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Available online at www.sciencedirect.com ScienceDirect Nuclear Physics B 923 (2017) 179–221 www.elsevier.com/locate/nuclphysb Long lived light scalars as probe of low scale seesaw models a b c,∗ P.S. Bhupal Dev , Rabindra N. Mohapatra , Yongchao Zhang a Department of Physics and McDonnell Center for the Space Sciences, Washington University, St. Louis, MO 63130, USA b Maryland Center for Fundamental Physics, Department of Physics, University of Maryland, College Park, MD 20742, USA c Service de Physique Théorique, Université Libre de Bruxelles, Boulevard du Triomphe, CP225, 1050 Brussels, Belgium Received 17 May 2017; received in revised form 9 July 2017; accepted 28 July 2017 Available online 5 August 2017 Editor: Tommy Ohlsson Abstract We point out that in generic TeV scale seesaw models for neutrino masses with local B–L symmetry breaking, there is a phenomenologically allowed range of parameters where the Higgs ﬁeld responsible for B–L symmetry breaking leaves a physical real scalar ﬁeld with mass around GeV scale. This particle (denoted here by H3) is weakly mixed with the Standard Model Higgs ﬁeld (h) with mixing θ1 ≲ mH3/mh, barring ﬁne-tuned cancellation. In the speciﬁc case when the B–L symmetry is embedded into the TeV scale left–right seesaw scenario, we show that the bounds on the h–H3 mixing θ1 become further strengthened due to low energy ﬂavor constraints, thus forcing the light H3 to be long lived, with displaced vertex signals at the LHC. The property of left–right TeV scale seesaw models are such that they make the H3 decay to two photons as the dominant mode. This is in contrast with a generic light scalar that mixes with the SM Higgs boson, which could also have leptonic and hadronic decay modes with comparable or larger strength. We discuss the production of this new scalar ﬁeld at the LHC and show that it leads to testable displaced vertex signals of collimated photon jets, which is a new distinguishing feature of the left–right seesaw model. We also study a simpler version of the model where the SU(2)R breaking scale is much higher than the O(TeV) U(1)B–L breaking scale, in which case the production and decay of H3 proceed differently, but its long lifetime feature is still preserved for a large range of parameters. Thus, the search * Corresponding author. E-mail addresses: [email protected] (P.S.B. Dev), [email protected] (R.N. Mohapatra), [email protected] (Y. Zhang). http://dx.doi.org/10.1016/j.nuclphysb.2017.07.021 0550-3213/© 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP3.

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180 P.S.B. Dev et al. / Nuclear Physics B 923 (2017) 179–221 for such long-lived light scalar particles provides a new way to probe TeV scale seesaw models for neutrino masses at colliders. © 2017 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY license 3 (http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP . 1. Introduction Seesaw mechanism seems to provide a very simple and elegant way to understand the small- ness of neutrino masses [1–5]. Two key ingredients of this mechanism are: (i) addition of the right-handed neutrinos (RHN) to the Standard Model (SM), and (ii) a large Majorana mass for the RHNs which breaks the accidental B–L symmetry of the SM. There exist a large class of well-motivated ultraviolet (UV)-complete seesaw models which necessarily employ local B–L symmetry, e.g. the TeV-scale left–right (LR) symmetric model [6–8]. It will therefore be an im- portant step to ﬁnd experimental evidence for the B–L symmetry and its breaking. If the local B–L is broken by a Higgs ﬁeld that carries this quantum number, there will be a remnant neutral scalar ﬁeld, denoted