# Restricted Permutations Enumerated by Inversions

Atli Fannar Franklín Department of Mathematics
University of Iceland
Reykjavík, Iceland<EMAIL_ADDRESS>Department of Mathematics
University of Iceland
Reykjavík, IcelandSchool of Computer Science and Mathematics
Keele University
Keele, United KingdonDepartment of Computer Science
Reykjavík University
Reykjavík, IcelandDepartment of Mathematical and Statistical Sciences
Marquette University
Milwaukee, WI, USA Anders Claesson Department of Mathematics
University of Iceland
Reykjavík, Iceland<EMAIL_ADDRESS>School of Computer Science and Mathematics
Keele University
Keele, United KingdonDepartment of Computer Science
Reykjavík University
Reykjavík, IcelandDepartment of Mathematical and Statistical Sciences
Marquette University
Milwaukee, WI, USA Christian Bean School of Computer Science and Mathematics
Keele University
Keele, United Kingdon<EMAIL_ADDRESS>Department of Computer Science
Reykjavík University
Reykjavík, IcelandDepartment of Mathematical and Statistical Sciences
Marquette University
Milwaukee, WI, USA Henning Úlfarsson Department of Computer Science
Reykjavík University
Reykjavík, Iceland<EMAIL_ADDRESS>Department of Mathematical and Statistical
Sciences
Marquette University
Milwaukee, WI, USA Jay Pantone Department of Mathematical and Statistical
Sciences
Marquette University
Milwaukee, WI, USA<EMAIL_ADDRESS>
###### Abstract

Permutations are usually enumerated by size, but new results can be found by
enumerating them by inversions instead, in which case one must restrict one’s
attention to indecomposable permutations. In the style of the seminal paper by
Simion and Schmidt [7], we investigate all combinations of permutation
patterns of length at most $3$.

##  1 Introduction

To enumerate permutations by their number of inversions we need the set of
permutations with a given number of inversions to be finite. This is generally
not the case, since in many cases one can add a new maximal element to the end
of the permutation to get a new one with equally many inversions. To get
around this we introduce the notion of decomposability. We say that a
permutation $\pi=\pi_{1}\dots\pi_{n}$ is decomposable if there exists an index
$i<n$ such that $\pi_{1}\dots\pi_{i}$ is a permutation of the elements
$1,\dots,i$. If we let $\rho=\pi_{1}\dots\pi_{i}$ and
$\tau=(\pi_{i+1}-i)\dots(\pi_{n}-i)$ we denote this $\pi=\rho\oplus\tau$. In
the same vein, permutations that are not decomposable are called
indecomposable. With this definition every permutation can be factored into a
set of indecomposable factors, and we call these factors its components.

An inversion in a permutation $\pi=\pi_{1}\dots\pi_{n}$ is a pair of indices
$(i,j)$ such that $i<j$ and $\pi_{i}>\pi_{j}$. The number of inversions in a
permutation $\pi$ will be denoted $\operatorname{inv}(\pi)$. The following
lemma highlights a relation between these two concepts.

###### Lemma 1

Let $\pi$ be a permutation on $n$ elements and $c$ components. Then
$\operatorname{inv}(\pi)\geq n-c$.

Crucially this means that if $c=1$ then $\operatorname{inv}(\pi)\geq n-1$.
Thus the set of indecomposable permutations with $k$ inversions is finite
since it is contained in the set of all permutations on $k+1$ elements or
fewer. With this in mind, we define $I_{k}$ to be the set of all
indecomposable permutations with exactly $k$ inversions.

Next we recall some notions related to patterns of permutations. We say two
sequences $a_{1},\dots,a_{m}$ and $b_{1},\dots,b_{m}$ are order-isomorphic if
$a_{i}<a_{j}$ holds if and only if $b_{i}<b_{j}$. For permutations
$\pi=\pi_{1}\dots\pi_{n}$, $\tau=\tau_{1}\dots\tau_{m}$ we say that $\pi$
contains the pattern $\tau$ if there exists a set of indices
$i_{1},\dots,i_{m}$ such that $\pi_{i_{1}}\dots\pi_{i_{m}}$ is order-
isomorphic to $\tau_{1}\dots\tau_{m}$. We call such a set of indices an
_occurrence_ of the pattern $\tau$. If $\pi$ has no occurrence of $\tau$ we
will say that $\pi$ _avoids_ the pattern $\tau$. We will denote the subset of
$I_{k}$ containing the permutations avoiding $\tau$ by $I_{k}(\tau)$.
Similarly we will denote the subset of $I_{k}$ containing permutations
avoiding several patterns $\tau_{1},\tau_{2},\dots,\tau_{r}$ by
$I_{k}(\tau_{1},\tau_{2},\dots,\tau_{r})$.

##  2 Single patterns

As examples we have that $I_{k}(1)$ and $I_{k}(21)$ have no non-empty elements
and $|I_{k}(12)|$ is the characteristic function of the triangular numbers,
listed in the OEIS as A010054. The remaining single patterns of length $3$ are
then $123,132,213,231,312$ and $321$. We consider the reverse complement of a
permutation: written out explicitly, this is
$\pi_{1}\dots\pi_{n}\mapsto(n+1-\pi_{n})(n+1-\pi_{n-1})\dots(n+1-\pi_{1})$.
Consider an inversion on $i<j$ in $\pi$. The values $\pi_{i}>\pi_{j}$ get
mapped to $n+1-\pi_{i},n+1-\pi_{j}$ at indices $n+1-i,n+1-j$ in the image.
Thus the size of the elements is inverted, but so is their order. Therefore
the number of inversions remains constant under reverse complement. We also
see that $\pi$ is decomposable if and only if its reverse complement is, so
the reverse complement is an involution on $I_{k}$ for every $k$. It maps the
pattern $132$ to the pattern $213$, so from this we see that
$|I_{k}(213)|=|I_{k}(132)|$, hence we only have to consider one of these
patterns. This is similar to what happens when counting pattern avoiding
permutations classically, however we must compose reversion and complement for
the argument to work. Normally this line of reasoning shows that the number of
$132$ and $231$ avoiding permutations are the same through reversion, but we
will see that this is no longer the case when enumerating by inversions. A
similar argument can be made to show that mapping to the inverse permutation
preserves inversions and indecomposability as well, and in fact the symmetries
are generated by these two maps.

This means we only have to consider the patterns $123,132,231$ and $321$. We
will start with $132$. To do this, we make use of a well known bijection on
permutations. The inversion table of a permutation $\pi=\pi_{1}\dots\pi_{n}$
is the sequence $b_{1}b_{2}\dots b_{n}$, where $b_{i}$ is the number of values
after $\pi_{i}$ in $\pi$ that are smaller than $\pi_{i}$. The image of this
bijection is given by the set of all sequences such that the first value is
$\leq n-1$, the next $\leq n-2$ and so on. We call such sequences subdiagonal.
Furthermore, elements in a subdiagonal sequence that are equal to their
maximum possible value are called diagonal elements.

###### Theorem 2

$|I_{k}(132)|$ counts partitions on $k$ elements, listed as A000041 on the
OEIS.

To work with $231$-avoiding permutations we define the _skew-sum_ of two
permutations $\pi_{1}\dots\pi_{n}$ and $\tau_{1}\dots\tau_{m}$, as the
permutation on $n+m$ elements given by
$\pi\ominus\tau=(\pi_{1}+m)\dots(\pi_{n}+m)\tau_{1}\dots\tau_{m}$. We will say
that $\pi$ is skew-decomposable if it can be written as
$\pi=\tau_{1}\ominus\tau_{2}$ for non-empty $\tau_{1},\tau_{2}$, and skew-
indecomposable otherwise.

###### Theorem 3

$|I_{k}(231)|$ counts fountains on $k$ coins. A fountain of coins is an
arrangement of coins in rows such that the bottom row is full (that is, there
are no “holes”), and such that each coin in a higher row rests on two coins in
the row below. This is listed as A005169 on the OEIS.

Enumerating $321$-avoiding permutations by size and inversions (allowing
decomposable permutations) has been investigated in [3]. A generating function
is derived, but as we need a bijective map from $I_{k}(321)$ to our target set
for what comes later, we will have to give a different proof.

###### Theorem 4

$|I_{k}(321)|$ counts parallelogram polyominoes with $k$ cells, listed as
A006958 on the OEIS.

This alternative view on parallelogram polyominoes opens up a formula for
efficiently computing new terms of the series, taking $\mathcal{O}(k^{2})$
time and space to compute $|I_{k}(321)|$.

###### Theorem 5

$|I_{k}(321)|=a_{k,1}$ where

$a_{n,m}=\begin{cases}1\text{ if }n=0\\\ \sum_{i=1}^{n}a_{n-i,i}\text{ if
}m=1\\\ a_{n,m-1}+\sum_{i=m}^{n}a_{n-i,i}\text{ otherwise}\\\ \end{cases}$

Not only can this view help with computing new terms, but it also gives rise
to new bijective correspondences. Consider fountains of coins where we only
count coins in even rows. We can still place coins as we like, but when
tallying the number we only count those in the bottom row, those 2 rows up, 4
rows up and so on. Such fountains with $n$ counted coins will be called even
fountains of size $n$. In [2] the problem of mapping parallelogram polyominoes
to such fountains is tackled. It is shown that there are equally many by an
algebraic argument, but it is left as an open question at the end whether
there is any bijective proof. Using $I_{k}(321)$, a bijective proof can be
found. Since Theorem 4 is proved by mapping parallelogram polyominoes
bijectively to $I_{k}(321)$, it suffices to map $I_{k}(321)$ bijectively to
even fountains of size $k$.

###### Theorem 6

$I_{k}(321)$ maps bijectively to even fountains of size $k$.

The proof of Theorem 5 is bijective, so it suffices to map the even fountains
to the sequences described there. We will describe a map taking even fountains
to such sequences.

Write out the fountain in the usual manner (see picture), calling coins in
even rows red and the ones in odd rows black for convenience. For each coin in
the bottom row, going from left to right, we do the following procedure
repeatedly:

  * •

If we are on a red coin, we remove it and move to the coin above and to the
right. If there is no such coin we stop.

  * •

If we are on a black coin, we remove it and move to the coin above and to the
right. If there is no such coin we move to the coin below and to the right
instead.

Once done with a coin in the bottom row, we write down the number of red coins
removed during this procedure. This produces a sequence of numbers, which we
then append a single zero to. We claim this produces a sequence of the desired
form.

$2$$3$$3$$0$$0$$1$$2$$0$

This leaves us only with $I_{k}(123)$, the only single pattern giving rise to
a sequence not in the OEIS.

###### Theorem 7

$|I_{k}(123)|$ counts indecomposable subdiagonal sequences where non-diagonal
elements are in decreasing order. Let

$c_{n,m,k}=\begin{cases}0\text{ if }n<0\text{ or }k<0\\\ 1\text{ if }n=k=0\\\
c_{n-1,m,l-n+1}+\sum_{i=0}^{\min(n-2,m-1)}c_{n-1,i,k-i}\text{ otherwise
}\end{cases}$

Then the total number of permutations with $k$ inversions (including
decomposable ones) avoiding $123$ can be calculated as
$\sum_{n=0}^{k+1}c_{n,n,k}$.

To obtain the number of such permutations that are indecomposable, subtract
the $k$-th coefficient of $\left(\sum_{i\geq 0}x^{i(i+1)/2}\right)^{2}$.

##  3 Several patterns

We now investigate permutations avoiding several patterns. Some groups of
patterns are restrictive enough to make all supersets of those patterns
trivially determined. For example $|I_{k}(123,321)|$ quickly decays to zero by
the Erdös-Szekeres theorem, making all supersets of $123,321$ easy to
determine. We have two more such pairs of patterns.

###### Theorem 8

$|I_{k}(231,321)|=1$ and the unique permutation with $k$ inversions is
$k12\dots(k-1)$.

###### Theorem 9

$|I_{k}(231,312)|=|I_{k}(12)|$.

By utilizing the symmetries we have, the only pairs of patterns left to
investigate are $123,231$ and pairs containing $132$.

