njpNJOPFMNew Journal of PhysicsNJPNew J. Phys.1367-2630IOP Publishingnjpaa18d010.1088/1367-2630/18/3/033033aa18d0NJP-104279.R1PaperElectroweak breaking and neutrino mass: ‘invisible’ Higgs decays at the LHC (type II seesaw)Electroweak breaking and neutrino mass: ‘invisible’ Higgs decays at the LHC (type II seesaw)BonillaCesar13RomãoJorge C2ValleJosé W F10000-0002-1881-5094
AHEP Group, Instituto de Física Corpuscular — C.S.I.C./Universitat de València Edificio de Institutos de Paterna, C/Catedratico José Beltrán, 2 E-46980 Paterna (València) - Spain
Departamento de Física and CFTP, Instituto Superior Técnico Universidade de Lisboa, Av. Rovisco Pais 1, 1049-001 Lisboa, Portugalcesar.bonilla@ific.uv.esjorge.romao@tecnico.ulisboa.ptvalle@ific.uv.es
C Bonilla et al
Author to whom any correspondence should be addressed.
Neutrino mass generation through the Higgs mechanism not only suggests the need to reconsider the physics of electroweak symmetry breaking from a new perspective, but also provides a new theoretically consistent and experimentally viable paradigm. We illustrate this by describing the main features of the electroweak symmetry breaking sector of the simplest type-II seesaw model with spontaneous breaking of lepton number. After reviewing the relevant ‘theoretical’ and astrophysical restrictions on the Higgs sector, we perform an analysis of the sensitivities of Higgs Boson searches at the ongoing ATLAS and CMS experiments at the LHC, including not only the new contributions to the decay channels present in the standard model (SM) but also genuinely non-SM Higgs Boson decays, such as ‘invisible’ Higgs Boson decays to majorons. We find sensitivities that are likely to be reached at the upcoming run of the experiments.
The electroweak breaking sector is a fundamental ingredient of the standard model (SM), many of whose detailed properties remain open even after the historic discovery of the Higgs Boson [1, 2]. The electroweak breaking sector is subject to many restrictions following from direct experimental searches at colliders [3, 4], as well as global fits [5, 6] of precision observables [7–9]. Moreover, its properties are may also be restricted by theoretical consistency arguments, such as naturalness, perturbativity and stability [10]. The latter have long provided strong motivation for extensions of the SM such as those based on the idea of supersymmetry.
Following the approach recently suggested in [11, 12] we propose to take seriously the hints from the neutrino mass generation scenario to the structure of the scalar sector. In particular, the most accepted scenario of neutrino mass generation associates the small size of the neutrino mass to their charge neutrality which suggests them to be of Majorana nature due to some, currently unknown, mechanism of lepton number violation. The latter requires an extension of the SU(3)c⨂SU(2)L⨂U(1)Y Higgs sector and hence the need to reconsider the physics of symmetry breaking from a new perspective. In broad terms this would provide an alternative to supersymmetry as a paradigm of electroweak breaking. Amongst its other characteristic features is the presence of doubly charged scalar bosons, compressed mass spectra of heavy scalars dictated by stability and perturbativity and the presence of ‘invisible’ decays of Higgs Bosons to the Nambu-Goldstone boson associated to spontaneous lepton number violation and neutrino mass generation [13].
In this paper we study the invisible decays of the Higgs Bosons in the context of a type-II seesaw majoron model [14] in which the neutrino mass is generated after spontaneous violation of lepton number at some low energy scale, ΛEW≲Λ∼(TeV) [15, 16]^{4}
The idea of the Majoron was first proposed in [17] though in the framework of the type I seesaw, not relevant for our current paper. On the other hand the triplet Majoron was suggested in [18] but has been ruled out since the first measurements of the invisible Z width by the LEP experiments. Regarding the idea of invisible Higgs decays was first given in [19], though the early scenarios have been ruled out.
. This scheme requires the presence of two lepton number carrying scalar multiplets in the extended SU(3)c⨂SU(2)L⨂U(1)Y model, a singlet σ and a triplet Δ under SU(2)—this seesaw scheme was called ‘123’-seesaw model in [14] and here we take the ‘pure’ version of this scheme, without right-handed neutrinos. The presence of the new scalars implies the existence of new contributions to ‘visible’ SM Higgs decays, such as the h→γγ decay channel, in addition to intrinsically new Higgs decay channels involving the emission of majorons, such as the ‘invisible’ decays of the CP-even scalar bosons. As a result, one can set upper limits on the invisible decay channel based on the available data which restrict the ‘visible’ channels.
The plan of this paper is as follows. In the next section we describe the main features of the symmetry breaking sector of the ‘123’ type II seesaw model. In section 3 we discuss the ‘theoretical’ and astrophysical constraints relevant for the Higgs sector. Taking these into account, we study the sensitivities of Higgs Boson searches at the LHC to Standard Modelscalar boson decays in section 4. Section 5 addresses the non-SM Higgs decays of the model. Section 6 summarizes our results and we conclude in section 7.
The type-II seesaw model
Our basic framework is the ‘123’ seesaw scheme originally proposed in [14] whose Higgs sector contains, in addition to the SU(3)c⨂SU(2)L⨂U(1)Y scalar doublet Φ, two lepton-number-carrying scalars: a complex singlet σ and a triplet Δ. All these fields develop non-zero vacuum expectation values (vevs) leading to the breaking of the Standard Model(SM) gauge group as well as the global symmetry U(1)L associated to lepton number. The latter breaking accounts for generation of the small neutrino masses.
