BARYON ASYMMETRY OF THE UNIVERSE
AND NEUTRINO PHYSICS
By
ABDEL G. BACHRI
Bachelor of Science
University of Hassan II
Casablanca, Morocco
1997
Master of Science
University of Hassan II
Casablanca, Morocco
1999
High Energy Physics Diploma
The Abdus Salam ICTP
Trieste, Italy
2000
Submitted to the Faculty of the
Graduate College of the
Oklahoma State University
in partial ful¯llment of
the requirements for
the Degree of
DOCTOR OF PHILOSOPHY
July, 2007
ii
BARYON ASYMMETRY OF THE UNIVERSE
AND NEUTRINO PHYSICS
Thesis Approved:
Kaladi S. Babu
Thesis Adviser
Satya Nandi
John Mintmire
John Chandler
A. Gordon Emslie
Dean of the Graduate College
ACKNOWLEDGMENTS
I would like to express my sincere appreciation and gratefulness to my thesis
advisor, Prof. Kaladi S. Babu for his guidance, motivation, ¯nancial support, inspira-
tion, and friendship. His valuable advice, criticism, and encouragement have greatly
helped me in the materialization of this thesis. I have bene¯ted much from his broad
range of knowledge, his scienti¯c approach and his warm personality. I am sure this
will have a positive in°uence on me for the rest of my scienti¯c career, and continue
to be a source of inspiration to me.
My deep appreciation extends to Prof. Satya Nandi for his assistance during
my stay at the Oklahoma State University, together with Prof. Kaladi S. Babu, he
created a pleasant productive atmosphere for the all High Energy students.
I want to thank the various individuals with whom I had the opportunity to
collaborate or discuss during these years. In particular, I want to acknowledge the
collaborations with Dr. Zurab Tavartkiladze and fruitful discussion with Dr. Ilia
Gogoladze, Dr. Tsedenbaljir Enkhbat, Dr. Cyril Anoka. Each of them has made
my experience at OSU unique. I am very grateful to my colleagues, all former and
present members of Prof. Babus and Prof. Nandi's research group, thank you for
providing such a pleasant and friendly working environment for the past few years.
I would also like to thank the faculty members that served on my advisory
committee; Profs J. Mintmire, J.P. Chandler and P. Westhaus for their constructive
comments and willingness to help.
Additionally, I want to thank all the faculty of the department of Physics and
sta® for their individual contributions to my education. I am deeply indebted to
Prof. Paul Westhaus for his unconditional help and regular guidance.
iii
I would like to acknowledge The Abdus Salam International Center for Theoret-
ical Physics in Trieste, Italy for my pre-Ph. D training. A special thank you goes to
Prof. Goran Senjanovic. I would like to acknowledge the generous ¯nancial support
that I have received during my stay from OSU Department of Physics, High Energy
Theory group and the US Department of Energy.
Although my father EL Hadj Hamid Bachri lives only in memory, I would like
to recognize his tremendous positive impact that helped shape my personality and
provided a big motivation to undertake Ph. D. May his soul rest in peace. I want
to especially thank my mother Rachida, my brothers and sisters; Lahcen, Ahmed,
Ali, Zakaria, Assia, Fatima, Malika, Khadija and Zohra for their unchanging moral
support during this undertaking.
Finally, I am deeply indebted to my wife, Shirley Bachri, for her unconditional
love, patience, care, and sacri¯ce. Thank you for your continuous assistance no matter
what the need was.
iv
TABLE OF CONTENTS
Chapter Page
1. INTRODUCTION : : : : : : : : : : : : : : : : : : : : : : : : : : : : 1
1.1. Matter Versus Anti-Matter : : : : : : : : : : : : : : 1
1.2. Sakharov criteria : : : : : : : : : : : : : : : : : : : : : 5
1.3. Boltzmann Equations : : : : : : : : : : : : : : : : : 8
1.4. Chemical potential, asymmetries relations
and Sphalerons : : : : : : : : : : : : : : : : : : : : : : : 10
2. LEPTOGENESIS IN MINIMAL LEFT-RIGHT SYMMET-
RIC MODELS : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15
2.1. Introduction : : : : : : : : : : : : : : : : : : : : : : : 15
2.2. Brief review of the minimal left-right sym-
metric model : : : : : : : : : : : : : : : : : : : : : : : : : 17
2.3. Leptogenesis in left-right symmetric framework : 20
2.4. CP violation and lepton asymmetry : : : : : : : : : 24
2.5. Numerical Boltzmann equations : : : : : : : : : : : 26
2.6. Results and discussion : : : : : : : : : : : : : : : : : 30
2.7. Gravitino Problem : : : : : : : : : : : : : : : : : : : : 32
2.8. Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : 36
3. BARYON ASYMMETRY VIA SOFT LEPTOGENESIS : : : : : : : 37
3.1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : 37
3.1.1. Soft Leptogenesis, a brief review : : : : : : : : : 37
3.2. The minimal left-right symmetric model : : : : : : 40
3.3. ~ºR decay mediated by SU(2)R gauge boson WR : : 43
3.4. Computing the two loop amplitude leading
to ~ºc ¡ ~ºcy mixing : : : : : : : : : : : : : : : : : : : : : : 48
3.5. SUSYLR RGEs e®ect on Soft Leptogenesis : : : : 49
3.6. Symmetry breaking contribution to r.h.n B¡term 52
3.7. Numerical result and estimation of BA : : : : : : 54
3.8. Conclusion : : : : : : : : : : : : : : : : : : : : : : : : : 56
iv
Chapter Page
4. PREDICTIVE MODEL OF INVERTED NEUTRINO MASS
HIERARCHY AND RESONANT LEPTOGENESIS : : : : : : : : : : 58
4.1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : 58
4.2. Predictive Framework for Neutrino Masses
and Mixings : : : : : : : : : : : : : : : : : : : : : : : : : 60
4.2.1. Improved µ12 with µ13 6= 0 : : : : : : : : : : : : 63
4.3. Resonant Leptogenesis : : : : : : : : : : : : : : : : : 66
4.4. Model with S3 £ U(1) Symmetry : : : : : : : : : : : 71
4.5. Conclusions : : : : : : : : : : : : : : : : : : : : : : : : 76
5. SUMMARY AND CONCLUSIONS : : : : : : : : : : : : : : : : : : : 78
BIBLIOGRAPHY : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 80
APPENDICES : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 85
APPENDIX A| : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 86
A.1. Basic Thermodynamics of The Expanding Universe : : : 86
A.2. FRW Universe and Boltzmann transport equations : : : 90
A.3. CP Violation in Neutral K-Meson System : : : : : : : : 92
A.4. Bessel Functions : : : : : : : : : : : : : : : : : : : : : : : 97
A.5. Loop Integrals : : : : : : : : : : : : : : : : : : : : : : : : 99
A.6. Mathematica Code : : : : : : : : : : : : : : : : : : : : : 100
v
LIST OF TABLES
Table Page
3.1. Particle assignment in SUSYLR gauge group SU(3)C £
SU(2)L £ SU(2)R £ U(1)B¡L. : : : : : : : : : : : : : : : : : : : : : : 41
3.2. Result: The left column of the table gives input values of the
parameters at the Gut scale, where the right column shows
the result of the Soft parameters at vR following RGE running.
The ¯nal estimation for the BA is also given. : : : : : : : : : : : : : 57
4.1. Transformation properties under S3 £ U(1). : : : : : : : : : : : : : : 73
A.1. The number density ni, energy density ½i and pressure pi of
the particle i, which is thermal equilibrium, in the limits of
T À mi and T ¿ mi. Where the following assumptions have
been made: j¹ij ¿ T and j¹ij < mi (no Bose-Einstein condensation). 87
vi
LIST OF FIGURES
Figure Page
2.1. Plots for CP asymmetry parameter "1 using analytical (dotted)
and numerical (solid) results as a function of the neutrino
oscillation angle µ13. The input parameters used are a12 =
1, b = 1, ¢m2
¯ = 2:5 £ 10¡5 eV 2, ¢m2
a = 5:54 £ 10¡3 eV 2
and f±; ®g = f¼=4; ¼=4g. Our model requires j"1j & 1:3 £
10¡7 to successfully generate an adequate number for the BA.
This criterium happens to be satis¯ed only in the region for
which 0:01 . µ13 . 0:07, this interval is not too sensitive to
variations in the input parameters. : : : : : : : : : : : : : : : : : : : 27
2.2. Various thermally averaged reaction rates ¡X contributing to
BE normalized to the expansion rate of the Universe H(z = 1).
The straight greyed line represents H(z)=H(z = 1), the
dashed line is for ¡D1=H(z = 1), the dotted-dashed line repre-
sents ¡¢L=1=H(z = 1) processes and the red curve represents
¡¢L=2=H(z = 1). : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 34
2.3. Evolution of YN1 (solid blue), Y eq
N1 (dot-dash) and the baryon
asymmetry ´B (dark solid line) in terms of z in the model.
The estimated value for the baryon asymmetry is ´B ' 6:03£
10¡10, with Y ini
N1 = 0 and assuming no pre-existing B ¡ L asymmetry. 35
3.1. Interfering ~N
¡ decay amplitudes for the fermionic ¯nal states.
The blob in the diagram contains a sum of all possible inter-
mediate states. The mixing between the two states ~N
¡ and
~N
+ leads to CP violation. : : : : : : : : : : : : : : : : : : : : : : : : 39
3.2. Diagrams Contributing to Leptogenesis: The lightest ^ºR decay
diagrams via SU(2)R gauge boson exchange that appear in
Left{Right models, corresponding to ^ºR ! ~e+u ¹ d(~e¡¹ud). The
lepton asymmetry can arise through ~ºR1 ¡ ~ºy
R1 mixing and
decay. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 45
vii
Figure Page
3.3. Two Loop Diagram Contributing to Leptogenesis: Feynman
diagram arising from ~ºc ! ec~ucdc decay, mediated by SU(2)R
gaugino (labeled ¸). Our results are based on the computation
of the corresponding decay amplitude. The lepton asymmetry
arises through the mixing of ~ºc ¡ ~ºcy
: : : : : : : : : : : : : : : : : : : 50
3.4. Two Loop Diagram Contributing to Leptogenesis: Feynman
diagram arising from ~ºc ! ec ~ dcuc decay, mediated by SU(2)R
gauge boson, it is simply the supersymmetric correspondent
of the previous Feynman amplitude. The lepton asymmetry
arises through the mixing of ~ºc ¡ ~ºcy
: : : : : : : : : : : : : : : : : : 50
3.5. Dependence of BA on B¡term: Two cases are shown above,
depending on the choice of A¡term and ¡2. In both cases
M1 = 6:9 £ 109GeV, for which the dilution is enhanced (d = 1). : : : 56
4.1. Correlation between µ12 and µ13 taken from Fogli et al. Three
sloped curves correspond to µ12 ¡ µ13 dependance (for three
di®erent values of CP phase ±) obtained from our model ac-
cording to Eq. (4.22) : : : : : : : : : : : : : : : : : : : : : : : : : : : 64
4.2. Curves (i) and (ii) respectively show the dependence of
pm¯¯
¢m2atm
's low and upper bounds on CP violating phase ±.
The shaded region corresponds to values of m¯¯ and ± realized
within our model. : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 66
4.3. Resonant leptogenesis for inverted mass hierarchical neutrino
scenario. In all cases nB
s = 9 £ 10¡11 and tan ¯ ' 2. Curves
(a), (b), (c), (d) correspond respectively to the cases with
M = (104; 106; 109; 1011) GeV and r = ¼=4. The curves with
primed labels correspond to same values of M, but with CP
phase r = 5 ¢ 10¡5. Bold dots stand for a maximized values of
CP asymmetry [see Eq. (4.38)]. The `cut o®' with horizontal
dashed line re°ects the requirement
¯¯
±¤N
+ ±
0
N
¯¯
<
» 0:1. Two
sloped dashed lines restrict low parts of the `ovals' of M =
1011 GeV, insuring the Yukawa coupling perturbativity. : : : : : : : 77
4.4. Diagram generating the ¯rst operator of Eq. (4.44) : : : : : : : : : : : 77
viii
CHAPTER 1
INTRODUCTION
1.1 Matter Versus Anti-Matter
One of the great mysteries of modern particle physics and cosmology is the lack
of antimatter in our surroundings. This is somewhat puzzling given that the rela-
tivistic ¯eld theories which underlie modern particle physics have built into them a
fundamental symmetry; which states that for every particle there is an antiparticle
degenerate in mass and with quantum numbers and charges of the opposite sign. In
the old language of Dirac, for example, the relativistic wave equation for the elec-
tron has both positive and negative energy solutions, which led Dirac to predict the
necessary existence of the positron. In more modern quantum ¯eld theory language,
we think of creation or annihilation operators acting on a ¯eld that respectively cre-
ate particles and destroy antiparticles or destroy particles and create antiparticles
associated with the ¯eld. The properties and dynamics of the particles and antipar-
ticles are fundamentally related. If we consider local, Lorentz-invariant ¯eld theory
equations such as currently used for the "Standard Model" of particle physics, and
we °ip the signs of all charges that appear in them, e®ectively turning particles into
antiparticles, and then perform a space reversal (¡!x
! ¡¡!x
) followed by a time re-
versal (t ! ¡t), we recover the same equations. This symmetry of the equations,
the so-called CPT (Charge-Parity-Time reversal) symmetry, implies, for example,
that particle and antiparticles should have exactly the same mass. Similarly, if the
protons exist, then anti-protons with the same characteristics should exist too (and
in fact, is being made now at CERN and FermiLab). Stretching our imagination
1
2
further, the Universe could then be ¯lled with antimatter stars and galaxies that
are indistinguishable from ordinary stars and galaxies if one studies them solely via
their light emission or their gravitational attraction on neighboring bodies. This of
course assumes that antimatter stars are spatially separated from matter stars, or
else the two will annihilate each other. The fact is that anti-matter on earth is very
rare, in fact, the only anti-protons ever observed were the ones produced at CERN or
Fermilab. Cosmic probes into planets conclude they are made out of matter. With
con¯dence, we can say that our entire solar system consists of matter only. One can
argue that there could be patches or regions at the larger scale containing antimatter,
but experiments 1 showed otherwise. There would be a strong detectable ° radia-
tion originating from nucleons-antinucleons reactions if there was a cluster out there
that contains one or many galaxies with both matter and antimatter. Furthermore,
the well-tested Standard Model also implies that total charge as well as quantum
numbers like baryon number and lepton number should be conserved in particle in-
teractions, excluding the notion that there could be a region elsewhere that does not
contain equal amounts of matter and antimatter. The most fundamental observation
we can make about the observed universe is that it is dominantly made out of mat-
ter (no-antimatter). Baryogenesis, or Baryon Asymmetry (BA), (matter-antimatter
asymmetry), explaining BA is one of the most challenging open questions in particle
physics as well as in cosmology. The subject has been of concern to particle physicists
since the discovery of microscopic CP violation, which encouraged the construction
of concrete Baryogenesis scenarios. The subject became a standard part of modern
cosmology with the introduction of grand uni¯ed theories (GUTs), introduced in the
1970s, which establish a possible source for baryon number violation, an essential
component of Baryogenesis. More recent ideas have attempted to link the baryon
asymmetry with details of models of electroweak symmetry breaking, and o®er the
possibility of testing models of Baryogenesis in future colliders such as the LHC. In
this dissertation however, we concentrate on three of the most recent and popular
3
mechanisms; realized in di®erent ways: Baryogenesis via Leptogenesis2, Soft Lep-
togenesis3;4 and resonant Leptogenesis5;6. The results of our study are reported in
7{9.
In the second chapter, we calculate 7 lepton asymmetry induced in the decay of
right{handed neutrinos in a class of minimal left{right symmetric models 10. In these
models, which assume low energy supersymmetry, the Dirac neutrino mass matrix
has a determined structure. As a result, lepton asymmetry is calculable in terms
of measurable low energy neutrino parameters. By solving the Boltzmann equations
numerically we show that adequate baryon asymmetry is generated in these models
in complete agreement with constrains by Big Bang Nucleosynthesis and the recent
high precision measurement by the NASA satellite mission WMAP experiment 11:
´B ´
nB
n°
= (6:5+0:4
¡0:3) £ 10¡10; (1.1)
where ´B is the baryon to photon ratio. Furthermore, we make predictions on the
light neutrino oscillation parameters, which can be tested in next generation neutrino
experiments.
In the third chapter of this thesis, we discuss a more recent idea, Soft Lep-
togenesis, which is an alternative and attractive mechanism to explain the baryon
asymmetry we are after. This time, we study the e®ect of the interactions of the
SU(2)R gauge boson WR on the generation of the primordial lepton asymmetry8.
B ¡ L violation occurs when Left{Right symmetry is broken by the vacuum expec-
tation value (VEV) vR of the B ¡ L = ¡2 triplet scalar ¯eld, which gives Majorana
masses to the r.h sneutrino, and lepton number is violated in their decay ~ºR1 ! ~eRu ¹ d
as well as ~ºR1 ! ~e¤
R¹ud, these decays are mediated by the right handed gauge boson
WR, and can dominate the traditional ºR ! LÁy, frequently used decay to explain
BA. Furthermore, by Renormalization Group Equations (RGE) analysis, we show
that the requirement of unconventionally small B¡term is no longer needed. In ad-
dition, we use RGE running and SUSY breaking e®ect to naturally account for the
complex O(1) phase as dictated by the success of the scenario. The mass of r.h
sneutrino can be M~º » MWR » (109 ¡ 1010) GeV .
4
In chapter 4 we present a new realization of inverted neutrino mass hierarchy
based on S3 £ U(1) °avor symmetry9. In this scenario, the deviation of the solar
oscillation angle from ¼=4 is correlated with the value of µ13, as they are both induced
by a common mixing angle in the charged lepton sector. We ¯nd several interesting
predictions: µ13 ¸ 0:13, sin2 µ12 ¸ 0:31, sin2 µ23 ' 0:5, 0 · cos ± · 0:7 for the neutrino
oscillation parameters and 0:01 eV <
» m¯¯
<
» 0:02 eV for the e®ective neutrino mass in
neutrinoless double ¯-decay. We show that the same scenario can naturally explain
the observed baryon asymmetry of the universe via resonant leptogenesis. The masses
of the decaying right{handed neutrinos can be in the range (103 ¡ 107) GeV, which
would avoid the generic gravitino problem of supersymmetric models.
In the appendix section, we brie°y review the basic thermodynamics of the ex-
panding universe, set up Boltzmann equations, review the formalism of CP violation
in the kaon system, and make some comments about the numerical methods.
5
1.2 Sakharov criteria
The Standard Model of Cosmology provides a very satisfactory picture that
accounts for variety of observational data, in particular, the observed 2:7oK back-
ground black-body radiation is in total agreement with the nucleosynthesis calcu-
lation of the primordial helium abundance. On the downside, the Standard Model
with only baryon-number conserving interactions does not ¯x baryon-number asym-
metry ratio as indicated earlier. It is desirable that, independent of any initial
conditions, such an asymmetry could be generated by underlying physical interac-
tions. To achieve this, we must postulate new particles interactions, beyond those of
SU(3)C SU(2)L U(1)Y Standard Model.
