PHYSICAL REVIEW D 100, 115031 (2019)

Disentangling the in the minimal left-right symmetric model

Goran Senjanović and Vladimir Tello International Centre for Theoretical Physics, Trieste, Italy

(Received 14 January 2019; published 18 December 2019)

In a recent paper we presented a systematic way of testing the seesaw origin of mass in the context of the minimal left-right symmetric model. The essence of the program is to exploit lepton number violating decays of doubly charged scalars, particles which lie at the heart of the Higgs-mechanism-based seesaw, to probe the Dirac neutrino mass term which in turn enters directly into a number of physical processes including the decays of right-handed into the W boson and left-handed charged leptons. In this longer version we discuss at length these and related processes, and we offer some missing technical details. We also carefully analyze the physically appealing possibility of a conserving Yukawa sector showing that the neutrino Dirac mass matrix can be analytically expressed as a function of light and heavy neutrino masses and mixing, without resorting to any additional discrete symmetries, a context in which the seesaw mechanism can be disentangled completely. When parity does get broken, we show that, in the general case, only the Hermitian part of the Dirac mass term is independent which substantially simplifies the task of testing experimentally the origin of neutrino mass. We illustrate this program through some physical examples that allow simple analytical expressions. Our work shows that the minimal left-right symmetric model is a self-contained theory of neutrino mass which can be in principle tested at the LHC or the next hadron collider.

DOI: 10.1103/PhysRevD.100.115031

I. INTRODUCTION currently being probed in low energy experiments, and MN, which could hopefully be determined at the LHC or a future Understanding the origin of neutrino mass is a central hadronic collider. Hereafter, this is what we will mean by task of the physics beyond the Standard Model. After all, a disentangling the seesaw, and though in general it is not nonvanishing neutrino mass is the only true failure of the always guaranteed and should not be taken for granted SM and thus provides a fundamental window into the new a priori, an effective disentangling would be analogous physics. Over the years, the seesaw mechanism emerged to the situation of charged fermions in which one can as the main scenario behind the smallness of neutrino determine their Yukawa couplings from the knowledge of mass [1–3]. By adding new neutral lepton singlets N their masses. (one per generation) to the SM, and allowing for their However, the situation arising from (1) is given by gauge invariant masses M , one gets for the light neutrino N (for an equivalent parametrization see [4]) mass matrix pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi ¼ ð Þ 1 MD i MNO Mν 2 T Mν ¼ −M M ð1Þ D M D N where O is an arbitrary complex orthogonal matrix: where M is the Dirac mass term between ν and N. The D OOT ¼ 1: ð3Þ above formula (1) is valid as long as MN ≫ MD, a natural assumption for the gauge singlets N. In order to probe the The arbitrariness of M in (2) provides a blow to the seesaw mechanism, ideally one would have to find M as a D D program of probing the seesaw origin of the neutrino mass. function of neutrino masses and mixing, namely of Mν, Ironically, the same arbitrariness is often used to make MD artificially large so that N could be produced at the colliders (when MD vanishes N are decoupled) but that is against Published by the American Physical Society under the terms of the spirit of the seesaw as a way of obtaining small neutrino the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to mass naturally. the author(s) and the published article’s title, journal citation, A clarification is called for at this point. When we speak and DOI. Funded by SCOAP3. here of the origin of neutrino mass, we mean the physical

2470-0010=2019=100(11)=115031(13) 115031-1 Published by the American Physical Society GORAN SENJANOVIĆ and VLADIMIR TELLO PHYS. REV. D 100, 115031 (2019)

Higgs mechanism origin, as in the case of charged fermions disentangle the seesaw [8]. This in turn allows us, for þ and gauge bosons. We do not have in mind the valid example, to predict the decay rates of N → W eL and question of the values of these parameters. Before asking N → hν, and thus probe the Higgs origin of neutrino mass. why the masses are what they are, we should first know The case of P is however highly nontrivial since the where they come from. For example, a program aiming to originally Hermitian Dirac mass matrices lose this essential solve for the mass hierarchies prior to the Higgs mechanism property after the symmetry breaking due to the emergence would have been obviously doomed from its very begin- of complex vacuum expectation values. Instead, we have ning. In that same spirit, we wish to have a theory that tells recently suggested an alternative approach of utilizing the us in a verifiable and predictive manner whether neutrinos decays of doubly charged scalars and heavy SM doublet owe their masses to the Higgs mechanism and in which Higgs to probe MD [9]. We have also shown how to manner. Only then can one finally address the issue of the determine MD in the Hermitian case (unbroken parity in the hierarchies of masses and mixings that we are all after. Dirac Yukawa sector), which in a simple case of the so- In the case of charged fermions the Higgs origin of called type I seesaw and same left and right leptonic mixing their masses implies the knowledge of Yukawa couplings matrices takes the following unique form [9]: ∝ from the values of masses yf mf=MW, and in turn allows pffiffiffiffiffiffiffiffiffiffiffiffi ¼ † ð Þ us to predict the Higgs decay rates into fermion-antifermion MD iVL mνmNVL: 6 pairs: This expression manifestly demonstrates the predictivity ¯ 2 Γðh → ffÞ ∝ mhðmf=MWÞ : ð4Þ of the theory—all ambiguities are gone from the Dirac neutrino mass matrix. The Hermitian case may not be of Understanding the origin of neutrino mass is comparable to pure academic interest only; for light WR it may be a must finding a theory that does for neutrinos what the SM does due to the constraint of strong CP violation as we discuss in for charged fermions, and in this sense the seesaw scenario the following section, below (21). by itself comes short. In this sequel of our paper [9], we describe at length how This should not come as a surprise. After all, the seesaw MD can be computed in the case of being Hermitian, and mechanism is an ad hoc extension of the SM and makes no we provide some simple appealing examples that can lead attempt for a dynamical explanation of the Vector-Axial to transparent analytic expressions. We also elaborate on theory of weak interactions. This is to be contrasted with its the phenomenological aspects of new particle decays and left-right (LR) symmetric extension [5,6] that attributes the on the subsequent determination of MD. The bottom line of left-handed nature of weak interactions to the spontaneous our work is that, independently of whether P is broken or breakdown of parity. The smoking gun signature of left- not in the Yukawa sector, the MLRSM is a self-contained right symmetry is the existence of RH neutrinos νR, leading theory of neutrino mass that allows for a direct probe of its to nonvanishing neutrino mass. Higgs origin. In the minimal version of the theory, coined the minimal We should stress an important, essential aspect of our left-right symmetric model (MLRSM) and based on extra work. We make no assumption whatsoever regarding the Higgs scalars that are left-handed (LH) and right-handed breaking scale of the MLRSM, or equivalently the masses (RH) triplets [1,2,7], the seesaw mechanism follows nat- of N’s. Needless to say, were they to be accessible at the urally from the spontaneous symmetry breaking with LHC or the next hadron collider, this would allow us in principle to verify this program, but our work is more ∝ ð Þ MN MWR 5 general than this scenario since it applies to any scale or any other imaginable way of knowing the masses and mixings ’ where MWR is the mass of the right-handed charged gauge of N s. boson. This offers a profound connection between the The rest of the paper is organized as follows. In the smallness of neutrino mass and the near maximality of following section we give the main features of the parity violation in weak interactions [2]. MLRSM, those which play an essential role in arriving In the MLRSM the left-right symmetry is broken at our main results, and set our formalism and notations. spontaneously and can be either a generalized parity P Its main purpose is to ease the reader’s pain in following or generalized charge conjugation C. The impact of this the technical aspects in later sections. In Sec. III we discuss symmetry on the properties of quarks and leptons is of the lepton masses, both charged and neutral, with a focus fundamental importance. The case of C is easier to deal with on the seesaw. This section plays a central role in under- since it leads to symmetric Dirac mass matrices of quarks standing neutrino mass in the MLRSM and it could be and leptons, even after the spontaneous symmetry break- useful even to the experts in the field. It has two sub- ing. It implies same LH and RH mixing angles in the quark sections, the first being devoted to the parity conserving T ¼ sector, while in the lepton sector the condition MD MD Yukawa sector that leads to Hermitian Dirac mass matrices. leads to the determination of MD which allows us to In this case we managed to solve analytically for MD as a