Therefore, the scalar sector is given byΦ=ϕ0ϕ−andΔ=Δ0Δ+2Δ+2Δ++with L = 0 and L=−2, respectively, and the scalar field σ with lepton number L = 2. Below we will consider the required vev hierarchies in the model.
Yukawa sector
Here we consider the simplest version of the seesaw scheme proposed in [14] in which no right-handed neutrinos are added, and only the SU(3)c⨂SU(2)L⨂U(1)Y electroweak breaking sector is extended so as to spontaneously break lepton number giving mass to neutrinos. Such ‘123’ majoron–seesaw model is described by the SU(3)c⨂SU(2)L⨂U(1)Y⨂U(1)L invariant Yukawa Lagrangian,Y=yijdQ¯iuRjΦ+yijuQ¯idRjΦ˜+yijℓLi¯ℓRjΦ+yijνLiTCΔLj+h.c.
In this model the neutrino mass (see figure 1) is given by,mν=yνκv1v22mΔ2where v_{1} and v_{2} are the vevs of the singlet and the doublet, respectively. Here κ is a dimensionless parameter that describes the interaction amongst the three scalar fields (see below), and mΔ is the mass of the scalar triplet Δ.
Diagram that generates non-zero neutrino mass in the model.
At this point we note that the smallness of neutrino mass i.e.mν≲1eVmay define interesting regions of the parameter space in any neutrino mass generation model where the new physics is expected to be hidden from direct observation. In particular, we are interested in spotting those regions accessible at collider searches such as the ongoing experiments at the LHC (see [20] and references therein).
In our pure type II seesaw model where lepton number is spontaneously violated at some low energy scale we havemν=yν〈Δ〉with the effective vev is given as 〈Δ〉=μ〈Φ〉2/MΔ2 where Δ is the isotriplet lepton–number–carrying scalar. Here 〈Φ〉 is fixed by the mass of the W boson andμ=κv1is the dimensionful parameter responsible of lepton number violation, see equation (3). Therefore if yν∼(1) and the mass MΔ lies at 1 TeV region then one has that 〈Δ〉∼mν and μ∼1eV.
Note that one may consider two situations: v1≫ΛEW (high-scale seesaw mechanism) in whose case the scalar singlet and the invisible decays of the Higgs are decoupled [15]; the second interesting case is when ΛEW≲v1≲ few TeV (low-scale seesaw mechanism). In this case the parameter κ is the range [10−14,10−16] for yν∼(1). In this case one has new physics at the TeV region including the ‘invisible’ decays of the Higgs Bosons.
Therefore, led by the smallness of the neutrino mass we can qualitatively determine that the analysis to be carried out is characterized by having a vev hierarchyv1≳v2≫v3and the smallness of the coupling κ, that is κ≪1.
The scalar potential
The scalar potential invariant under the SU(3)c⨂SU(2)L⨂U(1)Y⨂U(1)L symmetry is given by [15, 16] ^{5}
From now on we follow the notation and conventions used in [16].
As mentioned above the scalar fields σ, ϕ and Δ acquire non-zero vacuum expectation values, v_{1}, v_{2} and v_{3}, respectively, so that, they can be shifted as follows,σ=v12+R1+iI12,ϕ0=v22+R2+iI22,Δ0=v32+R3+iI32.
The minimization conditions of equation (4) are given by,μ12=−2β1v13−β2v1v22−β3v1v32+κv22v32v1,μ22=−12(2λ1v22+β2v12+(λ3+λ5)v32−2κv1v3),μ32=−2(λ2+λ4)v33−(λ3+λ5)v22v3−β3v12v3+κv1v222v3.and from these one can derive a vev seesaw relation of the typev1v3∼κv22,where κ is the dimensionless coupling that generates the mass parameter associated to the cubic term in the scalar potential of the simplest triplet seesaw scheme with explicit lepton number violationas proposed in [21] and recently revisited in [12].
Neutral Higgs Bosons
One can now write the resulting squared mass matrix for the CP-even scalars in the weak basis (R1,R2,R3) as follows,MR2=2β1v12+12κv22v3v1β2v1v2−κv2v3β3v1v3−12κv22β2v1v2−κv2v32λ1v22(λ3+λ5)v2v3−κv1v2β3v1v3−12κv22(λ3+λ5)v2v3−κv1v22(λ2+λ4)v32+12κv22v1v3.
The matrix M^{2}_{R} is diagonalized by an orthogonal matrix as follows, RMR2RT=diag(mH12,mH22,mH32), whereH1H2H3=RR1R2R3.
We use the standard parameterization R=R23R13R12 whereR12=c12s120−s12c120001,R13=c130s13010−s130c13,R23=1000c23s230−s23c23and cij=cosαij,sij=sinαij, so that the rotation matrix R is re-expressed in terms of the mixing angles in the following way:R=c12c13c13s12s13−c23s12−c12s13s23c23c12−s12s13s23c13s23−c12c23s13+s23s12−c23s12s13−c12s23c13c23.
On the other hand, the squared mass matrix for the CP-odd scalars in the weak basis (I1,I2,I3) is given as,MI2=κ12v22v3v1v2v312v22v2v32v1v3v1v212v22v1v212v22v1v3.The matrix M^{2}_{I} is diagonalized as, IMI2IT=diag(0,0,mA2), where the null masses correspond to the would-be Goldstone boson G^{0} and the Majoron J, while the squared CP-odd mass ismA2=κv22v12+v22v32+4v32v122v3v1.The mass eigenstates are linked with the original ones by the following rotation,A1A2A3≡JG0A=II1I2I3where the matrix I is given by,OI=cv1V2−2cv2v32−cv22v30v2/V−2v3/Vbv2/2v1bbv2/2v3,withV2=v22+4v32,c−2=v12V4+4v22v34+v24v32b2=4v12v32v22v12+v22v32+4v32v12.