###### Theorem 10

$|I_{k}(123,231)|$ counts fountains of $k$ coins where the missing coins with
respect to a full triangular fountain form a rectangle (removing no coins
counts as a rectangle). The generating function is given by $\sum_{i\geq
1}x^{\binom{i}{2}}+\sum_{i\geq 1}\sum_{j\geq
1}\sum_{\ell=0}^{\min(i,j)-1}x^{\binom{i+1}{2}+\binom{j+1}{2}-\binom{\ell+1}{2}}$.

This leaves us with four pairs, pairing $132$ with any of the patterns
$123,213,231$ and $321$. We now tackle them in that order.

###### Theorem 11

$|I_{k}(132,123)|$ enumerates the Pascal triangle with the first column
removed, which is listed as A135278 on the OEIS. It has generating function
$\sum_{n\geq 0}x^{n(n+3)/2}\left((x+1)^{n+2}-x^{n+2}\right)$.

Our next result involves a kind of partition called a Gorenstein partition.
Gorenstein partitions are partitions whose maximal chains are all of the same
size when regarded as order ideals of $\\{1,2,\dots\\}\times\\{1,2,\dots\\}$.
This definition is rather unwieldy for our purposes, so we first translate
this condition.

###### Lemma 12

A partition $\rho$ is Gorenstein if and only if $\rho_{i}+i$ is constant
across the indices $i$ that satisfy $\rho_{i}\neq\rho_{i+1}$, letting
$\rho_{|\rho|}=0$.

###### Theorem 13

$|I_{k}(132,213)$— counts Gorenstein partitions of $k$. This is listed as
A117629 on the OEIS.

###### Theorem 14

Let $\mu\vDash s$ denote that $\mu$ is a composition of $s$. Then
$|I_{k}(132,213)|$ has generating function $\sum_{s\geq 0}\sum_{\mu\vDash
s,|\mu|\neq 1}x^{\binom{s}{2}-\sum_{m\in\mu}\binom{m}{2}}$. This means
$|I_{k}(132,213)|$ also enumerates finite sequences of positive integers of
length $>1$ such that $k$ equals the second elementary symmetric function of
the values of the sequence, as noted in the OEIS entry.

The generating function is not useful for actually computing new terms in the
sequence, but we can use the following recurrence instead. By ignoring all but
the first and last $\sqrt{n}$ summands in the recurrence below, as they are
zero, we can compute the $n$-th value in $\mathcal{O}(n^{2.5})$ time and
$\mathcal{O}(n^{2})$ space.

###### Theorem 15

The number of Gorenstein partitions with sum $n$, and thus also the number of
elements in $|I_{n}(132,213)|$, is given by the sum $\sum_{d=0}^{n}f(n,d)$
where

$f(n,d)=\begin{cases}0\text{ if }n<0\\\ 1\text{ if }n=0\\\
\sum_{k=1}^{d}f(n-k(d+1-k),d-k)\text{ otherwise }\end{cases}$

###### Theorem 16

$|I_{k}(132,231)|$ counts partitions on $k$ elements with distinct parts, this
is listed as A000009 on the OEIS.

###### Theorem 17

$|I_{k}(132,321)|$ counts partitions on $k$ elements with equal values. This
is in turn equal to the number of divisors of $k$. This is listed as A000005
on the OEIS.

##  4 More than two patterns

Most of the remaining pattern combinations are trivially deduced as some
subset of the patterns forces the sequence to die out or contain only very
specific permutations. We consider here the complement of those cases.

###### Theorem 18

$|I_{k}(123,132,231)|=1$.

###### Theorem 19

$|I_{k}(123,132,213)|$ enumerates the Pascal triangle, read by diagonals,
offset by two elements. This means it reads the binomials $\binom{n}{k}$ in
increasing order by the sum $n+k$, with each set being read in increasing
order by $n$, and $|I_{0}(123,132,213)|$ starts at $\binom{1}{1}$. Furthermore
its generating function can be written as $1+\sum_{d\geq
3}x^{\binom{d-1}{2}}\sum_{n=2}^{d}\binom{n}{d-n}x^{n-2}$.

###### Theorem 20

$|I_{k}(132,213,231)|$ counts the odd divisors of $k$, which is listed as
A001227 on the OEIS.

###### Theorem 21

$|I_{k}(132,213,321)|=|I_{k}(132,321)|$.

###### Theorem 22

$|I_{k}(123,132,213,231)|$ enumerates the Pascal triangle, except all values
$>1$ are replaced by $0$. This is listed as on A103451 on the OEIS and has the
generating function $\sum_{i\geq 0}x^{i(i+1)/2}+x^{(i+1)(i+4)/2}$.

## Acknowledgements

The algorithm in [5] was used to generate elements of all the sequences above,
which helped tremendously in finding the formulas and other results in this
paper.

## References

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Domain walls and CP violation with left right supersymmetry :

implications for leptogenesis and electron EDM

Piyali Banerjee111<EMAIL_ADDRESS>and Urjit A. Yajnik222
<EMAIL_ADDRESS>
Department of Physics, Indian Institute of Technology Bombay, Mumbai 400076,
India

Abstract

Low scale leptogenesis scenarios are difficult to verify due to our inability
to relate the parameters involved in the early universe processes with the low
energy or collider observables. Here we show that one can in principle relate
the parameters giving rise to the transient $CP$ violating phase involved in
leptogenesis with those that can be deduced from the observation of electric
dipole moment (EDM) of the electron. We work out the details of this in the
context of the left right symmetric supersymmetric model (LRSUSY) which
provides a strong connection between such parameters. In particular, we show
that baryon asymmetry requirements imply the scale $M_{B-L}$ of $U(1)_{B-L}$
symmetry breaking to be larger than $10^{4.5}~{}\mathrm{GeV}$. Moreover the
scale $M_{R}$ of $\mathrm{SU}(2)_{R}$ symmetry breaking is tightly constrained
to lie in a narrow band significantly below $M_{B-L}^{2}/M_{EW}$. These are
the most stringent constraints on the parameter space of LRSUSY model being
considered.

PACS Numbers:

##  I Introduction

One of the three necessary Sakharov conditions for dynamical generation of
baryon asymmetry of the universe is CP violation Sakharov . The Standard Model
(SM) of particle physics has the ingredients to satisfy all the three Sakharov
conditions and could, in principle, generate some baryon asymmetryKuzmin .
However the requirement of a first order Electroweak Phase Transition (EWPT)
requires the mass of the Higgs to remain less than about 80 GeV Kajantie1 ;
Kajantie2 ; Csikor  which is far below the mass of the recently discovered SM
Higgs boson Exptmass1 ; Exptmass2 . Furthermore the CP violation in the CKM
matrix of SM is too small to produce a reasonably large baryon asymmetry
Gavela .

The left right symmetric model of Mohapatra and Senjanović MohapatraLR  (LRSM)
is a minimal extension of the Standard Model based on the gauge group
$\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}\times\mathrm{U}(1)_{B-L}$
augmented with the discrete $Z_{2}$ left-right symmetry. The usual
$\mathrm{SU}(2)_{L}$ Higgs doublet of SM is extended to a
$\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}$ bidoublet. The model naturally
accommodates the parity violation of SM as a result of spontaneous symmetry
breaking of $\mathrm{SU}(2)_{R}\times\mathrm{U}(1)_{B-L}$ and also elegantly
explains small neutrino masses via the seesaw mechanism. Both goals are
achieved by adding new heavy Higgs $\mathrm{SU}(2)$ triplets in the theory.

The supersymmetric extension of LRSM, which we call the Simplified Left Right
Symmetric Supersymmetric Standard Model (SLRSUSY), marries the desirable
features of both LRSM and supersymmetry. However it suffers from the drawback
that spontaneous parity violation cannot take place without violating R-parity
also Kuchimanchi . To remedy this, one can take the supersymmetry breaking
scale $M_{S}$ to be larger than the
$\mathrm{SU}(2)_{R}\times\mathrm{U}(1)_{B-L}$ breaking scale $M_{B-L}$ but
that would lose out on several desirable features of supersymmetry.
Alternatively, one can introduce additional heavy Higgs $\mathrm{SU}(2)$
triplets with appropriate gauge group charges. This was done in the two papers
Aulakh:1997ba ; Aulakh:1997fq , which we shall call the Left Right Symmetric
Supersymmetric Standard Model (LRSUSY). The LRSUSY model contains two distinct
high mass scales, $M_{R}$ the scale of $\mathrm{SU}(2)_{R}$ symmetry breaking,
and $M_{B-L}$ the scale of $U(1)_{B-L}$ symmetry breaking satisfying $M_{R}\gg
M_{B-L}$, an intermediate SUSY breaking scale $M_{S}$ and a low mass scale
$M_{H}$ where the bidoublet Higgs get vevs. Thus, the LRSUSY model has several
desirable features not possessed by SLRSUSY like spontaneous parity violation
without R-parity violation, low scale supersymmetry with exact R-parity and
existence of a stable lightest supersymmetric particle.

Breaking of the $Z_{2}$ discrete symmetry of left-right symmetric models in
the early universe ensures the occurrence of domain walls, but also begs for
the $Z_{2}$ symmetry to be not exact to avoid conflict with the observed
homogeneous universe. However an approximate $Z_{2}$ is adequate to generate
the domain walls which are a robust topological prediction independent of
details of parameters, and can play the same role as the phase transition
bubble walls for the purpose of leptogenesis.

In the context of non-supersymmetric LRSM, fate of the $CP$ violating phase
was studied in ClineYajnik  for the purpose of explaining leptogenesis. It was
shown that a spatially varying CP violating phase occurs inside the domain
walls separating the left handed and right handed domains. They showed that in
order to explain the observed baryon asymmetry of the universe, the Yukawa
coupling of the right handed neutrino to the $\mathrm{SU}(2)$-triplet Higgs
must be larger than $10^{-2}$. They also obtained some heuristic constraints
on the mass of the left handed neutrinos or alternatively, the temperature
scale of LR symmetry breaking. Since the LRSUSY model has additional Higgs
bosons as well as additional CP violating phases, it may be able to generate
the necessary conditions for successful baryogenesis in the early universe.
The first step in this direction was taken in Anjishnu  by showing the
possibility of having a spatially varying CP violating phase in the domain
walls in the context of LRSUSY. However they did not provide any quantitative
estimates and left open the question of whether LRSUSY is actually capable of
generating the baryon asymmetry of the universe.

The core idea of such proposals is that the spatially varying phase implies a
spatially varying complex mass for left handed neutrinos inside the wall.
Studying the diffusion equation for lepton number density with spatially
varying complex mass results in the preferential transmission of left handed
neutrinos across a slowly moving thick domain wall. The moving wall encroaches
upon the energetically disfavoured right handed domain. We solve the diffusion
equation numerically for various wall speeds and thicknesses, obtaining
excesses of almost massless left handed neutrinos inside the left handed
domain. After the wall disappears, electroweak sphaelerons convert a part of
the neutrino excess to baryon excess. We calculate the amount of baryon excess
that survives the washout processes. Requiring the surviving baryon excess to
be not much above the experimental limit of around $6\times 10^{-10}$ for the
baryon asymmetry to entropy ratio Canetti  allows us to constrain the
$(M_{R},M_{B-L})$ parameter space of LRSUSY.

A smoking gun signature of CP violation in a theory is the presence of a non-
zero EDM of the electron and neutron. The Standard Model predicts a non-zero
electron EDM at the three loop level but the effect is estimated to be very
small, around $1.9\times 10^{-39}~{}\mathrm{e~{}cm}$ YamaguchiYamanaka . This
is way below the current experimental upper bound of $1.1\times
10^{-29}~{}\mathrm{e~{}cm}$ obtained by the ACME II experiment ACME2 . Left
right symmetric theories contain many additional sources of CP violation as
compared to the Standard Model, and so predict larger electron and neutron
EDMs. Thus the experimental bound on EDMs serve to constrain the parameters of
these theories. Electron and neutron EDMs in the non-supersymmetric LRSM have
been studied in several earlier works e.g. Nieves ; Xu , obtaining lower
bounds on the scale $M_{B-L}$ of $\mathrm{SU}(2)_{R}\times U(1)_{B-L}$
symmetry breaking and on the mass $M_{H^{\prime}}$ of the heavier scalar Higgs
in the two Higgs doublets arising from breaking of left right symmetry in the
bidoublet Higgs of LRSM. Electron and neutron EDMs have been studied in the
SLRSUSY model by FrankE ; FrankN , obtaining bounds on the masses of certain
superpartners and some other parameters of SLRSUSY.

