R-parity violation in F-theory

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

Abstract

We discuss R-parity violation (RPV) in semi-local and local F-theory constructions. We first present a detailed analysis of all possible combinations of RPV operators arising from semi-local F-theory spectral cover constructions, assuming an SU(5) GUT. We provide a classification of all possible allowed combinations of RPV operators originating from operators of the form ......, including the effect of U(1) fluxes with global restrictions. We then relax the global constraints and perform explicit computations of the bottom/tau and RPV Yukawa couplings, at an SO(12) local point of enhancement in the presence of general fluxes subject only to local flux restrictions. We compare our results to the experimental limits on each allowed RPV operator, and show that operators such as LLe ^c, LQd ^c and u ^c d ^c d ^c may be present separately within current bounds, possibly on the edge of observability, suggesting lepton number violation or neutron-antineutron oscillations could constrain F-theory models.

Published for SISSA by Springer

Received: August 24, 2016

Accepted: November 7, 2016 Published: November 14, 2016

Miguel Crispim Romo,a Athanasios Karozas,b Stephen F. King,a George K. Leontarisb and Andrew K. Meadowcrofta

aPhysics and Astronomy, University of Southampton,

SO17 1BJ Southampton, U.K.

bPhysics Department, Theory Division, Ioannina University, GR-45110 Ioannina, Greece

E-mail: mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected] , mailto:[email protected]

Web End [email protected]

Abstract: We discuss R-parity violation (RPV) in semi-local and local F-theory constructions. We rst present a detailed analysis of all possible combinations of RPV operators arising from semi-local F-theory spectral cover constructions, assuming an SU(5) GUT. We provide a classi cation of all possible allowed combinations of RPV operators originating from operators of the form 10 [notdef]

5 [notdef] 5, including the e ect of U(1) uxes with global

restrictions. We then relax the global constraints and perform explicit computations of the bottom/tau and RPV Yukawa couplings, at an SO(12) local point of enhancement in the presence of general uxes subject only to local ux restrictions. We compare our results to the experimental limits on each allowed RPV operator, and show that operators such as LLec, LQdc and ucdcdc may be present separately within current bounds, possibly on the edge of observability, suggesting lepton number violation or neutron-antineutron oscillations could constrain F-theory models.

Keywords: F-Theory, Gauge Symmetry

ArXiv ePrint: 1608.04746

Open Access, c

[circlecopyrt] The Authors.

Article funded by SCOAP3. doi:http://dx.doi.org/10.1007/JHEP11(2016)081

Web End =10.1007/JHEP11(2016)081

R-parity violation in F-theory

JHEP11(2016)081

Contents

1 Introduction 1

2 R-parity violation in semi-local F-theory constructions 42.1 Multi-curve models in the spectral cover approach 42.2 Hypercharge ux with global restrictions and R-parity violating operators 6

3 Yukawa couplings in local F-theory constructions: formalism 83.1 The local SO(12) model 93.2 Wavefunctions and the Yukawa computation 12

4 Yukawa couplings in local F-theory constructions: numerics 174.1 Behaviour of SO(12) points 17

5 R-parity violating Yukawa couplings: allowed regions and comparison to data 20

6 Conclusions 24

A Semi-local F-theory constructions: R-parity violating couplings for the various monodromies 28A.1 2 + 1 + 1 + 1 29A.2 2 + 2 + 1 30 A.2.1 2 + 2 + 1 case 1 30 A.2.2 2 + 2 + 1 case 2 30

A.3 3 + 1 + 1 30A.4 3 + 2 31 A.4.1 3 + 2 case 1 31 A.4.2 3 + 2 case 2 31

B Local F-theory constructions: local chirality constraints on ux data andR-Parity violating operators 31B.1Y 0 32B.2Y > 0 33

1 Introduction

The quest for a uni ed theory of elementary particles has led to numerous extensions of the successful Standard Model (SM) of electroweak and strong interactions. During the last decades, string theory has been proven to be a powerful approach to describing gravity, which also enforces restrictions on the particle physics theory. Grand Uni ed Theories

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JHEP11(2016)081

(GUTs) [1] may be embedded in string scenarios, while supersymmetry (SUSY) is also incorporated in a consistent way, leading to a natural solution of the hierarchy problem. Although string theory does not provide a unique prediction for the precise GUT symmetry and matter content, it enables a classi cation of possible solutions in a well de ned and organised way. Moreover, it provides computational tools for various parameters such as the Yukawa couplings and potentials which would otherwise be left unspeci ed in more arbitrary extensions of the Standard Model.

Among other restrictions imposed by string theory principles, of particular importance are those on the massless spectrum. In many string constructions only small representations such as the fundamental and spinorial of the GUT group are available while the adjoint or higher ones are absent in the massless spectrum. In some cases this puts model building in a precarious position since the spontaneous breaking of most successful GUTs requires Higgs elds in the adjoint representation. But it was precisely this di culty which gave rise to the invention of new symmetry breaking mechanisms and other alternative ways to obtain the Standard Model. In the case of SU(5) for example [1], one manages to circumvent this obstacle by replacing it with the ipped -SU(5) [notdef] U(1)- version of the model [2, 3],

while in the case of Pati-Salam symmetry SU(4) [notdef] SU(2) [notdef] SU(2) [4] the adjoint Higgs

eld, which transforms under the gauge group as (15; 1; 1), is replaced by the vector-like Higgs pair of elds which transform as (4; 1; 2) + ( 4; 1; 2) [5, 6]. Analogously, a way out of this di culty in F-theory models [7{10], where the singularity is realised on a del Pezzo surface, is the use of uxes to break the GUT symmetry. Indeed, in the last decade or so, a considerable amount of work has been devoted to the possibility of successfully embedding GUTs such as SU(5) as well as exceptional E6,7,8 in an F-theory framework, leading to new features [11{14].

Recently, some of us have analysed various phenomenological aspects of F-theory effective models using the spectral cover description [15{17]. While, in F-constructions, R-parity conservation (RPC) can emerge either as a remnant symmetry of extra U(1) factors, or it can be imposed by appealing to some geometric property of the internal manifold and the ux [18], there is no compelling reason to assume this. Moreover, experimental bounds permit R-parity violating (RPV) interactions at small but non-negligible rates, providing a generic signature of F-theory models. In the eld theory context, RPV proved to be the Achilles heel of many SUSY GUTs. The most dangerous such couplings induce the tree-level operators QLdc; dcdcuc; ecLL and in the absence of a suitable symmetry or displacement mechanism, all of them appearing simultaneously can lead to Baryon and Lepton (B and L) violating processes at unacceptable rates [19]. On the other hand, in F-theory constructions, parts of GUT multiplets are typically projected out by uxes, giving rise only to a part of the above operators. In other cases, due to symmetry arguments, the Yukawa couplings relevant to RPV operators are identically zero. As a result, several B/L violating processes, either are completely prevented or occur at lower rates in F-theory models, providing a controllable signal of RPV. This observation motivates a general study of RPV in F-theory, which is the subject of this paper.

In the present paper, then, we consider RPV in local F-theory, trying to be as general as possible, with the goal of making a bridge between F-theory and experiment. An

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important goal of the paper is to compute the strength of the RPV Yukawas couplings, which mainly depend on the topological properties of the internal space and are more or less independent of many details of a particular model, enabling us to work in a generic local F-theory setting. We focus on F-theory SU(5) constructions, where a displacement mechanism, based on non-trivial uxes, renders several GUT multiplets incomplete. This mechanism has already been suggested to eliminate the colour triplets from the Higgs veplets, so that dangerous dimension-5 proton decay operators are not present. However, it turns out that, in several cases, not only the Higgs but also other matter multiplets are incomplete, while the superpotential structure is such that it implies RPV terms. In this context, it is quite common that not all of the RPV operators appear simultaneously, allowing observable RPV e ects without disastrous proton decay.

Our goal in this paper is twofold. Firstly, to present a detailed analysis of all possible combinations of RPV operators arising from a generic semi-local F-theory spectral cover framework, assuming an SU(5) GUT. This includes a detailed analysis of the classi cation of all possible allowed combinations of RPV operators, originating from the SU(5) term 10 [notdef]

5 [notdef] 5, including the e ect of U(1) uxes, with global restrictions, which are crucial in

controlling the various possible multiplet splittings. Secondly, using F-theory techniques developed in the last few years, we perform explicit computations of the bottom/tau and RPV Yukawa couplings, assuming only local restrictions on uxes, and comparing our results with the present experimental limits on the coupling for each speci c RPV operator. The ingredients for this study have already appeared scattered through the literature, which we shall refer to as we go along.

We emphasise that the rst goal is related to the nature of the available global Abelian uxes of the particular model and their restrictions on the various matter curves, hence, on its speci c geometric properties. The second goal requires the computation of the strengths of the corresponding Yukawa couplings. This in turn requires knowledge of the wavefunctions pro les of the particles participating in the corresponding trilinear Yukawa couplings and, as we will see, these involve the local ux data. Once such couplings exist in the e ective Lagrangian, we wish to explore the regions of the available parameter space where these couplings are su ciently suppressed and are compatible with the present experimental data.

Our aim in this dedicated study is to develop and extend the scope of the existing results in the literature, in order to provide a complete and comprehensive study, which make direct contact with experimental limits on RPV, enabling F-theory models to be classi ed and confronted with experiment more easily and directly than previously. We emphasise that this is the rst study of its kind in the literature which focusses exclusively on RPV in F-theory.