In 1967, Sakharov12 proposed a radical alternative: our physics is wrong! More
precisely, there is new physics beyond the Standard Model which, at higher energies
than can currently be tested with accelerators, allows for baryon number violation.
Assuming a highly symmetric state in the early Universe, a matter-antimatter asym-
metry can be dynamically generated in an expanding Universe if the particle interac-
tions and the cosmological evolution satisfy the so called Sakharov conditions, which
we enumerate below
(i) Underlying theory must have processes that violate B number
¢B 6= 0
where B is the baryon number. If the baryon number B was conserved by the interac-
tions, it would mean that the baryon number commutes with the Hamiltonian of the
system H: [B;H] = 0. Hence, if B(t0) = 0, we would have B(t) /
R t
t0
[B;H] dt0 = 0
at any subsequent time and no baryon number production would take place.
(ii) Both Charge Conjugation, and CP symmetry must be violated; otherwise,
one can never establish baryon-antibaryon asymmetry (since the action of C and CP
would transforms nB ! n¹B
). To see this, we de¯ne the following baryon number
operator,
^B
=
1
3
X
q
Z
d3x : qy(x; t)q(x; t) : ;
6
which is C-odd and CP-odd. This is evident from the action of P, C and T on the
quark ¯elds:
Pq(x; t)P¡1 = °0q(¡x; t); Pqy(x; t)P¡1 = qy(¡x; t)°0;
Cq(x; t)C¡1 = {°2qy(x; t); Cqy(x; t)C¡1 = {qy(x; t)°2;
Tq(x; t)T¡1 = ¡{q(x;¡t)°5°0°2; Tqy(x; t)T¡1 = ¡{°2°0°5qy(x;¡t)°0: (1.2)
Then
P : qy(x; t)q(x; t) : P¡1 =: qy(¡x; t)q(¡x; t) :;
C : qy(x; t)q(x; t) : C¡1 = ¡ : qy(x; t)q(x; t) :;
T : qy(x; t)q(x; t) : T¡1 =: qy(x;¡t)q(x;¡t) :; (1.3)
so that
P ^B
P¡1 = ^B
; C ^B
C¡1 = ¡^B
; (CP)^B
(CP)¡1 = ¡^B
:
A non-zero expectation value < B ^B
> requires that the Hamiltonian violates C and
CP. More intuitively, C symmetry would guarantee that ¡(i ! f) = ¡(iy ! fy),
while CP symmetry would guarantee that ¡(i ! f) = ¡(¹i
! ¹ f)¤. With CP alone it
might be possible to create baryon asymmetry in certain localized region of the phase
space, but integrating over all momenta and summing over all spins would leave a
vanishing asymmetry.
(iii) Departure from thermal equilibrium of X-particles mediating ¢B 6= 0
processes is necessary. This is because if all processes, including those which violate
baryon number, are in thermal equilibrium, the baryon asymmetry vanishes. This is
a direct consequence of the CPT invariance. To see this, de¯ne CPT ´ µ, and the
density matrix at time t for a system in thermal equilibrium as ½ (t) = e¡¯(t)H(t), then
from Eq (1.3) we obtain the equilibrium average of B,
hBiT = Tr
³
e¡¯H ^B
´
= Tr
³
µ¡1µe¡¯H ^B
´
¤xy has opposite charge but same chirality as x. ¹x has both opposite charge and
chirality.
7
= Tr
³
µe¡¯H ^B
µ¡1
´
= Tr
³
µe¡¯Hµ¡1µ ^B
µ¡1
´
= Tr
³
e¡¯H
³
¡^B
´´
= ¡Tr
³
e¡¯H ^B
´
= ¡hBiT (1.4)
where ¯ = 1
kBT , and we have used the fact that H commutes with the operator
CPT that we called µ above. Thus hBiT = 0. Whence, to establish asymmetry
dynamically, B violating processes must be out of equilibrium in the Universe. This
can be seen as follows:
d¢nB
dt
= ¡
·
°6Be
¡
m¡¹
kBT
¡ °6Be
¡
¹m¡¹
kBT
¸
(1.5)
where °6B denotes the rate for 6B and ¹ is the chemical potential, and ¹ = ¡¹. Since
m = ¹m by CPT theorem, e
¡ m
kBT is not relevant and we omit it. Then for kBT À ¹,
d¢nB
dt
=
¡2¹
kBT
°6B: (1.6)
On the other hand
¢nB =
2³ (3)
¼2 g0 (kBT)3
h
e
¹
kBT ¡ e
¡¹
kBT
i
'
2
¼2 g0 (kBT)3 2¹
kBT
: (1.7)
Thus eliminating 2¹
kBT
d¢nB
dt
= ¡
¼2
2
°6B
g0 (kBT)3¢nB
= ¡
¼2
2
¡6B¢nB (1.8)
where ¡6B = °6B
g0(kBT)3 = °6B
nB
gives the rate for 6B. The solution of above equation gives
¢nB = (¢nB)initial e¡¼2
2 ¡6Bt: (1.9)
What we learn from this result is that if B-violating processes are ever in equilibrium,
then these processes actually washes out any initial condition for ¡6Bt ¸ 1:
8
1.3 Boltzmann Equations
The processes of interest are active at high temperature while the universe is
expanding, when the system is far from thermodynamic equilibrium, and one needs
to follow evolution of a density while the particle species produces and collides with
many di®erent species. Boltzmann equations (BE) allow us to follow the e®ect of
di®erent interactions, in fact, all important calculations in Cosmology are done by
means of BE. In this section, we introduce the basic elements for setting up BE.
It is usually a good approximation to assume Maxwell-Boltzmann statistics, so
that the equilibrium number density of a particle i is given by
neq
i (T) =
gi
(2¼)3
Z
d3pi feq
i with feq
i (Ei; T) = e¡Ei=T : (1.10)
For a massive non relativistic particle one ¯nds
neq
i (T) =
giTm2i
2¼2 K2
³mi
T
´
; (1.11)
where K2 is bessel function of the second type. For a massless particle one gets
neq
i (T) =
giT3
¼2 : (1.12)
The universe expansion and di®erent interations modify the particle densities.
Since we are only interested in the e®ect of interactions, it is useful to scale out
the expansion. This is done by taking the number of particles per comoving volume
element, i.e. the ratio of the particle density ni to the entropy density s,
Yi =
ni
s
; (1.13)
as independent variable instead of the number density. In a radiation dominated
universe the entropy density reads
s = g¤
2¼2
45
T3 : (1.14)
In our case, elastic scatterings, which can only change the phase space distri-
butions but not the particle densities, occur at a much higher rate than inelastic
9
processes. Therefore, we can assume kinetic equilibrium, so that the phase space
densities are given by
fi(Ei; T) =
ni
neq
i
e¡Ei=T : (1.15)
In this framework the Boltzmann equation describing the evolution of a particle num-
ber Y´ in an isentropically expanding universe reads
dY´
dz
= ¡
z
sH (m´)
X
a;i;j;:::
·
Y´Ya : : :
Y eq
´ Y eq
a : : :
°eq (´ + a + : : : ! i + j + : : :)
¡
YiYj : : :
Y eq
i Y eq
j : : :
°eq (i + j + : : : ! ´ + a + : : :)
#
; (1.16)
where z = m´=T and H (m´) is the Hubble parameter at T = m´. The °eq are space
time densities of scatterings for the di®erent processes. For a decay one ¯nds
°D := °eq(´ ! i + j + : : :) = neq
´
K1(z)
K2(z)
¡ ; (1.17)
where K1 and K2 are modi¯ed Bessel functions and ¡ is the tree level decay width
in the rest system of the decaying particle. Neglecting a possible CP violation, one
¯nds the same reaction density for the inverse decay.
Calculation of Lepton Asymmetry will involve 2 body scattering. The reaction
density for a two body scattering is given by,
°eq(´ + a $ i + j + : : :) =
T
64¼4
Z1
(m´+ma)2
ds ^¾(s)
p
sK1
µp
s
T
¶
; (1.18)
where s is the squared center of mass energy and the reduced cross section ^¾(s) for
the process ´ + a ! i + j + : : : is related to the usual total cross section ¾(s) by
^¾(s) =
2¸(s;m2´
;m2
a )
s
¾(s); (1.19)
where ¸ is the usual kinematical function
¸(s;m2´
;m2
a ) ´
£
s ¡ (m´ + ma)2¤ £
s ¡ (m´ ¡ ma)2¤
: (1.20)
In order to compute the Baryon Asymmetry we will have to employ numerical
solution to the coupled Boltzmann Equation for the Lepton Asymmetry density and
the abundance of right handed neutrinos. We will shortly come back to this analysis
and discuss it in detail.
10
1.4 Chemical potential, asymmetries relations and Sphalerons
In the standard model, baryon number violating processes convert three baryons
to three antileptons. This violates conservation of baryon number and lepton number,
but the di®erence B ¡ L is conserved. This is because B ¡ L has no anomalies in
the Standard Model, while B (or L) has electroweak anomalies. A sphaleron is a
static (time independent) solution to the electroweak ¯eld equations of the Standard
Model, and it is involved in processes that violate baryon and lepton number. Such
processes cannot be represented by Feynman diagrams, and are therefore called non-
perturbative. This means that under normal conditions sphalerons are unobservably
rare. However, they would have been more common at the higher temperatures of
the early universe. In almost all theories of baryogenesis an imbalance of the number
of leptons and antileptons is formed ¯rst, and sphaleron transitions then recycle this
to an imbalance in the numbers of baryons and antibaryons. Below, we derive some
of the relations between various asymmetry densities, establishing the connection
between lepton asymmetry and baryon asymmetry.
As we will see, Sphaleron transitions lead to the baryon asymmetry by recycling
a lepton asymmetry. Further B + L asymmetry generated before EW transition i.e.
at T > TEW; will be washed out. However, since only left handed ¯elds couple to
sphalerons, a non zero value of B +L can persist in the high temperature symmetric
phase if there exist a non vanishing B ¡L asymmetry [see below]. In weakly coupled
plasma, one can assign a chemical potential ¹i to each of the quark, lepton and Higgs
¯eld.
ni ¡ ¹ni =
2
¼2 g0T3
µ
2¹i
T
¶
;
where g0 is the particle species e®ective degree of freedom, T is the temperature at
any given time. This also implies
nB = B
µ
4
¼2 g0T2
¶
nL = L
µ
4
¼2 g0T2
¶
(1.21)
11
where B and L are baryon and lepton asymmetries respectively. Note that in SM
qLi =
0
@ uLi
dLi
1
A B =
1
3
; L = 0
uRi; dRi
`Li =
0
@ ºLi
eLi
1
A B = 0; L = 1
ºRi; eRi
Thus in Eq. (1.21)
B = 3 £
1
3
X
i
(2¹qi + 2¹ui + 2¹di)
L =
X
i
(2¹li + 2¹ei) (1.22)
In high temperature plasma, quarks, leptons and Higgs interact via Yukawa and gauge
couplings and in addition, via the non perturbative sphaleron processes. In thermal
equilibrium all these processes yield constraints between various chemical potentials.
The e®ective interaction
OB+L = ¦i (qLiqLiqLi`Li)
yields
X
i
(3¹qi + ¹li) = 0: (1.23)
Another constraint is provided by vanishing of total charge of plasma
X
i
2
4 31
32¹qi + 34
3¹ui
+3
¡
¡2
3
¢
¹di + (¡1) 2¹li + (¡2) ¹ei + 1
N (1) ¹Á
3
5 = 0
where we have used
Yq =
1
3
; Yu =
4
3
; Yd = ¡
2
3
; Yl = ¡1; Ye¡ = ¡2; YÁ = 1
The above equation can be written as
X
i
µ
¹qi + 2¹ui ¡ ¹di ¡ ¹li ¡ ¹ei +
2
N
¹Á
¶
= 0: (1.24)
12
Furthermore, invariance of Yukawa couplings ¹qLiÁdRi, etc gives
¹qi ¡ ¹Á ¡ ¹dj = 0
¹qi ¡ ¹Á ¡ ¹uj = 0
¹li ¡ ¹Á ¡ ¹ej = 0 (1.25)
When all Yukawa interactions are in equilibrium, these interactions establish equilib-
rium in di®erent generations
¹li = ¹l; ¹qi = ¹q etc.
Thus we obtain from Eqs (1.23) and (1.24)
¹q = ¡
1
3
¹l
¹q + 2¹u ¡ ¹d ¡ ¹l ¡ ¹e +
2
N
¹Á = 0
giving
¡
4
3
¹l + 2¹u ¡ ¹d ¡ ¹e +
2
N
¹Á = 0: (1.26)
Furthermore, Eqs. (1.25) implies
¡
1
3
¹l ¡ ¹Á ¡ ¹d = 0
¡
1
3
¹l ¡ ¹Á ¡ ¹u = 0
¹l ¡ ¹Á ¡ ¹e = 0 (1.27)
Using the above equations, we can write (1.26) as
¡
4
3
¹l + 2
µ
¡
1
3
¹l + ¹Á
¶
¡
µ
¡
1
3
¹l ¡ ¹Á
¶
¡ (¡¹l ¡ ¹Á) +
2
N
¹Á = 0:
Thus ¯nally we can express ¹q, ¹u, ¹d, ¹e, and ¹Á in terims of ¹l:
¹Á =
8
3
N
1
4N + 2
¹l =
4N
6N + 3
¹l
¹d = ¡
1
3
¹l ¡ ¹Á
= ¡
1
3
¹l ¡
4N
6N + 3
¹l
13
= ¡
6N + 1
6N + 3
¹l
¹u = ¡
1
3
¹l + ¹Á
= ¡
1
3
¹l +
4N
6N + 3
¹l
=
2N ¡ 1
6N + 3
¹l
¹e = ¹l ¡ ¹Á
= ¹l ¡
4N
6N + 3
¹l
=
2N + 3
6N + 3
¹l (1.28)
Hence from Eqs. (1.22)
B = N
½
¡
2
3
¹l +
2N ¡ 1
6N + 3
¹l ¡
6N + 1
6N + 3
¹l
¾
= [¡4N ¡ 2 + 2N ¡ 1 ¡ 6N ¡ 1]
¹l
6N + 3
= ¡N
(8N + 4)
3 (2N + 1)
¹l
= ¡
4N
3
¹l (1.29)
L = N
µ
2¹l +
2N + 3
6N + 3
¹l
¶
=
14N2 + 9N
6N + 3
¹l (1.30)
B ¡ L = ¡
8N2 + 4N + 14N2 + 9N
6N + 3
¹l
= ¡
22N2 + 13N
6N + 3
¹l (1.31)
B
B ¡ L
=
8N2 + 4N
22N2 + 13N
=
8N + 4
22N + 13
=
8Ng + 4nH
22Ng + 13nH
´ a (1.32)
These relations hold for T À v. In general B=(B ¡ L) is a function of v=T . For SM,
Ng = 3, nH = 1 so that a = 28=79.
Thus ¯nally we obtain
YB( ´
nB ¡ n¹B
s
)
14
= aYB¡L =
a
a ¡ 1
YL (1.33)
From the relation between entropy density and photon number density, s ' ´°=7, we
¯nd
YB = ´
³´°
s
´
'
1
7
´
'
1
7
(6 § 3) £ 10¡10:
It is this number we try to explain via underlying physical process and in the
context or realistic physical model. As mentioned earlier, there are several Baryon
Asymmetry mechanisms that undertake the task of explaining this number, we con-
centrate on the 3 most popular Leptogenesis ideas. In speci¯c frameworks, we analyze
the mechanisms in details and derive interesting correlation with Leptogenesis and
the physics of neutrinos.
CHAPTER 2
LEPTOGENESIS IN MINIMAL LEFT-RIGHT
SYMMETRIC MODELS
2.1 Introduction
The discovery of neutrino °avor oscillations in solar, atmospheric, and reactor
neutrino experiments 13 may have a profound impact on our understanding of the
dynamics of the early universe. This is because such oscillations are feasible only if
the neutrinos have small (sub{eV) masses, most naturally explained by the seesaw
mechanism 14. This assumes the existence of super-heavy right{handed neutrinos Ni
(one per lepton family) with masses of order (108 ¡ 1014) GeV . The light neutrino
masses are obtained from the matrix Mº ' MDM¡1
R MD
T where MD and MR are
respectively the Dirac and the heavy Majorana right-handed neutrino (r.h.n) mass
matrices. The decay of the lightest right{handed neutrino N1 can generate naturally
an excess of baryons over anti-baryons in the universe 2 consistent with cosmological
observations. The baryon asymmetry parameter is an important cosmological ob-
servable constrained by Big Bang Nucleosynthesis and determined recently with high
precision by the WMAP experiment 11:
´B ´
nB
n°
= (6:5+0:4
¡0:3) £ 10¡10: (2.1)
The decay of N1 can satisfy all three of the Sakharov conditions 12 needed for suc-
cessful generation of ´B { it can occur out of thermal equilibrium, there is su±cient
C and CP violation, and there is also baryon number violation. The last condition
is met by combining lepton number violation in the Majorana masses of the right{
handed neutrinos with B + L violating interactions of the Standard Model arising
through the electroweak sphaleron processes 15. A compelling picture emerges, with
15
16
the same mechanism explaining the small neutrino masses and the observed baryon
asymmetry of the universe. ´B appears to be intimately connected to the observed
neutrino masses and mixings.
A more careful examination of the seesaw structure would reveal that, although
there is an underlying connection, the light neutrino mass and mixing parameters
cannot determine the cosmological baryon asymmetry, when the seesaw mechanism
is implemented in the context of the Standard Model (SM) gauge symmetry. It is
easy to see this as follows. Without loss of generality one can work in a basis where
the charged lepton mass matrix and the heavy right{handed neutrino Majorana mass
matrix MR are diagonal with real eigenvalues. The Dirac neutrino mass matrix would
then be an arbitrary complex 3 £ 3 matrix with 18 parameters (9 magnitudes and
9 phases). Three of the phase parameters can be removed by ¯eld rede¯nitions of
the left{handed lepton doublets and the right{handed charged lepton singlets. The
neutrino sector will then have 18 (= 15+3) parameters. 9 combinations of these will
determine the low energy observables (3 masses, 3 mixing angles and 3 phases), while
the lepton asymmetry (and thus ´B) would depend on all 18 parameters, leaving it
arbitrary.