The LRSUSY modelAulakh:1997fq  contains essentially only one major unknown,
the $M_{R}$ scale and a trilinear Higgs coupling parameter $\alpha$ that is
used to ensure that only two out of four SM type Higgs doublets arising after
$\mathrm{SU}(2)_{R}\times\mathrm{U}(1)_{B-L}$ symmetry breaking remain
lighter, with a relative phase between their vacuum expectation values. In
this paper we study the CP violating phase of the bidoublet fields in LRSUSY
by setting up the domain wall solutions. The parameters relevant for the
domain wall solutions are understood to involve the high temperature
corrections needed in the early Universe, at the temperature $T_{B-L}\sim
M_{B-L}$. We separately investigate the contribution of the lighter two mass
eigenstates of the bidoublets to the electron EDM at zero temperature, which
differ only by the $T_{B-L}$ corrections. We compute the contribution of this
low energy phase to the EDM at one loop and two loop levels as a function of
$\alpha$. The two loop computation follows along the lines of the seminal work
of Barr and Zee BarrZee  on the electron EDM in multi Higgs doublet models. A
similar calculation of electron EDM arising as a residual effect of domain
wall collapse in two Higgs doublet models was done in Chen . It turns out that
successful leptogenesis in LRSUSY requires $\alpha\gtrsim 0.1$. Combining this
with the requirement that the EDM obtained be less than the experimental limit
of $1.1\times 10^{-29}~{}\mbox{e cm}$, we get an allowed region in the
$(M_{B-L},M_{R})$-parameter space of LRSUSY.

It turns out that the limit on the parameter space arising from baryon
asymmetry is more stringent than the limit from electron EDM. Further
requiring that the observed baryon asymmetry be explained to within an order
of magnitude by LRSUSY puts stringent constraints on the
$(M_{B-L},M_{R})$-parameter space of LRSUSY. In particular
$M_{B-L}<10^{4.5}~{}\mathrm{GeV}$ is ruled out. These are the most stringent
constraints on the parameter space of LRSUSY by far.

##  II The LRSUSY model revisited

The gauge group of LRSUSY is the left-right symmetric group
$\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}\times\mathrm{U}(1)_{B-L}$.
The Higgs sector consists of two Higgs bidoublets $\Phi_{1}$ and $\Phi_{2}$,
three left handed Higgs triplets $\Delta$, $\bar{\Delta}$ and $\Omega$, and
three right handed Higgs triplets $\Delta_{c}$, $\bar{\Delta}_{c}$ and
$\Omega_{c}$. Following Kuchimanchi-Mohapatra Kuchimanchi , the gauge group
charges are

$\begin{array}[]{c c l c c l c c
l}\Phi_{1}&=&(1,2,2^{*},0),&\Phi_{2}&=&(1,2,2^{*},0),&\\\
\Omega&=&(1,3,1,0),&\Delta&=&(1,3,1,2),&\bar{\Delta}&=&(1,3,1,-2),\\\
\Omega_{c}&=&(1,1,3^{*},0),&\Delta_{c}&=&(1,1,3^{*},-2),&\bar{\Delta}_{c}&=&(1,1,3^{*},2).\end{array}$
(1)

In other words, the group action is given by

$\\{\Phi_{1},\Phi_{2}\\}\rightarrow
U_{L}\\{\Phi_{1},\Phi_{2}\\}U_{R}^{\dagger},~{}~{}~{}\\{\Delta,\bar{\Delta},\Omega\\}\rightarrow
U_{L}\\{\Delta,\bar{\Delta},\Omega\\}U_{L}^{\dagger},~{}~{}~{}\\{\Delta_{c},\bar{\Delta}_{c},\Omega_{c}\\}\rightarrow
U_{R}^{*}\\{\Delta_{c},\bar{\Delta}_{c},\Omega_{c}\\}U_{R}^{T}.$ (2)

The electric charge is given by $Q=T_{3L}+T_{3R}+\frac{B-L}{2}.$ From Aulakh
et al. Aulakh:1997fq  the Higgs part of the superpotential is given by

$\begin{array}[]{rcl}W&=&m_{\Delta}(\mathrm{Tr}\,\Delta\bar{\Delta}+\Delta_{c}\bar{\Delta}_{c})+\frac{m_{\Omega}}{2}(\mathrm{Tr}\,\Omega^{2}+\mathrm{Tr}\,\Omega_{c}^{2})+\mu_{ij}\mathrm{Tr}\,\tau_{2}\Phi_{i}^{T}\tau_{2}\Phi_{j}\\\
&&{}+a(\mathrm{Tr}\,\Delta\Omega\bar{\Delta}+\mathrm{Tr}\,\Delta_{c}\Omega_{c}\bar{\Delta}_{c})+\alpha_{ij}(\mathrm{Tr}\,\Omega\Phi_{i}\tau_{2}\Phi_{j}^{T}\tau_{2}+\mathrm{Tr}\,\Omega_{c}\Phi_{i}^{T}\tau_{2}\Phi_{j}\tau_{2}),\end{array}$
(3)

where $\alpha_{ij}$, $\mu_{ij}$ are complex numbers satisfying
$\mu_{12}=\mu_{21}$, $\alpha_{ij}=-\alpha_{ji}$. Define $\alpha=\alpha_{12}$.
Then $\alpha_{21}=-\alpha$ and $\alpha_{11}=\alpha_{22}=0$.

The charge zero condition forces the vevs of the Higgs fields to be

$\begin{array}[]{c c l c c l c c
l}\langle\Phi_{1}\rangle&=&\left(\begin{array}[]{c c}k_{1}&0\\\
0&k_{1}^{\prime}\end{array}\right),&\langle\Phi_{2}\rangle&=&\left(\begin{array}[]{c
c}k_{2}&0\\\ 0&k_{2}^{\prime}\end{array}\right),&\\\
\langle\Omega\rangle&=&\left(\begin{array}[]{c c}\omega&0\\\
0&-\omega\end{array}\right),&\langle\Delta\rangle&=&\left(\begin{array}[]{c
c}0&0\\\
d&0\end{array}\right),&\langle\bar{\Delta}\rangle&=&\left(\begin{array}[]{c
c}0&\bar{d}\\\ 0&0\end{array}\right),\\\
\langle\Omega_{c}\rangle&=&\left(\begin{array}[]{c c}\omega_{c}&0\\\
0&-\omega_{c}\end{array}\right),&\langle\Delta_{c}\rangle&=&\left(\begin{array}[]{c
c}0&0\\\
d_{c}&0\end{array}\right),&\langle\bar{\Delta}_{c}\rangle&=&\left(\begin{array}[]{c
c}0&\bar{d}_{c}\\\ 0&0\end{array}\right),\end{array}$ (4)

where the quantities above are in general complex numbers.

The D-terms are given by (where $m=1,2,3$ refer to the three generators of
$\mathrm{SU}(2)$) Kuchimanchi

$\begin{array}[]{rcl}D_{L,m}&=&\mathrm{Tr}\,(2\Omega^{\dagger}\tau_{m}\Omega+2\Delta^{\dagger}\tau_{m}\Delta+2\bar{\Delta}^{\dagger}\tau_{m}\bar{\Delta}+\Phi_{1}^{\dagger}\tau_{m}\Phi_{1}+\Phi_{2}^{\dagger}\tau_{m}\Phi_{2})\\\
D_{R,m}&=&\mathrm{Tr}\,(2\Omega_{c}^{\dagger}\tau_{m}\Omega_{c}+2\Delta_{c}^{\dagger}\tau_{m}\Delta_{c}+2\bar{\Delta}_{c}^{\dagger}\tau_{m}\bar{\Delta}_{c}+\Phi_{1}\tau_{m}^{T}\Phi_{1}^{\dagger}+\Phi_{2}\tau_{m}^{T}\Phi_{2}^{\dagger})\\\
D_{B-L}&=&\mathrm{Tr}\,(2\Delta^{\dagger}\Delta-2\bar{\Delta}^{\dagger}\bar{\Delta}-2\Delta_{c}^{\dagger}\Delta_{c}+2\bar{\Delta}_{c}^{\dagger}\bar{\Delta}_{c}).\end{array}$
(5)

After substituting the vevs into the D-term expressions we get

$\begin{array}[]{rcl}\langle
D_{B-L}\rangle&=&2(|d|^{2}-|\bar{d}|^{2}-|d_{c}|^{2}+|\bar{d}_{c}|^{2}),\\\
\langle D_{L,1}\rangle&=&\langle D_{L,2}\rangle=\langle D_{R,1}\rangle=\langle
D_{R,2}\rangle=0,\\\ \langle
D_{L,3}\rangle&=&2(-|d|^{2}+|\bar{d}|^{2})+|k_{1}|^{2}-|k_{1}^{\prime}|^{2}+|k_{2}|^{2}-|k_{2}^{\prime}|^{2},\\\
\langle
D_{R,3}\rangle&=&2(-|d_{c}|^{2}+|\bar{d}_{c}|^{2})+|k_{1}|^{2}-|k_{1}^{\prime}|^{2}+|k_{2}|^{2}-|k_{2}^{\prime}|^{2}.\end{array}$
(6)

Taking $|d|=|\bar{d}|$ and $|d_{c}|=|\bar{d}_{c}|$ ensures the vanishing of
$D_{B-L}$-term always. We can use the $B-L$ gauge invariance to ensure that
$d$, $\bar{d}$ have the same complex phase. Subsequently using
$\mathrm{SU}(2)_{L}$ invariance, we can ensure that $d=\bar{d}$ and real
positive.

Extending Aulakh et al, we see that the resulting F-terms are:

$\begin{array}[]{rcl}F_{\bar{\Delta}}&=&m_{\Delta}\Delta+a(\Delta\Omega-\frac{\mathrm{Tr}\,\Delta\Omega}{2}),\\\
F_{\Delta}&=&m_{\Delta}\bar{\Delta}+a(\Omega\bar{\Delta}-\frac{\mathrm{Tr}\,\Omega\bar{\Delta}}{2}),\\\
F_{\Omega}&=&m_{\Omega}\Omega+a(\bar{\Delta}\Delta-\frac{\mathrm{Tr}\,\bar{\Delta}\Delta}{2})+\alpha(\Phi_{1}\tau_{2}\Phi_{2}^{T}\tau_{2}-\Phi_{2}\tau_{2}\Phi_{1}^{T}\tau_{2}),\\\
F_{\bar{\Delta_{c}}}&=&m_{\Delta}\Delta_{c}+a(\Delta_{c}\Omega_{c}-\frac{\mathrm{Tr}\,\Delta_{c}\Omega_{c}}{2}),\\\
F_{\Delta_{c}}&=&m_{\Delta}\bar{\Delta_{c}}+a(\Omega_{c}\bar{\Delta_{c}}-\frac{\mathrm{Tr}\,\Omega_{c}\bar{\Delta_{c}}}{2}),\\\
F_{\Omega_{c}}&=&m_{\Omega}\Omega_{c}+a(\bar{\Delta_{c}}\Delta_{c}-\frac{\mathrm{Tr}\,\bar{\Delta_{c}}\Delta_{c}}{2})+\alpha(\Phi_{1}^{T}\tau_{2}\Phi_{2}\tau_{2}-\Phi_{2}^{T}\tau_{2}\Phi_{1}\tau_{2}),\\\
F_{\Phi_{1}}&=&2\mu_{11}\tau_{2}\Phi_{1}^{T}\tau_{2}+2\mu_{12}\tau_{2}\Phi_{2}^{T}\tau_{2}+\alpha(\tau_{2}\Phi_{2}^{T}\tau_{2}\Omega-\tau_{2}\Phi_{2}^{T}\Omega^{T}\tau_{2}+\tau_{2}\Omega_{c}\Phi_{2}^{T}\tau_{2}-\Omega_{c}^{T}\tau_{2}\Phi_{2}^{T}\tau_{2}),\\\
F_{\Phi_{2}}&=&2\mu_{12}\tau_{2}\Phi_{1}^{T}\tau_{2}+2\mu_{22}\tau_{2}\Phi_{2}^{T}\tau_{2}-\alpha(\tau_{2}\Phi_{1}^{T}\tau_{2}\Omega-\tau_{2}\Phi_{1}^{T}\Omega^{T}\tau_{2}+\tau_{2}\Omega_{c}\Phi_{1}^{T}\tau_{2}-\Omega_{c}^{T}\tau_{2}\Phi_{1}^{T}\tau_{2}).\end{array}$
(7)

