2. It can also be tested in the tt pair production in a possible future linear collider. 3.4.1 Improving the calculability To improve the calculability of the model, it seems necessary to raise (for given 1=g2D ) the strong coupling scale ¤, which would allow the appearance of more KK modes below the cuto®. In fact, it has recently been argued that the compacti¯cation of a 5D gauge theory on an orbifold S1=Z2 gives a cuto® which is by a factor of 2 larger than the NDA estimate obtained for an uncompacti¯ed space [48]. Let us now demonstrate this e®ect explicitly by repeating the NDA calculation of Ref. [49] on an orbifold following the methods of Refs. [35] and [54]. For this purpose, consider a 5D scalar ¯eld Á(x¹; y) (where we have de¯ned y = y1), propagating in an S1=Z2 orbifold extra dimension. The radius of the 5th dimension is R and periodicity implies y + 2¼R » y. As a consequence, the momentum in the ¯fth dimension is quantized as p5 = n=R for integer n. Under the Z2 action y ! ¡y the scalar transforms as Á(x¹; y) = §Á(x¹;¡y), where the + (¡) sign corresponds to Á being even (odd) under case C = [m]+1 and K = [m]0. 9The LEP/SLC ¯t of ¡b=¡had in Z decay requires the shift of the Z ! bLbL coupling to be . 0:3% [3]. 57 01 (a) (b) k p;05 p05k50 pk p5k5p;5 k05 k5 Figure 3.3: Oneloop diagram for ÁÁ scattering on S1=Z2. The total incoming momentum is (p; p0 5) and the total outgoing momentum is (p; p5). Generally, it is possible that jp0 5j 6= jp5j, since the orbifold ¯xed points break 5D translational invariance. Z2. The scalar propagator on this space is given by [35, 54] D(p; p5; p0 5) = i 2 ½ ±p5;p0 5 § ±¡p5;p0 5 p2 ¡ p25 ¾ ; (3.24) where the additional factor 1=2 takes into account that the physical space is only half of the periodicity. Consider now the oneloop ÁÁ scattering diagram in Fig. 3.3. The total incoming momentum is (p; p0 5) and the total outgoing momentum is (p; p5), which can in general be di®erent, since 5D translation invariance is broken by the orbifold boundaries. Locally, however, momentum is conserved at the vertices. The diagram then reads i§ = 1 4 ¸2 2 1 2¼R X k5;k05 Z d4k (2¼)4 ½ ±k5;k05 § ±¡k5;k05 k2 ¡ k2 5 ¾½ ±(p5¡k5);(p0 5¡k05 ) § ±¡(p5¡k5);(p0 5¡k05 ) (p ¡ k)2 ¡ (p5 ¡ k5)2 ¾ ; (3.25) where ¸ is the quartic coupling and the additional factor 1=4 results from working on S1=Z2. After summing over k05, the integrand can be written as F(k5) = 1 (k2 ¡ k2 5) [(p ¡ k)2 ¡ (p5 ¡ k5)2] © ±p5p0 5 + ±p5;¡p0 5 § ±2k5;(p5+p0 5) § ±2k5;(p5¡p0 5) ª : (3.26) In Eq. (3.26), the ¯rst two terms in the bracket conserve jp0 5j and contribute to the bulk kinetic terms of the scalar. The last two terms, on the other hand, violate jp0 5j conservation and thus lead to a renormalization of the brane couplings [35]. Note 58 that these brane terms lead in Eq. (3.25) to a logarithmic divergence. Applying, on the other hand, to the bulk terms the Poisson resummation identity 1 2¼R X1 m=¡1 F(m=R) = X1 n=¡1 Z 1 ¡1 dk 2¼ e¡2¼ikRnF(k); (3.27) we obtain a sum of momentum space integrals, where the \local" n = 0 term diverges linearly like in 5D uncompacti¯ed space. This term contributes a linear divergence to the diagram such that the scattering amplitude becomes under order one rescalings of the random renormalization point for the external momenta of the order i§ ! ¸2 4 Z d5k (2¼)5 [k2(p ¡ k)2]¡1 ' ¸2 2 ¤ (4¼)5=2¡(5=2) ; (3.28) where ¤ is an ultraviolet cuto®. On S1=Z2, we thus indeed obtain for the strong coupling scale ¤ ' 48¼3¸¡2, which is two times larger than the NDA value obtained in 5D uncompacti¯ed space. This is also in agreement with the de¯nition of ¤ for a 5D gauge theory on an interval given in Ref. [48]. Similarly, when the 5th dimension is compacti¯ed on S1=(Z2£Z02 ) [55], we expect a raising of ¤ by a factor of 4 with respect to the uncompacti¯ed case. Let us brie°y estimate how far this could improve the calculability of our 6D model. To this end, we assume, besides the two extra dimensions compacti¯ed on the rectangle, two additional