here by H3 (for reasons explained below), analogous to the Higgs boson h of the SM. So looking for signatures of H3 can provide invaluable clues to the nature of the new physics associated with neutrino mass generation. Clearly, such a search is realistic only if the B–L symmetry breaking scale is within the multi-TeV range. The mass and couplings of the new Higgs ﬁeld are still unrestricted to a large extent, mainly because it communicates to the SM sector only through its mixing with the SM Higgs and via the heavy gauge boson interactions. The mass range heavier than the SM Higgs boson has been discussed earlier [9–12]. The mass range near mh would generically lead to large h–H3 mixing, which is disfavored by the LHC Higgs data [13]. So we will focus here on the more interesting regime with mH 3 ≪ mh in which case, the H3–h mixing angle θ1 ≲ mH3/mh ≃ −3 8 × 10 (mH 3/GeV) from considerations of ﬁne tuning. Smaller values of θ1 can be analyzed as part of the allowed parameter range of generic B–L models. However, as we show here, this range is naturally dictated to us from low energy ﬂavor constraints, once the B–L symmetry is embedded into the minimal LR model. This leads to the H3 particle being necessarily long-lived, with interesting displaced vertex signals at the Large Hadron Collider (LHC). The important point is that even though a generic light scalar in a BSM theory that mixes with the SM Higgs can have leptonic, hadronic as well as photonic decay modes, the speciﬁc property of LR see- saw models makes the two photon decay mode exclusively dominant, as recently pointed out by us [14]. In this paper, we elaborate on the details of this scenario, including an in-depth discus- sion of all relevant high and low-energy constraints on the model parameter space, the production and decay of the new Higgs boson at the LHC and future colliders, as well as the experimental prospects for observing the displaced vertex collimated diphoton signal. We would like to em- phasize the complementarity of the new collider signal discussed here with existing and future low-energy probes of light neutral sectors at the intensity frontier. We will discuss two classes of UV-complete theories, based on (i) the full LR gauge group SU(2)L ×SU(2)R ×U(1)B–L, and (ii) a simpler B–L model with SU(2)L ×U(1)I 3R ×U(1)B–L local symmetry. The bulk of this paper deals with the class (i), where we consider a LR symmetric model where parity is broken at a much higher scale than the SU(2)R breaking. As a result, the SU(2)L Higgs triplet of the familiar left–right model is pushed to a much higher scale and also in neutrino mass formula, type I seesaw dominates. The heavy–light neutrino mixing is a separate √ −5.5 parameter in the model and is small i.e. ∼ mν/mN ≲ 10 in the kinematic region where

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P.S.B. Dev et al. / Nuclear Physics B 923 (2017) 179–221 181 mH 3 is less than 10 GeV and much less than the RH neutrino masses (assumed to be of order of 1 TeV), so its contribution to the H3 phenomenology can be ignored. The class (ii) scenario that we discuss can be thought of as an “effective” theory of the LR model, where the SU(2)R symmetry breaking scale is much higher than the U(1)B–L-breaking which is assumed to be at the TeV-scale to be accessible experimentally. Nevertheless, our results ′ for the U(1)B–L case are applicable to a wider class of Z -models, which have an associated Higgs boson that mixes with SM Higgs ﬁeld. The rest of the paper is organized as follows: in Section 2, we brieﬂy review the minimal LR seesaw model and set up our notation. In Section 3, we present the arguments for the light B–L breaking Higgs being weakly mixed with the SM Higgs h