After substituting the vevs, the expressions for the F-terms become

$\begin{array}[]{rcl}\langle F_{\bar{\Delta}}\rangle&=&\left(\begin{array}[]{c
c}0&0\\\ d(m_{\Delta}+a\omega)&0\end{array}\right),\\\ \langle
F_{\Delta}\rangle&=&\left(\begin{array}[]{c c}0&\bar{d}(m_{\Delta}+a\omega)\\\
0&0\end{array}\right),\\\ \langle F_{\Omega}\rangle&=&\left(\begin{array}[]{c
c}m_{\Omega}\omega+\frac{ad\bar{d}}{2}+\alpha(k_{1}k_{2}^{\prime}-k_{1}^{\prime}k_{2})&0\\\
0&-(m_{\Omega}\omega+\frac{ad\bar{d}}{2}+\alpha(k_{1}k_{2}^{\prime}-k_{1}^{\prime}k_{2}))\end{array}\right),\\\
\langle F_{\bar{\Delta}_{c}}\rangle&=&\left(\begin{array}[]{c
c}0&d_{c}(m_{\Delta}+a\omega_{c})\\\ 0&0\end{array}\right),\\\ \langle
F_{\Delta_{c}}\rangle&=&\left(\begin{array}[]{c c}0&0\\\
\bar{d}_{c}(m_{\Delta}+a\omega_{c})&0\end{array}\right),\\\ \langle
F_{\Omega_{c}}\rangle&=&\left(\begin{array}[]{c
c}m_{\Omega}\omega_{c}+\frac{ad_{c}\bar{d}_{c}}{2}+\alpha(k_{1}k_{2}^{\prime}-k_{1}^{\prime}k_{2})&0\\\
0&-(m_{\Omega}\omega_{c}+\frac{ad_{c}\bar{d}_{c}}{2}+\alpha(k_{1}k_{2}^{\prime}-k_{1}^{\prime}k_{2}))\end{array}\right),\\\
\langle F_{\Phi_{1}}\rangle&=&\left(\begin{array}[]{c
c}2\mu_{11}k_{1}^{\prime}+2\mu_{12}k_{2}^{\prime}+2\alpha
k_{2}^{\prime}(\omega-\omega_{c})&0\\\ 0&2\mu_{11}k_{1}+2\mu_{12}k_{2}-2\alpha
k_{2}(\omega-\omega_{c})\\\ \end{array}\right),\\\ \langle
F_{\Phi_{2}}\rangle&=&\left(\begin{array}[]{c
c}2\mu_{12}k_{1}^{\prime}+2\mu_{22}k_{2}^{\prime}-2\alpha
k_{1}^{\prime}(\omega-\omega_{c})&0\\\ 0&2\mu_{12}k_{1}+2\mu_{22}k_{2}+2\alpha
k_{1}(\omega-\omega_{c})\\\ \end{array}\right).\end{array}$ (8)

We now investigate what vevs ensure flatness conditions for all the F-terms
and all the D-terms. For generic values of $\mu_{11}$, $\mu_{12}$ and
$\mu_{22}$, the entries of $F_{\Phi_{1}}$ and $F_{\Phi_{2}}$ are linearly
independent. Hence flatness of $F_{\Phi_{1}}$ and $F_{\Phi_{2}}$ implies that
the vevs of $k_{1}$, $k^{\prime}_{1}$, $k_{2}$, $k^{\prime}_{2}$ are all zero.
This automatically makes $D_{L,3}$ and $D_{R,3}$ flat.

Proceeding ahead, we now see that the F-flatness conditions split into two
subsets viz. the left handed conditions and right handed conditions. This
allows to conclude, as noted first by Aulakh et al., that the complete
solution set for F-flatness and D-flatness is obtained by taking
$(\omega,d)=(0,0)$ or
$(\omega,d)=(\frac{m_{\Delta}}{-a},\frac{\sqrt{2m_{\Omega}m_{\Delta}}}{-a})$,
and $(\omega_{c},d_{c})=(0,0)$ or
$(\omega_{c},d_{c})=(\frac{m_{\Delta}}{-a},\frac{\sqrt{2m_{\Omega}m_{\Delta}}}{-a})$
and $(k_{1},k^{\prime}_{1},k_{2},k^{\prime}_{2})=(0,0,0,0)$ (note $a<0$).
Thus, the complete SUSY preserving solution set has, beside the trivial all
zero solution, three non-trivial solutions. Of these, the solution
$(\omega,d,\omega_{c},d_{c},k_{1},k_{1}^{\prime},k_{2},k_{2}^{\prime})=(\frac{m_{\Delta}}{-a},\frac{\sqrt{2m_{\Omega}m_{\Delta}}}{-a},\frac{m_{\Delta}}{-a},\frac{\sqrt{2m_{\Omega}m_{\Delta}}}{-a},0,0,0,0)$
is unphysical because it breaks down the gauge symmetry to
$\mathrm{SU}(3)_{c}$ even though it preserves SUSY. The other two solutions
are indeed physical and can be interpreted as breaking down the left-right
symmetric gauge group
$\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times\mathrm{SU}(2)_{R}\times
U(1)_{B-L}$ into either the left-handed Minimal Supersymmetric Standard Model
(MSSM) $\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times U(1)_{Y}$, or into
the right-handed MSSM $\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{R}\times
U(1)_{Y}$.

The expressions for the vevs must be related to the two physical scales in
LRSUSY. We need to set

$\displaystyle M_{R}$ $\displaystyle\cong$
$\displaystyle\frac{m_{\Delta}}{-a}$ (9) $\displaystyle M_{B-L}$
$\displaystyle\cong$ $\displaystyle\frac{\sqrt{2m_{\Omega}m_{\Delta}}}{-a}$
(10)

in the LH regions and likewise in the RH regions. A solution to the
proliferation of mass scales was sought in Aulakh:1997fq  by invoking an $R$
symmetry of the superpotential which forbids the terms $\Omega^{2}$ and
$\Omega_{c}^{2}$. The $R$ charge values can be set to

$\displaystyle\Delta,\bar{\Delta},\Delta_{c},\bar{\Delta_{c}},\Phi_{i}$
$\displaystyle\rightarrow$ $\displaystyle 1$ (11) $\displaystyle\Omega$
$\displaystyle\rightarrow$ $\displaystyle 0$ (12) $\displaystyle
L,L_{c},Q,Q_{c}$ $\displaystyle\rightarrow$ $\displaystyle\frac{1}{2}$ (13)

where the $L$, $Q$ etc are matter superfields which are not relevant to this
paper. The terms $\Omega^{2}$ and $\Omega_{c}^{2}$ can then be introduced only
as soft terms, with the coefficients $m_{\Omega}=m_{\Omega_{c}}$ determined by
SUSY breaking scale $\cong M_{EW}$. This leads to an elegant simplification
giving rise to the see-saw relation

$M_{B-L}^{2}\cong M_{R}M_{EW}$ (14)

From the cosmological viewpoint, the early universe enters an epoch with two
types of domains. In the left handed (LH) domains, the right handed vevs take
non-zero values and in the right handed (RH) domains, the left handed vevs
take non-zero values. The corresponding SUSY preserving vevs are

$\begin{array}[]{r c
l}(\omega,d,\omega_{c},d_{c},k_{1},k_{1}^{\prime},k_{2},k_{2}^{\prime})&=&(0,0,\frac{m_{\Delta}}{-a},\frac{\sqrt{2m_{\Omega}m_{\Delta}}}{-a},0,0,0,0)~{}~{}~{}\mbox{LH
domain},\\\
(\omega,d,\omega_{c},d_{c},k_{1},k_{1}^{\prime},k_{2},k_{2}^{\prime})&=&(\frac{m_{\Delta}}{-a},\frac{\sqrt{2m_{\Omega}m_{\Delta}}}{-a},0,0,0,0,0,0)~{}~{}~{}\mbox{RH
domain}.\end{array}$ (15)

The formation of the two types of domains also leads to topological domain
walls separating them. This is because together with the breaking of the gauge
symmetry, a discrete left-right symmetry is also broken. The presence of these
walls or energy barriers conflicts with current cosmology. Several earlier
works have discussed how such walls may be made to disappear fast enough so as
to be consistent with present day observations Mishra . The older study
Anjishnu  demonstrated the existence of domain walls containing a CP violating
phase in LRSUSY with implications to leptogenesis, but only as a proof-of-
concept study. In this paper, we extend their ideas greatly and come up with
quantitative estimates relating the parameter ranges that can give rise to the
required spatially varying CP violating phase within the domain wall with the
non-zero phase required for electron EDM in zero temperature translation
invariant theory. The next section provides the details.

To the SUSY scalar potential

$\begin{array}[]{r c
l}V_{\mathrm{SUSY}}&=&|F_{\bar{\Delta}}|^{2}+|F_{\Delta}|^{2}+|F_{\bar{\Delta}_{c}}|^{2}+|F_{\Delta_{c}}|^{2}+|F_{\Omega}|^{2}+|F_{\Omega_{c}}|^{2}+|F_{\Phi_{1}}|^{2}+|F_{\Phi_{2}}|^{2}\\\
&&{}+\sum_{m=1}^{3}(|D_{L,m}|^{2}+|D_{R,m}|^{2})+|D_{B-L}|^{2},\end{array}$
(16)

we add the following soft mass terms for the bidoublets

$\begin{array}[]{r c
l}V_{\mathrm{soft}}&=&-\mu_{1}^{2}\mathrm{Tr}\,(\Phi_{1}^{\dagger}\Phi_{1})-\mu_{2}^{2}\mathrm{Tr}\,(\Phi_{2}^{\dagger}\Phi_{2})\\\
&&{}-e^{i\beta_{3}}\mu_{3}^{2}\mathrm{Tr}\,(\Phi_{1}^{\dagger}\tau_{2}\Phi_{1}^{*}\tau_{2})-e^{i\beta_{4}}\mu_{4}^{2}\mathrm{Tr}\,(\Phi_{2}^{\dagger}\tau_{2}\Phi_{2}^{*}\tau_{2})-e^{i\beta_{5}}\mu_{5}^{2}\mathrm{Tr}\,(\Phi_{1}^{\dagger}\tau_{2}\Phi_{2}^{*}\tau_{2})+\mathrm{h.c.},\end{array}$
(17)

where $\mu_{i}^{2}>0$ and $\beta_{3}$, $\beta_{4}$, $\beta_{5}$ are explicit
CP phases. Substituting the vevs we get $\langle V\rangle=\langle
V_{\mathrm{SUSY}}\rangle+\langle V_{\mathrm{soft}}\rangle,$ where

$\displaystyle\langle V_{\mathrm{SUSY}}\rangle$ $\displaystyle=$
$\displaystyle|\langle F_{\bar{\Delta}}\rangle|^{2}+|\langle
F_{\Delta}\rangle|^{2}+|\langle F_{\bar{\Delta}_{c}}\rangle|^{2}+|\langle
F_{\Delta_{c}}\rangle|^{2}+|\langle F_{\Omega}\rangle|^{2}+|\langle
F_{\Omega_{c}}\rangle|^{2}+|\langle F_{\Phi_{1}}\rangle|^{2}+|\langle
F_{\Phi_{2}}\rangle|^{2}$ $\displaystyle{}+|\langle
D_{L,3}\rangle|^{2}+|\langle D_{R,3}\rangle|^{2}+|\langle
D_{B-L}\rangle|^{2},$

and

$\begin{array}[]{rcl}\langle
V_{\mathrm{soft}}\rangle&=&-\mu_{1}^{2}(|k_{1}|^{2}+|k^{\prime}_{1}|^{2})-\mu_{2}^{2}(|k_{2}|^{2}+|k^{\prime}_{2}|^{2})\\\
&&{}-4\mu_{3}^{2}\mathrm{Re}(e^{i\beta_{3}}k_{1}^{*}(k^{\prime}_{1})^{*})-4\mu_{4}^{2}\mathrm{Re}(e^{i\beta_{4}}k_{2}^{*}(k^{\prime}_{2})^{*})-2\mu_{5}^{2}\mathrm{Re}(e^{i\beta_{5}}(k_{1}^{*}(k^{\prime}_{2})^{*}+(k^{\prime}_{1})^{*}k_{2}^{*})).\end{array}$
(18)

We shall take the fine tuning condition
$\mu_{12}^{2}\approx\mu_{11}\mu_{22}+\alpha^{2}M_{R}^{2}$ of Aulakh et el.
Aulakh:1997fq  which ensures that out of the four neutral Higgs scalars that
arise from the bidoublets after the breaking of
$\mathrm{SU}(2)_{R}\times\mathrm{U}(1)_{B-L}$ symmetry, two of them have
masses near zero (the other two end up having mass near $M_{R}$). Our soft
masses $\mu_{1},\ldots,\mu_{5}$ are chosen to be around $\alpha^{2}M_{R}$, so
that at temperatures below the SUSY breaking scale, the two light Higgs
scalars eventually become the Higgs bosons of a two Higgs doublet model (2HDM)
satisfying $\mathrm{SU}(3)_{c}\times\mathrm{SU}(2)_{L}\times U(1)_{Y}$ gauge
symmetry. At even lower temperatures, the SM Higgs boson arises from the 2HDM.

