# Warped $R-$Parity Violation

B.C. Allanach<EMAIL_ADDRESS>DAMTP, CMS, Wilberforce Road,
University of Cambridge, Cambridge, CB3 0WA, United Kingdom A. M. Iyer
<EMAIL_ADDRESS>Department of Theoretical Physics, Tata Institute
of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005, India K.
Sridhar<EMAIL_ADDRESS>Department of Theoretical Physics, Tata
Institute of Fundamental Research, Homi Bhabha Road, Colaba, Mumbai 400 005,
India

###### Abstract

We consider a modified Randall-Sundrum (RS) framework between the Planck scale
and the GUT scale. In this scenario, RS works as a theory of flavour and not
as a solution to the hierarchy problem. The latter is resolved by
supersymmetrising the bulk, so that the minimal supersymmetric standard model
being the effective 4-dimensional theory. Matter fields are localised in the
bulk in order to fit fermion-mass and mixing-data. If $R$-parity violating
($\not{R}_{p}$) terms are allowed in the superpotential, their orders of
magnitude throughout flavour space are then predicted, resulting in rich
flavour textures. If the $\not{R}_{p}$ contributions to neutrino masses are
somewhat suppressed, then lepton-number violating models exist which explain
the neutrino oscillation data while not being in contradiction with current
experimental bounds. Another promising model is one where baryon number is
violated and Dirac neutrino masses result solely from fermion localisation. We
sketch the likely discovery signatures of the baryon-number and the lepton-
number violating cases.

###### pacs:

73.21.Hb, 73.21.La, 73.50.Bk

††preprint: DAMTP-2016-3 TIFR/TH/15-44

##  I Introduction

The discovery of the Higgs boson by the CMS Chatrchyan et al. (2012) and ATLAS
Aad et al. (2012); Chatrchyan et al. (2013) collaborations at the Large Hadron
Collider has validated the status of the Standard Model (SM) as the correct
theory of nature at the electroweak scale. The existence of a fundamental
scalar in the SM raises questions regarding the stability of the Higgs mass in
the face of radiative corrections. Supersymmetry emerges as one of the most
exciting prospects to address this problem due to its renormalizable nature
and its consistency with electroweak precision data. The model, however,
introduces a number of additional parameters which necessitates the study of
its phenomenological implications using simplified models.

However, supersymmetry in its minimal form viz the minimal supersymmetric
standard model (MSSM) does not give an explanation of the disparate couplings
of the Higgs boson to different generations of fermions. This is also referred
to as the fermion mass problem. In supersymmetry this problem can be addressed
by considering strong wave function renormalization of the matter fields
Nelson and Strassler (2000); Dudas et al. (2010). This is due to
renormalization group (RG) running from some fundamental renormalization scale
(where the Yukawa as well as the soft mass parameters are anarchic) to a low
scale where they develop a hierarchical structure due to renormalization
effects. The different running can be accounted by the different anomalous
dimensions of each matter field coupled to some conformal sector. After
canonically normalizing the kinetic terms, the superpotential terms are given
by

$\displaystyle\mathcal{W}$ $\displaystyle=$
$\displaystyle\epsilon^{q_{i}+h_{u}+u_{j}}\left(\hat{Y}^{ij}_{U}+\hat{A}^{ij}_{U}X\right)Q_{i}H_{u}U_{j}+\epsilon^{q_{i}+h_{u}+d_{j}}\left(\hat{Y}^{ij}_{D}+\hat{A}^{ij}_{D}X\right)Q_{i}H_{d}D_{j}$
(1) $\displaystyle+$
$\displaystyle\epsilon^{l_{i}+h_{d}+e_{j}}\left(\hat{Y}^{ij}_{e}+\hat{A}^{ij}_{e}X\right)L_{i}H_{d}E_{j}$

while the Kähler terms are given by

$\mathcal{K}=\sum_{F=Q,U,D,L,E}F^{\dagger}F+\hat{C}_{ij}\epsilon^{f_{i}+f_{j}}X^{\dagger}XF^{\dagger}F$
(2)

where $i$ and $j$ are generation indices and quantities with hatted quantities
denoting $\mathcal{O}(1)$ parameters. $X$ is the SUSY breaking spurion
parametrized as $X=\theta^{2}~{}F$. The expansion parameter $\epsilon\sim
0.02$ while the ‘charges’ $q_{i},h_{u,d}$ can be considered to be anomalous
dimensions of the matter field coupling to a strong sector. Alternatively,
they can be considered to be charges of the field under an extended gauge
group $U(1)_{FN}$ Froggatt and Nielsen (1979). The fermion mass matrix is then
given as

$m_{f}\sim\epsilon^{q_{i}+h_{u}+u_{j}}\frac{v}{\sqrt{2}}$ (3)

where $v\sim 246$ GeV is the vacuum expectation value (VEV) of the Higgs
field. The charges $q_{i}$ are determined with the requirement of reproducing
the correct pattern of fermion mass and mixing angles. Soft supersymmetry
breaking terms are generated when $F$ terms attains a VEV giving rise to the
gravitino mass $m_{3/2}=\frac{\langle F\rangle}{M_{Pl}}$. The mechanism which
fixes the fermion masses and mixing angles will also determine the soft
supersymmetry-breaking mass parameters, for example the squark mass-squared
terms:

$\tilde{m}^{2}_{ij}\sim{\mathcal{O}}(\epsilon^{q_{i}+q_{j}}m^{2}_{3/2}),$ (4)

and similarly for the other family-dependent soft supersymmetry-breaking
terms. This lends a certain level of predictivity to the orders of magnitude
of soft breaking terms.

The terms in Eqs. 1,2 conserve $R-$parity Farrar and Fayet (1978); Dimopoulos
and Georgi (1981), which is defined as

$R=(-1)^{3B+L+2s}$ (5)

where $s$ is spin of a particle and $B(L)$ is its corresponding baryon
(lepton) number (alternatively, the same terms conserve matter parity Sakai
and Yanagida (1982); Dimopoulos et al. (1982); Weinberg (1982)). While $R$
parity conservation has many useful features, predicting the stability of dark
matter and a stable proton, there is no a priori reason for it to be a
symmetry of the lagrangian111Scenarios in which R parity originated as a
discrete remnant of some extended gauge symmetries were considered in Krauss
and Wilczek (1989); Font et al. (1989); Mohapatra (1986); Martin (1992, 1996);
Allanach et al. (2004).. Thus in general, the super-potential terms in Eq.(1)
can also be extended to include terms which violate baryon and lepton-number,
and are referred to as R-parity violating ($\not{R}_{p}$) terms222For a
detailed review on ($\not{R}_{p}$) supersymmtery see Ref. Barbier et al.
(2005).. The most general $\not{R}_{p}$ terms are given by

$\displaystyle\mathcal{W}^{\Delta L=1}_{\not{R}_{p}}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{l_{i}+l_{j}+e_{k}}}{2}\lambda_{ijk}L_{i}L_{j}E_{k}+\frac{\epsilon^{l_{i}+q_{j}+d_{k}}}{2}\lambda^{\prime}_{ijk}L_{i}Q_{j}D_{k}+\epsilon^{l_{i}+h_{u}}\mu^{\prime}_{i}L_{i}H_{u},$
(6) $\displaystyle\mathcal{W}^{\Delta B=1}_{\not{R}_{p}}$ $\displaystyle=$
$\displaystyle\frac{\epsilon^{u_{i}+d_{j}+d_{k}}}{2}\lambda^{\prime\prime}_{ijk}U_{i}D_{j}D_{k},$

where we have omitted the gauge indices.