extra dimensions with radii R3 and R4, each of which has been compacti¯ed on S1=(Z2 £Z02 ). We assume that the gauge bosons are even under the actions of the Z2£Z02 groups. Moreover, we take for the bulk kinetic coe±cients in eight dimensions M4L = M4 Y and set R3 = R4 = R2 = R1= p 2. From the expression analogous to Eq. (3.21c), we then obtain the estimate ²3 ' g2(¼MLR2)4=3 p 2, where the relative factor (¼R2=2)2, arises from integrating over the physical space on each circle, which is only 1=4 of the circumference. With respect to the NDA value ¤4 ' (4¼)4¡(4)M4L in uncompacti¯ed space, the cuto® gets now modi¯ed as ¤4 ! 16 ¢ ¤4, implying that ²3 ' g2 192 p 2 (¤R2=4)4 ' 1:3 £ 10¡3 £ (¤R2=4)4: (3.29) 59 In agreement with EWPT, the loop expansion parameter could therefore assume here a value (¤R2)¡1 ' 0:25, corresponding to the appearance of 4 KK modes per extra dimension below the cuto®. Taking also a possible additional raising of ¤ by a factor of p 2 due to the reduced physical space on the rectangle into account, one could have (¤R2)¡1 ' 0:2 with 5 KK modes per extra dimension below the cuto®. In conclusion, this demonstrates that by going beyond ¯ve dimensions, the calculability of Higgsless models could be improved by factors related to the geometry. 60 CHAPTER 4 A New Two Higgs Doublet Model 4.1 Model and the Formalism Our proposed model is based on the symmetry group SU(3)c £ SU(2)L £ U(1) £ Z2. In addition to the usual SM fermions, we have three EW singlet righthanded neutrinos, NRi; i = 1 ¡ 3, one for each family of fermions. The model has two Higgs doublets, Â and Á. All the SM fermions and the Higgs doublet Â, are even under the discrete symmetry, Z2, while the RH neutrinos and the Higgs doublet Á are odd under Z2. Thus all the SM fermions except the lefthanded neutrinos, couple only to Â. The SM lefthanded neutrinos, together with the righthanded neutrinos, couple only to the Higgs doublet Á. The gauge symmetry SU(2) £ U(1) is broken spontaneously at the EW scale by the VEV of Â, while the discrete symmetry Z2 is broken by a VEV of Á, and we take hÁi » 10¡2 eV . Thus, in our model, the origin of the neutrino masses is due to the spontaneous breaking of the discrete symmetry Z2. The neutrinos are massless in the limit of exact Z2 symmetry. Through their Yukawa interactions with the Higgs ¯eld Á, the neutrinos acquire masses much smaller than those of the quarks and charged leptons due to the tiny VEV of Á. The Yukawa interactions of the Higgs ¯elds with the leptons are LY = ylª l LlRÂ + yºlª l LNR eÁ + h:c:; (4.1) where ª l L = (ºl; l)L is the usual lepton doublet and lR is the charged lepton singlet. The ¯rst term gives rise to the mass of the charged leptons, while the second term gives a tiny neutrino mass. The interactions with the quarks are the same as in the 61 Standard Model with Â playing the role of the SM Higgs doublet. Note that in our model, a SM lefthanded neutrino, ºL combines with a right handed neutrino, NR, to make a massive Dirac neutrino with a mass » 10¡2 eV, the scale of Z2 symmetry breaking. For simplicity, we do not consider CP violation in the Higgs sector. (Note that in this model, spontaneous CP violation would be highly suppressed by the small VEV ratio and could thus be neglected. However, one could still consider explicit CP violation). The most general Higgs potential consistent with the SM £Z2 symmetry is [56] V = ¡¹21 ÂyÂ ¡ ¹22 ÁyÁ + ¸1(ÂyÂ)2 + ¸2(ÁyÁ)2 + ¸3(ÂyÂ)(ÁyÁ) ¡ ¸4jÂyÁj2 ¡ 1 2 ¸5[(ÂyÁ)2 + (ÁyÂ)2]: (4.2) The physical Higgs ¯elds are a charged ¯eld H, two neutral scalar ¯elds h and ¾, and a neutral pseudoscalar ¯eld ½. In the unitary gauge, the two doublets can be written Â = 1 p 2 0 B@ p 2(VÁ=V )H+ h0 + i(VÁ=V )½ + VÂ 1 CA ; Á = 1 p 2 0 B@ ¡ p 2(VÂ=V )H+ ¾0 ¡ i(VÂ=V )½ + VÁ 1 CA ; (4.3) where VÂ = hÂi, VÁ = hÁi, and V 2 = V 2 Â + V 2 Á . The particle masses are m2 W = 1 4 g2V 2; m2 H = 1 2 (¸4 + ¸5)V 2; m2 ½ = ¸5V 2; m2 h;¾ = (¸1V 2 Â + ¸2V 2 Á ) § q (¸1V 2 Â ¡ ¸2V 2 Á )2 + (¸3 ¡ ¸4 ¡ ¸5)2V 2 