115031-2 DISENTANGLING THE SEESAW MECHANISM IN THE MINIMAL … PHYS. REV. D 100, 115031 (2019)   ϕ0 function of MN and Mν. We give here a detailed expos´eof i Φ ¼½ϕ1;iσ2ϕ ; ϕ ¼ ;i¼ 1; 2: ð11Þ this important result already found in [9]. The general case 2 i ϕ− is treated next in the second subsection, where we show that i only the Hermitian part of MD is independent and prepare The most general VEV of Φ can be written as the setup for phenomenological implications. The phenom- enological analysis is left for Sec. IV, where we use a hΦi¼vdiagðcos β; − sin βe−iaÞ: ð12Þ number of new processes, in particular the lepton number violation (LNV) decays of doubly charged scalars, to Under parity one has as determine MD and thus demonstrate manifestly that the MLRSM is a self-contained theory of neutrino mass. Our Δ ↔ Δ ; Φ → Φ†: ð13Þ conclusions and the outlook for future research are offered L R finally in Sec. V. The leptonic Yukawa interaction is given by

II. MINIMAL LEFT-RIGHT SYMMETRIC MODEL LY ¼ −lLðY1Φ − Y2σ2Φ σ2ÞlR The MLRSM is based on the following symmetry group: 1 − ðlT σ Δ l þ lT σ Δ l Þþ ð Þ 2 LYLi 2 L L RYRi 2 R R H:c: 14 G ¼ ð2Þ ð2Þ ð1Þ ð Þ LR SU L × SU R × U B−L 7 so that because of (13) one has where on top of the LR symmetric gauge group, a discrete generalized parity P ensures a LR symmetric world prior to † Y1;2 ¼ Y1 2;YL ¼ YR: ð15Þ spontaneous symmetry breaking. ; — We focus on the leptonic sector only for the quark sector These relations are central to our discussion and to our main see [10,11]. Under (7) the leptonic fields transform as results.   ¼ ν¯T ν Introduce next NL C R, which from (14) leads lL;R ¼ ð8Þ immediately to the heavy neutrino mass matrix e L;R ¼ ð Þ MN vRYR: 16 and under P as The SM Higgs doublet h and the new heavy doublet H l ↔ l : ð9Þ L R are the linear combination of ϕi:

The new Higgs sector consists of left and right SUð2Þ −ia ia h ¼ cβϕ1 þ e sβϕ2;H¼ −e sβϕ1 þ cβϕ2; ð17Þ triplets ΔLð3; 1; 2Þ and ΔRð1; 3; 2Þ, respectively, where the quantum numbers denote the representation content where cβ ≡ cosβ, sβ ≡ sinβ hereafter. The new doublet H is Δ under (7). The RH triplet R is responsible for the breaking basically decoupled, i.e., out of the LHC reach, since it of G down to the SM gauge symmetry, and its non- LR leads directly at the tree level to the flavor violation in vanishing vacuum expectation value (VEV) v (notice that R the K- and B-meson sectors. This implies a lower limit it can be made real) gives masses to the new heavy gauge m ≳ 20 TeV [12,13], which is far above the LHC reach bosons W and Z and the RH neutrinos N. H R R (for a recent study regarding the future hadron collider, The field decomposition of the triplets has the following see [14]). form: It is useful to rewrite the Yukawa interaction of the  pffiffiffi  Φ δþ 2 δþþ bidoublet as the function of the physical fields h and H L;R= L;R Δ ¼ pffiffiffi : ð10Þ and the charged lepton and neutrino Dirac mass matrices: L;R δ0 −δþ 2 L;R L;R=   † −ia † M þ e s2βM L ¼ −l¯ MD − e D δ δ Φ L h H NR Besides the usual singly charged fields L ( R gets eaten v vc2β by the W fields), the doubly charged states δþþ play an   R L;R † þ ia ¯ Me MD e s2βMe important role in lepton number violating decays and þ lL iσ2h − iσ2H eR; ð18Þ v vc2β in determining MD. As in the original work [6],at ¼hδ0 i¼0 the first stage of symmetry breaking vL L , ¼hδ0 i ≠ 0 where Me and MD are the charged lepton and neutrino vR R . On top of the new Higgs multiplets, there must also exist Dirac mass matrices, respectively, and are given by a SUð2Þ × SUð2Þ bidoublet, containing the usual SM L R ¼ − ð þ −ia ÞðÞ Higgs field, Φð2; 2; 0Þ, with the decomposition MD v cβY1 e sβY2 19

