Electroweak Breaking and Higgs Boson Profile in the Simplest Linear Seesaw Model
PPrepared for submission to JHEP
Electroweak Breaking and Higgs Boson Profile in theSimplest Linear Seesaw Model
Duarte Fontes, a Jorge C. Rom˜ao, a and J. W. F. Valle b a Departamento de F´ısica and CFTP, Instituto Superior T´ecnico, Universidade de Lisboa,Avenida Rovisco Pais 1, 1049-001 Lisboa, Portugal b AHEP Group, Institut de F´ısica Corpuscular – C.S.I.C./Universitat de Val`encia, Parc Cient´ıficde Paterna. C/ Catedr´atico Jos´e Beltr´an, 2 E-46980 Paterna (Valencia) - SPAIN
E-mail: [email protected] , [email protected] , [email protected] Abstract:
We examine the simplest realization of the linear seesaw mechanism withinthe Standard Model gauge structure. Besides the standard scalar doublet, there are twolepton-number-carrying scalars, a nearly inert SU(2) L doublet and a singlet. Neutrinomasses result from the spontaneous violation of lepton number, implying the existence ofa Nambu-Goldstone boson. Such “majoron” would be copiously produced in stars, leadingto stringent astrophysical constraints. We study the profile of the Higgs bosons in thismodel, including their effective couplings to the vector bosons and their invisible decaybranching ratios. A consistent electroweak symmetry breaking pattern emerges with acompressed spectrum of scalars in which the “Standard Model” Higgs boson can have asizeable invisible decay into the invisible majorons. a r X i v : . [ h e p - ph ] O c t ontents S, T, U P1 : BR inv ( h ) > .
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Unitarity Constraints 22
A.1 Q = 2 , Y = 2 23A.2 Q = 1 , Y = 2 23A.3 Q = 1 , Y = 1 23A.4 Q = 1 , Y = 0 23A.5 Q = 0 , Y = 2 24A.6 Q = 0 , Y = 1 24A.7 Q = 0 , Y = 0 24 Non-zero neutrino masses constitute one of the most robust evidences for new physics.Ever since the discovery [1, 2] and confirmation [3, 4] of neutrino oscillations took place,the efforts to underpin the origin of neutrino mass have been fierce. Yet the basic dynamicalunderstanding of the smallness of neutrino mass remains as elusive as ever. We have noclue as to what is the nature of the underlying mechanism and its characteristic energyscale. A popular approach to neutrino mass generation is the type-I seesaw mechanism,in which neutrinos get mass due to the exchange of heavy singlet mediators. FollowingRefs. [5–7] we assume here that the seesaw mechanism is realized using just the StandardModel (SM) gauge structure associated to the SU(3) c ⊗ SU(2) L ⊗ U(1) Y symmetry.In its standard high-scale realization, the seesaw mechanism hardly leads to any phe-nomenological implication besides those associated to the neutrino masses themselves.However, the seesaw can arise from low-scale physics [8, 9]. For example, the seesawmechanism can be realized at low scale in two different pathways, the inverse [10, 11]and the linear seesaw [12–14]. These low-scale seesaw schemes require the addition of asequential pair of isosinglet leptons, instead of just a single right-handed neutrino addedsequentially.In this work, we examine the simplest variant of the linear seesaw mechanism. Incontrast to the conventional formulations [12–14], here left-right symmetry is not imposed.In our setup the linear seesaw mechanism is realized in terms of the simplest StandardModel SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge structure, in which lepton number symmetry isungauged. Spontaneous breaking of the lepton number symmetry hence implies the exis-tence of a Nambu-Golstone boson, a variant of the so-called majoron [6, 7]. Such minimallyextended scalar boson sector contains, in addition to the Standard Model Higgs doublet,a second Higgs doublet, as well as a complex singlet scalar, both carrying lepton numbercharges. The singlet is required to ensure consistency with the LEP measurement of the– 1 –nvisible Z decay width. To prevent excessive stellar cooling by majoron emission, theirvacuum expectation values (vevs) must obey a stringent astrophysical bound on the vevof the second doublet. This nicely fits the generation of neutrino masses by the linearseesaw mechanism. Our “neutrino-motivated” singlet extension of the two-doublet Higgssector also leads to a peculiar benchmark for electroweak (EW) breaking studies at colliderexperiments.We perform a numerical study of the Higgs sector taking into account consistency withastrophysical bound, EW precision data as well as perturbative unitarity and vacuum sta-bility. These imply that the model nearly realizes the structure of the inert Higgs doubletmodel [15, 16]. Moreover, it also has an impact on the physics of the 125 GeV StandardModel Higgs boson discovered at the LHC. Indeed, the profile of the Higgs sector is modi-fied by the mixing of new CP-even states that affect its couplings and the presence of newCP-odd scalars. For example, it implies the existence of a new invisible Higgs decay chan-nel with majoron emission [8]. This has phenomenomenological implications for colliderexperiments [17–27] and has indeed been searched by LEP and LHC collaborations [28, 29].The new signals can be studied in proton proton collisions such as at the High-LuminosityLHC setup, as well as in the next generation of lepton collider experiments such as CEPC,FCC-ee, ILC and CLIC [30–33]. Moreover, the majoron can, in certain circumstances, playthe role of Dark Matter [34–41], in addition to having other potential astrophysical andcosmological implications [42, 43].The paper is organised as follows. In Sec. 2 we describe the basic theory setup of themodel, while the main features of the Higgs potential and EW breaking sector are describedin Sec. 3. The theoretical and experimental constraints are described in Sections 4 and 5,respectively. The physical profile of the Higgs boson spectra resulting from our numericalscans are presented in Sec. 6 while results for the invisible Higgs decay branching ratio aregiven in Sec. 7. Finally, a summary is presented in Sec.8. The Yukawa sector contains, besides the three Standard Model lepton doublets L i = (cid:34) ν i l i (cid:35) , (2.1)with lepton number 1, three lepton singlets ν ci with lepton number − ψ i with lepton number 1. The resulting Yukawa Lagrangian is given as − L Yuk = h ij L Ti Cν cj Φ + M ij ν ci Cψ j + f ij L Ti Cψ j χ L + h.c. (2.2)where h ij and f ij are dimensionless Yukawa couplings, M ij is an arbitrary matrix withdimensions of mass, and Φ and χ L are scalar doublets.