Whenever $R-$parity violation is introduced, one wonders where the apparent
relic density of dark matter might come from, given that it appears to be
stable on cosmological scales, and any MSSM fields will decay much too
quickly. One obvious answer is that massive hidden sector matter, might
provide dark matter. Unfortunately, this would result in no direct or indirect
signals for dark matter detection. Another possibility Takayama and Yamaguchi
(2000); Moreau and Chemtob (2002) is that the lightest supersymmetric particle
is the gravitino, which has Planck suppressed couplings anyway. With
additional smallish $\not{R}_{p}$ violating couplings, it is possible that its
lifetime is much longer than the age of the universe, resulting in a good dark
matter candidate. We leave this aspect of the model building to a future
paper.

The anomalous dimensions of the matter fields can also be considered dual to
the parameter which controls the localization of the field in an extra-
dimensional scenario with strong warping Gherghetta and Pomarol (2000);
Contino and Pomarol (2004); Gherghetta (2011). In this paper we consider the
effects of introducing all such terms in a supersymmetric model on a
gravitational background with strong warping also referred at as Randall-
Sundrum (RS) model Randall and Sundrum (1999). In Section II we briefly
introduce the model and set it up to understand the phenomenology. We review
the technique used to determine the RS-model parameters which fit the fermion-
mass and mixing-data at the high scale. The mathematical expressions used to
determine the soft- and $\not{R}_{p}$-parameters are presented. In section III
we discuss the implications of introducing $\not{R}_{p}$ couplings on various
low-energy processes. We find that if baryon-number and lepton-number
violating terms are simultaneously allowed, consistency with constraints from
proton decay require a slightly fine tuned choice of $~{}10^{-4}$ in some
undetermined parameters usually expected to be of $\mathcal{O}(1)$. We then
proceed to discuss simplified cases where either baryon- or lepton-number is
conserved separately. Scenarios with lepton-number violation present solutions
where the neutrinos can be Dirac-like even in the presence lepton number
violating terms. In each case, we briefly comment on the LHC phenomenology,
before presenting our conclusions.

##  II GUT-scale Randall-Sundrum model

We consider the following modified version of the original setup referred to
as ‘GUT-scale RS’ Marti and Pomarol (2001); Choi et al. (2004, 2003); Dudas et
al. (2010); Brummer et al. (2011); Iyer and Vempati (2013a). Like the original
RS model, it is a model of single extra-dimension compactified on a
$S_{1}/Z_{2}$ orbifold. The line-element is given as

$ds^{2}=e^{-2\sigma(y)}\eta_{\mu\nu}dx^{\mu}dx^{\nu}+dy^{2}$ (7)

where $\sigma(y)=k|y|$ with $k$ denoting the reduced Planck scale and $R\sim
1/k$ being the size of the extra spatial dimension $y$. There are two opposite
tension branes at each of the orbifold fixed points, $y=0$ and $y=\pi R$.
Assuming the scale of physics at the $y=0$ brane to be $M_{Pl}$, the effective
scale induced at the brane at $y=\pi R$ is given by

$M_{IR}=e^{-\sigma(\pi)}M_{Pl}=\epsilon M_{Pl}\sim M_{GUT}$ (8)

Thus, in comparison to the original proposal in Ref. Randall and Sundrum
(1999), the warp factor in this case is much larger and hence ab initio the
model is no longer a solution to the hierarchy problem. Hence, supersymmetry
is introduced into the bulk. With the GUT-scale Kaluza Klein (KK) modes
decoupled from the theory, the spectrum of the effective 4D theory is that of
MSSM.

We assume the two Higgs doublets to be localized on the infra-red (IR) brane
(i.e. on the GUT brane) while the matter and gauge multiplets are in the bulk.
The expressions for the fermion mass matrices are

$\displaystyle(m_{u})_{ij}$ $\displaystyle=$ $\displaystyle
v_{u}\,\hat{Y}_{u_{ij}}f(c_{Q_{i}})f(c_{u_{j}})$ $\displaystyle(m_{d})_{ij}$
$\displaystyle=$ $\displaystyle
v_{d}\,\hat{Y}_{d_{ij}}f(c_{Q_{i}})f(c_{d_{j}})$ $\displaystyle(m_{e})_{ij}$
$\displaystyle=$ $\displaystyle
v_{d}\,\hat{Y}_{e_{ij}}f(c_{L_{i}})f(c_{e_{j}})$ (9)

where $i$ and $j$ are generation indices, $v_{u}=\frac{v}{2}\sin\beta$ and
$v_{d}=\frac{v}{2}\sin\beta$ are the VEVs of the up-type and down-type Higgs’
of the MSSM, respectively and $c_{Z_{i}}$ are the dimensionless bulk mass
parameters of the matter multiplets $Z\in\\{Q,u,d,L,e\\}$. The corresponding
zero-mode wave-function $f$ is defined to be Chang et al. (2000); Gherghetta
and Pomarol (2000)

$f(c)=\sqrt{\frac{1-2c}{e^{(1-2c)\pi kR}-1}}e^{(0.5-c)kR\pi}.$ (10)

Using Eq. 9 and choosing $c_{Z_{i}}\sim\mathcal{O}(1)$ and
$\hat{Y}_{{U,D,E}}^{ij}\sim\mathcal{O}(1)$, one can explain the observed
hierarchy in the fermion masses and mixings Chang et al. (2000); Gherghetta
and Pomarol (2000); Grossman and Neubert (2000); Iyer and Vempati (2012,
2013a).

