New Paradigm for Baryon and Lepton Number Violation

Pavel Fileviez P´erez Particle and Astroparticle Physics Division Max Planck Institut fuer Kernphysik, 69117 Heidelberg, Germany

Abstract

The possible discovery of proton decay, neutron-antineutron oscillation, neutrinoless double beta decay in low energy experiments, and exotic signals related to the violation of the baryon and lepton numbers at collider experiments will change our understanding of the conservation of fundamental symmetries in nature. In this review we discuss the rare processes due to the existence of baryon and lepton number violating interactions. The simplest grand uniﬁed theories and the neutrino mass generation mechanisms are discussed. The theories where the baryon and lepton numbers are deﬁned as local gauge symmetries spontaneously broken at the low scale are discussed in detail. The simplest supersymmetric gauge theory which predicts the existence of lepton number violating processes at the low scale is investigated. The main goal of this review is to discuss the main implications of baryon and lepton number violation in physics beyond the Standard Model. arXiv:1501.01886v2 [hep-ph] 17 Aug 2015

Preprint submitted to Elsevier August 19, 2015 CONTENTS 2

Contents

1 Introduction 3

2 Particle Physics and the Great Desert Hypothesis4

3 Proton Stability 8

4 Neutron-antineutron oscillations9

5 Mechanisms for Neutrino Masses 12

6 Neutrinoless double beta decay 15

7 Grand Uniﬁed Theories 16

8 Theories for Local Baryon and Lepton Numbers 20 8.1 Theoretical Framework ...... 25 8.2 Phenomenological Aspects ...... 26 8.3 Cosmological Aspects ...... 27 8.4 Towards Low Scale Uniﬁcation ...... 32

9 Baryon and Lepton Number Violation in Supersymmetry 34 9.1 The Minimal Theory for Spontaneous R-parity Violation ...... 36 9.2 R-Parity Violating Interactions and Mass Spectrum ...... 40 9.3 Lepton Number Violation and Decays ...... 45 9.4 Signatures at the LHC ...... 47

10 Final Discussions 49 1. Introduction 3

1. Introduction

The discovery of the Higgs boson at the Large Hadron Collider did close a chapter in the history of particle physics. Now, the Standard Model of particle physics is completed and describes most of the experiments with a great precision. There are many open questions in particle physics and cosmology which we cannot answer in the context of the Standard Model. Some of these questions are related to the understanding of the baryon asymmetry and the dark matter in the Universe. There are even more fundamental questions related to the origin of all interactions in nature. One of the most appealing scenarios for high energy physics correspond to the case where a new theory is realized at the TeV scale and one could solve some of these enigmas in this context. The great desert hypothesis is very often considered as a well-defined scenario for physics beyond the Standard Model. In the great desert picture one has the Standard Model describing physics at the electroweak scale and a more fundamental theory at the high scale. This high scale is often related to the scale where there is a grand unified theory or even a theory where the unification of the gauge interactions and gravity is possible. From the low-energy point of view or bottom-up approach it is important to have an effective field theory which is not sensitive to the unknown physics at the high scale. It is well-known that in the Standard Model baryon and lepton numbers are conserved symmetries at the classical level. However, in our modern view the Standard Model is just an effective theory which describes physics at the electroweak scale and one should think about the impact of all possible higher-dimensional operators which could modify the Standard Model predictions. For example, in the Standard Model one can have the following higher-dimensional operators

cL 2 cB cF µ `L`LH + 2 qLqLqL`L + 2 (¯qLγ qL)(¯qLγµqL), (1) L ⊃ ΛL ΛB ΛF where the first operator violates lepton number, the second violates baryon and lepton numbers, and the third breaks the flavour symmetry of the Standard Model gauge sector. The 14 15 3 4 experimental bounds demand ΛL < 10 GeV, ΛB > 10 GeV [1], and ΛF > 10 − TeV [2], when cL, cB and cF are of order one. Since in any generic grand unified theory one generates the second operator due to new gauge interactions, it is impossible to achieve unification at the low scale without predicting an unstable proton and therefore one needs the great desert. One could imagine a theoretical framework where the second operator is absent, the proton is stable and the unification scale is only constrained by flavour violating processes. In this case 2. Particle Physics and the Great Desert Hypothesis 4 the unification scale must be above 104 TeV in order to satisfy all experimental constraints. In this review we present new theories for physics beyond the Standard Model where the effective theory at the TeV scale predicts a stable proton and the unification scale could be very low. In the first part of this review we present new extensions of the Standard Model where the baryon and lepton numbers are local gauge symmetries spontaneously broken at the low scale. In this context one can define a simple anomaly free theory adding a simple set of fields with baryon and lepton numbers which we call “lepto-baryons”. We show that in the spectrum of these theories one has a candidate to describe the cold dark matter in the Universe. We discuss the predictions for direct detection and the constraints from the allowed relic density. Using the relic density constraints we find an upper bound on the symmetry breaking scale which tells us that these theories can be tested or ruled out at current or future collider experiments. The relation between the baryon asymmetry and dark matter density is investigated. We show that even if the local baryon number is broken at the low scale one can have a consistent relation between the baryon asymmetry and the B L asymmetry generated through a mechanism − such as leptogenesis. Finally, we discuss the possibility to achieve the unification of gauge interactions at the low scale in agreement with the experiment. In the second part of this review we discuss the main issues in supersymmetric theories related to the existence of baryon and lepton number violating interactions. We discuss the possible origin of the R parity violating interactions and show that the simplest theories based − on the local B L gauge symmetry predicts that R parity must be spontaneously broken at − − the supersymmetric scale. We discuss the most striking signatures at the Large Hadron Collider which one can use to test the theory in the near future. The predictions for the lepton number violating channels with multi-leptons are discussed in detail. We discuss the main features of this generic mechanism for spontaneous R parity breaking and the consequences for cosmology. − The theories presented in this review can be considered as appealing extensions of the Standard Model of particle physics which could be tested in the near future.

2. Particle Physics and the Great Desert Hypothesis

The Standard Model of particle physics describes the properties of quarks, leptons and gauge bosons with great precision at the electroweak scale, Λ 100 200 GeV. Unfortunately, very EW ∼ − often when we try to understand the origin of the Standard Model interactions one postulates the existence of a grand uniﬁed theory or another fundamental theory which describes physics at the very high scale, M 1015 17 GeV. Therefore, typically one postulates a large energy GUT ∼ − 2. Particle Physics and the Great Desert Hypothesis 5

Figure 1: The Desert Hypothesis in Particle Physics and proton decay mediated by dimension six operators. The Standard Model describes physics at the electroweak scale and at the high scale one has some speculative ideas related to the uniﬁcation of fundamental forces. The grand uniﬁed theories could be realized at the 1015−16 GeV scale, while string theories could be realized at the Planck scale.

gap or desert between the electroweak scale and the new scale MGUT . See Fig. 1 for a simple representation of the desert hypothesis in particle physics. Notice that in the desert one could have some “animals” such as the fields needed for the seesaw mechanism of neutrino masses. However, in the simplest picture there are only two main energy scales where new theories are defined, the electroweak scale and the unification scale. Unfortunately, if the desert hypothesis is true there is no hope to test the unified theories at colliders or low-energy experiments. One of the main reasons to postulate the great desert is that in the Standard Model one can have new dimension six operators which mediate proton decay. For example, one can write down the following baryon and lepton number violating operator with three quarks and one lepton field,

qLqLqL`L SM cB 2 , (2) L ⊃ ΛB 32 34 and using the experimental bounds on the proton decay lifetime, τp > 10 − years, one obtains 15 16 a very strong bound on the scale ΛB, i.e. ΛB > 10 − GeV. One can understand the origin of these interactions in the context of grand uniﬁed theories. The simplest grand uniﬁed theory is based on the gauge group SU(5) and it was proposed by H. Georgi and S. Glashow in 1974 [3]. In this context one can understand the origin of the 2. Particle Physics and the Great Desert Hypothesis 6

Figure 2: The Desert Hypothesis in the case when we assume Low Energy Supersymmetry. We list the R−parity violating interactions and the dimension ﬁve operators mediating proton decay. In this case one could describe physics at the TeV scale in the context of the Minimal Supersymmetric Standard Model and at the high scale a possible supersymmetric grand uniﬁed theories or superstring theories could play a role.

Standard Model interactions since they are just different manifestations of the same fundamental force defined at the high scale. The desert hypothesis was postulated in the same article [3] when the authors realized that the new gauge bosons present in SU(5) mediate proton decay and generate the dimension six baryon number violating operator in Eq. (2). In the context of grand unified theories the Standard Model quarks and leptons are unified in the same multiplets. Therefore, the baryon number is explicitly broken at the high scale. One must mention that the desert hypothesis is also in agreement with the extrapolation of the gauge couplings up to the high scale. H. Georgi, H. R. Quinn and S. Weinberg [4] have shown also in 1974 that the Standard Model gauge couplings could be unified at the high scale if we assume no new physics between the electroweak and high scales. In 1974 they did not know the values of the gauge couplings with good precision but the main idea was correct.

The minimal supersymmetric Standard Model (MSSM) has been considered as one of the most appealing theories to describe physics at the TeV scale. In this context the desert hypothesis plays a main role because the unification of the Standard Model gauge couplings is realized with good precision. The authors in Refs. [5,6,7,8] studied the extrapolation of the gauge couplings assuming the great desert, showing that they meet without the need of large 2. Particle Physics and the Great Desert Hypothesis 7 threshold effects. This result has been quite influential in the particle physics community and very often is used as one of the main motivations for having supersymmetry at the low scale. Unfortunately, in the context of the minimal supersymmetric Standard Model one finds new interactions which violate the baryon and lepton numbers. In the MSSM superpotential one has the terms 0 00 LˆHˆ + λLˆLêˆc + λ QˆLˆdˆc + λ uˆcdˆcdˆc, (3) WMSSM ⊃ u where the first three terms violate the total lepton number and the last term breaks the baryon number. Here we use the standard superfield notation for all MSSM multiplets. It is well-known that the last two interactions together mediate the dimension four contributions to proton decay and one needs to impose a discrete symmetry by hand to forbid these interactions. Imposing the discrete symmetry called matter parity defined as

3(B L) M = ( 1) − , (4) − one can forbid these interactions. Here, B and L are the baryon and lepton numbers, respectively. Matter parity is defined in such way that it is equal to 1 for any matter superfield − and +1 for any gauge or Higgs superfield present in the MSSM. There is a very simple relation between R-parity and M-parity, R = ( 1)2SM, (5) − where S is the spin of the particle. Notice that the interactions in Eq. (3) break the R-parity discrete symmetry as well and often we refer to these terms as R-parity violating interactions. It is important to mention that even if we impose the conservation of M(R)-parity, still there are interactions which mediate proton decay. These are the dimension five operators such as

QˆQˆQˆLˆ MSSM c5 . (6) W ⊃ ΛB

In the context of grand unified theories these interactions are mediated by colored fermions and one needs an extra suppression mechanism to satisfy the experimental bounds on the proton decay lifetime. One can say that in the context of supersymmetric theories these interactions are problematic and even assuming the great desert we need a suppression mechanism. See Fig.2 for a naive representation of the desert hypothesis in the case when we have low energy supersymmetry. As we have discussed before, the desert hypothesis plays a major role in the particle physics community, and it is often assumed as true. This hypothesis is very naive and maybe we never will know if it is true because there is no way to test the theories at the high scale. The 3. Proton Stability 8 only theory we know which describes physics at the electroweak scale is the Standard Model of particle physics. Then, if we only worry about the theory defined at the low scale, the need to assume a large cutoff is uncomfortable. This issue tells us that one should think about effective theories defined at the low scale which are free of these baryon number violating operators mediating proton decay.