– 2 –fter symmetry breaking it will give the linear seesaw mass matrix, M ν = M D M L M TD MM TL M T , (2.3)where the × sub-matrices are given as M D = 1 √ v φ h, ( × ) , M L = 1 √ v L f, ( × ) , M = M ( × ) , (2.4)where v φ and v L represent the vevs of Φ and χ L , respectively. Here, the lepton number isbroken by the M L νS term. This leads to the effective light neutrino mass matrix given by M ν = M D ( M L M − ) T + ( M L M − ) M DT . (2.5)This matrix scales linearly with respect to the Dirac Yukawa couplings contained in M D ,hence giving name to this seesaw mechanism. It is clear that this vanishes as M L → c ⊗ SU(2) L ⊗ U(1) Y gauge structure, having the existence of the majoron as its characteristic feature. In thenext section we analyse the dynamical origin of the M L νS term from a scalar doublet Higgsvacuum expectation value, whose smallness is required by astrophysics and consistent withminimization of the potential.The presence of the heavy TeV-scale neutrinos needed to mediate neutrino massthrough the linear seesaw leads to a plethora of phenomenological signatures associatedto the heavy neutrinos. These include their signatures such as lepton flavor violation pro-cesses in high energy collisions [44, 45], as well as lepton flavor violation processes at lowenergies, such as µ → eγ [46, 47]. Many other aspects of this theory have also been dis-cussed in similar contexts, such as those associated with violation of unitarity of the leptonmixing matrix [48–51] and its possible impact upon neutrino oscillation experiments [52].In this paper we focus primarily on the profile of the Higgs sector. We consider two doublets, Φ, χ L and a singlet σ ,Φ = φ +1 √ ( v φ + R + i I ) , χ L = χ + L √ ( v L + R + i I ) , σ = 1 √ v σ + R + i I ) , (3.1)– 3 –nd choose the following lepton number assignments for the Higgs fields, L [Φ] = 0 , L [ χ L ] = − , L [ σ ] = 1 . (3.2)With these quantum number the most general Higgs potential that we can write thatrespects all the symmetries is V Higgs = − µ Φ † Φ + λ (cid:16) Φ † Φ (cid:17) − µ L χ † L χ L + λ L (cid:16) χ † L χ L (cid:17) − µ σ σ † σ + λ σ (cid:16) σ † σ (cid:17) + β Φ † Φ χ † L χ L + β Φ † Φ σ † σ + β χ † L χ L σ † σ + β Φ † χ L χ † L Φ − β (cid:0) Φ χ L σ + h.c. (cid:1) . (3.3)For definiteness we assume all couplings to be real. First we solve the minimization equations for the mass parameters µ, µ L , µ σ in the potential.We get µ = β v φ v L + β v φ v σ − β v L v σ + 2 λv φ + β v L v φ v φ ,µ L = β v φ v L + β v L v σ − β v φ v σ + 2 λ L v L + β v L v φ v L ,µ σ = β v φ v σ + β v L v σ − β v φ v L v σ + 2 λ S v σ v σ . (3.4) Substituting the minimization conditions we obtain the charged scalar mass matrix in thebasis ( φ + , χ + L ) as M = β v L v σ − β v L v φ v β v L v φ − β v σ β v L v φ − β v σ β v φ v σ − β v L v φ v L , (3.5)which we can easily see that has a zero eigenvalue corresponding to the charged Goldstoneboson, G + = 1 (cid:113) v φ + v L v φ v L = (cid:34) cos β sin β (cid:35) , (3.6)where we have defined, as usual, v = (cid:113) v φ + v L = 246GeV , tan β = v L v φ . (3.7)The physical charged Higgs has a mass given by m H + = ( β v σ − β v L v φ ) (cid:16) v φ + v L (cid:17) v φ v L = β v σ sin 2 β − β v . (3.8)– 4 – .3 Neutral Scalar Matrix The neutral scalar mass matrix is given by M = λv φ + β v L v σ v φ β v φ v L − β v σ + β v L v φ ( β v φ − β v L ) v σ β v φ v L − β v σ + β v L v φ λ L v L + β v φ v σ v L β v L v σ − β v φ v σ ( β v − β v L ) v σ β v L v σ − β v φ v σ λ σ v σ , (3.9)and we can check that has non-zero determinant, so there are three massive CP-evenscalars. The neutral pseudo-scalar mass matrix is given by M = β v L v σ v φ − β v σ − β v L v σ − β v σ β v φ v σ v L β v φ v σ − β v L v σ β v φ v σ β v φ v L . (3.10)It is easy to verify that it has zero determinant and two zero eigenvalues; their eigenvectorsare given by G = 1 (cid:113) v φ + v L vv L = v φ vv L v = cos β sin β , (3.11) J = v φ v L (cid:114)(cid:16) v φ + v L (cid:17) (cid:16) v φ (cid:0) v L + v σ (cid:1) + v L v σ (cid:17) v L v φ − v σ ( v φ + v L ) v φ v L = v φ v L vV v L − v φv σ v v φ v L = sin 2 β (cid:112) tan β (cid:48) + sin β sin β − cos β tan β (cid:48) sin 2 β , (3.12)where we have defined v = (cid:113) v φ + v L , V = (cid:113) v φ (cid:0) v L + v σ (cid:1) + v L v σ , tan β = v L v φ , tan β (cid:48) = v σ v . (3.13)The remaining pseudo-scalar is defined as A , and its squared mass is given by M A = β (cid:16) v φ v L + v φ v σ + v L v σ (cid:17) v φ v L = β (cid:20) v sin 2 β + v σ sin 2 β (cid:21) . (3.14)– 5 – .5 Parameters of the Lagrangian It is useful to write the parameters of the Lagrangian in terms of the physical masses, vevsand the relevant angles of rotation. We have already shown in Eq. 3.4 that the quadraticterms of the potential can be written in terms of the vevs and the other parameters. Thishas already been done in writing the mass squared matrices.
