11institutetext: Max-Planck-Institut für Physik (Werner-Heisenberg-Institut),
München 80805, Germany22institutetext: Centre for High Energy Physics, Indian
Institute of Science, Bangalore 560 012, India33institutetext: INFN, Sezione
di Padova and Dip. di. Fisica ‘Galileo Galilei’, Univ. di. Padova, Padova
35131, Italy44institutetext: Physical Research Laboratory, Navarangpura,
Ahmedabad 380 009, India

# Status of Supersymmetric Seesaw in SO(10) models

L. Calibbi<EMAIL_ADDRESS>2 D. Chowdhury<EMAIL_ADDRESS>3
A. Masiero<EMAIL_ADDRESS>4 K. M. Patel<EMAIL_ADDRESS>2 and
S. K. Vempati<EMAIL_ADDRESS>
###### Abstract

We report on the status of supersymmetric seesaw models in the light of recent
experimental results on $\mu\to e+\gamma$, $\theta_{13}$ and the light Higgs
mass at the LHC. SO(10)-like relations are assumed for neutrino Dirac Yukawa
couplings and two cases of mixing, one large, PMNS-like, and another small,
CKM-like, are considered. It is shown that for the large mixing case, only a
small range of parameter space with moderate $\tan\beta$ is still allowed.
This remaining region can be ruled out by an order of magnitude improvement in
the current limit on BR($\mu\to e+\gamma$). We also explore a model with non-
universal Higgs mass boundary conditions at the high scale. It is shown that
the renormalization group induced flavor violating slepton mass terms are
highly sensitive to the Higgs boundary conditions. Depending on the choice of
the parameters, they can either lead to strong enhancements or cancellations
within the flavor violating terms. Such cancellations might relax the severe
constraints imposed by lepton flavor violation compared to mSUGRA.
Nevertheless for a large region of parameter space the predicted rates lie
within the reach of future experiments once the light Higgs mass constraint is
imposed. We also update the potential of the ongoing and future experimental
searches for lepton flavor violation in constraining the supersymmetric
parameter space.

††arxiv:

##  1 Introduction

Current times are unprecedented in terms of experimental activity in high
energy physics. There have been several results of very high impact in the
recent times. The following three are the most relevant for the purpose of our
discussion.

  * •

Firstly, LHC experiments have reported the discovery of a new boson with mass
of about 125 GeV cmshiggs ; atlashiggs , compatible with the Standard Model
(SM) Higgs boson. In minimal supersymmetric standard model (MSSM), this would
imply a mass of this order and SM-like couplings for the lightest CP-even
Higgs boson strumia ; Ellis:2012hz ; Espinosa:2012im ; Carmi:2012in . In our
analysis, we will take the $2\sigma$ mass range obtained in strumia

$124.5~{}\text{GeV}\lesssim m_{h}\lesssim 126.5~{}\text{GeV},$ (1)

where $m_{h}$ stands for the mass of the lightest neutral Higgs. In addition,
LHC has also improved the limits on the spectrum of low-energy supersymmetry
(SUSY) CMS_2012 .

  * •

The limit on the lepton flavor violating (LFV) decay $\mu\to e+\gamma$ has
improved by one order of magnitude Adam:2011ch . The current limit is

$\text{BR}(\mu\to e+\gamma)<2.4\times 10^{-12}~{}(90\%~{}\textrm{CL}).$ (2)

  * •

Finally, the so-far unknown neutrino mixing angle, $\theta_{13}$ has been
experimentally determined An:2012eh ; Ahn:2012nd . The Daya Bay and RENO
experiments have measured $\theta_{13}$ with very good accuracy:

$\displaystyle\sin^{2}2\theta_{13}=$ $\displaystyle 0.092\pm 0.016({\rm
stat.})\pm 0.005({\rm syst.})$ Daya Bay (An:2012eh, )
$\displaystyle\sin^{2}2\theta_{13}=$ $\displaystyle 0.113\pm 0.013({\rm
stat.})\pm 0.019({\rm syst.})$ RENO (Ahn:2012nd, ) (3)

The implications of the measurement of the light Higgs mass for various
supersymmetric models have been studied in detail by various authors
Hall:2011aa ; Heinemeyer:2011aa ; Arbey:2011aa ; Arbey:2011ab ; Draper:2011aa
; Carena:2011aa ; Christensen:2012ei ; Baer:2011ab ; Kadastik:2011aa ;
Buchmueller:2011ab ; Aparicio:2012iw ; Ellis:2012aa ; Baer:2012uy .
Supersymmetric seesaw models have not been explored so far in the light of
these results111As this work was finished and being prepared for submission,
the following paper Ref.Hirsch:2012ti  appeared on arXiv.. Furthermore, flavor
violation is expected from SUSY seesaw models where the last two experimental
results would play a crucial role in constraining the parameter space. In the
present work, we consider the implications of all the three experimental
results on a class of SUSY seesaw models inspired by SO(10) GUTs.

SO(10) GUT Models typically relate up-quark Yukawa matrices with the Dirac
neutrino Yukawa couplings of the Type I seesaw mechanism. In fact, at least
one of the neutrino Yukawa couplings is expected to be as large as the top
Yukawa coupling, as a consequence of the undelying SO(10) symmetry
Masiero:2002jn . Concerning the mixing structure of the neutrino Yukawa
matrix, two extreme cases can be motivated by simple SO(10) models: large
PMNS-like mixing or small CKM-like mixing. Lepton flavor violation in this
class of models has been studied in Refs. Masiero:2002jn ; Calibbi:2006nq ;
Calibbi:2011dn . The present work can be considered as an update of these
studies.

