REVIEW published: 06 March 2018 doi: 10.3389/fphy.2018.00019

Neutrino Masses and Leptogenesis in Left–Right Symmetric Models: A Review From a Model Building Perspective

Chandan Hati 1*, Sudhanwa Patra 2, Prativa Pritimita 3 and Utpal Sarkar 1,2,3,4

1 Laboratoire de Physique Corpusculaire, Centre National de la Recherche Scientifique/IN2P3 UMR 6533, Aubière, France, 2 Department of Physics, Indian Institute of Technology Bhilai, Chhattisgarh, India, 3 Center of Excellence in Theoretical and Mathematical Sciences, Siksha ‘O’ Anusandhan University, Bhubaneswar, India, 4 Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur, India

In this review, we present several variants of left–right symmetric models in the context Edited by: of neutrino masses and leptogenesis. In particular, we discuss various low scale seesaw Alexander Merle, Max Planck Institute for Physics mechanisms like linear seesaw, inverse seesaw, extended seesaw and their implications (MPG), Germany to lepton number violating process like neutrinoless double beta decay. We also visit an Reviewed by: alternative framework of left–right models with the inclusion of vector-like fermions to Eduardo Peinado, analyze the aspects of universal seesaw. The symmetry breaking of left–right symmetric Instituto de Física, Universidad Nacional Autónoma de México, model around few TeV scale predicts the existence of massive right-handed gauge Mexico bosons WR and ZR which might be detected at the LHC in near future. If such signals are Bhupal Dev, Washington University in St. Louis, detected at the LHC that can have severe implications for leptogenesis, a mechanism to United States explain the observed baryon asymmetry of the Universe. We review the implications of *Correspondence: TeV scale left–right symmetry breaking for leptogenesis. Chandan Hati [email protected] Keywords: left-right symmetry, neutrino mass, leptogenesis, neutrinoless double beta decay, low scale seesaw

Specialty section: This article was submitted to 1. INTRODUCTION High-Energy and Astroparticle Physics, Although the Standard Model (SM) of particle physics is highly successful in explaining the low a section of the journal energy phenomenology of fundamental particles, the reasons to believe it is incomplete are not Frontiers in Physics less. The most glaring of them all is the issue of neutrino mass which has been confirmed by Received: 17 October 2017 the oscillation experiments. Some more unsolved puzzles like dark matter, dark energy, baryon Accepted: 14 February 2018 asymmetry of the universe strongly suggest that SM is only an effective limit of a more fundamental Published: 06 March 2018 theory of interactions. In addition to the fact that gravity is completely left out in the SM, the Citation: strong interaction is not unified with weak and electromagnetic interactions. In fact, even in the Hati C, Patra S, Pritimita P and electroweak “unification” one still has two coupling constants, g and g′ corresponding to SU(2)L Sarkar U (2018) Neutrino Masses and and U(1)Y . Thus, one is tempted to seek for a more complete theory where the couplings gs, g, and Leptogenesis in Left–Right Symmetric Models: A Review From a Model g′ unify at some higher energy scale giving a unified description of the fundamental interactions. Given that the ratio m /m is so large, where m 1.2 1019 GeV is the Planck scale, another Building Perspective. Pl W Pl = × Front. Phys. 6:19. major issue in the SM is the infamous “hierarchy problem.” The discovery of the Higgs boson with doi: 10.3389/fphy.2018.00019 a mass around 125 GeV has the consequence that, if one assumes the Standard Model as an effective

Frontiers in Physics | www.frontiersin.org 1 March 2018 | Volume 6 | Article 19 Hati et al. Neutrino Masses and Leptogenesis in LRSM theory, then λ O(0.1) and 2 (O(100) GeV)2 (including the al. [2], Das et al. [3], Chen et al. [4] and Mitra et al. [5] and ∼ ∼ effects of 2-loop corrections). The problem is that every particle for Higgs sector some relevant references are in Bambhaniya et that couples, directly or indirectly, to the Higgs field yields a al. [6], Dutta et al. [7], Dev et al. [8] and Mitra et al. [9]. correction to 2 resulting in an enormous quantum correction. The plan for rest of the review is as follows. In section 2 For instance, let us consider a one-loop correction to 2 coming we briefly review the standard seesaw and radiative mechanisms from a loop containing a Dirac fermion f with mass mf . If f for light neutrino mass generation. In section 3 we first couples to the Higgs boson via the coupling term ( λ φf f ), then introduce and then review the standard left–right symmetric − f the correction coming from the one-loop diagram is given by theories and the implementation of different types of low scale seesaw implementations. In section 4 we review an alternative 2 λf formulation of left–right symmetric theories which uses a 2 2 , (1) = 8π 2 UV + universal seesaw to generate fermion masses. We also discuss the implications of this model for neutrinoless double beta decay in where UV is the ultraviolet momentum cutoff and the ellipsis this case for the specific scenario of type II seesaw dominance. In 2 are the terms proportional to mf , growing at most logarithmically section 5 we give a brief introduction to leptogenesis and review with UV. Each of the quarks and leptons in the SM plays the some of the standard leptogenesis scenarios associated with role of f , and if UV is of the order of mPl, then the quantum neutrino mass generation. In section 6 we review the situation of correction to 2 is about 30 orders of magnitude larger than the leptogenesis in left–right symmetric theories and the implications required value of 2 92.9 GeV2. Since all the SM quarks, of a TeV scale left–right symmetry breaking for leptogenesis. = leptons, and gauge bosons obtain masses from φ , the entire Finally, in section 7 we make concluding remarks. mass spectrum of the Standard Model is sensitive to UV. Thus, one expects some new physics between mW and mPl addressing 2. NEUTRINO MASSES this problem. There are also other questions such as why the fermion families have three generations; is there any higher The atmospheric, solar and reactor neutrino experiments have symmetry that dictates different fermion masses even within each established that the neutrinos have small non-zero masses which generation; in the CKM matrix the weak mixing angles and the are predicted to be orders of magnitude smaller than the charged CP violating phase are inputs of the theory, instead of being lepton masses. However, in the SM the left handed neutrinos predicted by the SM. Finally, in the cosmic arena, the observed ν , i e, , τ, transform as (1, 2, 1) under the gauge group iL = − baryon asymmetry of the universe cannot be explained within SU(3) SU(2) U(1) . Consequently, one cannot write a gauge c× L× Y the SM. Also there are no suitable candidates for dark matter and singlet Majorana mass term for the neutrinos. On the other hand, dark energy in the SM. These also point toward the existence of there are no right handed neutrinos in the SM which would allow physics beyond the SM. a Dirac mass term. The simplest way around this problem is to In this review, we study several variants of left–right add singlet right handed neutrinos νiR with the transformation symmetric models which is one of the most popular candidates (1,1,0) under the SM gauge group. Then one can straightaway for physics beyond the standard model. We will review the left– write the Yukawa couplings giving Dirac mass to the neutrinos right symmetric models in the context of neutrino masses and leptogenesis. We will study various low scale seesaw mechanisms 1 Lmass hijψiLνjRφ, (2) in the context of left–right symmetric models and their − = 2 implications to lepton number violating process like neutrinoless such that once φ acquires a VEV, the neutrinos get Dirac double beta decay. We will also discuss an alternative framework mass m h υ. Here ψ stands for the SU(2) lepton of left–right models with the inclusion of vector-like fermions as Dij = ij iL L doublet. However, to explain the lightness of the neutrinos one proposed to analyze various aspects. Interestingly, the breaking of needs to assume a very small Yukawa coupling for neutrinos in left–right symmetry around few TeV scale predicts the existence comparison to charged leptons and quarks. However, we do not of massive right-handed gauge bosons W and Z in left–right R R have a theoretical understanding of why the Yukawa coupling symmetric models. These heavy gauge bosons might be detected should be so small. Moreover, the accidental B L symmetry of at the LHC in near future. If such signals are detected at the − the SM forbids Majorana masses for the neutrinos. One way out LHC that can have conclusive implications for leptogenesis, a is to consider the dimension-5 effective lepton number violating mechanism to explain the observed baryon asymmetry of the operator [10–13] of the form Universe. In this review we will also discuss the implications of such a detection of left–right symmetry breaking for leptogenesis 0 2 (νφ eφ+) in detail. Before closing this paragraph we would like to stress the L − , (3) dim-5 = fact that this review is far from comprehensive and covers onlya limited variety of topics from the vast choices of LRSM-related where is the scale corresponding to some new extension of the scenarios. For example, a detailed discussion of the relevant SM violating lepton number. This dimension-5 term can induce collider phenomenology of the right-handed gauge bosons WR small Majorana masses to the neutrinos after the eletroweak and ZR and the Higgs sector is beyond the scope of this review. symmetry breaking Some relevant references for the LHC phenomenology of WR and T 1 heavy neutrinos are in Keung and Senjanovic [1], Nemevsek et L mν ν C− ν , (4) − mass = iL jL

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2 0 0 with mν υ /. Here, C is the charge conjugation matrix. Now a non-zero VEV acquired by ξ ( ξ u) gives Majorana = = Consequently, lepton number violating new physics at a high masses to the neutrinos. Note that u has to be less than a few scale would naturally explain the smallness of neutrino masses. GeV to not affect the electroweak ρ-parameter. The most general In what follows, we discuss some of the popular mechanisms of Higgs potential with a doublet and a triplet Higgs has the form realizing the same. 2 † 2 † 1 † 2 1 † 2 V m φ φ m ξ ξ λ1(φ φ) λ2(ξ ξ) 2.1. Seesaw Mechanism: Type-I = φ + ξ + 2 + 2 1 † † T † The type-I seesaw mechanism [14–20] is the simplest λ3(φ φ)(ξ ξ) λ4φ ξ φ. (8) mechanism of obtaining tiny neutrino masses. In this + + mechanism, three singlet right handed neutrinos NiR are We assume λ 0, which manifests explicit lepton number 4 = added to the SM; and one can write a Yukawa term similar to violation and the mass of the triplet Higgs Mξ λ υ. The ∼ 4 ≫ Equation (2) and a Majorana mass term for the right handed mass matrix of the scalars √2 Imφ0 and √2 Imξ 0 is given by neutrinos since they are singlets under the SM gauge group. The relevant Lagrangian is given by 2 4λ4u 2λ4υ M − 2 , (9) = 2λ4υ λ4υ /u − 1 c T 1 c Ltype-I hiαNiRφlαLφ MijN C− N h.c.. (5) − = + 2 iLiL jL + which tells us that one combination of these fields remains massless, which becomes the longitudinal mode of the Z boson; Note that, the Majorana mass term breaks the lepton number while the other combination becomes massive with a mass of the explicitly and since the right handed neutrinos are SM gauge order of triplet Higgs and hence the danger of Z decaying into singlets, there is no symmetry protecting Mij and it can be very Majorons 2 is absent in this model. The minimization of the scalar large. Now after the symmetry breaking, combining the Dirac potential yields and Majorana mass matrices we can write 2 λ4υ T 1 c c T 1 c u 2 , (10) 2L m α ν C− N M N C− N h.c. = − M − mass = D i αL iL + i iL jL + ξ 0 m ν ν c T 1 Dαi α giving a seesaw mass to the left handed neutrinos α Ni L C− T c h.c. , (6) = mDαi Mi Ni L + 2 λ4υ where mDαi hDαiυ. Now assuming that the eigenvalues of mνij fiju fij 2 . (11) = = = − Mξ mD are much less than those of M one can block diagonalize the mass matrix to obtain the light Majorana neutrinos with 1 T Note that in the left–right symmetric extension of the SM, masses mν m α M− m and heavy neutrinos with mass ij = − D i i Dαi which we will discuss in the next subsection, both type-I m M . Note that if any of the right handed neutrino mass N = i and type-II seesaw mechanisms are present together. The eigenvalues (Mi) vanish then some of the left handed neutrinos type-II seesaw mechanism can also provide a very attractive will combine with the right handed neutrinos to form Dirac solution to leptogenesis, which we will discuss in the next neutrinos. For n generations, if the rank of M is r, then there will section. be 2r Majorana neutrinos and n r Dirac neutrinos. The type-I − seesaw mechanism not only generates tiny neutrino masses, but 2.3. Seesaw Mechanism: Type-III also provides the necessary ingredients for explaining the baryon In type-III seesaw mechanism [29, 30] the SM is extended to asymmetry of the universe via leptogenesis [21], which we will include SU(2)L triplet fermions to realize the effective operator discuss in length in the next section. given in Equation (3)3. The Yukawa interactions in Equation (5) are generalized straightforwardly to SU(2)L triplet fermions 2.2. Seesaw Mechanism: Type-II with hypercharge Y 0. The corresponding interaction = In type-II seesaw mechanism [18, 19, 22–28], the effective Lagrangian is given by operator given in Equation (3) is realized by extending the SM to include an SU(2)L triplet Higgs ξ which transforms under the SM 1 L τ φ cT 1c gauge group SU(3) SU(2) U(1) as (1,3,1). For simplicity type-III hiα iL α ˜ Mαβ α C− β h.c., c × L × Y − = + 2 + we assume that there are no right handed neutrinos in this model (12) and only one triplet scalar is present. The Yukawa couplings of where α 1,2,3. In exactly similar manner as in the case of = the triplet Higgs with the left handed lepton doublet (ν , l ) are type-I seesaw, one obtains for M υ, the left handed neutrino i i ≫ given by mass 2 1 T 0 mνij υ hiαMβα− hjβ . (13) L f ξ ν ν ξ +(ν l ν l )/√2 ξ ++l l . (7) = − − type-II = ij i j + i j + j i + i j 2Majorons correspond to Goldstone bosons associated with the spontaneous 1The seesaw mechanisms generically require a new heavy scale (as compared to the breaking of a global lepton number symmetry. electroweak scale) in the theory, inducing a small neutrino mass (millions of times 3Ma [30] established the nomenclature Types I, II, III, for the three and only three smaller than the charged lepton masses). Hence the name “seesaw.” tree-level seesaw mechanisms.