A SUSY-breaking spurion $X=\theta^{2}F$ is introduced on the GUT brane and IR
brane-localized contact interactions are introduced between the SM fields and
the SUSY breaking spurion $X$. The soft SUSY breaking terms are then generated
when the $F$-term attains a VEV and are given by

$\displaystyle m^{2}_{H_{u,d}}$ $\displaystyle=$
$\displaystyle\hat{h}_{u,d}~{}m_{3/2}^{2}$
$\displaystyle(m_{{\tilde{Z}}}^{2})_{ij}$ $\displaystyle=$ $\displaystyle
m_{3/2}^{2}~{}\hat{\beta}_{\tilde{Z}_{ij}}~{}f(c_{Z_{i}})f(c_{Z_{j}})$
$\displaystyle A_{U}^{ij}$ $\displaystyle=$ $\displaystyle
m_{3/2}\hat{A}_{U}^{ij}f(c_{Q_{i}})f(c_{u_{j}}),$ $\displaystyle A_{D}^{ij}$
$\displaystyle=$ $\displaystyle
m_{3/2}\hat{A}_{D}^{ij}f(c_{Q_{i}})f(c_{d_{j}}),$ $\displaystyle A_{E}^{ij}$
$\displaystyle=$ $\displaystyle
m_{3/2}\hat{A}_{E}^{ij}f(c_{L_{i}})f(c_{e_{j}}),$ $\displaystyle m_{\alpha}$
$\displaystyle=$ $\displaystyle\hat{g}_{\alpha}m_{3/2},$ (11)

where quantities denoted with a hat are fundamental dimensionless parameters,
which we assume are $\mathcal{O}(1)$. Here, $m^{2}_{H_{u,d}}$ are the up- and
down- Higgs mass squared soft SUSY-breaking parameters,
$(m_{\tilde{Z}}^{2})_{ij}$ the soft SUSY breaking mass squared matrix for
sfermion $\tilde{Z}$, $A_{(U,D,E)}^{ij}$ the matrix of trilinear soft SUSY-
breaking interactions for the up-quark, down-quark and charged leptons and
$m_{i}$ the $i^{th}$ gaugino mass (where $\alpha\in\\{3,2,1\\}$ denotes the
MSSM gauge group $SU(3)$, $SU(2)_{L}$, $U(1)_{Y}$), respectively.

$R$-parity violating interactions are introduced on this brane and so they are
considered to be generated at this scale333An equivalent description would
correspond to the Higgs doublets and the $\not{R}_{p}$ violating terms
localized on the ultra-violet (UV) brane.. Like the soft parameters, the
effective four-dimensional (4D) $\not{R}_{p}$ parameters can also be expressed
in terms of the bulk wavefunction of the fields. The effective 4D
$\not{R}_{p}$ violating superpotential in a warped background is written

$\displaystyle\mathcal{W}^{\Delta L=1}_{\not{R}_{p}}$ $\displaystyle=$
$\displaystyle\int dye^{-3ky}\delta(y-\pi
R)\left(\lambda^{(5)}_{ijk}L_{i}L_{j}E_{k}+\frac{1}{2}\lambda^{{}^{\prime}(5)}_{ijk}L_{i}Q_{j}D_{k}+\mu^{(5)}_{i}L_{i}H_{u}\right)$
(12) $\displaystyle\mathcal{W}^{\Delta B=1}_{\not{R}_{p}}$ $\displaystyle=$
$\displaystyle\int dye^{-3ky}\delta(y-\pi
R)\lambda^{{}^{\prime\prime}(5)}_{ijk}U_{i}D_{j}D_{k}.$

$\lambda^{(5)}_{ijk},\lambda^{{}^{\prime}(5)}_{ijk},\lambda^{{}^{\prime\prime}(5)}_{ijk}$
are 5D $\not{R}_{p}$ couplings which have mass dimension $-1$. Performing a KK
decomposition of the fields and retaining only the zero modes444Higher KK
modes in this model have mass $\sim M_{GUT}$ and are decoupled., the effective
4D $\not{R}_{p}$ couplings are written as

$\displaystyle\lambda_{ijk}$ $\displaystyle=$
$\displaystyle\hat{\lambda}_{ijk}f(c_{L_{i}})f(c_{L_{j}})f(c_{E_{k}})$
$\displaystyle\lambda^{\prime}_{ijk}$ $\displaystyle=$
$\displaystyle\hat{\lambda}^{\prime}_{ijk}f(c_{L_{i}})f(c_{Q_{j}})f(c_{D_{k}})$
$\displaystyle\mu_{i}$ $\displaystyle=$
$\displaystyle\mu\hat{\mu}_{i}f(c_{L_{i}})e^{-kR\pi}$ (13)

for the $\Delta L=1$ terms. $\mu$ is of order the electroweak scale and is
chosen here to be 100 GeV.
$\hat{\lambda}_{ijk},\hat{\lambda^{\prime}_{ijk}},\hat{\mu}_{i}$ are
dimensionless $\mathcal{O}(1)$ couplings, where as $\hat{\lambda}\equiv
k\lambda^{(5)}_{ijk}$, $\hat{\lambda}^{\prime}\equiv
k\lambda^{{}^{\prime}(5)}_{ijk}$ and $\hat{\mu}\equiv k\mu^{(5)}$. The $\Delta
B=1$ $\not{R}_{p}$ couplings are

$\lambda^{\prime\prime}_{ijk}=\hat{\lambda}^{\prime\prime}_{ijk}f(c_{U_{i}})f(c_{D_{j}})f(c_{D_{k}})$
(14)

with $\hat{\lambda}^{\prime\prime}_{ijk}=k\lambda^{\prime\prime(5)}_{ijk}$.
The supersymmetric parameters in Eqs. 11,13 and 14 are determined using the
set of same set of $c_{i}$ parameters that fit the fermion masses and mixing
at the GUT scale using Eqs. 9 and 10. This gives an order-of-magnitude level
of predictability for this framework, as these high-scale parameters can be
subsequently evolved to generate a characteristic spectrum at the low scale.
The set of $\mathcal{O}(1)$ parameters (which includes the $c_{i}$ parameters
as well as the $\mathcal{O}$(1) Yukawa parameters $\hat{Y}_{U,D,E}^{ij}$) is
determined by performing a $\chi^{2}$ fit of their GUT-scale values to the
data Iyer and Vempati (2013a). The $\chi^{2}$ function is defined as

$\chi^{2}=\sum_{i}\frac{\left(\mathcal{O}^{\textrm{theory}}_{i}(\\{c_{j}\\},\\{\hat{Y}_{U}^{ij}\\},\\{\hat{Y}_{D}^{ij}\\},\\{\hat{Y}_{E}^{ij}\\})-\mathcal{O}^{\textrm{expt}}_{i}\right)^{2}}{\sigma_{i}^{2}},$
(15)

where $O_{i}^{\textrm{theory}}$ denotes the theoretical prediction for
observable $O_{i}$, $O_{i}^{\textrm{expt}}$ denotes the empirical central
value and the experimental uncertainty is written $\sigma_{i}$.
$i\in\\{m_{u},m_{d},m_{c},m_{s},m_{t},m_{b},|V_{CKM}|_{ij}^{\textrm{ind}}\\}$
constitute the hadronic observables whereas
$i\in\\{m_{e},m_{\mu},m_{\tau},|V_{PMNS}|_{ij}^{\textrm{ind}}\\}$ constitute
the leptonic observables. Both are fit independently (‘${}^{\textrm{ind}}$’
indicates that the absolute values of a selection of independent entries - the
off-diagonal entries - of the CKM and PMNS matrices are fit, respectively). We
refer the interested reader to Ref. Iyer and Vempati (2013a) for further
details.