The previous works considered mSUGRA/CMSSM-type boundary conditions for the
SUSY-breaking soft terms. While universality of the soft terms at the GUT
scale is required for the fields belonging to the same 16 representation of
SO(10), the full universal boundary conditions of the CMSSM type are too
strong a condition. For example, there is no fundamental reason why the Higgs
doublets (that typically are in 10-dimensional representations) and the
sfermion soft masses should be degenerate. Thus, strict universality between
matter and Higgs fields can be relaxed Ellis:2002iu . These models are
typically dubbed Non-Universal Higgs Mass models (NUHM). The relaxation of the
universality has an important impact on the seesaw generated flavor violating
entries of the slepton mass matrix. The magnitude of the RG generated flavor
violating entries can either increase or decrease at the leading order due to
the interplay between the matter and Higgs mass terms at the GUT scale.
Cancellations in the flavor violating entry can indeed relax the LFV
constraints on the SUSY parameter space. However, as we are going to see, the
Higgs mass and LFV constraints are such that we have similar conclusions as in
the mSUGRA, for moderate/large values of $\tan\beta$: an improvement of one
order of magnitude in the BR($\mu\to e+\gamma$) bound is sufficient to rule
out significant amount of the parameter space. In the present work, we compare
and contrast the constraints on SUSY seesaw parameter space with CMSSM/mSUGRA
boundary conditions and NUHM boundary conditions.

We find that in the PMNS case, $\tan\beta$ is restricted between 4 to 20.
Lower values of $\tan\beta$ are disfavored by the light Higgs mass constraints
whereas higher values are strongly constrained by the present limit on
BR($\mu\to e+\gamma$) whose rates can be very large, as a consequence of the
sizable observed value of $\theta_{13}$. Furthermore, the one order of
magnitude improvement expected in the near future from the MEG experiment
would rule out most of the SUSY parameter space accessible at the LHC. This
situation is somewhat relaxed in the NUHM model where we considered the masses
of both the Higgs doublets to be the same (the so-called NUHM1 model). We have
further updated the future prospects for both the CKM and PMNS cases in
mSUGRA/CMSSM as well as in NUHM1.

The paper is organized as follows. In section 2, we recap the SUSY seesaw and
discuss the generation of flavor violation in mSUGRA and NUHM1. In section 3,
we discuss the details of our numerical analysis. In section 4, we present our
results. We conclude with a summary and outlook in section 5. Finally, in
appendix A we describe the proposed future experiments and their expected
sensitivity.

##  2 Seesaw in mSUGRA and NUHM

The phenomenology of SUSY Type I seesaw mechanism with universal boundary
conditions (mSUGRA/CMSSM) has been studied in many papers (see Masiero:2004js
; reviews1  for a set of recent works). Here we review some essential features
related to flavor violation for completeness and to do a comparison with the
case of non-universal Higgs masses. To set the notation, the Type I seesaw
mechanism is characterized by a superpotential containing the following terms

$\mathcal{W}\supset\mathbf{Y}_{e}Le^{c}H_{d}+\mathbf{Y}_{\nu}L\nu^{c}H_{u}+{1\over
2}M_{R}\nu^{c}\nu^{c}$ (4)

where $L$ ($e^{c}$) stands for the leptonic doublets (singlets) and $\nu^{c}$
are the right-handed (RH) neutrino superfields (with the generation indices
not explictely written). $\mathbf{Y}_{e}$ and $\mathbf{Y}_{\nu}$ are the
electron and neutrino (Dirac) Yukawa matrices.

In models like CMSSM/mSUGRA, the soft terms are assumed to be universal at the
Grand Unification (GUT) scale, $M_{\rm GUT}\sim 2\times 10^{16}$ GeV. At the
weak scale as is well known, the soft terms are no longer universal due to the
effects of the renormalization group (RG) running. The presence of the RH
neutrinos of eq. (4) at an intermediate scale contribute to the running and
generate flavor violating entries in the left-handed slepton mass matrix at
the weak scale Borzumati:1986qx . At the leading order these terms can be
estimated to be:

$(m^{2}_{\tilde{L}})_{i\neq j}\equiv\left(\Delta^{\ell}_{i\neq
j}\right)_{LL}\approx-\frac{3m_{0}^{2}+A_{0}^{2}}{8\pi^{2}}\sum_{k}\left(Y^{*}_{\nu}\right)_{ik}\left(Y_{\nu}\right)_{jk}\log\left(\frac{M_{X}}{M_{R_{k}}}\right),$
(5)

where $M_{X}$ represents the GUT scale and $M_{R_{k}}$, the scale of the
$k^{\text{th}}$ RH neutrino. $m_{0}$ and $A_{0}$ stand for the usual universal
soft mass and trilinear terms at the high scale. $\mathbf{Y}_{\nu}$, the Dirac
neutrino Yukawa couplings are free parameters in the Type I seesaw mechanism
which cannot be completely determined even after including the complete data
on the neutrino mass matrix Casas:2001sr .