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−ia Since parity is broken by the complex VEV hΦi,in Me ¼ vðe sβY1 þ cβY2Þ: ð20Þ general EL ≠ ER, and thus

The measure of spontaneous CP violation is provided by the † U ¼ E E ð24Þ small parameter sat2β which can be shown to satisfy [10] e R L

2mb provides a measure of parity breaking. s t2β ≲ : ð21Þ a m Equation (23) implies a redefinition of the Dirac neutrino t mass matrix It can be shown that the same parameter measures the → † ð Þ difference between the right- and left-handed quark mixing MD ERMDEL: 25 matrix and thus controls the weak contribution to the strong ¯ CP violating parameter θ. For light WR one has that sat2β is From (19) and (20) it follows that practically vanishing [15] in order to keep θ¯ acceptably − † ¼ ð ia þ ÞðÞ small. The point is that with the spontaneously broken parity MD UeMDUe isat2β e tβMD me 26 the strong CP parameter θ¯ is finite and calculable in − ¼ − ð þ −ia Þ ð Þ perturbation theory [16]. me UemeUe isat2β MD e tβme : 27 Once hΦi is turned on, the left-handed triplet ΔL gets a 2 small induced VEV vL ∝ v =vR providing a hierarchy of From the above equations it is clear that Ue is actually a ð2Þ function of M and vice versa as discussed below. SU L breaking [7]. It is important to stress that vL is D naturally small, since it is protected by a symmetry [7] (for From the Yukawa interaction and (15), one gets for the ðν Þ a recent discussion, see [17]). The small vL is thus a direct L;NL mass matrix source of neutrino mass, the so-called type II seesaw. It is ! vL T T interesting to contrast this situation with the usual type II v Ue MNUe MD ð2Þ R : ð28Þ SM scenario where one adds ad hoc a SU triplet in order M M to give the neutrino a nonvanishing mass [18]. The latter D N case is an example of a posteriori model building, while in This is a combination of both type II and type I seesaw the MLRSM this is a result of the underlying structure of matrices, with the important proviso of U entering in the the theory. Just as the type I seesaw emerges naturally in e direct type II mass term. In other words, in the MLRSM not this theory, since RH neutrinos are a must, the same only M enters the neutrino mass, but also U ; this point is mechanism that gives them Majorana masses leads auto- D e typically missed in the literature. matically to the direct type II contribution to light neutrino Under the usual seesaw assumption M ≫ M the masses. N D matrix (20) can be readily block-diagonalized, through the ν − N mixing, to the leading order in MD=MN given by III. LEPTON MASSES      ν 1 Θ† ν This is the central section of our work. We go here from → ð29Þ −Θ 1 the weak to the mass basis, which requires some care due N L N L to the common source of charged lepton and neutrino masses in the MLRSM. with The charged lepton mass matrix can be diagonalized by 1 performing unitary transformations EL and ER on the LH Θ ¼ MD: ð30Þ and RH charged lepton fields, respectively: MN ¼ † ð Þ Me ELmeER: 22 This in turn leads to the celebrated seesaw expression for the neutrino mass More precisely, one rotates the LH and RH doublets, v 1 ¼ L T − T ð Þ l → l l → l ð Þ Mν Ue MNUe MD MD: 31 L EL L; R ER R; 23 vR MN so that at this point the gauge interactions of charged gauge Thus, at the leading level Mν and MN stand for the bosons remain still diagonal. The leptonic mixings, i.e., the approximate (Majorana) mass matrices for the light and Pontecorvo-Maki-Nakagawa-Sakata (PMNS) matrix VL heavy neutrinos, respectively. The neutrino mass matrix is and its right-handed analog VR, will then be simply the given as a function of MD and MN. Together with (26) unitary transformations that diagonalize LH and RH and (27), the above equation serves to compute MD and neutrino mass matrices, respectively. thus to disentangle the seesaw.

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The next step is then to diagonalize these matrices by It is noteworthy that the Hermitian limit sat2β ≃ 0 is not the unitary rotations VL and VR respectively. From (23) one smooth and it has to be dealt with carefully, as we discuss gets immediately in the next section. This case exemplifies the fact that the theory predicts MD, and it does it even when the con- ¼ † ð Þ tribution of M to neutrino masses is small. Mν VLmνVL 32 D A. The tale of unbroken parity where mν stands for the diagonal matrix of the light neutrino masses and VL is the standard PMNS mixing What would happen if parity was not broken by the VEV matrix, which amounts to of Φ, i.e., what if the small parameter sat2β was to be negligible? Could MD be found analytically? The answer is νL → VLνL: ð33Þ yes as we show now. From (26) one would have Similarly, M is diagonalized by the unitary V : N R ¼ † ¼ I ð Þ ð Þ MD MD;Ue up to signs : 39 T MN ¼ VRmNV ð34Þ R By taking the complexpffiffiffiffiffiffiffiffi conjugate of (31) and dividing on the left and right by MN, one readily obtains an equation where mN stands for the diagonal matrix of the heavy between symmetric matrices neutrino masses. This means equivalently v 1 1 T ¼ L − pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ð Þ → ð Þ HH Mν 40 NL VRNL 35 vR MN MN when going from the weak to the mass basis. The apparent where H is a Hermitian matrix defined as difference in the rotations among the light and heavy 1 1 neutrino masses is due to the fact that N corresponds to ¼ pffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffi ð Þ H MD : 41 complex conjugate fields of the right-handed neutrinos νR, MN MN so that one has νR → VRνR in complete analogy with (33). It is easy to see that (33) and (35) give the usual form of Since ImTrðHHTÞn ¼ 0 for any n and Hermitian H, one the charged has the following conditions:   v 1 n L ¼ − pgffiffiffi ðν¯ † þ ¯ † Þþ ð Þ L − ¼ 0 ¼ 1 2 3 ð Þ W LVL=WLeL NRVR=WReR H:c: 36 ImTr Mν ;n ; ; : 42 2 vR MN