and the heavy CP-even Higgs H1, and study its decay lifetime and branching ratios. In Section 4, we derive a lower limit on the light scalar H3 mass from cosmological considerations at the nucleosynthesis epoch. In Section 5, we present an in-depth analysis of all available laboratory constraints on the light scalar param- eter space. In Section 6, we discuss the production of H3 at hadron colliders and its detection prospects at the LHC, as well as in future colliders. In Section 7, we analyze the LLP prospects of a simpler U(1)B–L model. Our conclusions are given in Section 8. In Appendix A, we col- lect the two-body partial decay widths of H3 in the LR model, and in Appendix B, the partial decay width of the loop-induced process Z → H3γ . Finally, in Appendix C, we study the LLP sensitivity of light RHNs in the LR model at the LHC. 2. Minimal left–right seesaw model For completeness and to set up our notation, we brieﬂy review the minimal LR model [6–8] in this section, based on the gauge group SU(3)c × SU(2)L × SU(2)R × U(1)B–L with parity broken at a higher scale than SU(2)R. The quarks and leptons are assigned to the following irreducible representations: ( ) ( ) ( ) ( ) uL 1 uR 1 QL,i = : 3, 2, 1, , QR,i = : 3, 1, 2, , (1) dL 3 dR 3 i i ( ) ( ) νL NR ψL,i = : (1, 2, 1,−1) , ψR,i = : (1, 1, 2,−1) , (2) eL eR i i where i = 1, 2, 3 represents the family index, and the subscripts L, R denote the left- and right- handed chiral projection operators PL,R = (1 ∓ γ5)/2, respectively. There are different ways to break the SU(2)R × U(1)B–L gauge symmetry and to understand the small neutrino masses in this model, depending on the choice of the Higgs ﬁelds. One of the simplest choices is to introduce SU(2)L × SU(2)R bi-doublet and SU(2)R triplet scalars ( ) ( √ ) 0 + + ++ φ φ / 2 = −1 20 : (1, 2, 2, 0) , R = R 0 +R √ : (1, 1, 3, 2) , (3) φ φ − / 2 1 2 R R respectively, with the neutral components of the above ﬁelds acquiring non-zero vacuum expec- tation values (VEV): 0 0 0 ′ ⟨ R⟩ = vR, ⟨φ1⟩ = κ, ⟨φ2⟩ = κ , (4) √ 2 ′ 2 with the electroweak (EW) VEV given by vEW = κ + κ ≃ 174 GeV. We assume the vR scale to be in the multi-TeV range for phenomenological purposes. This is the version of the LR seesaw that results when parity symmetry is broken at a much higher scale than SU(2)R [15]

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182 P.S.B. Dev et al. / Nuclear Physics B 923 (2017) 179–221 which, as remarked earlier, removes the L(1, 3, 1, 2) Higgs ﬁeld from the effective low energy theory. This makes it simpler to analyze and present our main results, although our conclusions remain unchanged in the fully parity symmetric version being a TeV scale theory. The fermion masses can be understood from the Yukawa Lagrangian: LY = hq,ijQ L,iQ R,j + h˜q,ijQL,iQ˜ R,j + hℓ,ijψL,iψ R,j +˜hℓ,ijψL,iψ˜ R,j T +fijψ R,iCiσ2 RψR,j + H.c. , (5) where ˜ = iσ2 ∗ (with σ2 being the second Pauli matrix) and C stands for charge conjugation. The standard type-I seesaw mass matrix for neutrinos follows from the leptonic part of the above couplings after symmetry breaking. The triplet VEV vR breaks lepton number and provides the Majorana mass for the heavy RHNs which goes into the seesaw matrix [2]. 3. Light scalar in TeV-scale LR seesaw In this section, we analyze the TeV-scale LR seesaw parameter space for a light scalar with mass in the (sub) GeV range and its couplings to determine the decay branching ratios and lifetime. The most general scalar potential for this model reads as follows: [ ] 2 † 2 † † 2 † V = −μ 1 Tr( ) − μ2 Tr(˜ ) + Tr(˜ ) − μ3 Tr( R R) { } [ ] [ ] [ ] 2 2 2 † † † +λ1 Tr( ) + λ2 Tr(˜ ) + Tr(˜ ) [ ] † † † † † +λ3 Tr(˜ )Tr(˜ ) + λ4 Tr( ) Tr(˜ ) + Tr(˜ ) (6) [ ] 2 † † † +ρ1 Tr( R R) + ρ2 Tr( R R)Tr( R R) [ ] +α1 Tr( †)Tr ( R † R) + α2eiδ2Tr(˜†)Tr ( R R† ) + H.c. † † +α3 Tr( R R) . 