In this section of the thesis we show that it is possible to quantitatively relate ´B
to light neutrino mass and mixing parameters by implementing the seesaw mechanism
in the context of a class of supersymmetric left{right models 10. We note that unlike
in the SM where the right{handed neutrinos appear as rather ad hoc additions, in the
left{right symmetric models they are more natural as gauge invariance requires their
existence. Supersymmetry has the well{known merit of solving the gauge hierarchy
problem. With the assumption of a minimal Higgs sector, it turns out that these
models predict the relation for the Dirac neutrino mass matrix, in a basis where the
charged lepton mass matrix is diagonal;
MD = c
0
BB@
me 0 0
0 m¹ 0
0 0 m¿
1
CCA
; (2.2)
17
where c ' mt=mb is determined from the quark sector, leaving only the Majorana
mass matrix MR to be arbitrary. 3 phases in MR can be removed, leaving a total
of 9 parameters which determine both the low energy neutrino masses and mixings
as well as the baryon asymmetry. It then becomes apparent that ´B is calculable in
terms of the neutrino observables. There have been other attempts in the literature
to relate leptogenesis with low energy observables 16;17. Such attempts often make
additional assumptions such as MD = Mup (which may not be fully realistic), or
speci¯c textures for lepton mass matrices.
While a lot has been learned from experiments about the light neutrino masses
and mixings, a lot remains to be learned. Our analysis shows that cosmology puts sig-
ni¯cant restrictions on the light neutrino parameters. Successful baryogenesis requires
within our model that three conditions be satis¯ed: tan2 µ12 ' m1=m2, ¯ ' ® + ¼=2
and µ13 = (0:01¡0:07). Here µ12 and µ13 are elements of the neutrino mixing matrix,
mi are the light neutrino mass eigenvalues and ®; ¯ are the Majorana phases entering
in the amplitude for neutrinoless double beta decay. Future neutrino experiments will
be able to either con¯rm or refute these predictions.
The rest of the chapter is organized as follows. In Sec. 2.2 we review brie°y
the minimal left{right symmetric model. In Sec. 2.3 we analyze leptogenesis in this
model. Here we derive constraints imposed on the model from the requirement of
successful leptogenesis. In Sec. 2.4 we calculate the lepton asymmetry parameter "1
generated in the model in N1 decay. Sec. 2.5 summarizes the relevant Boltzmann
equations needed for computing the baryon asymmetry parameter. Sec. 2.6 provides
our numerical results for ´B. We devote Sec. 2.7 for Gravition discussion Finally, in
Sec. 2.7 we conclude.
2.2 Brief review of the minimal left-right symmetric model
Let us brie°y review the basic structure of the minimal SUSY left{right sym-
metric model developed in Ref. 10. The gauge group of the model is SU(3)C £
SU(2)L £ SU(2)R £ U(1)B¡L. The quarks and leptons are assigned to the gauge
group as follows. Left{handed quarks and leptons (Q;L) transform as doublets
18
of SU(2)L [Q(3; 2; 1; 1=3) and L(1; 2; 1;¡1)], while the right{handed ones (Qc;Lc)
are doublets of SU(2)R [Qc(3¤; 1; 2;¡1=3) and Lc(1; 1; 2; 1)]. The Dirac masses of
fermions arise through their Yukawa couplings to a Higgs bidoublet ©(1; 2; 2; 0). The
SU(2)R £ U(1)B¡L symmetry is broken to U(1)Y by the VEV (vR) of a B ¡ L = ¡2
triplet scalar ¯eld ¢c(1; 1; 3;¡2). This triplet is accompanied by a left{handed triplet
¢(1; 3; 1; 2) (along with ¹¢
and ¢¹c ¯elds, their conjugates to cancel anomalies). These
¯elds also couple to the leptons and are responsible for inducing large Majorana
masses for the ºR. An alternative to these triplet Higgs ¯elds is to use B ¡ L = §1
doublets Â(1; 2; 1;¡1) and Âc(1; 1; 2; 1), along with their conjugates ¹Â and ¹ Âc. In
this case non{renormalizable operators will have to be invoked to generate large ºR
Majorana masses. For de¯niteness we shall adopt the triplet option, although our
formalism allows for the addition of any number of doublet Higgs ¯elds as well. The
superpotential invariant under the gauge symmetry involving the quark and lepton
¯elds is
W = YqQT ¿2©¿2Qc + YlLT ¿2©¿2Lc + (fLT i¿2¢L + fcLcT i¿2¢cLc) : (2.3)
Under left{right parity symmetry, Q $ Qc¤;L $ Lc¤;© $ ©y, ¢ $ ¢c¤, along
with WSU(2)L $ W¤
SU(2)R
, WB¡L $ W¤B
¡L and µ $ ¹µ. As a consequence, Yq = Yy
q,
Yl = Yy
l , and f = f¤
c in Eq. (3.7).¤ It has been shown in Ref. 10 that the hermiticity
of the Yukawa matrices (along with the parity constraints on the soft SUSY breaking
parameters) helps to solve the supersymmetric CP problem that haunts the MSSM.
Below vR, the e®ective theory is the MSSM with its Hu and Hd Higgs multi-
plets.y These are contained in the bidoublet © of the SUSY left-right model, but in
general they can also reside partially in other multiplets having identical quantum
numbers under the MSSM symmetry (such as the Â; Â doublet Higgs ¯elds alluded
to earlier). Allowing for such a possibility, the superpotential of Eq. (3.7) leads to
the relations for the MSSM Yukawa coupling matrices
Yu = °Yd; Y` = °YºD : (2.4)
¤We do not explicitly use these relations.
yThe right-handed gauge bosons have masses of order vR » 1014 GeV and thus
play no signi¯cant role in cosmology at T » M1 ¿ vR.
19
These relations have been called up{down uni¯cation 10. Here, the ¯rst relation of
Eq. (2.4) implies mt
mb
' ° tan ¯ ´ c where ° is a parameter characterizing how much
of Hu and Hd of MSSM are in the bidoublet ©. The case of Hu;d entirely in © will
correspond to ° = 1 and tan ¯ = mt=mb. At ¯rst sight the ¯rst of the relations in Eq.
(2.4) might appear phenomenologically disastrous since it leads to vanishing quark
mixings and unacceptable quark mass ratios. It was shown in the ¯rst paper of Ref.
10 that including the one{loop diagrams involving the gluino and the chargino and
allowing for a °avor structure for the soft SUSY breaking A terms, there exists a
large range of parameters (though not the entire range possible in the usual MSSM)
where correct quark mixings as well as masses can be obtained consistent with °avor
changing constraints.
It is the second of Eq. (2.4) that concerns us here. This relation would lead to
MD = cMl, with c ' mt=mb. The supersymmetric loop corrections for the leptonic
mass matrices are numerically small compared to similar corrections in the quark
sector, since no strongly interacting particles take part in these loops. Furthermore,
leptonic mixing angles are induced at tree level through the structure in the Majorana
neutrinos mass matrix, and any loop corrections to these will be subdominant. This
is especially true since two of the leptonic mixing angles are large to begin with. We
therefore ignore SUSY loop corrections to the lepton mass matrices.
One can thus go to a basis where the charged lepton and the Dirac neutrino mass
matrices are simultaneously diagonal. The heavy Majorana mass matrix MR = fvR
will then be a generic complex symmetric matrix. After removing three phases in
MR by ¯eld rede¯nitions, we are left with 9 parameters (6 magnitudes and 3 phases)
which determine the light neutrino spectrum as well as the heavy neutrino spectrum.
This in turns ¯xes the lepton asymmetry. The consequences of such a constrained
system for leptogenesis will be analyzed in the next section.
In principle the ¢(1; 3; 1; +2) Higgs ¯eld can also acquire a small VEV of order
eV 18. In this case the seesaw formula would be modi¯ed, as will the calculation of the
lepton asymmetry 18. We will assume such type II seesaw contributions proportional
to h¢i are zero in our analysis. This is consistent with the models of Ref. 10.
20
Leptogenesis in the context of more general left-right symmetric models has been
analyzed in Ref. 19.
2.3 Leptogenesis in left-right symmetric framework
The SU(2)R £ U(1)B¡L symmetry is broken down to U(1)Y by the VEV
h¢ci = vR » 1014 GeV . At least some of the right-handed neutrinos have masses
below vR. We thus focus on the neutrino Yukawa coupling in the context of MSSM.
The SU(2)L £ U(1)Y invariant Yukawa interactions are contained in the MSSM su-
perpotential
W = lHdY` ec + lHuYºD ºc +
1
2
ºcTCMRºc
| {z }; (2.5)
where l stands for the left-handed lepton doublet, and (ec; ºc) denote the conjugates
of the right-handed charged lepton and the right{handed neutrino ¯elds respectively.
Hu, Hd are the MSSM Higgs ¯elds with VEVs vu, vd. Ml = Y` vd, MD = YºD vu and
MR are respectively the charged lepton, the Dirac neutrino, and the Majorana r.h.n
mass matrices. Then one can generate light neutrino masses by the seesaw mechanism
14
Mº = ¡MDM¡1
R MD
T : (2.6)
There is mixing among generations in both MR and MD, the light neutrino mixing
angles will depend on both of these mixings. Within the SM or MSSM where MD is
an arbitrary matrix, the structure of the right-handed neutrino mass matrix can not
be fully determined even if the light matrix Mº were to be completely known from
experiments. As noted in Sec. 2, in the minimal version of the left-right symmetric
model one has
MD = cMl = c diag(me;m¹;m¿ ) (2.7)
where c ' mt
mb
. Here we have already gone to a basis where the charged lepton mass
matrix is diagonalized. In the three family scenario, the relations between the °avor
eigenstates (ºe; º¹; º¿ ) and the mass eigenstates (º1; º2; º3) can be expressed in terms
21
of observables as
Mº = U¤Mdiag
º Uy; (2.8)
where Mdiag
º ´ diag(m1;m2;m3), with mi being the light neutrinos masses and U
being the 3 £ 3 mixing matrix which we write as U = UPMNS:P . We parameterize
UPMNS
20 as
UPMNS =
0
BB@
Ue1 Ue2 Ue3
U¹1 U¹2 U¹3
U¿1 U¿2 U¿3
1
CCA
=
0
BB@
c12c13 s12c13 s13e¡{±
¡s12c23 ¡ c12s13s23e{± c12c23 ¡ s12s13s23e{± c13s23
s12s23 ¡ c12s13c23e{± ¡c12s23 ¡ s12s13c23e{± c13c23
1
CCA
(2.9)
where cij ´ cos µij , sij ´ sin µij and ± is the Dirac CP violating phase which appears
in neutrino oscillations. The matrix P contains two Majorana phases unobservable
in neutrino oscillation, but relevant to neutrinoless double beta decay 21:
P =
0
BB@
e{® 0 0
0 e{¯ 0
0 0 1
1
CCA
: (2.10)
Combining Eq. (3.21) with the seesaw formula of Eq. (3.18) and solving for the
right-handed neutrino mass matrix we ¯nd
MR = c2MlM¡1
º Ml
=
c2m2¿
m1
0
BB@
me
m¿
0 0
0 m¹
m¿
0
0 0 1
1
CCA
UPMNSP2
0
BB@
1 0 0
0 m1
m2
0
0 0 m1
m3
1
CCA
UT
PMNS
0
BB@
me
m¿
0 0
0 m¹
m¿
0
0 0 1
1
CCA
(2.11:)
This enables us to establish a link between high scale parameters and low scale ob-
servables.
We de¯ne a small expansion parameter ² as
² =
m¹
m¿
' 0:059;
in terms of which we have
me = ae²3m¿ ;
m1
m3
= a13²; µ13 = t13²; µ23 =
¼
4
+ t23²: (2.12)
22
Here ae, a13, t13 and t23 are . µ(1) parameters with ae = 1:400. These expansions
follow from low energy data assuming the picture of hierarchical neutrino masses.
We ¯nd that the requirement of generating adequate baryon asymmetry places
signi¯cant constraints on the neutrino mixing parameters. Speci¯cally, the following
expansions
m1
m2
= tan2 µ12 + a12² and ¯ = ® +
¼
2
+ b²; (2.13)
where a12 and b are . µ(1) parameters are required. To see this, we note that the
CP asymmetry parameter ²1 generated in the decay of N1 is too small, of order
"1 » ²6
8¼ » 2 £ 10¡9 if a12 or b are much greater than 1. This is because the heavy
neutrino masses would be strongly hierarchical in this case, M1 : M2 : M3 » ²6 :
²2 : 1. This can be altered to a weak hierarchy M1 : M2 : M3 » ²4 : ²2 : 1 by
observing that the elements of the 2-3 block of MR of Eq. (2.11) are all proportional
to fm1
m2
e2i¯ cos2 µ12 + e2i® sin2 µ12g and by demanding this quantity to be of order ².
Eq. (2.13) is just this condition. "1 » ²4
8¼ » 10¡6 in this case, which can lead to
acceptable baryon asymmetry, as we show.
An immediate consequence of Eq. (2.13) is that neutrinoless double beta decay
is suppressed in the model. The e®ective mass relevant for this decay is found to be
m¯¯ = j
X
i
U2
eimij ' jm2e2i®²(a12c2
12 ¡ 2ibs2
12) + m3s2
13e¡2i±j: (2.14)
This is of the order m3²2 » 10¡4 eV, which would be di±cult to measure. This ampli-
tude is small because of a cancelation between the leading contributions proportional
to m1 and m2 (see Eq. (2.13)).
In terms of these expansions, the r.h.n mass matrix becomes
MR =
0
BB@
A11²5 A12²3 A13²2
A12²3 A22²2 A23²
A13²2 A23² A33
1
CCA
; (2.15)
where
A11 =
M±² a2e
e2{® cos 2µ12
cos2 µ12
23
A12 = ¡
M±² aee2{® tan µ12 p
2
A13 =
M±² aee2{® tan µ12 p
2
A22 =
M±²
2
©
a13 ¡ a12e2{® cos2 µ12 ¡ 2{be2{® sin2 µ12 + 2e{(2®+±)t13 tan µ12
ª
A23 =
M±²
2
©
a13 + a12e2{® cos2 µ12 + 2{be2{® sin2 µ12
ª
A33 = ¡M±² e2{®
½
t13e{± tan µ12 + {b sin2 µ12 ¡
a13e¡2{®
2
+
a12 cos2 µ12
2
¾
: (2.16)
Here we de¯ned M± = c2m2¿
m1
. This hierarchical mass matrix is diagonalized by a series
of rotations U1, U2 and U3 such that;
(KU3U2U1)MR(KU3U2U1)T =
0
BB@
jM1j 0 0
0 jM2j 0
0 0 jM3j
1
CCA
(2.17)
where K = diag(k1; k2; k3) with ki = e¡{Ái=2 being phase factors which make each
r.h.n masses Mi real, Mi = jMijeÁi . V = (KU3U2U1)T is the matrix that diagonalizes
MR. The unitary matrix U1 is given by
U1 =
0
BB@
1 0 ¡A13
A33
²2
0 1 0
A?
13
A?
33
²2 0 1
1
CCA: (2.18)
Similarly, U2 and U3 are unitary matrices with o®-diagonal entries given by
(U2)23 = ¡
A23
A33
² ; (U3)12 = ¡
³
A12 ¡ A13A23
A33
´
²
A22 ¡ A2
23
A33
: (2.19)
The mass eigenvalues are found to be
M1 = M±k2
1²5 ¡
2a13a2ee2{® sin2 µ12
¢
£
¡
2t2
13e2{(®+±) sin2 µ12 + (a12 + 2{b + (a12 ¡ 2{b) cos 2µ12)a13 cos2 µ12
¢¡1
M2 = M±k2
2²3e2{® ¡
a13(a12 + 2{b + (a12 ¡ 2{b) cos 2µ12) + 2t2
13e{(±+®) tan2 µ12
¢
£
¡
¡a13 + {be2{® + e2{®(a12 cos2 µ12 ¡ {b cos 2µ12) + 2e{±t13 tan µ12
¢¡1
M3 =
M±k2
3²
2
¡
a13 ¡ {be2{® ¡ e2{®(a12 cos2 µ12 ¡ {b cos 2µ12 + 2t13e{± tan µ12)
¢
(2:.20)
We use these results in the next section to determine "1.
24
2.4 CP violation and lepton asymmetry
Now that we have developed our framework, we can turn attention to the eval-
uation of the CP asymmetry "1 generated in the decay of the lightest r.h.n N1. This
arises from the interference between the tree-level and one-loop level decay ampli-
tudes.¤ In a basis where the r.h.n mass matrix is diagonal and real, the asymmetry
in the decay of Ni is given by 22
"i = ¡
1
8¼À2 (My
DMD)ii
X
j=2;3
Im[(My
DMD)ij ]2
·
f
µ
M2
j
M2
i
¶
+ g
µ
M2
j
M2
i
¶¸
(2.21)
where f(x) and g(x) represent the contributions from vertex and self energy cor-
rections respectively. For the case of the non-supersymmetric standard model with
right-handed neutrinos, these functions are given by 22
fnon¡SUSY (x) =
p
x
·
¡1 + (x + 1) ln
µ
1 +
1
x
¶¸
; gnon¡SUSY (x) =
p
x
x ¡ 1
;(2.22)
while for the case of MSSM plus right-handed neutrinos, they are given by
fSUSY (x) =
p
x ln
µ
1 +
1
x
¶
; gSUSY (x) =
2
p
x
x ¡ 1
: (2.23)
Here À is the SM Higgs doublet VEV, À ' 174 GeV. For the case of MSSM, À in Eq.
(2.21) is replaced by À sin ¯. Hereafter, for de¯niteness in the numerical evaluation
of the Boltzmann equations, we assume the SM scenario. However, our result should
be approximately valid for the MSSM case as well.¤ Assuming a mass hierarchy
M1 ¿ M2 < M3 in the right-handed neutrino sector i.e., (x À 1), which is realized
in our model, see Eq. (2.15), the above formula is simpli¯ed to the following one:
"1 = ¡
3
16¼À2(My
DMD)11
X
k=2;3
Im[ (My
DMD)2 1k ]
M1
Mk
: (2.24)
¤We will assume M1 ¿ M2 < M3. In this case, even if the heavier right-handed
neutrinos N2 and N3 produce lepton asymmetry, it is usually erased before the decay
of N1.
¤The function f + g in MSSM is twice as big compared to the SM. However this
is compensated by the factor 1
g¤
that appears in ´B which in MSSM is half of the SM
value.