Charged Higgs Bosons
The squared mass matrix for the singly-charged scalar bosons in the original weak basis (ϕ±,Δ±) is given by,MH±2=κv1v3−12λ5v32122v2(λ5v3−2κv1)122v2(λ5v3−2κv1)14v3v22(−λ5v3+2κv1).We now defineG±H±=c±s±−s±c±ϕ±Δ±,and±MH±2±T=diag(0,mH±2).where c_{±} and s_{±} are given as c±=v2/v22+2v32 and s±=2v3/v22+2v32. The massless state corresponds to the would-be Golstone bosons G± and the massive state H± is characterized by,mH±2=14v3(2κv1−λ5v3)(v22+2v32).
On the other hand, the doubly-charged scalars Δ±± has massmΔ++2=12v3(κv1v22−2λ4v33−λ5v22v3).
Scalar boson mass sum rules
Notice that using the fact that the smallness of the neutrino mass implies that the parameters κ and v_{3} are very small one can, to a good approximation, rewrite equation (6) schematically in the form,MR2∼⋆⋆0⋆⋆000⋆sothatR∼c12s120−s12c120001,and equation (11) becomes,mA2∼κv22v12v3.As a result, the scalar H_{3} and the pseudo-scalar A are almost degenerate,mH3=(MR2)33≈mA2.
In the same way, by using equations (11), (18) and (19), one can derive the following mass relations,mA2−mH+2≈λ5v224and2mH+2−mA2−mΔ++2≈λ4v32,which can be rewritten in the form,mH+2−mΔ++2≈mA2−mH+2≈λ5v224.This sum rule is also satisfied in the type-II seesaw model with explicit breaking of lepton number. Imposing the perturbativity condition one finds that the squared mass difference between, say doubly and singly charged scalar bosons, cannot be too large [12]. Explicit comparison shows that λ5 in equation (4) corresponds to λHΔ′ in [12]. Therefore when the couplings of the singlet σ in equation (4) are small, λ5 is constrained to be in the range [−0.85,0.85], so that the remaining couplings are kept small up to the Planck scale and vacuum stability is guaranteed. See figure 4 in [12]. Likewise when one decouples the triplet one also recovers the results found in [11].
Theoretical constraints
Before analyzing the sensitivities of the searches for Higgs Bosons at the LHC experiments, we first discuss the restrictions that follow from the consistency requirements of the Higgs potential. We can rewrite the dimensionless parameters λ1,2,3 and β1,2,3 in equation (4) in terms of the mixing angles, αij and scalar the masses mH1,2,3 by solving RMR2RT=diag(mH12,mH22,mH32) and IMI2IT=diag(0,0,mA2). Hence one gets,λ1=12v22[mH12c132s122+mH22(c12c23−s12s13s23)2+mH32(c12s23+s12s13c23)2]λ2=12v32[mH12s132+c132(mH22s232+mH32c232)]−λ4+κv1v224v33λ3=c13v2v3[mH12s12s13+mH22s23(c12c23−s12s13s23)−mH32c23(c12s23+s12s13c23)]−λ5−κv1v3β1=12v12[mH12c122c132+mH22(s12c23+c12s23s13)2+mH32(s12s23−c12c23s13)2]−κv22v34v13β2=1v1v2[mH12c12s12c132−mH22(c23s12+c12s13s23)(c12c23−s12s13s23)−mH32(s23s12−c12s13c23)(c12s23+s12s13c23)]+κv3v1β3=c13v1v3[mH12c12s13−mH22s23(s12c23+c12s13s23)+mH32c23(s12s23−c12s13c23)]+κv222v1v3.
In addition, using equations (11), (18) and (19) we can write the dimensionless parameters λ4,5 and κ as functions of the vevs v1,2,3 and the masses of the pseudo-, singly- and doubly-charged scalar bosons (i.e. m_{A}, mH± and mΔ±±, respectively) as,λ4=1v322mH±2v22v22+2v32−mA2v12v22v22v32+v12(v22+4v32)−mΔ±±2λ5=−4mH±21v22+2v32+4mA2v12v22v32+v12(v22+4v32)κ=2mA2v1v3v22v32+v12(v22+4v32).
From the theoretical side we have to ensure that the scalar potential in the model is bounded from below (BFB).
Boundedness conditions
In order to ensure that the scalar potential in equation (4) is bounded from below we have to derive the conditions on the dimensionless parameters such the quartic part of the scalar potential is positive V(4)>0 as the fields go to infinity. We have that the parameter κ≪1 (due to the smallness of the neutrino mass) and non-negative. This follows fromκ≈2mA2v3v1v22.where we have used the last expression in equation (24) and the fact that v3≪v2,v1. Then κ is neglected with respect to the other dimensionless parameters λi and βj, i.e. λi,βj≫κ. As a result the quartic part of the potential V(4)∣κ=0 turns to be a biquadratic form λijφi2φj2 of real fields. Therefore, in this strict limit, the copositivity criteria described in [22] may be applied and the boundedness conditions for equation (4) are the following,λ1>0,β1>0,λ24>0,λˆ≡β2+2β1λ1>0,λ˜≡β3+2β1λ24>0,λ¯≡λ3+θ(−λ5)λ5+2λ1λ24>0,andβ1λ1λ24+[λ3+θ(−λ5)λ5]β1+β2λ24+β3λ1+λˆλ˜λ¯>0,where λ24≡λ2+λ4. In addition all the dimensionless parameters in the scalar potential are required to be less than 4π in order to fulfill the perturbativity condition.