SO(10) models with their matter representations being 16-dimensional provide a
natural setting for the seesaw mechanisms. Furthermore, they provide
information about the neutrino Yukawa couplings. For example, it is known that
as long as we restrict to renormalisable SO(10) models, at least one of the
neutrino Yukawa couplings should be as large as the top Yukawa coupling
Masiero:2002jn . Thus with suitable assumptions for the (left-handed) mixing
of the Dirac Yukawa Neutrino mass matrix, one can make predictions for the
flavor violation generated at the weak scale from eq. (5). Two extreme
scenarios for mixing are typically considered to be present in
$\mathbf{Y}_{\nu}$ Hisano:1995cp ; Masiero:2002jn ; Calibbi:2006nq  :

$\displaystyle\mathbf{Y}_{\nu}$
$\displaystyle=\mathbf{Y}_{u}\qquad\qquad\qquad\textrm{(CKM Case)}$
$\displaystyle\mathbf{Y}_{\nu}$ $\displaystyle=\mathbf{Y}_{u}^{\rm diag}\;{\bf
U}_{\rm PMNS}\quad\textrm{(PMNS Case)},$ (6)

where $\mathbf{Y}_{u}={\bf V}_{\rm CKM}\mathbf{Y}_{u}^{\rm diag}{\bf V}_{\rm
CKM}^{\dagger}$. Both these scenarios can be motivated from concrete models of
fermion masses within the SO(10) framework Masiero:2002jn ; Calibbi:2006nq .
The flavor violating off-diagonal entries at the weak scale, eq. (5), are then
completely determined by assuming $\mathbf{Y}_{\nu}$ as in eq. (6). The
dominant combinations of Yukawa couplings which enter the radiative generation
of $(\Delta^{\ell}_{i\neq j})_{LL}$ are shown in table (1). Notice that the
hierarchical structure of $\mathbf{Y}_{\nu}$ dictated by SO(10) determines
that the leading contribution corresponds to third generation particles
running in the loop. Hence, the flavor violating entries $(\Delta_{12})_{LL}$
responsible for $\mu\rightarrow e\gamma$ process and $(\Delta_{13})_{LL}$
responsible for $\tau\to e\gamma$ depend on $U_{e3}\sim\theta_{13}$ in the
PMNS case. The branching ratios of the LFV decays can be roughly estimated to
be

$\frac{{\rm BR}(l_{i}\rightarrow l_{j}\gamma)}{{\rm BR}(l_{i}\rightarrow
l_{j}\nu\bar{\nu})}\approx\frac{\alpha^{3}}{G^{2}_{F}}\,\frac{\left(\delta_{LL}\right)^{2}_{ij}}{m^{4}_{\rm
susy}}\,\tan^{2}\beta$ (7)

where $m_{\rm susy}$ is a typical SUSY mass and the flavor violation is as
usual parameterize by the following quantity

$\delta{{}^{f}_{ij}}\equiv\frac{\Delta^{f}_{ij}}{m^{2}_{\tilde{f}}}.$ (8)

Generations | PMNS | CKM
---|---|---
$\Delta_{12}$ | $Y_{t}^{2}U_{e3}U_{\mu 3}$ | $Y_{t}^{2}V_{td}V_{ts}$
$\Delta_{23}$ | $Y_{t}^{2}U_{\mu 3}U_{\tau 3}$ | $Y_{t}^{2}V_{tb}V_{ts}$
$\Delta_{31}$ | $Y_{t}^{2}U_{e3}U_{\tau 3}$ | $Y_{t}^{2}V_{td}V_{tb}$
Table 1: The dominant combination of neutrino Yukawa couplings which enter eq.
(5) in CKM and PMNS mixing case.

Let us now turn our attention to the NUHM1 boundary conditions.222Lepton
flavor violation in NUHM models has been previously studied in herrero , where
correlations between $\mu\to e+\gamma$ and $\mu\to e$ conversion rates have
been discussed. At the first sight one might expect that such a modification
has no significant effect on the LFV amplitudes, except for those due to the
modifications in the sparticle spectrum. However, it turns out that this is
not the only modification. The flavor mixing structure of the slepton mass
matrix can also be strongly affected. The radiatively generated flavor
violating entries in eq. (5) take the following form in NUHM models:

$\left(\Delta^{\ell}_{i\neq
j}\right)_{LL}\approx-\frac{2m_{0}^{2}+{m^{2}_{H_{u}}}+A_{0}^{2}}{8\pi^{2}}\sum_{k}\left(Y^{*}_{\nu}\right)_{ik}\left(Y_{\nu}\right)_{jk}\log\left(\frac{M_{X}}{M_{R_{k}}}\right),$
(9)

where $m_{H_{u}}$ is the soft mass of the up-type Higgs at the high scale. In
the present work, we consider the NUHM1 scenario, i.e. $m_{H_{u}}=m_{H_{d}}$
at the GUT scale. Furthermore, there can be a relative sign difference between
the universal mass terms for the matter fields (that we still call $m_{0}$
with abuse of notation) and the Higgs mass terms at the GUT scale. This can
clearly lead to cancellations (for $m^{2}_{H_{u}}\approx-2~{}m_{0}^{2}$) or
enhancements (for $m^{2}_{H_{u}}\gtrsim~{}m_{0}^{2}$)) in the magnitude of the
flavor violating entries at the weak scale compared to mSUGRA.

##  3 Parameter Range and Phenomenological Constraints

As mentioned earlier, we will consider two sets of boundary conditions for the
soft-terms in our numerical analysis. While the mSUGRA is characterized by the
standard ‘four and half’ parameters ($m_{0}$, $M_{1/2}$, $A_{0}$, $\tan\beta$,
sgn($\mu$)), we parametrize the NUHM1 case by $m_{H_{u}}=m_{H_{d}}\equiv
m_{0}-\Delta m_{H}$. Considering the present and future LHC accessible regions
as well as the reach of future flavor physics experiments, we scan the soft
parameter space in the following ranges:

$\displaystyle m_{0}$ $\displaystyle\in[0,\ 5]\ \text{TeV}$
$\displaystyle\Delta m_{H}$ $\displaystyle\in\begin{cases}0&\qquad{\rm
for~{}mSUGRA}\\\ [0,\ 5]&\qquad{\rm for~{}NUHM1}\\\ \end{cases}$
$\displaystyle m_{1/2}$ $\displaystyle\in[0.1,\ 2]\ \text{TeV}$ $\displaystyle
A_{0}$ $\displaystyle\in[-3m_{0},\ +3m_{0}]$ $\displaystyle\text{sgn}(\mu)$
$\displaystyle\in\left\\{-,+\right\\}$ (10)