25
"1 depends on the (1,1), (1,2) and (1,3) entries of My
DMD. These quantities can be
related to the light neutrino mass and mixing parameters measurable in low energy
experiments. In the basis where MR is diagonal, these elements are
(My
DMD)11 = (cm¿ )2 ¡
V31V ¤
31 + V21V ¤
21²2 + a2e
V11V ¤
11²6¢
(My
DMD)12 = (cm¿ )2 ¡
V31V ¤
32 + V21V ¤
22²2 + a2e
V11V ¤
12²6¢
(My
DMD)13 = (cm¿ )2 ¡
V31V ¤
33 + V21V ¤
23²2 + a2e
V11V ¤
13²6¢
; (2.25)
where V = KU3U2U1 is the unitary matrix diagonalizing MR. Straightforward calcu-
lations give, to leading order in ²,
(My
DMD)11 = 8a2e
c2m2¿
²4 cos2 µ12 sin2 µ12(a2
13 + t2
13 tan2 µ12)
£ 1=
©
8t4
13 sin4 µ12 + 32a13t2
13b cos2 µ12 sin4 µ12 sin 2(® + ±)
+ a13 cos4 µ12[4a13(a2
12 ¡ b2) cos 2µ12 + a13(a2
12 + 4b2)(3 + cos 4µ12)
+ 16a12t2
13 sin2 µ12 cos 2(® + ±)]
ª
(2.26)
(My
DMD)2
12 = 2a2e
c4m4¿
²6 tan2 µ12e¡{(Á1¡Á2)e¡2{(2®+±) ©
4(a2
13 ¡ t2
13) cos 2µ12 ¡ 2t13 sin 2µ12
(2a13e{(2®+±) ¡ (a12 + 2{b)e¡{±) + 4(a2
13 + t2
13) + t13 sin 4µ12e¡{±(a12 ¡ 2{b)
ª2
£ 1=
©
[{be{± ¡ a13e¡{(2®+±) + a12e¡{± cos2 µ12 ¡ {be¡{± cos 2µ12 + 2t13 tan µ12]2
£ [3a12a13 ¡ 2{a13b + 4t2
13e¡2{(®+±) + 4 cos 2µ12(a12a13 ¡ t2
13e¡2{(®+±)) +
a13(a12 + 2{b) cos 4µ12]2ª
(2.27)
(My
DMD)2
13 = 2a2e
c4m4¿²4 sin2 µ12e¡{(Á1¡Á3)(a13 cos µ12 + e¡{(2®+±)t13 sin µ12)2
£ 1=
©
a13 cos2 µ12(a12 ¡ 2{b + (a12 + 2{b) cos 2µ12) + 2t2
13 sin2 µ12e¡2{(®+±)ª2
(2.28)
These analytical expressions have been checked numerically. In Figure (1) we have
plotted j"1j as function of µ13 for ¯xed values of other observables. The solid line in
Fig (1) which corresponds to the exact numerical evaluation agrees very well with the
dashed line corresponding to the analytical expressions.
From Figure (1), it is apparent that µ13 is constrained in the model from cosmol-
ogy. If "1 < 1:3£10¡7, the induced baryon asymmetry would be too small to explain
26
observations. As can be seen from Figure (1), µ13 should lie in the range 0:01 ¡ 0:07
for an acceptable value of "1. This result does not change very much with variations
in the other input parameters. Electroweak sphaleron processes 15 will convert the
induced lepton asymmetry to baryon asymmetry. The ratio of baryon asymmetry to
entropy YB is related to the lepton asymmetry through the relation 23:
YB = C YB¡L =
C
C ¡ 1
YL (2.29)
where C = 8Nf+4N'
22Nf+13N'
, Nf = 3 and N' = 1; 2 in the case of the SM and MSSM
respectively. In either case C » 1
3 . In Eq. (2.29), YB = nB
s with s = 7:04 n°.
There has been considerable interest in obtaining approximate analytical ex-
pression for baryon asymmetry 24;25. In order to estimate this, the dilution factor,
often referred to as the e±ciency factor · that takes into account the washout pro-
cesses (inverse decays and lepton number violating scattering) has to be known. As
an example, · = (2 § 1) £ 10¡2
³
0:01 eV
em1
´1:1§0:1
has been suggested in Ref. 24 from
which ´B ' 0:96 £ 10¡2"N1· has been calculated. In our work we solve the cou-
pled Boltzmann equations numerically to estimate the baryon asymmetry without
referring to the e±ciency factor.
2.5 Numerical Boltzmann equations
In this section we set up the Boltzmann equations for computing the baryon
asymmetry ´B generated through the out of equilibrium decay of N1. In our model the
right-handed neutrino masses are not independent of the CP asymmetry parameter
"1. Therefore a self consistent analysis within the model is required.
In the early universe, at temperature of order N1 mass, the main thermal pro-
cesses which enter in the production of the lepton asymmetry are the decay of the
lightest r.h. neutrino,¤ its inverse decay, and the lepton number violation scattering,
¢L = 1 Higgs exchange plus ¢L = 2 r.h.n exchange 26. The production of the lepton
asymmetry via the decay of the r.h.n is an out-of-equilibrium process which is most
e±ciently treated by means of the Boltzmann equations (BE).
¤In our analysis we stick to the case where the asymmetry is due only to the decay
of the lightest r.h. neutrino N1.
27
0 0.05 0.1 0.15 0.2
13
0
2·10
-7
4·10
-7
6·10
-7
8·10
-7
1
Figure 2.1. Plots for CP asymmetry parameter "1 using analytical (dotted) and nu-
merical (solid) results as a function of the neutrino oscillation angle µ13.
The input parameters used are a12 = 1, b = 1, ¢m2
¯ = 2:5 £ 10¡5 eV 2,
¢m2
a = 5:54 £ 10¡3 eV 2 and f±; ®g = f¼=4; ¼=4g. Our model requires
j"1j & 1:3 £ 10¡7 to successfully generate an adequate number for the
BA. This criterium happens to be satis¯ed only in the region for which
0:01 . µ13 . 0:07, this interval is not too sensitive to variations in the
input parameters.
28
The ¯rst BE which describes the evolution of the abundance of the r.h. neutrino
and which corresponds to the source of the asymmetry is given byy
dYN1
dz
= ¡
z
Hs(z)
µ
YN1
Y eq
N
¡ 1
¶¡
°D1
+ °S1
¢
; (2.30)
where z = M1
T . Here s(z) is the entropy density and °D1
; °S1
are the interaction rates
for the decay and ¢L = 1 scattering contributions, respectively.
The second BE relevant to the lepton asymmetry is given by
dYB¡L
dz
= ¡
z
s(z)H(M1)
·
"1°D1
µ
YN1
Y eq
N
¡ 1
¶
+ °W
YB¡L
Y eq
L
¸
; (2.31)
where "1 is the CP violation parameter given by Eq. (2.21) and °W is the washout
factor which is responsible for damping of the produced asymmetry, see Eq. (2.49)
below. In Eqs. (2.30) and (2.31), Y eq
i is the equilibrium number density of a particle
species i, which has a mass mi, given by
Y eq
i (z) =
45
4¼4
gi
g¤
µ
mi
M1
¶2
z2K2
µ
miz
M1
¶
; (2.32)
where gi is the particle internal degree of freedom (gNi
= 2, g` = 4). At temperatures
far above the electroweak scale one has g¤ ' 106:75 in the standard model, and
g¤ ' 228:75 in MSSM. H, the Hubble parameter evaluated at z = 1, and s(z), the
entropy density, are given by
H =
r
4¼3g¤
45
M2
1
MP
; s(z) =
2¼2g¤
45
M3
1
z3 ; (2.33)
where MP = 1:22 £ 1019 GeV . We also have
°Sj
= 2°(1)
tj
+ 4°(2)
tj
: (2.34)
The decay reaction density °Dj
has the following expression:
°Dj
= neq
Nj
K1(z)
K2(z)
¡Nj ; (2.35)
yIn this section we follow the notation of the ¯rst paper of Ref. 16 to which we
refer the reader for further details.
29
where Kn(z) are the modi¯ed Bessel functions. ¡Nj of the r.h.n Nj is the tree level
total decay rate de¯ned as
¡Nj =
(¸y¸)jj
8¼
Mj ; (2.36)
where
neq
Ni(T) =
giTmi
2¼2 K2
³mi
T
´
: (2.37)
We used the de¯nition ¸ = MD=À. We de¯ne the reaction density °(i) of any process
a + b ! c + d by
°(i) =
M4
1
64¼4
1
z
Z 1
(Ma+Mb)2
M2
1
dx ^¾(i)(x)
p
x K1
¡p
xz
¢
; (2.38)
where ^¾(j)(x) are the reduced cross sections for the di®erent processes which con-
tribute to the Boltzmann equations. For the ¢L = 1 processes involving the quarks,
we have
^¾(1)
tj = 3®u
X3
®=1
¡
¸¤
®j¸®j
¢µ
x ¡ aj
x
¶2
; (2.39)
^¾(2)
tj = 3®u
X3
®=1
¡
¸¤
®j¸®j
¢µ
x ¡ aj
x
¶·
x ¡ 2aj + 2ah
x ¡ aj + ah
+
aj ¡ 2ah
x ¡ aj
ln
µ
x ¡ aj + ah
ah
¶¸
(2.40;)
where
®u =
Tr(¸y
u¸u)
4¼
'
m2t
4¼v2 ; aj =
µ
Mj
M1
¶2
; ah =
µ
¹
M1
¶2
; (2.41)
¹ is the infrared cuto® which we set to 800 GeV 26;27. For the ¢L = 2 r.h.n exchange
processes, we have
^¾(1)
N =
X3
®=1
X3
j=1
¡
¸¤
®j¸®j
¢ ¡
¸¤
®j¸®j
¢
A(1)
jj +
X3
®=1
X3
n<j;j=1
Re (¸¤
®n¸®j) (¸¤
®n¸®j)B(1)
nj (2.42)
^¾(2)
N =
X3
®=1
X3
j=1
¡
¸¤
®j¸®j
¢ ¡
¸¤
®j¸®j
¢
A(2)
jj +
X3
®=1
X3
n<j;j=1
Re (¸¤
®n¸®j) (¸¤
®n¸®j)B(2)
nj (2.43)
where
A(1)
jj =
1
2¼
·
1 +
aj
Dj
+
ajx
2D2
j
¡
aj
x
µ
1 +
x + aj
Dj
¶
ln
µ
x + aj
aj
¶¸
; (2.44)
A(2)
jj =
1
2¼
·
x
x + aj
+
aj
x + 2aj
ln
µ
x + aj
aj
¶¸
; (2.45)
30
B(1)
nj =
p
anaj
2¼
·
1
Dj
+
1
Dn
+
x
DjDn
+
³
1 +
aj
x
´µ
2
an ¡ aj
¡
1
Dn
¶
ln
µ
x + aj
aj
¶
(2.46)
+
³
1 +
an
x
´µ
2
aj ¡ an
¡
1
Dj
¶
ln
µ
x + an
an
¶¸
;
B(2)
nj =
p
anaj
2¼
½
1
x + an + aj
ln
·
(x + aj)(x + an)
ajan
¸
+
2
an ¡ aj
ln
µ
an(x + aj)
aj(x + an)
¶¾
(2.47;)
and
Dj =
(x ¡ aj)2 + ajcj
x ¡ aj
; cj = aj
X3
®=1
¡
¸¤
®j¸®j¸¤
®j¸®j
¢
64¼2 : (2.48)
Finally, °W that accounts for the washout processes in the Boltzmann equations is
°W =
X3
j=1
Ã
1
2
°Dj
+
YNj
Y eq
Nj
°(1)
tj
+ 2°(2)
tj
¡
°Dj
8
!
+ 2°(1)
N + 2°(2)
N : (2.49)
Here, we emphasize the so-called RIS (real intermediate states) in the ¢L = 2 interac-
tions which have to be carefully subtracted to avoid double counting in the Boltzmann
equations. This corresponds to the term ¡1
8°Dj
in Eq. (2.49). For more details see
Refs. 24;28 and the ¯rst paper of Ref. 29.
2.6 Results and discussion
We are now ready to present our numerical results. First we make several
important remarks. Even though our model is supersymmetric, we have considered
in our BE analysis only the SM particle interactions. This is a good approximation
(see footnote 7). The authors in Ref. 27 have demonstrated that SUSY interactions
do not signi¯cantly change the ¯nal baryon asymmetry. Furthermore, we have not
included in our analysis the e®ects of renormalization group on the running masses
and couplings. The ¯rst paper of Ref. 29 has studied these e®ects. This paper has
also included ¯nite temperature e®ects and ¢L = 1 scattering processes involving
SM gauge bosons, which we have ignored in our analysis. This should be a good
approximation since it is believed that these e®ects are signi¯cant in the weak washout
regime and our model parameters seem to favor the strong washout regime with
em1 = (My
DMD)11
M1
' 0:1 eV . Scattering processes involving gange bosons have also been
studied in Ref. 28 in the context of resonant leptogenesis where they have been shown
to be signi¯cant.
31
Our next step is to put this model to the test and check its predictions. In order
to compute the value of the baryon asymmetry we proceed to numerically solve the
Boltzmann equations. We scan the parameter space corresponding to the parameters
a12, b, the oscillation angle µ13, the CP phase ± and the Majorana phase ®. In order
to automatically satisfy the oscillation data, we input the following light neutrino
parameters:
¢m2
¯ = 2:5 £ 10¡5eV 2; ¢m2
a = 5:54 £ 10¡3eV 2; sin µ12 = 0:52: (2.50)
Using hierarchical spectrum, we see that the masses m1, m2 and m3 are ¯xed. On
the other hand we consider maximal mixing in the 2-3 sector of the leptonic mixing
matrix, i.e µ23 = ¼
4 +t23² with t23 being zero ( t23 » µ(1) has minimal impact on ´B).
The CP phase ± and the Majorana phase ® are allowed to vary in the intervals [0; 2¼]
and [0; ¼] respectively. We remind the reader that the second Majorana phase ¯ is
related to ® through ¯ ' ® + ¼
2 + b². µ13 will be allowed to vary in the interval [0;
0.2] as it is bounded from above by reactor neutrino experiments.
In Figure (2), for a given set of input parameters, we illustrate the di®erent
thermally averaged reaction rates ¡X = °X
neq
N1
contributing to BE as a function of
z = M1
T .
All rates at z = 1 ful¯ll the out of equilibrium condition (i.e. ¡X . H(z = 1)),
and so the expected washout e®ect due to the ¢L = 2 processes will be small. The
parameters chosen for this illustration are: ± = ¼=2, ® = ¼=2, a12 = 0:01, b = 0:9,
cm¿ = mt
³
m¿
mb
´
= 135 GeV and µ13 = 0:02. Eq. (2.50) ¯xes the light neutrino masses
to be: m1 = 0:00271292 eV , m2 = 0:00688186 eV and m3 = 0:0380442 eV : For this
choice we obtain j ²1 j' 2 £ 10¡7. The calculated r.h.n masses in this case are
M1 = 9 £ 109 GeV ; M2 = 8:7 £ 1011 GeV ; M3 = 2:6 £ 1014 GeV : (2.51)
The mass of the lightest r.h.n is consistent with lower bound derived in Ref. 29, M1 ¸
2:4£109 GeV, for hierarchical neutrino masses assuming that one starts with zero N1
initial abundance (which is what we assumed in our calculation). This mass is also
in accordance with the upper bound found in Ref. 30 following a model independent
32
study of the CP asymmetry, and the bound derived in Ref. 24 based on the estimation
of ºR production and the study of the asymmetry washout.
Figure (3) represents the solution of the BE, N1 abundance and the baryon
asymmetry both as functions of z for the same set of parameters mentioned above.
The ¯nal baryon asymmetry, in terms of the baryon to photon ratio, is (see dark,
solid curve in Fig. (3) for z À 1)
´B ' 6:03 £ 10¡10: (2.52)
This number is inside the observational range of Eq. (2.1). Our codes were tested to
reproduce the results in the ¯rst paper of Ref. 16 before being applied to this model.
2.7 Gravitino Problem
Leptogenesis scenario assumes the existence of heavy right handed neutrinos
which are thermally generated with su±ciently adequate abundance, during the re-
heating phase occurring right after in°ation. Therefore, the reheating temperature
TRH can not be much lower than 109 GeV, a bound on the right handed neutrino mass
30 necessary for the success of thermal Leptogenesis. This is already in con°ict with
a stringent upper bound on TRH , which may be as low as 106 ¡ 107 GeV, required
to avoid large Gravitino abundance which would upset the good predictions of BBN
31. In Supersymmetry, the Gravitino is the superpartner of the Graviton; with mass
of order natural SUSY scale; 1 TeV, therefore, the Gravitino is expected to be in the
range of 100GeV · m3=2 · 10 TeV. A combination of data and calculations of several
light elements abundance leads to the following recent upper bound 32
TRH · (1:9 ¡ 7:5)107 GeV;
which has been derived for m3=2 » 100 GeV. The standard thermal Leptogenesis with
normal hierarchical r.h neutrino seems to be at odds with the constraint above; one
has to invoke the BA in such way that these tensions are avoided. Thus M1 < TRH is
required, which for gravitino mass in the range 300 GeV to 3 TeV is in con°ict with
the predictions of Eq. (2.51).
33
There are several ways around this problem. (i) In gauge mediated SUSY
breaking scenario the gravitino is the lightest SUSY particle with mass in the range
10¡4 eV < m3=2 < 100 GeV . For m~g < 100 MeV, there are no cosmological or
astrophysical problems. In such a scenario the axion can serve as the dark matter.
(ii) In anomaly mediated SUSY breaking scenario, the gravitino mass is enhanced
by a loop factor compared to the squark masses and is naturally of order 100 TeV.
Such a gravitino would decay with a shorter lifetime without a®ecting big bang nu-
cleosynthesis. The gaugino is a natural dark matter candidate in this case. (iii) The
gravitino itself can be the LSP and dark matter with a mass of order 100 GeV, in
which case it does not decay 33. Other solutions include changing the dynamics of
the leptogenesis process by invoking (iii) non{thermal leptogenesis 34, (iv) resonant
leptogenesis 28;35, or (v) soft leptogenesis 36. In the following two chapters we invoke
Baryon Asymmetry via Resonant and Soft Leptogenesis. Especially, Our predictive
inverted neutrino hierarchy involving two nearly degenerate r.h.n, allows for the self-
energy contribution to the CP asymmetry to be resonantly enhanced, while the r.h.n
masses are low enough to be compatible with the reheating temperature bound. It
will be shown that baryon asymmetry can be maximized as long as M (4¡7)106 GeV
or above9.
34
0.01 0.1 1 10 100
z = M1 T
0.0001
10
1. ´ 106
1. ´ 1011
1. ´ 1016
1. ´ 1021
GX H Hz = 1L
H HzL
H Hz = 1L
GDL = 2
H Hz = 1L
GDL = 1
H Hz = 1L
GD1
H Hz = 1L
Figure 2.2. Various thermally averaged reaction rates ¡X contributing to BE nor-
malized to the expansion rate of the Universe H(z = 1). The
straight greyed line represents H(z)=H(z = 1), the dashed line is for
¡D1=H(z = 1), the dotted-dashed line represents ¡¢L=1=H(z = 1)
processes and the red curve represents ¡¢L=2=H(z = 1).
35
0.01 0.1 1 10 100
z = mN1 T
1. ´ 10-20
1. ´ 10-16
1. ´ 10-12
1. ´ 10-8
0.0001
abundancies
Figure 2.3. Evolution of YN1 (solid blue), Y eq
N1 (dot-dash) and the baryon asymmetry
´B (dark solid line) in terms of z in the model. The estimated value
for the baryon asymmetry is ´B ' 6:03 £ 10¡10, with Y ini
N1 = 0 and
assuming no pre-existing B ¡ L asymmetry.