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2.4. Radiative Models of Neutrino Mass broken symmetry; Gu-Sarkar model [41] with dark matter Small neutrino masses can also be induced via radiative candidates in the loop; Kanemura-Matsui-Sugiyama model [42] corrections. The advantage of these models is that without utilizing an extension of the two Higgs doublet model; Bonilla- introducing a very large scale into the theory the smallness of the Ma-Peinado-Valle model where the Dirac neutrino masses neutrino masses can be addressed. In fact, several of these models are generated at two-loops with dark matter in the loop can explain naturally the smallness of the neutrino masses with [43], etc. only TeV scale new particles. Thus, new physics scale in these models can be as low as TeV, which can be probed in current and 3. LEFT–RIGHT SYMMETRIC THEORIES next generation colliders. One realization of this idea is the so-called Zee model [31, 32], The SM gauge group G SU(3) SU(2) U(1) explains SM ≡ c × L × Y where one extends the SM to have two (or more) Higgs doublets the (V A) structure of the weak interaction and parity violation, − φ1 and φ2, and a scalar η+ which transforms under the SM gauge which is reflected by the trivial transformation of all right handed group SU(3)c SU(2)L U(1)Y as (1, 1, 1). The lepton number fields under SU(2) . However, the origin of parity violation × × L violating Yukawa couplings are given by is not explained within the SM, and it is natural to seek an explanation for parity violation starting from a parity conserved T 1 LZee fijψ C− ψjLη+ ε φaφ η− h.c., (14) theory at some higher energy scale. This motivated a left–right = iL + ab b + symmetric extension of the SM gauge theory, called the Left– where fij is antisymmetric in the family indices i, j and εab is the Right Symmetric Model (LRSM) [44–49], in which the Standard totally antisymmetric tensor. Now, the VEV of the SM Higgs Model gauge group is extended to doublet allows mixing between the singlet charged scalar and the charged component of the second Higgs doublet, resulting in a GLR SU(3)c SU(2)L SU(2)R U(1)B L neutrino mass induced through the one-loop diagram showed ≡ × × × − in Figure 1 (left). The antisymmetric couplings of η+ with the where B L is the difference between baryon (B) and lepton − leptons make the diagonal terms of the mass matrix vanish, with (L) numbers. The left–right symmetric theory, initially proposed the non-diagonal entries given by to explain the origin of parity violation in low-energy weak interactions has come a long way answering various other mν (i j) Af (m2 m2), (15) issues like small neutrino mass, dark matter as left by the ij = = ij i − j Standard Model. Originally suggested by Pati-Salam, the model where i, j e, , τ and A is a numerical constant. In the Zee has been studied over and over because of its versatility and = model, if the second Higgs doublet is replaced by a doubly many alternative formulations of the model have also been charged singlet scalar ζ ++, then one gets what is called Zee- proposed. The model stands on the foundation of a complete Babu Model [33, 34]. In this model a Majorana neutrino symmetry between left and right which means Parity is an explicit mass can be obtained through a two loop diagram shown symmetry in it until spontaneous symmetry breaking occurs. As in Figure 1(right). In fact, there are several other radiative evident from the gauge group, the natural inclusion of a right- models of Majorana neutrino mass such as the Ma model [35] handed neutrino in it makes the issue of neutrino mass an easy connecting the Majorana neutrino mass to dark matter at one- affair to discuss. Three new gauge bosons namely WR± that are loop; Krauss-Nasri-Trodden model [36] and Aoki-Kanemura- the heavier parity counterparts of WL± of the standard model Sato model [37] giving neutrino mass at the three loop level and a Z′ boson analogous to the Z boson also find place in with a dark matter candidate in the loop; Gustafsson-No- the framework. LRSM breaks down to Standard Model gauge Rivera model [38] involving a three loop diagram with a theory at low energy scales, SU(2)L SU(2)R U(1)B L × × − × dark matter candidate and the W boson; and Kanemura– SU(3) SU(2) U(1) SU(3) . It has been noticed that C −→ L × Y × C Sugiyama model [39] utilizing an extension of the Higgs triplet the choices of Higgs and their mass scales in the model offers rich model. There are also models for radiative Dirac neutrino phenomenology which can be verified at the current and planned masses such as the Nasri-Moussa model [40] utilizing a softly experiments.

FIGURE 1 | (Left) one-loop diagram diagram generating neutrino mass in Zee model. (Right) two loop diagram generating neutrino mass in Zee-Babu model.

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The basic framework and properties of Left–Right Symmetric TABLE 1 | LRSM representations of extended field content. Models are already discussed at length in various original Fields SU(2) SU(2) B − L SU(3) works [46–48], thus we only intend to study here various L R C seesaw mechanisms for the generation of neutrino mass and Fermions qL 2 1 1/3 3 its implications to leptogenesis in various Left–Right Symmetric qR 1 2 1/3 3 models. ℓL 2 1 1 1 A very brief sketch of the manifest left–right symmetric model − ℓR 1 2 1 1 is given here. The model is based on the gauge group, − Scalars 2 2 0 1 H 2 1 1 1 GLR SU(2)L SU(2)R U(1)B L SU(3)C . (16) L − ≡ × × − × H 1 2 1 1 R − The electric charge Q is difined as,

B L With usual quarks and leptons the Yukawa Lagrangian reads as, Q T3L T3R − T3L Y . (17) = + + 2 = + L q Y Y q ℓ Y Y ℓ h.c., − Yuk ⊃ L 1 + 2 R + L 3 + 4 R + Here, T and T are, respectively, the third components of (21) 3L 3R isospin of the gauge groups SU(2) and SU(2) , and Y is the where σ2∗σ2 and σ2 is the second Pauli matrix. When the L R = hypercharge. The particle spectrum of a generic LRSM can be scalar bidoublet () takes non-zero VEV , sketched as, υ1 0 , (22) = 0 υ2 υL υR ℓL (2, 1, 1, 1), ℓR (1, 2, 1, 1) ,(18) = eL ∼ − = eR ∼ − it gives masses to quarks and charged leptons in the following uR 1 uR 1 manner, qL (2, 1, 3 , 3), qR (1, 2, 3 , 3). (19) = dR ∼ = dR ∼ M Y υ1 Y υ∗ , M Y υ2 Y υ∗ , u = 1 + 2 2 d = 1 + 2 1 The spontaneous symmetry breaking of the gauge group Me Y3υ2 Y4υ∗ . (23) = + 1 which occurs in two steps gives masses to fermions including neutrinos. In the first step the gauge group SU(2) SU(2) It also yields Dirac mass for light neutrinos as L × R × U(1) SU(3) breaks down to SU(2) U(1) B L C L Y ν − × × × M M Y υ Y υ∗ . (24) SU(3)C i.e., the SM gauge group. This gauge group then D ≡ D = 3 1 + 4 2 breaks down to U(1) SU(3) . However these symmetry em × C breakings totally depend upon the choices of Higgs that we The only role that the Higgs doublets play here is helping in consider in the framework and their mass scales. Thus, in the spontaneous symmetry breaking of LRSM to SM. It is also this review we intend to discuss fermion masses emphasizing important to note that the breaking of SU(2)R by doublet Higgs on neutrino mass in possible choices of symmetry breakings leads to Dirac neutrinos. of LRSM. 3.2. LRSM With Bidoublet (B − L = 0) and B − L = 3.1. LRSM With Bidoublet (B − L = 0) and Triplets ( 2). Along with the bidoublet , here we use triplets L, R for the Doublets (B − L = −1). spontaneous symmetry breakings. Here we use Higgs bidoublet to implement the symmetry breaking of SM down to low energy theory leading to charged 0 φ1 φ2+ fermion masses. The symmetry breaking of LRSM to SM occurs 0 (2, 2, 0, 1), ≡ φ1− φ2 ∼ via RH Higgs doublet H (B L 1). We need the left- R − = − √ handed counterpart H to ensure left–right invariance. The δL+/ 2 δL++ L L 0 (3, 1, 2, 1), ≡ δ δ+/√2 ∼ fermions including usual quarks and leptons along with scalars L − L are presented in Table 1. δR+/√2 δR++ R 0 (1, 3, 2, 1), (25) The matrix structure of the scalar fields looks as follows, ≡ δ δ+/√2 ∼ R − R 0 φ1 φ2+ The particle content of the model is shown in Table 2. 0 (2, 2, 0, 1), ≡ φ− φ ∼ The Yukawa Lagrangian is given by 1 2 hL+ hR+ H (2, 1, 1, 1), H (1, 2, 1, 1). L qL Y1 Y2 qR ℓL Y3 Y4 ℓR L ≡ h0 ∼ − R ≡ h0 ∼ − − Yuk ⊃ + + + L R (20) f (ℓ )cℓ (ℓ )cℓ h.c., (26) + L L L + R R R +

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TABLE 2 | LRSM representations of extended field content. bidoublet with B L charge 0. The Yukawa Langrangian for − inverse seesaw mechanism is given by, Fields SU(2)L SU(2)R B − L SU(3)C

c c Fermions q 2 1 1/3 3 L ℓ Y Y ℓ F (ℓ )H S S S h.c. L − Yuk = L 3 + 4 R + R R L + L L + q 1 2 1/3 3 (30) R ℓL 2 1 1 1 c − MDνLNR MNRSL SSLSL h.c.. (31) ℓ 1 2 1 1 ⊃ + + + R −

Scalars 2 2 0 1 After spontaneous symmetry breaking the resulting neutral lepton mass matrix reads as follows, L 3 1 2 1 R 1 3 2 1

0 MD 0   M MT 0 M . (32) The scalar triplets L , R give Majorana masses to light left- = D   handed and heavy right-handed neutrinos. The neutral lepton    0 MT  mass matrix is given by  S   With the mass hierarchhy m , M , the light neutrino mass D ≫ S ML MD formula is given by, Mυ , (27) = MT M D R T 1 1 T mν MD(M )− M− MD . (33) Here M f f υ (M f f υ ) denoted as the = L = L L = L R = R R = R Majorana mass matrix for left-handed (right-handed) neutrinos 3.4. LRSM With Linear Seesaw and M Y υ Y υ is the Dirac neutrino mass matrix D = 3 1 + 4 2 Another interesting low scale seesaw type is linear seesaw connecting light-heavy neutrinos. The complete diagonalization mechanism [52, 53] which can be realized with the following results type-I+II seesaw formula for light neutrinos as, particle content in a LRSM.