Â V 2 Á : (4.4) 62 An immediate consequence of the scenario under consideration is a very light scalar ¾ with mass m2 ¾ = 2¸2V 2 Á [1 + O(VÁ=VÂ)]: (4.5) The mass eigenstates h; ¾ are related to the weak eigenstates h0; ¾0 by h0 = ch + s¾; ¾0 = ¡sh + c¾; (4.6) where c and s denotes the cosine and sine of the mixing angles, and are given by c = 1 + O(V 2 Á =V 2 Â ); s = ¡ ¸3 ¡ ¸4 ¡ ¸5 2¸1 (VÁ=VÂ) + O(V 2 Á =V 2 Â ): (4.7) Since VÁ » 10¡2 eV and VÂ » 250 GeV, this mixing is extremely small, and can be neglected. Hence, we see that h behaves essentially like the SM Higgs (except of course in interactions with the neutrinos). The interactions of the neutral Higgs ¯elds with the Z are given by Lgauge = g 2V (cVÁ + sVÂ)(½@¹h ¡ h@¹½)Z¹ + g 2V (sVÁ ¡ cVÂ)(½@¹¾ ¡ ¾@¹½)Z¹ + g2 4 (sVÁ ¡ cVÂ)hZ¹Z¹ + g2 4 (cVÁ + sVÂ)¾Z¹Z¹ + g2 8 (h2 + ¾2 + ½2)Z¹Z¹ (4.8) where g2 = g2 + g02, and VÂ and VÁ are the two VEV's. 4.2 Phenomenological Implications We now consider the phenomenological implications of this model. There are sev eral interesting phenomenological implications which can be tested in the upcoming 63 neutrino experiments and high energy colliders. The light neutrinos in our model are Dirac particles. So neutrinoless double beta decay is not allowed in our model. This is a very distinctive feature of our model for the neutrino masses compared to the traditional seesaw mechanism. In the seesaw model, light neutrinos are Majorana particles, and thus neutrinoless double beta decay is allowed. The current limit on the double beta decay is mee » 0:3 eV . This limit is expected to go down to about mee » 0:01 eV in future experiments [57]. If no neutrinoless double beta decay is observed to that limit, that will cast serious doubts on the seesaw model. In our model, of course, it is not allowed at any level. Next, we consider the implications of our model for high energy colliders. First we consider the production of the light scalar ¾ in e+e¡ collisions. The only possible decay modes of this particle are a diphoton mode, ¾ ! °° which can occur at the oneloop level and, if it has enough mass, a ¾ ! ºº mode. The one loop decay to two photons takes place with quarks, W bosons, or charged Higgs bosons in the loop. The largest contribution to this decay mode is » e8m5 ¾=mq 4. This gives the lifetime of ¾ to be » 1020 years, which is much larger than the age of the universe. Thus ¾ essentially behaves like a stable particle, and its production at the colliders will lead to missing energy in the event. The couplings of ¾ to quarks and charged leptons takes place only through mixing which is highly suppressed (proportional to the ratio VÁ=VÂ). Thus we need only consider its production via its interactions with gauge bosons. The ZZ¾ coupling is also highly suppressed, so that processes such as e+e¡ ! Z¤ ! Z¾ and Z ! Z¤¾ ! ff¾ are negligible. However, no such suppression occurs for the ZZ¾¾ coupling. Consider the Z decay process Z ! Z¤¾¾ ! ff¾¾. A direct calculation yields the width (neglecting the ¾ and fermion masses), 64 ¡(Z ! ff¾¾) = G3 Fm5 Z(g2V + g2A ) 2 p 2(2¼)5 Z mZ=2 0 dE1 Z mZ=2 0 dE2 £ Z 1 ¡1 d(cos µ) E2 1E2 2(3 ¡ cos µ) (2E1E2 ¡ 2E1E2 cos µ ¡ m2 Z)2 + m2 Z¡2 Z ; (4.9) where gV = T3 ¡ 2Qsin2 µW and gA = T3. This gives X f ¡(Z ! ff¾¾) ' 2:5 £ 10¡7 GeV: (4.10) For the 1:7 £ 107 Z's observed at resonance at LEP1 [58], this gives an expectation of only about two such events. Now we consider the production of the heavy Higgs particles in our model. Since the charged Higgs H§ and the pseudoscalar, ½ can be produced along with the light scalar ¾, there will be stricter mass bound on these particles than in a typical two Higgs doublet model. Let us consider the pseudoscalar ½, and assume m½ < mZ. Then the Z can decay via Z ! ¾½. Since ½ couples negligibly to all SM fermions except the neutrinos, here we need only consider its decay to ºº (or ¾¾ if we consider CP violation), so this process contributes to the invisible decay width of the Z. The width for this process is ¡ = GFm3 Z 24 p 2¼ µ 1 ¡ m2 ½ m2 Z ¶3 (4.11) This is less than the experimental uncertainty in the invisible Z width for m½ & 78 GeV . (The experimental value of the invisible Z width is 499:0 § 1:5 MeV [59].) For m½ > mZ, real pseudoscalar ½ can be produced via e+e¡ ! Z¤ ! ½¾. The total cross section for this process is ¾ = G2 Fm4 Z(g2V + g2A )s 24¼ µ 1 s ¡ m2 Z ¶2 µ 1 ¡ m2 ½ s ¶3 : (4.12) 65 For LEP2, p s ' 200 GeV , we ¯nd that less than one event is expected in ' 3000 pb¡1 [6] of data for m½ & 95 GeV . Note that the bound on the ½ mass we obtain is much less than the mass for which the Higgs potential becomes strongly coupled (¸5 · 2 p ¼ which gives m½ · 470 GeV). For m½ > mZ, the Z can still decay invisibly through Z ! ½¤¾ ! ºº¾. The width for this decay is invwidth¡ = GFm2 Zy2 ºl 3 p 2(2¼)3 Z mZ=2 0 dE E3(mZ ¡ 2E) (m2 Z ¡ 2mZE ¡ m2 ½)2 : (4.13) Summing over generations, this gives ¡(m½ = 100 GeV ) ' (0:1 MeV )( 1 3 X l y2 ºl) ¡(m½ = 200 GeV ) ' (4 £ 10¡3 MeV )( 1 3 X l y2 ºl): (4.14) Even if we take 1 3 P y2 ºl » 1, these values are well within the experimental uncertainty in the invisible Z width of 1:5 MeV . Note that if we allow explicit CP violation in the Higgs sector, the invisible decay Z ! ½¾ ! ¾¾¾ will also occur. Our model has very interesting implications for the discovery signals of the Higgs boson at the high energy colliders, such as the Tevatron and LHC. Note that since VÁ is extremely small compared to VÂ, the neutral Higgs boson, h is like the SM Higgs boson so far its decays to fermions and to W and Z bosons are concerned. However, in our model, h has new decay modes, such as h ! ¾¾ which is invisible. This could change the Higgs signal at the colliders dramatically. The width for this invisible decay mode h ! ¾¾ is given by ¡(h ! ¾¾) = (¸3 + ¸4 + ¸5)2V 2 Â 32¼mh : (4.15) 66 100 150 200 250 300 mh HGeVL 0 0.2 0.4 0.6 0.8 Br Hh s sL 0 0.2 0.4 0.6 0.8 1 l* 0 0.2 0.4 0.6 0.8 1 Br Hh s sL Figure 4.1: Left panel: Branching ratio for h ! ¾¾ as a function of mh for the value of the parameter, ¸¤ = 0:1. Right panel: Branching ratio for h ! ¾¾ as a function of ¸¤ for mh = 135 GeV . Using m2 h = 2¸1V 2 Â + O(V 2 Á =V 2 Â ); (4.16) this can be written ¡(h ! ¾¾) = (¸3 + ¸4 + ¸5)2mh 64¼¸1 : (4.17) Depending on the parameters, it is possible for the dominant decay mode of h to be this invisible mode. The branching ratios for the Higgs decay to this invisible mode are shown in Fig. 4 (left panel), for the Higgs mass range from 100 to 300 GeV , for the choice of the value of the parameter, ¸¤ equal to 0:1 where ¸¤ is de¯ned to be equal to (¸3+¸4+¸5)2 ¸1 . The right panel in Fig. 4 shows how this branching ratio depends on this parameter for a Higgs mass of 135 GeV . (The results for the branching ratio is essentially the same for other values of the Higgs mass between 120 and 160 GeV ). We see that for a wide range of this parameter, for the Higgs mass up to about 160 GeV , the invisible decay mode dominates, thus changing the Higgs search strategy at the Tevatron Run 2 and the LHC . The production rate of the neutral scalar Higgs h in our model are essentially the same as in the SM. This implies that the Higgs mass bound from LEP is not signi¯cantly altered . (The L3 collaboration set a bound of 67 mh ¸ 112:3 GeV for an invisibly decaying Higgs with the SM production rate [60]). However, because of the dominance of the invisible decay mode, it will be very di±cult to observe a signal at the LHC in the usual production and decay channels such as qqh ! qqWW, qqh ! qq¿ ¿ , h ! °°, h ! ZZ ! 