36
2.8 Conclusion
An attractive feature of the seesaw mechanism is that it can explain the origin of
small neutrino masses and at the same time account for the observed baryon asymme-
try in the universe by the out of equilibrium decay of the super-heavy right handed
neutrinos. It is then very tempting to seek a link between the baryon asymmetry
parameter ´B induced at high temperature and neutrino mass and mixing parame-
ters observable in low energy experiments. No quantitative connection can be found
between them in the SM. There have been several attempts in the literature 16;37;38
to establish a relationship between the two. In this paper we have addressed this
question in the context of a class of minimal left{right symmetric models.
In the models under consideration the minimality of the Higgs sector implies
that Ml and MD (charged lepton and Dirac neutrino mass matrices) are proportional.
As a result, the entire seesaw sector (including the heavy right{handed neutrinos and
the light neutrinos) has only 9 parameters. This is the same number as low energy
neutrino observables (3 masses, 3 mixing angles and 3 phases). As a result we are
able to link the baryon asymmetry of the universe to low energy neutrino observables.
This feature is unlike the SM seesaw which has too many arbitrary parameters. Our
numerical solution to the coupled Boltzmann equations shows that this constrained
system with Ml / MD leads to an acceptable baryon asymmetry. The requirement
of an acceptable baryon asymmetry restricts some of the light neutrino observables.
We ¯nd that tan2 µ12 ' m1=m2, 0:01 . µ13 . 0:07 and ¯ ' ® + ¼=2 are needed for
successful baryogenesis. Future neutrino oscillation experiments can directly probe
into the dynamics of the universe in its early stages.
CHAPTER 3
BARYON ASYMMETRY VIA SOFT
LEPTOGENESIS
3.1 Introduction
In this chapter we analyze lepton asymmetry induced in the right handed sneu-
trino ~ºR1 ¡ ~ºy
R1 mixing and decay through WR exchange in a class of SUSYLR mod-
els. Usual soft leptogenesis scenario requires small B¡term and relatively low heavy
neutrino mass. We include the e®ect of SUSY breaking contribution on the break-
ing parameters; and compute r.h.n soft parameters to show that Soft Leptogenesis
mechanism implemented in SUSYLR framework leads to adequate baryon number
asymmetry in the universe. We employ Renormalization Group Equations analysis
and show that one achieve this result with natural values of Soft breaking parameters;
B » 100 GeV . In this class of models; M~ºR1 » MWR » (109 ¡ 1010) GeV, is not
required to be small as originally proposed. There is no excessive CP violation in
these models even when we assume universality of parameters.
3.1.1 Soft Leptogenesis, a brief review
Recently, soft Supersymmetry breaking e®ects have been utilized to explain the
Baryon Asymmetry via the "Soft leptogenesis" mechanism 39;40. In these models;
lepton number violation occurs in the decay of the heavy right handed neutrino and
sneutrino, ºc ! LÁy, ~ºc ! L~Áy, etc. CP asymmetry needed for Leptogenesis the
mixing of ~ºc ¡ ~ ºcy
trough soft supersymmetric breaking terms. The relevant super-
potential is given by;
W = YD`ºcHu +
1
2
MRºcºc (3.1)
37
38
which generates small neutrino masses via the seesaw mechanism. Here, the light
neutrino masses are obtained from the matrix Mº ' MDM¡1
R MD
T where MD =
YD hHui and MR are respectively the Dirac and the heavy Majorana right-handed
neutrino (r.h.n) mass matrices. In supersymmetric models with seesaw mechanism,
Soft SUSY breaking e®ect involving ~ºc, should be taken into account for the study of
Leptogenesis. The corresponding soft SUSY breaking Lagrangian is;
¡Lsoft = ~m2 ~ ºcy
~ºc + (
1
2
BMR~ºc~ºc + AYD~`~ºcH + h:c:) (3.2)
The parameters A and B in Eq. (3) are complex in general. Their presence will
introduce mixing and CP¡violation in the ~ºc ¡ ~ºcy system, analogous to the well
known Ko ¡ K
o
system (see appendix A.2 for details). Successful Soft Leptogenesis
can occurs even with one family of neutrinos, so we focus on that case. The mass
matrix of the ~ºc ¡ ~ºcy system is given by,
m2
~ºc¡~ºcy =
Ã
jMRj2 BMR
B¤M¤R
jMRj2
!
(3.3)
Since the r.h.n mass MR is much larger than the SUSY breaking scale B, diagonliza-
tion of the mass matrix of Eq. (4) will lead to the mass eigenstates e N§ = 1
2 (~ºc § ~ºcy)
with masses eigenvalues,
M§ ' M1(1 §
jBj
2M1
) ; (3.4)
The mass and width di®erence of the two sneutrino mass eigenstates are given by
¢m = jBj; ¢¡ =
2jAj¡
MN
: (3.5)
After Sphaleron e®ect takes place the ¯nal Baryon asymmetry (BA) is determined to
be;
nB
s
' ¡10¡3 d
·
4¡jBj
4jBj2 + ¡2
¸
jAj
M1
sin Á : (3.6)
Á is a CP inducing phase desired to be of order 1; O(1), it would in general be con-
tained in the trilinear or bilinear couplings of r.h.n. d is an e±ciency parameter,
often referred to as dilution factor. In general, it depends on the production mech-
anism for the r.h. sneutrino. Soft leptogenesis can be successful for rather low ~ºR1
39
Figure 3.1. Interfering ~N
¡ decay amplitudes for the fermionic ¯nal states. The blob
in the diagram contains a sum of all possible intermediate states. The
mixing between the two states ~N
¡ and ~N
+ leads to CP violation.
masses which is favored from the Gravitino point of view, however, unconventionally
suppressed B¡term of order µ(1) GeV is required for this picture to succeed.
Supersymmetric left{right (SUSYLR) models based on the SU(3)C £SU(2)L£
SU(2)R £ U(1)B¡L gauge group naturally includes r.h.n and implements seesaw for
neutrino masses. In left{right models, parity symmetry imposes hermiticity on the
Yukawa matrices and constrains the Soft breaking parameters in a way that helps solve
the supersymmetric CP problem that hunts MSSM, leading to vanishing EDM, while
allowing su±cient CP violation in ~ºR mixing. It is therefore interesting to analyze the
idea of Soft leptogenesis in the context of Left{Right symmetry. Here, we study the
e®ect of the interactions of the SU(2)R gauge boson WR on the generation of the the
primordial lepton asymmetry via the Soft leptogenesis mechanism. B ¡ L violation
occurs when Left{Right symmetry is broken by the VEV vR of the B ¡ L = ¡2
triplet scalar ¯eld ¢c(1; 1; 3;¡2), which gives Majorana masses to the r.h sneutrino
and, lepton number is violated in their decays: ºR ! LÁy; ºR ! LcÁ and ~ºR1 ! ~eRu ¹ d
as well as ~ºR1 ! ~e¤
R¹ud, where this later is mediated by the right handed gauge boson
WR. We show that ~ºR1 decay through WR exchange can dominate the traditional
ºR ! LÁy frequently used decay to explain BA. Further more, by RGE analysis we
40
show that the requirement of unconventionally small B¡term is no longer needed, in
addition, we use RGE running and SUSY breaking e®ect to naturally account for the
complex O(1) phase as dictated by the scenario success. The mass of r.h sneutrino
can be » MWR » (109 ¡ 1010) GeV .
The rest of the chapter is organized as follows. In Sec. 3.2 we review the
minimal left{right symmetric model. In Sec. 3.3 we analyze leptogenesis in this
model. Here we review RGE and discuss their running e®ect of the soft breaking
parameters in the model from the requirement of successful soft leptogenesis. In Sec.
3.4 we calculate the main two loop amplitude responsible for the mixing of ~ºc ¡ ~ ºcy
.
In Sec. 3.5 we analyze SUSY breaking e®ect on these parameters. we calculate
the lepton asymmetry parameter "1 generated in the model in ~ºR1 decay. Sec. 3.6
provides our numerical results for ´B. Finally, in Sec. 3.7 we conclude.
3.2 The minimal left-right symmetric model
Let us brie°y review the basic structure of the minimal SUSY left{right sym-
metric model developed in Ref. 10. The gauge group of the model is SU(3)C £
SU(2)L £ SU(2)R £ U(1)B¡L. The quarks and leptons are assigned to the gauge
group are listed in the table.
Left{handed quarks and leptons (Q;L) transform as doublets of SU(2)L [Q(3; 2; 1; 1=3)
and L(1; 2; 1;¡1)], while the right{handed ones (Qc;Lc) are doublets of SU(2)R
[Qc(3¤; 1; 2;¡1=3) and Lc(1; 1; 2; 1)]. The Dirac masses of fermions arise through
their Yukawa couplings to a Higgs bidoublet ©(1; 2; 2; 0). The SU(2)R £ U(1)B¡L
symmetry is broken to U(1)Y by the VEV (vR) of a B ¡ L = ¡2 triplet scalar ¯eld
¢c(1; 1; 3;¡2). This triplet is accompanied by a left{handed triplet ¢(1; 3; 1; 2) (along
with ¹¢
and ¢¹c ¯elds, their conjugates to cancel anomalies). These ¯elds also couple
to the leptons and are responsible for inducing large Majorana masses for the ~ºR. An
alternative to these triplet Higgs ¯elds is to use B¡L = §1 doublets Â(1; 2; 1;¡1) and
Âc(1; 1; 2; 1), along with their conjugates ¹Â and ¹ Âc. In this case non{renormalizable
operators will have to be invoked to generate large neutrino Majorana masses. For
de¯niteness we shall adopt the triplet option, although our formalism allows for the
41
TABLE 3.1. Particle assignment in SUSYLR gauge group
SU(3)C £ SU(2)L £ SU(2)R £ U(1)B¡L.
SU(3)c SU(2)L SU(2)R U(1)B¡L
Q 3 2 1 ¡1
3
L 1 2 1 ¡1
3
Qc 3 1 2 -1
Lc 1 1 2 +1
¢ 1 3 1 +2
¹¢
1 3 1 -2
¢c 1 1 3 -2
¹¢
c 1 1 3 +2
© 1 2 2 0
addition of any number of doublet Higgs ¯elds as well. Also, in order to keep the
model general one has to allow for a number of singlet ¯elds S(1; 1; 1; 0), for simplicity
we only we assume one singlet. The most general superpotential and soft breaking
terms invariant under the gauge symmetry are
W = ihQ(QT ¿2©aQc) + ihL(LT ¿2©aLc) + if (LT ¿2¢L) + ifc(LcT ¿2¢cLc)
+ M¢ Tr
¡
¢¹¢
¢
+M¢c Tr
¡
¢c¹¢
c¢
+M©a Tr
¡
©Ta
¿2©a¿2
¢
+ ¹¢S Tr
¡
¢¹¢
¢
+ ¹¢cS Tr
¡
¢c¹¢
c¢
+ ¹©aS Tr
¡
©Ta
¿2©a¿2
¢
+
1
6
YSS3 + 1
2MSS2 + LSS ; (3.7)
and the corresponding soft breaking terms;
¡LSB = 1
2
³
MG
3 ~g~g +MG
L
W~L W~L +MG
R
W~R W~R +MG
1
~B
~B
+ h.c.
´
+
h
iAQ ~Q
T ¿2©a Q~c + iAL ~L
T ¿2©a L~c + iAf ~L
T ¿2¢~L
+ iAfc L~c
T
¿2¢cL~c + A¢S Tr
¡
¢¹¢
¢
+ A¢cS Tr
¡
¢c¹¢
c¢
+ A©aS Tr
¡
©Ta
¿2©a¿2
¢
+
1
6
ASS + h.c.
¸
42
+
h
B¢ Tr
¡
¢¹¢
¢
+ B¢c Tr
¡
¢c¹¢
c¢
+ B©a Tr
¡
©Ta
¿2©a¿2
¢
+ 1
2BSS2 + h.c.
i
+
h
m2
Q
~Q
T ~Q
¤ + m2
QcQ~c
y Q~c + m2
L
~L
T ~L
¤ + m2
Lc L~c
y L~c
+ m2
¢ Tr
¡
¢y¢
¢
+ m2
¹¢
Tr
¡
¹¢
y¹¢
¢
+ m2
¢c Tr
¡
¢c y¢c¢
+ m2
¹¢
c Tr
¡
¹¢
c y¹¢
c¢
+ m2
©a Tr
¡
©y
a©a
¢
+ m2
S j S j2
i
; (3.8)
Under left{right parity symmetry,
Q $ Qc¤; L $ Lc¤; ©a $ ©y
a; ¢ $ ¢c¤ (3.9)
WSU(2)L $ W¤
SU(2)R; WB¡L $ W¤
B¡L; and µ $ ¹µ (3.10)
By demanding parity invariance from this theory, we also ¯nd the following relations
among the parameters 41;42:
¹©a = ¹¤
©a M¢ = M¤¢
c M©a = M¤©
a MS = M¤
S
hQ = hy
Q hL = hy
L f = f¤
c ¹¢ = ¹¤
¢c
LS = L¤
S MG
1 = MG¤
1 MG
L = MG¤
R MG
3 = MG¤
3
gL = gR B¢ = B¤¢
c B©a = B¤©
a BS = B¤S
;
where gL and gR are the SU(2)L and SU(2)R coupling constants, respectively, andMG
i
are the gauge group masses. The correspondences, hQ = hy
Q, hL = hy
L, and f = f¤
c in
the above relations are very important feature of Left{Right symmetry. It has been
shown in Ref. 10 that the hermiticity of the Yukawa matrices (along with the parity
constraints on the soft SUSY breaking parameters) helps solve the supersymmetric
CP problem that haunts the MSSM. These constraints also lead to zero EDM at the
ºR scale. In fact, EDM for the neutron and electron is only induced by RGE, but
remains close to the current experimental limit. Notice that our B¡term for r.h.n is
contained in the term Afc(L~c
T
¿2¢cL~c), so in general Afc would induce r.h.n B¡term.
We will discuss this in great detail in the following section.
Below vR, the e®ective theory is the MSSM + r.h.n with its Hu and Hd Higgs
multiplets. These are contained in the bidoublet ©a of the SUSY left-right model, but
in general they can also reside partially in other multiplets having identical quantum
43
numbers under the MSSM symmetry (such as the Â; Â doublet Higgs ¯elds alluded
to earlier) ¤.
3.3 ~ºR decay mediated by SU(2)R gauge boson WR
The left-right supersymmetric potential SU(2)R£U(1)B¡L symmetry is broken
down to U(1)Y by the VEV h¢ci = vR » MWR. We assume the right-handed neutrino
ºR1 has masse below vR. We focus on a single generation sneutrino and discuss the
e®ect of RGE running on the soft leptogenesis mechanism. With SM gauge symmetry,
the e®ective superpotential involving r.h.n below vR is;
W = (fij
d h1 ~ d¤
Ri~qLj + fij
u h2~u¤
Ri~qLj + fij
l h1~e¤
Ri
~l
Lj + fij
º h2~º¤
Ri
~l
Lj + ::: + h:c:):
+(¹h1h2 +
1
2
Mij
º ~º¤
Ri~º¤
Rj + h:c:) (3.11)
and the analogous soft breaking Lagrangian
¡Lsoft = (Aij
d h1 ~ d¤
Ri~qLj + Aij
u h2~u¤
Ri~qLj + Aij
l h1~e¤
Ri
~l
Lj + Aij
º h2~º¤
Ri
~l
Lj + ::: + h:c:):
+(B¹h1h2 +
1
2
Bij
º Mij
º ~º¤
Ri~º¤
Rj + h:c:) (3.12)
These parameters satisfy the boundary condition at vR, Ad = Au = AQ. Mixing
between the sneutrino ~ºR1 and anti-sneutrino ~ºy
R1 in the Soft Lagrangian is introduced
via the soft SUSY breaking terms, giving a source for the CP violation in the ~ºR1¡~ºy
R1
system in a similar way it happens in the K0 ¡ K0 system. µ(1) non-vanishing CP
inducing phase Á would in general be contained in A¡term or B¡term of r.h.n. It is
this CP violation that is considered to be source of lepton number asymmetry. After
sphaleron e®ect take place, the ¯nal baryon number to entropy ratio is determined
to be
nB
s
= ¡10¡3 d
·
4¡jBºj
4jBºj2 + ¡2
¸
jAºj
M1
sin Á : (3.13)
¤Allowing for such a possibility, the superpotential of Eq. (3.7) leads to the re-
lations for the MSSM Yukawa coupling matrices fu = °fd, and, f` = °fºD These
relations have been called up{down uni¯cation 10. Here, the ¯rst relation implies
mt
mb
' ° tan ¯ ´ c where ° is a parameter characterizing how much of Hu and Hd of
MSSM are in the bidoublet ©. The case of Hu;d entirely in © will correspond to ° = 1
and tan ¯ = mt=mb. The consequences of such relations on Baryon asymmetry have
been analyzed in the context of thermal Leptogenesis 7. Leptogenesis in the context
of more general left-right symmetric models has been analyzed in Ref. 19
44
M1 is the lightest r.h.n mass and the decay width ¡ = (My
DMD)11
4¼À2 M1. d is e±ciency
factor; often referred to as a dilution factor, which takes into account the washout pro-
cesses (inverse decays and lepton number violating scattering).¤ The determination
of the dilution factor involves the integration of the full set of Boltzmann equations.
A simple approximated solution which has been frequently used is given by 43
d =
8>>>>>><
>>>>>>:
p
0:1 · exp
¡
¡4
3
4 p
0:1 ·
¢
; · & 106
0:24(· ln ·)¡3=5 ; 10 . · . 106
1=(2·) ; 1 . · . 10
1 ; 0 . · . 1
(3.14)
where the parameter ·, which measures the e±ciency in producing the asymmetry,
characterizes the wash-out e®ects due to the inverse decays and lepton number vio-
lating scattering processes together with the time evolution of the system, is de¯ned
as the ratio of the thermal average of the ºR1 decay rate and the Hubble parameter
at the temperature T = M1,
· =
¡
H
; where H =
r
4¼3g¤
45
M2
1
Mpl
(3.15)
Mpl ' 1:22 £ 1019 GeV is the Planck mass and, g¤ is the e®ective degree of freedom.
^ºR ! ~e+u ¹ d(~e¡¹ud)
As we pointed out before; once one considers Left{Right symmetry, a lepton number
violating decay arises via the SU(2)R gauge boson WR as it is indicated in the ¯gure
(1). If we call ¡2 the decay width of the process ^ºR ! ~e+u ¹ d(~e¡¹ud) and, ¡1 =
(Y y
º Yº)11
8¼ M1, being the decay width of ºR ! LÁy(LcÁ), then ¡2 become the leading
lepton violating decay and ¡2 dominates if ¡2 ¸ ¡1. In this case, BA will mainly be
driven by decays such as ^ºR ! ~e+u ¹ d. Given that
¡2 '
9G2
FM4w
L
192¼3
M5
1
M4w
R
; (3.16)
¤Recently there has been considerable e®ort in obtaining semi analytical expres-
sions for the e±ciency so one does not have to solve Boltzmann equations every time.