It turns out that Eqs. (26) and (27) allow for a direct so that VR is the right-handed counterpart of the PMNS determination of M . The crucial step is to decompose the matrix, making clear the choices of the VL and VR D rotations. symmetric matrix (40) as Before we enter into the nitty-gritty of our program, we T ¼ T ð Þ give a simple example where actually the canonical type I HH OsO 43 seesaw becomes negligible—and yet, interestingly enough, where O is a complex orthogonal matrix and s is known both MD and Ue are calculable, as it turns out, by the charged lepton masses and the mixing matrices V and V . as the symmetric normal form [19], to be determined from L R the knowledge of neutrino masses and mixing. This is an In this case, the term vL by definition dominates the contribution to neutrino masses and from (31) one gets unorthodox method for particle physicists, since one normally uses a unitary matrix instead of orthogonal, since j j it guarantees the diagonalization of a symmetric matrix we vL −iθ † mν ¼ m ;U¼ e L V V ð37Þ are dealing with. However, the unitary matrix approach is v N e R L R not suitable for our task. In this case, however, the form s is − θ not guaranteed to be diagonal. where we have used v ¼jv je i L . Thus, the Dirac L L From (43), when H is real and symmetric, it follows masses can be determined from (27) as a function of the pffiffiffi H O sOT difference of LH and RH leptonic mixing matrices and the immediately that becomes . It can be shown that in the general complex case this expression generalizes to charged lepton masses pffiffiffi † −2 θ † † H ¼ O sEO : ð44Þ i L − e VRVLmeVRVL me −ia M ¼ − e tβm : ð38Þ D e † isat2β The condition H ¼ H now becomes

115031-5 GORAN SENJANOVIĆ and VLADIMIR TELLO PHYS. REV. D 100, 115031 (2019) pffiffiffi pffiffiffiffiffi sE ¼ E s;ET ¼ E ¼ E−1: ð45Þ and rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Using (41) and (44), one can achieve the task of ¼ vL − mν † ð Þ MD VLmN VL 52 disentangling the seesaw by determining MD as vR mN pffiffiffiffiffiffiffiffi pffiffiffi pffiffiffiffiffiffiffiffi † which for v ¼ 0, i.e., in the type I seesaw limit, MD ¼ MNO sEO M : ð46Þ L N gives the simple form of (6). Matrix (52) is defined

Since O, s and E all follow from the knowledge of Mν and up to signs since, strictly speaking, Ue is defined up to signs. This example is to be contrasted with the MN, this manifestly shows how in the parity conserving situation in the SM seesaw scenario, in which MD is case MD can be determined from the knowledge of light and heavy neutrino masses and mixings. plagued by the arbitrariness of a complex orthogonal matrix, here fixed completely. The above expression is valid for any Mν and MN, i.e., any normal form s. It is illustrative to focus on the situation (ii) Case of two generations. It is always illustrative to when s takes a diagonal form, in which case the constraints imagine a two-generation world where one can offer (42) imply only two distinct possibilities: simple analytic formulas. We focus on a physically appealing situation of nondegenerate neutrinos, in which case the form s becomes diagonal. For sI ¼ diagðs1;s0;s2Þ;sII ¼ diagðs; s0;s Þð47Þ simplicity, choose again vL ¼ 0 so that the con- straints (42) become with s0;1;2 belonging to R. The matrix E is found to be 0 1 0 1 −1 ¼ 0 −1 ¼ 0 ð Þ 100 001 ImTrMN Mν ; Im det MN Mν : 53 B C B C ¼ 010 ¼ 010 ð Þ EI @ A;EII @ A 48 We can readily write down the eigenvalues of 001 100 s ¼ diagðs−;sþÞ: rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 for the respective values sI and sII. Equation (47) can be ¼ −1 ð −1 Þ2 − −1 generalized to any number of generations n: for n even, for s 2 TrMN Mν 4 TrMN Mν det MN Mν z z n every eigenvalue , there is also an eigenvalue .For odd ð54Þ there is on top one real eigenvalue. The matrix E in this case has a 1 in the diagonal for each corresponding real s ’ which shows explicitly that are either both real or eigenvalues and two 1 s symmetrically opposed in the form a complex conjugate pair. In turn, by writing antidiagonal for each corresponding complex eigenvalue   and its conjugate. cos θ sin θ A comment is called for. In [9] we chose a different O ¼ ð55Þ − sin θ cos θ decomposition of MD,

pffiffiffiffiffiffiffi pffiffiffiffiffiffiffi † one gets M ¼ V m H0 m V ; ð49Þ D R N N R pffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffi −1 −1 0 0T M Mν M which led to a different decomposition of H H in terms of N N 12 0 0 sin 2θ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : ð56Þ O and s in full analogy with (43). It is straightforward to 1 −1 2 −1 ðTrM MνÞ − det M Mν show that 4 N N 1 pffiffiffiffiffiffiffiffi 0 ¼ † 0 ¼ ð Þ O pffiffiffiffiffiffiffi VR MNO; s s: 50 We now turn to the general case, with one last comment mN regarding the Hermitian MD. In this case, the freedom in the RH quark mixing is all gone, and the constraints from Since in general M and Mν are arbitrary complex N the K- and B-meson system imply W too heavy to be matrices up to constraints (42), no general analytic expres- R accessible at the LHC [13]. The parity conserving situation sion can be offered for M . The following examples may D is then automatically postponed to a future hadron collider. help illustrate what is going on. (i) VR ¼ VL. Imagine an idealized situation with parity unbroken in the leptonic sector. Clearly, O0 ¼ 1 and B. Broken parity: Setting the stage thus We have seen in the previous section how a parity conserving situation with MD Hermitian allows for its 1 pffiffiffiffiffiffiffi v mν pffiffiffiffiffiffiffiffi L determination as a function of M and Mν.Thisisin O ¼ VL mN;s¼ − ð51Þ N MN vR mN complete analogy with the SM situation where a knowledge