2 All the 12 parameters μ 1,2,3, λ1,2,3,4, ρ1,2, α1,2,3 are chosen to be real, with an appropriate re- deﬁnition of the ﬁelds, and the only CP-violating phase δ2 is associated with the coupling α2, as explicitly shown in Eq. (6). There are 14 scalar degrees of freedom, six of which are Gold- ± ± stone modes eaten by the W , W R , Z, ZR gauge bosons, thus leaving 8 physical Higgs bosons, ± ±± denoted by h, H1, A1, H3, H 1 and H2 . The full scalar potential for this class of LR seesaw models was analyzed in detail in Refs. [10,11,16] and here we present only the results relevant for our purpose. We work in the natural parameter space, where the following parameters are small: ′ mb ′ ξ ≡ κ /κ = ≪ 1, ϵ ≡ κ /vR ≪ vEW/vR ≪ 1, (7) mt and also assume that there is no CP-violation in the scalar sector, i.e. the CP-violating phase δ2 is zero. There are 6 degrees of freedom in the neutral scalar sector: three dominantly CP-even and the other three CP-odd. This includes the two Goldstone modes eaten by the neutral Z and ZR gauge bosons. In the CP-even sector of neutral scalars, after applying the minimization ′ ′ conditions with respect to the three VEVs κ, κ , vR and the phase δ2 associated with the VEV κ ,

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P.S.B. Dev et al. / Nuclear Physics B 923 (2017) 179–221 183 0 Re 0 Re 0 Re we obtain the squared mass matrices in the basis of {φ , φ , } at the zeroth, ﬁrst and 1 2 R second order of the small parameters ξ and ϵ deﬁned in Eq. (7) (setting the phase δ2 to be zero), respectively, ⎛ ⎞ ( ) 0 0 0 (0) 2 2 M 0 = vR ⎝ 0 α3 0 ⎠ , (8) 0 0 4ρ1 ⎛ ⎞ ( ) 0 −α3ξ 2α1ϵ (1) 2 2 M 0 = vR ⎝−α3ξ 0 4α2ϵ ⎠ , (9) 2α1ϵ 4α2ϵ 0 ⎛ ⎞ 2 2 2 ( ) 4λ1ϵ + α3ξ 4λ4ϵ 4α2ϵξ (2) 2 2 ⎝ 2 2 2 ⎠ M 0 = vR 4λ4ϵ 4 (2λ2 + λ3) ϵ + α3ξ 2 (α1 + α3) ϵξ . (10) 4α2ϵξ 2 (α1 + α3) ϵξ 0 0 Re In the limit of ξ ≪ 1, one of the scalars, predominantly from φ 2 is heavy, with mass at the vR √ scale α3vR and decouples from the lower energy SM sector, thus leading to a reduced scalar 0 Re 0 Re mass matrix involving only φ 1 and R . If the quartic coupling ρ1 is large, say of order one, 0 Re then R (denoted here by H3) will also be heavy, close to the vR scale, and its mixing with the α1vEW SM Higgs boson h will be given by θ1 ≃ . However, there exists another possibility, mainly ρ1vR motivated by the lack of any direct experimental constraints on its mass, as well as supported by arguments based on radiative stability (see Section 3.2), where H3 could also be very light, and in particular, lighter than the SM Higgs boson. This happens when ρ1 ≪ 1, in which case, the 0 Re 0 Re sub-matrix in the basis of {φ , } becomes 1 R ( ) 2 2 4λ1ϵ 2α1ϵ 2 M ≃ v . (11) 0 R 2α1ϵ 4ρ1 2 2 In the limit of m ≪ m , Eq. (11) can be diagonalized in a seesaw-like approximation, and the H3 h two scalar masses are respectively given by 2 2 2 2 m h ≃ 4λ1ϵ vR = 4λ1vEW , (12) 2 2 2 2 m H3 ≃ 4ρ1vR − sin θ1 mh , (13) with the mixing angle α1 α1 vR sin θ1 ≃ = . (14) 2λ1ϵ 2λ1 vEW −1 Note that the mixing has an inverted dependence on the VEV ratio (vEW/vR) , compared to the case where mH 3 ≫ mh. This restricts the parameter α1 to be appropriately small in order to ensure that sin θ1 ≤ 1, as we show below. 3.1. Parameters for a light scalar The question that we address in this subsection is whether there is a range of parameters in the model where H3 could be light, i.e. in the (sub)-GeV range, which has important phenomeno- logical ramiﬁcations. From Eq. (13), it is clear that the lightness of H3 depends on how small the values of ρ1 and α1 can be:

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184 P.S.B. Dev et al. / Nuclear Physics B 923 (2017) 179–221 Fig. 1. Contours of mH 3 as a function of the quartic coupling parameters ρ1 and α1 deﬁned in the scalar potential (6). Here we have set the RH triplet VEV vR = 5 TeV. ( ) 2 α 2 1 2 m H3 ≃ 4ρ1 − vR . (15) λ1 The exact dependence of mH 3 on the parameters ρ1 and α1 is shown in Fig. 1 for a representative value of vR = 5 TeV, which is close to the smallest possible value allowed by the current di- rect [17–19] and indirect [20] constraints on the WR mass (with appropriate scaling for gR ≠ gL, −8 −5 where gL,R are the SU(2)L,R gauge couplings). We see that for ρ1 ∼ 10 and α1 ≲ 10 , we get mH 3 ∼ O(GeV), which is the region of great interest for long-lived particle (LLP) searches, as we will see below. For a small mixing sin θ1, the H3 mass is determined completely by ρ1 which is required to be very small for light H3. However, when the mixing is sizable, i.e. α1 is comparatively large, there can be large cancellation between the two terms in Eq. (13). 3.2. Radiative corrections In this subsection, we consider the one-loop corrections to the H3 mass to see whether mH 3 ∼ O(GeV) is radiatively stable. The one-loop renormalization group (RG) running of the gauge, quartic and Yukawa couplings in the LR model has been considered in Refs. [21–23]. Due to the large number of parameters in the scalar potential, some of the couplings would easily become non-perturbative before reaching up to the GUT or Planck scale, especially when the RH scale vR is relatively low [16]. There are also stability and perturbativity constraints on some of the √ quartic couplings [24], but the mass of H3, which is proportional to ρ1 at the leading order at tree level [cf. Eq. (15)], is not limited by these arguments. Moreover, even if the gauge symmetry SU(2)R is restored at the few-TeV scale, the LR model is far from being a UV-complete theory up to the GUT scale, and more ﬁelds have to be introduced at higher energy scales, which hardly leave any imprints at the TeV scale. Our aim here is not to analyze the ultimate UV-complete theory with parity, rather we remain at the TeV-scale, and using the Coleman–Weinberg effective potential approach [25], check how the mass of H3 would be affected by interacting with other particles in the TeV-scale LR model at the one-loop level. Our considerations are similar to what happens to the SM Higgs mass. It is well known that if we neglect the one-loop fermion contributions to the Coleman–Weinberg effective potential, there would be a lower limit of order of 5 GeV on the Higgs boson mass [26,27]. However,

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P.S.B. Dev et al. / Nuclear Physics B 923 (2017) 179–221 185 2 ± ±± Fig. 2. Dominant loop corrections to m H3 from interacting with the heavy scalars S (H1, A1, H1 , H2 ), the heavy gauge bosons VR (WR, ZR) and the heavy RHNs Ni . this bound goes away once the large top-quark Yukawa coupling is included. This approach for the LR models for the case of doublet Higgs but without singlet fermions was carried out in Ref. [28], where a lower bound on the heavy neutral Higgs of the order of 900 GeV was obtained. The crucial difference in our model is the fermion contribution, which enables us to avoid this lower bound on mH 3. In the Coleman–Weinberg effective potential, we ﬁnd that the dominant one-loop corrections 2 to m H3 arise from its gauge interaction with the heavy WR, ZR RH gauge bosons, the Yukawa ± ±± interaction with the RHNs, and the interactions with the heavy scalars H1, A1, H 1 and H2 . All the interactions with the SM ﬁelds are suppressed by the small mixing angles. The Feynman 2 diagrams of loop corrections to m from the heavy particle loops are collected in Fig. 2, which H3 sum up to [ ] ( ) loop 3 1 8 1 2 2 2 