1 T II I mν M m M− m m m , (28) : = L − D R D = ν + ν Fermions uL uR QL (2, 1, 1/3, 3), QR (1, 2, 1/3, 3), = dL ∼ = dR ∼ 3.3. LRSM With Inverse Seesaw In canonical seesaw mechanisms, the tiny mass of light neutrinos νL νR ℓL (2, 1, 1, 1), ℓR (1, 2, 1, 1), is explained with large value of seesaw scale thereby making it = eL ∼ − = eR ∼ − inaccessible to the ongoing collider experiments. On the other S (1, 1, 0, 1), hand, the light neutrino masses may arise from low scale seesaw ∼ Scalars : mechanisms like inverse seesaw [50, 51] where the seesaw scale h h can be probed at upcoming accelerators. The inverse seesaw H L+ (2, 1, 1, 1) H R+ (1, 2, 1, 1) L = h0 ∼ R = h0 ∼ mechanism in LRSM can be realized with the following particle L R 0 content; φ1 φ2+ 0 (2, 2, 0, 1). = φ− φ ∼ 1 2 Fermions : uL uR The scalars take non-zero vev as follows: qL (2, 1, 1/3, 3), qR (1, 2, 1/3, 3), = dL ∼ = dR ∼ k1, k2, HL υL, HR υR, νL νR = = = ℓL (2, 1, 1, 1), ℓR (1, 2, 1, 1), = eL ∼ − = eR ∼ − Let us write down the relevant Yukawa terms in the Lagrangian S (1, 1, 1, 0), ∼ that contribute to the fermion masses: Scalars :

hL+ hR+ LYuk hℓℓR ℓL hℓℓR ℓL fR S HR ℓR fL S HL ℓL HL 0 (2, 1, 1, 1), HR 0 (1, 2, 1, 1) = + + + = hL ∼ = hR ∼ c S (SL) SL h.c., (34) 0 + + φ1 φ2+ 0 (2, 2, 0, 1), (29) = φ1− φ2 ∼ where H iτ H with j L, R and τ τ . The singlet j = 2 j∗ = = 2 ∗ 2 Majorana field S in Equation (34) is defined as The fermion sector here comprises of the usual quarks and c leptons plus one extra fermion singlet per generation. The scalar SL (SL) sector holds the doublets H with B L charge 1 and the S + . (35) L,R − − = √2

Frontiers in Physics | www.frontiersin.org 6 March 2018 | Volume 6 | Article 19 Hati et al. Neutrino Masses and Leptogenesis in LRSM resulting in the neutral lepton mass matrix with B L 2 and doublets H H with B L 1. We call − = L ⊕ R − = − the model Extended LR model and thus the seesaw mechanism 0 hℓk hℓk f υ is called extended seesaw. Table 3 shows the complete particle 1 + 2 L L M T T spectrum. N hℓ k1 hℓ k2 0 fRυR = +T T The leptonic Yukawa interaction terms can be written as, f υL f υR S  L R    0 mD mL c c L ℓL Y3 Y4 ℓR f (ℓL) ℓLL (ℓR) ℓRR mT 0 M . (36) − Yuk = + + + ≡  D  mT MT ℓ c ℓ c L S F ( R)HRSL F′ ( L)HLSL SSLSL h.c.. (40)   + + + + M ν N M νc ν M Nc N The violation of lepton number by two units arises here through ⊃ D L R + L L L + R R R c c the combination m and . As a result, assuming m m < M, MNRSL Lν SL SS SL (41) L S L≪ D + + L + L one gets the light Majorana masses of the active neutrinos to be The neutral lepton mass matrix comes out to be; T 1 T 1 T mν mDM − mL mLM− mD = + ML MD L T 1 T 1 T ( 1k1 2k2) Mν M MR M , (42) m f f − f f − m + .(37) D D L R L R D η =  T T  = + × M′ P L M S   The last line in Equation (37) follows from the fact that in left– c in the basis νL, NR, SL after spontaneous symmetry breaking. right symmetric model where Parity and SU(2)R breaking occurs The individual elements of the matrix hold the following at different scales υL is given by meaning; M Y measures the light-heavy neutrino mixing D = and is usually called the Dirac neutrino mass matrix, MN (1k1 2k2)νR = v + , (38) f υR f R (ML f υL f L ) is the Majorana mass term L = = = ≃ − M′ηP for heavy (light) neutrinos, M F H is the N S mixing = R − matrix, L F′ HL stands for the small mass term connecting where 1, 2 are the trilinear terms arising in the Higgs potential = ν S and S is the Majorana mass term for the singlet fermion SL. involving Higgs bidoublet and Higgs doublets, ηP is the parity − Inverse Seesaw:- In Equation (42), following the mass hierarchy breaking scale and M is the SU(2) breaking scale. From ′ R M M and with the assumption that M , M , 0 one Equation (37) it is clear that the light neutrino mass is suppressed ≫ D≫ S L R L → obtains the inverse seesaw mass formula for light neutrinos [59] by the parity breaking scale η M . The f and f are Majorana P ≃ ′ L R couplings, k1, k2 being VEV of Higgs bidoublet while υL(υR) is T ν MD MD the VEV of LH (RH) scalar doublet. The smallness of L thus m s . ν = M M ensures the smallness of the observed sub-eV scale neutrino masses. The SU(2)R U(1)B L breaking scale υR can be as low × − as a few TeV. This is in contrast to the usual left–right symmetric Let us have a look at the model parameters of inverse seesaw model without D-parity, where the neutrino mass is suppressed framework and see how the light neutrino mass can be parametrized in terms of these. by vR and hence cannot be brought to TeV scales easily [54]. In addition we get two heavy pseudo-Dirac states, whose 2 2 mν M M − masses are separated by the light neutrino mass, given by D s . 0.1 eV = 100 GeV keV 104 GeV M M mν . (39) ≈ ± + In the above equation, the small masses of active neutrinos can TABLE 3 | LRSM representations of extended field content. arise through small values of mL/M. As a result of M around TeV and mD in the range of 100 GeV,sizable mixing between the light Fields SU(2)L SU(2)R B − L SU(3)C and heavy states arises, and the Pseudo-Dirac pair with mass M 4 can be probed at colliders . Fermions qL 2 1 1/3 3 qR 1 2 1/3 3 3.5. LRSM With Extended Seesaw ℓ 2 1 1 1 L − The LRSM here is extended with the addition of a neutral fermion ℓR 1 2 1 1 S per generation to the usual quarks and leptons.5 The scalar − L SL 1 1 0 1 sector consists of bidoublet with B L 0, triplets − = L ⊕ R Scalars 2 2 0 1 4Most often the linear seesaw is assumed to be realized with M m m ≫ D ∼ L ∼ HL 2 1 1 1 100 GeV, which results in the same expression for the m . This would result in − ν H 1 2 1 1 unobservable heavy fermions and negligible mixing. R − 5The discussion of extended seesaw mechanism can be found in Gavela et al. [55], L 3 1 2 1 Barry et al. [56], Zhang [57] and Dev and Pilaftsis[58]. R 1 3 2 1

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Testable collider phenomenology can be expected in such a The following transformation relates the intermediate block scenario because M lies at a few TeV scale which allows large left– diagonalised neutrino states to the flavor eigenstates. right mixing. For an extension of such a scenario which allows I O 1 large LNV and LFV one may refer the work [60]. ν′ MDMR− νL O I − 1 Linear Seesaw:- Alternatively, in Equation (42), the assumption S′ MMR− SL (50)   =  1 † 1 † − I   c  of M , M , 0 leads to the linear seesaw mass formula for N′ (MDMR− ) (MMR− ) NR L R S → light neutrinos given by Deppisch et al. [61]       In the mass matrix M′ the (2,2) entry is larger than other entries T 1 in the limit MR > M > MD ML. The same procedure can be mν MD M− L+transpose, (43) ≫ = repeated in Equation (48) and S′ can be integrated out. Now the mass formula for light neutrino is given by whereas the heavy neutrinos form a pair of pseudo-Dirac states with masses 1 T mν M M M− M = L − D R D M M m . (44) 1 ν 1 T 1 T − 1 T ± ≈ ± + M M− M MM− M MM− M − − D R − R − R D Type-II Seesaw Dominance:- On the other hand a type-II seesaw 1 T 1 T dominance can be realized with the assumption that , 0 ML MDMR− MD MDMR− MD L S → = − + in Equation (42).This allows large left–right mixing and thus II ML m , (51) leads to an interesting scenario. = = ν A natural type-II seesaw dominance can be realized from the and the physical block diagonalised states are following Yukawa interactions 1 ν νL MDM− SL ˆ = − 1 c 1 † c c S SL MM− N (MDM− ) SL (52) LYuk ℓL Y3 Y4 ℓR f (ℓL) ℓLL (ℓR) ℓRR ˆ R R − = + + + = − + F(ℓ )H Sc h.c. (45) with the corresponding block diagonalised transformation as + R R L + c c 1 MDνLNR MLν νL MRN NR MNRSL h.c.. ν I M M ν ⊃ + L + R + + ˆ − D − ′ (53) (46) S = (MM 1)† I S ˆ − ′ c Following this block diagonalization procedure the flavor The gauge singlet mass term SS S does not appear in the above Lagrangian since we have considered this to be zero or negligbly eigenstates can be related to mass eigenstates through the small to suppress the generic inverse seesaw contribution following transformation involving S. We have also assumed the induced VEV for HL to I 1 1 νL MDM− MDMR− ν′ be zero, i.e., HL 0. 1 † 1 → S (M M ) I MM− S (54) Now the complete 9 9 mass matrix for the neutral fermions  L  =  D − R   ′  × Nc O 1 † I N in flavor basis can be written as R (MMR− ) ′    −    Finally, the physical masses can be obtained by diagonalising the ν S Nc L L R final block diagonalised mass matrices by a 9 9 unitary matrix νL ML 0 MD × M   . (47) V9 9. The block diagonalised neutrino states can be expressed in = SL 0 0 M terms× of mass eigenstates as follows,  Nc MT MT M   R D R    ν U ν , S U S , N U N . (55) α ν αi i ˆα Sαi i ˆ α N αi i The heaviest right-handed neutrinos can be integrated out ˆ = = = following the standard formalism of seesaw mechanism. Using while the block diagonalised mass matrices for light left- handed neutrinos, heavy right-handed neutrinos and extra sterile mass hierarchy MR > M > MD ML one obtains ≫ neutrinos are

ML 0 MD 1 M M MT MT mν ML , ′ R− D = = 0 0 − M vR MN MR ML , M M M 1MT M M 1MT ≡ = v L D R− D D R− , (48) L − 1 T − 1 T 1 T = MMR− MD MMR− M M MM− M . (56) − S = − R where the intermediate block diagonalised neutrino states are Further these mass matrices can be diagonalised by respective modified as 3 3 unitarity matrices as, × diag † 1 c m U m U diag. m , m , m , ν′ ν M M− N , ν ν ν ν∗ 1 2 3 = L − D R R = = { } 1 c diag † S′ S M M− N , MS US MSUS∗ diag. MS1 , MS2 , MS3 , = L − D R R = = { } c 1 T 1 T diag † N′ N (M− M )∗ν (M− M )∗S . (49) M U M U∗ diag. M , M , M . (57) = R + R D L + R L N = N N N = { N1 N2 N3 }

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The complete block diagonalization results, 4. ALTERNATIVE FORMULATION OF LEFT–RIGHT SYMMETRIC MODEL: M † M W U † M W U V9 9 V9∗ 9 ( ) ( ) UNIVERSAL SEESAW = × × = diag. m1, m2, m3 MS1 , MS2 , MS3 MN1 , MN2 , MN3 , (58) = { ; ; } Among the various alternative formulations of left–right symmetric model that have been proposed so far, the model W U where is the block diagonalised mixing matrix and is the which includes isosinglet vector like fermions looks more unitarity matrix given by, upgraded. The advantages of this alternative formulation over the manifest one are the following: I 1 1 MDM− MDMR− W 1 † I 1 Due to the presence of vector like fermions and absence of (MDM− ) MMR− , • =  O 1 † I  the usual scalar bidoublet in it, the charged fermions get (MMR− ) their masses through a seesaw mechanism called the universal  −  Uν O O seesaw instead of standard Yukawa interaction. Thus, one does U O U O . (59) not need to finetune the Yukawa couplings. The universal =  S  O O UN seesaw is named as such since both quarks and leptons get their   masses through a common seesaw. Thus, the complete 9 9 unitary mixing matrix diagonalizing the Due to the absence of scalar bidoublet no tree level Dirac × • neutral leptons is as follows neutrino mass arises. However, the tiny Dirac neutrino mass is generated at two loop level while the right-

1 1 handed interactions still lie at TeV scale, as shown in Uν MDM− US MDMR− UN Figure 2. W U 1 † 1 V (MDM− ) Uν US MMR− UN The same set of symmetries offer the ambiance to address the = =  1 †  O (MM− ) U U • − R S N issue of weak and strong CP-violation.  (60) The scalar sector of the model is too simple which consists of U • Expressing Masses and Mixing in terms of PMNS and light two isodoublets. neutrino masses:- Usually, the light neutrino mass matrix is diagonalised by the UPMNS mixing matrix in the basis diag where the charged leptons are already diagonal i.e., mν † = 4.1. Left–Right Symmetry With Vector-Like UPMNSmνUPMNS∗ . The structure of the Dirac neutrino mass matrix MD which is a complex matrix in general can be Fermions and Universal Seesaw considered to be the up-quark type in LRSM. Its origin can be The fermion content of this model includes the usual quarks and motivated from a high scale Pati-Salam symmetry or SO(10) leptons, GUT. If we consider M to be diagonal and degenerate i.e., M m diag 1,1,1 , then the mass formulas for neutral leptons are= S { } given by uL uR qL (2, 1, 1/3, 3), qR (1, 2, 1/3, 3), = dL ∼ = dR ∼ diag T mν M f υ U mν U , νL νR = L = L = PMNS PMNS ℓL (2, 1, 1, 1), ℓR (1, 2, 1, 1), = eL ∼ − = eR ∼ − υR υR diag T MN MR f υR ML UPMNSmν UPMNS , ≡ = = υL = υL 1 1 T 2 υR diag T − M MM− M m U mν U , (61) S = − R = − S υ PMNS PMNS L