##  III Spatially varying Higgs vevs

The SUSY scalar potential and the soft mass terms for the bidoublets receive
temperature corrections determined by the scale $M_{B-L}$. The temperature
dependence of the squared mass term of a Higgs scalar has been evaluated at
the one loop level in earlier works Dolan . For this indicative study we shall
take the temperature correction to each mass matrix element to beCline2HDM ;
AndersonHall ; RamseyEWBG ,

$(\Delta m^{2})^{T}\sim O(g^{2}T^{2})$ (19)

The full temperature dependent mass matrix can be found in Appendix B. Thus,
going from temperature $M_{B-L}$ to zero temperature entails the lowering of
the mass matrix elements by $O(g^{2}T^{2})$. For the leptogenesis
calculations, we work at the high temperature $T=M_{B-L}$ with the mass matrix
arising from the SUSY scalar potential and the soft masses described above.
These choices for the mass parameters ensure that the mass matrix of the
bidoublets has a negative eigenvalue inside the wall whereas all its
eigenvalues are positive outside. This in turn means that it is energetically
favourable for the bidoublet fields to take non-zero vevs inside the wall
while continuing to take zero vevs outside, even though the soft terms have
negative squared masses. For the zero temperature electron EDM calculation
that we do later, we can work in a four Higgs doublet model Cline2HDM ; Fromme
with temperature corrections dropped from the mass matrix of Appendix B.

In this section we shall see that, for a certain choice of soft SUSY breaking
mass terms for the bidoublets, it becomes energetically favourable for the
bidoublet fields to take non-zero vevs within the wall while continuing to
take (almost) zero vevs outside it. Moreover, the generic $O(1)$ phases in the
soft mass terms entail a consequence that the bidoublet fields take on
spatially varying $CP$ violating phases inside the wall while returning to
constant non-zero phases outside the wall.

Let the spatial coordinate giving the distance from the wall, in units of
inverse temperature $1/T$, be denoted by $x$. The fields constituting the wall
have substantial variation only in a narrow region $\Delta x\sim(L/v)$ where
$v$ is a generic scalar vacuum expectation value and
$L^{-1}\sim\sqrt{\lambda}$ is derived from a generic quartic coupling
$\lambda$ of a renormalisable field theory. The SUSY preserving vev 8-tuples
inside the two domains are given by Equation 15. We want a smooth variation of
the vev 8-tuple as a function of $x$, going from the LH domain to the RH
domain passing through the wall on the way. By the argument in the previous
section, this necessarily entails breaking SUSY inside the wall. Thus in order
to get the shapes of the $x$-dependent vevs of the Higgs fields, we need to
first write down a functional for the energy per unit area of the wall and
then minimise the functional via Euler-Lagrange equations.

Let $\dot{f}$ denote the derivative of vev of field $f$ with respect to $x$.
Let $r_{1}$, $i_{1}$ be the real and imaginary parts of vev of $k_{1}$,
$r_{2},i_{2},\ldots,r^{\prime}_{2},i^{\prime}_{2}$ the real and imaginary
parts of the vevs of the respective bidoublet fields. We make the simplifying
assumptions that the non-bidoublet Higgs fields are real everywhere. The
finite temperature energy per unit area, which is the sum of gradient energies
and potential energies of all the fields, plus field dependent temperature
corrections can now be taken to be,

$\begin{array}[]{rcl}H^{T}&=&\int\,dx\,(\frac{1}{2}(\dot{\omega}^{2}+\dot{\omega_{c}}^{2}+\dot{d}^{2}+\dot{\bar{d}}^{2}+\dot{d_{c}}^{2}+\dot{\bar{d}_{c}}^{2}\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\dot{r_{1}}^{2}+\dot{i_{1}}^{2}+\dot{r^{\prime}_{1}}^{2}+\dot{i^{\prime}_{1}}^{2}+\dot{r_{2}}^{2}+\dot{i_{2}}^{2}+\dot{r^{\prime}_{2}}^{2}+\dot{i^{\prime}_{2}}^{2})\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+\langle V_{\mathrm{SUSY}}\rangle+\langle
V^{T}_{\mathrm{soft}}\rangle).\end{array}$ (20)

Here a superscript $T$ on $H$ and $V_{\mathrm{soft}}$ is a reminder of the
temperature dependence. We determine the domain wall solutions such that SUSY
is preserved asymptotically by the vevs upto relatively small temperature
correction. It is only in the narrow region of the wall where the omega fields
become small that the temperature dependent terms and soft terms become more
significant. In the equations below, all the vev are meant to be temperature
dependent, though for simplicity of notation we drop the superscript $T$.

$\begin{array}[]{c}\langle\omega(-\infty)\rangle=\langle
d(-\infty)\rangle=\langle\bar{d}(-\infty)\rangle=0,\\\
\langle\omega_{c}(-\infty)\rangle=M_{R},\langle
d_{c}(-\infty)\rangle=\langle\bar{d}_{c}(-\infty)\rangle=M_{B-L},\\\ \langle
r_{1}(-\infty)\rangle=\langle i_{1}(-\infty)\rangle=\langle
r^{\prime}_{1}(-\infty)\rangle=\langle i^{\prime}_{1}(-\infty)\rangle=0,\\\
\langle r_{2}(-\infty)\rangle=\langle i_{2}(-\infty)\rangle=\langle
r^{\prime}_{2}(-\infty)\rangle=\langle
i^{\prime}_{2}(-\infty)\rangle=0\end{array}$ (21)

in the left domain and

$\begin{array}[]{c}\langle\omega_{c}(\infty)\rangle=\langle
d_{c}(\infty)\rangle=\langle\bar{d}_{c}(\infty)\rangle=0,\\\
\langle\omega(-\infty)\rangle=M_{R},\langle
d(\infty)\rangle=\langle\bar{d}(\infty)\rangle=M_{B-L},\\\ \langle
r_{1}(\infty)\rangle=\langle i_{1}(\infty)\rangle=\langle
r^{\prime}_{1}(\infty)\rangle=\langle i^{\prime}_{1}(\infty)\rangle=0,\\\
\langle r_{2}(\infty)\rangle=\langle i_{2}(\infty)\rangle=\langle
r^{\prime}_{2}(\infty)\rangle=\langle
i^{\prime}_{2}(\infty)\rangle=0\end{array}$ (22)

in the right domain.

With all the parameters in place, we can now minimise the energy density per
unit area by solving the Euler-Lagrange equations arising from Equation 20.
The equations are given explicitly in Appendix A for completeness. In the next
section we describe how to solve them numerically in order to obtain the
shapes of the spatially varying vevs of the Higgs fields both within and
outside the wall. The vevs take the limiting values described in Equations 21,
22 outside the wall.

##  IV Solving the Euler-Lagrange equations for Higgs vevs

The Euler-Lagrange equations in Appendix A form a coupled system of second
order non-linear differential equations satisfying the boundary conditions of
Equation 21 in the left domain and Equation 22 in the right domain. Since the
derivatives of the vevs are zero in both left and right domains, these
equations are not well-suited for numerical solution by shooting methods.
Because of the large number of non-linear equations, their numerical solution
also faces difficulties under finite element or path deformation Wainwright
methods.

Naive attempts to solve the Euler-Lagrange equations as an initial value
system also run into problems. This is because if we take the initial
conditions at a point in the left domain, the algorithms give us the SUSY flat
left domain solution only as that solution minimises the energy density to
zero. A similar statement holds if we take the initial conditions at a point
in the right domain. We have to somehow model the loss of translation
invariance due to the domain wall in our solution.

Since $\Omega$, $\Omega_{c}$ have the heaviest vevs outside the wall, we fix a
natural ansatz for them that smoothly goes from the LH solution to the RH
solution while passing through the wall on the way. The wall is assumed to
extend from $-L$ to $L$ in units of inverse temperature. The ansatz takes the
form of kink functions:

$\begin{array}[]{rcl}\omega_{c}(x)&=&(1-\tanh(\frac{m_{\Delta}}{-2aL}x))\frac{m_{\Delta}}{-2a},\\\
\omega(x)&=&(1+\tanh(\frac{m_{\Delta}}{-2aL}x))\frac{m_{\Delta}}{-2a}.\\\
\end{array}$ (23)

The ansatz has the property that $\omega(x)$, $\omega_{c}(x)$ take the correct
limiting values outside the wall in both domains, but are non-zero within the
wall. In the example plots later on, we shall be taking
$L\sim(\sqrt{\lambda})^{-1}\sim 5$ for concreteness.

Fixing the ansatz for the vevs of $\Omega$, $\Omega_{c}$ models the effect of
the wall and reduces the Euler-Lagrange equations to a set of 12 coupled
second order differential equations for the four triplet Higgs vevs $d$,
$\bar{d}$, $d_{c}$, $\bar{d}_{c}$ and the eight vevs corresponding to the real
and imaginary parts of the bidoublet Higgs fields.

With this setting we solve the Euler-Lagrange equations as an initial value
problem numerically using the GSL 2.6 library, setting the stepping function
to be Runge-Kutta Dormand-Prince (8,9) with step size, absolute error and
relative error of $10^{-6}$. The obtained solutions for the bidoublet vevs are
non-zero and spatially varying inside the wall but become zero outside. For
the triplet vevs the obtained solutions are spatially varying inside the wall
and approach their constant SUSY determined values outside. The vevs of
$\Delta$, $\Delta_{c}$ are very sensitive to the values of the vevs of
$\Omega$, $\Omega_{c}$ and quickly drop to zero towards the centre of the wall
as that minimises the energy density.

Figures 1 and 2 show how the vacuum expectation values of the heavy triplet
Higgs fields $\Omega$, $\Omega_{c}$, $\Delta$, $\Delta_{c}$ vary as a function
of the distance $x$ from the wall for an ad hoc setting of parameters
$a=-1.5$, $\alpha=0.006$, $\mu_{11}=0.7M_{R}$, $\mu_{22}=0.7M_{R}$,
$\mu_{1}=0.2\alpha^{2}M_{R}$, $\mu_{2}=0.3\alpha^{2}M_{R}$,
$\mu_{3}=0.5\alpha^{2}M_{R}$, $\beta_{3}=-0.5$, $\mu_{4}=1.5\alpha^{2}M_{R}$,
$\beta_{4}=1.2$, $\mu_{5}=0.1\alpha^{2}M_{R}$, $\beta_{5}=-1.4$,
$M_{R}=10^{11}~{}\mathrm{GeV}$ and $M_{B-L}=10^{6.5}~{}\mathrm{GeV}$. Finally
the generic quartic coupling $\lambda$ which can be determined from those
appearing in the potential, is taken to be $\lambda\sim L^{-2}=0.04$.