Figure 1: Localization of fermion profiles in the bulk depends upon the $c$
parameter. The Higgs is assumed to be strongly localized on the IR brane, as
shown Raychaudhuri and Sridhar .

Because of the small value of the warp factor $\epsilon\sim 0.02$, the $c_{i}$
parameters for the lighter generations (eg. the electron or the up and down
quark) are close to $2.5$ while for the third generation of superfields
containing a top, they are close to -1. $c_{i}<0.5$ reflects a localization
more on the IR brane (where the Higgs doublets are localized), while
$c_{i}>0.5$ localizes the superfield closer to the UV brane, as heuristically
depicted in Fig. 1. The $c_{i}$ parameters for all charged matter superfields
except for $Q^{3}$ and $t_{R}$ are scanned in the range $[0,\ 3.5]$, while
$c_{Q^{3}}$ is scanned is scanned in $[0,\ 1.5]$ and $c_{t_{R}}$ is restricted
to $[-2.5,\ 0.5]$. This different choice for $c_{Q_{3}},c_{t_{R}}$ is to
facilitate a good fit to the top quark mass. We remind the reader that fits to
fermion mass and mixing data are done independently for the quark sector and
the leptonic sector, since to a good approximation (i.e. at tree-level), the
two sectors are decoupled.

The fits in the leptonic sector includes fitting the neutrino data by means of
introduction of three parameters $c_{N_{i}}$ corresponding to three right
handed neutrinos. To account for small neutrino masses at the sub eV level,
$c_{N_{i}}$ are scanned in the range $[5.5,\ 7]$ in order to imbue Dirac
neutrino masses via Eq. 9. The presence of lepton-number violating operators
gives rise to additional Majorana contributions to the neutrino masses. By
focussing on regions of the parameter space where these contributions are
suppressed, we will find that the dominant contribution to the neutrino mass
is from Eq. 9 and hence are primarily of Dirac type.

The fits are performed for two separate values of $\tan\beta=5,10$. Smaller
$\tan\beta$ facilitates a localization of the light down sector fields closer
to the UV brane owing to a larger value of $\cos\beta$ in the mass matrix in
Eq. 9. As we shall explain later, this helps in generating a smaller value for
the $\not{R}_{p}$ couplings, enabling them to satisfy experimental constraints
more easily , which are typically upper bounds. The $\mathcal{O}(1)$ model
parameters are determined by minimising the $\chi^{2}$ function in Eq. 15. The
minimisation is performed by MINUIT James and Roos (1975) which looks for a
minimum around a guess value of $c$ parameters and $\mathcal{O}$(1) Yukawa
parameters. The guess values are randomly generated in the ranges given above.
This is repeated for $10^{5}$ choices of guess values each constituting a
separate minimization. MINUIT has trouble searching our parameter space, and
finds many distinct local minima, depending upon which random guess we start
with. We view this as a sampling of the ‘good-fit’ parameter space, and all
points which satisfy $\chi^{2}<10$ are accepted as being a reasonable ‘fit’.
We remind the reader that this is not a fit to data in the usual statistical
sense: rather it is a fit to the orders of magnitude of the masses and
mixings. In addition to the $c_{i}$ parameters, the $R-$parity conserving
hatted $\mathcal{O}$(1) Yukawa parameters are all allowed to vary between 0.1
and 10, whereas the hatted $\not{R}_{p}$ violating parameters are set to 1 and
are not varied. With the $c_{i}$’s fixed in this manner, for each sampling, we
predict the orders of magnitude of the $\not{R}_{p}$ parameters.

##  III $\not{R}_{p}$ parameters

We now focus on the distribution of the various $\not{R}_{p}$ couplings which
are determined from the fermion mass fits. As given in Eq.I, $\not{R}_{p}$
terms include both baryon-number and lepton-number violating interactions. The
lepton-number violating interactions include the trilinear couplings
($\lambda_{ijk},\lambda^{\prime}_{ijk}$) and the bilinear operators $\mu_{i}$.
$\lambda_{ijk}$ is anti-symmetric in $i\leftrightarrow j$ because of the
$SU(2)_{L}$ structure, as is $\lambda^{\prime\prime}_{kij}$ because of the
implicit $SU(3)$ structure.

On account of the introduction of the $\not{R}_{p}$ operators on the same
brane as the Higgs superfields, their magnitude can be roughly understood from
the generation indices in these couplings. For instance, consider
$\lambda_{111}$ which is a product of the zero mode profiles of some first-
generation fermions and $\lambda_{333}$ is the corresponding product of third-
generation fermions. Since, as Fig. 1 illustrates, the lighter fermion
generations have a tendency to be localized away from the Higgs ($c_{i}>0.5$),
the corresponding value of the profile on the IR brane is small. The third
generation is relatively heavy and has a value of $c_{i}$ smaller than those
for the lighter generations. As a result the corresponding value of the
profile on the IR brane is relatively larger. This results in a larger value
for $\lambda_{333}$ as compared to $\lambda_{111}$. Similarly,
$\mu_{3}\geq\mu_{2}\geq\mu_{1}$. In evaluating the $\not{R}_{p}$ parameters,
the $\mathcal{O}(1)$ parameters
$\hat{\lambda}_{ijk},\hat{\lambda}^{\prime}_{ijk},\hat{\lambda}^{\prime\prime}_{ijk}$
and $\hat{\mu}_{i}$ were all chosen to be 1 (unless they are set to zero by
requiring baryon or lepton number conservation). We therefore should bear in
mind that we provide order of magnitude predictions, which will be multiplied
by some order one parameter.

The predictions thus obtained are then filtered against 2 $\sigma$ upper
bounds on $\not{R}_{p}$ violating parameters, for instance from the non-
observation of $\mu\rightarrow e\gamma$ de Gouvea et al. (2001), leptonic
decay of long lived neutral Kaon Choudhury and Roy (1996), bounds from
$n-\bar{n}$ oscillations and double neutron decay Goity and Sher (1995) or
constraints from the electroweak precision tests Bhattacharyya et al. (1995a,
b); Ledroit and Sajot (1998) etc. A complete list of constraints on the
various $\not{R}_{p}$ parameters that we use is given in Tables 6.1 to 6.5 of
Ref. Barbier et al. (2005), although in the first instance we do not apply
bounds from nucleon decay, upon which more later. The constraints do depend
upon the supersymmetric spectrum, for example the branching ratio of
$B\rightarrow\tau\nu$

$\lambda^{\prime}_{333}<0.32\left(\frac{m_{\tilde{b}_{R}}}{\textrm{100
GeV}}\right)$ (16)

depends upon the right-handed sbottom mass $m_{\tilde{b}_{R}}$. We shall
provide predictions for viable ranges of $\not{R}_{p}$ violating parameters
for soft masses $\tilde{m}\gtrsim 300$ GeV. If any one of the 2$\sigma$ bounds
is violated in the case with all hatted $\not{R}_{p}$ violating parameters
fixed to one and 300 GeV sparticles, the point is discarded. After this
filtering, we obtain 2203 good-fit parameter points to the quark mass and
mixing data and 848 to the lepton mass and mixing data. We combine each set of
good-fit quark parameters with each set of good-fit lepton parameters (since
they are approximately independent, as explained above) in order to determine
the possible ranges of the various $\not{R}_{p}$ violating couplings.