It is convenient to write the rotation matrix that connects the weak eigenstates to the masseigenstates. We get G JA = O I I I I , (3.15)with diag(0 , , m A ) = O I · M · O I T . (3.16)The matrix O I is given by O I = cos β sin β sin β sin(2 β ) √ sin (2 β )+tan ( β (cid:48) ) − β cos β √ sin (2 β )+tan ( β (cid:48) ) tan( β (cid:48) ) √ sin (2 β )+tan ( β (cid:48) ) − sin β tan( β (cid:48) ) √ sin (2 β )+tan ( β (cid:48) ) cos β tan( β (cid:48) ) √ sin (2 β )+tan ( β (cid:48) ) sin(2 β ) √ sin (2 β )+tan ( β (cid:48) ) . (3.17)Notice that the only parameters involved are again the vevs of the scalar multiplets in theHiggs potential. We also can invert Eq. (3.14) to obtain the parameter β as follows β = M A sin 2 βv (cid:0) sin β + tan β (cid:48) (cid:1) . (3.18) The physical mass of the charged scalar m H + and pseudo-scalar mass M A are relatedthrough β = 2 M A tan β (cid:48) v (cid:0) tan β (cid:48) + sin β (cid:1) − m H + v , (3.19)involving, again, the three vevs of the theory. The diagonalization of the neutral scalar mass matrix will give us six relations that canbe solved to get the parameters λ, λ L , λ σ and β , β , β describing the quartic couplings interms the physical masses, vevs and rotation angles. We define the scalar mass eigenstates h , h and h as h h h = O R R R R , (3.20)– 6 –uch that O R · M · O RT = diag( M , M , M ) , (3.21)where M , M and M are the squared masses of h , h and h , respectively. The matrix O R can be parameterized in terms of the angles θ i as O R = O R · O R · O R = c c s c s − c s s − s c c c − s s s c s − c s c + s s − c s − s s c c c , (3.22)where O R = c s − s c
00 0 1 , O R = c s − s c , O R = c s − s c , (3.23)and c i = cos α i , s i = sin α i . Using this parameterization one obtains, from Eq. (3.21), thefollowing relations for the quartic parameters: λ = − v φ (cid:20) β c v L v σ + 2 c (cid:0) β c s v L v σ − c (cid:0) M v φ − β s s v L v σ (cid:1) + s (cid:0) β c s s v L v σ + c (cid:0) β s s s v L v σ − M v φ (cid:1) + β s s s v L v σ − M s v φ (cid:1)(cid:1) + s (cid:0) β c s v L v σ + c (cid:0) β s s v L v σ − M v φ (cid:1) + β s s v L v σ − M s v φ (cid:1) − c c s s s v φ ( M − M ) (cid:21) , (3.24) λ L = − v L (cid:20) β c v φ v σ + 2 c (cid:0) β c s v φ v σ + c (cid:0) β s s v φ v σ − M v L (cid:1) + β s s v φ v σ − M s v L (cid:1) + s (cid:0) β c s v φ v σ − c (cid:0) M v L − β s s v φ v σ (cid:1) + s (cid:0) β c s s v φ v σ + c (cid:0) β s s s v φ v σ − M v L (cid:1) + β s s s v φ v σ − M s v L (cid:1)(cid:1) +4 c c s s s v L ( M − M ) (cid:21) , (3.25) λ σ = 12 v σ (cid:20) c (cid:0) c M + M s (cid:1) + M s (cid:21) , (3.26) β = − v L v φ (cid:20) c (cid:0) β v L v φ − β v σ (cid:1) + 2 c (cid:0) c s (cid:0) β v L v φ − β v σ (cid:1) +2 c s s (cid:0) β v L v φ − β v σ (cid:1) + s s (cid:0) β v L v φ − β v σ (cid:1) + c s s ( M − M ) (cid:1) + s (cid:0) c s (cid:0) β v L v φ − β v σ (cid:1) + 2 c s s (cid:0) β v L v φ − β v σ (cid:1) + s s (cid:0) β v L v φ − β v σ (cid:1) +2 c s s ( M − M ) (cid:1) + 2 c s (cid:0) c (cid:0) c M + M s (cid:1) − c (cid:0) c M + c (cid:0) M s − M s (cid:1) + M s − M s s (cid:1) + s (cid:0) c (cid:0) M s − M (cid:1) – 7 – s (cid:0) M s − M (cid:1)(cid:1)(cid:1) (cid:21) , (3.27) β = 1 v φ v σ (cid:20) β c v L v σ + s (cid:0) β c s v L v σ + β s s v L v σ + c c s ( M − M ) (cid:1) + c c s (cid:0) c M − c (cid:0) M − M s (cid:1) + M s − M s (cid:1) (cid:21) , (3.28) β = 1 v L v σ (cid:20) β c v φ v σ + 2 β c s v φ v σ + β s v φ v σ + c c c s ( M − M )+ c s (cid:0) c c s s ( M − M ) + c M s − c s (cid:0) M − M s (cid:1) + M s s − M s s (cid:1) (cid:21) . (3.29) In this section we study the theoretical constraints that must be applied to the modelparameters in order to ensure consistency of the electroweak symmetry breaking sector.