After some simplification the active LH neutrinos νL, active RH neutrinos NR and heavy sterile neutrinos SL in the flavor basis are related to their mass basis as

νν νS νN νL V V V νi Sν SS SN SL V V V Si  c  =  Nν NS NN    NR α V V V αi Ni       1 1 vL 1 diag.− UPMNS MDU∗ MDU− mν mS PMNS vR PMNS νi 1  1 † vL 1 diag.−  M UPMNS U∗ mSU− mν Si = mS D PMNS vR PMNS   1 N  O vL 1 diag.−  i  v mSUPMNS− mν UPMNS   R αi   FIGURE 2 | Two loop contributions to Dirac neutrino mass for light neutrinos.   (62)

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and the additional vector-like quarks and charged leptons [62– TABLE 4 | LRSM representations of extended field content. 70] Field SU(2)L SU(2)R B − L SU(3)C

UL,R (1, 1, 4/3, 3), DL,R (1, 1, 2/3, 3), ∼ ∼ − qL 2 1 1/3 3 E (1, 1, 2, 1). (63) L,R ∼ − qR 1 2 1/3 3 ℓL 2 1 1 1 To this new setup of left–right symmetric model, we add vector − ℓ 1 2 1 1 like neutral lepton in the fermion sector and a singlet scalar in R − U 1 1 4/3 3 the Higgs sector. The purpose behind the inclusion of vector like L,R D 1 1 2/3 3 neutral lepton is to allow seesaw mechanism for light neutrinos L,R − E 1 1 2 1 leading to Dirac neutrino mass. Similarly, the scalar singlet is L,R − N 1 1 0 1 introduced to give consistent vacuum stability in the scalar sector. L,R The particle content and the relevant transformations under the HL 2 1 1 1 LRSM gauge group are shown in Table 4. HR 1 2 1 1 We now extend the standard LRSM framework having S 1 1 0 1 isosinglet vector-like copies of fermions with additional neutral vector like fermions [71–75]. This kind of a vector-like fermion spectrum is very naturally embedded in gauged flavor groups L L 0 λEυL 0 λN υL with left–right symmetry [76] or quark-lepton symmetric models MeE R , MνN R , (65) = λ υR ME = λ υR MN [77]. E N The relevant Yukawa part of the Lagrangian is given by The mass eigenstates can be found by rotating the mass matrices L,R L (λ SXX M XX) (λL H q U λR H q U via left and right orthogonal transformations O (we assume all = − SXX + X − U ˜ L L R + U ˜ R R L X parameters to be real). For example, the up quark diagonalization LT R L R L R yields OU MuU OU diag(mu, MU ). Up to leading order in λDHLqLDR λDHRqRDL λEHLℓLER λEHRℓREL L = ˆ ˆ + + + + λU vL, the resulting up-quark masses are λL H ℓ N λR H ℓ N h.c.), (64) + N ˜ L L R + N ˜ R R L + L R (λ vL)(λ vR) where the summation is over X U, D, E, N and we suppress M M2 (λR v )2, m U U , (66) = ˆ U U U R u flavor and color indices on the fields and couplings. H denotes ≈ + ˆ ≈ MU ˜ L,R ˆ τ2HL∗,R, where τ2 is the usual second Pauli matrix. We would like to stress that Parity Symmetry is present in order to distinguish L,R L,R and the mixing angles θU parametrizing OU , between for instance NR and NL, otherwise extra terms in the Lagrangian Equation (64) would appear with the vector-like L R 2(λ vL)MU 2(λ vR)MU fermions Left and Right exchanged. tan(2θ L ) U , tan(2θ R) U . (67) U ≈ M2 (λR v )2 U ≈ M2 (λR v )2 The LRSM gauge group breaks to the SM gauge group when U + U R U − U R H (1,2, 1) acquires a VEV and the SM gauge group breaks R − to U(1) when H (2,1, 1) acquires a VEV. However, parity The other fermion masses and mixings are given analogously. EM L − can break either at TeV scale or at a much higher scale MP. For an order of magnitude estimate one may approximate the For the latter case the Yukawa couplings can be different for phenomenologically interesting regime with the limit λR v U R → right-type and left-type Yukawa terms (λR λL ) because of M in which case the mixing angles approach θ L m /M X = X U U → ˆ u ˆ U the renormalization group running below MP. Consequently, we and θ R π/4. This means that θ L is negligible for all fermions U → U will distinguish the left and right handed couplings explicitly but the top quark and its vector partner [72]. with the subscripts L and R. We use the VEV normalizations We here neglect the flavor structure of the Yukawa couplings T T L,R HL (0, υL) and HR (0, υR) . The scale of vR has to λ and λ which will determine the observed quark and = = X SXX lie between at around a few TeV (depending on the right-handed leptonic mixing. The hierarchy of SM fermion masses can be gauge coupling) to suit the experimental searches for the heavy generated by either a hierarchy in the Yukawa couplings or in right-handed WR boson at colliders and at low energies. the masses of the of the vector like fermions. Since the particle spectrum does not contain a bidoublet As described above, the light neutrino masses are of Dirac- Higgs, Dirac mass terms for the SM fermions can not be type as well, analogously given by written and the charged fermion mass matrices assume a seesaw structure. Alternatively, a Higgs bidoublet can be introduced L R λN λN vLvR along with HL,R. mν , (68) After symmetry breaking, the mass matrices for the fermions ˆ = MN are given by It is natural to assume that M v , as the vector like N is a N ≫ R 0 λL υ 0 λL υ singlet under the model gauge group. In this case, the scenario M U L , M D L , uU = λR υ M dD = λR υ M predicts naturally light Dirac neutrinos [76]. U R U D R D

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4.2. Left–Right Symmetry With Vector-Like where the scalar potential is given by Fermions and Type-II Seesaw for Neutrino † † Masses V (H , H , , ) 2(H H ) 2(H H ) L R L R = − 1 L L − 2 R R In Table 5, we present the field content of this model and their † † † † λ (H H )2 λ (H H )2 β (H H )(H H ) transformations under the LRSM gauge group. + 1 L L + 2 R R + 1 L L R R 2 † 2 † We implement a scalar sector consisting of SU(2)L,R doublets Tr( ) Tr( ) − 3 L L − 4 R R and triplets, however the conventional scalar bidoublet is absent. † † λ Tr( )2 λ Tr( )2 We use the Higgs doublets to implement the left–right and the + 3 L L + 4 R R 0 T † † electroweak symmetry breaking: HR (hR, hR−) [1,2, 1,1] β2Tr(LL)Tr(RR) ≡ ≡ 0 −T + breaks the left–right symmetry, while HL (hL, hL−) † † † † ≡ ≡ ρ1(Tr( L)(H HR) Tr( R)(H HL)) [2,1, 1,1] breaks the electroweak symmetry once they acquire + L R + R L − † † † † vacuum expectation values (VEVs), ρ Tr( )(H H ) Tr( )(H H ) + 2 L L L L + R R R R vR vL † † † † √ √ ρ3 HLLLHL HRRRHR HR 2 , HL 2 . (69) + + = 0 = 0 HTiσ H HTiσ H h.c. . (72) + L 2 L L + R 2 R R + Note that the present framework requires only doublet Higgs fields for spontaneous symmetry breaking. However, in the Assigning non-zero VEV to Higgs doublets H and H and absence of a Higgs bidoublet, we use the vector-like new fermions R L triplets and , to generate correct charged fermion masses through a universal R L seesaw mechanism. For the neutrinos we note that in the absence 0 √ 0 √ of a scalar bidoublet there is no Dirac mass term for light HL υL/ 2, HR υR/ 2, neutrinos and without scalar triplets no Majorana masses are ≡ ≡ 0 u /√2, 0 u /√2. (73) generated either. To remedy this fact we introduce additional L ≡ L R ≡ R scalar triplets L and R, the scalar potential takes the form, δL+,R/√2 δL++,R L,R 0 , (70) = δ δ+ /√2 L,R − L,R V H , H0 , 0 , 0 L R L R = which transform as L [3,1,2,1] and R [1,3,2,1], 1 1 1 1 ≡ ≡ 2υ2 2υ2 2u2 2u2 respectively. They generate Majorana masses for the light and − 2 1 L − 2 2 R − 2 3 L − 2 4 R heavy neutrinos although they are not essential in spontaneous 1 4 1 4 1 4 1 4 λ1υ λ2υ λ3u λ4u symmetry breaking here. The particle content of the model is + 4 L + 4 R + 4 L + 4 R shown in Table 5. In the presence of the Higgs triplets, the 1 2 2 1 2 2 1 2 2 β1υLυR β2uLuR υLuL υRuR manifestly Left–Right symmetric scalar potential has the form + 4 + 4 − 2√2 + † † 1 1 L (DHL) D HL (DHR) D HR 2 2 2 2 2 2 2 2 = + ρ1 vRuL vLuR ρ2 vLuL vRuR (74) † † + 4 + + 4 + + (D ) D (D ) D + L L + R R V (HL, HR, L, R) , (71) 0 − As non-zero VEV H vR breaks LRSM to SM at high scale R = and H0 v breaks SM down to low energy at electroweak L = L scale, we consider v v . We chose the induced VEVs for L = R TABLE 5 | Field content of the LRSM with universal seesaw. scalar triplets much smaller than VEVs of Higgs doublets, i.e., uL, uR vL, vR. Field SU(2) SU(2) B − L SU(3) ≪ L R C One can approximately write down the Higgs triplets induced VEVs as follows, QL 2 1 1/3 3 QR 1 2 1/3 3 2 2 ℓL 2 1 1 1 vL vR − uL , uR . (75) ℓ 1 2 1 1 2 2 R − = M 0 = M 0 δL δR UL,R 1 1 4/3 3 D 1 1 2/3 3 L,R − E 1 1 2 1 4.2.1. Fermion Masses via Universal Seesaw L,R − HL 2 1 1 1 As discussed earlier, in this scheme normal Dirac mass terms for − the SM fermions are not allowed due to the absence of a bidoublet HR 1 2 1 1 − Higgs. However, in the presence of vector-like copies of quark L 3 1 2 1 and charged lepton gauge isosinglets, the charged fermion mass R 1 3 2 1 matrices can assume a seesaw structure. The Yukawa interaction

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Lagrangian in this model is given by orthogonal transformations OL,R. For example, up to leading L order in YU vL, the SM and heavy vector partner up-quark L YL H Q U YR H Q U YL H Q D masses are = − U L L R + U R R L + D ˜ L L R R L R YDHRQRDL YE HLℓLER YE HRℓREL υ υ + ˜ + ˜ + ˜ L R L R 2 R 2 mu YU YU , MU MU (YU υR) , (78) 1 c c ≈ ˆ ≈ + f ℓ iτ2LℓL ℓ iτ2RℓR MU + 2 L + R ˆ MU ULUR MDDLDR MEELER h.c., (76) θ L,R OL,R − − − + and the mixing angles U in are determined as where we suppress the flavor and color indices on the fields and υ M tan(2θ L,R) 2YL,R L,R U . (79) couplings. HL,R denotes τ2H∗ , where τ2 is the usual second Pauli U U 2 R 2 ˜ L,R ≈ M (Y υR) matrix. Note that there is an ambiguity regarding the breaking of U ± U parity, which can either be broken spontaneously with the left– The other fermion masses and mixing are obtained in an right symmetry at around the TeV scale or at a much higher analogous manner. Note that here we have neglected the flavor scale independent of the left–right symmetry breaking. In the L,R structure of the Yukawa couplings YX which will determine latter case, the Yukawa couplings corresponding to the right- the observed quark and charged lepton mixings. The hierarchy type and left-type Yukawa terms can be different because of the of SM fermion masses can be explained by assuming either a renormalization group running below the parity breaking scale, hierarchical structure of the Yukawa couplings or a hierarchical YR YL. Thus, while writing the Yukawa terms above we X = X structure of the vector-like fermion masses. distinguish the left- and right-handed couplings explicitly with the subscripts L and R. 4.2.2. Neutrino Masses and Type II Seesaw After spontaneous symmetry breaking we can write the mass Dominance matrices for the charged fermions as [73] In the model under consideration there is no tree level Dirac mass term for the neutrinos due to the absence of a Higgs bidoublet. 0 YL υ 0 YL υ M U L , M D L , The scalar triplets acquire induced VEVs L uL and R uU = YR υ M dD = YRυ M = = U R U D R D uR giving the neutral lepton mass matrix in the basis (νL, νR) 0 YLυ given by M E L . (77) eE = YRυ M E R E fuL 0 Mν . (80) The corresponding generation of fermion masses is = 0 fuR diagrammatically depicted in Figure 3. Note that we are interested in a scenario where the VEVs of the Higgs doublets Thus, the light and heavy neutrino masses are simply mν fu = L ∝ are much larger than the VEVs of the Higgs triplets i.e., M fu . A Dirac mass term is generated at the two-loop N = R u υ , u υ . In the context of this work, we do not attempt level via the one-loop W boson mixing θ (see the next section) L ≪ L R ≪ R W to explain how the hierarchy between VEVs can be achieved. and the exchange of a charged lepton. It is of the order mD . 4 2 2 2 Assuming all parameters to be real one can obtain the mass g /(16π ) mτ m mt/M 0.1 eV for MW 5 TeV. This is L b WR ≈ R ≈ eigenstates by rotating the mass matrices via left and right intriguingly of the order of the observed neutrino masses; as long

FIGURE 3 | Generation of fermion masses through universal seesaw and induced triplet VEVs.