3. Proton Stability

We have discussed in the previous section that the main reason to assume the great desert between the weak and Planck scales is the proton stability. In the renormalizable Standard Model of particle physics the proton is stable since the baryon number is broken in three units by the SU(2) instantons. However, there are higher-dimensional operators [9] of dimension six which are allowed by the Standard Model gauge symmetry and violate baryon number in one unit. Those operators mediate proton decay and are highly constrained by the experimental bounds. The simplest dimension six baryon number violating operators are given by

= c (uc) γµq (ec) γ q , (7) O1 1 L L L µ L = c (uc) γµq (dc) γ ` , (8) O2 2 L L L µ L = c (dc) γµq (uc) γ ` . (9) O3 3 L L L µ L

Here q (3, 2, 1/6), ` (1, 2, 1/2), (ec) (1, 1, 1), (uc) (3¯, 1, 2/3) and (dc) L ∼ L ∼ − L ∼ L ∼ − L ∼ (3¯, 1, 1/3) are the Standard Model quarks and leptons. These operators are generated once we integrate out superheavy gauge bosons present in grand unified theories. The first two operators are mediated by the gauge bosons (X ,Y ) (3, 2, 5/6), while the last one is mediated by µ µ ∼ − 0 0 (X ,Y ) (3, 2, 1/6). These gauge bosons are present in SU(5) and flipped SU(5) grand µ µ ∼ unified theories, respectively.

The coefficients ci in the above equations are proportional to the inverse of the gauge boson 2 masses squared, MV− . Therefore, using the experimental bounds listed in Fig.3 one finds 15 16 naively a very strong lower bound on the mass of the gauge bosons, i.e. MV > 10 − GeV. Now, since these gauge bosons acquire mass once the grand unified symmetry is broken, the unified scale has to be very large. This simple result tells us that there is no way to test these theories at collider experiments. Even if proton decay is found in current and future experiments it will be very difficult to realize the test of a grand unified theory. See Ref. [1] for a review on grand unified theories and proton decay. For the possibility to find an upper bound on the total proton decay lifetime see Ref. [11]. 4. Neutron-antineutron oscillations 9

Figure 3: Experimental bounds on the proton decay lifetimes [10]. Here one can see the bounds from diﬀerent experimental collaborations.

As one can appreciate, these bounds have dramatic implications for particle physics because the simplest unified theories must describe physics at the high scale and the great desert is needed. A grand unified theory could be defined at the low scale. However, in this case baryon number should be a local symmetry spontaneously broken at the low scale. In the next section we will discuss the theories with gauged baryon number which predict that the proton is stable and define a new path to low scale unification.

4. Neutron-antineutron oscillations

In this section we discuss the simplified models for nn¯ oscillations studied in Ref. [12]. In Table1 we list the different scalar di-quarks relevant for this study. Now, we list the simplest models which give rise to processes with ∆B = 2 and ∆L = 0, but only the first three models contribute to nn¯ oscillations at tree-level due to the symmetry properties of the Yukawa 4. Neutron-antineutron oscillations 10

Interactions SU(3) SU(2) U(1) ⊗ ⊗ Y Xq q , Xu d (6¯, 1, 1/3) L L R R − Xq q (6¯, 3, 1/3) L L − XdRdR (3, 1, 2/3), (6¯, 1, 2/3) Xu u (6¯, 1, 4/3) R R −

Table 1: Possible interactions between the scalar di-quarks and the Standard Model fermions [12]. interactions. The sextet ﬁeld is deﬁned as   X˜ 11 X˜ 12/√2 X˜ 13/√2   αβ  ˜ 12 ˜ 22 ˜ 23  X =  X /√2 X X /√2  , (10)   X˜ 13/√2 X˜ 23/√2 X˜ 33 and the simplest models are discussed below.

Model 1: In this case the extra ﬁelds are X (6¯, 1, 1/3) and X (6¯, 1, 2/3), and the 1 ∈ − 2 ∈ relevant Lagrangian reads as

= habXαβ qa qb habXαβ(da db ) L1 − 1 1 Lα Lβ − 2 2 Rα Rβ ab αβ a b αα0 ββ0 γγ0 h0 X (u d ) + λ X X X 0 0 0 . (11) − 1 1 Rα Rβ 1 1 2 αβγ α β γ

Here the Greek indices α, β and γ are the color indices, while a and b are family indices. h1 must be antisymmetric in ﬂavor space but this antisymmetry is not retained upon rotation into the mass eigenstate basis. The coupling h2 must be symmetric because of the symmetric color structure in the second term.

Model 2: In the second model one has the ﬁelds X (6¯, 3, 1/3) and X (6¯, 1, 2/3) with 1 ∈ − 2 ∈

ab αβA a A b ab αβ a b αα0A ββ0A γγ0 = h X (q τ q ) h X (d d ) + λ X X X 0 0 0 . (12) L2 − 1 1 Lα Lβ − 2 2 Rα Rβ 1 1 2 αβγ α β γ

Here the matrix τ A is symmetric, while the ﬁrst and second terms have symmetric color structures, h1 and h2 must be symmetric in ﬂavor.

Model 3: One can have a simple scenario with the ﬁelds X (6¯, 1, 2/3) and X (6¯, 1, 4/3) 1 ∈ 2 ∈ − and the Lagrangian in this case is given by

ab αβ a b ab αβ a b αα0 ββ0 γγ0 = h X (d d ) h X (u u ) + λ X X X 0 0 0 . (13) L3 − 1 1 Rα Rβ − 2 2 Rα Rβ 1 1 2 αβγ α β γ 4. Neutron-antineutron oscillations 11

d d

u X2 u

X1 X1 d d

Figure 4: Feynman Graph for neutron-antineutron oscillations [13, 12].

Both terms have symmetric color structures and no weak structure, so h1 and h2 must be symmetric in ﬂavor.

Model 4: In the last scenario one has X (3, 1, 2/3) and X (6¯, 1, 4/3) with 1 ∈ 2 ∈ − = habX da db αβγ habXαβ(ua ub ) + λ X X Xαβ. (14) L4 − 1 1α Rβ Rγ − 2 2 Rα Rβ 1α 1β 2

Because of the antisymmetric color structure in the first term, h1 must be antisymmetric in flavor which prevents it from introducing meson-antimeson mixing. The antisymmetric structure of h1 also prevents the existence of six-quark operators involving all first-generation quarks, and one prevents nn¯ oscillations. In order to understand the constraints from the n n¯ oscillation experiments the authors − in Ref. [12] studied model 1. The transition matrix element

∆m = n¯ n , (15) h |Heff | i leads to a transition probability for a neutron at rest to change into an antineutron after time t 2 equal to Pn n¯(t) = sin ( ∆m t). Neglecting the coupling h1 in the Lagrangian given by Eq.(11) → | | the effective ∆B = 2 Hamiltonian that causes nn¯ oscillations reads as [12] | | 11 2 11 (h10 ) h2 λ α β γ δ λ χ eff = 4 2 dRidRi0 uRjdRj0 uRkdRk0 αβγδλχ H − 4M1 M2 0 0 0 + 0 0 0 + 0 0 0 + 0 0 0 + h.c. (16) × ijk i j k i jk ij k ij k i jk ijk i j k where Latin indices are color and Greek indices are spinor. It arises from the tree-level diagram in Fig.4. In the vacuum insertion approximation to Eq. (15) one finds [12, 14] that 11 2 11 2 (h10 ) h2 ∆m = 2λβ | 4 2 | . (17) | | 3M1 M2 Here the constant β 0.01 GeV3. Using the current experimental limit on ∆m [15], ' 33 ∆m < 2 10− GeV, (18) | | × 5. Mechanisms for Neutrino Masses 12

11 11 assuming the relation M1 = M2 = M, the values of the couplings h10 = h2 = 1, and λ = M one obtains

M & 500 TeV. (19)

The eﬀect on nn¯ oscillations is maximized if we choose M > M because ∆m can be larger. 2 1 | | Assuming M1 = 5 TeV and λ = M2 experiments in the future may be able to probe nn¯ oscillations with increased sensitivity of ∆m 7 10 35 GeV. If no oscillations are observed | | ' × − the new lower bound in the case of equal masses will be

M & 1000 TeV . (20)

In this way one can see the possible bounds that can be achieved if nn¯ oscillations are not discovered. It is important to say that nn¯ experiments cannot probe the grand uniﬁed scale but the discovery of this process will change the way we think about physics beyond the Standard Model. More experiments are badly needed in this area. See Ref. [16] for a recent review on nn¯ oscillations.

5. Mechanisms for Neutrino Masses

The discovery of massive neutrinos have motivated the experts in the ﬁeld to think about the possible mechanisms to generate neutrino masses. The neutrinos could be Majorana or Dirac particles. In the Majorana case the total lepton number must be broken in two units, ∆L = 2, as in the seesaw mechanisms. In this section we discuss the simplest mechanisms for generating neutrino masses at tree and one-loop level. The simplest mechanisms at tree level are the following: Type I Seesaw [17, 18, 19, 20, 21]: This is perhaps the simplest mechanism for generating neutrino masses. In this case one adds a SM singlet, νc (1, 1, 0), and using the interactions ∼ 1 I = Y l H νc + M νc νc + h.c., (21) −Lν ν 2 in the limit M Y v one ﬁnds ν 0

I 1 1 T 2 = Y M − Y v , (22) Mν 2 ν ν 0 where M is typically deﬁned by the B L breaking scale. Then, one understands the smallness of − the neutrino masses due to the existence of a mass scale, M Y v m . Here, if we assume ν 0 ν Y 1 the scale M 1014 15 GeV. Now, in general it is not possible to make predictions for ν ∼ ∼ − 5. Mechanisms for Neutrino Masses 13

the neutrino masses and mixing in this framework since we do not know the matrices Yν and M. Then, one should look for a theory where one could predict these quantities. In the context of SO(10) grand unified theories one can relate the charged fermion and neutrino masses in a consistent way. Once the right-handed neutrinos are present in the theory the B L symmetry can be defined − as a local symmetry. In this scenario the same Higgs breaking B L can generate Majorana − masses for the right-handed neutrinos. Therefore, the scale M is defined by the B L breaking − scale. In this case if the scale is in the TeV range one can produce the right-handed neutrinos with large cross sections through the B L gauge boson −

pp ZB∗ L N1N1 ei±ej±W ∓W ∓, → − → → and one can have signatures with two same-sign dileptons at the LHC. Here ei = e, µ, τ and N1 is the lightest physical νc-like state. See Ref. [22] for a recent study of these collider signals. For the original idea of looking for lepton number violation at colliders see Ref. [23]. Type II Seesaw [24, 25, 26, 27, 28]: In this scenario one introduces a new Higgs boson, ∆ (1, 3, 1), which couples to the leptonic doublets and the SM Higgs boson ∼

II = Y ` ∆ ` + µ H ∆† H + h.c., (23) −Lν ν L L and when the neutral component in ∆ = (δ0, δ+, δ++) acquires a vacuum expectation value, v∆, one ﬁnds II = √2 Y v = µ Y v2/M 2 . (24) Mν ν ∆ ν 0 ∆ Notice that if µ M and M 1014 15 GeV the vev v should be of order 1 eV. However, ∼ ∆ ∆ ∼ − ∆ in general the triplet mass can be around the TeV scale and µ can be small. Unfortunately, in this context one cannot make predictions for neutrino masses because in general the matrix

Yν and v∆ are unknown. Then, as in the previous case, one should look for a theory where one can predict these quantities. We would like to mention that the field ∆ can be light in agreement with the unification constraints in a simple grand unified theory based on the SU(5) gauge symmetry [29, 30, 31]. In this scenario one can have spectacular signatures at the LHC if the Higgs triplet is light. When M 1 TeV one could produce at the LHC the doubly and singly charged Higgses ∆ ≤ present in the model. When v < 10 4 GeV the dominant decays are H++ e+e+ and ∆ − → i j H+ e+ν¯, and we could learn about the neutrino spectrum. These signatures have been → i investigated in great detail in Refs. [32, 33, 34, 35, 36]. 5. Mechanisms for Neutrino Masses 14

Type III Seesaw [37, 38, 39, 40, 41, 42]: In the case of Type III seesaw one adds new fermions, ρ (1, 3, 0), and the neutrino masses are generated using the following interactions L ∼ III = Y ` ρ H + M Tr ρ2 + h.c., (25) −Lν ν L L ρ L 0 + where ρL = (ρ , ρ , ρ−). Integrating out the neutral component of the fermionic triplet one ﬁnds III 1 1 T 2 = Y M − Y v . (26) Mν 2 ν ρ ν 0 Here, as in the case of Type I seesaw, if Y 1 one needs M 1014 15 GeV. One faces the ν ∼ ρ ∼ − same problem, if we want to make predictions for neutrinos masses and mixings, a theory where

Yν and Mρ can be predicted is needed. The existence of a light fermionic triplet is consistent with unification of gauge interactions in the context of SU(5) theories. See Refs. [39, 43, 44] for details. The testability of the Type III seesaw mechanism at the LHC has been investigated in detail in Refs. [45, 46, 47]. We have mentioned the simplest mechanisms at tree level. Now, if Supersymmetry is realized in nature one has the extra possibility to generate neutrino masses through the R-parity violating couplings. One could say that in this case we use a combination of the different seesaw mechanisms. We will discuss this possibility in the next sections. There are several mechanisms for generating neutrino masses at one-loop level. In this case one assumes that the mechanisms discussed above are absent and only through quantum corrections one generates neutrino masses. This possibility is very appealing since the neutrino masses are very tiny and the seesaw scale can be low. Zee Model [48]: In the so-called Zee model one introduces two extra Higgs bosons, h ∼ 0 (1, 1, 1) and H (1, 2, 1/2). In this case the relevant interactions are ∼ 2 0 X c = Y ` h ` + µ H H h† + Y e H† ` + h.c., (27) −LZee L L i i L i=1 where in general both Higgs doublets couple to the matter fields. Using these interactions one can generate neutrino masses at one-loop level. See Ref. [48] for details. Now, it is important to mention that in the simple case where only one Higgs doublet couples to the leptons [49] it is not possible to generate neutrino masses in agreement with neutrino data. See for example Refs. [50, 51, 52, 53, 54] for details. Colored Seesaw [55]: Now, suppose that one looks for the simplest mechanism for neutrino masses at one-loop level where we add only two types of representations, a fermionic and a scalar one, and with no extra symmetry. All the possibilities were considered in Ref. [55] where we found that only two cases are allowed by cosmology. In this case one has two possible scenarios: 6. Neutrinoless double beta decay 15

Figure 5: Colored Seesaw Mechanism [55]. The ﬁeld S is a colored scalar octet, while ρ is a fermionic colored octet.