In order to look at the stability or bounded from below (BFB) conditions, we start byconsidering only the neutral vacuum. Defining x , y and z such that:Φ = √ xe iθ , χ L = √ ye iθ , σ = √ ze iθ , (4.1)we can write the quartic terms of the potential 3.3 as V q = V + V , (4.2)where V = λx + λ L y + λ σ z + 2 αxz + 2 βyz + 2 γxy, (4.3)with α = β , β = β , γ = β , (4.4)and V = 2 | β |√ x √ yz cos( δ ) , (4.5)where δ is some combination of phases. For the potential of the form V , the conditionsfor stability have been given in Ref.[53]. The problem is the extra piece V . However, wecan always say that V > V a = − | β |√ x √ yz. (4.6)Now, note that for any positive x, y , we always have − √ x √ y > − x − y . (4.7)– 8 –herefore, we can bound our potential in the following way: V > V a > V b = − | β | xz − | β | yz, (4.8)which can be joined into V to give V > ˆ V = λx + λ L y + λ σ z + 2 α (cid:48) xz + 2 β (cid:48) yz + 2 γxy, (4.9)with α (cid:48) = α − | β | , β (cid:48) = β − | β | . (4.10)Now, from Ref.[53] we get the conditions for the potential to be BFB as follows, (cid:110) λ > , λ L > , λ σ > α (cid:48) > − (cid:112) λ σ λ ; β (cid:48) > − (cid:112) λ σ λ L ; γ > − (cid:112) λλ L ; α (cid:48) ≥ − β (cid:48) (cid:112) λ/λ L (cid:111) ∪ (cid:110) λ > , λ L > , λ σ > (cid:112) λ σ λ L ≥ β (cid:48) > − (cid:112) λ σ λ L ; − β (cid:48) (cid:112) λ/λ L ≥ α (cid:48) > − (cid:112) λ σ λ ; λ σ γ > α (cid:48) β (cid:48) − (cid:112) ∆ α ∆ β (cid:111) , (4.11)where ∆ α = α (cid:48) − λ σ λ, ∆ β = β (cid:48) − λ σ λ L . (4.12)These conditions are sufficient, although they might be more restrictive than the necessaryand sufficient conditions, because of the method of bounding the potential we have used. In order to discuss the unitarity constraints, we follow the procedure developed in Ref.[54].As explained there, we have to obtain all the coupled channel matrices for the scattering oftwo scalars into two scalars, and bound the highest of their eigenvalues. Since the electriccharge and the hypercharge are conserved in this high energy scattering, we can separatethe states according to these quantum numbers. For this purpose, and because we are inthe very high-energy limit, it is better to work in the unbroken phase. It is convenient thento use the following notation for the Higgs fields.Φ = (cid:34) w +1 n (cid:35) , Φ † = (cid:34) w − n ∗ (cid:35) T ; χ L = (cid:34) w +2 n (cid:35) , χ † L = (cid:34) w − n ∗ (cid:35) T ; σ = s , σ ∗ = s ∗ . (4.13)The relevant two body states are given in the entries of Table 1, and their complex conju-gates. It is important to note that the index α is a compound index; it refers to a set of { i, j } indices for the two body states. Also note that in Table 1 the two body states withequal particles have a normalization of 1 / √ Y State Number of states2 2 S ++ α = { w +1 w +1 , w +1 w +2 , w +2 w +2 }
31 2 S + α = { w +1 n , w +1 n , w +2 n , w +2 n }
41 1 T + α = { w +1 s, w +1 s ∗ , w +2 s, w +2 s ∗ }
41 0 U + α = { w +1 n ∗ , w +1 n ∗ , w +2 n ∗ , w +2 n ∗ }
40 2 S α = { n n , n n , n n }
30 1 T α = { n s, n s ∗ , n s, n s ∗ }
40 0 U α = { w +1 w − , w +1 w − , w +2 w − , w +2 w − , n n ∗ , n n ∗ , n n ∗ , n n ∗ , s ∗ s, ss, s ∗ s ∗ } Table 1 : List of two body scalar states separated by (
Q, Y ). We will give the full results in the Appendix A, but let us illustrate with the simplestexample, the state S ++ α . With the notation of Eq. (4.13), the quartic part of the potantialwill read V = λw +1 w − w +1 w − + λ L w +2 w − w +2 w − + β w +1 w − w +2 w − + β w +1 w − w +2 w − + · · · (4.14)Now consider the scattering w +1 w +2 → w +1 w +2 (4.15)This will proceed through the quartic vertex in Fig. 1. Therefore the amplitude M ( w +1 w +2 → replacements w +1 w +1 w +2 w +2 i ( β + β ) Figure 1 : Quartic coupling for w +1 w +2 → w +1 w +2 . w +1 w +2 ) will be given by β + β . Now consider the scattering w +1 w +1 → w +1 w +1 (4.16)The Feynman rule for the quartic term would be 4 λ , but remembering the factor of 1 / √ M ( w +1 w +1 √ → w +1 w +1 √ λ √ √ λ (4.17)– 10 –nd similarly for w +2 w +2 → w +2 w +2 . Therefore for the coupled channel states in S ++ α we get,16 πa ++0 = M ++ = λ β + β
00 0 2 λ L , (4.18)where | a | < /
2, is the partial wave to be bounded, requiring the eigenvalues of Eq. (4.18)to obey, Λ i < π . (4.19)In the Appendix A we present all the coupled channel matrices for the sates in Table 1and give their eigenvalues. Then the limits implied in Eq. (4.19) were aplied in the code. S, T, U
In order to discuss the effect of the oblique
S, T, U parameters, we use the results ofRef.[55, 56]. To apply their expressions we have to find the matrices U and V that we nowdefine explicitly. For the matrix U , we have (cid:34) φ + χ + L (cid:35) = U (cid:34) G + H + (cid:35) , (4.20)which gives U = (cid:34) cos β − sin β sin β cos β (cid:35) . (4.21)The matrix V is a 2 × (cid:34) R + iI R + iI (cid:35) = V G JA h h h . (4.22)Using the rotation matrices O R and O I we get V = (cid:34) i O I , i O I , i O I , O R , O R , O R i O I , i O I , i O I , O R , O R , O R (cid:35) . (4.23)To apply the expressions for S, T, U , we need the following matrices: U † U = (cid:34) (cid:35) , (4.24)– 11 – ( V † V ) = (cid:34) × A × − A × × (cid:35) , (4.25)where A × = −O I O R − O I O R −O I O R − O I O R −O I O R − O I O R −O I O R − O I O R −O I O R − O I O R −O I O R − O I O R −O I O R − O I O R −O I O R − O I O R −O I O R − O I O R , (4.26) U † V = i O R c β + O R s β O R c β + O R s β O R c β + O R s β − i sin 2 β √ sin (2 β )+tan β (cid:48) i tan β (cid:48) √ sin (2 β )+tan β (cid:48) O R c β − O R s β O R c β − O R s β O R c β − O R s β , (4.27)with c β = cos β, s β = sin β . We also need the diagonal elements of V † V :Diag( V † V ) = (cid:104) , sin 2 β (sin β +tan β (cid:48) ) , tan β (cid:48) (sin β +tan β (cid:48) ) , O R + O R , O R + O R , O R + O R (cid:105) . (4.28)We have implemented a numerical code to take all of the above constraints into account. In this section we study the constraints that must be applied to the scalar potential pa-rameters and which follow from various experimental considerations.