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H of charged fermion masses fixes their Yukawa couplings and This is an important expression that says that only MD is allows us to predict branching ratios for the Higgs boson physical, halving the effective degrees of freedom in the H decays. In the present case one needs to know both Mν and task of probing the seesaw. The elements of MD can be MN due to their Majorana nature, but still, MD is then determined numerically from the seesaw, and we address uniquely fixed and we can determine associated decay rates this in a forthcoming publication. such as N → hν, N → Zν and N → Wl. When parity gets broken, however, we have not (yet) C. Broken parity: General situation managed to compute M . Not all is lost though, as we D As we argued, the expression for U in (57) shows argued in [9]. The crucial point is the existence of a number e manifestly its dependence on M . The trouble is that this of physical processes, in particular the same sign leptonic D δ Ue is not automatically Hermitian and thus becomes really decays of doubly charged scalars L;R, that depend crucially useful only when expanded in small sat2β. What one needs on MD and can serve to verify the Higgs seesaw origin of is an explicit unitary expression for U , valid to all orders in neutrino mass. We discuss these processes in the next e s t2β which we now provide. section. a It turns out to be useful to rewrite Eqs. (26) and (27) as Now, although we found no way of computing MD analytically in the general case, we show that only its † U M U − M ¼ 0 ð62Þ Hermitian part is independent, which makes the predictions e e significantly easier to test. As a first step, we compute the UemeUe − me ¼ isat2βM ð63Þ matrix Ue as a function of MD. Multiplying (27) with me and taking the square root gives where M is given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 −ia 2 −ia 2 M ¼ M þ e tβm : ð64Þ Ue ¼ me þ isat2βðtβe me þ meMDÞ: ð57Þ D e me We leave as an exercise for the reader to show that the 2 We keep a small term sat2βme in order to emphasize above equations can be written as † that (57) is exact. Notice the fact that UeUe ¼ 1 reduces by pffiffiffiffiffiffiffiffiffiffiffi † half the number of independent elements of MD, implying M ¼ MM Ue ð65Þ A ¼ 1 ð − † Þ that the anti-Hermitian part MD 2 MD MD becomes a   1 † 1 is t2β function of the Hermitian part MH ¼ ðM þ M Þ. This ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ a D 2 D D Ue me 2 M : can be shown from (26). isat2β isat2β † ðme þ 2 MÞðme − 2 M Þ Here we give the leading expression in s t2β, based on a ð Þ the following expansion of the square root of a matrix 66 (see Appendix A in [11]): The last equation is what we were after and what we pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A promised: an explicitly unitary form of Ue as a function of 2 þ ϵ 2 ¼ δ þ ϵ ij þ ðϵ2Þ ð Þ m i A mi ij i O : 58 MD. It becomes explicitly unity when sat2β vanishes. ij m þ m i j There is actually more in these equations; they tell us finally what is really going on. From (65) the matrix Ue Using this in (57) gives canbeviewedasthe“phase” of M,and(66) shows that   due to parity this phase is determined in terms of M itself. δ tβ ij 2 2 In other words, in spite of being broken spontaneously, P ðU Þ ¼ δ þ is t2β þðHDÞ þ Oðs t2βÞð59Þ e ij ij a 2 ij a still acts and sets the phase of M equal to the phase i of me þ 2 sat2βM. where we have defined This is seen nicely in the one-generation toy example when the matrix M is just a complex number. From (64) H iθ −ia ðM Þ one then has m ¼ ρe − e tβm and (66) shows that θ ðH Þ ¼ D ij ð Þ D e D ij þ : 60 is not arbitrary, but actually a function of ρ itself: mei mej sin θ ¼ sat2βρ=2me. Parity does its job by halving the From (26) and (59) it follows readily that number of independent degrees of freedom. Notice also that mD is bounded from above and below, as expected from (21) and an analogy with the quark system. A isat2β H H H M ¼ ðm þ 2tβM þ HDM þ M HDÞ D 2 e D D D Here we give a few words on the parametrization of MD. 3 3 Instead of parametrizing the Hermitian degrees of freedom þ Oðsat2βÞ: ð61Þ of MD by its Hermitian part as we did above in Sec. III B,

115031-7 GORAN SENJANOVIĆ and VLADIMIR TELLO PHYS. REV. D 100, 115031 (2019) pffiffiffiffiffiffiffiffiffiffiffi one could have used the Hermitian matrix M ¼ MM† as IV. PHENOMENOLOGICAL IMPLICATIONS well, without any loss of generality. One has then from (65) We now turn our attention to the phenomenological and (66) issues, and we do it in full generality without assuming Hermitian MD. As stressed before, besides eliminating the Mij ð Þ ¼ δ þ þ ð 2 2 ÞðÞ freedom in MD, the MLRSM offers a number of new Ul ij ij isat2β þ O sat2β 67 mei mej physical processes which can pave the road for the probing of MD and the origin of neutrino mass. M M First of all, there is an exciting possibility of observing −ia ik kj 2 2 ðM þ e tβm Þ ¼ M þ is t2β þ Oðs t Þ: D e ij ij a m þ m a 2β direct lepton number violation in the WR decays due to the ek ej Majorana nature of RH neutrinos N. Namely, once pro- ð Þ 68 duced WR decays either into two jets or charged leptons and N’s which then further decay into charged leptons and two The task now becomes to determine M from the seesaw. jets. The main semileptonic decays consist then equally of Again, we have found no analytic form yet, but one can same and opposite sign charged lepton pairs accompanied always use a numerical procedure. It can however be by the pairs of jets. The former provide a direct LNV, and illustrated in our toy one-generation example, where one together with the latter allow for a unique direct test of the iθD has readily for mD ¼jmDje Majorana nature of N. The same sign process [20], coined pffiffiffiffiffiffiffiffiffiffiffiffi Keung-Senjanovic (KS), is a high-energy hadron collider mD ¼ i mνmNVR: ð69Þ analog of the neutrinoless double beta decay, with a clear signature that the outgoing charged leptons have RH The memory of parity provides an important constraint, chiralities. It has been argued that the charged lepton written here to the leading order in sat2β can actually be measured at the LHC [21,22]. In the MLRSM there is also a deep connection between the  pffiffiffiffiffiffiffiffiffiffiffiffi s t2β mνm neutrinoless double beta decay and the high-energy KS θ ¼ a 2 þ me þ N ð Þ D tβ pffiffiffiffiffiffiffiffiffiffiffiffi : 70 process, studied in [23]. 2 mνmN me