1. The extra ﬁelds are a fermionic ρ (8, 1, 0) and the scalar S (8, 2, 1/2). 1 ∼ ∼ 2. One adds the fermion ρ (8, 3, 0) and the scalar ﬁeld S (8, 2, 1/2). 2 ∼ ∼ In both cases one generates neutrino masses through the loop in Fig.5. The relevant interactions in this case are given by

2 2 = Y ` S ρ + M Tr ρ + λ Tr S†H + h.c. (28) −LCS 2 L 1 ρ1 1 2

Using as input parameters, Mρ1 = 200 GeV, v0 = 246 GeV and MS = 2 TeV we find that in order to get the neutrino scale, 1 eV, the combination of the couplings, Y 2λ 10 8. ∼ 2 2 ∼ − The experimental bounds on the colored octet masses depend of their decays. See Ref. [56] for a detailed discussion. This mechanism could be easily tested at the LHC. In this case the seesaw fields can be produced with very large cross sections through the QCD interactions. See Ref. [56] for a detailed analysis of the collider signatures in this context. We would like to mention that this mechanism can be realized in the context of grand unified theories.

6. Neutrinoless double beta decay

The total lepton number can be broken in two units, ∆L = 2, and one can have the exotic process A A X Y + 2e−, (29) Z → Z+2 A which is called neutrinoless double beta decay. Here Z X is a nuclei with atomic number Z and atomic mass A. For a recent review on neutrinoless double beta decay see Ref. [57]. This process proceeds through the Majorana neutrino mass insertion as one can see in Fig.6 and the amplitude of these processes is proportional to the quantity 7. Grand Unified Theories 16

Figure 6: Feynman Graph for neutrinoless double beta decay [57]. This contribution is present in any model for Majorana neutrino masses.

X 2 mββ = Uei mi, (30) i where U is the PMNS mixing matrix [58, 59] in the leptonic sector and mi are the neutrino eigenvalues. Unfortunately, the neutrino spectrum is unknown and one cannot predict the value of mββ and the lifetime for this process. The most optimistic scenario corresponds to the case when the neutrino spectrum has an inverted hierarchy. In this scenario the lower bound on mββ is about 2 10 2 eV and there is a hope to discover this process at current or future experiments. × − See Ref. [57] for more details and Refs. [60, 61, 62] for the status of the experimental searches. In many extensions of the standard model one can have extra contributions for neutrinoless double beta decay. For example, in supersymmetric models with R-parity violation one has extra terms mediated by the trilinear and bilinear interactions breaking lepton number. In models with Type II seesaw the double charged Higgses give rise to new contributions and the same in other scenarios. One can say that if neutrinoless double beta decay is discovered one can establish that the neutrino is a Majorana particle but we cannot learn too much about the mechanism for neutrino masses.

7. Grand Uniﬁed Theories

The grand unified theories (GUTs) have been considered the most appealing extensions of the Standard Model where one can understand the origin of the Standard Model interactions. In this context the gauge interactions of the SM are just different manifestations of the fundamental force defined at a different scale. The simplest GUT was proposed in 1974 by H. Georgi and S. Glashow in Ref. [3]. This theory is based on the SU(5) gauge symmetry and the matter is 7. Grand Unified Theories 17 unified in two representations in agreement with anomaly cancellation

 c   c c 1 1  d1 0 u3 u2 u d    − − −   c   c 1 2 2   d2   u3 0 u u d     − − −   c   c c 3 3  5 =  d3  , 10 =  u2 u1 0 u d  . (31)    − − −     1 2 3 c   e   u u u 0 e      ν d1 d2 d3 ec 0 − L − L The Higgs sector is composed of only two Higgs ﬁelds   T 1    2   T        Σ8 Σ 1 2 0 3 (3,2) 5H =  T  , 24H =   +   Σ24. (32)   √   Σ∗ Σ3 2 15 0 3  +  (3,2)  H  −   H0

The gauge bosons live in the adjoint representation 24G which contains the Standard Model gauge bosons and the Vµ gauge bosons mediating proton decay,     Gµ Vµ 1 2 0 24G =   +   Bµ. (33) V W 2√15 0 3 µ∗ µ − The new gauge bosons transform as V (3, 2, 5/6) and mediate the dimension six operators µ ∼ − discussed in the previous section. Unfortunately, the Georgi-Glashow model is not realistic because it fails to explain the values of the gauge couplings at the low scale. The idea of deﬁning a simple and realistic grand uniﬁed theory based on SU(5) has been investigated by many groups. There are two simple scenarios which are consistent with the experiment and we discuss their main features.

Type II-SU(5) [29, 30, 31]: • In this context the neutrino masses are generated through the Type II seesaw mechanism and one can have a consistent relation between the charged lepton and quark masses using higher-dimensional operators allowed by the gauge symmetry. The neutrino masses are generated using the following interactions

V Y 5 5 15 + µ 5∗ 5∗ 15 + h.c., (34) ⊃ ν H H H H

where the 15 field contains three fields, Φ (1, 3, 1) is the field needed for Type H a ∼ II seesaw, Φ (3, 2, 1/6) is a scalar leptoquark and Φ (6, 1, 2/3). In order to b ∼ c ∼ − 7. Grand Unified Theories 18

= 11 MFb 4.3 10 GeV

L M M P3 F a 8 F = M a 8.83

1GeV =

Ú 130 GeV b 3 = F 1.69 10 M

H 8

10 GeV 6 10 10

Log M M Ú GeV F M 3 = a Ú M = 3 = Z 10 10 4 4 4 GeV GeV = P2 P1 MFb 242 GeV 2 1 1.5 2 2.5 14 MGUT H10 GeVL

Figure 7: Constraints from gauge coupling unification in Type II-SU(5) [31]. Here we show the allowed masses for the fields Φa,Φb, and Σ3 in agreement with gauge coupling unification at the one-loop level. The vertical line defines the bound from proton decay. The points P1, P2, and P3 define the allowed area by unification and proton decay.

understand the possible predictions in this model, the constraints coming from unification have been investigated in detail in Refs. [29, 30, 31]. In Fig.7 we show the constraints on the spectrum of the theory assuming gauge coupling unification at the one-loop level. As one can appreciate, in order to find the maximal value of the unification scale the

leptoquark Φb must be light. Here in order to achieve a proton lifetime consistent with the experiment one needs to use the suppression mechanism proposed in Ref. [11]. This model has two main features: a) Predicts the existence of a light lepto-quark which could be discovered at the LHC, b) The total lifetime of the proton is τ 1036 years. The p ≤ main phenomenological aspects of the leptoquark Φb were investigated in Ref. [63].

Type III-SU(5) [39, 43, 44]: • The neutrino masses can be generated through the Type III seesaw mechanism. In this case one needs to include an extra fermionic representation in the adjoint representa-

tion, 24 = (ρ8, ρ3, ρ(3,2), ρ(3¯,2), ρ24). This mechanism can be realized using the following interactions

5¯ V Y 5¯ 24 5 + (Y 24 24 + Y 24 24 + Y Tr(24 24 )) 5 + h.c. (35) ⊃ 1 H Λ 2 H 3 H 4 H H 7. Grand Unified Theories 19

2.05 Λ= 1.2 1019 GeV

M =MGUT 2.04 Σ8 M 5 ρ GeV 8 Z M 2.03 = 3 M Σ / 1 GeV) / 3 2.02 ρ M ( 10 2.01 Log M =100 GeV ρ3 2 GeV 30 GeV

GeV =1 12 13 Σ 3 1.99 M = 10 = = 5 10 = (3,2) ρ (3,2) M ρ

1.98 M 15.5 15.55 15.6 15.65 15.7 15.75 M Log10( GUT / 1 GeV)

Figure 8: Constraints from gauge coupling unification in Type III-SU(5) [43]. Here we show the allowed masses for the ρ3,Σ3 and ρ(3,2) fields in agreement with gauge coupling unification at one-loop level. The dashed and blue vertical lines show the allowed parameter space by perturbativity. See Ref. [43] for more details.

Notice that 24 contains two relevant ﬁelds for the seesaw mechanism, ρ3 for Type III

seesaw and ρ24 for Type I seesaw. The uniﬁcation constraints have been investigated in great detail in this model [39, 43, 44, 64]. In Fig.8 we show the allowed parameter space in agreement with uniﬁcation and the values of the gauge couplings at the low

scale. As one can appreciate, the ﬁeld ρ3 needed for Type III seesaw must be light in the theory. Therefore, one could hope to test the Type III seesaw mechanism at the LHC. See Refs. [45, 46, 47] for the testability of this mechanism. This model also predicts a 36 37 lifetime of the proton equal to τp < 10 − years [39, 43, 44].

As we have discussed above, there are two simple theories based on SU(5) which are consistent with the experiment. Theories based on the SO(10) gauge symmetries are very appealing because one has the uniﬁcation of the Standard Model fermions in only one representation. In the 16 spinor representation one has SM fermions for each family and the right-handed neutrinos. Unfortunately, the Higgs sector of these theories are often very involved and since we have several symmetry breaking paths to the Standard Model one cannot make good predictions for proton decay. See Ref. [65] for a review on grand uniﬁed theories. 8. Theories for Local Baryon and Lepton Numbers 20

The supersymmetric version of SU(5) has been proposed by H. Georgi and S. Dimopou- los [66], and independently by N. Sakai [67]. In this context one needs to impose by hand the discrete symmetry R parity in order to forbid the dimension four contributions to proton − decay coming from the term 10ˆ 5¯ˆ 5.¯ˆ The dimension ﬁve operators are also very important in this context. However, in general it is very diﬃcult to predict the lifetime of the proton without knowing the full spectrum of supersymmetric particles. It is well-known, that the minimal renormalizable supersymmetric SU(5) is ruled out because the relation Yd = Ye is in disagreement with the experiment. Here Yd and Ye are the Yukawa matrices for down quarks and charged leptons, respectively. The simplest solution to this problem is to include the allowed higher-dimensional operators suppressed by the Planck scale. However, in this case one can have even less predictivity for proton decay. See Ref. [1] for a review on proton decay and Refs. [68, 69, 70, 71] for a detailed discussion of this issue.