Frontiers in Physics | www.frontiersin.org 12 March 2018 | Volume 6 | Article 19 Hati et al. Neutrino Masses and Leptogenesis in LRSM as the right-handed neutrinos are much heavier than the left- neutrino masses are directly proportional to the light neutrino handed neutrinos, the type-II seesaw dominance is preserved and masses. the induced mixing mD/MN is negligible. The mixing between In the present analysis, we discuss 0νββ decay due to exchange 2 2 2 charged gauge bosons θW g /(16π )m mt/M is generated of light neutrinos via left-handed currents, right-handed ≈ L b WR through the exchange of bottom and top quarks, and their vector- neutrinos via right handed currents as shown in Figure 4.0νββ like partners. This yields a very small mixing of the order θW decay can also be induced by a right handed doubly charged 7 ≈ 6 10− for TeV scale WR bosons. scalar as shown in Figure 5 . The half-life for a given isotope for Incorporating three fermion generations leads to the mixing these contributions is given by matrices for the left- and right-handed matrices which we take to 0ν 1 2 2 be equal [T ]− G Mνην (M′ η M η) , (83) 1/2 = 01 | | +| N N + N | V Vν U , (81) where G01 corresponds to the standard 0νββ phase space factor, N = ≡ the Mi correspond to the nuclear matrix elements for the where U is the phenomenological PMNS mixing matrix. Thus, different exchange processes and ηi are dimensionless parameters the unmeasured mixing matrix for the right-handed neutrinos determined below. is fully determined by the left-handed counterpart. The present Light Neutrinos framework gives a natural realization of type-II seesaw providing The lepton number violating dimensionless particle physics a direct relation between light and heavy neutrinos, Mi mi, ∝ parameter derived from 0νββ decay due to the standard i.e., the heavy neutrino masses Mi can be expressed in terms of the light neutrino masses m as M m (M /m ), for a normal mechanism via the exchange of light neutrinos is i i = i 3 3 and Mi mi(M2/m2) for a inverse hierarchy of light and heavy = 3 ν neutrino masses. 1 2 mee ην U mi . (84) = m ei = m e i 1 e 4.3. Implication to Neutrinoless Double =

Beta Decay Here, me is the electron mass and the effective 0νββ mass is As discussed earlier, there is no tree level Dirac neutrino mass explicitly given by term connecting light and heavy neutrinos. Consequently, the ν 2 2 2 2 iα 2 iβ mixing between light and heavy neutrinos is vanishing at this mee c12c13m1 s12c13m2e s13m3e , (85) order. Also, the mixing between the charged gauge bosons = + + is vanishing at the tree level due to the absence of a scalar with the sine and cosine of the oscillation angles θ12 and θ13, bidoublet. c cos θ , etc. and the unconstrained Majorana phases 0 12 = 12 ≤ The charged current interaction in the mass basis for the α, β < 2π. leptons is given by Right-Handed Neutrinos 3 The contribution to 0νββ decay arising from the purely right- gL gR Uei ℓLγνiWL ℓRγNiWR h.c. (82) handed currents via the exchange of right-handed neutrinos √2 + gL + i 1 generally results in the lepton number violating dimensionless = particle physics parameter The charged current interaction for leptons leads to 0νββ decay via the exchange of light and heavy neutrinos. There are 4 4 3 2 gR MWL UeiMi additional contributions to 0νββ decay due to doubly charged ηN mp . (86) = g M p 2 M2 triplet scalar exchange. While the left-handed triplet exchange is L WR i 1 i = | | + suppressed because of its small induced VEV, the right-handed The virtual neutrino momentum p is of the order of the triplet can contribute sizeably to 0νββ decay. | | nuclear Fermi scale, p 100 MeV. m is the proton mass Before numerical estimation, let us point out the mass ≈ p and for the manifest LRSM case we have g g , or else the relations between light and heavy neutrinos under natural type- L = R II seesaw dominance. For a hierarchical pattern of light neutrinos new contributions are rescaled by the ratio between these two the mass eigenvalues are given as m < m m . The lightest couplings. We in general consider right-handed neutrinos that 1 2 ≪ 3 neutrino mass eigenvalue is m1 while the other mass eigenvalues can be either heavy or light compared to nuclear Fermi scale. are determined using the oscillation parameters as follows, m2 2 = 6 m2 m2 , m2 m2 m2 m2 . On the other hand, A detailed discussion of 0νββ decay within LRSMs can be found e.g., in 1 + sol 3 = 1 + atm + sol Mohapatra and Senjanovic [19], Mohapatra and Vergados [78], Hirsch et al. [79], for the inverted hierarchical pattern of the light neutrino masses Tello et al. [80], Chakrabortty et al. [81], Patra [75], Awasthi et al. [60], Barry m3 m1 m2 where m3 is the lightest mass eigenvalue while and Rodejohann [82], Bhupal Dev et al. [83], Ge et al. [84], Awasthi et al. [85], ≪ ≈ 2 2 2 other mass eigenvalues are determined by m1 m3 matm, Huang and Lopez-Pavon [86], Bhupal Dev et al. [87], Borah and Dasgupta [88], 2 2 2 2 = + Bambhaniya et al. [89], Gu [90], Borah and Dasgupta [91] and Awasthi et al. [92] m2 m3 msol matm. The quasi-degenerate pattern of light = + + and for an early study of the effects of light and heavy Majorana neutrinos in neutrinos is m m m m2 . In any case, the heavy neutrinoless double beta decay see in Halprin et al. [93]. 1 ≈ 2 ≈ 3 ≫ atm

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FIGURE 4 | Feynman diagrams for 0 ν ββ decay due to light left-handed and right-handed neutrinos.

FIGURE 5 | Feynman diagrams for 0 ν ββ decay due to doubly charged scalar triplets.

If the mass of the exchanged neutrino is much higher than its because both currents are right-handed. As a result, the 0νββ momentum, M p , the propagator simplifies as decay contribution leads to the dimensionless parameter i ≫| |

Mi 1 4 4 3 , (87) mp gR MW 2 p2 M2 ≈ −M ηN U Mi ην . (90) i i = p 2 g M ei ∝ − L WR i 1 | | = and the effective parameter for right-handed neutrino exchange yields This is proportional to the standard parameter ην but in the case of very light right-handed neutrinos, e.g., M m , i ≈ i the contribution becomes negligible because of the strong 4 4 3 2 gR MW Uei 1 suppression with the heavy right-handed W boson mass. ηN mp ην (m− ), (88) = g M M ∝ i L WR i 1 i In general, we consider right-handed neutrinos both lighter = and heavier than 100 MeV and use (86) to calculate the 1 contribution. In addition, the relevant nuclear matrix element where in the expression for ην (m− ) the individual neutrino i changes; for M 100 MeV it approaches M M whereas masses are replaced by their inverse values. Such a contribution i ≫ N′ → N for M 100 MeV it approaches M Mν. For intermediate clearly becomes suppressed the smaller the right-handed i ≪ N′ → values, we use a simple smooth interpolation scheme within the neutrino masses are. regime 10 MeV – 1 GeV,which yields a sufficient accuracy for our On the other hand, if the mass of the neutrino is much less purposes. than its typical momentum, M p , the propagator simplifies i ≪| | in the same way as for the light neutrino exchange, Right-Handed Triplet Scalar Finally, the exchange of a doubly charged right-handed triplet p/ Mi Mi PR + PR , (89) scalar shown in Figure 5 (where doubly charged left-handed p2 M2 ≈ p2 − i triplet scalar contributes negligible and thus, neglected from the

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TABLE 6 | Phase space factor G01 and ranges of nuclear matrix elements for light and heavy neutrino exchange for the isotopes 76Ge and 136Xe [95].

−1 Isotope G01 (yr ) Mν MN

76Ge 5.77 10 15 2.58–6.64 233–412 × − 136Xe 3.56 10 14 1.57–3.85 164–172 × − present discussion) gives

3 m g 4 M 4 η p R W 2 η 2 UeiMi ν . (91) = M gL MWR ∝ δ−− i 1 R =

This expression is also proportional to the standard ην because the relevant coupling of the triplet scalar is proportional to the right-handed neutrino mass. Numerical Estimate In the following, we numerically estimate the half-life for 0νββ decay of the isotope 136Xe as shown in Figure 6. We use the current values of masses and mixing parameters from neutrino FIGURE 6 | 0νββ decay half-life as a function of the lightest neutrino mass in the case of normal hierarchical (NH) and inverse hierarchical (IH) light neutrinos oscillation data reported in the global fits taken from Gonzalez- in red and green bands respectively. We defined m m such that m is lightest ≃ i 1 Garcia et al. [94]. For the 0νββ phase space factors and nuclear the lightest neutrino mass for NH and m3 for IH pattern. The other parameters matrix elements we use the values given in Table 6. In Figure 6, are fixed as MW 5 TeV, M 5 TeV and the heaviest right-handed R = δR−− ≈ νββ neutrino mass is 1 TeV. The gauge couplings are assumed universal, gL gR, we show the dependence of the 0 decay half-life on the = and the intermediate values for the nuclear matrix elements are used, lightest neutrino mass, i.e., m1 for normal and m3 for inverse Mν 4.5, M 270. The bound on the sum of light neutrino masses from = N = hierarchical neutrinos. The other model parameters are fixed as the KATRIN and Planck experiments are represented as vertical lines. The bound from KamLAND-Zen experiment is presented in horizontal line for heaviest gR gL, MWR Mδ 5 TeV, MN 1TeV. (92) Xenon isotope. The bands arise due to 3σ range of neutrino oscillation = = R−− ≈ = parameters and variation in the Majorana phases from 0 2π. − The lower limit on lightest neutrino mass is derived to be m< 0.9 meV, 0.01 meV for NH and IH pattern of light ≈ neutrino masses respectively by saturating the KamLAND-Zen decay half-life. The red-shaded area is already excluded with a experimental bound. predicted half life of 1026 yr or faster. As expected, this sets an As for the experimental constraints, we use the current best upper limit on the lightest neutrino mass m . 1 eV, but limits at 90% C.L., T0ν (136Xe) > 1.07 1026 yr and T0ν (76Ge) > lightest 1/2 × 1/2 it also puts stringent constraints on the mass scale of the right- 2.1 1025 yr from KamLAND-Zen [96] and the GERDA Phase × handed neutrinos. For an inverse hierarchy, the range 50 MeV . I [97], respectively. Representative for the sensitivity of future M2 . 5 GeV is excluded whereas in the normal hierarchy case, 0νββ experiments, we use the expected reach of the planned large M3 can be excluded if there is a strong hierarchy, m1 0. nEXO experiment, T0ν (136Xe) 6.6 1027 yr [98]. As for the → 1/2 ≈ × This is due the large contribution of the lightest heavy neutrino other experimental probes on the neutrino mass scale, we use N1 in such a case. the future sensitivity of the KATRIN experiment on the effective single β decay mass mβ 0.2 eV [99] and the current limit ≈ 5. LEPTOGENESIS on the sum of neutrino masses from cosmological observations, imi . 0.7 eV [100]. Cosmological observations (studies of the cosmic microwave For a better understanding of the interplay between the left- background radiation, large scale structure data, the primordial and right-handed neutrino mass scales, we show in Figure 7 the abundances of light elements) indicate that our visible universe 0νββ decay half-life as a function of the lightest neutrino mass is dominated by matter and there is very little antimatter. The and the heaviest neutrino mass for a normal (left) and inverse baryon asymmetry normalized to number density of photons (right) neutrino mass hierarchy. The other model parameters are (nγ ) can be extracted out of these observations, which gives fixed, with right-handed gauge boson and doubly-charged scalar masses of 5 TeV. The oscillation parameters are at their best fit nB nB 10 η(t present) − 10− . (93) values and the Majorana phases are always chosen to yield the = = nγ ∼ smallest rate at a given point, i.e., the longest half life. The nuclear matrix employed are at the lower end in Table 6. This altogether The astrophysical observations suggest that at an early epoch yields the longest, i.e., most pessimistic, prediction for the0νββ before the big-bang nucleosynthesis this asymmetry was