Note that we use the convention in which $m_{H_{u}}^{2}=$ sgn($m_{H_{u}}$)
$\left|m_{H_{u}}\right|^{2}$. For this range of the parameter space the first
two generations squarks have masses up to $m_{\tilde{q}_{1,2}}\simeq 7$ TeV
and the first two generations sleptons up to $m_{\tilde{\ell}_{1,2}}\simeq 5$
TeV. We include in our scan such spectra beyond the reach of direct SUSY
searches at the LHC, in order to check the capability of the flavor violating
observables in constraining the parameter space.

The numerical analysis is carried out using the SUSEFLAV package
Chowdhury:2011zr . It evaluates 2-loop MSSM RGEs with full $3\times 3$ flavor
mixing effects and also incorporates one-loop SUSY threshold corrections in
all the MSSM parameters. It checks for consistent Radiative Electroweak
Symmetry Breaking (REWSB) by minimizing the one-loop corrected effective
superpotential. The program incorporates the effect of RH neutrinos on MSSM
RGEs and calculates the branching ratios of various LFV processes induced by
such RGE effects. The program also calculates BR($b\rightarrow s\gamma$) in
the minimal flavor violation assumptions. We also calculate the
BR($B_{s}\rightarrow\mu^{+}\mu^{-}$) using ISABMM subroutine of ISAJET
Paige:2003mg . The light Higgs mass is computed using the full two loop
corrections of Degrassi:2001yf ; Brignole:2001jy ; Dedes:2002dy ; slavich .
First, we collect the points which (a) successfully give REWSB, (b) have no
any tachyonic sfermions at the weak scale and (c) have the lightest neutralino
as Lightest Supersymmetric Particle (LSP). Then we calculate all the LFV
observables, BR($b\rightarrow s\gamma$) using the SUSY spectrum evaluated for
each point. Finally, we impose the following experimental constraints on the
data points we collected.

$\displaystyle 121.5~{}\text{GeV}\leq m_{h}$ $\displaystyle\leq\,129.5~{}{\rm
GeV}$ $\displaystyle
m_{\tilde{\chi}^{\pm}}~{}{\rm(lightest~{}Chargino~{}mass)}$
$\displaystyle\geq\,103.5~{}{\rm GeV}$ Nakamura:2010zzi  $\displaystyle{\rm
BR}(B_{s}\rightarrow\mu^{+}\mu^{-})$ $\displaystyle<\,4.5\times 10^{-9}$
Aaij:2012ac  $\displaystyle 2.85\times 10^{-4}\leq{\rm BR}(b\rightarrow
s\gamma)$ $\displaystyle\leq\,4.24\times 10^{-4}$
$\displaystyle(2\sigma)\text{\cite[cite]{\@@bibref{Authors
Phrase1YearPhrase2}{Asner:2010qj}{\@@citephrase{(}}{\@@citephrase{)}}}}.$ (11)

In comparing our predictions for $m_{h}$ with the experimental range of
eq.(1), we take into account 3 GeV of theoretical uncertainty (for a recent
discussion see Arbey:2012dq ). We have not considered the Supersymmetric
solution to $(g-2)_{\mu}$ discrepancy in the present work.

LFV Process | Present bound | Near future sensitivity
---|---|---
|  | of ongoing experiments
BR($\mu\rightarrow e\gamma$) |  $2.4\times 10^{-12}$ Adam:2011ch  |  $10^{-13}$ Hewett:2012ns
BR($\mu\rightarrow eee$) |  $1.0\times 10^{-12}$ Bellgardt:1987du  | $-$
CR($\mu\rightarrow e$ in Ti) |  $4.3\times 10^{-12}$ Dohmen:1993mp  | $-$
BR($\tau\rightarrow e\gamma$) |  $3.0\times 10^{-8}$ Asner:2010qj  | $-$
BR($\tau\rightarrow eee$) |  $3.0\times 10^{-8}$ Asner:2010qj  | $-$
BR($\tau\rightarrow\mu\gamma$) |  $4.5\times 10^{-8}$ Asner:2010qj  |  $10^{-8}$ Hewett:2012ns
BR($\tau\rightarrow\mu\mu\mu$) |  $2.0\times 10^{-8}$ Asner:2010qj  |  $3\times 10^{-9}$ Hewett:2012ns
Table 2: Present bounds and expected sensitivities on LFV processes.

In our study, we assume normal hierarchy in the light neutrino mass spectrum
and set

$\displaystyle m_{\nu_{1}}=0.001~{}\text{eV},~{}~{}m_{\nu_{2}}$
$\displaystyle=\sqrt{\Delta m^{2}_{\rm
sol}+m_{\nu_{1}}^{2}}~{}\text{and}~{}m_{\nu_{3}}=\sqrt{\Delta m^{2}_{\rm
atm}+m_{\nu_{1}}^{2}}$ (12)

where $\Delta m^{2}_{\rm sol}$ and $\Delta m^{2}_{\rm atm}$ are the solar and
atmospheric squared mass differences respectively and we use the central
values obtained from recent global fits on neutrino data Tortola:2012te :

$\displaystyle\Delta m^{2}_{\rm sol}=7.62$ $\displaystyle\times 10^{-5}\,{\rm
eV}^{2}\ {\rm and}\ \Delta m^{2}_{\rm atm}=2.53\times 10^{-3}\,{\rm eV}^{2}$
(13)