Figure 2:  Pattern of $\not{R}_{p}$ couplings. The vertical bars give the
range of couplings that result from good fits to fermion masses and mixings
for $\tan\beta=5$, and that respect experimental bounds on $\not{R}_{p}$
couplings (if either only the baryon number violating or lepton-number
violating couplings are allowed). The points correspond to a pattern from one
particular fit (see sections III.2,III.3 for details).

Fig. 2 gives the ranges of the $\not{R}_{p}$ couplings predicted from the
good-fit scanned points, and constitutes the main result of the present paper.
Dimensionless $\not{R}_{p}$ couplings that are larger than around $10^{-6}$
result in prompt decays of the lightest supersymmetric particle at colliders,
whereas if all couplings are smaller than $10^{-6}$, displaced couplings
result. We note that the smallest couplings are always predicted to be larger
than this lower limit and so $\not{R}_{p}$ decays are prompt.

We note from Fig. 2 that the $\lambda^{\prime}_{ijk}$ couplings have a
possibility to be smaller than the $\lambda_{ijk}$ and the
$\lambda^{\prime\prime}_{ijk}$ couplings. This can be attributed to the fact
that, in the latter case, the couplings are separately determined by the fits
to the lepton and quark sector. As a result the individual $c_{i}$ parameters
in each sector are interlinked so as to reproduce the correct hierarchy in the
mass matrix. For instance, given a choice $c_{L_{1}}$, there is less freedom
in the choice of $c_{L_{2,3}}$. The $\lambda^{\prime}_{ijk}$ couplings on the
other hand, depend on $c_{L_{i}},c_{Q_{j}}$ and $c_{D_{k}}$. Thus, for a given
choice of $c_{Q_{j}}$, $c_{D_{k}}$, which are related from the quark mass
fits, there is freedom in the choice of $c_{L_{i}}$ which are determined from
the fits to leptonic sector and are decoupled from the quark sector at tree
level. We now proceed to discussing the phenomenological implications of the
presence of these $\not{R}_{p}$ parameters.

###  III.1 Nucleon decay

Figure 3: Possible $\not{R}_{p}$ violating process
($p\rightarrow\pi^{0}e^{+}$) yielding a non-zero decay rate for non-zero
$\lambda^{\prime}_{11j}\lambda^{\prime\prime}_{11j}$.

The presence of both lepton- and baryon-number violating terms in the
lagrangian can give rise to small proton-decay lifetimes for baryon and lepton
number violating couplings being simultaneously non-zero. For instance, a
combination of $\lambda^{\prime\prime}_{ijk},\lambda^{\prime}_{ijk}$ can give
rise to the contribution to proton decay shown in Fig. 3. This leads to
particularly stringent constraints on the sizes of the couplings. Some of the
strongest constraints come from searches for the following decays Eidelman et
al. (2004):

$\displaystyle|\lambda^{\prime}_{l1k}\lambda^{{}^{\prime\prime}*}_{11k}|$
$\displaystyle\leq$ $\displaystyle 2\times
10^{-25}\left(\frac{\tilde{m}}{\textrm{1
TeV}}\right)^{2}~{}(l=1,2)~{}p\rightarrow[\pi^{0}l^{+}]$
$\displaystyle|\lambda^{\prime}_{31k}\lambda^{{}^{\prime\prime}*}_{11k}|$
$\displaystyle\leq$ $\displaystyle 7\times
10^{-25}\left(\frac{\tilde{m}}{\textrm{1
TeV}}\right)^{2}~{}n\rightarrow[\pi^{0}\bar{\nu}]$
$\displaystyle|\lambda^{\prime}_{i2k}\lambda^{{}^{\prime\prime}*}_{11k}|$
$\displaystyle\leq$ $\displaystyle 3\times
10^{-25}\left(\frac{\tilde{m}}{\textrm{1
TeV}}\right)^{2}~{}p\rightarrow[K^{+}\bar{\nu}].$ (17)

There exist similar bounds on the product of lepton and baryon number
violating couplings from other decay modes of the proton and neutron Rajagopal
et al. (1991); Chang and Keung (1996); Bhattacharyya and Pal (1998, 1999).

We note that, even with a relatively heavy sparticle spectrum at around 1 TeV,
the product of the minimum values of the couplings in Fig. 2 will violate the
bounds on nucleon decay. The violation of the bounds is more severe for the
couplings involving third generation fermions. If one insists on simultaneous
lepton- and baryon-number violation, then a choice of
$\hat{\lambda}_{ijk},\hat{\lambda}^{\prime}_{ijk},\hat{\lambda}^{\prime\prime}_{ijk}\sim\mathcal{O}(1)$
is no longer viable. Assuming one has a common scaling factor for each
$\not{R}_{p}$ coupling, putting it equal to at most $\delta=2\times 10^{-4}$
is necessary to completely evade all the bounds from nucleon decay. Thus the
nucleon decay problem is vastly ameliorated, but not solved, by warping. As a
result, we do not pursue this further and instead focus on cases where either
baryon-number or lepton-number is violated, predicting a stable proton and
none of the dangerous lepton and baryon number violating nucleon decay
channels

###  III.2 Lepton Number Violation Only

Since baryon-number is conserved, the proton does not decay in this case.
However in this scenario there are additional contributions to the neutrino
mass: tree level contributions originating from $\mu_{i}$ Joshipura and
Nowakowski (1995), and loop-induced contributions from
$\lambda_{ijk},\lambda^{\prime}_{ijk}$ Hall and Suzuki (1984); Babu and
Mohapatra (1990). While it may be possible to generate $\mathcal{O}(0.1)$ eV
neutrino masses with these couplings, it is very difficult to satisfy the
solar and atmospheric neutrino data, which require neutrino mass splittings
one or two orders of magnitude smaller than this. As a result, we focus on the
parameter space where these contributions are suppressed in comparison to
Dirac neutrino mass terms generated by Eq. 9. Non-supersymmetric Randall-
Sundrum scenarios in which lepton-number violating effects could be hidden
have been considered in Refs. Gherghetta (2004); Iyer and Vempati (2013b).