4l, tth (with h ! bb) and h ! WW ! lºlºl [61]. However, a signal with such an invisible decay mode of the Higgs (as in our model) can be easily observed at the LHC through the weak boson fusion processes, qq ! qqW+W¡ ! qqH and qq ! qqZZ ! qqH [62] if appropriate trigger could be designed for the ATLAS and CMS detector. For example, with only 10 fb¡1 of data at the LHC, such a signal can be observed at the 95 percent CL with an invisible branching ratio of 31 percent or less for a Higgs mass of upto 400 GeV [62]. Thus our model can be easily tested at the LHC for a large region of the Higgs mass. Of course, establishing that this signal is from the Higgs boson production will be very di±cult at the LHC. For the Higgs search at the Tevatron, the usual signal from the Wh production, and the subsequent decays of h to WW¤ or bb will be absent. The most promising mode in our model will be the production of ZH, with Z decaying to l§l§ (l = e; ¹) and the Higgs decaying invisibly. There will be a peak in the missing energy distribution in the ¯nal state with a Z. We urge the Tevatron collaborations to look for such a signal. 4.3 Cosmological Implications Our model has several interesting astrophysical and cosmological implications. Firstly, there is a problem with primordial nucleosynthesis [63]. This occurs because the relatively strong interactions between left and righthanded neutrinos and the light scalar ¾ will keep righthanded neutrinos and ¾ in thermal equilibrium with left handed neutrinos during nucleosynthesis. So, the e®ective number of light degrees of freedom, g¤ = gb + 7 8gf (gb and gf are the numbers of bosonic and fermionic spin degrees of freedom respectively), is 68 g¤ = (g¤)SM + 1 + 7 8 (6) = 17: (4.18) (Equivalently, the e®ective number of neutrinos is Nº = 6 + 4 7 .) This increases the expansion rate of the universe, which is proportional to p g¤. As a result, reactions which interconvert protons and neutrons freeze out of thermal equilibrium at a higher temperature, increasing the ratio of neutrons to protons during nucleosynthesis. This increase alters the abundances of light elements produced in subsequent nucleosyn thesis reactions, most notably, helium4 is greatly overproduced. The mass fraction of helium4 obtained here is ' 0:3 compared to the observed fraction ' 0:25. To solve this problem, our model requires a nonstandard nucleosynthesis scenario. One possibility is a large neutrino degeneracy. It is assumed in standard nucleosynthe sis that the chemical potential of neutrinos ¹º ' 0. However, since relic neutrinos are not observed, this is not required by observation. A large value of ¹º alters the equilibrium ratio of neutrons to protons, n p = e¡¹º=T µ n p ¶ ¹º=0 ; (4.19) leading to an alteration of light element abundances. Our problem can be solved with ¹º » 0:1 MeV . In depth studies have been conducted, where the e®ective number of neutrinos, neutrino degeneracy and the density of baryons are allowed to vary, in order to ¯nd the most general values consistent with BBN and WMAP [65](as well as studies which ¯x Nº = 3, leading to much stronger bounds on neutrino degeneracy [66]). These studies ¯nd upper bounds on Nº from 7:1 to 8:7, depending on how conservative an interpretation of the data is used. Another possible solution could be the existence of massive particle species that decay after nucleosynthesis. Energetic decay products of these particles interact with background nuclei, causing nonthermal nuclear reactions, such as helium4 dissociation, that reset light element 69 abundances [64]. (We also note that in the above analysis, we have taken three right handed neutrinos. For the oscillation experiments, as well as for direct measurements, the lightest neutrino mass can be zero. So, only two righthanded neutrinos are strictly required. This could make the Big Bang nucleosynthesis problem somewhat milder.) There are also bounds on the e®ective number of neutrinos coming from astro physical observations other than light element abundances. For example, data from WMAP and the Sloan Digital Sky Survey (SDSS) power spectrum of luminous red galaxies, give a bound 0:8 < Nº < 7:6 [67]. The authors of [68] claim that data from the SDSS Lyman® forest power spectrum, along with cosmic microwave background, supernova, and galaxy clustering data, seem to require Nº > 3. Additionally, the ºº¾ interaction can a®ect supernova explosion dynamics,and since this interaction can be fairly strong it may bind ºº, giving rise to the possibility of ºº atoms and a new kind of star formation. Also, the spontaneous breaking of the discrete global symmetry Z2 will lead to the formation of cosmological domain walls. These walls will have energy per unit area ´ » V 3 Á , so their e®ect will be small. The resulting temperature anisotropies are ±T T ' G´H¡1 0 » 10¡20; (4.20) where G is Newton's gravitational constant and H0 is the present Hubble parameter. The observed level of CMB temperature anisotropies is 10¡5 [59], so this is not a problem. 