Regarding the mixing angles, we take the most conservative value for the
reactor mixing angle and set $|U_{e3}|=0.11$, that corresponds to the lower
limit of the 3$\sigma$ range given by RENO Ahn:2012nd  as well as by the
global fits Tortola:2012te . The remaining two angles of the PMNS matrix are
set to their global fit values $\theta_{12}=33.5^{\circ}$ and
$\theta_{23}=45^{\circ}$. The masses of heavy neutrinos which we use in our
analysis are:

$M_{R_{1}}=10^{6}~{}\text{GeV},~{}~{}M_{R_{2}}=10^{9}~{}\text{GeV},~{}~{}\text{and}~{}~{}M_{R_{3}}=10^{14}~{}\text{GeV}.$
(14)

The recent results from the MEG collaboration Adam:2011ch  have improved the
existing bound on BR($\mu\rightarrow e\gamma$) by one order of magnitude. The
present limits on different LFV observables are summarized in table 2. In the
following subsections, we discuss in detail the results of numerical analysis
carried out in the PMNS and CKM cases with mSUGRA and NUHM1 boundary
conditions and present a quantitative comparisons between them.

##  4 LFV in SUSY SO(10)

Let us start considering the PMNS case, where there is a direct link between
LFV processes and neutrino parameters. Besides the CP violating phases,
$\theta_{13}$ was the only unknown parameter in the leptonic mixing sector
till some time ago. There have been various theoretical models based on the
idea that $\theta_{13}$ could be very small, close to zero. In such a case,
the resulting $\mu\to e+\gamma$ and $\tau\to e+\gamma$ rates could have been
suppressed Antusch:2006vw ; Calibbi:2006ne . Recent experiments prove the
contrary. Both the experiments measuring $\theta_{13}$ are in good agreement
with each other and indicate a sizeable value of $\theta_{13}$. The recent
global fit analysis Tortola:2012te  also leads to similar value of
$\theta_{13}$. All these results indicate that the smallest value of
$|U_{e3}|$ is 0.11 at $3\sigma$. The resulting rates of BR$\left(\mu\to
e\gamma\right)$ are significantly enhanced for such a value of
$\left|U_{e3}\right|$. As we are going to see, the updated MEG limit together
with a large $\theta_{13}$ puts significant constraints on mSUGRA for moderate
as well as large value of $\tan\beta$.333The interplay between large
$\theta_{13}$ and BR$\left(\mu\to e\gamma\right)$ in the context of discrete
flavor groups have been recently discussed in Altarelli:2012bn .

Figure 1: The figure in the left panel shows the BR($\mu\rightarrow e\gamma$)
obtained by scanning the mSUGRA (in red color) and NUHM (in green color)
parameters in the ranges given in eq. (3) and for fixed $\tan\beta=10$ and
$U_{e3}=0.11$ (the lowest value allowed at $3\sigma$ by recent RENO
observation) and satisfy all the constraints in eq. (3). Different horizontal
lines correspond to present and future bounds on BR($\mu\rightarrow e\gamma$).
The figure in the right panel shows the allowed space in the $m_{0}-m_{1/2}$
plane which satisfy the current MEG bound and eq. (3). The region below the
red line is excluded by the direct searches for SUSY at the LHC (CMS_2012, ).

Figure 2: The same as figure 1 for $\tan\beta=40$.

In figures 1 and 2 we present the constraints from BR$\left(\mu\to
e\gamma\right)$ on mSUGRA and NUHM1 parameter space for $\tan\beta=10$ and 40
respectively. As can be seen, while only small part of the paramater space
survives for $\tan\beta=10$ in mSUGRA, it is completely ruled out for
$\tan\beta=40$. The allowed regions for low $\tan\beta$ require very heavy
spectra, i.e. $m_{0}\gtrsim 4$ TeV for small $M_{1/2}$ or $M_{1/2}\gtrsim 2$
TeV for small $m_{0}$. What is surprising is that the constraint on the NUHM1
parameter space is not as weak as one might expect form eq. (9). As we can see
from the figures even in the presence of partial cancellations, most of the
NUHM1 parameter space is going to be explored by MEG. If one removes the light
Higgs mass constraint, points with stronger cancellations would be allowed,
even with $\mu\to e\gamma$ rates below the MEG sensitivity. Thus points
compatible with the Higgs mass bound, eq. (3), do not allow strong
cancellations in the flavor violating entry in eq. (9). For the large
$\tan\beta$ case, the $\mu\to e\gamma$ constraint is so strong that only few
points with $M_{1/2}\gtrsim 800$ GeV are allowed. In the section 4.1, we will
discuss in more detail about the impact of the constraint on $m_{h}$ in mSUGRA
and NUHM1.

Figure 3: The figure in the left panel shows the BR($\mu\rightarrow e\gamma$)
obtained by scanning the mSUGRA parameters in the ranges given in eq. (3) and
for fixed $\tan\beta=10$ and $U_{e3}=0.11$ (the lowest value allowed at
$3\sigma$ by recent RENO observation). The red (blue) colored points
correspond to PMNS (CKM) case. Different horizontal lines correspond to
present and future bounds on BR($\mu\rightarrow e\gamma$). The figure in the
right panel shows the allowed space in the $m_{0}-m_{1/2}$ plane which satisfy
the current MEG bound. The region below the red line is excluded by the
current LHC searches CMS_2012 . Both the plots satisfy all the constraints in
eq. (3).