8. Theories for Local Baryon and Lepton Numbers

In this section we discuss the simplest realistic theories where the baryon and lepton numbers are defined as local gauge symmetries spontaneously broken at the low scale. Here we summarize the main results presented in Refs. [72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82]. The original idea related to the spontaneous breaking of local baryon number with the use of the Higgs mechanism was proposed by A. Pais in Ref. [83]. In this article the author did not build a theory based on local baryon number but he proposed to use the Higgs mechanism to understand the spontaneous breaking of this symmetry. See also Refs. [84, 85, 86] for previous studies in theories with local baryon number. In the Standard Model the baryon and lepton numbers are accidental global symmetries of the Lagrangian but they are not free of anomalies. In order to define a consistent theory where baryon and lepton numbers are local gauge symmetries, all relevant anomalies need to be cancelled. Therefore, the SM particle content has to be extended by additional fermions. In our notation, the SM fermionic fields and their transformation properties under the gauge group SU(3) SU(2) U(1) U(1) U(1) ⊗ ⊗ Y ⊗ B ⊗ L are listed in Table 2. Here, the right-handed neutrinos are part of the SM fermionic spectrum. The baryonic anomalies are the following

SU(3)2 U(1) , SU(2)2 U(1) , U(1)2 U(1) , A1 ⊗ B A2 ⊗ B A3 Y ⊗ B U(1) U(1)2 , (U(1) ) , U(1)3 . A4 Y ⊗ B A5 B A6 B 8. Theories for Local Baryon and Lepton Numbers 21

Field SU(3) SU(2) U(1)Y U(1)B U(1)L

qL 3 2 1/6 1/3 0

uR 3 1 2/3 1/3 0 d 3 1 1/3 1/3 0 R − ` 1 2 1/2 0 1 L − e 1 1 1 0 1 R − νR 1 1 0 0 1

Table 2: The Standard Model fermionic content and the right-handed neutrinos.

In the Standard Model the only non-zero anomalies are SM = SM = 3/2, while leptonic A2 −A3 anomalies which must be cancelled are

SU(3)2 U(1) , SU(2)2 U(1) , U(1)2 U(1) , A7 ⊗ L A8 ⊗ L A9 Y ⊗ L U(1) U(1)2 , (U(1) ) , U(1)3 . A10 Y ⊗ L A11 L A12 L

The anomalies which are non-zero in the Standard Model with right-handed neutrinos are SM = SM = 3/2. One also has to think about the cancellation of the mixed anomalies for A8 −A9 the Abelian symmetries

U(1)2 U(1) , U(1)2 U(1) , (U(1) U(1) U(1) ) , A13 B ⊗ L A14 L ⊗ B A15 Y ⊗ L ⊗ B which of course vanish in the Standard Model. Various solutions to the equations which deﬁne the cancellation of anomalies were studied in Refs. [72, 73, 74, 75, 76, 77, 78]. Here we discuss the simplest solutions to understand how one can write down a realistic version which could be tested at future experiments.

Sequential Family: In Refs. [72, 73, 74] a sequential (chiral) family was added to the • spectrum. In this case the new quarks have baryon number 1 and the leptons have − lepton number 3. This solution is ruled out today because the new quarks are chiral − and change the gluon fusion Higgs production by a large factor, approximately a factor 9. Therefore, this is in clear disagreement with the recent experimental results at the Large Hadron Collider.

Mirror Family: In Refs. [72, 73, 74] the possibility to use mirror fermions was considered. • However, as in the previous case, this scenario is ruled out by the Standard Model Higgs properties. 8. Theories for Local Baryon and Lepton Numbers 22

Vector-Like Fermions: In Ref. [74] the anomalies are cancelled using vector-like fermions. • In this case, anomaly cancellation requires that the diﬀerence between the baryon numbers of the new quarks is equal to 1, while the diﬀerence between the lepton numbers of − the new leptons is 3. In this scenario the neutrino masses are generated through the − Type I seesaw mechanism and the new charged leptons acquire masses only from the SM Higgs vacuum expectation value. Therefore, the lepton number is broken by two units and one does not have proton decay. Unfortunately, the new charged leptons change dramatically the Higgs branching ratio into gamma gamma, reducing it by about a factor of 3. Therefore, this scenario is also ruled by the Higgs results at the Large Hadron Collider.

One can modify this model adding a new Higgs boson with lepton number and generate vector-like masses for charged leptons, but one will generate dimension nine operators mediating proton decay, e.g.,

c9 0 = (u u d e ) S S† S . (36) O9 Λ5 R R R R B L L

Here the Higgs bosons transform as

0 S (1, 1, 0, 1, 0),S (1, 1, 0, 0, 2), and S (1, 1, 0, 0, 3). B ∼ − L ∼ − L ∼ −

0 Now, assuming that c 1, and the vacuum expectation values for S ,S , and S around 9 ∼ B L L a TeV, one ﬁnds that Λ 107 8 GeV. This means that one still has to postulate half of ≥ − the desert in order to satisfy the proton decay bounds or assume a small coupling c9.

As one can appreciate the simplest solutions for anomaly cancellation are ruled out by the collider experiments or require an extra mechanism to suppress proton decay. The possibility to deﬁne an anomaly free theory with gauged baryon and lepton numbers using lepto-baryons has been investigated. In this context lepto-baryons are fermions with baryon and lepton numbers. There are two simple models which are in agreement with all bounds from cosmology and collider physics:

In the first case adding only six vector-like fields one can define an anomaly free theory. • See Table 3 for the particle content and Ref. [76] for details. In this context the lightest field could be a neutral Dirac fermion which is a good candidate for the cold dark matter in the universe. We will discuss the properties of this model in the next section. 8. Theories for Local Baryon and Lepton Numbers 23

Field SU(3) SU(2) U(1)Y U(1)B U(1)L

ΨL N 2 Y1 B1 L1

ΨR N 2 Y1 B2 L2

ηR N 1 Y2 B3 = B1 L3 = L1

ηL N 1 Y2 B4 = B2 L4 = L2

χR N 1 Y3 B5 = B1 L5 = L1

χL N 1 Y3 B6 = B2 L6 = L2

Table 3: Fermionic Lepto-baryons in the model proposed in Ref. [76].

There is a second simple model with only four representations. See Table 5 for the particle • context and Ref. [78] for the detailed discussion. In this case one uses ﬁelds in the adjoint and fundamental representation of SU(2) to cancel the non-trivial anomalies. The dark matter candidate in this case is a Majorana fermion.

In order to ﬁnd realistic scenarios one considers the particle content listed in Table3, where the new fermionic ﬁelds are singlets or live in the fundamental representation of SU(2). The SU(2)2 U(1) anomaly can be cancelled if one imposes the condition A2 ⊗ B 3 B B = . (37) 1 − 2 −N Now, to cancel the SU(3)2 U(1) anomaly when N = 1 one needs to impose the relation A1 ⊗ B 6

2(B B ) (B B ) (B B ) = 0, (38) 1 − 2 − 3 − 4 − 5 − 6 which can be cancelled using

B1 = B3 = B5, and B2 = B4 = B6. (39)

See Table3 for more details. There is a “duality” between the cancellation the baryonic and leptonic anomalies. Both types of anomalies can be cancelled in the same way. The anomaly 8. Theories for Local Baryon and Lepton Numbers 24

SU(2)2 U(1) can be cancelled using the condition A2 ⊗ L 3 L L = , (40) 1 − 2 −N for the leptonic charges and the others imposing the relation

L1 = L3 = L5, and L2 = L4 = L6 . (41)

Finally, one has to think about the anomalies with weak hypercharge. Once one uses the above assignment of baryon and lepton numbers, and are always cancelled and do not provide A4 A10 a condition for the hypercharges. In order to cancel the anomaly U(1)2 U(1) , one needs Y ⊗ B 1 Y 2 + Y 2 2Y 2 = . (42) 2 3 − 1 2

A simple set of solutions for this equation is given by

1 1 2 1 1 1 (Y ,Y ,Y ) ( , 1, 0), ( , , ), (0, , ) . (43) 1 2 3 ⊂ ±2 ± ±6 ±3 ±3 ±2 ±2

It is important to mention that since the new particles are vector-like with respect to the Standard Model gauge group, the SM anomalies do not pose a problem. One can prove that the U(1) U(1)2 , U(1)2 U(1) and U(1) U(1) U(1) anomalies are cancelled automatically. L⊗ B L⊗ B Y ⊗ L⊗ B Now, one needs to discuss the possibility to satisfy all cosmological and collider constraints. In these scenarios one must avoid a stable electrically charged or colored field. Therefore, the new fields should have a direct coupling to the SM fermions or the lightest particle in the new sector must be stable and neutral. Here we discuss the possible scenarios for different values of N. N = 1: In this case the new fields do not feel the strong interaction, and the only solution which allows for a stable field in the new sector is the one with Y = 1/2, Y = 1, and Y = 0. 1 ± 2 ± 3 Then, when the lightest field is neutral one can have a dark matter candidate. The stability of the dark matter candidate in these models is an automatic consequence coming from symmetry breaking. This is a very appealing solution because as a bonus one has a candidate to describe the cold dark matter in the Universe. N = 3: In this scenario one can use the weak hypercharges Y = 1/6, Y = 2/3, and 1 ± 2 ± Y = 1/3, and a stable colored field can be avoided. In order to generate vector-like masses 3 ± for the new fields one needs a scalar S (1, 1, 0, 1, 1), and one generates dimension seven BL ∼ − − 3 operators such as qLqLqL`LSBL/Λ mediating proton decay. Then, in this case the great desert must be postulated. 8. Theories for Local Baryon and Lepton Numbers 25

Fields SU(3) SU(2) U(1)Y U(1)B U(1)L

1 ΨL 1 2 - 2 B1 L1

1 ΨR 1 2 - 2 B2 L2

ηR 1 1 - 1 B1 L1

ηL 1 1 - 1 B2 L2

χR 1 1 0 B1 L1

χL 1 1 0 B2 L2

Table 4: Lepto-baryons in the model presented in Ref. [76].

N = 8: This scenario could be interesting but in order to couple the new lepto-baryons to the SM fermions we need to include extra colored scalar ﬁelds. One way is to add color octet scalars that let the new fermions couple to leptons. The new colored scalars can decay at one loop to a pair of gluons after spontaneous symmetry breaking. This scenario is possible but we will focus our discussions on the simplest case which corresponds to N = 1.

8.1. Theoretical Framework

The main goal here is to discuss a realistic anomaly free theory based on the gauge group [72]

SU(3) SU(2) U(1) U(1) U(1) . ⊗ ⊗ Y ⊗ B ⊗ L

The simplest solution discussed in the last section is the one with colorless fermions. The new fermion ﬁelds of this model are given in Table 4 assuming N = 1, Y = 1/2, Y = 1, and 1 ± 2 ± Y3 = 0. For simplicity, in this section we discuss only the baryonic sector of the theory. In this case the relevant Lagrangian is given by

= + , (44) L LSM LB 8. Theories for Local Baryon and Lepton Numbers 26 where is the Lagrangian for the ﬁelds present in the SM. Since all quarks have baryon LSM number their kinetic term is modiﬁed. In the above equation reads as LB 1 = BB Bµν,B B BB Bµν + + , (45) LB −4 µν − 2 µν Lf LSB where B = ∂ B ∂ B is the U(1) strength tensor, BB = ∂ BB ∂ BB is the U(1) µν µ ν − ν µ Y µν µ ν − ν µ B strength tensor, and is Lf

= iΨ D/Ψ + iΨ D/Ψ + iη Dη/ + iη Dη/ + iχ Dχ/ + iχ Dχ/ Lf L L R R L L R R L L R R Y Ψ Hη Y Ψ Hχ˜ Y Ψ Hη Y Ψ Hχ˜ − 1 L R − 2 L R − 3 R L − 4 R L λ Ψ Ψ S λ η η S λ χ χ S + h.c. (46) − Ψ L R B − η R L B − χ R L B

The term is deﬁned by LSB

µ 2 2 = (D S )† D S m S† S λ (S† S ) λ (H†H)(S† S ). (47) LSB µ B B − B B B − B B B − HB B B

As we have discussed above the baryon numbers B1 and B2 of the new fermions are constrained by the condition B B = 3. (48) 1 − 2 −

In this context the scalar sector is composed of the SM Higgs and the new boson SB with the following quantum numbers

H (1, 2, 1/2, 0), and S (1, 1, 0, 3). (49) ∼ B ∼ −

Notice that the need to generate vector-like masses for the new fermions deﬁne the baryon number of the new scalar. Once S gets a vacuum expectation value we will have only ∆B = 3 B | | interactions and proton decay never occurs. Therefore, the great desert is not needed. For the application of this idea in low energy supersymmetry see Ref. [87].