Astrophysical constraints
In our type-II seesaw model there are some constraints on the magnitude of SU(2) triplet’s vev 〈Δ〉=v3, that one must take into account. First of all, v_{3} is constrained to be smaller than a few GeVs due to the ρ parameter (ρ=1.0004±0.00024 [23]).
On the other hand, the presence of the Nambu-Goldstone boson associated to spontaneous lepton number violationand neutrino mass generation implies that there is a most stringent constraint on v_{3} coming from astrophysics, due to supernova cooling. If the majoron is a strict Goldstone boson (or lighter than typical stellar temperatures) one has an upper bound for the Majoron-electron coupling∣gJee∣≲10−13,This is discussed, for example, in [24] and references therein. This implies∣gJee∣=∣12Ime/v2∣.Taking into account the profile of the Majoron [14]^{6}
This is derived either by explicit analysis of the scalar potential or simply by symmetry, using Noether’s theorem [14].
one can translate this as a bound on the projection of the Majoron onto the doublet as follows [16]∣〈J∣ϕ〉∣=2∣v2∣v32v12(v22+4v32)2+4v22v34+v24v32≲10−7.
Notice that this restriction on the triplet’s vev is stronger that the one stemming from the ρ parameter. The shaded region in figure 2 corresponds to the allowed region of v_{3} as function of v_{1}.
The shaded region represents the allowed region of v_{3} as function of v_{1}.
To close this section we mention that our phenomenological analysis remains valid if the Nambu-Goldstone boson picks up a small mass from, say, quantum gravity effects.
Type-II seesaw Higgs searches at the LHC
We now turn to the study of the experimental sensitivities of the LHC experiments to the parameters characterizing the ‘123’ type-II majoron seesaw Higgs sector, as proposed in [14]. In the following we will assume that mH1<mH2<mH3 where 1, 2, 3 refer to the mass ordering in the CP even Higgs sector. Therefore, there are two possible cases that can be considered^{7}
Recall that mH3≈mA, equation (22), which implies that the mass of H_{3} must be close to that of the doubly-charged scalar mass. Therefore, as we will see in the next section, the existing bounds on searches of the doubly-charged scalar exclude the case where mH3 is lighter than the other CP-even mass eigenstates.
:
mH1<mH and mH2=mH;
mH1=mH,
where m_{H} is the mass of the Higgs reported by the ATLAS [2] and CMS [25] collaborations, i.e. mH=125.09±0.21(stat.)±0.11(syst.)GeV[26]. For case (i), we have to enforce the constraints coming from LEP-II data on the lightest CP-even scalar coupling to the SM and those coming from the LHC Run-1 on the heavier scalars. Such situation has been discussed by us in [13] in the simplest ‘12-type’ seesaw Majoron model. In case (ii), only the constraints coming from the LHC must be taken into account.
The neutral component of the Standard Model Higgs doublet couplings get modified as follows,ϕ0→C1H1+C2H2+C3H3where we have defined Ci≡i2R and ijR are the matrix elements of R in equation (9).
LEP constraints on invisible Higgs decays
The constraints on H_{1}, when mH1<125GeV, stem from the process e+e−→Zh→Zbb¯ which is written as [27]σhZ→bb¯Z=σhZSM×RhZ×BR(h→bb¯)=σhZSM×CZ(h→bb¯)2,where σhZSM is the SM hZ cross section, R_{hZ} is the suppression factor related to the coupling of the Higgs Boson^{8}
The Feynman rules for the couplings of the Higgs Bosons H_{i} to the Z are the following: ig22cW2(i2Rv2+i3Rv3)gμν.
to the gauge boson Z. Since v3≪v2, we have that the factor RHiZ≈Ci2 where C1=cosα13sinα12, equation (28). Notice that C1≈sinα12 for the limit α13≪1 and then one obtains the same exclusion region depicted in figure 1 in [13].
LHC constraints on the Higgs signal strengths
In addition, we have to enforce the limits coming from the Standard Modeldecay channels of the Higgs Boson. These are given in terms of the signal strength parameters,μf=σNP(pp→h)σSM(pp→h)BRNP(h→f)BRSM(h→f),where σ is the cross section for Higgs production, BR(h→f) is the branching ratio into the Standard Modelfinal state f, the labels NP and SM stand for New Physics and Standard Model, respectively. These can be compared with those given by the experimental collaborations. The most recent results of the signal strengths from a combined ATLAS and CMS analysis [28] are shown in table 1.
Current experimental results of ATLAS and CMS, [28].
channel
ATLAS
CMS
ATLAS+CMS
μγγ
1.15−0.25+0.27
1.12−0.23+0.25
1.16−0.18+0.20
μWW
1.23−0.21+0.23
0.91−0.21+0.24
1.11−0.17+0.18
μZZ
1.51−0.34+0.39
1.05−0.27+0.32
1.31−0.24+0.27
μττ
1.41−0.35+0.40
0.89−0.28+0.31
1.12−0.23+0.25
μbb
0.62−0.36+0.37
0.81−0.42+0.45
0.69−0.27+0.29
One can see with ease that the LHC results indicate that μVV∼1. In our analysis, we assume that the LHC allows deviations up to 20% as follows,0.8≤μXX≤1.2
Scalar mass eigenstates in the model. c±=v2/v22+2v32, s±=2v3/v22+2v32.