In the context of the updated MEG limit on BR($\mu\rightarrow e\gamma$), it is
now worthwhile to see what is the situation with the small mixing CKM case.
Here we compare the CKM case and the PMNS case with mSUGRA boundary
conditions. As above, red points correspond to the PMNS case while we use the
blue color for CKM case. The CKM case has highly suppressed branching
fractions due to the smallness of CKM angles (see table (1)) as has been
detailed in Calibbi:2006nq . Though there has been no strong improvements in
the experimental sensitivity compared to the analyses of Calibbi:2006nq , we
update the result with the light Higgs mass constraint. In figure 3 we show
the results for $\tan\beta=10$. As we can see, some part of the parameter
space of the CKM case can be probed by the proposed Project-X experiment444In
appendix A we present a brief summary of all the future experimental
facilities related to the flavor violating observables discussed in the text.
for $\mu\rightarrow e\gamma$. At present the main constraint to this scenario
is simply provided by the $m_{h}$ range of eq. (1), that excludes the regions
with lighter SUSY spectra: $m_{0}\lesssim 2$ TeV for small $M_{1/2}$,
$M_{1/2}\lesssim 1$ TeV for small $m_{0}$, as we can see from the right panel
of the figure. We can also notice that the LHC limits on the mSUGRA parameter
space has already started to constrain regions of the parameter space
otherwise allowed by the bounds in eq. (3).

Let us now turn our attention to other observables like $\mu\to eee$, $\mu\to
e$ conversion in nuclei and $\tau\to\mu\gamma$, which is independent of
$\theta_{13}$. In figures 4, 5 and 6, we show the predicted rates for
$\tau\to\mu\gamma$, $\mu\to eee$ and $\mu\to e$ conversion in the Titanium
nucleus versus the BR$(\mu\to e\gamma)$ (that is at present the most
constraining LFV observable), for the PMNS case in mSUGRA (red points) and in
NUHM1 (green points) as well as for the CKM case (blue points).

As can be seen from figure 4, in the PMNS case, the present MEG limit on
BR$\left(\mu\to e\gamma\right)$ implies BR$(\tau\rightarrow\mu\gamma)\lesssim
10^{-12}$, beyond the reach of the proposed experiments. This is a direct
consequence of the large value of $\theta_{13}$ measured by Daya Bay and RENO.
In fact, from eq.(7,8) and table 1, we have:

$\frac{{\rm BR}(\tau\rightarrow\mu\gamma)}{{\rm BR}(\mu\rightarrow
e\gamma)}\approx\frac{|U_{\tau 3}U_{\mu 3}|^{2}}{|U_{\mu
3}U_{e3}|^{2}}\times{\rm
BR}(\tau\rightarrow\mu\nu\bar{\nu})\approx\mathcal{O}(1).$ (15)

In the CKM case (blue points), the small mixing angle and the $m_{h}$ bound
are such that BR$(\tau\rightarrow\mu\gamma)\lesssim 10^{-10}$. Thus, the
scenarios discussed here allow possible signals of LFV in $\mu-e$ transition
only and evidence for ${\rm BR}(\tau\rightarrow\mu\gamma)$ at future
experiments would strongly disfavor them.

Figure 4: BR($\tau\rightarrow\mu\gamma$) versus BR$(\mu\to e\gamma)$ for the
PMNS case in mSUGRA (red) and NUHM (green), and for the CKM case (blue) for
$\tan\beta$=10 (left), for $\tan\beta$=40 (right). The different horizontal
and vertical lines correspond to present and future limits on the respective
observables.

Figure 5: BR($\mu\rightarrow eee$) versus BR$(\mu\to e\gamma)$ for the PMNS
case in mSUGRA (red) and NUHM (green), and for the CKM case (blue) for
$\tan\beta$=10 (left), for $\tan\beta$=40 (right). The different horizontal
and vertical lines correspond to present and future limits on the respective
observables.

Figure 6: CR($\mu\rightarrow e$ in Ti) versus BR$(\mu\to e\gamma)$ for the
PMNS case in mSUGRA (red) and NUHM (green), and for the CKM case (blue) for
$\tan\beta$=10 (left), for $\tan\beta$=40 (right). The different horizontal
and vertical lines correspond to present and future limits on the respective
observables.

Let us now consider the decay $\mu\rightarrow eee$. It is known (see e.g.
Hisano:2009ae ) that in SUSY (with conserved R-parity) the dominant
contribution to this process arises from the same dipole operator responsible
for $\mu\to e\gamma$, hence the correlation of the two processes is striking:

${\rm BR}(\mu\to eee)\sim\alpha_{\rm em}\times{\rm BR}(\mu\to e\gamma).$ (16)

Such prediction is consistent with our results shown in figure 5 for
$\tan\beta=10$ and 40. The present bound on $\mu\rightarrow eee$ comes from
the SINDRUM experiment at PSI. At present, MEG sets a stronger bound, as
expected. However the future sensitivity reach of MUSIC and Project-X
experiments will be able to go beyond the reach of MEG, testing most of the
NUHM parameter space of our scan.

Let us now discuss $\mu\to e$ conversion in Nuclei, that will represent one of
the most important probes of LFV in the future. Project X and J-PARC are
envisaging facilities where $\mu$ conversion on various Nuclei can be studied
(for a review, please see Kuno:1999jp ). In the present work, we have computed
the $\mu\to e$ conversion rate in Titanium. The conversion rate is as usual
normalized by the capture rate of the muon by the nucleus.

In figure 6, we present our results for CR($\mu\to e~{}{\rm~{}in~{}Ti}$) with
respect to $\mu\to e+\gamma$. We find that there is a significant spread in
the parameter space. This spread is due to the existence of cancellations
between the penguin contributions at low $M_{1/2}$ and in low tan$\beta$
regions, which has been noted earlier in the literature Hisano:1995cp . Still,
we can see that the future experiments will be able to test most of the PMNS
parameter space and will start to constrain the small-mixing scenario (CKM
case).