8.2. Phenomenological Aspects

The class of theories discussed above predict the existence of a new neutral gauge boson associated to the local baryon number. The interactions of the lepto-phobic gauge boson are given by

1 1 X g χ¯ (B P + B P ) γµχ ZB + M 2 ZBZB,µ g q¯ γµq ZB, (50) LB ⊃ − B 1 L 2 R µ 2 ZB µ − 3 B i i µ i neglecting the kinetic mixing between the two Abelian symmetries, the term B in Eq. (45), such that ZB = BB. Here P = (1 γ5)/2 and P = (1 + γ5)/2 are the usual projection µ µ L − R 8. Theories for Local Baryon and Lepton Numbers 27

Fields SU(3) SU(2) U(1)Y U(1)B U(1)L

1 3 3 ΨL 1 2 2 2 2

1 3 3 ΨR 1 2 2 - 2 - 2

3 3 ΣL 1 3 0 - 2 - 2

3 3 χL 1 1 0 - 2 - 2

Table 5: Lepto-baryons in the model presented in Ref. [78].

operators, while B1 and B2 are the baryon numbers of the dark matter candidate, see Table 4 for the quantum numbers. The mass of the leptophobic gauge boson is given by MZB = 3gBvB, where vB is the vacuum expectation value of the SB boson. The Dirac spinor, χ = χL + χR, is stable and it is the cold dark matter candidate.

Using the decay of ZB into two top quarks the ATLAS collaboration has set bounds on this type of gauge bosons, see Ref. [88]. Here the relevant production channel is pp Z tt¯ . → B∗ → The numerical results for the cross section are shown in Fig.9 for diﬀerent values of the gauge coupling gB, we use the values 0.1, 0.5, and 1.0. The black curve in Fig.9 shows the experimental bounds from the ATLAS collaboration [88]. Then, since the area above the black curve is ruled out by the experiment one can say that the gauge coupling must be smaller than 0.5 to be consistent with the experiment in most of the parameter space. In Fig. 10 we show the bounds on the gauge couplings and the gauge boson mass in a large range. As one can appreciate the gauge coupling can be large in the low mass region. See Refs. [89, 90, 91] for a detailed discussion of the experimental bounds. The properties of the Higgs sector of this models have been investigated in detail in Ref. [82]. Here the new physical Higgs boson can decay with a large branching ratio into dark matter when the mixing between the two physical Higgses is small. This scenario is preferred by the recent discovery of a SM-like Higgs boson.

8.3. Cosmological Aspects

We have mentioned above that in the context of the simplest theories with gauged baryon and lepton numbers one has a candidate for the cold dark matter in the Universe. In the model 8. Theories for Local Baryon and Lepton Numbers 28

√s = 8 TeV, Mχ = 500 GeV 105 gB =1.0 104 gB =0.5 gB =0.1 ATLAS bound 103 ) [fb] ¯ tt 102 → ∗ B Z 101 →

pp 0 ( 10 σ

1 10−

2 10− 500 1000 1500 2000 2500 3000

MZB [GeV]

Figure 9: Decay of the leptophobic gauge boson ZB into two top quarks. Here we show the experimental bounds from the ATLAS collaboration [88] (solid black) and the theoretical predictions for diﬀerent values of the gauge √ couplings (gB = 1 in red, gB = 0.5 in blue, and gB = 0.1 in green) when s = 8 TeV. For more details see Ref. [82].

discussed in the previous section the dark matter candidate is a Dirac fermion, χ = χL + χR. Since the local baryon number symmetry is spontaneously broken at the low scale one must understand the possibility to have a consistent scenario for baryogenesis and explain the possible relation between the baryon asymmetry and the cold dark matter density. This issue has been investigated in Ref. [80] and we outline the main results. See Ref. [93] for a review of asymmetric dark matter mechanisms. Baryon and Dark Matter Asymmetries: The model discussed in the previous section enjoys three anomaly-free symmetries [80]:

B L in the Standard Model sector is conserved in the usual way. We will refer to this • − symmetry as (B L) . − SM

The accidental η global symmetry in the new sector. Under this symmetry the new ﬁelds • transform in the following way:

Ψ eiηΨ , η eiηη , χ eiηχ . L,R → L,R L,R → L,R L,R → L,R

Above the symmetry breaking scale, total baryon number B carried by particles in both, • T the standard model and the new sectors is also conserved. Here we will assume that the symmetry breaking scale for baryon number is low, close to the electroweak scale. 8. Theories for Local Baryon and Lepton Numbers 29

2.5

CDF Run I CMS CMS 5 fb-1 2.0 0.13 fb-1

-1 UA2 ATLAS 1 fb 1.5 ATLAS B CMS -1 g 13 fb 4 fb-1 1.0 CDF 1.1 fb-1

CMS 20 fb-1 0.5

0.0 0 500 1000 1500 2000 2500 M (GeV) Z'B

Figure 10: Experimental bounds on the properties of a leptophobic gauge boson in the plane gauge coupling– gauge boson mass. For more details see Ref. [89].

We recall the thermodynamic relationship between the number density asymmetry n+ n − − and the chemical potential in the limit µ T , ∆n n+ n 15 g µ = − − = , (51) s s 2π2g ξ T ∗ where g counts the internal degrees of freedom, s is the entropy density, and g is the total ∗ number of relativistic degrees of freedom. Here ξ = 2 for fermions and ξ = 1 for bosons. In order to derive the relationship between the baryon asymmetry and the dark matter relic density, one deﬁnes the densities associated with the conserved charges in this theory. Following the notation of Ref. [92], the B L asymmetry in the SM sector is deﬁned in the usual way − 15 ∆(B L) = 3(µ + µ + µ + µ µ µ µ ), (52) − SM 4π2g T uL uR dL dR − νL − eL − eR ∗ and the η charge density is given by 15 ∆η = 2µ + 2µ + µ + µ + µ + µ . (53) 4π2g T ΨL ΨR χL χR ηL ηR ∗ Isospin conservation T3 = 0 implies

µuL = µdL , µeL = µνL , µ+ = µ0 . (54)

Standard Model interactions and the Yukawa interactions present in the theory give us other equilibrium conditions for the chemical potentials. The conservation of electric charge Qem = 0 in this model implies

6(µ + µ ) 3(µ + µ ) 3(µ + µ ) + 2µ µ µ µ µ = 0. (55) uL uR − dL dR − eL eR 0 − ΨL − ΨR − ηL − ηR 8. Theories for Local Baryon and Lepton Numbers 30

The electroweak sphalerons in this context must satisfy the conservation of total baryon number, and the associated ‘t Hooft operator is

3 (qLqLqL`L) ΨRΨL . (56)

Therefore, the sphalerons give us an additional equilibrium condition between the standard model particles and new degrees of freedom

3(3µ + µ ) + µ µ = 0. (57) uL eL ΨL − ΨR This result is very unique. It is important to emphasize that the baryon number in the SM sector is broken but the total baryon number is conserved. Notice that in this model the sphalerons play the crucial role of transferring the asymmetry from the standard model sector to the dark matter sector. The total baryon number is conserved above the symmetry breaking scale. Therefore the requirement of BT = 0 gives 15 h B = 3(µ + µ + µ + µ ) + B (2µ + µ + µ ) + B (2µ + µ + µ ) T 4π2g T uL uR dL dR 1 ΨL ηR χR 2 ΨR ηL χL ∗ i 6µ ) = 0. (58) − SB Using all conditions on the chemical potential in the theory, one can write the ﬁnal baryon asymmetry in the SM sector in terms of ∆(B L) and ∆η [80], − SM 15 BSM (12µ ) = C ∆(B L) + C ∆η, f ≡ 4π2g T uL 1 − SM 2 ∗ with 32 (15 14B ) C = and C = − 2 . (59) 1 99 2 198 Requiring that the dark matter asymmetry is bounded from above by the observed dark matter density, one ﬁnds the following upper bound on the dark matter mass

ΩDM C2 Mp Mχ . (60) ≤ ΩB C1 ΩB L | − − | Here ΩB L = s ∆(B L)SM Mp/ρc, Mp is the proton mass, ρc is the critical density and s is − − the entropy. As one can appreciate, this bound is a function of the baryon number B2, and the B L asymmetry generated through a mechanism such as leptogenesis. Therefore, in general one − could say that there is a consistent relation between the baryon and dark matter asymmetries. In this model one does not have a mechanism to explain the primordial asymmetry in the dark matter sector. Therefore, using ∆η = 0 we ﬁnd the following relation 32 ΩB = ΩB L, (61) 99 − 8. Theories for Local Baryon and Lepton Numbers 31

Ω h2 =0.1199 0.0027 DM ±

43 10−

XENON100 44 10− ] 2 LUX

[cm 45 10− SI χN σ

XENON1T 46 10− B =1/2 B =2 47 10− 500 1000 1500 2000

Mχ [GeV]

2 Figure 11: Prospects for DM direct detection, assuming the value of the DM relic density ΩDMh = 0.1199±0.0027 SI measured by Planck [94]. The plot shows the spin-independent elastic DM–nucleon cross section σχN as a function of the DM mass Mχ. The exclusion limits of XENON100 [95] and LUX [96] are given, as well as the projected limit for XENON1T [97]. The gauge coupling is varied inside gB ∈ [0.1, 0.5], and the gauge boson mass is varied inside MZB = 0.5 − 5.0 TeV. Blue dots are for B = 1/2, red triangles are for B = 2. See Ref. [82] for more details.

where ΩB L is generated through a mechanism such as leptogenesis. In this case one has only − thermal dark matter relic density and a simple relation between the B L asymmetry and the − ﬁnal baryon asymmetry.

Thermal Baryonic Dark Matter: In Refs. [79, 82] the possibility to achieve the correct relic density and the constraints from direct detection have been investigated. Assuming that the total cold dark matter relic density is explained with the Dirac spinor χ, the predictions for direct detection are shown in Fig. 11. One can see that there are many allowed scenarios when one can satisfy the relic density and direct detection constraints from dark matter experiments. In this case the dark matter annihilation cross section into SM particles and the nucleon–dark matter elastic cross section proceed through the leptophobic ZB. A nice feature of this model is that the direct detection cross section is independent of the matrix element because baryon number is a conserved current. See Refs. [79, 82] for more details.

Upper Bound on the Symmetry Breaking Scale: The existence of a non-zero relic density can be used to find an upper bound on the symmetry breaking scale on models with gauged baryon 8. Theories for Local Baryon and Lepton Numbers 32 number when the dark matter candidate is a Dirac fermion. Here we summarize the results presented in Ref. [82]. Using the upper bound on the dark matter relic density, Ω h2 0.12, DM ≤ and the fact that the dark matter and the gauge boson masses are generated through the same √ Higgs mechanism, i.e. Mχ = λχvB/ 2 and MZB = 3gBvB in these models, one finds an upper bound on the symmetry breaking scale given by 4 2 2 9 2 g λ (B1 + B2) 1.77 10 GeV v2 B χ × , (62) B ≤ 168π (2λ2 9g2 )2 + 9 g8 x χ − B 4π2 B f for a given value of the freeze-out temperature xf . Therefore, it is possible to find an upper bound on the gauge boson mass using the above equation

(B1 + B2) MZB 316.1 TeV. (63) ≤ √xf

Now, using xf = 20 and B1 + B2 = 1/2 as example, the upper bound on the gauge boson mass reads as M 35.3 TeV, (64) ZB ≤ and M 17.7 TeV. Notice that this bound is true when Z is heavier than the dark matter χ ≤ B candidate. This result is very interesting because one can say that there is a hope to test or rule out this model at the current or future collider experiments. Of course, the LHC cannot test the theory if the new gauge boson has a mass close to the upper bound. However, one could rule out this type of spectrum at future 100 TeV colliders. Notice that this bound is much smaller than the one coming from unitarity constraints [98].