Spontaneous breaking of a global symmetry such as lepton number leads to the existenceof a Nambu-Goldstone boson, dubbed “majoron”. This would be copiously produced instars, leading to new mechanisms of stellar cooling. If the majoron is strictly massless (orlighter than typical stellar temperatures), one has an upper bound for the majoron-electroncoupling [57, 58] | g Jee | < ∼ − . (5.1)where | g Jee | = | (cid:104) J | φ (cid:105) | m e v φ , (5.2)with (cid:104) J | φ (cid:105) denoting the majoron projection into the Standard Model doublet. This can beobtained in a model-independent way by using Noether’s theorem [7] and checked explicitlyfrom the form of the pseudo-scalar mass matrix, Eq. (3.12). In our case, it leads to theconstraint | (cid:104) J | φ (cid:105) | = 2 v φ v L (cid:113) ( v φ + v L )( v φ (4 v L + v σ ) + v L v σ (cid:46) − . (5.3)Note, however, that the majoron can, in certain circumstances, acquire a nonzeromass as a result of interactions explicitly breaking the global lepton number symmetry.– 12 –hese could arise, say, from quantum gravity effects. Unfortunately, we have no way ofproviding a reliable estimate of their magnitude. If the majoron is massive, it may playthe role of warm [34–39] or cold Dark Matter [40, 41], in addition to having other potentialastrophysical and cosmological implications [42, 43]. If the majoron mass exceeds thecharacteristic temperatures of stellar environments, then the bound in Eq. (5.3) need notapply. While the missing energy signature associated to the light majoron would remain,there would be important changes in the phenomenological analysis of the scalar sector.In what follows, we stick to the validity of Eq. (5.3), which amounts to having a (nearly)massless majoron. We have to enforce the LHC constraints on the 125 GeV scalar Higgs boson. These aregiven in terms of the so-called signal strength parameters, µ f = σ NP ( pp → h ) σ SM ( pp → h ) BR NP ( h → f ) BR SM ( h → f ) , (5.4)where σ ( pp → h ) is the cross section for Higgs production, and BR ( h → f ) is the branchingratio into the Standard Model final state f , with the labels NP and SM denoting NewPhysics and Standard Model, respectively. These can be compared with those given bythe experimental collaborations. For the 8 TeV data, the signal strengths from a combinedATLAS and CMS analysis [59] are shown in Table 2. For the 13 TeV of Run-2, the datachannel ATLAS CMS ATLAS+CMS µ γγ . +0 . − . . +0 . − . . +0 . − . µ W W . +0 . − . . +0 . − . . +0 . − . µ ZZ . +0 . − . . +0 . − . . +0 . − . µ ττ . +0 . − . . +0 . − . . +0 . − . Table 2 : Combined ATLAS and CMS results for the 8 TeV data, Ref. [59]. is separated by production process. We took the recent results from ATLAS [60] shownin Table 3. Finally, we have also enforced the LHC constraints in the other neutral andcharged Higgs. This was done using the
HiggBounds-4 package [61].In practice, in order to optimize our scans, we started by imposing just the simplerequirement that the coupling of the 125 GeV Higgs scalar boson with the vector bosons, k V ( h ), lies in the range k V ( h ) ∈ [0 . , . (5.5)– 13 –ecay Production ProcessMode ggF VBF ZH ttH H → W W . +0 . − . . +0 . − . . +0 . − . . +0 . − . H → ZZ . +0 . − . . +0 . − . . +0 . − . . +0 . − . H → τ τ . +0 . − . . +0 . − . . +0 . − . . +0 . − . H → γγ . +0 . − . . +0 . − . . +0 . − . . +0 . − . H → bb . +0 . − . . +0 . − . . +0 . − . . +0 . − . Table 3 : ATLAS results for the 13 TeV data, Ref. [60].