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FIGURE 7 | Half-life of 0νββ decay in Xe as a function of the lightest and the heaviest neutrino mass for a normal (left) and inverse (right) neutrino mass hierarchy. The contours denote the half-life in years. Best-fit oscillation data are used and the Majorana phases are chosen to yield the longest half-life. Likewise, the smallest values of the nuclear matrix elements in Table 6 are employed. The other model parameters are chosen as g g and M M 5 TeV. R = L WR = = generated. Thus, it is natural to seek an explanation for this Consequently, the CP violation in the SM is too small to explain asymmetry from the fundamental particle interactions within or the observed baryon asymmetry of the universe. Furthermore, beyond the SM of particle physics. There are three conditions, the electroweak phase transition is not first order; but just a often called Sakharov’s conditions [101], that must be met in smooth crossover. order to generate a baryon asymmetry dynamically: Thus, to explain the baryon asymmetry of the universe one must go beyond the SM, either by introducing new sources of CP 1. baryon number violation, violation and a new kind of out-of-equilibrium situations (such 2. C and CP violation, and as the out-of-equilibrium decay of some new heavy particles) 3. departure from thermal equilibrium. or modifying the electroweak phase transition itself. One such In principle, the SM has all the ingredients to satisfy all three alternative is leptogenesis. Leptogenesis is a mechanism where conditions. a lepton asymmetry is generated before the electroweak phase transition, which then gets converted to baryon asymmetry of the 1. In the SM baryon number B and lepton number L are violated universe in the presence of sphaleron induced anomalous B L due to the triangle anomaly, leading to 12-fermion processes + violating processes, which converts any primordial L asymmetry, involving nine left handed quarks (three of each generation) and hence B L asymmetry, into a baryon asymmetry. A and three left handed leptons (one from each generation) − realization of leptogenesis via the decay of out-of-equilibrium obeying the selection rule (B L) 0. These processes have − = heavy neutrinos transforming as singlets under the SM gauge a highly suppressed amplitude proportional to e 4π/α ( where − group was proposed in Fukugita and Yanagida [21]. The Yukawa α α / sin2 θ , with α being the fine structure constant = EM W EM couplings provide the CP through interference between tree and θ being the weak mixing angle) at zero temperature. W level and one-loop decay diagrams. The departure from thermal However, at high temperature this suppression is lifted and equilibrium occurs when the Yukawa interactions are sufficiently these processes can be very fast. slow7. The lepton number violation in this scenario comes from 2. The weak interactions in the SM violate C in a maximal way. CP is also violated via the CKM phase δCKM. 7The out of equilibrium condition can be understood as follows. In thermal 3. The electroweak phase transition can result in the departure equilibrium the expectation value of the baryon number can be written as B βH βH = from thermal equilibrium if it is sufficiently strongly first Tr[Be− ]/Tr[e− ], where β is the inverse temperature. Since particles and anti order. particles have opposite baryon number, B is odd under C operation, while it is even under P and T operations. Thus, CPT conservation implies a vanishing However, in practice it turns out that only the first Sakharov total baryon number since B is odd and H is even under CPT, unless there is a condition is fulfilled in a satisfactory manner in the SM. The non-vanishing chemical potential. Assuming a non-vanishing chemical potential CP violation coming from the CKM phase is suppressed by implies that the above equation for the expectation value of the baryon number a factor T12 in the denominator, where T 100 GeV is no longer valid and the baryon number density departs from the equilibrium EW EW ∼ distribution. This is achieved when the interaction rate is very slow compared to is the temperature during the electroweak phase transition. the expansion rate of the universe.

Frontiers in Physics | www.frontiersin.org 16 March 2018 | Volume 6 | Article 19 Hati et al. Neutrino Masses and Leptogenesis in LRSM the Majorana masses of the heavy neutrinos. The generated An account of the B L symmetry getting converted to a − lepton asymmetry then gets partially converted to baryon baryon asymmetry via an analysis of the chemical potential can asymmetry in the presence of sphaleron induced anomalous B L be found in Khlebnikov and Shaposhnikov [109], Harvey and + violating interactions before the electroweak phase transition. In Turner [110] and Sarkar [111]. The baryon asymmetry in terms what follows, we will discuss the sphaleron processes and few of the B L number density can be written as − of the most popular scenarios of leptogenesis in some detail to set the stage before discussing leptogenesis in LRSM scenarios in 24 4m B(T > TEW) + (B L), particular. = 66 13m − 32+ 4m B(T < TEW) + (B L). (99) 5.1. Anomalous B + L Violating Processes = 98 13m − and Relating Baryon and Lepton + Thus, the primordial B L asymmetry gets partially converted Asymmetries − into a baron asymmetry of the universe after the electroweak In the SM both B and L are accidental symmetries and at the phase transition. tree level these symmetries are not violated. However, the chiral nature of weak interactions gives rise to equal global anomalies 5.2. Leptogenesis With Right Handed for B and L, giving a vanishing B L anomaly, but a non- − Neutrinos vanishing axial current corresponding to B L, given by t Hooft + In section 2, we have discussed how adding singlet right handed [102, 103] neutrinos NRi to the SM can generate tiny seesaw masses [14– 2N 20] for light neutrinos. Beyond the generation of light neutrino ∂ f α a aν α ν j(B L) 2Wν W 1Bν B , (94) masses, the interaction terms + = 8π ˜ − ˜ a c where Wν and Bν are the SU(2)L and U(1)Y field strength L hα l αφN M (N ) N , (100) int = iL Ri + i Ri Ri tensors and Nf is the number of fermion generations. The corresponding B L violation can obtained by integrating the can also provide all the ingredients necessary for realizing + divergence of the B L current, which is related to the change in leptogenesis. We will work on a basis where the right handed + the topological charges of the gauge field neutrino mass matrix is real and diagonal. Furthermore we assume a hierarchical mass spectrum for the right handed 4 (B L) (B L) d x∂ j + 2N N , (95) neutrinos M3 > M2 > M1. The Majorana mass term gives rise to + = = f cs lepton number violating decays of the right handed neutrinos where N 1, 2, corresponds to the topological charge cs = ± ± N l φ, of gauge fields, called the Chern-Simons number. In the SM there Ri iL → c+ are three generations of fermions (N 3), leading to B liL φ, (101) f = = → + L 3N , thus the vacuum to vacuum transition changes B = cs and L by multiples of 3 units. At the lowest order, one has the which can generate a lepton asymmetry if there is CP violation B L violating effective operator and the decay is out of equilibrium [21]. This lepton asymmetry + (equivalently B L asymmetry) then gets converted to baryon − O(B L) (q q q l ), (96) asymmetry in presence of anomalous B L violating processes + = Li Li Li Li + i 123 before the electroweak phase transition. = In the original proposal [21] and few subsequent works [112– which gives rise to 12-fermion sphaleron induced transitions, 116], only the CP violation coming from interference of tree such as level and one-loop vertex diagrams, shown in Figure 8. was considered. This is somewhat analogous to the CP violation in K- vac [u u d e− c c s − t t b τ −]. (97) | → L L L L + L L L L + L L L L physics coming from the penguin diagram. The CP asymmetry 4π/α parameter corresponding to the vertex type CP violation is given At zero temperature the transition rate is suppressed by e− 165 = by O(10− ) [102, 103]. However, when the temperature is larger than the barrier height, this Boltzmann suppression disappears Ŵ(N lφ†) Ŵ(N lcφ) and B L violating transitions can occur at a significant rate ευ → − → + ≡ Ŵ(N lφ†) Ŵ(N lcφ) [104]. In the symmetric phase, when the temperature is grater → + → than the electroweak phase transition temperature, T TEW, Im α(h∗ hαi)β (h∗ hβi) 2 ≥ 1 α1 β1 Mi the transition rate per unit volume is [105–108] 2 fv 2 ,(102) = −8π α hα M i 2,3 1 1 = | | ŴB L 5 1 4 − α ln α− T , (98) V ∼ where the loop function fv is defined by

2 where α αEM/ sin θW, with αEM being the fine structure 1 x = fv (x) √x 1 (1 x) ln + . (103) constant and θW being the weak mixing angle. = − + x

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FIGURE 8 | Tree level and one-loop vertex diagrams contributing to the vertex type CP violation in models with right handed neutrinos.

In the limit M M , M the asymmetry simplifies to where the loop function f is defined by 1 ≪ 2 3 s √x 3 M Im α(hα∗1hαi)β (hβ∗1hβi) fs (x) . (106) ε 1 = 1 x v 2 . (104) ≃ −16π M α hα − i 2,3 i | 1| = When the mass difference between the right handed neutrinos is very large compared to the width, M M 1 Ŵ , the It was later pointed out in Flanz et al. [117] and Flanz et al. [118] 1 − 2 ≫ 2 N1,2 CP asymmetries coming from vertex and self-energy diagrams and confirmed rigorously in Pilaftsis [119], Pilaftsis and Resonant are comparable. However, when two right handed neutrinos are [120], Roulet et al. [121], Buchmuller and Plumacher [122], nearly degenerate, such that their mass difference is comparable Flanz and Paschos [123], Hambye et al. [124] and Pilaftsis and to their width, then CP violation contribution coming from the Underwood [125], that there is another source of CP violation self-energy diagram becomes very large (orders of magnitude coming from interference of tree level diagram with one-loop larger than the CP asymmetry generated by the vertex type self-energy diagram shown in Figure 9. This CP violation is diagram). This is often referred to as the resonance effect. similar to the CP violation due to the box diagram, entering the To ensure that the lightest right handed neutrino decays out- mass matrix in K K mixing in K-physics. If the heavy neutrinos − of-equilibrium so that an asymmetry is generated, the out-of- decay in equilibrium, the CP asymmetry coming from the self- equilibrium condition given by energy diagram due to one of the heavy neutrinos may cancel with the asymmetry from the decay of another heavy neutrino 2 hα1 T to preserve unitarity. However, in out-of-equilibrium decay of M1 < 1.66√g at T M1. (107) heavy neutrinos the number densities of the two heavy neutrinos 16π ∗ mPl = differ during their decay and consequently, this cancellation is must be satisfied, where g correspond to the effective number of no longer present. This can be understood as the right handed relativistic degrees of freedom.∗ This gives a lower bound m > neutrinos oscillating into antineutrinos of different generations, N1 108 GeV [127]. Though this gives us a rough estimate, in an which under the condition Ŵ[particle antiparticle] → = actual calculation of the asymmetry one solves the Boltzmann Ŵ[antiparticle particle], can create an asymmetry in right → equation, which takes into account both lepton number violating handed neutrinos before they decay. An elementary discussion as well as lepton number conserving processes mediated by heavy regarding how the CP violation enters in Majorana mass matrix, neutrinos. The Boltzmann equation governing lepton number which then generates a lepton asymmetry can be found in asymmetry n n n c , is given by Sarkar [111] and Langacker et al. [126]. The basic idea is to L ≡ l − l treat the particles and the antiparticles independently. The CP c dnL eq eigenstates Ni and N are no longer physical eigenstates, 3HnL (εv εs)Ŵψ (nψ n ) | | i 1 1 ψ1 which evolves with time. Consequently, the physical states, which dt + = + − n eq c L n Ŵ 2n n σ v , (108) are admixtures of Ni and Ni , can decay into both leptons ψ1 ψ1 γ L | | −nγ − | | and antileptons, giving rise to a CP violation. The CP asymmetry parameter coming from the interference of tree level and one- eq where Ŵψ is the decay rate of the physical state ψ1 , n is the loop self-energy diagram is given by 1 | ψ1 equilibrium number density of ψ1 given by

1 † sg T m c eq − ψ1 Ŵ(N lφ N l φ) 3/2∗ ≫ ε n mψ mψ (109) s → † − → c ψ1 s 1 1 ≡ Ŵ(N lφ N l φ) = exp T mψ1 , → + → g∗ T − T ≪ Im α(h∗ hαi)β (h∗ hβi) 2 1 α1 β1 Mi where s is the entropy density. The first term on the right 2 fs 2 ,(105) = 8π α hα M i 2,3 1 1 hand side of Equation (108) corresponds to the CP violating = | |