The above plots have been obtained for the Titanium Nuclei. As in the case of
$\mu\to eee$, it has been noted that the dominant contributions are from the
dipole operators. In such a limit, where only the dipole operators contribute,
one can easily estimate the conversion rate for the new nuclei, X, by knowing
its effective charge, $Z_{eff}$, the form factor $F(q)$ and the atomic number
$Z$ and multiplying the conversion rates presented in the above plots by the
ratio:

$R=\frac{\left[Z_{eff}^{4}\left|F(q)\right|^{2}Z\right]_{\rm
X}}{\left[Z_{eff}^{4}\left|F(q)\right|^{2}Z\right]_{\rm Ti}}$ (17)

The Form factors and $Z_{eff}$ for various Nuclei can be found in
Kitano:2002mt .

###  4.1 Combined effect of the Higgs mass and BR$\left(\mu\rightarrow
e\gamma\right)$ bounds

Figure 7: Here we present combined regions of parameter space allowed by
BR$\left(\mu\rightarrow e\gamma\right)$ and the light Higgs mass ($m_{h}$),
eq. (3), on the PMNS case in mSUGRA and NUHM1.

Given the strong constraints from both BR$\left(\mu\rightarrow e\gamma\right)$
and the light Higgs mass one would wonder how much of the total parameter
space from eq. (3) survives in the PMNS case. In figure 7 we plot the lightest
Higgs mass as a function of $m_{0}$, $M_{1/2}$, $A_{0}$ and $\tan\beta$ in the
three left (right) panels for the mSUGRA (NUHM1) case. In particular, the left
panel of the third row shows the asymmetric regions in $A_{0}$ required by the
Higgs mass range.555We remind that the gluino-driven radiative effects gives a
negative contribution to the stop A-term $A_{t}$, so that, if $A_{0}>0$ only
very large values could still provide a sizeable $|A_{t}|$ at low energy.
Flavor constraints instead are approximately symmetric in $A_{0}$. The last
row in the left panel shows the constraint on $\tan\beta$. As we can see in
the mSUGRA case both low $\tan\beta$ $(\lesssim 5)$ and high $\tan\beta$
$(\gtrsim 20)$ are ruled out. The constraints in the low $\tan\beta$ are due
to $m_{h}$ whereas those at high $\tan\beta$ are from BR$\left(\mu\rightarrow
e\gamma\right)$.

In mSUGRA/CMSSM models, a light Higgs mass around $\sim$ 125 GeV requires
either very heavy stops ($\tilde{t}_{1,2})\sim 4$ TeV or a large stop mixing
triggered by a large low-energy value of the stop $A$-term, $|A_{t}|$. In
either of these cases, one can easily convince oneself, using eq.(5) that the
flavor violating parameter $\left(\delta^{\ell}_{i\neq j}\right)_{LL}$ is not
suppressed. Thus a light Higgs mass $\sim 125$ GeV does not necessarily mean a
suppressed flavor violating entry in spite of the largeness of stops
($\tilde{t}_{1,2}$) or $A$-terms required. In fact flavor violation
constraints are still very strong.

In NUHM1 case this correlation is somewhat lost due to partial cancellation in
the flavor violating entry666In NUHM1 case the cancellations are constrained
by the parameter choice of eq.(3).. All values of $\tan\beta$ are now allowed
and $A_{0}$ is slightly more symmetric compared to the mSUGRA case. The
surprising thing is that imposing the light Higgs mass constraint restricts
the parameter space to be within the reach of MUSIC and Project-X proposals.

##  5 Summary and Outlook

The discovery of a Higgs-like boson at the LHC with a mass close to 125 GeV is
one of the most significant achievements in particle physics of all times. In
the present work, we assumed that the particle seen at the LHC is the lightest
neutral Higgs scalar of the MSSM. We then studied the implications of the
observed mass range on SUSY seesaw models along with recent improvements in
$\text{BR}(\mu\to e+\gamma)$ and measurements of $\theta_{13}$. We looked at
Type I seesaw model assuming SO(10) relations between the top Yukawa and the
heaviest RH neutrino Dirac mass. We assuming two extreme cases of mixing to be
present in the Dirac Yukawa matrix (a) small CKM like mixing and (b) PMNS like
mixing.

We find that there is a strong complementarity in the PMNS case between the
light higgs mass constraint and BR($\mu\to e+\gamma$). The lower $\tan\beta$
regions are strongly constrained by the recent measurement of the light Higgs
mass and high $\tan\beta$ regions by present limit of MEG. What is surprising
is that $\tan\beta>20$ is already ruled out.

The NUHM1 boundary conditions is one more interesting framework where the
interplay between the light Higgs mass constraint and the LFV constraints
comes to play. To relax the LFV constraints one would need strong
cancellations in the flavor violating entry, however regions with large
cancellations are not favored by a light Higgs mass of around 125 GeV. Partial
cancellations are however allowed which put these regions within the reach of
MEG (Project X) for $\mu\to e\gamma$ ($\mu\to eee$).

###### Acknowledgements.

AM would like to acknowledge support from the MIUR PRIN project “Matter-
Antimatter Asymmetry, Dark Matter and Dark Energy in the LHC Era” and the EU
project “Unification in the LHC Era” contract PITN-GA-2009-237920 (UNILHC).
SKV thanks INFN, Padova section and Dipartimento di Fisica ‘Galileo Galilei’,
University of Padova, for supporting his visit. He also thanks DST Ramanujan
Fellowship SR/S2/RJN-25/2008 of Government of India for support. LC and KMP
are grateful to CHEP, IISc for hospitality and support during their visits.
KMP also thanks Anjan S. Joshipura for partial support for this visit.