The remainder of the paper divides into two parts: in the rst part, we consider semi-local F-theory constructions where global restrictions are imposed on the uxes, which imply that they take integer values. In section 2 we show that RPV is a generic expectation of semi-local F-theory constructions. In section 2.1 we classify F-theory SU(5) models in the spectral cover approach according to the type of monodromy which dictates the di erent curves on which the matter and Higgs elds can lie, with particular attention of

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5 operators, involving complete SU(5) multiplets, focussing on which multiplets contain the Higgs elds Hu and

Hd. In section 2.2 we introduce the notion of ux, quantised according to global restrictions, which, when switched on, leads to incomplete SU(5) multiplets in the low energy (massless) spectrum, focussing on missing components of the multiplets projected out by the ux, and tabulating the type of physical process (RPV or proton decay) can result from particular operators involving di erent types of incomplete multiplets. Appendix A details all possible sources of R-parity violating couplings for all models classi ed with respect to the monodromies in semi-local F-theory constructions.

In the second part of the paper, we relax the global restrictions of the semi-local constructions, and allow the uxes to take general values, subject only to local restrictions. In section 3 we describe the calculation of a Yukawa coupling originating from an operator 10[notdef]

5[notdef]

5 [notdef]

the possibility for RPV operators in each case at the level of 10 [notdef]

5 at an SO(12) local point of enhancement in the presence of general local uxes, with only local (not global) ux restrictions. In section 4 we apply these methods to calculate the numerical values of Yukawa couplings for bottom, tau and RPV operators, exploring the parameter space of local uxes. In section 5 we nally consider RPV coupling regions and calculate ratios of Yukawa couplings from which the physical RPV couplings at the GUT scale can be determined and compared to limits on these couplings from experiment. Section 6 concludes the paper. Appendix B details the local F-theory constructions and local chirality constraints on ux data and RPV operators.

2 R-parity violation in semi-local F-theory constructions

2.1 Multi-curve models in the spectral cover approach

In the present F-theory framework of SU(5) GUT, third generation fermion masses are expected to arise from the tree-level superpotential terms 10f [notdef]

5f [notdef]

5 H , 10f [notdef] 10f [notdef] 5H and

5H [notdef]

5f [notdef] 1f, where the index f stands for fermion, H for Higgs and we have introduced

the notation

10f = (Q; uc; ec); 5f = (dc; L); 1f = c; 5H = (D; Hu); 5 = (

D; Hd) (2.1)

The lighter generations receive masses from higher order terms, involving the same invariants, although suppressed by powers of [angbracketleft] i[angbracketright]=M, with i representing available singlet

elds with non-zero vacuum expectation values (vevs), while M is the GUT scale. The 4-d RPV couplings are obtained similarly with the replacements 5 H !

5f (provided that the symmetries of the theory permit the existence of such terms). At the level of the minimal supersymmetric standard model (MSSM) superpotential the RPV couplings read [20]:

W 10f [notdef]

5f [notdef]

5f ! iHuLi +

dc (2.3)

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1

2 [prime][prime]ijkucidcjdck (2.2)

in the conventional notation for matter multiplets Qi; uci; dci; Li; eci where i = 1; 2; 3 is a avour index. Notice that in the presence of vector-like pairs, 5f + 5f, additional RPV couplings appear from the following decompositions

W 10f [notdef] 10f [notdef] 5f ! Quc

1

2 ijkLiLjeck + [prime]ijkLiQjdck +

L + [prime]uc

dcec + 12 [prime][prime]QQ

{ 4 {

where we have introduced the notation 5f = (

dc;

L) and dropped the avour indices here for simplicity. However, as we will analyse in detail, Abelian uxes and additional continuous or discrete symmetries which are always present in F-theory models, eliminate several of these terms. We will perform the analysis in the context of the spectral surfaces whose covering group is SU(5)? (dubbed usually as perpendicular) and is identi ed as the

commutant to the GUT SU(5) in the chain

E8 SU(5) [notdef] SU(5)? ! SU(5) [notdef] U(1)4?

where E8 is assumed to be the highest singularity in the elliptically bred compact space. Then, a crucial r^ole on the RPV remaining terms in the e ective superpotential is played by the speci c assignment of fermion and Higgs elds on the various matter curves and the remaining perpendicular U(1)?s after the monodromy action.

A classi cation of the set of models with simple monodromies that retain some perpendicular U(1)? charges associated with the weights ti has been put forward in [21{23],

where we follow the notation of Dudas and Palti [23] . In the following, we categorize these models in order to assess whether tree-level, renormalizable, perturbative RPV is generic if matter is allocated in di erent curves. More speci cally, we present four classes, characterised by the splitting of the spectral cover equation. These are:

2 + 1 + 1 + 1-splitting, which retains three independent perpendicular U(1)?. These

models represent a Z2 monodromy (t1 $ t2), and as expected we are left with seven

5 curves, and four 10 curves.

2+2+1-splitting, which retains two independent perpendicular U(1)?. These models

represent a Z2 [notdef] Z2 monodromy (t1 $ t2, t3 $ t4), and as expected we are left with

ve 5 curves, and three 10 curves.

3+1+1-splitting, which retains two independent perpendicular U(1)?. These models

represent a Z3 monodromy (t1 $ t2 $ t3), and as expected we are left with ve 5

curves, and three 10 curves.

3 + 2-splitting, which retains a single perpendicular U(1)?. These models represent a

Z3 [notdef] Z2 monodromy (t1 $ t2 $ t3, t4 $ t5), and as expected we are left with three

5 curves, and two 10 curves.

In appendix A we develop the above classes of models, identifying which curve contains the Higgs elds and which contains the matter elds, in order to show that RPV is a generic phenomenon in semi-local F-theory constructions. Of course, if all the RPV operators are present, then proton decay will be an inevitable consequence. In the next subsection we show that this is generally avoided in semi-local F-theory constructions when uxes are switched on, which has the e ect of removing some of the RPV operators, while leaving some observable RPV in the low energy spectrum.

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2.2 Hypercharge ux with global restrictions and R-parity violating operators

In F-theory GUTs, when the adjoint representation is not found in the massless spectrum, the alternative mechanism of ux breaking is introduced to reduce the GUT symmetry down to the SM gauge group. In the case of SU(5) this can happen by turning on a nontrivial ux along the hypercharge generator in the internal directions. At the same time, the various components of the GUT multiplets living on matter curves, interact di erently with the hypercharge ux. As a result, in addition to the SU(5) symmetry breaking, on certain matter curves we expect the splitting of the 10 and 5; 5 representations into di erent numbers of SM multiplets.

In a minimal scenario one might anticipate that the hyper ux is non-trivially restricted only on the Higgs matter curves in such a way that the zero modes of the colour triplet components are eliminated. This would be an alternative to the doublet-triplet scenario since only the two Higgs doublets remain in the light spectrum. The occurrence of this minimal scenario presupposes that all the other matter curves are left intact by the ux. However, in this section we show that this is usually not the case. Indeed, the common characteristic of a large class of models derived from the various factorisations of the spectral cover are that there are incomplete SU(5) multiplets from di erent matter curves which comprise the three known generations and eventually possible extraneous elds. Interestingly, such scenarios leave open the possibility of e ective models with only a fraction of RPV operators and the opportunity of studying exciting new physics implications leading to suppressed exotic decays which might be anticipated in the LHC experiments.

To analyse these cases, we assume that m10; m5 integers are units of U(1) uxes, with nY representing the corresponding hyper ux piercing the matter curves. The integer nature of these uxes originates from the assumed global restrictions [21{23]. Then, the tenplets and veplets split according to:

Representation ux units n(3,2)1/6 n(

3,2)1/6 = m10

n(3,1)2/3 n(3,1)2/3 = m10 nY n(1,1)+1 n(1,1)1 = m10 + nY

Representation ux units n(3,1)1/3 n(

3,1)+1/3 = m5

n(1,2)+1/2 n(1,2)1/2 = m5 + nY

The integers m10,5; nY may take any positive or negative value, leading to di erent numbers of SM representations, however, for our purposes it is enough to assume the cases1 m; nY =

1; 0. Then, substituting these numbers in eqs. (2.4), (2.5) we obtain the cases of table 1. Depending on the speci c choice of m; nY integer parameters, we end up with incomplete

SU(5) representations. For convenience we collect all distinct cases of incomplete SU(5) multiplets in table 1.

1Of course there are several combinations of (m, nY ) values which do not exceed the total number of three generations. Here, in order to illustrate the point, we consider only the cases with m, nY = [notdef]1, 0.

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10ti =

8

>

>

>

<

>

>

>

:

(2.4)

5ti =

(2.5)

8

>

<

>

:

10 Flux units 10 content 5 Flux units 5 content 101 m10 = 1; nY = 0 [notdef]Q; uc; ec[notdef]

51 m5 = 1; nY = 0 [notdef]dc; L[notdef] 102 m10 = 1; nY = 1 [notdef]Q; ; 2ec[notdef]

52 m5 = 1; nY = 1 [notdef]dc; 2L[notdef] 103 m10 = 1; nY = 1 [notdef]Q; 2uc; [notdef]

53 m5 = 1; nY = 1 [notdef]dc; [notdef] 104 m10 = 0; nY = 1 [notdef];

uc; ec[notdef]

54 m5 = 0; nY = 1 [notdef]; L[notdef] 105 m10 = 0; nY = 1 [notdef]; uc;

ec[notdef]

55 m5 = 0; nY = 1 [notdef];

L[notdef]

Table 1. Table of MSSM matter content originating from 10; 10; 5; 5 of SU(5) for various uxes.

SU(5)-invariant matter content operators Dominant =

R-process

101 [notdef]

51 [notdef]

51 (Q; uc; ec)(dc; L)2 All proton decay 101 [notdef]

52 [notdef]

52 (Q; uc; ec)(dc; 2L)2 All proton decay 101 [notdef]

53 [notdef]

53 (Q; uc; ec)(dc; )2 ucdcdc n

n-osc.