For e.g; see 24;25. Rigorous derivations, however, have to include °avor e®ects on
leptogenesis 43.
45
ˆ R
ˆeR
W+
R
dR
uR
ˆ R
ˆeR
W+
R
ˆuR
ˆ dR
ˆ R
eR
ˆW
+
R
dR
ˆuR
ˆ R
eR
ˆW
+
R
ˆ dR
uR
Figure 3.2. Diagrams Contributing to Leptogenesis: The lightest ^ºR decay diagrams
via SU(2)R gauge boson exchange that appear in Left{Right models,
corresponding to ^ºR ! ~e+u ¹ d(~e¡¹ud). The lepton asymmetry can arise
through ~ºR1 ¡ ~ºy
R1 mixing and decay.
46
the condition translate into
(Y y
º Yº)11 . 1:55 £ 10¡4
µ
M1
MwR
¶4
(3.17)
On the other hand, if 0 . (¡2=H) . 1, the dilution parameter d can enhanced to
equal 1, which puts a constraint on the mass M1. A natural value for ¡2 follows from
SUSY breaking scale and preferred to be ¡2 » 100 GeV. For optimal e±ciency, i.e,
¡2 » H » 100 GeV, we ¯nd M1 ' 6:92£109 GeV. From Eq. (3.16) we then compute
MwR » 4:45 £ 1010 GeV. The condition on the Dirac Yukawa coupling in Eq. (3.17)
can be easily realized in Left{Right symmetry in way that is not in con°ict with light
neutrinos masses. Working a basis where the charged lepton mass matrix is diagonal
M` = D`, there is mixing among generations in both MR and MD, where MD = vYº,
the light neutrino mixing angles will depend on both of these mixings. While there is
some arbitrariness in the forms for MD and MR, one simple possibility consistent with
Soft Leptogenesis is as follows. As noted before, due left-right symmetry and assuming
the existence of two or more bidoublet ©a, the dirac mass matrix is hermitian and
can be diagonlized as MD = UDUy and r.h.n mass matrix as MR = V DRV T , where
U and V are unitary matrices. One can then generate light neutrino masses via the
seesaw mechanism ¤ 14
Mº = MDM¡1
R MD
T : (3.18)
Employing Eq. (3.18) to solve for Mº,
Mº = UDUyV ¤D¡1
R V yU¤DUT (3.19)
we explicitly make the simple choice U = V ¤, so thatMº becomesMº = UDD¡1
R DUT ,
where D ´ diag(d1; d2; d3) and DR ´ diag(M1;M2;M3). Mº is then found to be,
Mº = U
0
BB@
d21
=M1 0 0
0 d22
=M2 0
0 0 d23
=M3
1
CCA
UT (3.20)
¤In principle the ¢(1; 3; 1; +2) Higgs ¯eld can also acquire a small VEV . µ(eV ).
In this case the seesaw formula would be modi¯ed 18, as will the calculation of the
lepton asymmetry. We will assume such type II seesaw contributions proportional to
h¢i are zero in our analysis. This is consistent with the models of Ref. 10.
47
In the three family scenario, the relations between the °avor eigenstates (ºe; º¹; º¿ )
and the mass eigenstates (º1; º2; º3) can be expressed in terms of observables as
Mº = U¤
PMNSMdiag
º Uy
PMNS; (3.21)
where Mdiag
º ´ diag(m1;m2;m3), with mi being the light neutrinos masses and
UPMNS being the 3 £ 3 mixing matrix, we simply chosen U such that U¤ = UMNPS.
We get the following identity;
0
BB@
m1 0 0
0 m1 0
0 0 m3
1
CCA
=
0
BB@
d21
=M1 0 0
0 d22
=M2 0
0 0 d23
=M3
1
CCA
(3.22)
In a basis where the r.h.n mass matrix is diagonal, the Dirac mass matrix is
^M
D = MDV ¤ = MDU = MDU¤
PMNS. The condition of Eq. (3.17) then reads
( ^M
y
D
^M
D)11 = D2
11 = m1M1 . 1:16 £ 10¡2 GeV2; (3.23)
therefore, the lepton number violating right handed sneutrino decay via the SU(2)R
gauge boson dominance can be easily realized, as long as m1 . 1:67£10¡3 eV, which
is consistent with neutrino experiments.
It was concluded before that in order for Soft leptogenesis to succeed, the value
of M1 has to be very small; much smaller than the value naturally predicted by seesaw
of (109 ¡1010)GeV. Seesaw scale is also favorable by the traditional thermal leptoge-
nesis, however, it makes M1 borderline compatible with bounds derived on reheating
temperature as imposed by Gravitino production, but not conclusively excluded 31.
Furthermore, it is believed that the Soft bilinear coupling has to be signi¯cantly be-
low the MSUSY for this mechanism to provide viable leptonic asymmetry. In the
following we show that the above requirements do not hold in Left{Right symme-
try. In fact, Soft leptogenesis can proceed in Left-Right model with natural values of
M1 » (109 ¡1010)GeV and natural scale for the bilinear coupling B » ¡ » 100 GeV.
Also, by employing SUSY breaking e®ects on the running of RGE, we are able to
naturally generate the µ(1) complex phase that drives leptogenesis.
48
3.4 Computing the two loop amplitude leading to ~ºc ¡ ~ºcy mixing
In our analysis in the previous section, we have left out an important detail,
the A-term appearing in Eq (3.13) was conveniently assumed to have the right order
of magnitude for our estimate of Baryon Asymmetry to have the right order. Since
we are introducing a new decay; ~ºc ! ec~ucdc(ecuc ~ dc), to be the potentially dominant
decay, leading to adequate baryon asymmetry, such statement has to enforced by
computing the corresponding decay amplitude exactly. Our idea is that the mixing
~ºc ¡ ~ºcy in Left-Right symmetry is introduced and mediated by the SU(2)R gauge
boson (WR). The Feynman Diagram leading to this picture has been depicted in Fig
(3.4).
A = ¡g3R
m1=2M¸
f
p
2
(3 £ 3)
Z Z
d4k
(2¼)4
d4q
(2¼)4
£ Tr
"
k=
¡1+°5
2
¢
k2 ¡ m2`
+ {²
(k= + p= +M¸)
(k + p)2 ¡M2
¸ + {²
q=
¡1¡°5
2
¢
q2 ¡ m2
d + {²
(k= + p= +M¸)
¡1+°5
2
¢
(k + p)2 ¡M2
¸ + {²
£
M¸
¡1¡°5
2
¢
(k + p)2 ¡M2
¸ + {²
#
1
(k + p ¡ q)2 ¡ m2
~uc + {²
(3.24)
= M1=2Nf
Z Z
d4k
(2¼)4
d4q
(2¼)4
£
k(k + p):q(k + p) ¡ (k:q)(k + p)2
(k2 ¡ m2e
+ {²)(q2 ¡ m2
d + {²) [(k + p)2 ¡M2
¸ + {²]3 [(k + p ¡ q)2 ¡ m2
~uc + {²]
:
The 8-dimensional integral has to be done, notice the topology of our denomi-
nator with 6 propagators. The small masses can be set to zero, i.e, m2e
= m2
d = m2
~uc ,
without introducing any infrared divergence. The denominator then takes the form;
Den = (k2 + {²)
£
(k + p ¡ q)2 + {²
¤ £
(k + p)2 ¡M2
¸ + {²
¤3
(q2 + {²); (3.25)
we carry out the d4q-integral ¯rst, but to do this, the denominator has to be sim-
pli¯ed to become of the form Den = qn ¡ f (k; p;M¸; {²). First we employ Feynman
parametrization on the denominator;
1
a b c d3 =
¡(6)
2
Z 1
0
dX1
Z 1
0
dX2
Z 1
0
dX3
Z 1
0
dX4
X2
4 ± (1 ¡ X1 ¡ X2 ¡ X3 ¡ X4)
(aX1 + bX2 + cX3 + dX4)6 ;
(3.26)
49
where a = (k2 +{²); b = (q2 +{²); c = ((k + p ¡ q)2 + {²) ; d = ((k + p)2 ¡M2
¸ + {²)3
and p2 = M2
~ºc . After manipulating a shift on the momentum q, the d4q part of Eq
(3.24) becomes of regular form;
J =
Z +1
¡1
d4q
(2¼)4
1
[q4 + f (k; p;M¸; {²)]6 ; (3.27)
for which we can use the known dimensional regularization formulas. We are able to
perform the d4k integral part of Eq (3.24) in similar fashion. The resulting quantity
after integrating out q and k takes the form;
A =
6
p
2(16¼2)2
fg3R
m1=2M¸
Z 1¡X3¡X4
0
dX1
Z 1¡X4
0
dX3
Z 1
0
dX4
X3X2
4
(1 ¡ X1 ¡ X4)7
1
A4
£
2
6 4ln¤2 ¡ ln
µ
B2p2
A2
¡
C
A
¶
¡
¡p2
2
¡
1 ¡ B
A
¢ ¡
1 ¡ 2B
A
¢
³
B2p2
A2 ¡ C
A
´ ¡
¡p4
6
B
A
¡
1 ¡ B
A
¢3
³
B2p2
A2 ¡ C
A
´2 ¡
11
6
3
75
(3.28)
where A; B; C are functions of Xi's, M¸ and {² as follows;
A = 1 +
X2
3
(1 ¡ X1 ¡ X4)2 ¡
1 + X3
1 ¡ X1 ¡ X4
B =
X2
3
(1 ¡ X1 ¡ X4)2 ¡
X3 + X4
1 ¡ X1 ¡ X4
C = BM2
~ºc +M2
¸
X4
1 ¡ X1 ¡ X4
¡
{²
1 ¡ X1 ¡ X4
(3.29)
dXi integrals are carried our numerically and A / Âm1=2, where Â is order one
parameter. In table 3:2 we give an estimate of BA based on this numerical integration
and the running of soft parameters as we discuss below.
3.5 SUSYLR RGEs e®ect on Soft Leptogenesis
In order to generate a baryon asymmetry consistent with the observed number
of Eq. (2.1) one obtains the following constraints from Eq. (3.13):
A » 1 TeV ; M1 » (109 ¡ 1010) GeV; B » ¡ » 100 GeV ; and Á » 1; (3.30)
50
−
+
< c
0
>
˜uc
ec
−
˜ c
˜ c
dc
c
−
Figure 3.3. Two Loop Diagram Contributing to Leptogenesis: Feynman diagram
arising from ~ºc ! ec~ucdc decay, mediated by SU(2)R gaugino (labeled
¸). Our results are based on the computation of the corresponding
decay amplitude. The lepton asymmetry arises through the mixing of
~ºc ¡ ~ºcy
:
−
< c
0
>
ec
˜ c
˜ c
c
−
+
˜ dc
+
uc
Figure 3.4. Two Loop Diagram Contributing to Leptogenesis: Feynman diagram
arising from ~ºc ! ec ~ dcuc decay, mediated by SU(2)R gauge boson, it
is simply the supersymmetric correspondent of the previous Feynman
amplitude. The lepton asymmetry arises through the mixing of ~ºc¡~ºcy
51
assuming optimal e±ciency from Eq. (3.14). It is our purpose in this paper to
accommodate for these constraints in a Left-Right symmetric framework. It is also
desirable to have su±cient BA where the Soft parameters and the r.h.n mass assume
their natural values.
Above ÀR, the breaking scale of B ¡ L, the spectrum is that of Left-Right
Symmetry and the gauge group is SU(3)C £ SU(2)L £ SU(2)R £ U(1)B¡L. The full
set of one loop RGE Corresponding to the parameters 8>>>>>>>>>>>>>>>>>>>>><
>>>>>>>>>>>>>>>>>>>>>:
AQ AL Af Afc
A¢ A¢c A© AS
hQ hL f fc
B¢ Bc¢
B© BS
¹¢ ¹c
¢ ¹© YS
g1 gL gR g3
MG
1 MG
L MG
R MG
3
M¢ Mc¢
M© MS
LS CS
(3.31)
Most of the RGEs for these parameters can be found in 41. In SUGRA, it is allowed
to set all A¡terms to zero at Mpl, then soft breaking trilinear coupling like Afc would
be induced at ÀR. The evolution of Afc is given by
16¼2 d
dt
Afc = Afc
·
6fy
c fc + 2hy
LhL + 2Tr
¡
fy
c fc
¢
+ ¹¤
¢c¹¢c ¡
9
2
g2
1 ¡ 7g2R
¸
+ fc
h
12fy
cAfc + 4hy
LAL + 4Tr
¡
fy
cAfc
¢
+ 2¹¤
¢cA¢c + 9g2
1M1 + 14g2R
MR
i
+
£
6fcfy
c + 2hT
Lh¤
L
¤
Afc +
£
12Afcfy
c + 4AT
Lh¤
L
¤
fc (3.32)
Above ÀR there is no B¡term for r.h.n, but it will be induced by A¡terms like Afc .
In SUSYLR there is a proportionality between A¡ and B¡terms. In fact we can
approximately estimate Bind. The relevant term in this case is
¡LÃ SB = iAfcL~c
T
¿2¢cL~c + ::: (3.33)
when ¢c acquires VEV we get the following term
ÃL = Afc ~ ºc ~ ºc < ¢c >
´ Bind ~ ºc ~ ºc (3.34)
52
B¡term is then estimated to be, Bind ' (AfcÀR)=M1 = Afc=fc. From Eq (3.32),
setting Ai = 0 and analytically solving for Afc and ¯nding its value at ÀR then
estimate Bind induced at ÀR;
Bind ' ¡
(fc)11ÀR
16¼2M1
n
9g2
1MG
1 + 14g2R
MG
R
o
Log
µ
Mpl
ÀR
¶
; (3.35)
with ÀR = MwR
gR
' 6:35 £ 1010 GeV, Mpl » 1018 GeV where g1; gR; M1 and MR
have natural values, it is possible to generate the right order of magnitude for the
r.h. sneutrino B¡term of µ(50 ¡100) GeV. It is not possible however to explain the
complex phase necessary for the Soft Leptogenesis, for that we employ supersymmetry
breaking e®ect which has to be included anyways, otherwise, the result would be
misleading. In the result section we numerically compute the B¡term by including
all the RGEs that enter in the calculation of the soft breaking parameters in addition
to implementing SUSY breaking a®ect.
3.6 Symmetry breaking contribution to r.h.n B¡term
In this section we analyze the e®ects of supersymmetry breaking on the bilinear
coupling B which have to be included to get the correct magnitude. It turns out that
the µ(1) phase needed has it's origin from the F¡term of ¢c. From Eq. (3.7), the
part of the superpotential of interest to us;
W = M¢c Tr
¡
¢c¹¢
c¢
+ ¹¢cS Tr
¡
¢c¹¢
c¢
(3.36)
+
1
6
YSS3 + 1
2MSS2 + LSS
For simplicity we denote X = S, a = LS, b = 1
2MS, c = 1
6YS and d = ¹¢c . Then the
corresponding soft potential is
Vsoft = ~aX +~bX2 + ~cX3 + ~ dX¢c¹¢
c + ~M
¢c¢c¹¢
c
+ m2
X j X j2 + m2
¢c j ¢c j2 + m2
¹¢
c j ¹¢
c j2 (3.37)
where one can write the D¡term
VD =
1
4
g2B
¡
j ¢c j2 ¡ j ¹¢
c j2 ¢
(3.38)
53
and
VF =
¯¯¯¯¯
X
i
@W
@Ái
¯¯¯¯¯
2
=
¯¯
a + 2bX + 3cX2 + d¢c¹¢
c
¯ ¯2
+ jdX +M¢c j2 ¡
j ¢c j2 + j ¹¢
c j2 ¢
= jdj2 j ^X
j2 ¡
j ¢c j2 + j ¹¢
c j2¢
+
¯¯¯
¡a0 + b0 ^X
+ 3c ^X
2 + d¢c¹¢
c
¯¯¯
2
; (3.39)
where in the last step we shifted X by X = ^X
¡ M¢c=d and de¯ned a0 =
¡
¡
a ¡ 2b
d M¢c + 3c
d2M2¢
c
¢
, b0 =
¡
2b ¡ 6c
d M¢c
¢
. In the supersymmetric limit;
< ^X
> = 0 and
¢c¹¢
c®
= a0=d (3.40)
¢c is of order the breaking scale of Left-Right symmetry; < ¢c >= ÀR, then < ¹¢c >=
ÀRe{Á where Á = arg(a0=d) and jÀRj =
¯¯
a0
d
¯¯
1=2
. Now if one includes SUSY breaking
that we parameterize by small ²X, ² and ¹² as follow
< ^X
> = ²X (3.41)
¢c =
¯¯¯¯
a0
d
¯¯¯¯
1=2
+ ²
¹¢
c =
¯¯¯¯
a0
d
¯¯¯¯
1=2
e{Á + ¹² e{Á ;
with this, after computing D ¡ term and F¡term Eq. (3.37) and the potential
become;
Vsoft = ~a0 ^X
+
¯¯¯¯
a0
d
¯¯¯¯
1
2
e{Á
Ã
~M
¢c ¡M¢c
~ d
d
!
(² + ¹²) + h.c. (3.42)
V = 2 ja0dj j²Xj2 +
¯¯¯¯¯
b0²X +
¯¯¯¯
a0
d
¯¯¯¯
1
2
e{Á (² + ¹²)
¯¯¯¯¯
2
+ g2B
¯¯¯¯
a0
d
¯¯¯¯
(Re(² ¡ ¹²))2
+
"
~a0²X +
¯¯¯¯
a0
d
¯¯¯¯
1
2
e{Á
Ã
~M
¢c ¡M¢c
~ d
d
!
(² + ¹²) + h.c.