4 4 2 2 2 2 m ≃ α + ρ − 8f + g + (g + g ) v , (16) H3 2 3 2 R R BL R 2π 3 3 2 where gBL is the SU(2)R and U(1)B–L gauge coupling strength, f is the RHN Yukawa coupling as deﬁned in Eq. (5), and α3 and ρ2 are the scalar quartic couplings deﬁned in Eq. (6). Without any tuning of the scalar, gauge and Yukawa couplings, the loop correction to mH 3 is expected to be of order vR/4π. In the above expression, we have kept only the contribution of the scalar coupling α3 since that is the only coupling which is expected to be of order one to satisfy the ﬂavor changing neutral current (FCNC) constraints on bidoublet Higgs mass, as we will discuss below in Section 5. The important point here is that with the minus sign for the f contribution in Eq. (16), the bosonic and fermionic contributions can be made to cancel each other keeping H3 vR −2 mass light even in the presence of radiative corrections. With a tuning of order GeV/ ∼ 10 4π (with vR at the TeV-scale) for the parameters in Eq. (16), we could easily obtain a light scalar H3 at or below the GeV scale. 3.3. Couplings of H3 In order to study the collider phenomenology of light H3, it is essential to delineate its cou- plings to various particles in the theory. In the limit of zero CP phase δ2 in Eq. (6), H3 could only mix with the scalars h and H1:

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186 P.S.B. Dev et al. / Nuclear Physics B 923 (2017) 179–221 Table 1 The couplings of the light scalar H3, up to the leading order in ϵ and ξ . Couplings Values 1 H3hh √ α1vR 2 √ hH3H3 − 2α1vEW √ 0 H3hH 1 2 2α2vR 0 0 1 H3H 1 H1 √2 α3vR 0 0 1 H3A 1A1 √2 α3vR √ + − H3H 1 H1 2α3vR √ ++ −− H3H 2 H2 2 2 (ρ1 + 2ρ2) vR ( ) H3u¯u √1 2 ŶU sin θ˜1 − √12 VLŶDVR† sin θ˜2 ( ) H3d¯d √1 2 ŶD sin θ˜1 − √12 VL†ŶUVR sin θ˜2 H3e¯e √1 ŶE sin θ˜1 − √1 YνN sin θ˜2 2 2 H3NN √mN 2vR √ + − 1 2 2 2 H3W W √ 2gL sin θ1 vEW + 2gR sin ζW vR √ + − 2 H3W W R 2gR sin ζW vR √ + − 2 H3W R WR 2gRvR √ 2 2 2 H3ZZ g 2L√s2incθo1s2vθEwW + 2gRcosisn2 φζZ vR √ 2 H3ZZR −gLgR √sin 2θc1oscoθws φ vEW + 2 2gcRos2inφζZ vR √ 2 2g RvR H3ZRZ R cos2 φ + − 1 H3H 1 W 2 gL(sin θ2 − sin θ1ξ) + − 1 H3H 1 WR 2 gRϵ igL(sin θ2−sin θ1ξ) H3A1Z − 2 cos θw i H3A1ZR 2 gR(sin θ2 − sin θ1ξ) cosφ ⎛ ⎞ ⎛ ⎞⎛ ⎞ 1 2 0 Re h 1 − 2ξ ξ −sin θ1 φ1 ⎝H 1 ⎠ = ⎝ −ξ 1 − 1 2ξ2 − sin θ2 ⎠⎝ φ20 Re ⎠ , (17) 0 Re H3 sin θ 1 sin θ2 1 R where sin θ1 is the h–H3 mixing already deﬁned in Eq. (14) and 4α2ϵ 4α2 vEW sin θ2 ≃ = (18) α3 α3 vR is the mixing between H3 and H1. Both θ1 and θ2 are expected to be small for a GeV-scale H3. The “effective” mixing angles of H3 responsible for the ﬂavor conserving and violating couplings to the SM quarks and charged leptons can be deﬁned as ˜ sin θ1 ≡ sin θ1 + ξ sin θ2 , (19) sin θ˜2 ≡ sin θ2 + ξ sin θ1 , (20) which will be used in our subsequent discussion. For completeness we collect all the coupling of light H3 in the minimal LR models in Table 1, which is based on the calculation of Ref. [11] and up to the leading order in the small parameters

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P.S.B. Dev et al. / Nuclear Physics B 923 (2017) 179–221 187 ξ and ϵ deﬁned in Eq. (7). Here the RH gauge mixing φ is deﬁned as tanφ ≡ gBL/gR, and the W–WR and Z–ZR mixings are respectively given by ( ) 2 2gRξ mW tan ζW ≃ − , (21) gL mW R [ ( ) ] 1/2 ( ) 2 2 2 g R gR 2 mZ tan ζZ ≃ − 1 + sin θw , (22) 2 2 g L gL mZR where θw is the weak mixing angle. In Table 1, the couplings to the charged leptons depend on the neutrino sector via the Dirac coupling matrix YνN = mD/vEW, which can be parameterized through the Casas–Ibarra form [29] 1/2 1/2 mD = im N Omν (23) with O an arbitrary complex orthogonal matrix, mν and mN are the light neutrino and RHN mass matrix, respectively. Without ﬁne-tuning, the Dirac Yukawa