$\langle\Omega_{c}\rangle=M_{R}$$\langle\Omega\rangle=M_{R}$

$\langle\Omega_{c}\rangle=M_{R}$$\langle\Omega\rangle=M_{R}$

Figure 1:  Above: Plots of vacuum expectation values $\omega$, $\omega_{c}$ as
a function of the distance $x$, in units of $1/T$, from the domain wall,
plotted for $T=M_{B-L}=10^{6.5}~{}\mathrm{GeV}$,
$M_{R}=10^{11}~{}\mathrm{GeV}$, $L=5$, $\lambda\sim L^{-2}=0.04$ and the
choice of the other parameters, $a=-1.5$, $\alpha=0.006$, $\mu_{11}=0.7M_{R}$,
$\mu_{22}=0.7M_{R}$, $\mu_{1}=0.2\alpha^{2}M_{R}$,
$\mu_{2}=0.3\alpha^{2}M_{R}$, $\mu_{3}=0.5\alpha^{2}M_{R}$, $\beta_{3}=-0.5$,
$\mu_{4}=1.5\alpha^{2}M_{R}$, $\beta_{4}=1.2$, $\mu_{5}=0.1\alpha^{2}M_{R}$,
$\beta_{5}=-1.4$. The domain wall stretches from $-L$ to $L$. Below: The same
plot magnfied, showing the gradual drop of the vevs of $\omega$, $\omega_{c}$
just inside the wall around $x=\pm L$.
$\langle\Delta_{c}\rangle=M_{B-L}$$\langle\Delta\rangle=M_{B-L}$ Figure 2:
Plots of vacuum expectation values $d$, $d_{c}$ obtained for
$T=M_{B-L}=10^{6.5}~{}\mathrm{GeV}$, $M_{R}=10^{11}~{}\mathrm{GeV}$, and for
the ansatz and and parameters as in Fig. 1.

Figure 3 exhibits the spatial variation of the real and imaginary parts of the
vacuum expectation values of the bidoublets. It turns out that the vevs of the
fields $k_{1}$ and $k_{2}$ are real throughout while $k^{\prime}_{1}$ and
$k^{\prime}_{2}$ do take spatially varying complex vevs. In Figure 4, we plot
the complex phases of vevs of $k^{\prime}_{1}$ and $k^{\prime}_{2}$ as a
function of $x$. Observe that the phases vary inside the wall but converge to
a constant non-zero value outside.

$\langle\Phi\rangle$ Figure 3: Left: Real and imaginary parts of the bidoublet
Higgs fields, $r_{1}$, $r^{\prime}_{1}$, $i^{\prime}_{1}$, $r_{2}$,
$r^{\prime}_{2}$, $i^{\prime}_{2}$, as a function of the distance $x$, in
units of $1/T$, from the domain wall, plotted for
$T=M_{B-L}=10^{6.5}~{}\mathrm{GeV}$, $M_{R}=10^{11}~{}\mathrm{GeV}$, and other
parameters as in Fig. 1.  $\delta_{\mathrm{CP}}$ Figure 4: The phases of
$k^{\prime}_{1}$, $k^{\prime}_{2}$ plotted as a function of $x$ for the same
parameters as in Fig. 1 and Fig. 3.

These considerations show that a spatially varying CP violating phase can
indeed be produced by the bidoublet Higgs vevs inside the domain wall in the
early universe. This phase persists as a constant non-zero quantity outside
the wall. In the next two sections, we investigate the implications of this
phenomenon for the electric dipole moment of the electron and the baryon
asymmetry of the universe.

##  V Electron EDM constraints on LRSUSY

The zero temperature mass matrix of the neutral components of the bidoublet
Higgs fields is given in Appendix B. As there are two neutral complex
components in each bidoublet, we get in total eight real neutral fields and so
the mass matrix is $8\times 8$. It turns out that the neutral mass eigenstates
induce complex phases for the bidoublet Higgs fields relative to each other.
This is true both within the wall as well as outside it. This feature gives
rise to an electric dipole moment (EDM) for the electron at one loop and two
loop levels. The maximum effect on the electron EDM is exerted by the lightest
mass eigenstate.

The one loop contribution to electron EDM $d_{e}$ is given by RamseyEWBG

$(d_{e}/e)|_{\mathrm{one~{}loop}}\sim\frac{\alpha m_{e}}{4\pi
M_{h}^{2}}\sin\delta,$ (24)

where $M_{h}$ is the mass of the lightest eigenstate of the bidoublet Higgs,
$\alpha$ is the fine structure constant evaluated at the scale $M_{h}$ and
$\delta$ is the complex relative phase between the neutral scalars induced by
the lightest mass eigenstate. Surprisingly however, for large values of
$M_{B-L}$ and $M_{R}$ two loop effects arising from the neutral scalars
dominate the one loop effect. This was first realised by Barr and Zee BarrZee
and then refined by several other authors. We use the formulas of Chang, Keung
and Yuan Chang  in order to compute the two loop contribution. The two loop
contribution is a sum of contributions from several diagrams. Most important
amongst those are four diagrams coming from Figure 5 arising from the choice
of top quark or $W$ boson in the inner loop, and the choice of $H\gamma\gamma$
or $HZ\gamma$ as the bosons interacting with this inner loop.

$e^{-}$$e^{-}$$e^{-}$$H_{0}$$W^{-},t$$\gamma,Z$$\gamma$

Figure 5:  The two loop diagram giving the maximum contribution to the
electron EDM in our model. Four such diagrams have to be calculated,
corresponding to the choice of top quark or $W$ boson in the inner loop, and
the choice of $H_{i}\gamma\gamma$ or $H_{i}Z\gamma$ as the bosons interacting
with this inner loop. Here $H_{0}$ denotes the lightest mass eigenstate of the
bidoublet Higgs.

Their total contribution is of the form

$\begin{array}[]{r c
l}(d_{e}/e)|_{\mathrm{two~{}loop}}&=&\frac{G_{F}m_{e}\alpha\sin\delta}{\pi^{3}\sqrt{2}}(f_{W,H\gamma\gamma}(M_{W}^{2}/M_{h}^{2})+f_{W,HZ\gamma}(M_{W}^{2}/M_{h}^{2})\\\
&&~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}{}+f_{t,H\gamma\gamma}(M_{t}^{2}/M_{h}^{2})+f_{t,HZ\gamma}(M_{t}^{2}/M_{h}^{2})).\end{array}$
(25)

where the functions $f$ are certain logarithmically growing functions defined
in BarrZee ; Chang .

We calculate the electron EDM numerically as a function of the LRSUSY model
parameters $M_{B-L}$ and $M_{R}$ using ROOT version 6.16 libraries. We let
$M_{B-L}$ range from $10^{4}$ to $10^{10}~{}\mathrm{GeV}$. For a given
$M_{B-L}$, we let $M_{R}$ range from a low of $10^{2}M_{B-L}$ to a high of
$M_{B-L}^{2}/M_{\mathrm{EW}}$. The lower bound on $M_{R}$ allows us to safely
break parity and left-right symmetry before breaking R-parity, and the upper
bound on $M_{R}$ allows the $\Omega$ fields in the left handed domain, which
have a mass of about $M_{B-L}^{2}/M_{R}$, to stay heavier than the electroweak
scale or the supersymmetry breaking scale Aulakh:1997fq . The lower bound on
$M_{B-L}$ ensures that it is above any reasonable supersymmetry breaking scale
$M_{S}$ and so one can comfortably break the $U(1)_{B-L}$ gauge symmetry to
reduce to the MSSM. In other words, as argued in more detail in Aulakh:1997fq
, the low energy effective theory of LRSUSY turns out to be the MSSM with
strictly unbroken R-parity, and so the lightest supersymmetric particle is
stable. The upper bound on $M_{B-L}$ follows from the consideration that for
$M_{B-L}\geq 10^{10}~{}\mathrm{GeV}$ we have to take $M_{R}\geq
10^{12}~{}\mathrm{GeV}$ which is rather high for parity breaking. The
experimentally allowed region, where the electron EDM, is less than $1.1\times
10^{-29}~{}\mathrm{e~{}cm}$ ACME2 , is plotted as the green hatched region in
the $(M_{B-L},M_{R})$-plane in Figure 6.

##  VI Leptogenesis constraints on LRSUSY

The non-supersymmetric LR model was studied in the context of conventional
electroweak baryogenesis mechanisms in MohapatraZhang ; Frere , and in a
domain wall mediated baryogenesis via leptogenesis mechanism in ClineYajnik .
The possibility of extending the latter mechanism to LRSUSY was indicated in
Anjishnu , but no concrete numerical calculations were performed there. In
this paper, we address this deficiency.

Adequate amount of CP violation as well as strong loss of equilibrium
conditions have been major challenges for low energy baryogenesis. The
presence of a moving domain wall, a topological defect, towards the
energetically disfavoured right handed domain immediately guarantees a strong
loss of equilibrium. The main LRSUSY model per se does not explain why this
happens, but we assume that this occurs because of tiny effects like soft SUSY
terms Anjishnu  or Planck suppressed non-renormalisable terms Mishra  breaking
exact left-right symmetry. A similar calculation has recently been done
SO10DomainWall  showing how Planck suppressed non-renormalisable terms can
remove a pseudo domain wall in supersymmetric $\mathrm{SO}(10)$ GUT without
conflicting with standard cosmology. Given that the domain wall in LRSUSY has
somehow disappeared early enough so as not to conflict with present day
cosmology, we can exploit it to obtain leptogenesis and consequent
baryogenesis consistent with experimental bounds on baryon asymmetry. This is
done as follows.

The lepton-Higgs Yukawa part of the superpotential of LRSUSY is Aulakh:1997fq

$W_{Y}=\mathbf{h}_{l}^{(j)}L^{T}\tau_{2}\Phi_{j}\tau_{2}L_{c}+i\mathbf{f}(L^{T}\tau_{2}\Delta
L+L_{c}^{T}\tau_{2}\Delta_{c}L_{c}),$ (26)

where $j=1,2$ and $\mathbf{h}$, $\mathbf{f}$ are $3\times 3$ real symmetric
matrices. The Majorana mass terms above corresponding to the Yukawa coupling
matrix $\mathbf{f}$ are a source of lepton number violation. However, in
LRSUSY they do not favour conventional thermal leptogenesis because at the
usual scale of thermal leptogenesis, the $B-L$ gauged symmetry is unbroken
Anjishnu . That is why we have to resort to domain wall mediated leptogenesis
in LRSUSY. The lepton number violating decay of the heavy Majorana RH neutrino
will instead give rise to a lepton asymmetry washout which will dilute any
lepton asymmetry mediated by the domain wall. Nevertheless, as we will see
below, it is possible to obtain baryon asymmetry consistent with experimental
data for a certain region of the LRSUSY parameter space.

Consider a domain wall moving slowly with speed $v_{w}$ in the $+x$ direction
i.e. encroaching upon the energetically disfavoured RH domain. The wall is
assumed to stretch from $-L$ to $+L$ in the $x$ direction and be flat in the
$yz$ plane. Slow speed means that $v_{w}<1/\sqrt{3}$, the speed of sound in
the hot plasma, allowing one to get a solution to the chemical potential of
the neutrinos in terms of a fluid approximation Joyce . Since the wall is
assumed to move due to tiny energy differences between the LH and RH domain,
this is a reasonable assumption. We consider the case of thick walls i.e.
$2L>1/T$, the de Broglie wavelength of the neutrinos at temperature $T$ Joyce
. This is a reasonable assumption ensuring that the mean free path of the
neutrinos is smaller than the wall thickness, leading to multiple interactions
between neutrinos and the CP violating condensate in the wall and allowing a
classical WKB treatment of the neutrinos. A wall thickness of $10/T$ will be
typical in our analysis. We assume that the neutrino diffusion coefficient
$D<2L/(3v_{w})$ Joyce , allowing us to do a thermalised fluid analysis of the
chemical potential of the LH neutrinos. The expression for $D$ below easily
satisfies this constraint.