Applying this first restriction is useful in order to optimize the size of our data pointssample. Notice that our model has the same structure of the Higgs coupling to the vectorbosons as any two Higgs doublet model. This means we can, for instance, use the resultsof Ref. [62] to obtain k V ( h i ) = O Ri c β + O Ri s β . (5.6)As in the case of any multi-doublet Higgs model, NHDM [54], the couplings of the CP evenHiggs bosons to the vector bosons obey a sum rule, (cid:88) i =1 | k V ( h i ) | = (cid:88) i =1 (cid:2) O Ri c β + O Ri s β (cid:3) = 1 (5.7)where we have used the properties of the orthogonal matrix O R . We have started byenforcing Eq. (5.5) in our scans and then, to this optimized data set, we applied theconstraints on Eq. (5.4) using the results from Table 2 and Table 3. In this section we present a study of the impact of the previous constraints on the parameterspace of the scalar potential in the linear seesaw model. In all plots we have imposed thetheoretical constraints plus the LHC constraints on k V . We fix the h mass to 125 GeVand vary other the model parameters in the following way: M =125 GeV , M , M , M A , M H + ∈ [125 , , α , α , α ∈ [ − π , π For technical reasons one can not put m J = 0, since there appear logarithms of mass ratios in theevaluation of the S,T,U parameters. However, the limit m J → m J = 1 eV, keeping the stellar cooling argument. – 14 – L ∈ [10 − , ] GeV , v σ ∈ [10 , . × ] GeV . (6.1)This first general scan is useful to see the impact of the stellar cooling constraint, Eq. (5.3),on the parameter space. In fact, to comply more easily with this constraint we have sampled v σ values above 1 TeV, aware that lower values could be possible. The resulting allowedregion in the ( v σ , v L ) plane is shown in Fig. 2. The yellow region is the set of values of( v σ , v L ) that satisfy Eq. (5.3). From here we immediately see that the allowed range of v L is much smaller than what we start with. This requires a stringent restriction on v L , i.e. v L < . W mass, M W = 12 g (cid:113) v L + v φ , (6.2)which explains the boundary in the plane ( v σ , v L ). In Fig. 2 we also plot the allowed pointsonce all the constraints on the model are implemented. The color code is as follows: thepoints in dark green have v L > . v L ∈ [0 . , v L < .
01 GeV. We have imposed a lower bound on the value of v σ which is visible in the figure. Figure 2 : v σ as a function of v L . The yellow region is the set of values of ( v σ , v L ) that satisfyEq. (5.3) without any other constraint. The other points, in shades of green, satisfy all the othertheoretical and experimental constraints. For these, the color code is as follows: the points in darkgreen have v L > . v L ∈ [0 . , v L < .
01 GeV. In our scans a cut v σ > – 15 – .2 Constrained Scan Using the above result, we have performed a constrained scan where all points satisfythe astrophysical constraint, Eq. (5.3). We have also required that the points satisfy theLHC constraints on the signal strengths given in Table 2 and Table 3, at the 3 σ level. Asexplained above, we have separated all the points in three bins, according to the value ofthe vev v L as follows: v L > . ,v L ∈ [0 . , . , (6.3) v L < . . We have further restricted the scan to scalar masses values below 600 GeV, which might beexplored in the next generation of collider experiments. How the allowed model parametersare constrained is shown in Fig. 3 and Fig. 4. The mixing angles in the neutral scalarrotation matrix are shown in Fig. 3. Their values are restricted by the LHC constraint
Figure 3 : α and α versus α . The color code is as in Fig. 2. See text for the explanation. on the signal strengths, which we imposed at the 3 σ level. The color code is defined inEq. (6.3). We see from Eq. (3.25) that smaller values of v L require the the numeratorof that equation to be small, otherwise perturbative unitarity on λ L would be violated.One can verify that the smallness of the numerator occurs for α close to zero and for acompressed spectrum, as we will discuss below. We should also note that, as the allowedvalues of v L are quite small, β in Eq. 3.13, is a very small angle, as shown in Fig. 4; here,we show in the left panel the relation of β with v L , and in the right panel the correlationbetween k V ( h ) and α . The color code is the same. We see that the lower limit on k V ( h ),set in Eq. (5.5), corresponds indeed to the signal strengths at 3 σ , as we have points for allvalues of k V ( h ) in that range. If we enforced the signal strength constraints at 2 σ , therange would be reduced as we will show below.– 16 – igure 4 : Left panel: Correlation between β and v L . Right panel: Correlation between k V ( h )and α . The color code is as in Fig.. 2. See text for the explanation. We have found that, in order to ensure very small values of the lepton number breakingvev v L , the spectrum of Higgs bosons tends to be compressed. The reason can be tracedto the perturbative unitarity constraints on the quartic coulings, in particular those on λ L .This can be easily understood by considering Eq. (3.25). As λ L is inversely proportional tothe third power of v L , for small values of v L the numerator must be very small, to prevent λ L violating the perturbative unitarity constraints. It turns out that this is achieved bya compressed spectrum and a value of the mixing angle α close to zero. In fact, one canshow that, for v L (cid:28) v φ , the numerator of Eq. (3.25) vanishes for α = α = 0 and forequal masses. We did not impose these last conditions on our scan, it just turned out thethat the good points have this profile. This is shown in the plots in Fig. 5. Notice that Figure 5 : Correlation between M A /M (left), M H + /M (middle), M /M A (right) and v L . Thecolor code as in Fig. 2. See text for explanation. this compression is much stronger than that usually imposed by the oblique parameters.One can see that, for larger values of v L , the points satisfying all the constraints, includingthose from the oblique parameters, can have a sizeable splitting.– 17 – Invisible Higgs decays at the LHC