While it may be simple to suppress the masses of the electron and muon
neutrinos, $\lambda^{\prime}_{333}$ (being relatively large compared to the
other $\not{R}_{p}$ couplings) and give rise to too heavy a tau neutrino as
compared to data. We focus on the following region of parameter space, which
leads to $\not{R}_{p}$ contributions to neutrino masses that are not larger
than the observed mass splittings:

$\displaystyle\lambda^{\prime}_{133}<10^{-6},\qquad\lambda^{\prime}_{233}<5\times
10^{-6},\qquad\lambda^{\prime}_{333}<5\times 10^{-6},\qquad\mu_{3}\lesssim
0.01~{}\text{GeV}.$ (18)

The condition Eq. 18 requires the lepton doublets to be far away from the IR
brane. However in order to fit the required charged lepton masses, the SM
singlets must then be localized relatively close to the IR brane. Picking one
particular ‘good-fit’ point, we have:

$\displaystyle
c_{L_{1}}=2.32,~{}~{}~{}c_{L_{2}}=2.27,~{}~{}~{}c_{L_{3}}=1.61,~{}~{}~{}c_{E_{1}}=1.74,~{}~{}~{}c_{E_{2}}=0.5,~{}~{}~{}c_{E_{3}}=0.5.$
(19)

The corresponding values of the lepton-number violating couplings in this case
are represented by the points in Fig. 2. One may push the lepton doublets to
be further away from the IR brane by choosing $c_{E_{i}}<0.5$. However, this
choice is not ideal as this may induce large off-diagonal elements in the
slepton mass matrix, potentially leading to large (and excluded) flavour
violation. Along with Eq. 19, one example of a good-fit point includes the
following choices:

$\displaystyle
c_{Q_{1}}=0.68,~{}~{}c_{Q_{2}}=1.04,~{}~{}c_{Q_{3}}=0.77,~{}~{}c_{D_{1}}=2.89,~{}~{}c_{D_{2}}=2.07,~{}~{}c_{D_{3}}=1.48,$
$\displaystyle
c_{U_{1}}=3.5,~{}~{}c_{U_{2}}=1.98,~{}~{}c_{U_{3}}=0.47,~{}~{}M_{1}=M_{2}=2.5\text{~{}TeV},~{}~{}M_{3}=1.2~{}\text{TeV}.$
(20)

With this choice, the masses of the neutrinos at the low scale can be
determined using SOFTSUSY Allanach (2002); Allanach et al. (2012) and are
predicted to be:

$\displaystyle m_{\nu_{1}}=1.6\times
10^{-6}~{}\text{eV};~{}~{}~{}m_{\nu_{2}}=7.4\times
10^{-6}~{}\text{eV};~{}~{}~{}m_{\nu_{3}}=0.8~{}\text{eV}.$ (21)

These masses do respect the direct constraints upon neutrino masses, however
they do not respect oscillation data, which require $\Delta
m^{2}_{\text{atm}}\sim 2\times 10^{-3}$ eV2 and $\Delta m^{2}_{\text{sol}}\sim
7.5\times 10^{-5}$ eV2 to be the values of differences in the neutrino masses
squared Maltoni et al. (2004).

In order to suppress the $\not{R}_{p}$ contribution to the neutrino masses, we
make the following choices:

$\displaystyle\hat{\lambda}^{\prime}_{133}=0.1;~{}~{}~{}\hat{\lambda}^{\prime}_{233}=0.1;~{}~{}~{}\hat{\lambda}^{\prime}_{333}=0.2;~{}~{}~{}\hat{\mu}_{3}=0.1$
(22)

With this choice, the $\not{R}_{p}$ violating contributions to the neutrino
masses are then

$\displaystyle m_{\nu_{1}}=1.0\times
10^{-8}~{}\text{eV},~{}~{}~{}m_{\nu_{2}}=4.0\times
10^{-6}~{}\text{eV},~{}~{}~{}m_{\nu_{3}}=0.008~{}\text{eV},$ (23)

smaller than the values required to satisfy oscillation data. (We could also
have suppressed the $\not{R}_{p}$ contribution by further raising the gaugino
masses $M_{1},M_{2}$ from the values in Eq. 20 at the expense of making the
supersymmetric spectrum heavy, thus worsening the supersymmetric solution to
the hierarchy problem).

In addition to the operators in Eq. I, operators of the form
$(L_{i}H_{u})(L_{j}H_{u})$ can also contribute to neutrino masses. This
operator violates lepton number by 2 units. The superpotential term is given
by

$\mathcal{W}_{\Delta
L=2}=\frac{\kappa_{ij}}{M_{Pl}}(L_{i}H_{u})\cdot(L_{j}H_{u}),$ (24)

where $\kappa_{ij}$ are 5D Yukawa couplings with mass dimension $M^{-1}$. The
neutrino mass matrix entry generated from this operator is given by

$(m_{\nu})_{ij}=\hat{\kappa}_{ij}\frac{v_{u}^{2}}{2M_{Pl}}e^{kR\pi}f(c_{L_{i}})f(c_{L_{j}})$
(25)

where $\hat{\kappa}_{ij}=k\kappa_{ij}$ is a dimensionless $\mathcal{O}$(1)
parameter and the function $f$ is defined in Eq. 10. For
$c_{L_{i}}=c_{L_{j}}>0.5$, this expression can be simplified to

$(m_{\nu})_{ij}\sim\frac{v_{u}^{2}}{2M_{Pl}}e^{(2-2c_{L_{i}})kR\pi},$ (26)

which for $c_{L_{i}}=c_{L_{j}}=1.6$ comes out to be around $10^{-5}$ eV, much
smaller than the Dirac mass contribution from Eq. 9. The
$(L_{i}H_{u})(L_{j}H_{u})$ contribution is generally negligible in our model.

|
---|---
Figure 4: The left hand plot shows the Dirac neutrino mass eigenvalues (eV)
predicted by wave function overlap in RS models (a normal hierarchy is
assumed). The right hand plot displays the predicted mixing angles. The
vertical axis is effectively arbitrary, showing the frequency of the
prediction in a large number of scanned models.

We are now free, after the addition of right-handed neutrino superfields, to
arrange for dominant Dirac contributions to the neutrino masses. The
oscillation parameters are determined from the fits to the leptonic data as
outlined in Section II. The $c_{N_{i}}$ parameters (for the right handed
neutrinos) which pass the filtering criteria give rise to specific forms of
neutrino mass textures leading to a determination of the neutrino parameters.
The mixing angles and the mass eigenvalues can be determined by using the
$c_{N_{i}}$ in Eq.(9). Corresponding to the set in Eq.(19), the set of
$c_{N_{i}}$ parameters is:

$c_{N_{1}}=6.26~{}~{}~{}~{}~{}~{}c_{N_{2}}=5.99~{}~{}~{}~{}~{}~{}c_{N_{3}}=8.72$
(27)

Fig. 4 shows the results of the fit to the neutrino oscillation data obtained
from the model parameters. The left-handed plot shows the predicted
distribution of neutrino mass eigenvalues using different sets of $c_{N_{i}}$
parameters satisfying both $\delta m^{2}_{12}\sim 7.5\times
10^{-5}~{}\text{eV}^{2}$ and $\delta m^{2}_{32}\sim 2.32\times
10^{-3}~{}\text{eV}^{2}$. In addition, the corresponding PMNS mixing angle
predictions are shown in the right hand plot of Fig. 4, and they are close to
the values inferred from experiment Maltoni et al. (2004). Thus we see that,
predictions in line with oscillation data are easy to achieve in a RS model
that generates the masses purely from wave function overlap in the Dirac
masses.