101 [notdef]

54 [notdef]

54 (Q; uc; ec)(; L)2 LLec Le,, -violation

101 [notdef]

102 [notdef]

51 [notdef]

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55 [notdef]

55 (Q; uc; ec)(;

L)2 None None

51 (Q; ; ec)(dc; L)2 QLdc; LLec Le,, -violation

102 [notdef]

52 [notdef]

52 (Q; ; ec)(dc; 2L)2 QLdc; LLec Le,, -violation

102 [notdef]

53 [notdef]

53 (Q; ; ec)(dc; )2 None None

102 [notdef]

54 [notdef]

54 (Q; ; ec)(; L)2 LLec Le,, -violation

102 [notdef]

55 [notdef]

55 (Q; ; ec)(;

L)2 None None

103 [notdef]

51 [notdef]

51 (Q; 2uc; )(dc; L)2 QLdc; dcdcuc proton decay

103 [notdef]

52 [notdef]

52 (Q; 2uc; )(dc; 2L)2 QLdc; dcdcuc proton decay

103 [notdef]

53 [notdef]

53 (Q; 2uc; )(dc; )2 dcdcuc n

n-osc.

103 [notdef]

54 [notdef]

54 (Q; 2uc; )(; L)2 None None

103 [notdef]

L)2 None None

Table 2. Fluxes, incomplete representations and =

R-processes emerging from the trilinear coupling 10a 5b 5c for all possible combinations of the incomplete multiplets given in table 1.

We now examine all parity violating operators formed by trilinear terms involving incomplete representations. Table 2 summarises the possible cases emerging form the various combinations 10a 5b 5c of the incomplete representations shown in table 1.

In the last column of table 2 we also show the dominant RPV processes, which lead to baryon and/or lepton number violation. We notice however, that there exist other rare processes beyond those indicated in the tables which can be found in reviews (see for example [20].) We have already stressed, that in addition to the standard model particles, some vector-like pairs may appear too. For example, when uxes are turned on, we have seen in several cases that the MSSM spectrum is accompanied in vector like states such as:

uc +

uc; L +

L; d + dc; Q + Q : : :

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55 [notdef]

55 (Q; 2uc; )(;

Of course they are expected to get a heavy mass but if some vector-like pairs remain in the light spectrum they may have signi cant implications in rare processes, such as contributions to diphoton events which are one of the primary searches in the ongoing LHC experiments.

3 Yukawa couplings in local F-theory constructions: formalism

In this section (and subsequent sections) we relax the global constraints on uxes, and consider the calculation of Yukawa couplings, imposing only local ux restrictions. The motivation for doing this is to calculate the Yukawa couplings associated with the RPV operators in a rather model independent way, and then compare our results to the experimental limits. Flavour hierarchies and Yukawa structures in F-theory have been studied in a large number of papers [24]{[42]. In this section we shall discuss Yukawa couplings in F-theory, following the approach of [35{37].

In the previous section we assessed how chirality is realised on di erent curves due to ux e ects. These considerations take into account the global ux data and are therefore called semi-local models. The ux units considered in the examples above are integer valued as they follow from the Dirac ux quantisation

1

2

Z S

F = n (3.1)

where n is an integer, a matter curve (two-cycle in the divisor S), and F the gauge eld-strength tensor, i.e. the ux. In conjugation with the index theorems, the ux units piercing di erent matter curves will tell us how many chiral states are globally present in a model.

While the semi-local approach de nes the full spectrum of a model, the computation of localised quantities, such as the Yukawa couplings, requires appropriate description of the local geometry. As we will see below, a crucial quantity in the local geometry is the notion of local ux density, understood as follows.

First we notice that the uni cation gauge coupling is related to the compacti cation scale through the volume of the compact space

1G = m4 [integraldisplay]S

2! ^ ! = m4 [integraldisplay]

dVolS = Vol(S)m4 (3.2)

where G is the uni cation gauge coupling, m is the F-Theory characteristic mass, S the

GUT divisor with Kahler form

! = i

2(dz1 ^ d

z1 + dz2 ^ d

z2) (3.3)

z2: (3.4)

As the volume of is bounded by the volume of S, we assume that

Vol( ) [similarequal]

z1 ^ d

pvol(S); (3.5)

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that de nes the volume form

dVolS = 2! ^ ! = dz1 ^ dz2 ^ d

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and if we now consider that the background of F is constant, we can estimate the values that F takes in S by

F [similarequal] 2p Gm2 n: (3.6) This means that, in units of m , the background F is an O(1) real number. Since in

the computation of Yukawa couplings its the local values of F | and not the global quantisation constraints | that matter, we will from now on abuse terminology and refer to ux densities, F , as uxes. Furthermore, as we will see later, the local values of F also de ne what chiral states are supported locally. This will be crucial to study the full plenitude of RPV couplings in di erent parts of the parameter space.

Before dealing with the particular rare reaction, it is useful to recall a few basic facts about the Yukawa couplings.

3.1 The local SO(12) model

In F-theory matter is localised along Riemann surfaces (matter curves), which are formed at the intersections of D7-branes with the GUT surface S. Yukawa couplings are then realised when three of these curves intersect at a single point on S, while, at the same time, the gauge symmetry is enhanced. The computation relies on the knowledge of the pro le of the wavefunctions of the states participating in the intersection. When a speci c geometry is chosen for the internal space (and in particular for the GUT surface) these pro les are found by solving the corresponding equations of motion [31]{[37]. Their values are obtained by computing the integral of the overlapping wavefunctions at the triple intersections.

In SU(5) two basic Yukawa terms are relevant when computing the Yukawa matrices and interactions. These are yu10 [notdef] 10 [notdef] 5 and yd10 [notdef]

5 [notdef]

5. The rst one generates the top Yukawa coupling while the symmetry at this intersection enhances to the exceptional group E6. The relevant couplings that we are interested in, are related to the second coupling.

This one is realised at a point where there is an SO(12) gauge symmetry enhancement.2 To make this clear, next we highlighted some of the basic analysis of [37].

The 4-dimensional theory can be obtained by integrating out the e ective 8-dimensional one over the divisor S

W = m4 [integraldisplay]S

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Tr(F ^ ) (3.7)

where F = dA iA ^ A is the eld-strength of the gauge vector boson A and is a

(2; 0)-form on S.

From the above superpotential, the F-term equations can be computed by varying A and . In conjugation with the D-term

D =

ZS ! ^ F +12[ ; ]; (3.8)

where ! is the Kahler form of S, a 4-dimensional supersymmetric solution for the equations of motion of F and can be computed.

2For a general E8 point of enhancement that containing both type of couplings see [33, 39]. Similar, an E7 analysis is given in [40].

Both A and , locally are valued in the Lie algebra of the symmetry group at the Yukawa point. In the case in hand, the bre develops an SO(12) singularity at which point couplings of the form 10 [notdef]

5 arise. Away from the enhancement point, the background breaks SO(12) down to the GUT group SU(5). The r^ole of [angbracketleft]A[angbracketright] is to provide a 4d chiral

spectrum and to break further the GUT gauge group.

More systematically, the Lie-Algebra of SO(12) is composed of its Cartan generators Hi with i = 1; : : : ; 6, and 60 step generators E. Together, they respect the Lie algebra

[Hi; E] = iE (3.9)

where i is the ith component of the root . The E generators can be completely identi ed by their roots

([notdef]1; [notdef]1; 0; 0; 0; 0; 0) (3.10) where underline means all 60 permutations of the entries of the vector, including di erent sign combinations. To understand the meaning of this notation it is su cient to consider a simpler example:

(0; 1; 0; 0; 0; 0; 0) [notdef](0; 1; 0; 0; 0; 0); (0; 0; 1; 0; 0; 0); (0; 0; 0; 1; 0; 0)[notdef] (3.11) The background of will break SO(12) away from the SO(12) singular point. In order to see this consider it takes the form

= z1z2dz1 ^ dz2 (3.12) where its now explicit that it parametrises the transverse directions to S. The background we are considering is

h z1z2[angbracketright] = m2 (z1Qz1 + z2Qz2) (3.13)

where m is related to the slope of the intersection of 7-branes, and

Qz1 = H1 (3.14)

Qz2 = 12

Xi

Hi: (3.15)

The unbroken symmetry group will be the commutant of [angbracketleft] z1z2[angbracketright] in SO(12). The

commutator between the background and the rest of the generators is

[[angbracketleft] z1z2[angbracketright]; E] = m2q ()E (3.16)

where q () are holomorphic functions of the complex coordinates z1, z2. The surviving symmetry group is composed of the generators that commute with [angbracketleft] [angbracketright] on every point of

S. With our choice of background, the surviving step generators are identi ed to be

E : (0; 1; 1; 0; 0; 0); (3.17) which, together with Hi, trivially commute with [angbracketleft] [angbracketright], generating SU(5) [notdef] U(1) [notdef] U(1).

{ 10 {

JHEP11(2016)081

Curve Roots q SU(5) irrep qz1 qz2

a[notdef] ([notdef]1; 1; 0; 0; 0; 0) z1

10 0 [notdef]1 c[notdef] ( 1; 1; 0; 0; 0; 0) [notdef](z1 z2)

5=5 [notdef]1 1

Table 3. Matter curves and respective data for an SO(12) point of enhancement model with a background Higgs given by equation 3.13. The underline represent all allowed permutations of the entries with the signs xed.

When q () = 0 in certain loci we have symmetry enhancement, which accounts for the presence of matter curves. This happens as at these loci, extra step generators survive and furnish a representation of SU(5) [notdef] U(1) [notdef] U(1). For the case presented we identify

three curves joining at the SO(12) point, these are

a = [notdef]z1 = 0[notdef] (3.18) b = [notdef]z2 = 0[notdef] (3.19)

c = [notdef]z1 = z2[notdef]; (3.20) and de ning a charge under a certain generator as

[Qi; E] = qi()E (3.21)

all the data describing these matter curves are presented in table 3. Since the bottom and tau Yukawas come from such an SO(12) point, in order to have such a coupling the point must have the a+, b+, and c+.