It has long been noticed that theories of neutrino mass where spontaneous violation ofungauged lepton number symmetry takes place at collider-accessible scales all lead to thephenomenon of invisibling Higgs decay bosons [8]. Collider implications have been widelydiscussed in the literature in various theory contexts and collider setups [17–27].In our model, the new decay channels of the CP-even scalars that contribute to theinvisible decays are just h i → J J and h i → h j (when M i > M j ). The latter alsocontributes as h i → h j → J . The Higgs-majoron coupling is given by (no summation on a ), g h a JJ = − (cid:18) ( O I ) v φ O Ra + ( O I ) v L O Ra + ( O I ) v σ O Ra (cid:19) M a , (7.1)where O Iij are the elements of the rotation matrix in Eq. (3.17), and the decay width isgiven by Γ( h a → J J ) = 132 π g h a JJ M a . (7.2)Following our conventions, the trilinear coupling h h h are giving by: g h h h = − (cid:18) λ ( O R ) O R v φ + β ( O R ) O R v φ + β ( O R ) O R v φ + 2 β O R O R O R v φ − β ( O R ) O R v φ + 2 β O R O R O R v φ − β O R O R O R v φ + 2 β O R O R O R v L − β ( O R ) O R v L + β ( O R ) O R v L + 6 λ L ( O R ) O R v L + β ( O R ) O R v L − β O R O R O R v L + 2 β O R O R O R v L + β (cid:0) ( O R ) O R v φ + ( O R ) O R v L +2 O R O R (cid:0) O R v φ + O R v L (cid:1)(cid:1) + 2 β O R O R O R w − β O R O R O R w − β O R O R O R w + 2 β O R O R O R w + β ( O R ) O R w − β O R O R O R w + β ( O R ) O R w + 6 λ σ ( O R ) O R w (cid:19) , (7.3) and hence, for example when 2 M < M , we have the decay width h → h h given byΓ ( h → h h ) = g h h h πM (cid:18) − M M (cid:19) / . (7.4)We have computed all the decay channels of the neutral and charged scalars in order toobtain their branching fractions. For finding the new Feynman rules and computing theamplitudes and decay rates we used the new software FeynMaster [63], that makes useof
FeynRules [64]
QGRAF [65], and
FeynCalc [66, 67]. For the computation of the decaywidths h → γγ and h → Zγ , we used the expressions and conventions given in Ref. [62].The results obtained for the invisible branching ratios of the 125 GeV Higgs boson areshown in Fig. 6. In the left panel we show the BR( h → invisible) as a function of v L andin the right panel as a function of the mass of the second scalar Higgs boson. The points– 18 –n dark green satisfy the LHC constraints on the signals strengths at 3 σ level. The pointsin yellow green satisfy those constraints at 2 σ . Figure 6 : Left panel: BR inv versus v L . Right panel: BR inv versus M . In dark green the LHCconstraints are imposed at 3 σ , while in yellow green is shown the result of requiring 2 σ LHCconstraints.
We see that an invisible branching ratio around 20%, close to the present upper bound[28, 29], is possible within this model. This is consistent with all the LHC constraintsincluding those on the signal strengths at the 3 σ level. However, if 2 σ limits are applied,the ratio reduces to a maximum of 10%. This is better illustrated on the left panel ofFig. 7, where we plot the invisible branching ratio of the 125 GeV Higgs boson, versus thesignal strength of the Higgs boson produced via gluon fusion and decaying in the ZZ finalstate. Figure 7 : Left panel: BR inv versus µ ggFzz . Right panel: BR inv ( h ) versus BR inv ( h ). In darkgreen the LHC constraints are imposed at 3 σ , while in yellow green is shown the result ofrequiring 2 σ LHC constraints.
If it is found at the LHC that this signal strength becomes closer to one, then the– 19 –nvisible branching ratio in this model will be smaller, which can be important for LHCsearches. On the right panel of Fig.(7) we show the correlation between the invisiblebranching ratios of the two lightest CP even Higgs bosons. We see that the invisiblebranching ratio of the Higgs boson lying above the one with m H = 125GeV can have awide range of values, from very small (hence visible) to close to an 100% invisible branchingratio. To better understand the different possibilities, we have also plotted in the left panelof Fig. 8 the correlation between the invisible branching ratios of the second and thirdCP even Higgs bosons. On the right panel of Fig. 8 we show the absolute value of their Figure 8 : Left panel: BR inv ( h ) versus BR inv ( h ). Right panel: The effective couplings to vectorbosons with respect to that of the SM, k V ( h i ) of the second and third CP even Higgs boson.Notice that these couplings are bounded due to the sum rule of Eq. (5.7). In dark green the LHCconstraints are imposed at 3 σ , while in yellow green (on top) is shown the result of requiring 2 σ LHC constraints. couplings to vector bosons, which gives a measure of the probability that they are producedand observed. We notice that the effective couplings of the second and third CP even Higgsboson to vector bosons are limited to a maximum value below 1, due to the sum rule inEq. (5.7) and to the fact that the coupling of the lightest Higgs boson is bounded frombelow to be in agreement with LHC data, as shown in Eq. (5.5). From these figures wesee that the model has a very rich structure with many possibilities. In order to betterillustrate this we selected three benchmark points with quite different characteristics. inv ( h ) > . , BR inv ( h ) < . , BR inv ( h ) < . M H + < M h , M h , thus allowing the second and third CPeven Higgs bosons to decay visibly into the charged one. One such example is given inTable 4.We see that all the CP even Higgs bosons have non-negligible couplings to vectorbosons, although these are bounded due to the sum rule of Eq. (5.7). In contrast, only the– 20 – h M h M h M H + M A v L BR inv BR inv BR inv | k V | | k V | | k V | (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) h h h h h h .
11 5 × − × − .
95 0 .
16 0 . Table 4 : Parameters for point P1. lightest one (the 125 GeV SM Higgs) has an observable invisible branching ratio. inv ( h ) > . , BR inv ( h ) > . , BR inv ( h ) < . M H + > M h , M h , so that the second and third CP evenHiggs bosons do not decay visibly into the charged one. One such example is given inTable 5. M h M h M h M H + M A v L BR inv BR inv BR inv | k V | | k V | | k V | (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) h h h h h h .
13 0 .
18 0 .
13 2 × − .
95 0 .
31 0 . Table 5 : Parameters for point P2.
We see that the second CP even Higgs boson can have a sizable invisible branchingratio, while its coupling to the vector bosons is still large, therefore allowing, in principle,to be searched at the LHC. In this case, due to Eq. (5.7), the coupling to vector bosons ofthe third CP even Higgs boson is very small, making it very difficult to produce. inv ( h ) < . , BR inv ( h ) < . , BR inv ( h ) < . Point M h M h M h M H + M A v L BR inv BR inv BR inv | k V | | k V | | k V | (GeV) (GeV) (GeV) (GeV) (GeV) (GeV) h h h h h h P3a 125.0 391.4 391.5 493.4 391.5 6 × − × − × − × − .