8.4. Towards Low Scale Unification The theories with local baryon number open up a new possibility for physics beyond the Standard Model. Since the great desert is not needed in this picture one can imagine that the unification of gauge forces can be realized at the low scale. Using a bottom-up approach we will discuss the possibility to have the unification of the gauge interactions at the low scale [81]. In order to investigate the unification of gauge interactions at the low scale it is important to make sure that the proton is stable or the baryon number violating operators are highly suppressed. This theory based on the gauge group

SU(3) SU(2) U(1) U(1) U(1) , ⊗ ⊗ Y ⊗ B ⊗ L has been discussed in the previous sections. In one of the models the new fermions called “lepto-baryons” needed for anomaly cancellation are

Ψ (1, 2, 1/2,B,L), Ψ (1, 2, 1/2, B, L), L ∼ R ∼ − − Σ (1, 3, 0, B, L), and χ (1, 1, 0, B, L), L ∼ − − L ∼ − − 8. Theories for Local Baryon and Lepton Numbers 33

60 SM -1 Α1 50 - Α SM 1 40 2 1 I M - i SM -1 Α Α -1 3 30 Α2 I M -1 20 Α1 -1 I M HΑ3 L 10

H 4L 6 8 10 12 14 16

H L Log10 M GeV

Figure 12: Evolution of the gauge couplings in the Standard Model represented by dashed lines and in the new model where the uniﬁcation scale is realized at 104 TeV.H See Ref.L [81] for details.

where B = L = 3/(2nF ), and nF is the number of copies. Notice that these fields change only the evolution of the SU(2) and U(1)Y gauge couplings. The Higgs sector of this theory is composed of the fields S (1, 1, 0, 2B, 2L) and S (1, 1, 0, 0, 2). When U(1) is spontaneously broken B ∼ L ∼ B and n = 3n (n is an integer number) the proton is stable because one only has ∆B = 3/n F 6 ± F interactions. The evolution of the Standard Model gauge couplings at one-loop level is described by the equation 1 1 Bi M kiαi− (M) = αi− (MZ ) Log , (65) − 2π MZ where ki = (k1, k2, k3) are the normalization factors. In the simplest grand unified theory based on SU(5) one has ki = (5/3, 1, 1). However, in general the k1 value depends of the embedding in a given theory. The coefficients

SM new Log(M/MF ) Bi = bi + θ(M MF ) nF bi , (66) − × Log(M/MZ ) contain all possible contributions from the low scale, MZ , to the unification scale, MU . Here, M is the mass scale of the new fermions, θ(x) is the step function, and bSM = (41/6, 19/6, 7) F i − − are the coefficients in the Standard Model. In the model discussed above the new coefficients new are bi = (2/3, 2, 0). In Fig. 12 one has the evolution of the gauge couplings in the Standard Model and in the model with lepto-baryons. The lepto-baryons at the low scale change the evolution of α2 and α1 dramatically without affecting α3. Here one assumes four copies of lepto-baryons with baryon and lepton numbers equal to 3/8. Since they have different baryon and lepton numbers than 9. Baryon and Lepton Number Violation in Supersymmetry 34 the Standard Model fields one never induces new sources of flavor violation and the unification 4 scale MU is larger than 10 TeV in order to suppress the flavor violating effective operators. The evolution of the new couplings is defined by the equation 1 1 BX MU αX− (MZ ) = kX αX− (MU ) + Log , (67) 2π MZ where X = B,L. The BX coefficients are given by

SM new Log(MU /MF ) BB = bB + θ(MU MF ) bB , (68) − × Log(MU /MZ ) and

SM new Log(MU /MF ) BL = bL + θ(MU MF ) bL . (69) − × Log(MU /MZ )

Notice that the values of these gauge couplings at different scales and the kX are barely constrained. However, one can envision that it is possible to define a theory where all gauge couplings are unified, the Standard Model couplings, α1, α2, α3, and the new couplings αB and

αL.Therefore, assuming unification at the MU scale, one can find the values of αB and αL at the MZ scale for given values of kB and kL. The contribution of the Standard Model fields to SM the running of the baryonic and leptonic couplings is given by the coefficients bB = 8/3 and SM bL = 6. The new fields contribute as follows new 16 4 2 (4nF + 1) bB = nF + B = 3 2 , (70) 3 3 nF 10 bnew = bnew + . (71) L B 3 In Fig. 13 one shows the numerical results for the running of all gauge couplings. As one can appreciate, these results tell us that it is possible to have consistent unification of the gauge interactions using a bottom-up approach. Of course, the embedding of this model in a grand unified theory is very important. Theories with local baryon number and left-right symmetry have been investigated in Ref. [77]. These theories are based on the gauge group SU(3) SU(2) SU(2) U(1) U(1) ⊗ L⊗ R⊗ L⊗ B and define a possible new path towards the unification of gauge interactions.

9. Baryon and Lepton Number Violation in Supersymmetry

In the minimal supersymmetric Standard Model [99, 100, 101] one has interactions violating the total lepton and baryon numbers at the renormalizable level. These interactions are given by 0 00 = LˆHˆ + λLˆLˆeˆc + λ QˆLˆdˆc + λ uˆcdˆcdˆc, (72) WRpV u 9. Baryon and Lepton Number Violation in Supersymmetry 35

35 -1 ΑL 30 -1 Α2 25 -1 HΑ1 L -1 ΑB 1 - i 20 H L Α

15 H L H L Α -1 10 3

5 3 4 5 6 7 H L Log10 M GeV

4 Figure 13: Evolution of the gauge couplings with lepto-baryons where the uniﬁcation scale is MU ∼ 2 × 10

TeV. Assuming for simplicity kB = kL = k1 for the runningH ofL the αB and αL. At the MZ scale one ﬁnds −1 −1 αB (MZ ) = 25.17 and αL (MZ ) = 28.69 [81]. breaking B L and the discrete symmetry called matter parity, M = ( 1)3(B L). There is a − − − simple relation between R-parity and M-parity, R = ( 1)2SM [102, 103] where S is the spin. − Now, even if the conservation of M-parity is imposed by hand there are also dimension ﬁve operators which give rise to proton decay. These operators are

λ λ λ c 5 = L QˆQˆQˆLˆ + R uˆcdˆcuˆceˆc + ν uˆcdˆcdˆcνˆc. (73) WRpC Λ Λ Λ

Notice that these interactions conserve B L and in order to satisfy the experimental bounds on − proton decay one needs to assume a large cutoﬀ scale, i.e. Λ > 1017 GeV when we assume low- energy supersymmetry and the coeﬃcients are of order one. The simplest way to understand the origin of the B and L violating interactions in the MSSM is to consider a theory where B L − is inside the algebra. The relation between M-parity and B L has been investigated by many − groups [104, 105, 106, 107, 108, 109, 110, 111]. It is important to say that most of the people in the SUSY community assume the conservation of matter parity because the lightest neutralino can describe the cold dark matter of the Universe. Recently, this issue was investigated in Refs. [112, 113, 114, 115, 116] and it has been shown that the minimal theory based on B L − predicts that R parity must be spontaneously broken. These results are interesting because − the minimal theory for spontaneous R parity violation could tells us what we should expect at − the Large Hadron Collider if low-energy supersymmetry is realized in nature. In this section we discuss in great detail the minimal theory for spontaneous R parity violation and the testability − at the Large Hadron Collider. 9. Baryon and Lepton Number Violation in Supersymmetry 36

10 10 NH: Bino LSP IH: Bino LSP

1 1 L L mm mm H 0.1 H 0.1

Length 0.01 Length 0.01 Decay Decay 0.001 0.001

10-4 10-4 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 LSP Mass HGeVL LSP Mass HGeVL (a) (b)

Figure 14: Decay length in millimeters versus LSP mass for a dominantly bino LSP in (a) for a NH and in (b) for an IH. See Ref. [126] for details.

9.1. The Minimal Theory for Spontaneous R-parity Violation

The simplest gauge theory for spontaneous R-parity breaking was proposed in Ref. [113]. In this context one can understand dynamically the origin of the R-parity violating terms in the MSSM. Here we discuss the structure of the theory and the full spectrum. This theory is based on the gauge group

SU(3) SU(2) U(1)Y U(1)B L, ⊗ ⊗ ⊗ − and the diﬀerent matter chiral superﬁelds are given by

         uˆ   νˆ      Qˆ =   (2, 1/6, 1/3), Lˆ =   (2, 1/2, 1), (74)   ∼   ∼ − −      dˆ   eˆ 

uˆc (1, 2/3, 1/3), dˆc (1, 1/3, 1/3), eˆc (1, 1, 1). (75) ∼ − − ∼ − ∼

In order to cancel the B L anomalies one introduces three chiral superﬁelds for the right- − handed neutrinos,

νˆc (1, 0, 1). (76) ∼ 9. Baryon and Lepton Number Violation in Supersymmetry 37

10 10 NH: Wino LSP IH: Wino LSP

1 1 L L mm mm H 0.1 H 0.1

Length 0.01 Length 0.01 Decay Decay 0.001 0.001

10-4 10-4 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 LSP Mass HGeVL LSP Mass HGeVL (a) (b)

Figure 15: Decay length in millimeters versus LSP mass for a dominantly wino LSP in (a) for a NH and in (b) for an IH. See Ref. [126] for details.

The Higgs sector is composed of two Higgs chiral superﬁelds as in the MSSM    

 ˆ +   ˆ 0   Hu   Hd      Hˆu =   (2, 1/2, 0), Hˆd =   (2, 1/2, 0). (77)   ∼   ∼ −      ˆ 0   ˆ  Hu Hd−

In this content the superpotential reads as

= + Y Lˆ Hˆ νˆc, (78) WBL WMSSM ν u where

= Y Qˆ Hˆ uˆc + Y Qˆ Hˆ dˆc + Y Lˆ Hˆ eˆc + µ Hˆ Hˆ . (79) WMSSM u u d d e d u d In addition to the superpotential, the model is also speciﬁed by the soft terms

2 2 c 2 2 2 c 2 2 2 2 2 V = m c ν˜ + m L˜ + m c e˜ + m H + m H soft ν˜ | | L˜ e˜ | | Hu | u| Hd | d| 1 + M B˜0 B˜0 + A LH˜ ν˜c + Bµ H H + h.c. + V MSSM , (80) 2 BL ν u u d soft where the terms not shown here correspond to terms in the soft MSSM potential. Since we have a new gauge symmetry in the theory we need to modify the kinetic terms for all MSSM matter superﬁelds, and include the kinetic term for right-handed neutrino superﬁelds Z c 2 2 c g Vˆ c (ν ) = d θd θ¯ (ˆν )†e BL BL νˆ . (81) LKin 9. Baryon and Lepton Number Violation in Supersymmetry 38

0.010 IH: Higgsino LSP IH: Higgsino LSP 0.005 0.01 L L 0.001 mm mm H H 0.001 5 ´ 10-4 Length Length

´ -4 -4 1 10 Decay

10 Decay 5 ´ 10-5

10-5 1 ´ 10-5 100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 LSP Mass HGeVL LSP Mass HGeVL (a) (b)

Figure 16: Decay length in millimeters versus LSP mass for a dominantly Higgsino LSP in (a) for a NH and in (b) for an IH. See Ref. [126] for details.

Here Vˆ is the B L vector superﬁeld. Using these interactions we can study the full spectrum BL − 0 0 of the theory. As in the MSSM, electroweak symmetry is broken by the vevs of Hu and Hd , while U(1)B L is broken due to the vev of right-handed sneutrinos. Notice that this is the only − ﬁeld which can break local B L and give mass to the new neutral gauge boson in the theory. − Therefore, the theory predicts spontaneous R-parity violation. It is important to mention that the B L and R-parity breaking scales are determined by the soft supersymmetric breaking − scale, and one must expect lepton number violation at the LHC.