In order to both induce chirality on the matter curves and break the two U(1) factors, we have to turn on uxes on S valued along the two Cartan generators that generate the extra factors.

We rst consider the ux

hF1[angbracketright] = i(Mz1dz1 ^ d z1 + Mz2dz2 ^ d z2)QF ; (3.22)

with

QF = Qz1 Qz2 =

Z a, b

F1 [negationslash]= 0 ) Induced Chirality: (3.24)

This ux does not induce chirality in c[notdef] curves as qF = 0 for all roots in c[notdef]. To induce chirality in c[notdef] one needs another contribution to the ux

hF2[angbracketright] = i(dz1 ^ d z2 + dz2 ^ d z1)(NaQz1 + NbQz2) (3.25)

that does not commute with the roots on the c[notdef] sectors for Na [negationslash]= Nb.

{ 11 {

5=5 1 0 b[notdef] (0; [notdef]1; [notdef]1; 0; 0; 0) z2 10=

JHEP11(2016)081

1

2

H1 6

Xj=2

Hj

: (3.23)

Its easy to see that the SU(5) roots are neutral under QF , and therefore this ux does not break the GUT group. On the other hand, the roots on a, b sectors are not neutral. This implies that this ux will be able to di erentiate

5 from 5 and 10 from

10

Breaking the GUT down to the SM gauge group requires ux along the Hypercharge. In order to avoid generating a Green-Schwarz mass for the Hypercharge gauge boson, this ux has to respect global constraints. Locally we may de ne it as

hFY [angbracketright] = i[(dz1 ^ d z2 + dz2 ^ d z1)NY + (dz2 ^ d z2 dz1 ^ d z1)Y ]QY (3.26)

and the Hypercharge is embedded in our model through the linear combination

QY = 13(H2 + H3 + H4)

1

2(H5 + H6): (3.27)

Since this contribution to the ux does not commute with all elements of SU(5), only with its SM subgroup, distinct SM states will feel this ux di erently. This known fact is used extensively in semi-local models as a mechanism to solve the doublet-triplet splitting problem. As we will see bellow, it can also be used to locally prevent the appearance of certain chiral states and therefore forbid some RPV in subregions of the parameter space.

The total ux will then be the sum of the three above contributions. It can be expressed as

hF [angbracketright] = i(dz2 ^ d z2 dz1 ^ d z1)QP

+ i(dz1 ^ d

z2 + dz2 ^ d

z1)QS

JHEP11(2016)081

+ i(dz2 ^ d

z2 + dz1 ^ d

z1)Mz1z2QF (3.28)

with the de nitions

QP =MQF +Y QY (3.29)

QS =NaQz1 + NbQz2 + NY QY (3.30)

and

M = 12(Mz1 Mz2) (3.31)

Mz1z2 = 12(Mz2 + Mz1): (3.32)

As the Hypercharge ux will a ect SM states di erently, breaking the GUT group, we will be able to distinguish them inside each curve. The full split of the states present in the di erent sectors, and all relevant data, is presented in table 4.

3.2 Wavefunctions and the Yukawa computation

In general, the Yukawa strength is obtained by computing the integral of the overlapping wavefunctions. More precisely, according to the discussion on the previous section one has to solve for the zero mode wavefunctions for the sectors a; b and c presented in table (4). The physics of the D7-Branes wrapping on S can be described in terms of a twisted 8-dimensional N = 1 gauge theory on R1,3[notdef]S, where S is a Kahler submanifold of elliptically

bered Calabi-Yau 4-fold X. One starts with the action of the e ective theory, which was given in [10]. The next step is to obtain the equations of motion for the 7-brane fermionic

{ 12 {

Sector Root SM qF qz1 qz2 qS qP

a1 (1; 1; 0; 0; 0; 0) (

3; 1)

13 1 1 0 Na 13NY M 13

Y

a2 (1; 0; 0; 0; 1; 0) (1; 2)

1

2 1 1 0 Na + 12NY M + 12

Y

b1 (0; 1; 1; 0; 0; 0) (3; 1)2

3 1 0 1 Nb + 23NY M + 23

Y

b2 (0; 1; 0; 0; 1; 0) (3; 2)

1 6

1 0 1 Nb 16NY M 16

Y

b3 (0; 0; 0; 0; 1; 1) (1; 1)1 1 0 1 Nb NY M

Y

c1 (1; 1; 0; 0; 0; 0) (

3; 1)

13 0 1 1 Na Nb 13NY 13

Y

c2 (1; 0; 0; 0; 1; 0) (1; 2)

1

2 0 1 1 Na Nb + 12NY

1

2Y

Table 4. Complete data of sectors present in the three curves crossing in an SO(12) enhancement point considering the e ects of non-vanishing uxes. The underline represent all allowed permutations of the entries with the signs xed.

zero modes. This procedure has been performed in several of papers including [33, 36, 37] and we will not repeat it here in detail. In order for this paper to be self-contained we highlight the basic computational steps.

The equations for a 4-dimensional massless fermionic eld are of the Dirac form:

DA = 0 (3.33)

where

DA =

0

B

B

B

@

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0 D1 D2 D3

D1 0 D3 D2

D2 D3 0 D 1

D3 D2 D1 0

1

C

C

C

A

; = E =

0

B

B

B

@

p2

1

2

~12

1

C

C

C

A

: (3.34)

The indices here are a shorthand notation instead of the coordinates z1; z2; z3. The

components of are representing 7-brane degrees of freedom. Also the covariant derivatives are de ned as Di = @i i[[angbracketleft]Ai[angbracketright]; : : :] for i = 1; 2;

1; 2 and as D3 = i[[angbracketleft] 12[angbracketright]; : : :] for the coordinate z3. It is clear from equations (3.33), (3.34) that we have to solve the equations for each sector. According to the detailed solutions in [37] the wavefunctions for each sector have the general form

f(az1 + bz2)eMijzizj (3.35) where f(az1 + bz2) is a holomorphic function and Mij incorporates ux e ects. In an appropriate basis this holomorphic function can be written as a power of its variables fi (az1 + bz2)3i and in the case where the generations reside in the same matter curve,

the index-i can play the r^ole of a family index. Moreover the Yukawa couplings as a triple wavefunction integrals have to respect geometric U(1) selection rules. The coupling must be invariant under geometric transformations of the form: z1,2 ! ei z1,2. In this case the only non-zero tree level coupling arises for i = 3 and by considering that, the

{ 13 {

Figure 1. Intersecting matter curves, Yukawa couplings and the case of RPV.

index in the holomorphic function fi indicates the fermion generation we obtain a nonzero top-Yukawa coupling. Hierarchical couplings for the other copies on the same matter curve can be generated in the presence of non commutative uxes [31] or by incorporating non-perturbative e ects [36]{[40].

The RPV couplings under consideration emerge from a tree level interaction. Hence, its strength is given by computing the integral where now the r^ole of the Higgs 5H is

replaced by 5M. We consider here the scenario where the generations are accommodated in di erent matter curves. In this case the two couplings, the bottom/tau Yukawa and the tree level RPV, are localised at di erent SO(12) points on SGUT, (see gure 1). In this approach, at rst approximation we can take the holomorphic functions f as constants absorbed in the normalization factors.

As a rst approach, our goal is to calculate the bottom Yukawa coupling as well as the coupling without hypercharge ux and compare the two values. So, at this point we write down the wavefunctions and the relevant parameters in a more detailed form as given in [37] but without the holomorphic functions. The wavefunctions in the holomorphic gauge have the following form

~ (b)hol

10M = ~v(b)~(b)hol10M = ~v(b) (b)10M e bz2(z2 bz1) (3.36)

~ (a)hol

5M = ~v(a)~(a)hol5M = ~v(a) (a)5M e az1(z1 az2) (3.37)

~ (c)hol

5H = ~v(c)~(c)hol5H = ~v(c) (c)5He(z1z2)( cz1( c c)z2) (3.38)

~ (c)hol

5M = ~v(c)~(c)hol5H = ~v(c) (c)5M e(z1z2)( cz1( c c)z2): (3.39)

where

a =

qS(b)

b + qP (b) (3.41)

c = c( c qP (c) qS(c)

2( c qS(c))

{ 14 {

JHEP11(2016)081

qS(a)

a qP (a)

(3.40)

b =

(3.42)

and is the smallest eigenvalue of the matrix

m =

0

B

@

qP qS im2qz1 qS qP im2qz2

im2qz1 im2qz2 0

1

C

A

: (3.43)

To compute the above quantities we make use of the values of qi from table 4. It is important to note that the values of the ux densities in this table depend on the SO(12) enhancement point. This means that one can in principle have di erent numerical values for the strength of the interactions at di erent points.

The column vectors are given by

~v(b) =

JHEP11(2016)081

0

B

B

@

i bm2 b

i b m2

1

C

C

A

; ~v(a) =

0

B

B

@

i am2

i a m2 a

1

C

C

A

; ~v(c) =

0

B

B

@

i cm2

i( c c)

m2

1

C

C

A

: (3.44)

Finally, the coe cients in equations (3.36){(3.37) are normalization factors. These factors are xed by imposing canonical kinetic terms for the matter elds. More precisely, for a canonically normalized eld ~i supported in a certain sector (e), the normalization condition for the wavefunctions in the real gauge is

1 = 2m4 [notdef][notdef]~v(e)[notdef][notdef]2 [integraldisplay]

(~(e)real) i~(e)realidVolS (3.45)

where ~(e)reali are now in the real gauge, and in our convention TrE E = 2 . The wavefunctions in real and holomorphic gauge are related by

real = ei hol (3.46)

where

= i

2

[bracketleftBig]

Mz1[notdef]z1[notdef]2 + Mz2[notdef]z2[notdef]2

1

1

1

[parenrightbig]

QF

Y [notdef]z1[notdef]2 [notdef]z2[notdef]2

[parenrightbig]

QY + (z1

z2 + z2

z1) QS

[bracketrightBig]

; (3.47)

which only transforms the scalar coe cient of the wavefunctions, ~, leaving the ~v part invariant.