Consider the interaction of LH neutrinos with the domain wall. The LH
neutrinos $\nu_{L}$ are massive in the RH domain due to the Majorana Yukawa
coupling to the $\mathrm{SU}(2)_{L}$-triplet Higgs field $\Delta$ which takes
a vev in the RH domain. Conversely they are massless in the LH domain where
$\Delta$ has a vanishing vev. Due to the existence of a spatially varying CP
violating condenstate arising from the complex bidoubet vevs inside the wall,
one get an asymmetry between $\nu_{L}$ and its antiparticle $\bar{\nu}_{L}$ in
terms of their reflection and transmission coefficients with respect to the
wall. There will be a preference for transmission of, say, $\nu_{L}$ from RH
domain to LH domain through the wall. The LH neutrinos reflected back into RH
domain from the wall quickly equilibrate with their antiparticles around them
because of the high rate of helicity flipping of LH neutrinos owing to their
large Majorana mass in the RH domain. Thus the RH domain continues to have
almost no particle antiparticle asymmetry. In contrast, the transmitted excess
of LH neutrinos into LH domain survives because they are almost massless in LH
domain and hardly flip helicity. This leads to a particle antiparticle
asymmetry in LH domain. Eventually as the domain wall encroaches totally upon
RH domain and destroys it, we end up with an excess of leptons in our left
handed Universe. Weak sphaeleron processes convert a part of the early lepton
excess into a baryon excess till they go out of equilibrium as our left handed
Universe cools. Thus, one is left with a baryon excess in the present day
Universe.

The diffusion equation for the chemical potential $\mu$ of the LH neutrino in
the wall rest frame is given by ClineYajnik

$-D\mu^{\prime\prime}+v_{w}\mu^{\prime}+\Gamma\Theta(x)\mu=S(x),$ (27)

where $S(x)$ is the so-called source term defined below, $D$ is the neutrino
diffusion coefficient, $v_{w}$ is the speed of the wall taken to be moving in
the $+x$ direction, $\Theta(x)$ is the step function which takes value one if
$x$ is positive and zero otherwise, $\Gamma$ is the rate of helicity flipping
interactions at temperature $T$ which is very high in the RH domain due to the
heavy Majorana mass of the LH neutrino in the RH domain and zero in the LH
domain since the LH neutrino is almost massless in the LH domain. Observe that
the LH neutrino mass $m_{\nu}(x)$ is spatially dependent and complex inside
the wall since the neutrino couples to the spatially varying complex bidoublet
Higgs and the triplet Higgs fields $\Delta$, $\Delta_{c}$ quickly vanish
inside the wall (see Figures 2, 3).

The source term, which is a CP-violating non-zero force if the neutrino mass
$m_{\nu}(x)$ and the domain wall CP phase $\delta(x)$ are spatially varying,
is given by ClineSUSY

$S(x)=-\frac{v_{w}D}{2\Gamma}\langle\frac{|p_{x}|}{E^{2}\tilde{E}}\rangle(m_{\nu}(x)\delta^{\prime}(x))^{\prime\prime},$
(28)

where $p_{x}$ is the $x$-component of the LH neutrino’s momentum, $E$ is the
neutrino energy, $\tilde{E}$ is the related quantity
$\sqrt{m_{\nu}(x)^{2}+p_{x}^{2}}$, and the angular brackets indicate thermal
averages. It was shown in ClineSUSY  that

$\langle\frac{|p_{x}|}{E^{2}\tilde{E}}\rangle=\frac{e^{-a}-aE_{1}(a)}{2T^{2}a^{2}K_{2}(a)},$
(29)

where $a=m_{\nu}(\infty)/T$, $E_{1}$ is the error function, $K_{2}$ is the
modified cylindrical Bessel function of the second kind and the LH neutrino
mass is evaluated deep inside the RH domain. The source term is zero outside
the wall but non-zero inside.

The helicity flipping rate can be calculated by Joyce

$\Gamma=\frac{\alpha_{w}^{2}m_{\nu}(\infty)^{2}}{T},$ (30)

where $\alpha_{w}$ is the weak coupling constant evaluated at temperature $T$.
The diffusion coefficient has the expression ClineSUSY ,

$D=\frac{\langle v_{x}^{2}\rangle}{\Gamma},~{}~{}~{}\langle
v_{x}^{2}\rangle=\frac{3a+2}{a^{2}+3a+2},$ (31)

where $v_{x}$ is the $x$-component of the neutrino velocity and $a$ is defined
above.

We solve the diffusion equation (Equation 27) numerically the using GSL
version 2.6 library under the same settings as before, and take the limiting
value $\mu(-\infty)$ in the LH domain as the steady state chemical potential
of the LH neutrino. Then, the steady state neutrino-antineutrino asymmetry in
the LH domain becomes ClineYajnik

$\Delta_{\nu_{L}}=\frac{T^{2}}{6}\mu(-\infty).$ (32)

To obtain the raw lepton asymmetry to entropy density ratio
$\eta^{\mathrm{raw}}$, we need to divide the above quantity by
$\frac{2\pi^{2}g_{*}T^{3}}{45}$, where $g_{*}\approx 110$ is the number of
relativistic degrees of freedom. Doing so gives us

$\eta^{\mathrm{raw}}_{L}=0.0035\frac{\mu(-\infty)}{T}.$ (33)

Since the heavy neutrinos in this model have mass less than the temperature
$M_{B-L}$, they can easily decay violating lepton number. This process washes
out most of the raw lepton asymmetry $\eta^{\mathrm{raw}}_{L}$. The surviving
lepton asymmetry by entropy density ratio becomes ClineYajnik

$\eta_{L}=\eta^{\mathrm{raw}}\cdot 10^{-4\cdot
10^{-4}m_{\nu}M_{\mathrm{Planck}}v^{-2}}=3.054\cdot
10^{-11}\frac{\mu(-\infty)}{T},$ (34)

where $m_{\nu}$ is the mass of the heaviest light neutrino and
$v=174~{}\mathrm{GeV}$ is the SM Higgs vev. We take
$m_{\nu}=0.05~{}\mathrm{eV}$ from the Nu-FIT Group NuFit .

Finally electroweak sphaelerons convert part of the lepton asymmetry to baryon
asymmetry starting from the temperature $M_{B-L}$ till the universe cools to
the sphaeleron scale of about one TeV. This gives the steady state baryon
asymmetry to entropy density ratio ClineYajnik

$\eta_{B}=\frac{28}{51}\eta_{L}=1.677\cdot 10^{-11}\frac{\mu(-\infty)}{T}.$
(35)

The yellow strip in Figure 6 shows the $(M_{R},M_{B-L})$ tuples where the
final baryon asymmetry to entropy density ratio is between $10^{-11}$ and
$10^{-8}$, which can thus provide a good explanation of the experimentally
observed baryon asymmetry of $6\cdot 10^{-10}$ Canetti . The parameter space
of the yellow strip is consistent with the experimental upper bound on
electron EDM indicated by the green hatched region in Figure 6.

$M_{R}=\frac{M_{B-L}^{2}}{M_{\mathrm{EW}}}$$M_{R}=100M_{B-L}$$<10^{-29}$ e
cm$10^{-11}$—$10^{-8}$ Figure 6:  The regions in the $(M_{B-L},M_{R})$-plane
allowed by the experimental bounds on baryon asymmetry in yellow and electron
EDM hatched green, plotted for the same parameter set as Fig. 1-4. The upper
blue line denotes the setting $M_{R}=M_{B_{L}}^{2}/M_{EW}$ and the lower blue
line $M_{R}=100M_{B-L}$. The baryon asymmetry allowed region is the narrow
region entirely subsumed within that allowed by the EDM constraint and
overlaid on the latter. The two regions run almost parallel at upper boundary
with EDM allowed region somewhat larger. To be specific,
$10^{6.5}<M_{B-L}<10^{10}\,\mathrm{GeV}$ and bears out the philosophy of Eq.
(14) only towards the lowest end of the range.

##  VII Discussion and conclusion

Armed with our numerical calculation tools, we have made a strategic
exploration of the parameter space of LRSUSY to find the subregion consistent
with the current experimental bounds on electron EDM and baryon-antibaryon
asymmetry. We have varied the trilinear $\Omega\Phi\Phi$ coupling parameter
$\alpha$ from $0.001$ to $0.1$, and the mass parameters $\mu$ from $0.5$ to
$1$ in order to study the shape of the Higgs fields inside the wall and have
obtained similar results as described above. These ranges of the parameters
were chosen from the following considerations. Values of $\alpha$ greater than
$0.1$ make the soft terms not so soft anymore, and values of mass parameters
$\mu$ outside the above range either make the bidoublets heavier than the
largest mass scale $M_{R}$ or make our fine tuning ineffective. The
manifestation of the limiting values of the parameters shows up in the
observation that for $\mu_{11}$, $\mu_{22}$ smaller than $0.5M_{R}$, or for
$\alpha>0.1$, there is no value of $M_{B-L}$ between $10^{4}$ to
$10^{10}\,\mathrm{GeV}$ consistent with both electron EDM and baryon asymmetry
experiments.

For parameter values in the above ranges, we see that the lowest $M_{B-L}$
consistent with the two experiments is $10^{4.5}\,\mathrm{GeV}$. The lowest
allowed value of $M_{B-L}$ is most sensitive to $\alpha$, and is a decreasing
function of $\alpha$. Figure 7 shows its variation when $\alpha$ ranges from
$0.001$ to $0.1$.

Figure 7: The lowest value of $M_{B-L}$ consistent with both electron EDM and
baryon asymmetry experiments plotted as a function of $\alpha$.

It turns out that the lowest allowed value of $M_{B-L}$ is hardly sensitive to
the mass parameters and variations in wall thickness and wall velocity. Thus,
by scanning the parameter space we can conclude that for consistency with both
electron EDM and baryon asymmetry experiments, the scale $M_{B-L}$ of
$B-L$-symmetry breaking must be larger than $10^{4.5}\,\mathrm{GeV}$, and over
the whole allowed range must be significantly less that $M_{B-L}^{2}/M_{EW}$.
These novel bounds provide the most stringent constraints on the parameter
space of LRSUSY by far.

More interestingly, our implication that $M_{R}$ must be significantly less
than $M_{B-L}^{2}/M_{EW}$ accords with the simplifying proposal of Aulakh et
al. Aulakh:1997fq  viz. $M_{R}\gtrsim M_{B-L}^{2}/M_{EW}$, as discussed at
(14), only at the lowest allowed value of $M_{B-L}$. Aulakh et al.’s proposal
arose from gravity mediated TeV scale SUSY breaking where the gravitino is not
much heavier than $M_{EW}$. Within the validity of this proposal this analysis
makes a specific prediction of the $M_{B-L}$ and hence the $M_{R}$. Since LHC
has not discovered any signatures of SUSY, TeV scale SUSY breaking is now a
highly unlikely possibility though an exciting one to confirm if true.

On the other hand we can reject the $R$ charge proposed in Aulakh:1997fq , in
order to allow the fact that the scale of parity breaking and $(B-L)$-symmetry
breaking are not maximally far apart. The parameter $m_{\Omega}$ could then be
intrinsic to the superpotential. In this case our results are perfectly
consistent with PeV scale supersymmetry Wells  also considered in the recent
work Dhuria , where the gravitino can be much heavier. This places LRSUSY
outside the experimental reach of colliders in the near future. On the other
hand, the discovery of a non-zero electron EDM value can be taken to be a hint
to a narrow range for the $M_{R}$ scale assuming a world with a renormalisable
supersymmetric left-right model. Finally it is interesting that two very
different low-energy probes viz. baryon asymmetry and electron EDM provide a
strong constraint on the allowed parameter space of LRSUSY.

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##  Appendix A Euler-Lagrange equations for spatially varying Higgs vevs

The Euler-Lagrange equations are written explicitly below, where $\ddot{f}$
represents the double derivative of vev of field $f$ with respect to $x$ and
$F_{(\Phi_{i})_{j}}$ denotes the $(j,j)$ matrix element of the vev $\langle
F_{\Phi_{i}}\rangle$. The expressions for the vevs of the F-terms and D-terms
can be found in Equations 8 and 6 respectively. There are eight bidoublet
field vevs $r_{1},i_{1},\ldots,r_{2}^{\prime},i_{2}^{\prime}$ which are the
real and imaginary parts of the vev $k_{1},\ldots,k_{2}^{\prime}$
respectively. There are 4 triplet fields vevs viz. $d$, $\bar{d}$, $d_{c}$,
$\bar{d}_{c}$. All these vevs vary as a function of distance $x$ from the
centre of the wall, but we write field $f$ in the equations below and not
$f(x)$ for clarity of notation. The vevs of the fields $\omega(x)$,
$\omega_{c}(x)$ below take their values from the ansatz in Equation (23).