The neutral ﬁelds are deﬁned as

0 1 i H = (vu + hu) + Au, (82) u √2 √2 0 1 i H = (vd + hd) + Ad, (83) d √2 √2 1 i ν˜i = vi + hi + Ai , (84) √2 L L √2 L 1 i ν˜c = vi + hi + Ai , (85) i √2 R R √2 R

and the relevant scalar potential reads as

V = VF + VD + Vsoft, (86) 9. Baryon and Lepton Number Violation in Supersymmetry 39

100 100 NH: Bino LSP Ν Z e-W+ + e+W- IH: Bino LSP Ν Z e-W+ + e+W- 70 Ν h Μ-W+ + Μ+W- 70 Ν h Μ-W+ + Μ+W- Τ-W+ + Τ+W- Τ-W+ + Τ+W- 50 50

30 30 L L % % H H 20 20 Brs Brs 15 15

10 10

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 LSP Mass HGeVL LSP Mass HGeVL (a) (b)

Figure 17: LSP branching ratios versus LSP mass for a dominantly bino LSP in (a) for a NH and in (b) for an IH. See Ref. [126] for details.

with

X X V = µ 2 H0 2 + µH0 +ν ˜ Y ijν˜c 2 + Y ijν˜c 2 H0 2 + ν˜ Y ij 2 H0 2, (87) F | | | u| | − d i ν j | | ν j | | u| | i ν | | u| i j !2 !2 (g2 + g2) X g2 X V = 1 2 H0 2 H0 2 ν˜ 2 + BL ( ν˜c 2 ν˜ 2) , (88) D 8 | u| − | d | − | i| 8 | i | − | i| i i c 2 c 2 2 0 2 2 0 2 ij c 0 0 0 c Vsoft = (˜νi )†mν˜ ν˜j +ν ˜i†m˜ ν˜j + mHu Hu + mH Hd + ν˜iaν ν˜j Hu BµHuHd + h.c. . ij Lij | | d | | − (89)

i i In order to have phenomenologically allowed solutions the vL have to be small, and the vR have i to be much larger than vu, vd and vL. These vacuum expectation values must be small in order to have small neutrino masses generated through the R-parity violating terms. Up to negligibly small terms the right-handed sneutrino acquire a vev in only one family. A possible solution is i vR = (0, 0, vR). In this case

2 2 8(mν˜c )33 (vR) 2 , (90) ≈ − gBL k3 k3 v µvdY a vu vk R ν − ν . (91) L h (g2+g2) g2 i ≈ √2 (m2 ) 1 2 (v2 v2) BL (v )2 L˜ kk − 8 u − d − 8 R Notice that the vacuum expectation value for the right-handed sneutrino is determined by the soft term. In this way we understand why the B L and R-parity violating scales are deﬁned − by the SUSY breaking scale. In the MSSM, the large top Yukawa coupling drives the up-type soft Higgs mass squared parameter to negative values for generic boundary conditions leading 9. Baryon and Lepton Number Violation in Supersymmetry 40

100 100 NH: Wino LSP Ν Z e-W+ + e+W- IH: Wino LSP Ν Z e-W+ + e+W- 70 Ν h Μ-W+ + Μ+W- 70 Ν h Μ-W+ + Μ+W- Τ-W+ + Τ+W- Τ-W+ + Τ+W- 50 50

30 30 L L % % H H 20 20 Brs Brs 15 15

10 10

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 LSP Mass HGeVL LSP Mass HGeVL (a) (b)

Figure 18: LSP branching ratios versus LSP mass for a dominantly wino LSP in (a) for a NH and in (b) for an IH. See Ref. [126] for details.

to radiative electroweak symmetry breaking [117, 118]; a celebrated success of the MSSM. A valid question is then if the same success is possible in achieving a tachyonic right-handed sneutrino mass in this B L model as required by Eq. (90). Unfortunately, this is not possible − through a large Yukawa coupling since the Yukawa couplings of the right-handed neutrino are all dictated to be small by neutrino masses. However, there is an alternate possibility whereby a positive mass squared parameter for the right-handed sneutrino at the high scale will run to a tachyonic value at the low scale. This is due to the presence of the so-called S-term (due to D-term contributions to the RGE) in the soft mass RGE, as discussed for this B L model in − Refs. [119, 120, 121]. For the implementation of the radiative mechanism in the non-minimal model where the right-handed neutrino masses are generated through the Higgs mechanism see Ref. [116].

9.2. R-Parity Violating Interactions and Mass Spectrum

After symmetry breaking lepton number is spontaneously broken in the form of bilinear R-parity violating interactions. There are no trilinear R-parity violating interactions at the renormalizable level. These bilinear interactions mix the leptons with the Higgsinos and gaug- inos

1 c 0 1 i 0 1 i + gBLvR(ν B˜ ), g2v (νiW˜ ), g2v (eiW˜ ), 2 3 2 L √2 L

1 i 1 i3 T 1 ij i 0 c 1 i i c g1v (νiB˜), Y vR(L iσ2 H˜u), Y v (H˜ ν ), Y v (H˜ − e ). 2 L √2 ν i √2 ν L u j √2 e L d i 9. Baryon and Lepton Number Violation in Supersymmetry 41

100 100 NH: Higgsino LSP Ν Z e-W+ + e+W- IH: Higgsino LSP Ν Z e-W+ + e+W- 70 Ν h Μ-W+ + Μ+W- 70 Ν h Μ-W+ + Μ+W- Τ-W+ + Τ+W- Τ-W+ + Τ+W- 50 50

30 30 L L % % H H 20 20 Brs Brs 15 15

10 10

100 200 300 400 500 600 700 800 100 200 300 400 500 600 700 800 LSP Mass HGeVL LSP Mass HGeVL (a) (b)

Figure 19: LSP branching ratios versus LSP mass for a dominantly Higgsino LSP in (a) for a NH and in (b) for an IH. See Ref. [126] for details.

The ﬁrst term is new and is the only term not suppressed by neutrino masses. The ﬁfth term corresponds to the so-called term, and second, third and fourth terms are small but important for the decay of neutralinos and charginos. There are also lepton number violating interactions coming from the soft terms and the B L D-term. From V one gets − soft

i3 vR T A L˜ iσ2 Hu, ν √2 i while from the D-term one ﬁnds 2 c 1 1 g v ν˜ q˜† q˜ ˜l† ˜l . BL R 6 − 2

As one can expect these terms are important to understand the scalar sector of the theory. The neutral gauge boson associated to the B L gauge group is Z . Using the covariant − BL c c i 0 c derivative for the right-handed sneutrinos, Dµν˜ = ∂µν˜ + 2 gBLBµν˜ , the mass term for ZBL is g M = BL v . (92) ZBL 2 R Now, using the experimental collider constraint [122]

M ZBL 3 TeV, (93) gBL ≥ and Eq. (21) one ﬁnds the condition

(m c > 2.12 g TeV. (94) | ν˜ )33 | BL 9. Baryon and Lepton Number Violation in Supersymmetry 42

Then, if gBL = 0.1 the soft mass above has to be larger than 200 GeV. This condition can be easily satisfied without assuming a very heavy spectrum for the supersymmetric particles. As in any supersymmetric theory where R-parity is broken all the fermions with the same quantum numbers mix and form physical states which are linear combinations of the original fields. The neutralinos in this theory are a linear combination of the fields, c ˜ ˜ ˜ 0 ˜ 0 ˜ 0 νi, νj , B0, B, W , Hd , Hu .

Then, the neutralino mass matrix is given by  

 1 ij 1 i 1 i 1 i 1 ij j   0 Yν vu 2 gBL vL 2 g1 vL 2 g2 vL 0 Yν vR  √2 − − √2         1 ij 1 j 1 ij i   Yν vu 0 gBL v 0 0 0 Yν v   √2 2 R √2 L          1 g vi 1 g vj M 0 0 0 0  − 2 BL L 2 BL R BL          N =  1 g vi 0 0 M 0 1 g v 1 g v  . M  2 1 L 1 2 1 d 2 1 u   − −         1 i 1 1   2 g2 vL 0 0 0 M2 2 g2vd 2 g2vu   −         1 1   0 0 0 g1vd g2vd 0 µ   − 2 2 −         1 ij j 1 ij i 1 1  Yν v Yν v 0 g v g v µ 0 √2 R √2 L 2 1 u − 2 2 u − (95) i We have discussed above that only one right-handed sneutrinos get a vev, vR = (0, 0, vR). Now, integrating out the neutralinos one can ﬁnd the mass matrix for the light neutrinos. In this case one has three active neutrinos and two sterile neutrinos, and the mass matrix in the basis c c (νe, νµ, ντ , νe, νµ) is given by   h i  i j i3 j j3 i i3 j3 1 iβ A vLvL + B Yν vL + Yν vL + CYν Yν vuYν   √2  Mν =   , (96)      1 αj  vuYν O2 2 √2 × where 2 ! 2 2 2 2µ vu √2µvdvR 2MBLvu vdvR A = 3 ,B = + 3 ,C = 2 2 + 3 , (97) m˜ √2vR m˜ gBLvR m˜ 9. Baryon and Lepton Number Violation in Supersymmetry 43

2 2 2 3 4 µvuvd g1M2 + g2M1 2M1M2µ m˜ = 2 2 − . (98) g1M2 + g2M1

Here α and β take only the values 1 and 2. From the experimental limits on active neutrino 12 5 masses we obtain (Yν)iα . 10− . This can be compared to (Yν)i3 . 10− , which is less constrained because of the TeV scale seesaw suppression. It has been pointed out in Refs. [123, 124] (and earlier in a diﬀerent context [125]) that this theory predicts the existence of two light sterile neutrinos which are degenerate or lighter than the active neutrinos, a so-called 3 + 2 neutrino model. c ˜ + ˜ + ˜ ˜ In this theory the chargino mass matrix, in the basis ej, W , Hu and ei, W −, Hd− , is given by

     0 MC    ± =   , (99) Mχ˜      T  MC 0 with

   1 ij 1 ij j   √ Ye vd 0 √ Ye vL − 2 2        MC =  1 i 1  . (100)  g2v M2 g2vd   √2 L √2         1 ij j 1  Yν v g v µ − √2 R √2 2 u

In the sfermion sector, the mass matrices 2 , and 2 for squarks, and 2 for charged Mu˜ Md˜ Me˜ c sleptons, in the basis f,˜ f˜ ∗ , are given by

 

 2 2 1 2 2 2 1 1   m ˜ + mu + sW MZ c2β + DBL (au vu Yu µ vd)   Q 2 − 3 3 √2 −  2 =   , Mu˜      1 2 2 2 2 2 1  (a v Y µ v ) m c + m + M c s D √2 u u − u d u˜ u 3 Z 2β W − 3 BL (101) 9. Baryon and Lepton Number Violation in Supersymmetry 44

 

 2 2 1 1 2 2 1 1   m ˜ + md sW MZ c2β + DBL (Yd µ vu ad vd)   Q − 2 − 3 3 √2 −  2 =   , Md˜      1 (Y µ v a v ) m2 + m2 1 M 2 c s2 1 D  √2 d u − d d d˜c d − 3 Z 2β W − 3 BL (102)  

 2 2 1 2 2 1   m˜ + me sW MZ c2β DBL (Ye µ vu ae vd)   L − 2 − − √2 −  2 =   , Me˜      1 2 2 2 2  (Y µ v a v ) m c + m M c s + D √2 e u − e d e˜ e − Z 2β W BL (103)

where c2β = cos 2β, sW = sin θW and

1 1 D = g2 v2 = M 2 . (104) BL 8 BL R 2 ZBL mu, md and me are the respective fermion masses and au, ad and ae are the trilinear a-terms corresponding to the Yukawa couplings Yu,Yd and Ye. Regardless, the physical states are related to the gauge states by        ˜     ˜  f1  cos θ ˜ sin θ ˜  f     f f      =     . (105)             f˜   sin θ cos θ  f˜c  2 − f˜ f˜ ∗

The masses in the sneutrino sector are given by

2 2 1 2 1 2 Mν˜ = m˜ + MZ cos 2β MZ , (106) i Li 2 − 2 BL 2 2 M c = M , (107) ν˜3 ZBL 2 2 M c = m c + D , (108) ν˜α ν˜α BL and α = 1..2. For simplicity we listed the mass matrices in the limit vi , a ,Y 0. It is L ν ν → important to mention that all sfermion masses are modified due to the existence of the B L − D-term. See Ref. [126] for a detailed study of the spectrum in this model. At first glance, finding a dark matter in R-parity violating theories seems hopeless. But while the traditional neutralino LSP case is no longer valid, the situation is not lost. As first discussed in [127, 128], such models can have an unstable LSP gravitino, with a lifetime longer than the age of the universe. The strong suppression on its lifetime is due to both the Planck 9. Baryon and Lepton Number Violation in Supersymmetry 45

e+ i ej± q 0 W ∓ + χ˜1 e˜i γ,Z,ZBL

0 e˜i− χ˜1 q W ∓

e± ei− k

Figure 20: Topology of the signals with multi-leptons [126].

mass suppression associated with its interaction strength and bilinear R-parity violation which is small due to neutrino masses and must facilitate the decay of the LSP. In the mass insertion approximation, this can be understood as the gravitino going into a photon and neutralino which then has some small mixing with the neutrino due to R-parity violation (mχ ν), thereby allowing G˜ γν. Adopting approximations made in [127], the lifetime for the gravitino decaying into a → photon and neutrino (in years) is about

m 3 2 ˜ 10 3/2 − mχν/mχ − τ(G γν) 2 10 6 years, (109) → ∼ × 100 GeV 10− which for appropriate values of the gravitino mass leads to a long enough life time. Unlike in R-parity conserving models with a gravitino LSP, there are no issues with big bang nucleosyn- thesis from slow NLSP decay since the NLSP decays more promptly through R-parity violating interactions. Several interesting studies have been conducted on the signatures and constraints of unstable gravitino dark matter, see for example [127, 129, 130].