With the above considerations, one can nd the normalization factors to be

| (a)5M [notdef]2 = 4gs2 [notdef]

qP (a)(2 a + qP (a)(1 + 2a))

a(1 + 2a) + m4 ; (3.48)

| (b)10M [notdef]2 = 4gs2 [notdef]

qP (b)(2 b + qP (b)(1 + 2b))

b(1 + 2b) + m4

; (3.49)

| (c)5H[notdef]2 = 4gs2 [notdef]

2(qP (c) + c)(qP (c) + 2 c 2 c) + (qS(c) + c)2

2c + ( c c)2 + m4

; (3.50)

| (c)5M [notdef]2 = 4gs2 [notdef]

2(qP (c) + c)(qP (c) + 2 c 2 c) + (qS(c) + c)2

2c + ( c c)2 + m4

: (3.51)

{ 15 {

where we used the relation m

m

2 = (2)3/2g1/2s, making use of the dimensionless quantity = (m=mst)2, where mst the string scale. The expressions (3.48){(3.51) above can be shown numerically to be always positive.

The superpotential trilinear couplings can be taken to be in the holomorphic gauge. For the bottom Yukawa, we consider that 10M and 5M contain the heaviest down-type quark generations. In this case the bottom and tau couplings can be computed:

yb, = m4 tabc [integraldisplay]S

det(~ (b)hol

10M ; ~

(a)hol

5M ; ~

(c)hol

5H )dVolS

= m4 tabc det(~v(b);~v(a);~v(c)) [integraldisplay]S

~(b)hol10M~(a)hol5M~(c)hol5HdVolS: (3.52)

The bottom and tau Yukawa couplings di er since they have di erent SM quantum numbers and arise from di erent sectors, leading to di erent qS and qP as shown in table 4.

A similar formula can be written down for the RPV coupling

yRPV =m4 tabc [integraldisplay]S

det(~ (b)hol

10M ; ~

JHEP11(2016)081

(a)hol

5M ; ~

(c)hol

5[prime]M )dVolS

=m4 tabc det(~v(b);~v(a);~v(c)) [integraldisplay]S

~(b)hol10M~(a)hol5M~(c)hol5MdVolS: (3.53)

Here this RPV Yukawa coupling can in principle refer to any generations of squarks and sleptons, and may have arbitrary generation indices (suppressed here for simplicity).

The factor tabc represents the structure constants of the SO(12) group. The integral in the last term can be computed by applying standard Gaussian techniques. Computing the determinant and the integral, the combined result of the two is a ux independent factor and the nal result reads:

yb, = 2

m m

4tabc (b)10M (a)5M (c)5H: (3.54)

This is a standard result for the heaviest generations. As we observe the ux dependence is hidden on the normalization factors.

We turn now our attention in the case of a tree-level RPV coupling of the form 10M [notdef]

5M [notdef]

5M. This coupling can be computed in a di erent SO(12) enhancement point p. As a rst approach we consider that the hypercharge ux parameters are zero in the vicinity of p. From a di erent point of view, 5M replaces the Higgs matter curve in the previous computation. The new wavefunction ( (c)5M ) can be found by setting all the Hypercharge ux parameters on (c)5H, equal to zero. The RPV coupling will be given by an equation

similar to that of the bottom coupling:

yRPV = 2

4tabc (b)10M (a)5M (c)5M : (3.55)

and we notice that family indices are understood and this coupling is the same for every type of RPV interaction, depending on which SM states are being supported at the SO(12) enhancement point. Notice that the s in equations (3.54), (3.55) are the modulus of the normalization factors de ned in equations (3.48){(3.51).

{ 16 {

m m

In the next section, using equations (3.54) and (3.55), we perform a numerical analysis for the couplings presented above with emphasis on the case of the RPV coupling. We notice that in our conventions for the normalization of the SO(12) generators, the gauge invariant coupling supporting the above interactions has tabc = 2.

4 Yukawa couplings in local F-theory constructions: numerics

Using the mathematical machinery developed in the previous section, we can study the behaviour of SO(12) points in F-theory - including both the bottom-tau point of enhancement and RPV operators. The former has been well studied in [37] for example. The coupling is primarily determined by ve parameters - Na, Nb, M, NY andY . The parameters Na and Nb give net chirality to the c-sector, while NY andY are components of hypercharge ux, parameterising the doublet triplet splitting. M is related to the chirality of the a and b-sectors. There is also the Nb = Na 13NY constraint, which ensures the elimination of

Higgs colour triplets at the Yukawa point. This can be seen by examining the text of the previous section, based on the work found in [37].

For a convenient and comprehensive presentation of the results we make the following rede nitions. In eq. (3.54) and (3.55), one can factor out 4gs2 from inside eq. (3.48), (3.49), and (3.50). In addition by noticing that m

2 = (2)3/2g1/2s, we obtain

yb, = 2g1/2s y[prime]b, (4.1)

yRPV = 2g1/2s y[prime]RPV (4.2)

where y[prime]b, and y[prime]RPV are functions of the ux parameters. Furthermore, we set the scale m = 1 and as such the remainder mass dimensions are given in units of m. The presented values for the strength of the couplings are then in units of 2g1/2s.

Figure 2 shows the ratio of the bottom and tau Yukawa couplings at a point of SO(12) in a region of the parameter space with reasonable values. These results are consistent with those in [37]. Note that the phenomenological desired ratio of the couplings at the GUT scale is Y=Yb = 1:37 [notdef] 0:1 [notdef] 0:2 [43], which can be achieved within the parameter ranges

shown in Figure 2. Having shown that this technique reproduces the known results for the bottom to tau ratio, we now go on to study the behaviour of an RPV coupling point in SO(12) models.

4.1 Behaviour of SO(12) points

The simplest scenario for an SO(12) enhancement generating RPV couplings, would be the case where all three of the types of operator, QLD, UDD, and LLE arise with equal strengths, which would occur in a scenario with vanishing hypercharge ux, leading to an entirely \unsplit" scenario. This assumption sets NY andY to vanish, and we may also ignore the condition Nb = Na 13NY . The remaining parameters determining are then Na,

Nb and M. Figure 3 shows the coupling strength in the Na plane for di ering Nb and M values. The general behaviour is that the coupling strength is directly related to M, while the coupling vanishes at the point where Na = Nb. This latter point is due to the ip in

{ 17 {

JHEP11(2016)081

m

3.0

3.0

1.400

1.600

1.800

2.000

2.200

2.400

2.600

2.5

2.5

1.400

2.0

2.0

M

M

1.5

1.600

1.800

2.000

2.200

2.400

1.5

1.0

1.0

0.5

0.5

2.800

3.000

2.800

2.600

3.000

0.0

0.0

0.0 0.2 0.4 0.6 0.8 1.0

NY

0.0 0.2 0.4 0.6 0.8 1.0

NY

3.0

3.0

2.5

1.200

2.5

2.0

2.0

M

M

1.400

1.600

1.800

2.000

2.2002.400

JHEP11(2016)081

1.5

1.5

1.400

1.600

1.800

1.0

1.0

0.5

2.000

2.200

0.5

2.600

2.800

0.0

0.0

0.0 0.2 0.4 0.6 0.8 1.0

NY

0.0 0.2 0.4 0.6 0.8 1.0

NY

Figure 2. Ratio between bottom Yukawa and tau Yukawa couplings, shown as contours in the plane of local uxes. The requirement for chiral matter and absence of coloured Higgs triplets xes Nb = Na 13NY .

10 5 0 5 10

Na

M =0.5

M =1

3.5

3.5

N =2 N =1 N =0 N =1

N =2

3.0

2.5

2.5

N =2 N =1 N =0 N =1

3.0

y[prime]RPV

2.0

y[prime]RPV

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

10 5 0 5 10

Na

M =2

M =3

3.5

3.5

3.0

N =2 N =1

3.0

2.5

2.5

y[prime]RPV

2.0

y[prime]RPV

2.0

1.5

1.5

1.0

1.0

0.5

0.5

0.0

0.0

10 5 0 5 10

Na

10 5 0 5 10

Na

Figure 3. Dependency of the RPV coupling (in units of 2g1/2s) on Na in the absence of hypercharge uxes, for di erent values of M and Nb.

net chirality for the c-sector at this point in the parameter space - Na > Nb gives the c+ part of the spectrum.

Figure 4 and Figure 5 also demonstrate this set of behaviours, but for contours of the coupling strength. Figure 4, showing all combinations of the three non-zero parameters, shows that in the Na Nb plane there is a line of vanishing coupling strength about the

Na = Nb, chirality switch point for the c-sector. The gure also reinforces the idea that small values of M correspond to small values of the coupling strength, as close to the point of M = 0 the coupling again reduces to zero. Figure 5 again shows this behaviour, with

{ 18 {

10

10

10

3.000

5

5

5

1.800

0.600

1.200

1.500

3.000

1.000

2.000

Nb

M

M

0

0

0

2.400

1.500

5

5

5

JHEP11(2016)081

3.600

2.000

10 10 5 0 5 10

Na

10 10 5 0 5 10

Na

10 10 5 0 5 10

Nb

Figure 4. Dependency of the RPV coupling (in units of 2g1/2s) on di erent ux parameters, in absence of Hypercharge uxes. Any parameter whose dependency is not shown is set to zero.