70 CHAPTER 5 CONCLUSIONS We have presented several scenarios that alter the Higgs sector from that of the SM. First, we presented a renormalizable nonsupersymmetric model based on the ¯nite symmetry G = (G1 £G2)oA4, with G1 = S3 £S3 £S3 £S3 and G2 = Z2 £Z2 £Z2, with SM leptons assigned to representations of A4. Neutrino masses are generated by a Higgs ¯eld Á belonging to a 16dimensional representation of G1 o A4 while chargedlepton masses are generated by a Higgs ¯eld Â belonging to a 6dimensional representation of G2 oA4. The additional symmetries, G1 and G2, prevent quadratic and cubic interactions between Á and Â and allow only a trivial quartic interaction that does not cause an alignment problem, addressing the alignment problem without altering the desired properties of the family symmetry. In this way, we are able to explain all aspects of neutrino mixing using only symmetries which are spontaneously broken by the Higgs mechanism. Next, we have considered a 6D Higgsless model for EWSB based only on the SM gauge group SU(2)L £ U(1)Y . The model is formulated in °at space with the two extra dimensions compacti¯ed on a rectangle of size » (TeV )¡2. EWSB is achieved by imposing consistent BC's on the edges of the rectangle. The higher KK resonances of W§ and Z decouple below » 1 TeV through the presence of a dominant 4D brane induced gauge kinetic term at the point where SU(2)L £ U(1)Y remains unbroken. The ½ parameter is arbitrary and can be set exactly to unity by appropriately choosing the bulk gauge couplings and compacti¯cation scales. The resulting gauge couplings 71 in the e®ective 4D theory arise essentially from the brane couplings, slightly modi¯ed (at the level of one percent) by the bulk interaction. Thus, the main role played by the bulk interactions is to break the electroweak gauge symmetry. We calculate the treelevel oblique corrections to the S, T, and U parameters and ¯nd them to be consistent with current data. Finally, we have presented a simple extension of the Standard Model supplemented by a discrete symmetry, Z2. We have also added three righthanded neutrinos, one for each family of fermions, and one additional Higgs doublet. While the electroweak symmetry is spontaneously broken at the usual 100 GeV scale, the discrete symmetry, Z2 remains unbroken to a scale of about 10¡2 eV . The spontaneous breaking of this Z2 symmetry by the VEV of the second Higgs doublet generates tiny masses for the neutrinos. The neutral heavy Higgs in our model is very similar to the SM Higgs in its couplings to the gauge bosons and fermions, but it also couples to a very light scalar Higgs present in our model. This light scalar Higgs, ¾, is essentially stable, or decays to ºº. Thus the production of this ¾ at the high energy colliders leads to missing energy. The SMlike Higgs, for a mass up to about 160 GeV dominantly decays to the invisible mode h ! ¾¾. Thus the Higgs signals at high energy hadron colliders are dramatically altered in our model. Our model also has interesting implications for astrophysics and cosmology. 