9.3. Lepton Number Violation and Decays

At this point, the relevant pieces of this model have been laid out and the question of interesting signals can now be tackled. Since lepton number is violated, same-sign dileptons and multijets are possible final states. Such signatures are interesting since they have no SM background. However, the final states depend critically on the nature of the LSP and since R-parity is violated the possibilities are more numerous than normal, i.e. colored and charged fields. In this section we will discuss only the decays of the lightest neutralino through the R-parity violating couplings. 9. Baryon and Lepton Number Violation in Supersymmetry 46

= MZBL 3 TeV 100 gBL = 0.3

10 14 TeV

L 1 fb H Σ 8 TeV 0.1 7 TeV

0.01

0.001 200 400 600 800 1000 H L meL GeV

± ∓ Figure 21: Cross sections for p p → γ, Z, ZBL → e˜i e˜i with LHC center of mass energies of 7 TeV (red), 8 TeV

(green) and 14 TeV (blue), assuming MZBL = 3 TeV and gBL = 0.3 versus the left-handed slepton mass. See Ref. [131] for details.

0 Neutralino Decays: The leading decay channels for the lightest neutralino,χ ˜1, include

0 0 0 0 χ˜ e±W ∓, χ˜ ν Z, χ˜ ν h , χ˜ e±H∓. (110) 1 → i 1 → i 1 → i k 1 → i

The amplitudes for the two ﬁrst channels are proportional to the mixing between the leptons and neutralinos, while the last one is proportional to the Dirac-like Yukawa terms. While decays to all the MSSM Higgses are possible, typically, only the lightest MSSM Higgs, h (k = 1), is light enough for the scenario we consider here and so we will only take it into account. A naive estimation of the decay width yields

g2 Γ(˜χ0) 2 V 2M , (111) 1 ∼ 32π | νχ| χ p where Vχν is the mixing between the neutralino and neutrino which is proportional to mν/Mχ. Assuming that m < 0.1 eV for the decay length one ﬁnds L(˜χ0) 0.6 mm. Therefore, even ν 1 without making a detailed analysis of the decays of the lightest neutralino one expects signals with lepton number violation and displaced vertices in part of the parameter space. In Figs. 14-16 we show the decay lengths versus LSP mass resulting from a scan over all the possible values of 1 and 2 and over the parameters and ranges speciﬁed in Ref. [126]. The points are divided according to the largest component of the LSP and the neutrino hierarchy with a dominantly bino, wino and Higgsino LSP in the case when the neutrino spectrum has a Normal Hierarchy (NH) shown in (a) and for an Inverted Hierarchy (IH) in (b), respectively. 9. Baryon and Lepton Number Violation in Supersymmetry 47

1.0 1.0 14 TeV LHC 5 10 = 20 50 MZBL 3 TeV 10 20 gBL = 0.3 0.8 0.8 5 10 100 1 5 50 L L 0 1 0.6

0 1 0.6

1 Χ Χ ± i ± i e

e 20 ® i ® i e H e 10 H 0.4 0.4 Br Br 5

± ¡ ± ± -1 0.2 Num. of ei ei ej ej 4j Events20 fb 0.2 ± ¡ ± ± -1 14 TeV LHC Num. of ei ei ej ej 4j Events20 fb 0 ± ¡ BrHΧ1®ej W L=0.5 M = 200 GeV MZ = 3 TeV ei 0 ± ¡ BL BrHΧ1®ej W L=0.1 M = 400 GeV gBL = 0.3 ei 0.0 0.0 200 400 600 800 1000 0.0 0.2 0.4 0.6 0.8 1.0 H L 0 ± ¡ Mei GeV BrHΧ1® ei W L

± ± 0 √ Figure 22: Number of events in the Br(˜ei → ei χ˜1) vs Me˜i plane in the left panel assuming s = 14 TeV, 0 ± ∓ −1 Br(˜χ1 → ek W ) = 0.1(0.5), when MZBL = 3 TeV and L = 20 fb . In the right panel the number of events are ± ± 0 0 ± ∓ plotted in the plane Br(˜ei → ei χ˜1)-Br(˜χ1 → ek W )[131].

The relevant decay lengths can be understood by studying the mixings in Eq. (95). Since the higgsino-neutrino decay strength is the largest, Y v , the Higgsino LSP has the shortest ∼ ν R decay length. It is followed by the wino LSP with mixing g v and ﬁnally the bino with ∼ 2 L coupling g v and therefore the largest possible decay lengths. Displaced vertices associated ∼ 1 L with the lifetime of the LSP will only be discernible in a very limited part of the parameter space.

The LSP branching ratios into the various possible channels versus the LSP mass are dis- played in Figs. 17-19, plotting the dominantly bino, wino and Higgsino LSP in (a) for a NH and in (b) for an IH. Although it is not obvious from the Figs. 17-19, the branching ratio to the electron W ± channel is always smaller then either the µ∓W ± or the τ ∓W ± in the NH.

9.4. Signatures at the LHC

We have mentioned in the previous section that the minimal supersymmetric B L theory − predicts lepton number violation at the LHC and the relevant couplings are small. Therefore, the production mechanisms for the supersymmetric particles at the LHC are not modiﬁed but the LSP is not stable and decays via the lepton number violating couplings. In this case one could have the productions of several SM particles together with two LSPs

p p Ψ ... Ψ LSP LSP Ψ ... Ψ Ψ Ψ , → 1 n → 1 n i j 9. Baryon and Lepton Number Violation in Supersymmetry 48

where Ψi is a SM particle, and n = 2, 4, 6, ... One example is the scenario where the stop is the LSP and decays as a leptoquark:

+ p p t˜∗ t˜ b ¯b τ τ −. → 1 1 →

In Table6 we list the possible LSP scenarios and their main decays, focusing only on light third generation sfermions. As discussed in Ref. [126], the most exotic signals in this theory

LSP Scenario Decays

˜ + + t1 t ν,¯ j ν,¯ b ei , j ei

˜ b1 b ν,¯ j ν,¯ t ei−, j ei−

0 0 χ˜1 ei± W, ν Z, ei± H∓, ν Hi

0 0 χ˜1± ei± Z, ν W ±, ei± Hi , ν Hi

τ˜± ei±ν, q¯ q0, hW ±

ν˜3 qq,¯ e¯iej, W W, ZZ, hh, HH

Table 6: Lepton number violating LSP decays.

correspond to a neutralino LSP. In this case we can produce two charged sleptons through the photon, the Z and the B L gauge boson: −

+ p p γ, Z, ZB L e˜i e˜i−. (112) → − →

The sleptons subsequently decay ase ˜ e χ˜0 and finally the neutralinos decay through i → i 1 0 χ˜ e±W . In Ref. [126] has been shown that both of these branching ratios are typically 1 → j ∓ large. Therefore, one can expect a significant number of events with four leptons and two W’s. When the W gauge bosons decay hadronically, one is left with the unambiguous lepton number violating final states:

ei± ei∓ ej± ek± 4j. (113) 10. Final Discussions 49

where ei = e, µ, τ. See Fig. 20 for the topology of these events. The number of events for these channels with multileptons is given by

N = σ(pp e˜±e˜∓) C , (114) ijk L × → i i × jk where is the integrated luminosity, and C is given by L jk

0 2 0 0 2 C = 2(2 δ ) Br(˜e± e±χ˜ ) Br(˜χ e±W ∓) Br(˜χ e±W ∓) Br(W jj) . (115) jk − jk × → 1 × 1 → j × 1 → k × →

In order to analyze the testability of the channels with multi-leptons we need to estimate the cross section using the production mechanism mentioned above and focusing on left-handed sleptons, as motivated above. This study was done in Ref. [131] and here we discuss the main results. In Fig. 21 we show the production cross section for left-handed sleptons assuming that the mass of the B L gauge boson is M = 3 TeV and the corresponding coupling is g = 0.3. − ZBL BL In this ﬁgure we show the numerical results for σ(pp e˜±e˜∓) when the center mass energy is → i i √s = 7 TeV, √s = 8 TeV, and √s = 14 TeV. It is important to mention that when √s = 14 TeV the cross section is above 1 fb when the selectron mass is below 400 GeV. For selectron masses in the TeV range, the signals discussed above will be very diﬃcult to test. The number of events with four leptons and four jets are estimated for the scenario: √s = 14 1 TeV and 20 fb in Fig. 22. The left panel in Fig. 22 shows the number of events in the Br(˜e± − i → 0 0 e±χ˜ )–M plane assuming an optimistic (pessimistic) branching ratio forχ ˜ e±W of 0.5 i 1 e˜i 1 → i ∓ 0 (0.1). In the left panel, the number of events is shown in the Br(˜e± e±χ˜ )–Br(˜χ e±W ) i → i 1 1 → i ∓ for a light (heavy) selectron mass of 200 GeV (400 GeV) for the 14 TeV run. Here one assumes

MZBL = 3 TeV and gBL = 0.3. As one can see, only when the sleptons are light with mass below 300 GeV we expect a large number of events. See Ref. [131] for a more detailed study. For recent phenomenological studies in these theories see Refs. [132, 133, 134].

10. Final Discussions

In this review we have discussed the desert hypothesis in particle physics and its role in physics beyond the Standard Model. As we have mentioned, the main reason to assume the great desert between the weak and Planck scales is the proton stability. The simplest grand uniﬁed theories predict proton decay and in order to satisfy the experimental bounds on the proton decay lifetimes one needs to assume that these theories describe physics at the very high scale. This approach has been accepted by the particle physics community even if there is no 10. Final Discussions 50 way to test these theories. The desert hypothesis is very naive and we have shown in this review the possibility to deﬁne the simplest theories where the great desert is not needed.

In the first part of the review we have discussed simple theories where the baryon and lepton numbers are defined as local symmetries. In this context one can define an anomaly free theory adding new fermions with lepton and baryon number which we call “lepto-baryons”. After symmetry breaking one finds that the baryon number is broken in three units. Therefore, the proton is stable and there is no need to assume the great desert. We have shown that the simplest theories with local B and L contain a candidate for the cold dark matter in the Universe. Using the relic density constraints we found an upper bound on the symmetry breaking scale. Therefore, one can test or rule out these theories in current or future colliders. We have discussed the relation between the baryon asymmetry and cold dark matter density in these theories, showing that these theories are consistent with cosmology. We have shown that these theories open the possibility of having unification at the low scale. These theories are appealing extensions of the Standard Model of particle physics.

In the second part of the review we have presented the simplest supersymmetric gauge theory which predicts that R parity must be spontaneously broken at the supersymmetric breaking − scale. As we have discussed, in the context of supersymmetric theories one needs to assume the great desert to suppress the dimension ﬁve and six operators mediating proton decay. At the same time the uniﬁcation of gauge couplings is realized at the high scale. We have discussed the main features of the simplest gauge theory for spontaneous R parity violation. This theory − predicts lepton number violating signatures at collider. We have investigated the spectrum and the signatures with multi-leptons which could help us to test the theory at the Large Collider Collider. The most important result in this section is that the simplest theory for R parity − predicts that this symmetry must be broken. Maybe this is the key element needed to discover supersymmetry at colliders. As it has been discussed, in this case one does not expect signatures with missing energy and one should look for signals with multi-leptons with the same electric charge and multi-jets.

We would like to emphasize that the origin of baryon and lepton number (non)conservation provides a unique guide to understand the different theories for physics beyond the Standard Model. We have discussed two main type of these theories which could be tested in current and future experiments. In our opinion the theories with local baryon and lepton numbers define a new way to think about the unification of gauge interactions and cosmology. 10. Final Discussions 51

Acknowledgments

I would like to thank my collaborators Jonathan M. Arnold, Borut Bajc, Vernon Barger, Ilja Dorsner, Michael Duerr, Bartosz Fornal, Manfred Lindner, Pran Nath, Sebastian Ohmer, Hiren H. Patel, German Rodrigo, Goran Senjanovic, Sogee Spinner and Mark B. Wise for great collaborations in diﬀerent projects related to this review. I would like to thank the Max Planck Institut fuer Kernphysik (MPIK) and the California Institute of Technology (Caltech) for supporting my work. This work has been partly supported by the Gordon and Betty Moore Foundation through Grant 776 to the Caltech Moore Center for Theoretical Cosmology and Physics. REFERENCES 52

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