#
(3.43)
where ~a0 =
³
~a ¡ 2~b
M¢c
d + 3~c (M¢c
d )2 + ~ d
¯¯
a0
d
¯¯
1
2 e{Á
´
. Minimizing this potential with
respect to ²X and (² + ¹²), i.e, solving for @V
@²X
= @V
@(²+¹²) = 0 we ¯nd;
²¤
X = ²1
X + ²2
X + ²3
X (3.44)
54
where upon expressing everything in term of the notation of Eq. (3.36);
²1
X =
µ
MS ¡
YSM¢c
¹¢c
¶µ
B¢c ¡
M¢cA¢c
¹¢c
¶
£
½
2 j¹¢c j
¯¯¯¯
LS ¡
MSM¢c
¹¢c
+
YSM2¢
c
2¹2
¢c
¯¯¯¯
¾¡1
(3.45)
²2
X = ¡
µ
CS ¡
BSM¢c
¹¢c
+
YSM2¢
c
2¹2
¢c
¶
£
½
2 j¹¢c j
¯¯¯¯
LS ¡
MSM¢c
¹¢c
+
YSM2¢
c
2¹2
¢c
¯¯¯¯
¾¡1
(3.46)
²3
X =
A¢c
2¹¢c j¹¢c j
£ exp
½
{ arg
µ
LS ¡
MSM¢c
¹¢c
+
YS
2
M2¢
c
¹2
¢c
¶¾
(3.47)
² + ¹² is not of interest to this calculation of the B¡term contribution coming from
SUSY breaking; therefore we do not write its solution. It turns out, however, that ²¤
X
which is a complex quantity, enters the contribution of F¢c¡term to the r.h. sneutrino
B¡term at ÀR. We therefore compute ²¤
X at ÀR from the running of RGEs. We ¯nd
the F¡ term for ¢c to be;
jF¢c j2 = ¡
f
2
~ºR~ºR
¹¤
¢c²¤
X
j¹¢c j
1
2
¯¯¯¯
LS ¡
MSM¢c
¹¢c
+
YS
2
M2¢
c
¹2
¢c
¯¯¯¯
1
2
£ exp
½
{ arg
µ
¡
LS
¹¢c
+
MSM¢c
¹2
¢c
¡
YS
2
M2¢
c
¹3
¢c
¶¾
(3.48)
and so ¯nally
B =
f
2
¹¤
¢c²¤
X
j¹¢c j
1
2
¯¯¯¯
LS ¡
MSM¢c
¹¢c
+
YS
2
M2¢
c
¹2
¢c
¯¯¯¯
1
2
e
¡{ arg
¡ LS
¹¢c
+
MSM¢c
¹2
¢c
¡YS
2
M2¢
c
¹3
¢c
(3.49)
The ²¤
X parameter appearing in B carries just the right order of the complex phase
alluded to earlier as required for the soft leptogenesis. In the next section we show the
result of numerical computation of RGEs and the e®ect of SUSY breaking discussed
in this section.
3.7 Numerical result and estimation of BA
In This section, we report the result of our analysis in the table 3.2 and display a
particular case in the ¯gure 3.5, where the A-term is found to be between 700 GeV ¡
¡ 1 TeV . We have taken into account all RGEs that enter the calculation of the
B¡term and A¡terms for the r.h. sneutrino. Below, we write down some of the
RGEs not available in the literature.
55
²Soft breaking terms below vR
16¼2 d
dt
Al = Al
½
¡
9
5
g2
1 ¡ 3g2
2 + 3Tr(fy
dfd) + Tr(fy
l fl)
¾
+ 2fl
½
¡
9
5
g2
1M1 ¡ 3g2
2M2 + 3Tr(fy
dAd) + Tr(fy
l Al)
¾
+ 4(flfy
l Al) + 5(Alfy
l fl) + 2(flfy
ºAº) + (Alfy
ºfº); (3.50)
16¼2 d
dt
Aº =
·
Aº
½
¡
3
5
g2
1 ¡ 3g2
2 + 3Tr(fy
ufu) + Tr(fy
ºfº)
¾
+2fº
½
¡
3
5
g2
1M1 ¡ 3g2
2M2 + 3Tr(fy
uAu) + Tr(fy
ºAº)
¾
+4(fºfy
ºAº) + 5(Aºfy
ºfº) + 2(fºfy
l Al) + (Aºfy
l fl)
i
; (3.51)
16¼2 d
dt
Bº =
h
2(Bºf¤
º fT
º ) + 2(Bºfºfy
º ) + 4(Mºf¤
ºATº
) + 4(Aºfy
ºMT
º )
i
(3.52)
²The soft parameter corresponding the the linear term in singlet ¯eld S
16¼2 d
dt
CS =
£
CS
©1
2YSY ¤
S + 3¹¢¹¤
¢ + 3¹¢c¹¤
¢c + 8¹©¹¤
©
ª
+ LS
n
6¹¤
¢A¢ + 6¹¤
¢cA¢c + 2YSAS + 16¹¤
©A©
o
(3.53)
+ MS
n
2(YSMSBS) + 6(¹¤
¢M¢B¢) + 6(¹¤
¢cM¢cB¢c) + 16(¹¤
©M©B©)
oi
²Yukawa Couplings
16¼2 d
dt
fl =
·
fl
½
¡
9
5
g2
1 ¡ 3g2
2 + 3Tr(fdfy
d ) + Tr(flfy
l )
¾
+3(flfy
l fl) + (flfy
ºfº)
i
; (3.54)
16¼2 d
dt
fº =
·
fº
½
¡
3
5
g2
1 ¡ 3g2
2 + 3Tr(fufy
u) + Tr(fºfy
º )
¾
+3(fºfy
ºfº) + (fºfy
l fl)
i
: (3.55)
56
25 50 75 100 125 150 175 200
È B È
2·10-11
4·10-11
6·10-11
8·10-11
1·10-10
1.2·10-10
hB s
A = 1 TeV, G2 = 100 GeV
A = 700 GeV, G2 = 200 GeV
Figure 3.5. Dependence of BA on B¡term: Two cases are shown above, depending
on the choice of A¡term and ¡2. In both cases M1 = 6:9 £ 109GeV,
for which the dilution is enhanced (d = 1).
3.8 Conclusion
Soft Leptogenesis is an attractive mechanism to explain the baryon asymmetry.
In this paper we have addressed this question in the context of a class of minimal left{
right symmetric models. We analyze lepton asymmetry induced in the right handed
sneutrino ~ºR1 ¡ ~ºy
R1 mixing and decay due to soft SUSY breaking parameters in a
class of minimal left{right symmetric models (SUSYLR). Successful soft leptogenesis
scenario requires small B¡term and relatively low heavy neutrino mass. We study
the e®ect of full RGE running on the breaking parameters; this combined with the
contribution SUSY breaking we compute r.h.n soft parameters and show that Soft
Leptogenesis mechanism can indeed be fully implemented in SUSYLR framework
leading to adequate baryon number asymmetry in the universe. We also discuss the
bene¯ts of working in the context of Left-Right Symmetry.
57
TABLE 3.2. Result: The left column of the table gives input values of the parameters
at the Gut scale, where the right column shows the result of the Soft
parameters at vR following RGE running. The ¯nal estimation for the
BA is also given.
Input of model at MGut = 1018GeV Output at vR = MwR
gR
= 6:35 1010GeV
g1 = g2 = gR = gL ' 0:7
MG
1 = MG
2 = MG
L = MG
R = 300GeV j BAf
ind j= (Af )11vR
M1
» 50GeV; Á = 0
M¢ » M¢c » MS » M© ' vR
¹¢ = ¹¤
¢c » ¹© » YS ' µ(1) ²¤
X = ¡211:804 ¡ 355:125{
(hL)ii ' mli=(v £ Cos¯), (hL)ij;i6=j » 0
(hQ)33 ' mt=(v £ Sin¯), (hQ)ij;i;j6=3 » 0 j BF¢c
ind j» 100GeV; Á = µ(1)
tan(¯) = 20, fij = (f¤
c )ij =
MRij
vR
(AL)ij = A0(hL)ij ; (AQ)ij = A0(hQ)ij M1 = 6:92 £ 109GeV
(Af )ij = A0fij ; (Afc)ij = A0fc
ij d » 1 for 0 . (¡2=H) . 1
A¢ = A¤
¢c = A0¹¢; A© = A0¹©; AS = A0YS ¡2 » 100 GeV
B¢ = B0M¢; B¢c = B0M¢c A » 1 TeV
B© = B0M©; BS = B0MS; LS » v2R
; CS = C0LS
A0 » (300 ¡ 500)GeV;B0 = C0 » 100GeV nB=s ' 1 £ 10¡10
(universality condition)
me;¹;¿;t = f0:35 10¡3; 75:67 10¡3; 1:22; 82:43gGeV
CHAPTER 4
PREDICTIVE MODEL OF INVERTED
NEUTRINO MASS HIERARCHY AND
RESONANT LEPTOGENESIS
4.1 Introduction
A lot has been learned about the pattern of neutrino masses and mixings over
the past decade from atmospheric 44 and solar 45 neutrino oscillation experiments.
When these impressive results are supplemented by results from reactor 46, 47 and
accelerator 48 neutrino oscillation experiments, a comprehensive picture for neutrino
masses begins to emerge. A global analysis of these results gives rather precise deter-
mination of some of the oscillation parameters 49, 50:
j¢m2
atmj = 2:4 ¢
¡
1+0:21
¡0:26
¢
£ 10¡3 eV 2 ; sin2 µ23 = 0:44 ¢
¡
1+0:41
¡0:22
¢
;
¢m2
sol = 7:92 ¢ (1 § 0:09) £ 10¡5 eV 2 ; sin2 µ12 = 0:314 ¢
¡
1+0:18
¡0:15
¢
;
µ13
<
» 0:2 : (4.1)
While these results are impressive, there are still many important unanswered ques-
tions. One issue is the sign of ¢m2
atm = m23
¡m22
which is presently unknown. This is
directly linked to nature of neutrino mass hierarchy. A positive sign of ¢m2
atm would
indicate normal hierarchy (m1 < m2 < m3) while a negative sign would correspond
to an inverted mass hierarchy (m2
>
» m1 > m3). Another issue is the value of the
leptonic mixing angle µ13, which currently is only bounded from above. A third issue
is whether CP is violated in neutrino oscillations, which is possible if the phase angle
± in the MNS matrix is nonzero. Forthcoming long baseline experiments 48, NOºA
51, T2K 52 and reactor experiments double CHOOZ and DaiBay will explore some
58
59
or all these fundamental questions. Answers to these have the potential for revealing
the underlying symmetries of nature.
While there exists in the literature a large number of theoretical models for
normal neutrino mass hierarchy, such is not the case with inverted hierarchy. A large
number of models for inverted hierarchy based on symmetries 53{55 that were proposed
a few years ago are now excluded by the solar and Kamland data, which proved that
µ12 is signi¯cantly away from the maximal value of ¼=4 predicted by most of these
models. As a result, there is a dearth of viable inverted neutrino mass hierarchy
models. In this chapter, we attempt to take a step towards remedying this situation.
Here we suggest a class of models for inverted neutrino mass hierarchy based
on S3 £ U(1) symmetry. S3 is the non-Abelian group generated by the permutation
of three objects, while the U(1) is used for explaining the mass hierarchy of the
leptons. This U(1) symmetry is naturally identi¯ed with the anomalous U(1) of
string origin. In our construction, the S3 permutation symmetry is broken down to
an Abelian S2 in the neutrino sector, whereas it is broken completely in the charged
lepton sector. Such a setup enables us to realize e®ectively a º¹ $ º¿ interchange
symmetry in the neutrino sector (desirable for an inverted hierarchical spectrum),
while having non-degenerate charged leptons. The U(1) symmetry acts as leptonic
Le¡L¹¡L¿ symmetry, which is also desirable for an inverted neutrino mass spectrum.
The breaking of S2 symmetry in the charged lepton sector enables us to obtain µ12
signi¯cantly di®erent from ¼=4.
Interestingly, we ¯nd that the amount of deviation of µ12 from ¼
4 is determined
by µ13 through the relation
sin2 µ12 '
1
2
¡ tan µ13 cos ± : (4.2)
When compared with the neutrino data, the relation (4.2) implies the constraints (see
Fig. 1):
µ13 ¸ 0:13 ; 0 · ± · 43o : (4.3)
At the same time, the model gives
sin2 µ23 '
1
2
(1 ¡ tan2 µ13) ; (4.4)
60
which is very close to 1/2. These predictions will be tested in forthcoming experi-
ments.
Our models have the right ingredients to generate the observed baryon asym-
metry of the universe via resonant leptogenesis. The U(1) symmetry which acts on
leptons as Le ¡ L¹ ¡ L¿ symmetry guarantee that two right{handed neutrinos are
quasi-degenerate. This feature leads to a resonant enhancement in the leptonic CP
asymmetry, which in turn admits low right{handed neutrino masses, as low as few
TeV. With such light right-handed neutrinos (RHN) generating lepton asymmetry,
there is no cosmological gravitino problem when these models are supersymmetrized.
The class of neutrino mass models and leptogenesis scenario that we present here
will work well in both supersymmetric and non-supersymmetric contexts. However,
since low energy SUSY has strong phenomenological and theoretical motivations, we
stick here to the supersymmetric framework for our explicit constructions.
4.2 Predictive Framework for Neutrino Masses and Mixings
In order to build inverted hierarchical neutrino mass matrices which are pre-
dictive and which lead to successful neutrino oscillations, it is enough to introduce
two right{handed neutrino states N1;2. Then the superpotential relevant for neutrino
masses is
Wº = lTYºNhu ¡
1
2
NTMNN ; (4.5)
where hu denotes the up{type Higgs doublet super¯eld, while Yº and MN are 3 £ 2
Dirac Yukawa matrix and 2£2 Majorana mass matrix respectively. Their structures
can be completely determined by °avor symmetries. In order to have predictive
models of inverted hierarchy, the Le¡L¹¡L¿ ´ L symmetry can be used 53{56. This
symmetry naturally gives rise to large µ23 and maximal µ12 angles. At the same time,
the mixing angle µ13 will be zero. In order to accommodate the correct solar neutrino
mixing angle, the L-symmetry must be broken. The pattern of L-symmetry breaking
will determine the relations and predictions for neutrino masses and mixings. As a
starting point, in the neutrino sector let us impose ¹¡¿ symmetry S2: l2 ! l3; l3 ! l2,
which will lead to maximal º¹¡º¿ mixing, consistent with atmospheric neutrino data.
61
The leptonic mixing angles receive contributions from both the neutrino sector
and the charged lepton sector. As an initial attempt let us assume that the charged
lepton mass matrix is diagonal. We will elaborate on altering this assumption in the
next subsection.
For completeness, we will start with general couplings respecting the S2 sym-
metry. Therefore, we have
N1 N2
Yº =
l1
l2
l3
0
BBBBB@
® 0
¯0 ¯
¯0 ¯
1
CCCCCA
;
N1 N2
MN =
N1
N2
0
@ ¡±N 1
1 ¡ ±
0
N
1
AM :
(4.6)
Note that setting (1; 2) element of Yº to zero can be done without loss of generality.
This can be achieved by proper rede¯nition of N1;2 states. The couplings ®; ¯ and
(1; 2); (2; 1) entries in MN respect L symmetry, while the couplings ¯0; ±N and ±
0
N
violate it. Therefore, it is natural to expect that j¯0j ¿ j®j; j¯j, j±Nj; j±
0
Nj ¿ 1.
Furthermore, by proper ¯eld rede¯nitions all couplings in Yº can be taken to be real.
Upon these rede¯nitions ±N and ±
0
N entries in MN will be complex.
Integration of the heavy N1;2 states leads to the following 3 £ 3 light neutrino
mass matrix:
mº =
0
BBB@ 2±
0
º
p
2
p
2
p
2 ±º ±º
p
2 ±º ±º
1
CCCA
m
2 ;
(4.7)
where
m =
hh0
ui2
M(1 ¡ ±N± 0
N)
p
2®
³
¯ + ¯0±
0
N
´
;
±º =
p
2
®
2¯¯0 + ¯2±N + (¯0)2±
0
N
¯ + ¯0± 0
N
; ±
0
º =
®
p
2
±
0
N
¯ + ¯0± 0
N
: (4.8)
The entries ±º, ±
0
º in (4.7) are proportional to the L-symmetry breaking couplings
and therefore one naturally expects j±ºj; j±
0
ºj ¿ 1. These small entries are responsible
for ¢m2
sol 6= 0, i.e. for the solar neutrino oscillation. The neutrino mass matrix
62
is diagonalized by unitary transformation UT
º mºUº = Diag (m1;m2; 0), were Uº =
U23U12 with
U23 =
0
BBBBB@
1 0 0
0 p1
2
¡p1
2
0 p1
2
p1
2
1
CCCCCA
; U12 '
0
BBB@
¹c ¡ ¹sei½ 0
¹se¡i½ ¹c 0
0 0 1
1
CCCA
;
(4.9)
where ¹c = cos ¹µ, ¹s = sin ¹µ and
tan ¹µ ' 1 §
1
2
· ; · =
j±ºj2 ¡ j±
0
ºj2
j±¤
º + ± 0
ºj
: (4.10)
The phase ½ is determined from the equation
j±ºj sin(!º ¡ ½) = j±
0
ºj sin(!
0
º + ½) ; !º = Arg(±º) ; !
0
º = Arg(±
0
º) ; (4.11)
and should be taken such that
j±ºj cos(!º ¡ ½) + j±
0
ºj cos(!
0
º + ½) < 0 : (4.12)
This condition ensures ¢m2
sol = m22
¡ m21
> 0 needed for solar neutrino oscillations.
For ¢m2
atm and the ratio ¢m2
sol=j¢m2
atmj we get
j¢m2
atmj ' jmj2 ;
¢m2
sol
j¢m2
atmj
' ¡2
³
j±ºj cos(!º ¡ ½) + j±
0
ºj cos(!
0
º + ½)
´
= 2
¯¯ ¯±¤
º + ±
0
º
¯¯¯
:
(4.13)
With no contribution from the charged lepton sector, the leptonic mixing matrix
is Uº. From (4.9), (4.10) for the solar mixing angle we will have sin2 µ12 = 1
2 § ·
4 . In
order to be compatible with experimental data one needs · ¼ 0:7. On the other hand
with j±ºj » j±
0
ºj and no speci¯c phase alignment from (4.13) we estimate j±ºj » j±
0
ºj »
10¡2. Thus we get the expected value · » 10¡2, but with the µ12 mixing angle nearly
maximal, which is incompatible with experiments. This picture remains unchanged
with the inclusion of renormalization group e®ects. Therefore, we learn that it is hard
to accommodate the neutrino data in simple minded inverted hierarchical neutrino
mass scenario. In order for the scenario be compatible with the experimental data
we need simultaneously
¯¯ ¯±¤
º + ±
0
º
¯¯¯
=
¢m2
sol
2j¢m2
atmj
' 0:016 ;
j±ºj2 ¡ j±
0
ºj2
j±¤
º + ± 0
ºj
= ¨(0:52 ¡ 0:92) : (4.14)
63
Therefore, one combination of ±º and ±
0
º must be » 50-times larger than the other.
This is indeed unnatural and no explanation for these conditions is provided at this
stage. To make this point more clear let's consider the case with ±º = 0¤. In this
case from (4.13) we have j±
0
ºj ' 0:016. Using this in (4.10) we obtain sin2 µ12 ¸ 0:496,
which is excluded by the neutrino data (4.1).
Summarizing, although the conditions in (4.14) can be satis¯ed, it remains a
challenge to have a natural explanation of these hierarchies. This is a shortcoming
of the scenario. Below we present a possible solution to this conundrum which looks
attractive and maintains predictive power without ¯ne tuning.