A comment is called for. The reader familiar with the RH A. Decays and the probe of MD quark mixing in the MLRSM can recognize here the The KS signature, if accessible at the LHC or the next expression given in the formula (A11) of [11], with the hadron collider, would allow for the determination of MN understanding that here we have used the seesaw formula (for phenomenological studies, see e.g., [24]), i.e., mN and for jm j. The reason is simple: were neutrinos Dirac D the leptonic RH mixing matrix VR, the first step towards particles, one would have a complete analogy between the probe of the seesaw origin of neutrino mass. Next, the leptonic and quark sectors and θ would correspond to D through ν − N mixing induced by the nonvanishing MD the conjugate of the RH quark phase. An interested reader elements, N can decay into LH charged leptons too with the θ can also find an exact expression for D analogous to (C1) following rate: of Appendix C in [11], true to all orders in sat2β. m † It is worthwhile to confront the MLRSM one-generation ΓðN → Wþe Þ ∝ Ni jðV M Þ j2: ð71Þ case with the SM seesaw scenario. There the phase of m is i L Lj M2 R D ij D WL simply not physical since its impact can be countered by the phase of the electron field. Here, on the contrary, the phase In the above and hereafter, we will not worry about the of mD is physical [see e.g., (82) in the next section] and precise rates; we give flavor and mass dependence up to related to the phase of VR—and furthermore it can be overall dimensionless constants. computed as a function of electron and neutrino masses, In the same manner, decays such as N → Zν and and the VEVs of the bidoublet. N → hν (we are assuming N to be also heavier than Z The final message is clear. Whatever parametrization one and h, otherwise the opposite happens) are less exciting chooses to use, the important point is that P fixes n2 since they involve missing energy. We give anyway their elements of MD. We can say that the spontaneous breaking decay rates for the sake of completeness: of parity plays an equally important role as the seesaw itself mN † in determining MD. ΓðN → Zν Þ ∝ ΓðN → hν Þ ∝ i jðV M V Þ j2: ð72Þ i j i j M2 R D L ij The crucial point is all this is the softness of the WL spontaneous symmetry breaking which makes the theory þ remember that parity was there to start with. One is tempted These processes, especially N → W eL, can serve in to agree with Coleman on calling it a hidden, rather than the determination of MD; for phenomenological studies spontaneously broken symmetry. see [8,25].

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In the SM seesaw scenario these decays would of course This expression is of great importance, since it also happen [26], with an important difference being that MD directly and manifestly probes, through the asym- is ambiguous there. Worse, the production of N in the SM metry or left and right doubly charged leptonic H seesaw can only be achieved through MD, which requires MD decays, the Hermitian part MD of the Dirac mass to be large, contrary to the seesaw philosophy of MN ≫ MD matrix. It is straightforward to obtain higher orders as an explanation of the smallness of neutrino masses. of the above expression. Moreover, in the MLRSM there are other fundamental It cannot be overstressed: it is not just MD,as processes which can probe MD, discussed in what follows. usually assumed, that enters into the seesaw formula. δþþ (i) Decays of the doubly charged scalars L;R. The matrix Ue also plays an essential role and there Doubly charged scalars are produced pairwise by is a deep connection between these quantities which the Z-boson and the photon, and unless they are very can in principle be probed through the above doubly light, are expected to be less accessible than the WR. charged scalar decays. Nonetheless, their lepton number violating decays Of course, in the Hermitian limit sat2β ≃ 0, one play an important role in disentangling the seesaw. simply ends up with a LR symmetric prediction From (12), the relevant Yukawa interaction is of the same LH and RH doubly charged scalar   decay rates. 1 þ þþ T T MN δ −Lδ ¼ δ e U U e (ii) Decays of the singly charged scalars L . 2 L L e v e L δ− → −  R The decay L eLN which proceeds through 1 M the ν − N mixing Θ is of similar importance as the þ δþþeT N e : ð73Þ → l 2 R R v R N LW decay since they both proceed through R δþ MD. The L interaction is given in (14), and it is The matrix Ue provides the expected mismatch easily shown to lead to the decay rate δ−− → between LH and RH states. Notice first that R − e e decays measure, complementary to the KS mδ † −1 2 R R Γ − ∝ L jð T Þ j ð Þ δ →e N 2 UeMNUeM M VR 77 process, the masses and mixings of N’s: L Li j M D N ij WR