Mass eigenstate ϕ
Mass squared mϕ2
Composition
H_{i}(i=1,2,3)
m_{i}^{2}
i1RR1+i2RR2+i3RR3
J
0
11II1+12II2+13II3
G^{0}
0
22II2+23II3
A
κv22v12+v22v32+4v32v122v3v1
31II1+32II2+33II3
G±
0
c±ϕ±+s±Δ±
H±
14v3(2κv1−λ5v3)(v22+2v32)
−s±ϕ±+c±Δ±
Δ±±
12v3(κv1v22−2λ4v33−λ5v22v3)
Δ±±
LHC bounds on the heavy neutral scalars
In our study we will impose the constraints on the heavy scalars from the recent LHC scalar boson searches. Therefore, we use the bounds set by the search for a heavy Higgs in the H→WW and H→ZZ decay channels in the range [145−1000]GeV [29] and in the h→ττ decay channel in the range range [100−1000]GeV [30]. We also adopt the constraints on the process h→γγ in the range [65−600]GeV [31] and the range [150,850]GeV [32]. Besides, we impose the bounds in the A→Zh decay channel in the range [220−1000]GeV [33].
Summary of the searches of charged scalars
The type-II seesaw model with explicit breaking of lepton number contains seven physical scalars: two CP-even neutral scalars H_{1} and H_{2}, one CP-odd scalar A and four charged scalars Δ±± and H±. Such a scenario has been widely studied in the literature and turns out to be quite appealing because it could be tested at the LHC [34–44]. For instance, the existence of charged scalar bosons provides additional contributions to the one-loop decays of the Standard Model Higgs Boson. Indeed, they could affect the one-loop decays h→γγ [39, 40] and h→Zγ [40] in a substantial way. In this case the signal strength μγγ can set bounds on the mass of the charged scalars, Δ±± and/or H±.
The doubly-charged scalar boson has the following possible decay channels: ℓ±ℓ±, W±W±, W±H± and H±H±. However, it is known that for an approximately degenerate triplet mass spectrum and vev v3≲10−4GeV the doubly charged Higgs coupling to W± is suppressed (because it is proportional to v_{3} as can be seen from table 3) and hence Δ±± predominantly decays into like-sign dileptons [41, 44, 45]. In this case, CMS [46] and ATLAS [47] have currently excluded at 95% C.L., depending on the assumptions on the branching ratios into like-sign dileptons, doubly-charged masses between 200 and 460 GeV^{9}
From doubly-charged scalar boson searches performed by ATLAS and CMS one can also constrain the lepton number violation processes pp→Δ±±Δ∓∓→ℓ±ℓ±W∓W∓ and pp→Δ±±H∓→ℓ±ℓ±W∓Z [41]. This may also shed light on the Majorana phases of the lepton mixing matrix [34–36].
. For v3≳10−4GeV, the Yukawa couplings of triplet to leptons are too small so that Δ±± dominantly decays to like-sign dibosons, in which case the collider limits are rather weak [43, 48–50].
Feynman rules for the couplings of the Higgs Bosons H_{i} to the gauge bosons.
Vertex
Gauge Coupling
1
H1Wμ+Wν−
ig22(12Rv2+213Rv3)gμν
2
H2Wμ+Wν−
ig22(22Rv2+223Rv3)gμν
3
H3Wμ+Wν−
ig22(32Rv2+233Rv3)gμν
4
Δ±±Wμ∓Wν∓
i2g2v32gμν
5
H±Wμ∓Zν
ig2cWc±v32gμν
6
G±Wμ∓Zν
ig2cWv22sW2c±+v32(1+sW2)s±gμν
7
G±Wμ∓Aν
−iemWgμν
8
Δ++Δ−−Wμ+Wν−
ig2gμν
9
H+H−Wμ+Wν−
ig22(1+3c±2)gμν
10
G+G−Wμ+Wν−
ig22(1+3s±2)gμν
11
H1H1Wμ+Wν−
ig22(12R2+213R2)gμν
12
H2H2Wμ+Wν−
ig22(22R2+223R2)gμν
13
H3H3Wμ+Wν−
ig22(32R2+233R2)gμν
14
JJWμ+Wν−
ig22(12I2+213I2)gμν
15
G0G0Wμ+Wν−
ig22(22I2+223I2)gμν
16
AAWμ+Wν−
ig22(32I2+233I2)gμν
17
Δ±±H∓Wν∓
∓igc±(p1-p2)μ
18
Δ±±G∓Wν∓
∓igs±(p1-p2)μ
19
H1H±Wμ∓
±ig2(s±12R−2c±13R)(p1-p2)μ
20
H2H±Wμ∓
±ig2(s±22R−2c±23R)(p1-p2)μ
21
H3H±Wμ∓
±ig2(s±32R−2c±33R)(p1-p2)μ
22
H±JWμ∓
g2(s±12I+2c±13I)(p1-p2)μ
23
G0H±Wμ∓
−g2(s±22I+2c±23I)(p1-p2)μ
24
AH±Wμ∓
−g2(s±32I+2c±33I)(p1-p2)μ