4.2.1 Improved µ12 with µ13 6= 0
Let us now include the charged lepton sector in our studies. The relevant
superpotential is
We = lTYEechd ; (4.15)
where YE is 3£3 matrix in the family space. In general, YE has o®{diagonal entries.
Being so, YE will induce contributions to the leptonic mixing matrix. We will use this
contribution in order to ¯x the value of µ12 mixing angle. It is desirable to do this in
such a way that some predictivity is maintained. As it turns out, the texture
YE =
0
BBB@
0 a0 0
a ¸¹ 0
0 0 ¸¿
1
CCCA
;
(4.16)
gives interesting predictions. In the structure (4.16) there is only one unremovable
complex phase and we leave it in (1,2) entry. Thus, we make the parametrization
a0 = ¸¹µeei!, while all the remaining entries can be taken to be real. In order to
get the correct value of the electron mass for µe ¿ 1, we should take the coupling
a = ¸e=µe. For ¯nding the unitary matrix which rotates the left{handed charged
¤This case is realized within the model with S3 £U(1) °avor symmetry presented
in section 4.4.
64
Figure 4.1. Correlation between µ12 and µ13 taken from Fogli et al. Three sloped
curves correspond to µ12 ¡µ13 dependance (for three di®erent values of
CP phase ±) obtained from our model according to Eq. (4.22)
lepton states, upon diagonalization of YE, we need to diagonalize the product YEY y
E.
Namely, with UeYEY y
EUy
e =
³
Y diag
E
´2
, it is easy to see that
Ue =
0
BBB@
c sei! 0
¡se¡i! c 0
0 0 1
1
CCCA
;
(4.17)
where c ´ cos t, s ´ sin t and tan t = ¡µe . Finally, the leptonic mixing matrix takes
the form
Ul = U¤
e Uº ; (4.18)
where Uº = U23U12 can be derived from Eq. (4.9). Therefore, for the corresponding
mixing elements we get
Ul e3 = ¡
s
p
2
e¡i(!+½) ; jUle
2j =
1
p
2
¯¯¯¯
c ¡
s
p
2
e¡i(!+½)
¯¯¯¯
; jUl ¹3j =
c
p
2
: (4.19)
65
Comparing these with those written in the standard parametrization we obtain the
relations
s13 = ¡
s
p
2
; ! + ½ = ± + ¼ ; (4.20)
s12c13 = jUle
2j ; s23c13 = jUl ¹3j : (4.21)
Using (4.20) and (4.19) in (4.21) leads to the prediction:
sin2 µ12 =
1
2
¡
p
1 ¡ tan2 µ13 tan µ13 cos ± ;
sin2 µ23 =
1
2
¡
1 ¡ tan2 µ13
¢
: (4.22)
Since the CHOOZ bound is s13
<
» 0:2, the ¯rst relation in (4.22), with the help of the
solar neutrino data provides an upper bound for the CP violating phase: ± <
» ±max ¼
48o. However, this estimate ignores the dependence of µ12 on the value of µ13 in the
neutrino oscillation data. Having µ13 6= 0, this dependence shows up because one
deals with three °avor oscillations. This has been analyzed in Ref. 50 and is shown
in and Fig. 1 (borrowed from Ref. 50) along with the constraints arising from our
model. We have shown three curves corresponding to (4.22) for di®erent values of ±.
Now we see that maximal allowed value for ± is ±max ' 43o. Moreover, for a given
± we predict the allowed range for µ13. In all cases the values are such that these
relations can be tested in the near future. An interesting result from our scenario is
that we obtain lower and upper bounds for µ13 and ± respectively
µ13 ¸ 0:13 ; 0 · ± · 43o : (4.23)
Finally, the neutrino-less double ¯-decay parameter in this scenario is given by
m¯¯ ' 2
q
¢m2
atm tan µ13
p
1 ¡ tan2 p µ13
1 + tan2 µ13
: (4.24)
We have neglected the small contribution (of order ¢m2
solar=¢m2
atm) arising from
the neutrino mass matrix diagonalization. Since the value of µ13 is experimentally
constrained (<
» 0:2), to a good approximation we have m¯¯ ¼ 2
p
¢m2
atm tan µ13.
Using this and the atmospheric neutrino data (4.1) we ¯nd m¯¯
<
» 0:02 eV. Knowledge
of µ13-dependence on ± (see Fig. 1) allows us to make more accurate estimates for the
66
(i)
(ii)
Figure 4.2. Curves (i) and (ii) respectively show the dependence of pm¯¯
¢m2atm
's low and
upper bounds on CP violating phase ±. The shaded region corresponds
to values of m¯¯ and ± realized within our model.
range of m¯¯ for each given value of ±. The dependence of m¯¯ on ± is given in Fig.
2. We have produced this graph with the predictive relations (4.22), (4.24) using the
neutrino data 50. Combining these results we arrive at
0:011 eV <
» m¯¯
<
» 0:022 eV : (4.25)
As we see the predicted range, depending on the value of ±, is quite narrow. Future
measurements of CP violating phase ± together with a discovery of the neutrino-less
double ¯-decay will be another test for the inverted hierarchical scenario presented
here.
4.3 Resonant Leptogenesis
Neutrino mass models with heavy right{handed neutrinos provide an attractive
and natural framework for explaining the observed baryon asymmetry of the universe
through thermal leptogenesis 57. This mechanism takes advantage of the out-of-
equilibrium decay of lightest right-handed neutrino(s) into leptons and the Higgs
boson. In the scenario with hierarchical RHNs, a lower bound on the mass of decaying
RHN has been derived: MN1 ¸ 109 GeV 58. The reheating temperature can not be
67
much below the mass of N1. In low energy SUSY models (with m3=2 » 1 TeV)
this is in con°ict with the upper bound on reheating temperature obtained from
the gravitino problem 59. This con°ict can be naturally avoided in the scenario of
`resonant leptogenesis' 5, 60. Due to the quasi-degeneracy in mass of the RHN states,
the needed CP asymmetry can be generated even if the right{handed neutrino mass
is lower than 109 GeV.
Our model of inverted hierarchical neutrinos involves two quasi-degenerate RHN
states and has all the needed ingredients for successful resonant leptogenesis. This
makes the scenario attractive from a cosmological viewpoint as well. Now we present
a detailed study of the resonant leptogenesis phenomenon in our scenario.
68
The CP asymmetry is created by resonant out of equilibrium decays of N1;N2
and is given by 60
²1 =
Im( ^ Y y
º
^ Yº)2
21
( ^ Y y
º ^ Yº)11( ^ Y y
º ^ Yº)22
(M2
2 ¡M2
1 )M1¡2
(M2
2 ¡M2
1 )2 +M2
1¡22
; (4.26)
with a similar expression for ²2. The asymmetries ²1 and ²2 correspond to the decays
of N1 and N2 respectively. Here M1;M2 are mass the eigenvalues of the matrix MN in
(4.6), while ^ Yº = YºUN is the Dirac Yukawa matrix in a basis where RHN mass matrix
is diagonal. The tree{level decay width of Ni is given as ¡i = ( ^ Y y
º
^ Yº)iiMi=(8¼). The
expression (4.26) deals with the regime M2 ¡ M1 » ¡1;2=2 (relevant for our studies)
consistently and has the correct behavior in the limit M1 ! M2
60. From (4.6) we
have
UTN
MNUN = Diag (M1;M2) ; UN ' p1
2
0
@ 1 ¡ eir
e¡ir 1
1
A ;
(4.27)
with
M2
2 ¡M2
1 = 2M2
¯¯ ¯±¤
N + ±
0
N
¯¯¯
; tan r =
Im
¡
±N ¡ ±
0
N
¢
Re
¡
±N + ± 0
N
¢ : (4.28)
Introducing the notations
®
¯
= x ;
¯0
¯
= x0 ; (4.29)
we can write down the appropriate matrix elements needed for the calculation of
leptonic asymmetry:
( ^ Y y
º
^ Yº)11 =
1
2
¯2 ¡
2 + x2 + 2(x0)2 + 4xx0 cos r
¢
;
( ^ Y y
º
^ Yº)22 =
1
2
¯2 ¡
2 + x2 + 2(x0)2 ¡ 4xx0 cos r
¢
;
Im( ^ Y y
º
^ Yº)2
21 = ¡
1
4
¯4 ¡
2 ¡ x2 ¡ 2(x0)2 + 4xx0 cos r
¢2
sin 2r : (4.30)
In terms of these entries the CP asymmetries are give by
²1 =
Im( ^ Y y
º
^ Yº)2
21
( ^ Y y
º ^ Yº)11
j±¤N
+ ±
0
Nj
16¼j±¤N
+ ± 0
Nj2 + ( ^ Y y
º ^ Yº)2
22=(16¼)
; ²2 = ¡²1(1 $ 2) : (4.31)
69
We have ¯ve independent parameters and in general one should evaluate the lepton
asymmetry as a function of x; x0; j±Nj; j±
0
Nj and r. Below we will demonstrate that
resonant decays of N1;2 can generate the needed CP asymmetry.
It turns out that for our purposes we will need j±¤N
+ ±
0
Nj ¿ 1. This, barring
precise cancelation, implies j±Nj; j±
0
Nj ¿ 1. From the symmetry viewpoint and also
from further studies, it turns out that
¯¯
x0
x
¯¯
¿ 1 is a self consistent condition. Taking
these and the results from the neutrino sector, to a good approximation we have
¯2 =
p
¢m2
atmM
p
2xhh0
ui2
; atm2j ' 6 ¢ 10¡3 (4.32)
and
²1 ' ²2 '
Im( ^ Y y
º
^ Yº)2
12
( ^ Y y
º ^ Yº)11
j±¤N
+ ±
0
Nj
16¼j±¤N
+ ± 0
Nj2 + ( ^ Y y
º ^ Yº)2
11=(16¼)
'
¡
(2 ¡ x2)2
2(2 + x2)
¯2 j±¤N
+ ±
0
Nj
16¼j±¤N
+ ± 0
Nj2 + (2 + x2)2¯4=(64¼)
sin 2r ; (4.33)
where in the last expression we have ignored x0 contributions. This approximation
is good for all practical purposes. The combination j±¤N+ ±
0
Nj is a free parameter
and since we are looking for a resonant regime, let us maximize the expression in
(4.33) with respect to this variable. The maximum CP asymmetry is achieved with
j±¤N
+ ±
0
Nj = ( ^ Y y
º
^ Yº)11=(16¼). Plugging this value back in (4.33) and taking into
account (4.30), (4.32) we arrive at
¹²1 ' ¹²2 ' ¡
(2 ¡ x2)2
2(2 + x2)2 sin 2r ; (4.34)
where ¹²1;2 indicate the maximized expressions, which do not depend on the scale of
right{handed neutrinos. We can take these masses as low as TeV! The expression in
(4.34) reaches the maximal values for x ¿ 1 and x À 1. However, the ¯nal value
of x will be ¯xed from the observed baryon asymmetry. The lepton asymmetry is
converted to the baryon asymmetry via sphaleron e®ects 61 and is given by nB
s '
¡1:48 ¢ 10¡3(·(1)
f ²1 +·(2)
f ²2), where ·(1;2)
f are e±ciency factors given approximately by
62
·(1;2)
f =
Ã
3:3 ¢ 10¡3 eV
~m1;2
+
µ
~m1;2
0:55 ¢ 10¡3 eV
¶1:16
!¡1
;
70
with ~m1 =
hh0
ui2
M1
( ^ Y y
º
^ Yº)11 ; ~m2 =
hh0
ui2
M2
( ^ Y y
º
^ Yº)22 : (4.35)
In our model, with
¯¯
x0
x
¯¯
¿ 1 we have
~m1 ' ~m2 '
p
¢m2a
tm
2
p
2x
(2 + x2) ' 0:017 eV £
2 + x2
x
: (4.36)
This also gives ·(1)
f ' ·(2)
f ´ ·f and as a result we obtain
nB
s
¯¯¯
²=¹²
' 1:48 ¢ 10¡3·f (x)
(2 ¡ x2)2
(2 + x2)2 sin 2r : (4.37)
With sin 2r = 1 in order to reproduce the experimentally observed value
¡nB
s
¢exp
=
9¢10¡11 we need to take x = 3:8¢10¡5, x = 5:3¢104, x =
p
2¡0:0047 or x =
p
2+0:0047.
For these values of x we have respectively
¯¯¯
±¤
N + ±
0
N
¯¯¯
²=¹²
'
2 + x2
32
p
2¼x
p
¢m2
atmM
hh0
ui2
'
¡
6 ¢ 10¡7 ; 6 ¢ 10¡7 ; 3:2 ¢ 10¡11 ; 3:2 ¢ 10¡11¢
£
1 + tan2 ¯
tan2 ¯
M
106GeV
(4.38)
(¯xed from the condition of maximization). The MSSM parameter tan ¯ should not
be confused with Yukawa coupling in (4.32)). Note that these results are obtained
at the resonant regime jM2 ¡ M1j = ¡1;2=2. If we are away from this point, then
the baryon asymmetry will be more suppressed and we will need to take di®erent
values of x. In Fig. 4.3 we show
¯¯
±¤N
+ ±
0
N
¯¯
¡ x dependence corresponding to baryon
asymmetry of 9 ¢ 10¡11. The curves are constructed with Eqs. (4.32), (4.33). We
display di®erent cases for di®erent values of the mass M and for two values of CP
violating phase r. For smaller values of r the `ovals' shrink indicating that there is less
room in
¯¯
±¤N
+ ±
0
N
¯¯
¡ x plane for generating the needed baryon asymmetry. We have
limited ourselves to
¯¯
±¤N
+ ±
0
N
¯¯
<
» 0:1. Above this value the degeneracy disappears and
the validity of our expression (4.26) breaks down¤. Also, in this regime the inverted
mass hierarchical neutrino scenario becomes unnatural. The dashed horizontal line
in Fig. 4.3 corresponds to this `cut{o®'. This limits cases with larger masses [case
(d) in Fig. 4.3, of M = 1011 GeV]. The sloped dashed cut{o® lines appear due to
¤There will be another contributions to the CP asymmetry, the vertex diagram,
which would be signi¯cant in the non-resonant case.
71
the requirement that the Yukawa couplings be perturbative (®; ¯ <
» 1). As one can
see from (4.32), for su±ciently large values of M, with x À 1 or x ¿ 1, one of the
Yukawa couplings becomes non-perturbative.
As we see, in some cases (especially for suppressed values of r) the degeneracy
in mass between N1 and N2 states is required to be very accurate, i.e.
¯¯
±¤N
+ ±
0
N
¯¯
¿ 1.
In section 4.4 we discuss the possibility for explaining this based on symmetries.
4.4 Model with S3 £ U(1) Symmetry
In this section we present a concrete model which generates the needed textures
for the charged lepton and the neutrino mass matrices. It also blends well with
the leptogenesis scenario investigated in the previous section. We wish to have an
understanding of the appropriate hierarchies and the needed zero entries in the Dirac
and Majorana neutrino couplings. Also, the values of masses MN1;2 ' M <
» 108 GeV
and their tiny splitting must be explained. Note that one can replace L = Le¡L¹¡L¿
symmetry by other symmetry, which will give approximate L. For this purpose the
anomalous U(1) symmetry of string origin is a good candidate 63. However, in our
scenario the charged lepton sector also plays an important role. In particular, the
structure (4.16) is crucial for the predictions presented above. We wish to understand
this structure also by symmetry principles. For this a non Abelian discrete °avor
symmetries can be very useful 64, 65, 66. Therefore, in addition, we introduce S3
permutation symmetry. The S3 will be broken by two steps S3 ! S2 ! 1. Since in
the neutrino sector we were using S2 symmetry, we will arrange for that sector to feel
only the ¯rst stage of breaking, i.e. S2 will be unbroken in the neutral lepton sector.
Thus, the model we present here is based on S3 £ U(1) °avor symmetry. The
S3 permutation group has three irreducible representations 1; 10 and 2, where 10 is
an odd singlet while 1 and 2 are true singlet and doublet respectively. With doublets
denoted by two component vectors, it is useful to give the product rule
72
0
B@
x1
x2
1
CA
2
£
0
BB@
y1
y2
1
CCA
2
= (x1y1 + x2y2)1 © (x1y2 ¡ x2y1)10 ©
0
BB@
x1y2 + x2y1
x1y1 ¡ x2y2
1
CCA
2
(4.39)
where subscripts denote the representation of the corresponding combination. The
other products are very simple. For instance 1 £ 1 = 1, 10 £ 1 = 10, etc.
As far as the U(1) symmetry is concerned, a super¯eld Ái transforms as
U(1) : Ái ! eiQiÁi ; (4.40)
where Qi is the U(1) charge of Ái. The U(1) symmetry will turn out to be anomalous.
The anomalous U(1) factors can appear in e®ective ¯eld theories from string theory
upon compacti¯cation to four dimensions. The apparent anomaly in this U(1) is
canceled through the Green-Schwarz mechanism 67. Due to the anomaly, a Fayet-
Iliopoulos term ¡»
R
d4µVA is always generated 68 and the corresponding DA-term
has the form 69
g2A
8
D2A
=
g2A
8
³
¡» +
X
QijÁij2
´2
; » =
g2A
M2
P
192¼2 TrQ : (4.41)
In SUSY limit one of the VEVs should set DA-term to be zero.
For S3£U(1) breaking we introduce the MSSM singlet scalar super¯elds ~S
; ~T;X,
where vector symbols will denote S3 doublets. The transformation properties - the
S3 `membership' and U(1) charges - of these and other ¯elds are given in Table 4.1.
In the table we do not display MSSM pair of higgs doublet super¯elds hu; hd, noting
that they are invariant under S3 £ U(1).
Further we will use the following VEV con¯guration:
h~S
i = (0; V ) ; h~Ti = ~ V ¢ (1; i) ; hXi = VX : (4.42)
These structures can be obtained in verious simple ways. With »;QX < 0, in Eq.
(4.41) the VEV of the scalar component of X is ¯xed as VX =
p
»=QX. The direction
for ~T can be obtained from its bi-linear coupling with some neutral singlet Y 65.
Namely with superpotential coupling Y ~T2, the F-°atness condition gives the solution
73
TABLE 4.1. Transformation properties under S3 £ U(1).
~S
~T X ec
1 ~e c l1 ~l
N1 N2
S3 2 2 10 10 2 10 2 1 1
U(1) 0 0 ¡1 4 ¡ n ¡n n + 2 n ¡(n+1) 2m¡(n + 1)
in (4.42) and hY i = 0. Similarly with couplings Y 0
³
~S
2 ¡ V 2
´
we get VEV solution
for ~S
given in (4.42) and hY 0i = 0. We just mentioned this simple minded examples
in order to demonstrate that desirable VEVs can be obtained self-consistently (of
course many other possibilities can be discussed).
Further we will use the following parametrization
VX
MPl
»
V
MPl
´ ² : (4.43)
All non renormalizable operators that we consider below will be cut o® by appropriate
powers of the Planck scale MPl and therefore in those ope