mδ−− −− R 2 Γðδ → e e Þ ∝ jðM Þ j : ð74Þ which could in principle probe MD if MN was to be R Ri Rj M2 N ij WR determined. It is a rather complicated expression in general and obviously hard to verify. It is illustrative These decays have been studied recently at the to see what happens in the P conserving situation lepton colliders in [27]. and the same LH and RH mixing matrices VL ¼ VR. The LH analog decays play an even more im- Using (52) the above decaypffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rate obtains a simple portant role in verifying the seesaw mechanism due jð − Þ j2 flavor structure VLmN vL=vR mν=mN ij . to the presence of the Ue matrix, since one has δ0 (iii) Decays of the neutral scalars L;R 0 mδ−− The decays δ → νN also proceed through Θ. −− L † 2 L;R Γðδ → e e Þ ∝ jðUeM U Þ j : ð75Þ L Li Lj M2 N e ij The relevant interactions are given in (14), which WR δ0 then gives the decay rate for R: Up to a proportionality factor, this is precisely the mδ0 R † † 2 direct type II contribution to neutrino mass, i.e., the Γ 0 ∝ jð Þ j ð Þ δ →ν N 2 VLMDVR ij : 78 upper-left block of the neutrino mass matrix in (28). R i j M WR These decays can thus directly probe the pure type II seesaw as discussed in [28] since in that case one has It has the same flavor dependence as the N → hν −− 2 Γðδ → e e Þ ∝ jðMνÞ j . In our case it is M δ0 L Li Lj ij N decay. This is to be expected since h and R mix in that gets probed, together with the Ue matrix. general and the above decay can also proceed through Using the formulas (57) and (60), one obtains to this mixing and the direct decay in (71). Since the the leading order in sat2β mixing is proportional to the ratio of WL=WR masses, the final result is naturally of the same order of Γδ−−→ L eLi eLj magnitude. δ0 Γ Similarly, one has for the left-handed L decay δR→eR eR i j   T mδ−− ðH M þ M H Þ mδ0 L D N N D ij L † † −1 2 ≃ 1 þ 2 Γ 0 ∝ jð T Þ j sat2βIm : δ →ν N 2 VLUeMNUeMDMN VR ij : mδ−− ðM Þ L i j M R N ij WR ð76Þ ð79Þ

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As expected, this decay has a rather different measuring MN as in the original KS case. If these form due to the impact of Ue and is quite messy, states were to be accessible at the LHC or the next not easy to probe. It becomes illustrative in the hadron collider, one would have a clear shot of VL ¼ VR limit, since by using (52) the complicated verifying or disproving the MLRSM. Here we have flavorpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi structure above becomes diagonal: given the list and the associated decay rates for these jð − Þ j2 processes; their phenomenology shall be dealt with mN vL=vR mν=mN ij . (iv) Decay of the heavy charged scalars Hþ in detail in a forthcoming publication. − The relevant decays here are H → eLN and H− → e ν; they can be easily computed from (18) R B. The limits on particle masses to be Before we turn to the conclusions it may be useful to the mH −ia † 2 reader to have an idea of the limits on the masses of the Γ −→ ∝ j½ðm þ e s2βM ÞV j ð80Þ H eLiNj M2 e D R ij relevant gauge boson and scalar particles. WL (i) Limits on the masses of heavy charged and neutral and gauge bosons WR and ZR. The best limit on the WR mass comes from the mH † † 2 Γ − ∝ j½ ð þ ia Þ j ð Þ dijet final state. It is thus independent of the detailed H →eRiνj 2 VL MD e s2βme ij : 81 MW properties of the RH neutrinos and amounts to L ≥ 4 MWR TeV [29]. The limit from the KS process depends on the mass of the RH neutrinos and it It is interesting to check what happens on the varies from 3.5 to 5 TeV for mN in the range from ≃ 0 Hermitian limit sat2β . The result depends on 0.1 to 1.8 TeV [30]. ≃ 0 how sat2β vanishes: if a nothing new happens, The limit on the ZR mass is MZ ≥ 4.6 TeV [30]. β ≃ 0 R unlike the opposite possibility . In this case the We should stress that the MLRSM prediction MZ ≃ − → R decay H eLN measures directly VR, while the 1.7MW implies that the LHC cannot see the ZR, and − → ν R H eR decay probes MD,which,asweknow thus, if it were to be seen, it would automatically from Sec. III A, gets predicted from Mν and MN. 0 invalidate the MLRSM. (v) Decays of the heavy neutral scalar H (ii) Limit on the mass of the heavy doublet H from the The relevant interactions can be easily determined bidoublet Φ. 0 from (18). The decay H → ee¯ is then given by As discussed in Sec. II, since the couplings of the H, given in (18), are determined by the structure of the mH † 2 Γ 0 ∝ jð þ ia Þ j ð Þ H →e e¯ 2 M e s2βme : 82 bidoublet, the flavor conservation in neutral currents i j M D ij ≥ 20 WL sets a stringent limit mH TeV [12,13].This raises the question of the perturbativity of the theory The decay H0 → νN is probably of secondary for a WR accessible at the LHC, i.e., for MW ≲ importance, but nonetheless we give the relevant R 8 TeV, since H and WR get the mass at the same decay rate large stage of symmetry breaking, and thus a particular scalar coupling must be large enough to ensure the mH † −ia † 2 Γ 0 ∝ j½V ðm þ e s2βM ÞV j : H →νiNj 2 L e D R ij heaviness of H.Thisimpliesstringentlimitsonother MW L scalar masses to be discussed below. For a heavy WR, ð Þ ≳ 20 83 with MWR TeV, so that the H mass does not require a large coupling, these limits go away and the As before, the limit β ≃ 0 simplifies matters. The theory is of course highly perturbative. 0 Δ H → ee¯ decay would then directly probe MD, (iii) Limits on the masses of the heavy scalar triplet L. while the H0 → νN decay depends on the product The best direct limit is on the mass of the doubly charged component and is roughly mδþþ ≥ 400 GeV, of VL and VR, which makes it harder to disentangle. L We should stress once again that β ≃ 0 requires but is flavor dependent [31]. Since the T parameter is heavy WR, beyond the LHC reach. Furthermore, the sensitive to the mass splittings in the ΔL multiplet, same flavor constraints indicate [13] that light WR one obtains a better limit from the high precision T in the LHC energy reach basically fixes (modulo constraints; see Fig. 6 in [31]. For a relatively light ≃ 0 02 uncertainties) sat2β . . This in return reduces WR accessible at the LHC, this implies a limit on the substantially the freedom in the quark RH mixing (basically degenerate) multiplet mass on the order and MD. of TeV. In short, there are a plethora of high-energy As discussed above, the heaviness of the second physical processes that probe MD, on top of doublet H brings the issue of perturbativity and a