72 BIBLIOGRAPHY [1] S. Weinberg, Phys. Rev. Lett. 19 (1967) 1264; A. Salam, p.367 of Elementary Particle Theory, ed. N. Svartholm (Almquist and Wiksells, Stockholm, 1969); S.L. Glashow, J. Iliopoulos, and L. Maiani, Phys. Rev. D 2 (1970) 1285. [2] Particle Data Group Collaboration, K. Hagiwara et al., Phys. Rev. D 66 (2002) 010001. [3] The ElectroWeak Working Group, http://lepewwg.web.cern.ch/LEPEWWG/ [4] F. Englert and R. Brout, Phys. Rev. Lett. 13 (1964) 321; P.W. Higgs, Phys. Lett. 12 (1964) 132 and Phys. Rev. Lett. 13 (1964) 508; T.W. Kibble, Phys. Rev. 155 (1967) 1554. [5] C. H. Llewellyn Smith, Phys. Lett. B 46 (1973) 233; D.A. Dicus, V.S. Mathur, Phys. Rev. D 7 (1973) 3111; J.M. Cornwall, D.N. Levin, and G. Tiktopoulos, Phys. Rev. D 10 (1974) 1145; B.W. Lee, C. Quigg, and H.B. Thacker, Phys. Rev. D 16 (1977) 1519; M.J.G. Veltmann, Acta Phys. Polon. B 8 (1977) 475. [6] ALEPH, DELPHI, L3, and OPAL Collaborations, The LEP working group for Higgs boson searches, G. Abbiendi et al., Phys. Lett. B565 (2003) 61. [7] T. Schwetz, M. Tortola, and J.W.F. Valle, New J. Phys. 10, 113011 (2008). [8] G.L. Fogli, E. Lisi, A. Mirizzi, D. Montanino, and P.D. Serpico, Phys. Rev. D74, 093004 (2006). [9] P.F. Harrison, D.H. Perkins and W.G. Scott, Phys. Lett. B458, 79 (1999); Phys. Lett. B530, 167 (2002). 73 [10] Z.Z. Xing, Phys. Lett. B533, 85 (2002); X.G. He and A. Zee, Phys. Lett. B560, 87 (2003); Phys. Rev. D68, 037302 (2003). [11] A. Aranda, C.D. Carone, and R.F. Lebed, Phys. Rev. D62, 016009 (2000). [12] J. Kubo, A. Mondragon, M. Mondragon, and E. RodriguezJauregui, Prog. Theor. Phys. 109, 795 (2003). [13] C. Hagedorn, M. Lindner, and F. Plentinger, Phys. Rev. D74, 025007 (2006). [14] I. de Medeiros Varzielas, S.F. King, and G.G. Ross, Phys. Lett. B648, 201 (2007). [15] M.C. Chen and K.T. Mahanthappa, Phys. Lett. B652, 34 (2007). [16] F. Feruglio, C. Hagedorn, Y. Lin, and L. Merlo, Nucl. Phys. B775, 120 (2007). [17] P. Frampton and T. Kephart, JHEP 0709, 110 (2007). [18] A. Blum, C. Hagedorn, and M. Lindner, Phys. Rev. D77, 076004 (2008). [19] W. Grimus and L. Lavoura, JHEP 0904, 013 (2009). [20] E. Ma and G. Rajasekaran, Phys. Rev. D64, 113012 (2001) [21] E. Ma, Mod. Phys. Lett. A17, 2361 (2002). [22] K.S. Babu, E. Ma, and J.W.F. Valle, Phys. Lett. B552, 207 (2003). [23] G. Altarelli and F. Feruglio, Nucl. Phys. B720, 64 (2005); Nucl. Phys. B741, 215 (2006). [24] K. S. Babu and X.G. He, hepph/0507217. [25] X.G. He, Y.Y. Keum, and R.R. Volkas, JHEP 0604, 039 (2006). [26] E. Ma, Phys. Lett. B671, 366 (2009). 74 [27] J. Scherk and J.H. Schwarz, Phys. Lett. B 82 (1979) 60; Nucl. Phys. B 153 (1979) 61; E. Cremmer, J. Scherk, and J.H. Schwarz, Phys. Lett. B 84 (1979) 83; Y. Hosotani, Phys. Lett. B 126 (1983) 309 (1983); Phys. Lett. B 129 (1983) 193; An nals Phys. 190 (1989) 233; for a recent review see M. Quiros, hepph/0302189. [28] T. Kaluza, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys. ) 1921 (1921) 966; O. Klein, Z. Phys. 37 (1926) 895. [29] R. Sekhar Chivukula, D.A. Dicus, and H.J. He, Phys. Lett. B 525 (2002) 175; R.S. Chivukula, D.A. Dicus, H.J. He, and S. Nandi, Phys. Lett. B 562 (2003) 109. [30] C. Csaki, C. Grojean, H. Murayama, L. Pilo, and J. Terning, Phys. Rev. D 69 (2004) 055006. [31] C. Csaki, C. Grojean, J. Hubisz, Y. Shirman, and J. Terning, hepph/0310355. [32] R. Barbieri, A. Pomarol, and R. Rattazzi, Phys. Lett. B 591 (2004) 141. [33] L. Randall and R. Sundrum, Phys. Rev. Lett. 83 (1999) 4690; Phys. Rev. Lett. 83 (1999) 3370. [34] E.A. Mirabelli and M.E. Peskin, Phys. Rev. D 58 (1998) 065002. [35] H. Georgi, A.K. Grant, and G. Hailu, Phys. Lett. B 506 (2001) 207. [36] W.D. Goldberger and M.B. Wise, Phys. Rev. D 65 (2002) 025011. [37] See for example, Higgs Hunters Guide, by J.F. Gunion, H. E. Haber, G. L. Kane and S. Dawson, AddisonWesley Publishing, New York, 1990. [38] R. Barbieri, L.J. Hall, and V.S. Rychkov, Phys. Rev. D74 (2006) 015007. [39] S.K. Kim, Group Theoretical Methods: And Applications to Molecules and Crys tals, Cambridge University Press, New York, 1999. 