25
G±H1Wμ∓
±ig2(c±12R+2s±13R)(p1-p2)μ
26
G±H2Wμ∓
±ig2(c±22R+2s±23R)(p1-p2)μ
27
G±H3Wμ∓
±ig2(c±32R+2s±33R)(p1-p2)μ
28
G±JWμ∓
−g2(c±12I−2s±13I)(p1-p2)μ
29
G0G±Wμ−
g2(c±22I−2s±23I)(p1-p2)μ
30
AG±Wμ−
g2(c±32I−2s±33I)(p1-p2)μ
Vertex
Gauge Coupling
31
H1ZμZν
ig22cW2(12Rv2+413Rv3)gμν
32
H2ZμZν
ig22cW2(22Rv2+423Rv3)gμν
33
H3ZμZν
ig22cW2(32Rv2+433Rv3)gμν
34
Δ++Δ−−ZμZν
i2g2cW2(cW2−sW2)2gμν
35
H+H−ZμZν
ig22cW2(s±2(cW2−sW2)2+4sW4c±2)gμν
36
G+G−ZμZν
ig22cW2(c±2(cW2−sW2)2+4sW4s±2)gμν
37
H1H1ZμZν
ig22cW2(12R2+413R2)gμν
38
H2H2ZμZν
ig22cW2(22R2+423R2)gμν
39
H3H3ZμZν
ig22cW2(32R2+433R2)gμν
40
JJZμZν
ig22cW2(12I2+413I2)gμν
41
G0G0ZμZν
ig22cW2(22I2+423I2)gμν
42
AAZμZν
ig22cW2(32I2+433I2)gμν
43
Δ++Δ−−Zμ
−igcW(cW2−sW2)(p1-p2)μ
44
H−H+Zμ
ig2cW(s±2(cW2−sW2)−2sW2c±2)(p1-p2)μ
45
G−G+Zμ
ig2cW(c±2(cW2−sW2)−2sW2s±2)(p1-p2)μ
46
H1JZμ
−g2cW(12R12I−213R13I)(p1-p2)μ
47
G0H1Zμ
g2cW(12R22I−213R23I)(p1-p2)μ
48
AH1Zμ
g2cW(12R32I−213R33I)(p1-p2)μ
49
H2JZμ
−g2cW(22R12I−223R13I)(p1-p2)μ
50
G0H2Zμ
g2cW(22R22I−223R23I)(p1-p2)μ
51
AH2Zμ
g2cW(22R32I−223R33I)(p1-p2)μ
52
H3JZμ
−g2cW(32R12I−233R13I)(p1-p2)μ
53
G0H3Zμ
g2cW(32R22I−233R23I)(p1-p2)μ
54
AH3Zμ
g2cW(32R32I−233R33I)(p1-p2)μ
55
G∓H±Zμ
∓g2cWc±s±(p1-p2)μ
56
Δ++Δ−−AμAμ
8ie2gμν
57
H−H+AμAμ
i2e2gμν
58
G−G+AμAμ
i2e2gμν
59
Δ++Δ−−Aμ
−2ie(p1-p2)μ
60
H+H−Aμ
ie(p1-p2)μ
61
G+G−Aμ
ie(p1-p2)μ
62
Δ++Δ−−AμZν
4iegcW(cW2−sW2)gμν
63
H+H−AμZν
iegcW(s±2(cW2−sW2)−2c±2sW2)gμν
64
G+G−AμZν
iegcW(c±2(cW2−sW2)−2s±2sW2)gμν
In the present ‘123’ type-II seesaw model there are two additional physical scalars, a massive CP-even scalar H_{3} and the massless majoron J. The latter, associated to the spontaneous breaking of lepton number, provides non-standard decay channels of other Higgs Bosons as missing energy in the final state^{10}
These include, for example, Hi→JJ and H±→JW∓. Here we focus mainly on the first, the decays of H± deserve further study but it is beyond the scope of this work and will be considered elsewhere.
.
Invisible Higgs decays at the LHC
We now turn to the case of genuinely non-standard Higgs decays. We focus on investigating the LHC sensitivities on the invisible Higgs decays. In so doing we take into account how they are constrained by the available experimental data. In the previous section we mentioned that in our study the CP- even scalars obey the following mass hierarchy mH1<mH2<mH3. Furthermore, we will also assume that the masses mH3, m_{A}, mH+ and mΔ++ are nearly degenerate.
As a consequence, the decay of any CP-even Higgs H_{i} into the pseudo-scalar A is not kinematically allowed. Therefore, the new decay channels of the CP-even scalars are just, Hi→JJ and Hi→2Hj (when mHi<mHj2 for i≠j). The latter contributing also to the invisible decay channel of the Higgs as, Hi→2Hj→4J.
The Higgs-Majoron couplings are given by,gHaJJ=(12I)2v2a2R+(13I)2v3a3R+(11I)2v1a1RmHa2,where ijI are the elements of the rotation matrix in equation (13) and the decay width is given byΓ(Ha→JJ)=132πgHaJJ2mHa.Following our conventions we have that the trilinear coupling H2H1H1 turns out to be,gH2H1H12=3λ1(12R)222Rv2+3(λ2+λ4)(13R)223Rv3+(λ3+λ5)2[(13R)222Rv2+(12R)223Rv3+212R13R(23Rv2+22Rv3)]+3β1(11R)221Rv1+β22[(12R)221Rv1+(11R)222Rv2+211R12R(22Rv1+21Rv2)]+β32[(13R)221Rv1+(11R)223Rv3+211R13R(23Rv1+21Rv3))]+κ2[−211R13R22Rv2−(12R)2(23Rv1+21Rv3)−212R(13R(22Rv1+21Rv2)+11R(23Rv2+22Rv3))].and hence, for example when mH1<2mH2, the decay width H2→H1H1 is given byΓ(H2→H1H1)=gH2H1H1232πmH21−4mH12mH221/2.