10 10 5 0 5 10

Na

10

10

0.425

5

5

Nb

Nb

0

0

5

5

0.200

0.200

10 10 5 0 5 10

Na

10

10

5

5

Nb

Nb

0

0

5

5

10 10 5 0 5 10

Na

10 10 5 0 5 10

Na

Figure 5. Dependency of the RPV coupling (in units of 2g1/2s) on the (Na; Nb)-plane, in absence of hypercharge uxes and for di erent values of M. Top: left M = 0:5, right M = 1:0. Bottom: left M = 2:0, right M = 3:0.

the smallest values of M giving the smallest values of the coupling. From this we can infer that an RPV SO(12) point is most likely to be compatible with experimental constraints if M takes a small value.

Figure 6(a) (and Figure 6(b)) shows the RPV coupling strength in the absence of ux for the Na (Nb) plane, along with the \bottom" coupling strength for corresponding values.

The key di erence is that the Hypercharge ux is switched on at the bottom SO(12) point, with values of NY = 0:1 andY = 3:6. The gures show that for the bottom coupling, the uxes always push the coupling higher, similarly to increasing the M values.

Figure 6(c) plots out the two couplings in the M-plane, showing that the bottom Yukawa goes to zero for two values of M, while the RPV point has only one. Considering

{ 19 {

Bottom: M =1 Bottom: M =2 Bottom: M =3 RPV: M =1 RPV:M =2 RPV: M =3

Bottom: M =1 Bottom: M =2 Bottom: M =3 RPV: M =1 RPV:M =2 RPV: M =3

Bottom RPV

y'

y'

y'

Na

(c) Varying M with xed Na=1, Nb = 29/30, NY = 0.1,Y = 3.6

Figure 6. Dependency of the RPV and bottom Yukawa couplings (in units of 2g1/2s) on di erent

parameters at di erent regions of the parameter space.

the form of equation (3.54), we can see that the factors 5M and 10M are proportional to the parameter qp. Referring to table 4, one can see which values these take for each sector - namely, qp(a1) = M 13

Y and qp(b2) = M 16

Y . Solving these two equations shows trivially that zeros should occur when M = 13Y and 16

Y , which is the exact behaviour

exhibited in Figure 6(c).

5 R-parity violating Yukawa couplings: allowed regions and comparison to data

In this section we focus on calculating the RPV Yukawa coupling constant at the GUT scale, which may be directly compared to the experimental limits, using the methods and results of the previous two sections. As a point of notation, we have denoted the RPV Yukawa coupling at the GUT scale to be generically yRPV, independently of avour or operator type indices. This coupling may be directly compared to the phenomenological RPV Yukawa couplings at the GUT scale ijk, [prime]ijk and [prime][prime]ijk as de ned below.

Recall that, in the weak/ avour basis, the superpotential generically includes RPV couplings, in particular those from eq. (2.2):

W

1

2 ijkLiLjeck + [prime]ijkLiQjdck +

(a) Varying Na with xed M

Nb

(b) Varying Nb with xed M

M

JHEP11(2016)081

1

2 [prime][prime]ijkucidcjdck (5.1)

In the local F-theory framework, each of the above Yukawa couplings (generically denoted as yRPV) is computable through eq. (3.55). What distinguishes di erent RPV couplings, say from [prime], are the values of the ux densities, namely the hypercharge ux. This is because the normalization of matter curves depends on the hypercharge ux density. As such, di erent SM states will have di erent hypercharges and consequently di erent respective normalization coe cient.

Even though a given SO(12) enhancement point can in principle support di erent types of trilinear RPV interactions, the actual e ective interactions arising at such point depend on the local chiral spectrum present at each curve. For example, in order to have an LLec interaction, both a and c curves need to have chiral L states, and the b curve an ec

{ 20 {

state at the enhancement point. In gure 7 we show contours on the (Na,Nb) plane for the di erent types of trilinear RPV couplings.

The local spectrum is assessed by local chiral index theorems [33]. In appendix B we outline the results for the constraints on ux densities such that di erent RPV points are allowed at a given SO(12) enhancement point. These results are graphically presented in gure 8 and may be compared to the operators presented in table 2 in the semi-local approach. Thus, the green coloured region is associated with the 103 51 51 operator of this table, the blue colour with 101 53 53, the pink with 102 54 54 and so on. Thus di erent regions of the parameter space can support di erent types of RPV interactions at a given enhancement point. We can then infer that in F-theory the allowed RPV interactions can, in principle, be only a subset of all possible RPV interactions.

In the limiting cases where only one coupling is turned on, one can derive bounds on its magnitude at the GUT scale from low-energy processes [44]. In order to do so, one nds the bounds at the weak scale in the mass basis, performs a rotation to the weak basis and then evaluates the couplings at the GUT scale with the RGE. Since the e ects of the rotation to the weak basis in the RPV couplings requires a full knowledge of the Yukawa matrices, we assume that the mixing only happens in the down-quark sector as we are not making any considerations regarding the up-quark sector in this work. Table 5 shows the upper bounds for the trilinear RPV couplings at the GUT scale.

The bounds presented in table 5 have to be understood as being derived under certain assumptions on mixing and points of the parameter space [20, 45]. For example, the bound on 12k can be shown to have an explicit dependence on

~mek,R

100 GeV (5.2) where ~mek,R refers to a right-handed selectron soft-mass. The values presented in table 5, as found in [44], were obtained by setting the soft-masses to 100 GeV, which are ruled out by more recent LHC results [46{51] . By assuming heavier scalars, for example around 1 TeV, we would then get the bounds in table 5 to be relaxed by one order of magnitude.

The results show that the type of coupling, corresponding to the LLec interactions, is bounded to be < 0:05 regardless of the indices taken. The red regions of gures 11(a) and 9 show the magnitude of the coupling where it is allowed. A similar analysis can be carried out for the remaining couplings. The [prime] coupling, which measures the strength of the LQdc type of interactions, can be seen in the yellow regions of gure 10. Finally, the derived values for [prime][prime] coupling, related to the ucdcdc type of interactions, are shown in the blue regions of gures 10 and 11(b). However these couplings shown are all expressed in units of 2g1/2s, and so cannot yet be directly compared to the experimental limits.

In order to make contact with experiment we must eliminate the 2g1/2s coe cient. We do this by taking ratios of the couplings computed in this framework where the 2g1/2s

coe cient cancels in the ratio. The ratio between any RPV coupling and the bottom Yukawa at the GUT scale is given by

r = yRPVyb =

{ 21 {

JHEP11(2016)081

y[prime]RPV

y[prime]b

; (5.3)

ijk ijk [prime]ijk [prime][prime]ijk 111 | 1:5 [notdef] 104 |

112 | 6:7 [notdef] 104 4:1 [notdef] 1010 113 | 0:0059 1:1 [notdef] 108 121 0:032 0:0015 4:1 [notdef] 1010 122 0:032 0:0015 |

123 0:032 0:012 1:3 [notdef] 107 131 0:041 0:0027 1:1 [notdef] 108 132 0:041 0:0027 1:3 [notdef] 107 133 0:0039 4:4 [notdef] 104 |

211 0:032 0:0015 | 212 0:032 0:0015 (1:23) 213 0:032 0:016 (1:23) 221 | 0:0015 (1:23) 222 | 0:0015 -223 | 0:049 (1:23) 231 0:046 0:0027 (1:23) 232 0:046 0:0028 (1:23) 233 0:046 0:048 | 311 0:041 0:0015 | 312 0:041 0:0015 0:099 313 0:0039 0:0031 0:015 321 0:046 0:0015 0:099 322 0:046 0:0015 | 323 0:046 0:049 0:015 331 | 0:0027 0:015 332 | 0:0028 0:015 333 | 0:091 |

Table 5. Upper bounds of RPV couplings (ijk refer to avour/weak basis) at the GUT scale under the assumptions: 1) Only mixing in the down-sector, none in the Leptons; 2) Scalar masses ~m = 100 GeV; 3) tan (MZ) = 5; and 4) Values in parenthesis refer to non-perturbative bounds, when these are stronger than the perturbative ones. This table is reproduced from [44].

as de ned in equation (4.1) and equation (4.2). This ratio can be used to assess the absolute strength of the RPV at the GUT scale as follows.

First we assume that the RPV interaction is localised in an SO(12) point far away from the bottom Yukawa point. This allows us to use di erent and independent ux densities at each point. We can then compute y[prime]b at a point in the parameter space where the ratio yb=y

takes reasonable values, following [37]. Finally we take the ratio, r. In certain regions of the parameter space, r is naturally smaller than 1. This suppression of the RPV coupling in respect to the bottom Yukawa is shown in gures 12(a), 12(b), 12(c), and 12(d), for di erent regions of the parameter space that allows for distinct types of RPV interactions.

{ 22 {

JHEP11(2016)081

Qb La Dc

Qb Lc Da

10

10

1.000

0.300

5

Nb

5 1.500

1.250

Nb

0.500

0.400

0

0

0.750

0.200

5

5

10 10 5 0 5 10

10 10 5 0 5 10

Na

Na

Ub Da Dc

La Lc Eb

10

10

JHEP11(2016)081

0.450

1.500

5

5

Nb

0.600

0.450

Nb

1.800

1.800

0

0

0.300

1.200

5

5

10 10 5 0 5 10

10 10 5 0 5 10

Na

Na

Figure 7. Strength of di erent RPV couplings (in units of 2g1/2s) in the (Na; Nb)-plane in the presence of Hypercharge uxes NY = 0:1,Y = 3:6, and with M = 1. The scripts a, b, c refer to which sector each state lives.

Since r is the ratio of both primed and unprimed couplings, respectively unphysical and physical, at the GUT scale, we can extend the above analysis to nd the values of the physical RPV couplings at the GUT scale. To do so, we use low-energy, experimental, data to set the value of the bottom Yukawa at the weak scale for a certain value of tan . Next, we follow the study in [43] to assess the value of the bottom Yukawa at the GUT scale through RGE runnings.