Parameter | Mass/TeV | Parameter | Mass/TeV | Parameter | Mass/TeV | Parameter | Mass/TeV
---|---|---|---|---|---|---|---
$\tilde{t}_{1}$ | 1.8 | $\tilde{b}_{1}$ | 2.2 | $\tilde{\tau}_{1}$ | 1.1 | $\tilde{\nu}_{\tau}$ | 1.6
$\tilde{t}_{2}$ | 2.3 | $\tilde{b}_{2}$ | 2.3 | $\tilde{\tau}_{2}$ | 1.6 | $\tilde{\nu}_{\mu}$ | 1.6
$\tilde{c}_{1}$ | 2.2 | $\tilde{s}_{1}$ | 2.2 | $\tilde{\mu}_{R}$ | 1.2 | $\tilde{\nu}_{e}$ | 1.6
$\tilde{c}_{2}$ | 2.7 | $\tilde{s}_{2}$ | 2.7 | $\tilde{\mu}_{L}$ | 1.6 | $\tilde{g}$ | 2.6
$\tilde{u}_{1}$ | 2.2 | $\tilde{d}_{1}$ | 2.2 | $\tilde{e}_{R}$ | 1.1 | $\chi^{\pm}_{1}$ | 2.0
$\tilde{u}_{2}$ | 2.7 | $\tilde{d}_{2}$ | 2.7 | $\tilde{e}_{L}$ | 1.6 | $\chi^{\pm}_{2}$ | 2.3
$m_{A^{0}}$ | 3.1 | $m_{H}^{\pm}$ | 3.1 | $m_{h}$ | 0.121 | $m_{H}$ | 3.1
$\chi^{0}_{1}$ | 1.1 | $\chi^{0}_{2}$ | 2.0 | $\chi^{0}_{3}$ | 2.3 | $\chi^{0}_{4}$ | 2.4
Table 1: Example supersymmetric spectrum for the lepton number violating case
and $\tan\beta=5$.

The supersymmetric spectrum corresponding to the choice of GUT scale
parameters555It was shown in Cohen et al. (2007) that the running of soft
masses may depend on physics in the hidden sector which breaks SUSY. These
effects are likely to be relevant only for third generation squarks and are
not included here because they add additional model dependence  obtained from
Eqs. 19,20 is given in Table 1. The lightest CP-even Higgs mass $m_{h}$ is
predicted a little on the low side, but the discrepancy with the experimental
measurement of 0.125 TeV can be explained by missing higher order corrections
in its prediction. The spectrum is heavy enough to have not yet been ruled out
by LHC constraints, but light enough to expect a discovery in future runs.
Indeed, the model predicts that there will be many multi-lepton rich signals
from all of the lepton-number violating couplings that are switched on:
strongly interacting sparticles are likely to be detected first. These then
undergo cascade decay via $R_{p}$ conserving processes until the lightest
supersymmetric particle - in this case the $\tilde{e}_{R}$ \- is reached: this
is because the $R_{p}$ preserving dimensionless couplings such as gauge
couplings and third family Yukawa couplings are larger than the $\not{R}_{p}$
ones that are shown in Fig. 2. This particle then decays via $\not{R}_{p}$:
the predominant decay in this case is via $\lambda^{\prime}_{133}$ into a
bottom quark and an anti-top. Thus, SUSY events are $b$-rich (predicting 4 $b$
quarks) and may produce leptons from top decays, or be susceptible to top
taggers. There is a non-zero branching ratio for
$\tilde{e}_{R}\rightarrow\mu\nu_{\tau}$ via a similar sized coupling
$\lambda_{132}$, and the additional muons may also aid detection strategies in
multi-lepton channels.

We note that recent $\not{R}_{p}$ explanations of an apparent excesses in LHC
data Allanach et al. (2015a, b, c); Ding et al. (2015); Allanach et al.
(2015d) are not naturally accommodated in this set-up. They all are based on
resonant slepton production and require a large order 0.1 coupling
$\lambda^{\prime}_{i11}$, which is not possible in our set-up. One would need
additional flavour symmetry in order to fix some $\not{R}_{p}$ couplings to
zero and reduce the effect of various bounds on products of them in order to
be able to accommodate such a coupling.

###  III.3 Baryon-number violation only

We now consider a scenario where only baryon-number violating terms are
included in the lagrangian. Since lepton number is perturbatively conserved in
this case, the superpotential terms proportional to $\lambda_{ijk}$,
$\lambda^{\prime}_{ijk}$ or $\mu_{i}$ are absent. Proton decay is forbidden as
it requires the presence of both baryon- and lepton-number violating terms in
the Lagrangian. The neutrinos in this case must be purely Dirac type and their
masses are determined using Eq. 9, just as for the other charged fermions. The
results of the fit to the neutrino oscillation data is given in Fig. 4.

We illustrate the spectrum for the following parameter choice:

$\displaystyle c_{Q_{1}}$ $\displaystyle=$ $\displaystyle
2.2,~{}~{}c_{Q_{2}}=1.7,~{}~{}c_{Q_{3}}=0.7,~{}~{}c_{D_{1}}=1.8,~{}~{}c_{D_{2}}=1.2,~{}~{}c_{D_{3}}=1.4,$
$\displaystyle c_{U_{1}}$ $\displaystyle=$ $\displaystyle
2.3,~{}~{}c_{U_{2}}=1.3,~{}~{}c_{U_{3}}=0.3,~{}~{}c_{L_{1}}=2.2,~{}~{}c_{L_{2}}=1.8,~{}~{}c_{L_{3}}=1.4,$
(28) $\displaystyle c_{E_{1}}$ $\displaystyle=$ $\displaystyle
1.7,~{}~{}c_{E_{2}}=0.9,~{}~{}c_{E_{3}}=0.5,~{}~{}M_{1}=1.0\text{~{}TeV},~{}M_{2}=1.0\text{~{}TeV},~{}M_{3}=1.4\text{~{}TeV}.$

The choice of the corresponding $\not{R}_{p}$ couplings is represented by the
blue points in Fig. 2. Note that the lepton doublet fields need not be so
strongly localized towards the UV as in the lepton number violating case
because there are no bilinear couplings which can contribute to the neutrino
masses. This is reflected in the values of $c_{L_{i}}$. Table 2 gives the low
energy spectrum corresponding to the choice of GUT scale parameters by Eq.
III.3.