As we already mentioned, a salient feature of adding an isotriplet to the Standard Modelis that some visible decay channels of the Higgs receive further contributions from the charged scalars, namely the one-loop decays h→γγ and h→Zγ. That is, the scalars H± and Δ±± contribute to the one-loop coupling of the Higgs to two-photons and to Z-photon, leading to deviations from the Standard Modelexpectations for these decay channels. The interactions between CP-even and charged scalars are described by the following vertices,HaH+H−:igHaH+H−HaΔ++Δ−−:igHaΔ++Δ−−wheregHaH+H−=12(v22+2v32)[8λ1a2Rv2v32+4(λ2+λ4)a3Rv22v3+2λ3(a2Rv23+2a3Rv33)+λ5v2[−2a3Rv2v3+a2R(v22−2v32)]+4β2a1Rv1v32+2β3a1Rv1v22+4κv2v3(a2Rv1+a1Rv2)]gHaΔ++Δ−−=2λ2a3Rv3+λ3a2Rv2+β3a1Rv1.
Note that the contributions of H± and Δ±± to the decays h→γγ and h→Zγ are functions of the singlet’s vev v_{1}, this is in contrast to what happens in the type-II seesaw model with explicit violation of lepton number. According to equation (26) the dimensionless parameters λi and βi can change the sign of the couplings of gHaH+H− and gHaΔ++Δ−−, hence the contribution of the charged scalars to h→γγ and h→Zγ may be either constructive or destructive.
For the computation of the decay widths h→γγ and h→Zγ we use the expressions and conventions given in [51]. The decay width Γ(Ha→γγ) turns out to beΓ(Ha→γγ)=GFα2mHa31282π3∣XFγγ+XWγγ+XHγγ∣2where G_{F} is the Fermi constant, α is the fine structure constant and the form factors X_{i}^{j} are given by^{11}
We have taken into account that v_{3} is very small so that any contribution involving the triplet’s vev is neglected. Then for instance the Feynman rule for the vertex HaWμ+Wν−:ig22(Oa2Rv2+2Oa3Rv3)gμν, is approximated as ∼ig22(Oa2Rv2)gμν (see table 3).
,XFγγ=−2Ca∑fNcfQf2τf[1+(1−τf)f(τf)],XWγγ=Ca[2+τW+3τW(2−τW)f(τW)]XHγγ=−gHaH+H−v2mH±2τH±[1−τH±f(τH±)]−4gHaΔ++Δ−−v2mΔ±±2τΔ±±[1−τΔ±±f(τΔ±±)].where τx=4mx2/mZ2. Here N_{c}^{F} and Q_{F} denote, respectively, the number of colors and electric charge of a given fermion. The one-loop function f(τ) is defined in appendix B. The parameters C_{a} correspond to the Standard Model Higgs couplings in equation (28).
The decay width Γ(Ha→Zγ), using the notation in [51], is expressed as followsΓ(Ha→Zγ)=GFα2mHa3642π31−mZ2mHa23∣XFZγ+XWZγ+XHZγ∣2where the form factors X_{i}^{j} are given by^{12}
Here we have also assumed v3≪1 so as to make the following approximation, H+H−Zμ:−igsinθWtanθW(p+−p−)μ.
,XFZγ=−4Ca∑fNcfgVfQfmf2sWcW2mZ2(mHa2−mZ2)2ΔB0f+1mHa2−mZ2×[(4mf2−mHa2+mZ2)C0f+2]gViQfmi2sWcWXWZγ=CatanθW1(mHa2−mZ2)2[mHa2(1−tan2θW)−2mW2(−5+tan2θW)]mZ2ΔB0W+1(mHa2−mZ2)[mHa2(1−tan2θW)−2mW2(−5+tan2θW)+2mW2[(−5+tan2θW)(mHa2−2mW2)−2mZ2(−3+tan2θW)]C0W]XHZγ=−2gHaH+H−vtanθW(mHa2−mZ2)mZ2mHa2−mZ2ΔB0±+(2mH±2C0±+1)−4gHaΔ++Δ−−vtanθW(1−tan2θW)(mHa2−mZ2)mZ2mHa2−mZ2ΔB0±±+(2mΔ±±2C0±±+1)where C^{b}_{0} and ΔB0b are defined in appendix B.
Type-II seesaw neutral Higgs searches at the LHC
We stated above that in our study we are assuming mH1<mH2<mH3 and v1≳v2. Furthermore, because of the ρ parameter and the astrophysical constraint on the triplet’s vev we also have that v3≪v1,v2. We found that the smallness of v_{3} and the perturbativity condition of the potential lead to a very small mixing between the mass eigenstate H_{3} and the CP-even components of the fields, σ and Φ, in other words, the angles α13 and α23 must lie close to 0 or π. As a result, we obtain the following relation,mH32−mA2≃2λ2v32⟹mH3≃mA.This extra mass relation is derived from equation (24), by using equation (25) and the fact that α13,23∼0(π).
In addition, also as a result of α13,23∼0(π), we find that the coupling of H_{3} to the Standard Modelstates is negligible,gH3ffghffSM=gH3VVghVVSM=C3∼0.
In figure 12 of appendix A we give a schematic illustration of the mass profile of the Higgs Bosons in our model. The mass spectrum and composition are summarized in table 2, and provide a useful picture in our following analyses.