In order to make a connection with the bounds in table 5, we pick tan = 5 and we nd yb(MGUT) [similarequal] 0:03. The results for the value of the RPV couplings in di erent regions in the

parameter space at the GUT scale are presented in gures 13(a), 13(b), 13(c), and 13(d). These results show that, for any set of avour indices, the strength of the coupling related to an LLec interaction is within the bounds. This means that this purely leptonic RPV operator, which violates lepton number but not baryon number, may be present with a su ciently suppressed Yukawa coupling, according to our calculations. Therefore in the future lepton number violating processes could be observed.

By contrast, only for a subset of possible avour index assignments for baryon number violating (but lepton number conserving) ucdcdc couplings are within the bounds in table 5. The constraint on the rst family up quark coupling [prime][prime]1jk for the uc1dcjdck interaction is so stringent, that this operator must only be permitted for the cases uc2dcjdck and uc3dcjdck (corresponding to the two heavy up-type quarks cc; tc), assuming no up-type quark mixing.

However, if up-type quark mixing is allowed, then such operators could lead to an e ective uc1dcjdck operator suppressed by small mixing angles, in which case it could induce n

n

oscillations [16].

{ 23 {

Y = +1 and NY = +1

Y = +1 and NY =1

M

M

JHEP11(2016)081

LLEQLD +LLE ALL

UDDLLEQLD +LLE ALL

Na

Na

Y =1 and NY = +1

Y =1 and NY =1

M

M

UDDQLDUDD +QLD QLD +LLE ALL

UDDUDD +QLD ALL

Na

Na

Figure 8. Allowed regions in the parameter space for di erent RPV couplings. These gures should be seen in conjunction with the operators presented in table 2.

Finally the LQdc operator with Yukawa coupling [prime] apparently must be avoided, since according to our calculations, the value of [prime] that we predict exceeds the experimental limit by about an order of magnitude for all avour indices, apart from [prime]333 coupling corresponding to the L3Q3dc3 operator. This implies that we should probably eliminate such operators which violate both baryon number and lepton number, using the ux mechanism that we have described. However in some parts of parameter space, for certain avour indices, such operators may be allowed leading to lepton number violating processes such as K+ ! e+e+ and D+ ! Ke+e+.

6 Conclusions

In this paper we have provided the rst dedicated study of R-parity violation (RPV) in F-theory semi-local and local constructions based on the SU(5) grand uni ed theory (GUT)

{ 24 {

UDD

LLE

2.0

2.0

1.500

1.250

0.800

1.5

0.750

0.500

0.200

0.400

1.000

1.5

1.000

1.200

1.0

1.0

0.5

0.5

M

M

0.0

0.0

0.600

0.5

0.5

UDDLLEQLD +LLE ALL

UDDLLEQLD +LLE ALL

1.0

1.0

JHEP11(2016)081

1.5

1.5

2.0

2.0

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

Na

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

Na

Figure 9. Allowed regions in the parameter space for di erent RPV couplings withY = NY = 1. We have also include the corresponding contours for the ucdcdc operator (left) and LLec (right).

UDD

Qb La Dc

Qb Lc Da

2.0

0.200

2.0

2.0

1.200

1.000

0.800

0.600

0.400

0.200

1.5

1.5

1.5

0.400

0.600

0.800

1.250

1.0

1.0

1.0

0.500

0.750

1.000

0.200

0.5

0.5

0.5

M

M

M

0.250

0.0

0.0

0.0

0.400

0.5

1.0

UDDQLDUDD +QLD QLD +LLE ALL

UDDQLDUDD +QLD QLD +LLE ALL

0.5

0.5

1.0

1.0

UDDQLDUDD +QLD QLD +LLE ALL

1.5

1.5

1.5

2.0

2.0

2.0

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

Na

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

Na

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

Na

Figure 10. Allowed regions in the parameter space for di erent RPV couplings with NY =

Y =

1. We have also include the corresponding contours for the ucdcdc operator (left) and QLdc (middle and right). The scripts a, b and c refer to which sector each state lives.

contained in the maximal subgroup SU(5)GUT[notdef]SU(5)? of an E8 singularity associated with

the elliptic bration. Within this framework, we have tried to be as general as possible, with the primary aim of making a bridge between F-theory and experiment.

We have focussed on semi-local and local F-theory SU(5) constructions, where a nontrivial hypercharge ux breaks the GUT symmetry down to the Standard Model and in addition renders several GUT multiplets incomplete. Acting on the Higgs curves this novel mechanism can be regarded as the surrogate for the doublet-triplet splitting of conventional GUTs. However, from a general perspective, at the same time the hyper ux may work as a displacement mechanism, removing certain components of GUT multiplets while accommodating fermion generations on other matter curves.

In the rst part of the paper we considered semi-local constructions, focussing on F-theory SU(5)GUT models which are classi ed according to the discrete symmetries |

{ 25 {

LLE

UDD

2.0

2.0

1.400

1.200

1.000

0.800

0.600

1.5

1.5

1.500

1.750

1.0

0.250

0.750

1.250

1.0

0.200

0.400

0.5

0.5

M

1.000

M

0.0

0.0

0.500

0.5

0.5

LLEQLD +LLE ALL

UDDUDD +QLD ALL

1.0

1.0

1.5

1.5

JHEP11(2016)081

2.0

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

Na

2.0

2.0 1.5 1.0 0.5 0.0 0.5 1.0 1.5 2.0

Na

(a) LLec regions with ~

NY = NY = 1

(b) ucdcdc regions with ~

NY = NY = 1

Figure 11. Allowed regions in the parameter space for di erent RPV couplings.

Na

0.150

0.010

0.050

0.100

M

M

0.040

0.050

0.075

0.020

0.030

0.025

(a) LLec region with NY = 10,Y = 0.1

Na

(b) LLec region with NY = 10,

Y = 0.1

0.100

0.300

0.400

0.500

0.150

0.450

0.600

0.750

0.600

0.900

M

M

0.700

0.300

0.200

Na

Y = 10

Figure 12. yRPV=yb ratio. The bottom Yukawa was computed in a parameter space point that returns a reasonable yb=y ratio [37].

acting as identi cations on the SU(5)? representations | and appearing as a subgroup of

the maximal SU(5)? Weyl group S5. Furthermore, we considered phenomenologically ap

pealing scenarios with the three fermion generations distributed on di erent matter curves

{ 26 {

(c) QLdc region with NY = 0.1,Y = 10

Na

(d) ucdcdc region with NY = 0.1,

0.004

0.0015

0.003

0.0005

M

M

0.0010

0.002

0.001

Na

Na

JHEP11(2016)081

(a) LLec region with NY = 10,Y = 0.1

(b) LLec region with NY = 10,

Y = 0.1

0.003

0.009

0.012

0.015

0.018

0.004

0.012

0.016

0.020

M

M

0.024

0.006

0.008

Na

Na

(c) [prime]QLdc region with NY = 0.1,Y = 10

(d) [prime][prime]ucdcdc region with NY = 0.1,

Y =

10

Figure 13. yRPV at GUT scale for tan = 5. The values here can be compared directly to the bounds presented in table 5.

and showed that RPV couplings are a generic feature of such models. Upon introducing the ux breaking mechanism, we classi ed all possible cases of incomplete GUT multiplets and examined the implications of their associated RPV couplings. Then we focused on the induced MSSM plus RPV Yukawa sector which involves only part of the MSSM allowed RPV operators as a consequence of the missing components of the multiplets projected out by the ux. Next, we tabulated all distinct cases and the type of physical process (RPV or proton decay) that can arise from particular operators involving di erent types of incomplete multiplets.

In the second part of the paper we computed the strength of the RPV Yukawa couplings, which mainly depend on the topological properties of the internal space and are more or less independent of many details of a particular model, enabling us to work in a generic local F-theory setting. Due to their physical relevance, we paid special attention to those couplings originating from the SU(5) operator 10 [notdef]

5 [notdef]

5 in the presence of general uxes, which is realised at an SO(12) point of enhancement. Then, we applied the already developed F-theory techniques to calculate the numerical values of Yukawa couplings for bottom, tau and RPV operators. Taking into account ux restrictions, which

{ 27 {

limit the types of RPV operators that may appear simultaneously, we then calculated ratios of Yukawa couplings, from which the physical RPV couplings at the GUT scale can be determined. We have explored the possible ranges of the Yukawa coupling strengths of the 10 [notdef]

5 [notdef] 5-type operators in a ve-dimensional parameter space, corresponding to the

number of the distinct ux parameters/densities associated with this superpotential term. Varying these densities over a reasonable range of values, we have observed the tendencies of the various Yukawa strengths with respect to the ux parameters and, to eliminate uncertainties from overall normalization constants, we have computed the ratios of the RPV couplings to the bottom Yukawa one. This way, using the experimentally determined mass of the bottom quark, we compared our results to limits on these couplings from experiment.

The results of this paper show rstly that, in semi-local F-theory constructions based on SU(5) GUTs, RPV is a generic feature, but may occur without proton decay, due to ux e ects. Secondly, our calculations based on local F-theory constructions show that the value of the RPV Yukawa couplings at the GUT scale may be naturally suppressed over large regions of parameter space. Furthermore, we found that the existence of LLec type of RPV interactions from F-Theory are expected to be within the current bounds. This implies that such lepton number violating operators could be present in the e ective theory, but simply below current experimental limits, and so lepton number violation could be observed in the future. Similarly, the baryon number violating operators ccdcjdck and tcdcjdck could also be present, leading to n

n oscillations. Finally some QLdc operators could be present leading to lepton number violating processes such as K+ ! e+e+ and

D+ ! Ke+e+. In conclusion, our results suggest that RPV SUSY consistent with proton

decay and current limits may be discovered in the future, shedding light on the nature of F-theory constructions.