Parameter | Mass/TeV | Parameter | Mass/TeV | Parameter | Mass/TeV | Parameter | Mass/TeV
---|---|---|---|---|---|---|---
$\tilde{t}_{1}$ | 2.3 | $\tilde{b}_{1}$ | 2.3 | $\tilde{\tau}_{1}$ | 0.3 | $\tilde{\nu}_{\tau}$ | 0.3
$\tilde{t}_{2}$ | 2.7 | $\tilde{b}_{2}$ | 2.8 | $\tilde{\tau}_{2}$ | 1.0 | $\tilde{\nu}_{\mu}$ | 0.3
$\tilde{c}_{1}$ | 2.8 | $\tilde{s}_{1}$ | 2.8 | $\tilde{\mu}_{1}$ | 0.3 | $\tilde{\nu}_{e}$ | 0.3
$\tilde{c}_{2}$ | 2.7 | $\tilde{s}_{2}$ | 2.8 | $\tilde{\mu}_{2}$ | 0.9 | $\tilde{g}$ | 3.2
$\tilde{u}_{1}$ | 2.8 | $\tilde{d}_{1}$ | 2.8 | $\tilde{e}_{1}$ | 0.3 | $\chi^{\pm}_{1}$ | 0.8
$\tilde{u}_{2}$ | 2.6 | $\tilde{d}_{2}$ | 2.8 | $\tilde{e}_{2}$ | 0.9 | $\chi^{\pm}_{2}$ | 3.1
$m_{A^{0}}$ | 3.3 | $m_{H}^{\pm}$ | 3.3 | $m_{h}$ | 0.121 | $m_{H}$ | 3.3
$\chi^{0}_{1}$ | 0.1 | $\chi^{0}_{2}$ | 1.0 | $\chi^{0}_{3}$ | 2.0 | $\chi^{0}_{4}$ | 2.1
Table 2: Example supersymmetric spectrum for $\tan\beta=5$ in the baryon
number violating case.

We find the spectrum has a nice feature wherein the coloured sparticles are
grouped together in a small mass window. The sleptons in this case have a
tendency to be lighter that the lepton number violating case as the there are
no constraints coming from upper bounds on the neutrino masses. The light
smuon and neutralino gives a non-negligible contribution to the anomalous
magnetic moment of the muon $(g-2)_{\mu}$, which may explain the apparent
3.6$\sigma$ discrepancy between measurements and SM predictions:
$\delta(g-2)_{\mu}/2=(29\pm 8)\times 10^{-10}$ Bennett et al. (2006). SUSY
loops with smuons and neutralinos running in the loop yield
$\delta(g-2)_{\mu}/2\approx 13\times
10^{-10}\left(100\text{~{}GeV}/\text{max}(m_{{\tilde{\mu}}_{L}},\
m_{\chi_{1}^{0}})\right)^{2}\tan\beta$ Czarnecki and Marciano (2001) 666For
recent work on explaining $g-2$ in $\not{R}_{P}$ SUSY see Chakraborty and
Chakraborty (2015). Thus, it appears that by increasing $\tan\beta$ (which may
go as high as 50) one may fit $(g-2)_{\mu}/2$. Again, the spectrum presented
is allowed by previous collider constraints, but should be covered in coming
LHC runs. Again, production of the strongly interacting particles will proceed
via $R-$parity conserving decays, and usually end in the lightest neutralino
$\chi_{1}^{0}$. This will then decay via $\lambda^{\prime\prime}_{323}$ into a
top, a strange and a bottom so we again expect bottom-rich events (at least
four), but now there is no obvious source of missing energy unless leptons
come from the top decay with an associated neutrino. The ‘golden’ decay chain
$\tilde{q}\rightarrow\chi_{2}^{0}q\rightarrow\tilde{e}eq\rightarrow\chi_{1}^{0}e^{+}e^{-}q$
is also open, which may lead to interesting invariant mass edges between the
leptons (golden decays with $e$ replaced by $\mu$ in the preceding decay
should also be present).

##  IV Conclusions

In a general supersymmetric extension of the SM, lepton and baryon number are
not necessarily perturbatively conserved, unless a symmetry such as $R-$parity
is invoked. As a result, the most general supersymmetric lagrangian includes
terms which violate both these symmetries. This however increases the number
of free parameters in the form of undetermined values of the $\not{R}_{p}$
couplings.

In this work we propose a scenario by embedding the MSSM in a higher
dimensional warped framework. Following the aesthetic that all dimensionless
parameters should be of order 1 in a fundamental theory the warped dimensional
set-up provides the flavour structure, while supersymmetry resolves the
technical hierarchy problem. All of the supersymmetric parameters at the GUT
scale including the $\not{R}_{p}$ couplings are determined by the same set of
parameters which fix the fermion masses and mixings at this scale. This lends
a certain level (order of magnitude-wise) of predictability to the framework,
and we present, in Fig. 2, predictions of ranges of $R-$parity violating
parameters. The couplings involving the third family tend to be the largest
because of the warping structure. The predictions typically range over several
orders of magnitude but are dependent on the flavour indices of the coupling.

In the most general scenario which includes both baryon and lepton number
violating terms, the nucleon decays too quickly for $\sim{\mathcal{O}}(1)$
dimensionless $\not{R}_{p}$ parameters, although if instead they are all set
to be $\mathcal{O}(10^{-4})$, the lifetime may be long enough to evade current
experimental bounds (for superpartner masses of around 2 TeV). Following our
initial idea of the aesthetic, it appears though that one needs to forbid
either the lepton-number or baryon-number violating terms, in which case
plenty of parameter space exists where current experimental bounds on the
couplings are respected.

For the case where only lepton number is violated we find points in parameter
space where the neutrino masses are predominantly Dirac-like nature, even in
the presence of various lepton number violating operators contributing to the
neutrino masses. The neutrino masses and mixings are fit to oscillation data
just as the charged fermions are fit. In the baryon number violating case, the
sleptons have a tendency to be lighter making it more appealing from the
collider searches point of view: leptons may appear more often in
supersymmetric decay chains, providing clean objects with low backgrounds to
search for. In addition, the lighter smuons mean that a supersymmetric
explanation for the discrepant anomalous magnetic moment of the muon is
viable. In either the lepton-number or baryon-number violating cases, LHC
signals consist of prompt hard jets, and $b-$rich events (at least four per
event are predicted) containing tops. In the lepton-number violating case
there may also be a modest amount of missing transverse momentum coming from
neutrino production. We illustrate points in parameter space where current
collider limits are respected but where the LHC should be able to discover
sparticles in future runs, which we eagerly await.

##  V Acknowledgements

This work was partially supported by STFC grant ST/L000385/1. BCA was adjunct
faculty in DTP TIFR during December 2014 and would like to thank the
department for the hospitality extended during the early stages of this work
and the Cambridge SUSY Working Group for helpful suggestions. BCA, AI and KS
would like to thank Sudhir Vempati for the motivation and useful discussions.
AI and KS also thank the organisers at WHEPP 2013 for the hospitality where
the idea was conceived. AI and KS would also like to thank the CERN theory
division for hospitality where part of the work was completed. AI would like
to thank Tuhin Roy for discussions on contribution to renormalization of soft
masses due to hidden sector effects.

AI would like to dedicate this paper to the memory of close friend and fellow
physicist
Dr. Senti Imsong, who passed away recently.

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