Frontiers in Physics | www.frontiersin.org 18 March 2018 | Volume 6 | Article 19 Hati et al. Neutrino Masses and Leptogenesis in LRSM contribution to the asymmetry and is the only term that generates leptogenesis are given by asymmetry when ψ decays out-of-equilibrium, while the second 1 a † L f ξψ f ξ ++l l ξ φφ. (112) term corresponds to inverse decay of ψ1, and the last term int = ij Li = ij a i j + a a corresponds to 2 2 lepton number violating scattering process ↔ From these interactions we have the decay modes of the triplet such as l φ† lc φ, with σ v being the thermally + ↔ + | | Higgs averaged cross section. The number density of ψ1 is governed by the Boltzmann equation li+lj+ ξa++ (113) → φ+φ+, dnψ1 eq 3Hnψ Ŵψ (nψ n ). (110) dt + 1 = − 1 1 − ψ1 The CP violation is obtained through the interference between the tree level and one-loop self-energy diagrams shown in One often defines a parameter K Ŵψ1 (T mψ1 )/H(T = 1/2= 2 = Figure 10. There are no one-loop vertex diagrams in this case. mψ ), where the Hubble rate H 1.66g (T /MPl), which 1 = ∗ gives a measure of the deviation from thermal equilibrium. For One needs at least two ξ’s. To see how this works, we will follow K 1 one can find an approximate solution for Equation (108) the mass-matrix formalism [22], in which the diagonal tree-level ≪ given by mass matrix of ξa is modified in the presence of interactions to

s 1 † 2 1 † 2 nL (εv εs). (111) ξ M ab ξb ξa∗ M ab ξb∗, (114) = g + 2 + + 2 − ∗ The Yukawa couplings are constrained by the required amount where of primordial lepton asymmetry required to generate the 2 2 M1 iŴ11M1 iŴ12± correct baryon asymmetry of the universe, while the lightest M − 2 − , (115) ± = iŴ21± M1 M2 iŴ22M2 right handed neutrino mass is constrained from the out-of- − − equilibrium condition. In the resonant leptogenesis scenario, with Ŵ+ Ŵ and Ŵ− Ŵ . From the absorptive part of the ab = ab ab = ab∗ the CP violation is largely enhanced, making the constrains on one-loop diagram for ξ ξ we obtain a → b Yukawa couplings relaxed. Consequently the scale of leptogenesis can be considerably lower, making it possible to realize a TeV 1 a b Ŵ M a∗ MaM f ∗f . (116) scale leptogenesis, which can be put to test at the LHC [128, 129]. ab b = 8π  b + b kl kl k,l   5.3. Leptogenesis With Triplet Higgs 2 Assuming Ŵa Ŵaa Ma, the eigenvalues of M are given by In section 2, we have discussed how small neutrino masses can be ≡ ≪ ± generated by adding triplet Higgs ξa to the SM [22, 26–28, 130– 1 2 2 132]. The interactions of these triplet Higgs that are relevant for λ1,2 (M M √S), (117) = 2 1 + 2 ±

FIGURE 9 | Tree level and one-loop self-energy diagrams contributing to the CP violation in models with right handed neutrinos.

FIGURE 10 | Tree level and one-loop self-energy diagrams contributing to the CP violation in a model with triplet Higgs.

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2 2 2 2 where S (M M ) 4 Ŵ M and M > M . The right-handed charged gauge boson W±. In this section, we will = 1 − 2 − | 12 2| 1 2 R physical states, which evolves with time, can be written as linear discuss the aspect that if WR± is detected at the LHC with a combinations of the CP eigenstates as mass of a few TeV then it can have profound implications for leptogenesis. If indeed WR± is detected at the LHC then that ψ+ a+ ξ1 b+ ξ2 , ψ− a− ξ ∗ b− ξ ∗ , (118) will give rise to an excess in the dilepton + dijet channel as 1,2 = 1,2 + 1,2 1,2 = 1,2 1 + 1,2 2 reported sometime back by the CMS collaboration. A signal of 2 2 2.8 σ level was reported in the mass bin 1.8 TeV < M < where a1± b2± 1/ 1 Ci± , b1± C1±/ 1 Ci± , lljj = = +| | = +| | 2.1TeV in the di-lepton + di-jet channel at the LHC by the 2 a± C±/ 1 C± with 2 = 2 +| i | CMS collaboration [136]. One of the popular interpretations of this signal was WR± decay in the framework of LRSM with 2iŴ M2 C C − 12∗ , gL gR via an embedding of LRSM in SO(10) [137, 138]. 1+ 2− 2 2 = = − = M M √S Another popular interpretation was for the case gL gR which 1 − 2 + = 2iŴ12M2 utilized the CP phases and non-degenerate mass spectrum of C1− C2+ − . (119) the heavy neutrinos [139]. Around the same time the ATLAS = − = M2 M2 √S 1 − 2 + collaboration had also reported a resonance signal decaying into a pair of SM gauge bosons. They found a local excess signal of The physical states ψ± evolve with time and decay into lepton 1,2 3.4σ (2.5σ global) in the WZ channel at around 2 TeV [140]. This and antilepton pairs. Assuming (M2 M2)2 4 Ŵ M 2, the 1 − 2 ≫ | 12 2| signal was shown to be explained by a W in LRSM framework CP asymmetry is given by Ma [22] R for a coupling g 0.4 in Brehmer et al. [141]. Some other R ∼ 1 Mi notable work along this direction can be found in Dobrescu ε Im f 1f 2∗ . (120) i 2 2 2 2 1 2∗ kl kl and Liu Bhupal [142], Dev and Mohapatra [143] and Das et al. ≃ 8π (M1 M2) Ŵi − k,l [144]. However, these interesting signals were either washed out by more accumulated data or reduced significantly below their For M > M , when the temperature drops below M , ψ decays 1 2 1 1 initial reported levels. Nevertheless such signals have intrigued away to create a lepton asymmetry. However, this asymmetry several studies concerning the impact of a TeV scale W on is washed out by lepton number violating interactions of ψ ; R± 2 leptogenesis. and the subsequent decay of ψ at a temperature below M 2 2 As discussed earlier, the Higgs sector in one of the popular sustains. The generated lepton asymmetry then gets converted to versions of LRSM consists of one bidoublet Higgs and the baryon asymmetry in the presence of the sphaleron induced two triplet complex scalar fields . The relevant gauge anomalous B L violating processes before the electroweak phase L,R + transformations are as follows transition. The approximate final baryon asymmetry is given by

n ε (2, 2, 0, 1), L (3, 1, 2, 1), R (1, 3, 2, 1). (122) B 2 , (121) ∼ ∼ ∼ s ∼ 3g K(ln K)0.6 ∗ Here one breaks the left–right symmetry in a spontaneous where K Ŵ (T M )/H(T M ) is the parameter manner to reproduce the Standard Model. On the other hand ≡ 2 = 2 = 2 measuring the deviation from thermal equilibrium when, H the smallness of the neutrino masses is realized using the seesaw 1/2 2 = 1.66g (T /MPl) is the Hubble rate, and g∗ corresponds to the mechanism [14–20]. ∗ number of relativistic degrees of freedom. In another variant of LRSM one has only the doublet Higgs In a more rigorous estimation of the baryon asymmetry, which are employed to break all the relevant symmetries. Here in addition to the decays and the inverse decays of triplet the Higgs sector consists of doublet scalars with the gauge scalars, one needs to incorporate the gauge scatterings ψψ transformations ↔ FF, φφ, GG (F corresponds to SM fermions and G corresponds to gauge bosons) and L 2 scattering processes ll φ φ (2, 2, 0, 1), H (2, 1, 1, 1), H (1, 2, 1, 1), (123) = ↔ ∗ ∗ L R and lφ lφ into the Boltzmann equation analysis of the ∼ ∼ ∼ ↔ ∗ asymmetry. Including these washout processes, one finds a lower In addition there is one fermion gauge singlet SR (1, 1, 0, 1). 11 ∼ limit on Mξ , Mξ & 10 GeV [133]. For a quasi-degenerate The Higgs doublet HR acquires a VEV to break the left–right spectrum of scalar triplets the resonance effect can enhance the symmetry which results in the mixing of S with right-handed CP asymmetry by a large amount and a successful leptogenesis neutrinos. This gives rise to a light Majorana neutrino and a scenario can be attained for a much smaller value of triplet scalar heavy pseudo-Dirac neutrino or alternatively a pair of Majorana mass. In Strumia [134] and Aristizabal Sierra et al. [135], an neutrinos. absolute bound of Mξ & 1.6 TeV is obtained for a successful Historically, in LRSM, the left–right symmetry was broken at 10 resonant leptogenesis scenario with triplet Higgs. a fairly high scale, MR > 10 GeV. This serves two purposes– firstly, the requirement of gauge coupling unification implies 6. LEPTOGENESIS IN LRSM this scale to be high, and secondly, thermal leptogenesis in this scenario gives a comparable bound. To get around this In Left–Right Symmetric Model (LRSM) [17, 19, 44–49] the left– problem one often introduces a parity odd scalar which is right parity symmetry breaking implies the existence of a heavy then given a large VEV. This is often called D-parity breaking.

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Consequently, one can have g g even before the left– violating processes at the colliders were studied in the context of L = R right symmetry breaking. This in turn allows the possibility of high-scale thermal leptogenesis. In Flanz et al. [117, 118], Pilaftsis a gauge coupling unification with TeV scale WR. This is true [119],Rouletetal.[121], Buchmuller and Plumacher [122], Flanz for both triplet and doublet models of LRSM. Embedding the and Paschos [123], Hambye et al. [124], Pilaftsis and Underwood LRSM in an SO(10) GUT framework, the violation of D-parity [125] resonant leptogenesis has been discussed in the context of a [54] at a very high scale helps in explaining the CMS TeV considerably low mass WR. In Frere et al. [148] it was pointed scale W signal for g 0.6g as shown in Deppisch et al. out that one requires an absolute lower bound of 18 TeV on R R ≈ L [137, 138]. the WR mass in order to have successful low-scale leptogenesis ± with a quasi-degenerate right-handed neutrinos. Recently, it was 6.1. Can a TeV-Scale WR at the LHC Falsify found that just the correct lepton asymmetry can be obtained Leptogenesis? by utilizing relatively large Yukawa couplings, for WR mass scale For a TeV scale WR±, all leptogenesis scenarios may be broadly higher than 13.1TeVin Bhupal Dev et al. [149, 150]. Note that in classified into two groups: Frere et al. [148] and Bhupal Dev et al. [150], the lepton number violating gauge scattering processes such as N e u d , At a very high scale a leptonic asymmetry is generated. It R R → R R • NRuR eRdR, NRdR eRuR and NRNR eReR have been can be either in the context of LRSM with D-parity breaking → → → analyzed in detail. However, lepton number violating scattering or through some other interactions (both thermal and non- processes with external WR were ignored because for a heavy thermal). m /m W there is a relative suppression of e WR NR in comparison At the TeV scale a lepton asymmetry is generated with R − • to the processes where there is no WR in the external legs. resonant enhancement, when the left–right symmetry Now, if indeed the mass of W is around a few TeV, as was breaking phase transition is taking place. R suggested by an excess signal reported by the CMS experiment The following discussions hold for the LRSM variants with a then one has to take the latter processes seriously. In Dhuria et Higgs sector consisting of triplet Higgs as well as a Higgs sector al. [151], it was pointed out that the lepton number violating with exclusively doublet Higgs. We will often refer to these washout processes (e±e± W±W± and e±W∓ e∓W±) can R R → R R R R → R R two broad classes of the LRSM mentioned above to discuss the be mediated via the doubly charged Higgs in the conventional lepton number violating washout processes and point out how LRSM. In Bhupal Dev et al. [149] it was shown that in a parity- all these possible scenarios of leptogenesis are falsifiable for a WR asymmetric type-I seesaw model with relatively small MNR one of TeV scale. In the case where high-scale leptogenesis happens obtains a small contribution from this process which is expected 9 at T > 10 GeV, the low energy B L breaking gives rise for a large MW /MN . However, in this scenario some other − R R to gauge interactions which depletes all the baryon asymmetry relevant gauge scattering processes are efficient in washing out very rapidly before the electroweak phase transition is over. the lepton asymmetry. Including these washout processes one Now, these same lepton number violating gauge interactions will obtains a lower bound of 13.1 TeV on the WR mass [149]. Here significantly slow down the generation of the lepton asymmetry we will mainly discuss R++ and NR mediated lepton number for resonant leptogenesis at around TeV scale. Consequently, it violating scattering processes in a much more general context to is not possible to generate the required baryon asymmetry of the establish their importance as washout processes which can falsify universe for TeV scale WR± in this case. the possibility of leptogenesis depending on WR mass [152]. One For the case M M M M , severe of the vertices in the ++ mediated process is gauge vertex while N3R ≫ N2R ≫ N1R = NR R constraints on the WR± mass for a successful scenario of high- the other one is a Yukawa vertex. On the other hand for NR scale leptogenesis come from the SU(2)R gauge interactions as mediated lepton number violating scattering processes both the pointed out in Ma [145]. To have successful leptogenesis in the vertices are gauge vertices. Consequently, these lepton number case MNR > MWR , the condition that the gauge scattering process violating scattering processes are very rapid as compared to the e− W+ N e+ W− goes out-of-equilibrium yields scattering processes involving only Yukawa vertices. It turns R + R → R → R + R out that NR and R++ mediated scattering process eR±WR∓ 16 → MNR & 10 GeV (124) eR∓WR± does not go out of equilibrium till the electroweak phase transition if the mass of WR is around TeV scale. Consequently, with mWR /mNR & 0.1. For the scenario where MWR > MNR these lepton number violating scattering processes continue to 8 leptogenesis happens either at T > MWR after the breaking of wash out or slow down the generation of lepton asymmetry . B L gauge symmetry or at T M , the out-of equilibrium In the scenario of LRSM involving only doublet Higgs in the − ≃ NR condition for the scattering process e±e± W±W± through Higgs sector the doubly charged Higgs is absent. Nevertheless, R R → R R NR exchange leads to the constraint the NR mediated lepton number violating scattering processes will be present and will wash out the lepton asymmetry in such a M & 3 106 GeV(M /102 GeV)2/3. (125) scenario. WR × NR

Thus, a WR with mass in the TeV range (in the case of a 8In passing we would like to note that the other relevant lepton number violating hierarchical neutrino mass spectrum) rules out the high-scale scattering process is doubly phase space suppressed for a temperature below the leptogenesis scenario. In Deppisch et al. [146, 147], neutrinoless WR mass scale. Consequently, we will neglect such a process for leptogenesis double beta decay and the observation of the lepton number occurring at T . MWR .