Acknowledgments

SFK and GKL are grateful to the Mainz Institute for Theoretical Physics (MITP) for its hospitality and its partial support during the completion of this work. SFK, AKM, and MCR are grateful to the University of Ioannina for its hospitality and its partial support during the completion of this work. SFK acknowledges support from the STFC Consolidated grant ST/L000296/1 and the European Union Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreements InvisiblesPlus RISE No. 690575 and Elusives ITN No. 674896. AKM is supported by STFC studentship 1238679. MCR acknowledges support from the FCT under the grant SFRH/BD/84234/2012.

A Semi-local F-theory constructions: R-parity violating couplings for the various monodromies

In this appendix we examine the semi-local F-theory models in detail in order to demonstrate that RPV couplings are generic or at least common. To this end we note that:

{ 28 {

JHEP11(2016)081

Curve : 5Hu 51 52 53 54 55 56 10M 102 103 104

Charge : 2t1 t1t3 t1t4 t1t5 t3t4 t3t5 t4t5 t1 t3 t4 t5

Table 6. Matter curves and the corresponding U(1) charges for the case of a 2 + 1 + 1 + 1 spectral cover split. Note that because of the Z2 monodromy we have t1 ! t2.

1. We want models with matter being distributed on di erent curves. This setup we call multi-curve models, in contrast to the models presented section 4 of [23] and usually considered in other papers that compute Yukawa couplings.

2. The models de ned in this framework \choose" the Hu assignment for us, since a tree-level, renormalizable, perturbative top-Yukawa requires the existence of the coupling

10a10a5b (A.1)

such that the perpendicular charges cancel out. As such, all the models listed above will have a de nite assignment for the curve supporting Hu, and we do not assign the remaining MSSM states to curves, i.e. all the remaining 5 curves will be called 5a, making clear that they are either supporting some 5M or Hd. Furthermore, we will refer to the 10 curve containing the top quark as 10M.

3. The indication for existence of tree-level, renormalizable, perturbative RPV is given by the fact we can nd two couplings of the form

10a5b5c (A.2)

10d5e5f (A.3)

for (b; c) [negationslash]= (e; f), and a; d unconstrained. This happens as Hd cannot be both

supported in one of the 5b, 5c and at the same in one of the 5e, 5f.

4. We do not make any comment on ux data. The above criteria can be evaded by switching o the uxes such that the RPV coupling (once the assignment of Hd to a curve is realised) disappears.

With this in mind we study the possible RPV realisations in multi-curve models.

A.1 2 + 1 + 1 + 1

In this case the spectral cover polynomial splits into four factors, three linear terms and a quadratic one. Also, due to the quadratic factor we impose a Z2 monodromy. The bestiary of matter curves and their perpendicular charges (ti) is given in the table 6.

In this model RPV is expected to be generic as we have the following terms

1045152; 1035153; 10M5156; 1025253; 10M5255; 10M5354 (A.4)

{ 29 {

JHEP11(2016)081

case 1

Curve 5Hu 51 52 53 54 10M 102 103

Charge 2t1 t1t3 t1t5 t3t5 2t3 t1 t3 t5

case 2

Curve 5Hu 51 52 53 54 10M 102 103

Charge 2t3 t1t3 t1t5 t3t5 2t1 t3 t1 t5

Table 7. The scenario of a 2 + 2 + 1 spectral cover split with the corresponding matter curves and

U(1) charges. Note that we have two possible cases.

A.2 2 + 2 + 1

Here the spectral cover polynomial splits into three factors, it is the product of two quadratic terms and a linear one. We can impose a Z2 [notdef] Z2 monodromy which leads

to the following identi cations between the weights,(t1 $ t2) and (t3 $ t4) . In this case

there are two possible assignments for Hu (and 10M), as we can see in table 7.

A.2.1 2 + 2 + 1 case 1

The bestiary of matter curves and their perp charges is given in the upper half table of table 7.

In this model RPV is expected to be generic as we have the following terms

1025152; 10M5153; 10M5254; 1035151 (A.5)

Notice that if 51 contains only one state, then the last coupling is absent due to antisymmetry of SU(5) contraction.

A.2.2 2 + 2 + 1 case 2

The bestiary of matter curves and their perp charges is given in the lower half table of table 7.

In this model RPV is expected to be generic as we have the following terms

10M5152; 1025153; 10M5354; 1035151 (A.6)

Notice that if 51 contains only one state, then the last coupling is absent due to antisymmetry of SU(5) contraction.

A.3 3 + 1 + 1

In this scenario the splitting of the spectral cover leads to a cubic and two linear factors. We can impose a Z3 monodromy for the roots of the cubic polynomial. The bestiary of matter curves and their perpendicular charges is given in table 8.

In this model R-parity violation is not immediately generic as we only have

1025152; 10M5153 (A.7)

and as such assigning Hd to 51 avoids tree-level, renormalizable, perturbative RPV.

{ 30 {

JHEP11(2016)081

Curve 5Hu 51 52 53 10M 102 103 Charge 2t1 t1t4 t1t5 t4t5 t1 t4 t5

Table 8. Matter curves and the corresponding U(1) charges for the case of a 3 + 1 + 1 spectral cover split. Note that we have impose a Z3 monodromy.

case 1

Curve 5Hu 52 53 10M 102

Charge 2t1 t1t3 2t3 t1 t3

case 2

Curve 5Hu 52 53 10M 102

Charge 2t3 t1t3 2t1 t3 t1

Table 9. The two possible cases in the scenario of a 3 + 2 spectral cover split, the matter curves and the corresponding U(1) charges.

A.4 3 + 2

These type of models are in general very constrained because of the large monodromies which leads to a low number of matter curves.

In this case there are two possible assignments for Hu (and 10M), as described in table 9.

A.4.1 3 + 2 case 1

The matter curves content is given in the upper half of table 9 (case 1).Possible RPV couplings are

10M5253 ; 1025252 (A.8)

Notice that if 52 contains only one state, then the last coupling is absent due to antisymmetry of SU(5) contraction.

A.4.2 3 + 2 case 2

This second scenario is referred as case 2 in the lower half of table 9.Only one coupling

10M5252 (A.9)

which is either RPV or is absent. Notice that if 52 contains only one state, then the last coupling is absent due to anti-symmetry of SU(5) contraction.

B Local F-theory constructions: local chirality constraints on ux data and R-Parity violating operators

The chiral spectrum of a matter curve is locally sensitive to the ux data. This is happens as there is a notion of local chirality due to local index theorems [33, 38]. The presence of

{ 31 {

JHEP11(2016)081

Y

M <

Y 3

3 < M < Y6

Y

6 < M <

Y

Y < M

(Na Nb) < NY2 None None None None

NY2 < (Na Nb) < NY3 None None QLdc QLdc, LLec

NY

3 < (Na Nb) None ucdcdc QLdc, ucdcdc All

Table 10. Regions of the parameter space and the respective RPV operators supported forY 0, NY > 0.

a chiral state in a sector with root is given if the matrix

m =

0

B

@

JHEP11(2016)081

qP qS im2qz1 qS qP im2qz2

im2qz1 im2qz2 0

1

C

A

with qi presented in table 4, has positive determinant

det m > 0: (B.1)

As such, if we want a certain RPV coupling to be present, then the above condition has to be satis ed for the three states involved in the respective interaction at the SO(12) enhancement point. For example, in order for the emergence of an QLdc type of RPV interaction, locally the spectrum has to support a Q, a L, and a dc states. The requirement that at a single point equation (B.1) hold for each of these states imposes constraints on the values of the ux density parameters.

Therefore, while RPV e ects in general include all three operators - QLdc, ucdcdc,

LLec - there are regions of the parameter space that allow for the elimination of some or all of the couplings. These are in principle divided into four regions, depending on the sign of the parametersY and NY . In the appendix we present the resulting regions of the parameter space and which operators are allowed in each.

B.1Y 0ForY 0, the conditions on the ux density parameters for which each RPV interaction

is turned on are

QLdc : M >

Y

6

Na Nb >

NY

2

Y 3

Na Nb >

NY 3

ucdcdc : M >

LLec : M >

Y

Na Nb >

NY

2

Depending on the sign of NY , the above conditions de ne di erent regions of the ux density parameter space. These are presented in tables 10 and 11.

{ 32 {

Y

M <

Y 3

3 < M < Y6

Y

6 < M <

Y

Y < M

(Na Nb) < NY3 None None None None

3 < (Na Nb) < NY2 None ucdcdc ucdcdc ucdcdcNY

2 < (Na Nb) None ucdcdc QLdc, ucdcdc All

Table 11. Regions of the parameter space and the respective RPV operators supported forY 0, NY < 0.

M < Y2 Y2 < M <

NY

Y 3

Y

3 < M < 2Y3

2Y

3 < M

(Na Nb) < NY2 None None None None

NY2 < (Na Nb) < NY3 None LLec QLdc, LLec QLdc, LLec

NY

3 < (Na Nb) None LLec QLdc, LLec All

Table 12. Regions of the parameter space and the respective RPV operators supported forY > 0,

NY > 0.

M < Y2 Y2 < M <

Y 3

Y

3 < M < 2Y3

2Y

3 < M

(Na Nb) < NY3 None None None None

JHEP11(2016)081

3 < (Na Nb) < NY2 None None None ucdcdc

NY

2 < (Na Nb) None LLec QLdc, LLec All

Table 13. Regions of the parameter space and the respective RPV operators supported forY > 0,

NY < 0.

B.2Y > 0

ForY > 0, the conditions on the ux density parameters for which each RPV interaction is turned on are

QLdc : M >

Y 3

Na Nb >

NY

NY

2

ucdcdc : M > 2

Y 3

Na Nb >

NY 3

LLec : M >

Y

2

Na Nb >

NY

2

Depending on the sign of NY , the above conditions de ne di erent regions of the ux density parameter space. These are presented in tables 12 and 13.

{ 33 {

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