couplings YνN , and hence, the light–heavy neutrino mixing, are expected to be small for TeV-scale RH neutrinos. So we can safely ignore those couplings involving higher powers of YνN , such as H3νν. 3.4. Decay lifetime and branching ratios Through the mixing with the SM Higgs and the heavy scalar H1 via respectively the mixing angles sin θ1 and sin θ2 in Eq. (14) and (18), a light H3 could decay at tree level into the SM 1 fermions, through the Yukawa couplings in Table 1. Note that the couplings to quarks could be ˜ ﬂavor-changing, depending on the magnitude of effective mixing angle sinθ2, while the lepton ˜ ﬂavor violating (LFV) couplings are proportional to the Dirac coupling YνN and sin θ2. At the one-loop level, the Yukawa couplings to the SM fermions induce the decay into digluon and diphoton, i.e. H3 → gg, γ γ , analogous to the SM Higgs. Considering the ﬂavor limits on the mixing angles sin θ1,2 below in Section 5, the fermion loops for both the γ γ and gg channels are highly suppressed. However, for the decay H3 → γ γ , there are extra contributions from ± ± ±± the heavy W R , H1 and H2 loops. In the low mass limit mH3 ≪ vR, the diphoton channel is −2 sensitive only to the RH scale Ŵγγ ∝ v R [32] and dominated by the WR loop for TeV range vR, as the scalar loops are comparatively suppressed by the loop function 5A0(0)/A1(0) = −5/21 [cf. Eq. (A.3)]. Similarly, the SM W loop is highly suppressed by the small W–WR mixing sin ζW [cf. Eq. (21)]. The dominant two-body tree- and loop-level decay channels of H3 covering all the parameter space of interest for a light scalar mH 3 ≲ mh are presented in Appendix A, while the ∗ ∗ other decay channels, such as H3 → h h → bbbb, are suppressed either kinematically or by the multi-particle phase space. + − The decay branching ratios (BR) of H3 to the qq, ℓ ℓ , γ γ and gg channels are presented in Fig. 3 as functions of its mass and mixing with h (top panel) and H1 (bottom panel). For 1 We assume here that all the three RHNs (typically at the TeV scale) are heavier than the light scalar H3, and there- fore, the H3 decay into RHNs is kinematically forbidden. Allowing for the H3 → NN decay could lead to additional lepton number violating signatures, as discussed in Refs. [30,31]. In addition, H3 could in principle decays into the light −1 neutrinos (or one light neutrino plus one heavy neutrino) through the heavy–light neutrino mixing VνN ≃ mDM N . However, for the TeV-scale type-I seesaw without any ﬁne-tuning in the seesaw mass matrix, the mixing VνN turns out −6 4 to be ≲ 10 [cf. Eq. (23)]. So the width Ŵ(H3 → νν) ∝ V νN is completely negligible.

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188 P.S.B. Dev et al. / Nuclear Physics B 923 (2017) 179–221 Fig. 3. Color-coded branching ratios of H3 as functions of its mass and mixing with h (top) and H1 (bottom) for different decay modes. Here we have set the RH scale vR = 5 TeV and the RHN masses at 1 TeV. concreteness, we have made the following reasonable assumptions: (i) The RH scale vR = 5 TeV, which is close to the smallest value allowed by the current constraints on WR. (ii) In the minimal LR model, the RH quark mixing matrix VR is very similar to the CKM matrix VL, up to some additional phases [33,34]. For simplicity, we adopt VR = VL in the calculation. Thus with the experimental values of the SM quark masses and CKM mixing, we obtain the numerical values of the H3 couplings to the SM quarks, including the FCNC couplings, arising from its mixing with the heavy scalar H1 [cf. Table 1]. (iii) The couplings of H3 to the charged leptons depend on the heavy and light neutrino masses and their mixings via the Yukawa coupling matrix YνN.

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