75 [40] C. Hagedorn, M. Lindner, and R.N. Mohapatra, JHEP 0606, 042 (2006). [41] C.S. Lam, Phys. Rev. D79, 073015 (2008); W. Grimus, L. Lavoura, and P.O. Ludl, 0906.2689 [hepph]. [42] B.A. Dobrescu and E. Ponton, JHEP 0403 (2004) 071 (2004). [43] A. Muck, A. Pilaftsis, and R. Ruckl, Phys. Rev. D 65 (2002) 085037. [44] M.E. Peskin and T. Takeuchi, Phys. Rev. D 46 (1992) 381 (1992). [45] G. Altarelli and R. Barbieri, Phys. Lett. B 253 (1991) 161; G. Altarelli, R. Barbieri, and S. Jadach, Nucl. Phys. B 369 (1992) 3; Erratumibid. B 376 (1992) 444. [46] B. Holdom and J. Terning, Phys. Lett. B 247 (1990) 88; M. Golden and L. Randall, Nucl. Phys. B 361 (1991) 3. [47] M. Carena, E. Ponton, T.M.P. Tait, and C.E.M.Wagner, Phys. Rev. D 67 (2003) 096006. [48] R. Barbieri, A. Pomarol, R. Rattazzi, and A. Strumia, hepph/0405040. [49] A. Manohar and H. Georgi, Nucl. Phys. B 234 (1984) 189. [50] H. Georgi and L. Randall, Nucl. Phys. B 276 (1986) 241. [51] Z. Chacko, M.A. Luty, and E. Ponton, JHEP 0007 (2000) 036. [52] K. Agashe, A. Delgado, M.J. May, and R. Sundrum, JHEP 0308 (2003) 050. [53] B.A. Dobrescu and E. Poppitz, Phys. Rev. Lett. 87 (2001) 031801; N. Arkani Hamed, H.C. Cheng, B.A. Dobrescu, and L.J. Hall, Phys. Rev. D 62 (2000) 096006. [54] H.C. Cheng, K.T. Matchev, and M. Schmaltz, Phys. Rev. D 66 (2002) 036005. 76 [55] R. Barbieri, L.J. Hall, and Y. Nomura, Phys. Rev. D 63 (2001) 105007. [56] S. Nandi, Phys. Lett. B202 (1988) 385, Erratumibid, B207 (1988) 520. [57] L. Baudis et al. Phys. Rev. Lett. 83 (1999) 41; IGEX Collaboration, C.E. Aalseth et al., Phys. Rev. D65 (2002) 092007; I. Abd et al., hepex/0404039. [58] ALEPH, DELPHI, L3, and OPAL Collaborations, The LEP EW working group, The SLD EW and heavy °avour groups, Phys. Rept. 427 (2006) 257. [59] W.M. Yao et al., J. Phys. G33 (2006) 1. [60] L3 Collaboration, P. Achard et al., Phys. Lett. B609 (2005) 35. [61] K. Crammer, B. Mellado, W. Quayle, and S. L. Wu, (ATLAS Collaboration), ATLPHYS2004034. [62] O.J.P. Eboli and D. Zepppenfeld, Phys. Lett. B495 (2000) 147. [63] The Early Universe, by E.W. Kolb and M.S. Turner, AddisonWesley Publishing, New York, 1990. [64] See for example, M. Kawasaki, K. Kohri, and T. Moroi, Phys. Rev. D71 (2005) 083502; S. Dimopoulos, R. Esmailzadeh, L.J. Hall, and G.D. Starkman, Astro phys. J. 330 (1988) 545. [65] V. Barger, J.P. Kneller, P. Langacker, D. Marfatia, and G. Steigman, Phys. Lett. B569 (2003) 123; A. Cuoco, F. Iocco, G. Mangano, G. Miele, O. Pisanti, and P.D. Serpico, Int. J. Mod. Phys. A19 (2004) 4431; G. Steigman, Phys. Scripta T121 (2005) 142. [66] P.D. Serpico and G.G. Ra®elt, Phys. Rev. D71 (2005) 127301. [67] K. Ichikawa, M. Kawasaki, and F. Takahashi, arXiv: astroph/0611784v1. 77 [68] U. Seljak, A. Slosar, and P.McDonald, JCAP 0610 (2006) 014. [69] A. Vilenkin, Phys. Rept. 121 (1985) 263. 78 VITA Steven Gabriel Candidate for the Degree of Doctor of Philosophy Dissertation: NEW IDEAS IN HIGGS PHYSICS Major Field: Physics Biographical: Personal Data: Born in Snellville, Georgia, United States on March 2, 1980. Education: Received the B.S. degree from Georgia State University, Atlanta, Georgia, United States, 2002, in Physics Completed the requirements for the degree of Doctor of Philosophy with a major in Physics, Oklahoma State University in May, 2010. Name: Steven Gabriel Date of Degree: May, 2010 Institution: Oklahoma State University Location: Stillwater, Oklahoma Title of Study: NEW IDEAS IN HIGGS PHYSICS Pages in Study: 78 Candidate for the Degree of Doctor of Philosophy Major Field: Physics The Higgs mechanism, which is responsible for electroweak symmetry breaking and unitarization of massive W§ and Z scattering, is a fundamental ingredient of the Standard Model (SM). However, there is as yet no direct evidence of the Higgs boson, so that the details of the Higgs sector, if it even exists, remain a mystery. Here, we explore several scenarios that alter the Higgs sector from that of the SM. The ¯rst uses additional symmetries of the Higgs sector to address certain issues of neutrino mixing, the second uses extra dimensional boundary conditions to avoid the need for a Higgs entirely, and the last uses additional Higgs ¯elds to provide an alternative explanation for tiny neutrino masses. ADVISOR'S APPROVAL: 



A 

B 

C 

D 

E 

F 

I 

J 

K 

L 

O 

P 

R 

S 

T 

U 

V 

W 