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In LRSM the right handed leptonic charged current (s u)2 8M4 interaction is given by eRWR (s, t, u) 4g4( t) + + WR M 2 eRWR R  2 NR ++ = − R  t M2 1  − R L N gRJRWR− h.c. (126) = 2√2 + (s u) 2 s u + 2 MNR 2 2 +t M s M + u M where JR eRγ (1 γ5) NR. The relevant interactions of the − R − NR − NR = + right-handed Higgs triplet are given by 4M4 WR 2 1 1 2 MNR 2 2 , (133) † +t M s M + u M R NR NR L D D , (127) − − − R ⊃ R R R R where we neglect any mixing between WL and WR. On the right- 0 where ++, +, . The covariant derivative is given hand side of Equation (133), the first term corresponds to the R = R R R j j j Higgs exchange. The last two terms are due to the interference by DR ∂ igR TRAR ig′B, where AR and B are gauge = − − between Higgs and NR exchange. The Mandelstam variables s fields corresponding to SU(2)R and U(1)B L gauge groups with 2 2 2 = the associated gauge couplings given by g− and g , respectively. p k , t p p′ and u p k′ are related by the R ′ + = θ − = − When the neutral Higgs field 0 acquires a VEV 0 1 v scattering angle as follows R R = √2 R SU(2) , the interaction between the gauge boson W and the R R st 1 2 doubly charged Higgs is given by Doi [153] s M2 (1 cos θ) . (134) su M4 = −2 − WR ∓ − WR v L R 2 R gRW− W− ++ h.c. (128) The differential scattering cross section for the process ⊃ −√2 R R R + e±(p)e±(p ) W±(k)W±(k ) is given by Doi [153] R R ′ → R R ′ The Yukawa interaction between the lepton doublet ψeR T = eReR (N , e ) and the components of triplet Higgs are given by dσW W 1 R R R R R eReR (s, t, u), (135) dt = 512πM4 s2 WRWR WR L hR (ψ )c iτ τ. ψ h.c., (129) Y = ee eR 2 R eR + where where τ’s are the Pauli matrices. After the Higgs triplet field eReR (s, t, u) eReR (s, t, u) eReR (s, t, u) . acquires a VEV, the relevant Yukawa coupling can be written as WRWR WRWR WRWR = NR + ++ R MNR R h where MN is the Majorana mass of NR. ee = 2vR R (136) The relevant Feynman diagrams for the lepton number eReR The expressions for W W (s, t, u) can be obtained after violating processes induced by these interactions are depicted in R R interchanging s t in eRWR (s, t, u): eReR (t, s, u)= Figure 11. ↔ eRWR WRWR eRWR (s, t, u). The Mandelstem variables t p k 2 and Using the interactions given in Equations (126)–(129), one can − eRWR = − estimate the differential scattering cross section for the process u p k 2 can be written in terms of s p p 2 and = − ′ = + ′ e∓(p)W±(k) e±(p′)W∓(k′) to obtain [153] scattering angle θ as follows R R → R R eRWR dσe W 1 e W 2 2 2 R R R R (s, t, u), (130) t s 2M 2M dt = 2 eRWR 1 WR 1 1 WR cos θ . 384πM4 s M2 u  2  WR WR = −2 − s ∓ − s 2MW −  − R  where (137)   eRWR (s, t, u) eRWR (s, t, u) eRWR (s, t, u) (131) eRWR eRWR eRWR = NR + R++ 6.1.1. Wash Out of Lepton Asymmetry for T > M WR and During the period when the temperature is such that vR > T > MWR , the lepton number violating washout processes are very 2 s u rapid in the absence any suppression. To have a quantitative eRWR (s, t, u) g4 t M eRWR R NR 2 2 estimate of the strength of these scattering processes in NR = − s M + u M  − NR − NR depleting the lepton asymmetry one can estimate the parameter 2 defined as  2 4 2 MNR 4MW su MW (s u) n σ v − R − R − 2 2 K | |, (138) s M u M ≡ H − NR − NR 2 2 for both the processes during v > T > M , where n m M R WR 4 NR NR corresponds to the number density of relativistic species and 4MW t , (132) − R  2 + 2  3ζ (3) s M u M is given by n 2 T3. H corresponds to the Hubble − NR − NR  = × 4π2    Frontiers in Physics | www.frontiersin.org 22 March 2018 | Volume 6 | Article 19 Hati et al. Neutrino Masses and Leptogenesis in LRSM

FIGURE 11 | Feynman diagrams for e−W+ e+W− scattering mediated by N and ++ fields. The Feynman diagrams for e−e− W−W− can be obtained by R R → R R R R R R → R R appropriately changing the direction of the external legs.

FIGURE 12 | Plot showing the behavior of K as a function of temperature T FIGURE 13 | Plot showing the dependance of out of equilibrium temperature for the processes e±W∓ e∓W± and e±e± W±W± (including both R R → R R R R → R R and N mediated diagrams) for υ > T > M . The right handed (T) on MW for the process e±W∓ e∓W± (mediated via ++ and NR R++ R R WR R R R → R R R M M charged gauge boson mass is taken to be M 3.5 TeV. fields) for NR WR . The three different lines correspond to three different WR = ∼ values of MR .

1/2 2 rate H 1.7g T /MPl, where g 100 corresponds to ≃ ∗ ∗ ∼ the relativistic degrees of freedom. The thermally averaged cross section is denoted by σ υ . To choose a rough estimate of vR, function of temperature in Figure 12. The plot corresponds to a | | let us compare the situation with the Standard Model, where we temperature range 3MWR > T > MWR and right handed charged vL have φ where vL 246 GeV, and MWL 80 GeV. Now gauge boson mass M 3.5 TeV. = √2 = ∼ WR = in case of LRSM 0 υR breaks the left–right symmetry and In Figure 12, the large values of K for both the processes R = √2 φ 0 indicates that high wash out efficiency of these scattering R MWR gRυR. Taking gR gL, we have M M 3. processes for T & M . For the LRSM variant with its Higgs = ∼ WL = WR ≈ WR Making use of the differential scattering cross sections in sector consisting of only doublet Higgs the doubly charged Equations (130) and (135), we plot the behavior of K as a Higgs mediated channels are absent for these processes and the

Frontiers in Physics | www.frontiersin.org 23 March 2018 | Volume 6 | Article 19 Hati et al. Neutrino Masses and Leptogenesis in LRSM right handed neutrino will mediate these processes, which will B L charge, we have introduced scalar triplets with small − washout the lepton asymmetry for T & MWR . induced VEVs such that they give Majorana masses to light as well as heavy neutrinos. We have also discussed how the Majorana nature of these neutrinos leads to 0νββ decay. < 6.1.2. Wash Out of Asymmetry for T MWR Interestingly, the right-handed currents play an important role During the period when the temperature is such that T < MWR , in discriminating between the mass hierarchy as well as the the process eR±WR∓ eR∓WR± is of more importance. Let us absolute scale of light neutrinos. To summarize the situation for → > now estimate a lower bound on T below T M till which leptogenesis, in the high-scale leptogenesis scenario (T MW ), = WR R this process stays in equilibrium and continues to deplete lepton in all the variants of LRSM the lepton number violating processe∼ s asymmetry. The scattering rate can be written as9 Ŵ n σ v . e±e± W±W± and e±W∓ e∓W± are highly efficient = rel R R → R R R R → R R For T < MWR the Boltzmann suppression of the scattering rate in washing out the lepton asymmetry. In the case of resonant TM 3/2 M leptogenesis scenario at around TeV scale we found that the stems from the number density n g WR exp WR . = 2π − T latter process stays in equilibrium until the electroweak phase Now, the scattering process stays in thermal equilibrium when transition, making the generation of lepton asymmetry for T < the condition Ŵ > H is satisfied. MWR significantly weaker. Thus, if the LHC discovers a TeV In Figure 13 we show the temperature until which the scale WR± then one needs to look for some post-electroweak scattering process eR±WR∓ eR∓WR± stays in equilibrium as a phase transition mechanism to explain the baryon asymmetry → < function of MWR for three values of MR and taking MNR MWR of the Universe. To this end the observation of the neutron- ∼ and vrel 1. We have taken the lowest value of MR to be 500 antineutron oscillation [155, 156] or (B L) violating proton = − GeV to be consistent with the recent collider limits on the doubly decay [157] will play a guiding role in confirming such scenarios. charged Higgs mass [154]. The plot shows that unless MWR is Complementing these results, the low-energy subgroups of the significantly heavier than a few TeV, the process e±W∓ R R → superstring motivated E6 model have also been explored which eR∓WR± will continue to be in equilibrium till a temperature can also give rise to left–right symmetric gauge structures but similar to the electroweak phase transition. Consequently, this with a number of additional exotic particles as compared to process will continue to washout or slow down the generation the conventional LRSM. Interestingly, one of the low-energy of lepton asymmetry until the electroweak phase transition. supersymmetric subgroups of E6, also known as the Alternative In the LRSM variant with doublet Higgs, the heavy neutrinos Left–Right Symmetric Model, gives a model alternative to will mediated lepton number violating scattering processes will successfully realize high-scale leptogenesis in the absence of washout or slow down the generation of lepton asymmetry until the dangerous gauge washout processes [158]. The vector-like the electroweak phase transition. Thus, the lower limit on the fermions added to the minimal framework of LRSM to realize WR mass for a successful leptogenesis scenario is significantly a universal seesaw can pave new ways to realize baryogenesis as higher a few TeV. This was also confirmed by explicitly discussed in Deppisch et al. [73]. solving the relevant Boltzmann equations in Bhupal Dev et al. [149, 150]. AUTHOR CONTRIBUTIONS

All authors listed have made a substantial, direct and intellectual 7. CONCLUDING REMARKS contribution to the work, and approved it for publication. We have reviewed the standard left–right symmetric theories and the implementation of different types of low scale seesaw ACKNOWLEDGMENTS mechanisms in the context of neutrino masses. We have also discussed a left–right symmetric model with additional vector- The authors would like to thank Frank F. Deppisch for many like fermions in order to simultaneously explain the charged helpful discussions and encouragement. The authors are also fermion and Majorana neutrino masses. In this model the thankful to both the reviewers for many valuable suggestions quark and charged lepton masses and mixings are realized via and CH and US are thankful to Mansi Dhuria and Raghavan a universal seesaw mechanism while spontaneous symmetry Rangarajan for several wonderful collaborations which have breaking is achieved with two doublet Higgs fields with non-zero resulted in some of the interesting findings covered in this review. The work of US is supported partly by the JC Bose National 9We ignore any finite temperature effects to simplify the analysis. Fellowship grant under DST, India.

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156. Phillips DG II, Snow WM, Babu K, Banerjee S, Baxter DV, Berezhiani Z, Conflict of Interest Statement: The authors declare that the research was et al. Neutron-antineutron oscillations: theoretical status and experimental conducted in the absence of any commercial or financial relationships that could prospects. Phys Rept. (2016) 612:1–45. doi: 10.1016/j.physrep.2015. be construed as a potential conflict of interest. 11.001 157. Pati JC, Salam A, Sarkar U. Delta B = - Delta L, neutron —> e- pi+, e- K+, Copyright © 2018 Hati, Patra, Pritimita and Sarkar. This is an open-access article mu- pi+ and mu- K+ DECAY modes in SU(2)-L X SU(2)-R X SU(4)-C or distributed under the terms of the Creative Commons Attribution License (CC SO(10). Phys Lett B (1983) 133:330–6. BY). The use, distribution or reproduction in other forums is permitted, provided 158. Dhuria M, Hati C, Rangarajan R, Sarkar U. Explaining the CMS the original author(s) and the copyright owner are credited and that the original eejj and e missing pT jj excess and leptogenesis in superstring publication in this journal is cited, in accordance with accepted academic practice. inspired E6 models. Phys Rev D (2015) 91:055010. arXiv:1501. No use, distribution or reproduction is permitted which does not comply with these 04815. terms.

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