$\begin{array}[]{r c
l}\ddot{r}_{1}&=&4\alpha\mathrm{Re}(F_{\Omega}+F_{\Omega_{c}})r_{2}^{\prime}+8\alpha\mathrm{Im}(F_{\Omega})i_{2}^{\prime}+8\mu_{11}\mathrm{Re}(F_{(\Phi_{1})_{2}})+8(\mu_{12}+\alpha(\omega-\omega_{c}))\mathrm{Re}(F_{(\Phi_{2})_{2}})\\\
&&{}+8g^{2}r_{1}D_{L,3}-2\mu_{1}^{2}r_{1}-4\mu_{3}^{2}(r_{1}^{\prime}\cos\beta_{3}+i_{1}^{\prime}\sin\beta_{3})-2\mu_{5}^{2}(r_{2}^{\prime}\cos\beta_{5}+i_{2}^{\prime}\sin\beta_{5}),\\\
\ddot{i}_{1}&=&-4\alpha\mathrm{Re}(F_{\Omega}+F_{\Omega_{c}})i_{2}^{\prime}+8\alpha\mathrm{Im}(F_{\Omega})r_{2}^{\prime}+8\mu_{11}\mathrm{Im}(F_{(\Phi_{1})_{2}})+8(\mu_{12}+\alpha(\omega-\omega_{c}))\mathrm{Im}(F_{(\Phi_{2})_{2}})\\\
&&{}+8g^{2}i_{1}D_{L,3}-2\mu_{1}^{2}i_{1}-4\mu_{3}^{2}(-i_{1}^{\prime}\cos\beta_{3}+r_{1}^{\prime}\sin\beta_{3})-2\mu_{5}^{2}(-i_{2}^{\prime}\cos\beta_{5}+r_{2}^{\prime}\sin\beta_{5}),\\\
\ddot{r}_{2}&=&-4\alpha\mathrm{Re}(F_{\Omega}+F_{\Omega_{c}})r_{1}^{\prime}-8\alpha\mathrm{Im}(F_{\Omega})i_{1}^{\prime}+8(\mu_{12}-\alpha(\omega-\omega_{c}))\mathrm{Re}(F_{(\Phi_{1})_{2}})+8\mu_{22}\mathrm{Re}(F_{(\Phi_{2})_{2}})\\\
&&{}+8g^{2}r_{2}D_{L,3}-2\mu_{2}^{2}r_{2}-4\mu_{4}^{2}(r_{2}^{\prime}\cos\beta_{4}+i_{2}^{\prime}\sin\beta_{4})-2\mu_{5}^{2}(r_{1}^{\prime}\cos\beta_{5}+i_{1}^{\prime}\sin\beta_{5}),\\\
\ddot{i}_{2}&=&4\alpha\mathrm{Re}(F_{\Omega}+F_{\Omega_{c}})i_{1}^{\prime}-8\alpha\mathrm{Im}(F_{\Omega})r_{1}^{\prime}+8(\mu_{12}-\alpha(\omega-\omega_{c}))\mathrm{Im}(F_{(\Phi_{1})_{2}})+8\mu_{22}\mathrm{Im}(F_{(\Phi_{2})_{2}})\\\
&&{}+8g^{2}i_{2}D_{L,3}-2\mu_{2}^{2}i_{2}-4\mu_{4}^{2}(-i_{2}^{\prime}\cos\beta_{4}+r_{2}^{\prime}\sin\beta_{4})-2\mu_{5}^{2}(-i_{1}^{\prime}\cos\beta_{5}+r_{1}^{\prime}\sin\beta_{5}),\\\
\ddot{r}^{\prime}_{1}&=&-4\alpha\mathrm{Re}(F_{\Omega}+F_{\Omega_{c}})r_{2}-8\alpha\mathrm{Im}(F_{\Omega})i_{2}+8\mu_{11}\mathrm{Re}(F_{(\Phi_{1})_{1}})+8(\mu_{12}-\alpha(\omega-\omega_{c}))\mathrm{Re}(F_{(\Phi_{2})_{1}})\\\
&&{}-8g^{2}r_{1}^{\prime}D_{L,3}-2\mu_{1}^{2}r_{1}^{\prime}-4\mu_{3}^{2}(r_{1}\cos\beta_{3}+i_{1}\sin\beta_{3})-2\mu_{5}^{2}(r_{2}\cos\beta_{5}+i_{2}\sin\beta_{5}),\\\
\ddot{i}^{\prime}_{1}&=&4\alpha\mathrm{Re}(F_{\Omega}+F_{\Omega_{c}})i_{2}-8\alpha\mathrm{Im}(F_{\Omega})r_{2}+8\mu_{11}\mathrm{Im}(F_{(\Phi_{1})_{1}})+8(\mu_{12}-\alpha(\omega-\omega_{c}))\mathrm{Im}(F_{(\Phi_{2})_{1}})\\\
&&{}-8g^{2}i_{1}^{\prime}D_{L,3}-2\mu_{1}^{2}i_{1}^{\prime}-4\mu_{3}^{2}(-i_{1}\cos\beta_{3}+r_{1}\sin\beta_{3})-2\mu_{5}^{2}(-i_{2}\cos\beta_{5}+r_{2}\sin\beta_{5}),\\\
\ddot{r}^{\prime}_{2}&=&4\alpha\mathrm{Re}(F_{\Omega}+F_{\Omega_{c}})r_{1}+8\alpha\mathrm{Im}(F_{\Omega})i_{1}+8(\mu_{12}+\alpha(\omega-\omega_{c}))\mathrm{Re}(F_{(\Phi_{1})_{1}})+8\mu_{22}\mathrm{Re}(F_{(\Phi_{2})_{1}})\\\
&&{}-8g^{2}r_{2}^{\prime}D_{L,3}-2\mu_{2}^{2}r_{2}^{\prime}-4\mu_{4}^{2}(r_{2}\cos\beta_{4}+i_{2}\sin\beta_{4})-2\mu_{5}^{2}(r_{1}\cos\beta_{5}+i_{1}\sin\beta_{5}),\\\
\ddot{i}^{\prime}_{2}&=&-4\alpha\mathrm{Re}(F_{\Omega}+F_{\Omega_{c}})i_{1}+8\alpha\mathrm{Im}(F_{\Omega})r_{1}+8(\mu_{12}+\alpha(\omega-\omega_{c}))\mathrm{Im}(F_{(\Phi_{1})_{1}})+8\mu_{22}\mathrm{Im}(F_{(\Phi_{2})_{1}})\\\
&&{}-8g^{2}i_{2}^{\prime}D_{L,3}-2\mu_{2}^{2}i_{2}^{\prime}-4\mu_{4}^{2}(-i_{2}\cos\beta_{4}+r_{2}\sin\beta_{4})-2\mu_{5}^{2}(-i_{1}\cos\beta_{5}+r_{1}\sin\beta_{5}).\end{array}$
(36)
$\begin{array}[]{rcl}\ddot{d}&=&2a\mathrm{Re}(F_{\Omega})\bar{d}+2F_{\bar{\Delta}}(m_{\Delta}+a\omega),\\\
\ddot{\bar{d}}&=&2a\mathrm{Re}(F_{\Omega})d+2F_{\Delta}(m_{\Delta}+a\omega),\\\
\ddot{d}_{c}&=&2a\mathrm{Re}(F_{\Omega_{c}})\bar{d}_{c}+2F_{\bar{\Delta_{c}}}(m_{\Delta}+a\omega_{c}),\\\
\ddot{\bar{d}}_{c}&=&2a\mathrm{Re}(F_{\Omega_{c}})d_{c}+2F_{\Delta_{c}}(m_{\Delta}+a\omega_{c}).\end{array}$
(37)

##  Appendix B Temperature dependent mass matrix of neutral bidoublet Higgs

Here we display the temperature dependent mass matrix corresponding to
Equations 16, 17. This is a symmetric $8\times 8$ matrix in the fields
$r_{1}$, $i_{1}$, $r_{2}$, $i_{2}$, $r^{\prime}_{1}$, $i^{\prime}_{1}$,
$r^{\prime}_{2}$, $i^{\prime}_{2}$. The mass parameters $\mu$ are assumed to
be corrected for the temperature of $T=M_{B-l}$. Following Cline2HDM , we can
write the matrix with the zero temperature Debye mass corrections as shown
below. In obtaining the domain wall profiles we use $T=0$ and for the electron
EDM calculation we use $T=M_{B-L}$.

$\begin{array}[]{rcl}M_{r_{1},r_{1}}&=&M_{i_{1},i_{1}}=\mu_{11}^{2}+(\mu_{12}-\alpha
M_{R})^{2}-\mu_{1}^{2}-\frac{g^{2}T^{2}}{6}\\\
M_{r_{2},r_{2}}&=&M_{i_{2},i_{2}}=(\mu_{12}+\alpha
M_{R})^{2}+\mu_{22}^{2}-\mu_{2}^{2}-\frac{g^{2}T^{2}}{6}\\\
M_{r^{\prime}_{1},r^{\prime}_{1}}&=&M_{i^{\prime}_{1},i^{\prime}_{1}}=\mu_{11}^{2}+(\mu_{12}+\alpha
M_{R})^{2}-\mu_{1}^{2}-\frac{g^{2}T^{2}}{6}\\\
M_{r^{\prime}_{2},r^{\prime}_{2}}&=&M_{i^{\prime}_{2},i^{\prime}_{2}}=(\mu_{12}-\alpha
M_{R})^{2}+\mu_{22}^{2}-\mu_{2}^{2}-\frac{g^{2}T^{2}}{6}\\\ &&\\\
M_{r_{1},i_{1}}&=&M_{r_{2},i_{2}}=M_{r^{\prime}_{1},i^{\prime}_{1}}=M_{r^{\prime}_{2},i^{\prime}_{2}}=M_{r_{1},i_{2}}=M_{i_{1},r_{2}}=M_{r^{\prime}_{1},i^{\prime}_{2}}=M_{i^{\prime}_{1},r^{\prime}_{2}}=0\\\
&&\\\ M_{r_{1},r_{2}}&=&\mu_{11}(\mu_{12}+\alpha M_{R})+(\mu_{12}-\alpha
M_{R})\mu_{22}\\\ M_{i_{1},i_{2}}&=&\mu_{11}(\mu_{12}+\alpha
M_{R})+(\mu_{12}-\alpha M_{R})\mu_{22}\\\
M_{r^{\prime}_{1},r^{\prime}_{2}}&=&\mu_{11}(\mu_{12}-\alpha
M_{R})+(\mu_{12}+\alpha M_{R})\mu_{22}\\\
M_{i^{\prime}_{1},i^{\prime}_{2}}&=&\mu_{11}(\mu_{12}-\alpha
M_{R})+(\mu_{12}+\alpha M_{R})\mu_{22}\\\ &&\\\
M_{r_{1},r^{\prime}_{1}}&=&-M_{i_{1},i^{\prime}_{1}}=-2\mu_{3}^{2}\cos\beta_{3}\\\
M_{r_{1},i^{\prime}_{1}}&=&-M_{i_{1},r^{\prime}_{1}}=-2\mu_{3}^{2}\sin\beta_{3}\\\
&&\\\
M_{r_{1},r^{\prime}_{2}}&=&-M_{i_{1},i^{\prime}_{2}}=M_{r_{2},r^{\prime}_{1}}=-M_{i_{2},i^{\prime}_{1}}=-\mu_{5}^{2}\cos\beta_{5}\\\
M_{r_{1},i^{\prime}_{2}}&=&M_{i_{1},r^{\prime}_{2}}=M_{r_{2},i^{\prime}_{1}}=M_{i_{2},r^{\prime}_{1}}=-\mu_{5}^{2}\sin\beta_{5}\\\
&&\\\
M_{r_{2},r^{\prime}_{2}}&=&-M_{i_{2},i^{\prime}_{2}}=-2\mu_{4}^{2}\cos\beta_{4}\\\
M_{r_{2},i^{\prime}_{2}}&=&-M_{i_{2},r^{\prime}_{2}}=-2\mu_{4}^{2}\sin\beta_{4}\end{array}$
(38)