##  Appendix A Description of Future Experiments and Prospects: Circa 2020

LFV process | Experiment | Future limits | Year (expected)
---|---|---|---
BR($\mu\rightarrow e\gamma$) | MEG Adam:2011ch  | ${\cal O}(10^{-13})$ |  $\sim$ 2013
| Project X projectx  | ${\cal O}(10^{-15})$ |  $>$ 2021
BR($\mu\rightarrow eee$) | Mu3e mu3e  | ${\cal O}(10^{-15})$ |  $\sim$ 2017
|  ” | ${\cal O}(10^{-16})$ |  $>$ 2017
| MUSIC Kuno:2011zz  | ${\cal O}(10^{-16})$ |  $\sim$ 2017
| Project X projectx  | ${\cal O}(10^{-17})$ |  $>$ 2021
CR($\mu\rightarrow e$) | COMET Kuno:2011zz  | ${\cal O}(10^{-17})$ |  $\sim$ 2017
| Mu2e mu2e  | ${\cal O}(10^{-17})$ |  $\sim$ 2020
| PRISM/PRIME Barlow:2011zz ; Kuno:2011zz  | ${\cal O}(10^{-18})$ |  $\sim$ 2020
| Project X projectx  | ${\cal O}(10^{-19})$ |  $>$ 2021
BR($\tau\rightarrow\mu\gamma$) | Belle II Aushev:2010bq  | ${\cal O}(10^{-8})$ |  $>$ 2020
BR($\tau\rightarrow\mu\mu\mu$) | Belle II Aushev:2010bq  | ${\cal O}(10^{-10})$ |  $>$ 2020
BR($\tau\rightarrow e\gamma$) | Super B Hewett:2012ns  | ${\cal O}(10^{-9})$ |  $>$ 2020
BR($\tau\rightarrow\mu\gamma$) | Super B Hewett:2012ns  | ${\cal O}(10^{-9})$ |  $>$ 2020
BR($\tau\rightarrow\mu\mu\mu$) | Super B Hewett:2012ns  | ${\cal O}(10^{-10})$ |  $>$ 2020
Table 3: Future sensitivities of next-generation experiments.

PRISM Barlow:2011zz ; Kuno:2011zz  or Phase Rotated Intense Slow Muon source
is an upcoming facility at J-PARC. It will accelerates muon beam using a
magnetic field inside a muon storage ring and will deliver $10^{12}$
$\mu/$second at an energy of few 10s of MeV. PRIME Barlow:2011zz ; Kuno:2011zz
is a $\mu-e$ conversion detector for PRISM designed to stop clumps of $\mu$ in
a thin foil, having an energy of about 20 MeV. Because of high
monochromaticity, large pulse rate and very low background level it would
reach a sensitivity of detecting branching ratios of ($\mu\rightarrow e$)
conversion in nuclei around $10^{-18}$ within few years of running. The PRIME
will improve the sensitivities to $\mu-e$ conversions by two orders of
magnitude compared to the next-generation experiment COMET (Coherent Muon to
Electron Transition) experiment.

Project X projectx , proposed at Fermilab, is a next-generation experiment
which has the potential to deliver very high power unprecedentedly intense
$\mu$-beams for precise measurements of the rare muon decays. Project X will
use 0.5 MW beam of $\mu$ accelerated at 3 GeV which will be generated from
high-power primary proton beam. With such an intense continuous muon beam,
Project X can look for $\mu\rightarrow e\gamma$ with sensitivity of ${\cal
O}(10^{-15})$ which is two order of magnitude improvement over the future
reach of MEG. However, sensitivity below $10^{-15}$ appears beyond the reach
of Project X unless innovative ideas regarding the detectors emerge. Like
$\mu\rightarrow e\gamma$, searches for $\mu\rightarrow 3e$ also need
continuous muon beam. Experiments at Project X will improve the sensitivity to
$\mu\rightarrow 3e$ decays by at least three to four orders of magnitude,
assuming significant efforts to develop new detector technologies. The Project
X proposals for studying the $\mu-e$ conversion in nuclei are particularly
interesting. Two distinct scenarios has been proposed. If the next-generation
round of experiments (COMET and Mu2e) observe any signal of $\mu-e$
conversion, then the available muon beams at Project X would allow further
precision studies with several $\mu-e$ events in different nuclei. If no
signal of $\mu-e$ will be found in next-generation experiments, Project X
could reach the sensitivities ${\cal O}(10^{-19})$ or beyond given some
improvements in beam technologies. To achieve these goals, a proposal to use
muon storage ring installed in the muon beam line is under consideration.

The Belle II (Aushev:2010bq, ) is an upgrade of the existing Belle detector at
the KEK B-factory in Japan. It will make use of the upgraded Super KEKB
accelerator. The Belle II detector is expected to collect 40 times more
luminosity ($8\times 10^{35}\,{\rm cm}^{-2}\,{\rm s}^{-1}$) than the previous
generation Belle detector and by 2022 it would collect an integrated
luminosity of $\sim 50\,{\rm ab}^{-1}$. The sensitivity for the $\tau$ two
body decays is expected to improved by an order of magnitude.

Super B factory Hewett:2012ns  is a proposed high luminosity electron-positron
collider. With an integrated luminosity of 75 ab-1, Super B would be able to
explore a significant portion of parameter space of new physics scenarios by
searching for LFV in $\tau$ decays. It is going to improve the sensitivities
to different channels of $\tau$ decays by at least an order of magnitude or
two. The future sensitivities of all these experiments are summarized in the
table 3.

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