90 0 .
32 0 . .
44 1 × − × − × − .
98 0 .
10 0 . Table 6 : Parameters for points P3a and P3b.
The first one (P3a) has a very compressed spectrum as a result of a low value of v L ,as we have discussed before (see Fig. 5). For the second point (P3b), with a larger v L , ithas a broader spectrum, but the couplings to vector bosons are smaller (see Eq. (5.7)).– 21 – Summary and conclusions
In this work we have examined the simplest realization of the linear seesaw mechanismwithin the context of the Standard Model SU(3) c ⊗ SU(2) L ⊗ U(1) Y gauge symmetry struc-ture. In addition to the standard scalar doublet, we employ two scalar multiplets chargedunder lepton-number. One of these is a nearly inert doublet, while the other is a sin-glet. Neutrino mass generation through spontaneous violation of lepton number impliesthe existence of a Nambu-Goldstone boson. The existence of such “majoron” would lead tostringent astrophysical constraints. A consistent electroweak symmetry breaking patternrequires a compressed mass spectrum of scalar bosons in which the Standard Model Higgsboson can have a large invisible decay into the invisible majorons.The Higgs boson production rates are similar to what is expected in two-doublet Higgsschemes. Indeed, we saw in Eq. (5.7) that the couplings of the CP even Higgs bosons to thevector bosons obeys a simple sum rule, characteristic of two-doublet Higgs boson schemes.In contrast, no sum rule holds concerning their visible decay branching, of course, so muchso that, with current experimental precision, values of an invisible branching ratio up to20% are allowed. However, future lepton colliders may play a decisive role here; in fact, it isexpected that BR inv may be measured with precision better than 1% level [31], which willimpose severe constraints on these decay modes. All in all, the model provides interestingand peculiar benchmarks for electroweak breaking studies at collider experiments. Wethink these deserve a dedicated experimental analysis that lies beyond the scope of thispaper. Acknowledgments
Work supported by the Spanish grants SEV-2014-0398 and FPA2017-85216-P (AEI/FEDER,UE), PROMETEO/2018/165 (Generalitat Valenciana) and the Spanish Red ConsoliderMultiDark FPA2017-90566-REDC. . D. F. and J. C. R are supported by projects CFTP-FCT Unit 777 (UID/FIS/00777/2013 and UID/FIS/00777/2019), and PTDC/FIS-PAR/29436/2017 which are partially funded through POCTI (FEDER), COMPETE, QRENand EU. D.F. is also supported by the Portuguese
Funda¸c˜ao para a Ciˆencia e Tecnologia under the project SFRH/BD/135698/2018.
A Unitarity Constraints
In this appendix we list all the coupled channel matrices for the states in Table 1. As wediscussed before, these matrices can be separated by charge Q and hypercharge Y of theinitial and final state, as the states with different values of Q, Y will not mix.– 22 – .1 Q = 2 , Y = 2We start with the highest charge combination. We have already discussed this case. Weget the following matrix 16 π a ++0 = λ β + β
00 0 2 λ L , (A.1)with eigenvalues 2 λ, β + β , λ L . (A.2) A.2 Q = 1 , Y = 2We obtain the following matrix16 π a +0 ( Y = 2) = λ β β β β
00 0 0 2 λ L , (A.3)with eigenvalues 2 λ, λ L , β + β , β − β . (A.4) A.3 Q = 1 , Y = 1We obtain the following matrix16 π a +0 ( Y = 1) = β − β β β − β β , (A.5)with eigenvalues β , β , (cid:18) β + β + (cid:113) β + ( β − β ) (cid:19) , (cid:18) β + β − (cid:113) β + ( β − β ) (cid:19) . (A.6) A.4 Q = 1 , Y = 0We obtain the following matrix16 π a +0 ( Y = 0) = λ β β β β λ L , (A.7)with eigenvalues β , β , λ + λ L + (cid:113) β + ( λ − λ L ) , λ + λ L − (cid:113) β + ( λ − λ L ) . (A.8)– 23 – .5 Q = 0 , Y = 2In this case we get 16 π a ( Y = 2) = λ β + β
00 0 2 λ L , (A.9)with eigenvalues 2 λ, β + β , λ L . (A.10) A.6 Q = 0 , Y = 1In this case we get the following matrix16 π a +0 ( Y = 1) = β − β β β − β β , (A.11)with eigenvalues β , β , (cid:18) β + β + (cid:113) β + ( β − β ) (cid:19) , (cid:18) β + β − (cid:113) β + ( β − β ) (cid:19) . (A.12) A.7 Q = 0 , Y = 0Finally we get the last coupled channel matrix,16 πa ( Y = 0) = λ β λ β β β β −√ β β β −√ β β λ L β λ L β λ β λ β β β β −√ β β β −√ β β λ L β λ L β β β β β λ σ −√ β −√ β λ σ −√ β −√ β λ σ , (A.13)where we have defined β = β + β . The different eigenvalues are β , λ + λ L ± (cid:113) β + ( λ − λ L ) , (cid:18) β + 2 β + 2 λ σ ± (cid:113) β + 4 β β − β λ σ + 16 β + 4 β − β λ σ + 4 λ σ (cid:19) , (A.14)– 24 –lus the cubic roots of the equation z + az + bz + c = 0 , (A.15)where a = − λ − λ L − λ σ ,b = − β − β β − β − β − β + 36 λλ L + 24 λλ σ + 24 λ L λ σ , (A.16) c =16 β λ σ − β β β + 16 β β λ σ + 12 β λ L − β β β + 12 β λ + 4 β λ σ − λλ L λ σ . We have implemented all the constraints that the eigenvalues should satisfy [54]: | Λ i | < π . (A.17) References [1] T. Kajita, “Nobel Lecture: Discovery of atmospheric neutrino oscillations,”
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