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H much stronger limit emerges. By asking that the the Hermitian part MD is independent, simplifying thus

perturbative cutoff be not below 10 MWR ,for considerably the experimental verification of the program ¼ 6 ≥ 9 MWR TeV one gets mΔL TeV [17],farfrom of disentangling the seesaw. the LHC reach. For a WR at the LHC energies, seeing In this longer followup of our original work, we have the ΔL multiplet would invalidate the MLRSM. We filled in the missing gaps in obtaining our results. We have should stress though that this is not a generic prediction given a pedagogical expos´e of the Hermitian case, illus- of the theory. For a heavy WR above, say 20 TeV, the trating it with a few illuminating examples, and providing only limit that remains is the direct one. the relevant derivations. We have also provided a list of δþþ δ0 (iv) Limits on the masses of the R and R. relevant decays that depend on MD, and can thus serve to δþþ The direct limit on the mass of R is similar to probe the seesaw origin of neutrino mass. A number of the one of its left-handed counterpart, of roughly these processes can test directly the Majorana nature of N’s 400 GeV, also flavor dependent [31]. However, and are complementary to the neutrinoless double beta similar to the situation regarding ΔL, the perturba- decay. This is true in particular of the so-called KS process ’ tivity bound becomes huge for a WR at the LHC of the direct production of N s, but it applies also to the and in this case mδþþ ≥ 12 TeV [17]. Once again, decays of doubly and singly charged scalars from the heavy R Higgs triplets Δ . the limit disappears for heavy WR above 20 TeV. L;R On the other hand, δ0 can be arbitrarily light. In principle also the neutral scalar decays can play the R δ0 ’ One could hope for a cosmological limit from the same role, especially the decay of R into the pairs of N s, stability of the scalar potential, but we may be living and possibly even through the same decays of the SM − δ0 in a metastable vacuum. Higgs boson, in the case of appreciable h R mixing (for a recent phenomenological study, see [32]). All these V. SUMMARY AND OUTLOOK particles can also decay into final states that probe MD, as discussed in Sec. V. Moreover, the decays of the heavy The origin and nature of neutrino mass is arguably one scalar doublet H from the bidoublet Φ also probe directly of the central issues in the quest for the theory beyond the MD, but due to the large limit on its mass are out of the Standard Model. Over the years, the seesaw mechanism has LHC’s reach. emerged as the main scenario behind the smallness of Before closing it is worthwhile to compare the MLRSM neutrino mass, but by itself falls short of providing a full- with other gauge theories of neutrino mass. As we argued fledged theory. First of all, the SM seesaw cannot be from the beginning it is the LR symmetry that leads to the disentangled, and furthermore, the heavy RH neutrinos can existence of RH neutrinos N which then lead to the seesaw be produced at the hadron colliders, such as the LHC, only and the rest of the story told here. In turn the global B − L through the Dirac mass terms. This obviously forces the symmetry of the SM is gauged automatically, and the Dirac mass terms to be incomparably larger than their cancellation of new induced anomalies provides an addi- natural seesaw values. tional raison d’être for N besides the LR symmetry. What The situation changes dramatically in the context of the happens then in the more modest approach where one ð1Þ LR symmetric theory, the same one that led originally to gauges just the GSM × U B−L subgroup of the nonvanishing neutrino mass and to the seesaw mechanism MLRSM? First of all, N can be produced pairwise through itself. The SM singlet RH neutrinos can be easily and the new neutral gauge boson Z0, an improvement over the naturally produced at the hadron colliders due to their simple SM seesaw. Moreover, if one were to observe the þ gauge interactions with the WR boson, the RH counterpart N → W eL process, this would require the ν − N mixing, of the SM W boson. Equally important, the Dirac mass and in a renormalizable theory that mixing would require the terms stop being plagued by ambiguities allowing for a Yukawa Dirac coupling with the SM Higgs. Thus one could verifiable seesaw based on the Higgs mechanism as the argue that the above decay would probe the Higgs origin of origin of neutrino mass. This is manifest in the case of LR neutrino mass, just as in the MLRSM. However, there is a ð1Þ symmetry being charge conjugation, but it turns out to be profound difference. In the U B−L gauged model MD is quite subtle in the case of parity. arbitrary as in the SM seesaw; i.e., one has the situation C Whereas for it is straightforward to compute MD as a described in (2). It is precisely here where the MLRSM function of Mν and MN in the minimal LR Symmetric stands out by having M structurally predicted from the P D Model, the same was never achieved for the case, at least knowledge of Mν and MN, which is per se a great motivation not in full generality. Instead, we recently suggested an for studying MLRSM, besides the desire to understand the alternative approach using the exciting LNV violating origin of the breaking of parity in weak interactions. decays of doubly charged scalars into the pair of the same One could also be more ambitious and embed the sign charged leptons as a probe of MD. Moreover we MLRSM in the SOð10Þ which besides managed to solve analytically the parity conserving case providing the N’s and the seesaw, also links quark and ¼ † MD MD, and to show that, when parity gets broken, only charged lepton masses. This would be extremely exciting if

115031-11 GORAN SENJANOVIĆ and VLADIMIR TELLO PHYS. REV. D 100, 115031 (2019) the scale of LR symmetry breaking were accessible to unknown parameters, such as the masses and couplings of present day or near future colliders. Unfortunately, in the the superpartners, impedes the predictivity. minimal predictive version of the SOð10Þ theory the scale All in all, our work offers yet another conclusive ð2Þ of breaking of SU R is predicted to be huge, above evidence in favor of the MLRSM standing out by being 1010 GeV or so, and thus hopelessly out of reach. The a self-contained theory of neutrino mass, in complete beauty of predicting MD or using MD to determine MN analogy with the SM as a theory of the origin of charged ends up being more a question of aesthetics than of the fermion masses. directly verifiable physics. Another natural theory of neutrino mass is the minimal ACKNOWLEDGMENTS supersymmetric Standard Model without the artificial assumption of R-parity conservation. It provides the We acknowledge the collaboration with Miha Nemevšek mixture of both the seesaw and the radiative origin of at the initial stages of this work. We thank Alessio Maiezza, neutrino masses, and it offers the connection between the Alejandra Melfo, Fabrizio Nesti and Juan Carlos Vasquez neutrinoless double beta decay and lepton number violation for useful comments and discussions, and a careful reading at hadronic colliders [33]. Unfortunately the proliferation of of the manuscript.

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