Analysis (i)
In this case we have taken the isotriplet vev v3=10−5GeV, automatically safe from the constraints stemming from astrophysics and the ρ parameter. We have also considered the following mass spectrum,mH1=[15,115]GeV,mH2=125GeV,mH3≃mA≃mH±≃mΔ±±=500GeV,and varied the parameters asv1∈[100,2500]GeV,α12∈[0,π]andα13,23=δα(π−δα)where 0≤δα<0.1. As described in section 4 we must enforce the LEP constraints on the lightest CP-even Higgs H_{1} and LHC constraints on the heavier scalars. The near mass degeneracy of H3,A,H± and Δ±± ensures that the oblique parameters are not affected. In analogy to the type-II seesaw model with explicit lepton number violationwe expect that, because of v3<10−4GeV, the doubly-charged scalar predominantly decays into same sign dileptons [41, 44, 45] and that mΔ±±=500GeV is consistent with current experimental data, see section 4.4.
We show in figure 3 the mass of the lightest CP-even scalar as a function of the absolute value of its coupling to the Standard Modelstates, ∣C1∣ in equation (28). The blue region corresponds to the LEP exclusion region and the green(red) one is the LHC allowed(exclusion) region provided by the signal strengths 0.8<μXX<1.2.
Analysis (i). The mass of the lightest CP-even scalar as a function of the absolute value of its coupling to Standard Model states. The blue region corresponds to the LEP exclusion region and the green (red) one is the LHC allowed (exclusion) region.
The presence of light charged scalars can enhance significantly the diphoton channel of the Higgs [39]. Figure 4 shows the correlation between μZZ and μγγ(μZγ) on the left(right) with μγγ≲1.2 for charged Higgs Bosons of 500GeV.
Analysis (i). On the left, we show correlation between μZZ and μZγ. On the right, correlation between μZZ and μγγ. The color code as in figure 3.
The correlation between the signal strength μZZ and the signal strengths μγγ and μZγ is shown in figure 4. Note that the former may exceed one due to the new contributions of the singly and doubly charged Higgs Bosons.
The invisible decays of the Higgs Bosons, characteristic of the model, turn out to be correlated to the visible channels, represented in terms of the signal strengths, as shown in figure 5. Note that the upper bound on the invisible decays of a Higgs Boson with a mass of 125 GeV has been found to be BR(H2→Inv)≲0.2. This limit is stronger than those provided by the ATLAS [52] and the CMS [53] collaborations^{13}
The ATLAS collaboration has set an upper bound on the BR(H→Inv) at 0.28 while the CMS collaboration reported that the observed (expected) upper limit on the invisible branching ratio is 0.58(0.44), both results at 95% C.L.
.
Analysis (i). On the left: the signal strength μγγ versus BR(H2→Inv). On the right: μZZ versus BR(H2→Inv). The color code as in figure 3.
In figure 6 we depict the correlation between the invisible branching ratios of H_{2} with the one of the lightest scalar boson H_{1}. And, as can be seen, H_{1} can decay 100% into the invisible channel (majorons).
Analysis (i). Correlation between the invible branchings BR(H2→Inv) and BR(H1→Inv). The color code as in figure 3.
Finally, as we have mentioned we obtained that the reduced coupling of H_{3} to the Standard Modelstates is C3∼(10−7) so that it is basically decoupled. As a result its invisible branching is essentially unconstrained, 10−5≲BR(H3→Inv)≤1. On the other hand we find that the constraint coming from the LHC on the pseudo-scalar A with a mass of 500GeV is automatically satisfied as well, since from the LHC, σ(gg→A)BR(A→ZH2)BR(H2→ττ)≲10−2 while for mA=500GeV we obtain σ(gg→A)BR(A→ZH2)BR(H2→ττ)≲10−15.
Analysis (ii)
We now turn to the other case of interest, namelymH1=125GeV,mH2=[150,500]GeV,mH3≃mA≃mH±≃mΔ±±=600GeV,with v3=10−5GeV, as before. Now we scanned overv1∈[100,2500]GeV,α12∈[0,π]andα13,23=δα(π−δα)where 0≤δα<0.1. As we already mentioned in this case we only have to take into account the constraints coming from Run 1 of the LHC at 8TeV, see table 1.
In practice we assume μXX=1.0−0.2+0.2. We show in figure 7 the correlation between μZZ and μγγ(μZγ) on the left(right). As before, the allowed region is in green while the forbidden one is in red. We can see that μγγ≲1.2 for mH±≃mΔ±±=600GeV.
Analysis (ii). On the left, μZZ versus μZγ. On the right, μZZ versus μγγ. The allowed region (in green) is the region inside the range μXX=1.0−0.2+0.2 while the forbidden one (in red) is the one outside that range.
On the left(right) of figure 8 is depicted the correlation between the signal strength μZZ (μγγ) and the branching ratio of the channel H1→JJ. We can see in figures 8–10 that BR(H1→Inv)≲0.2. One can see from figure 9 that BR(H1→Inv)≲0.1 for v1≳2500GeV.
Analysis (ii). On the left: the signal strength μZZ versus Br(H1→Inv). On the right: μγγ versus Br(H1→Inv). The color code as in figure 7.
Analysis (ii). BR(H1→Inv) versus v_{1} (on the left) and BR(H1→Inv) versus the Higgs-majoron coupling gH1JJ (on the right). The color code as in figure 7.
Analysis (ii). Correlation between BR(H2→Inv) and BR(H1→Inv). The color code as in figure 7.
In this case we find that equation (32) (for α13,23∼0(π) and v3≪v1,v2) at leading order is given by,gH1JJ∼cosα12v1mH12,where mH1=125GeV. BR(H1→Inv) versus the Higgs-majoron coupling gH1JJ is shown on the right of figure 9. Note also from the left panel in figure 9 that BR(H1→Inv) is anti-correlated with v_{1}, as expected.
In figure 10 we show the correlation between the invisible branching ratio of H_{2} (the Higgs with a mass in the range 150GeV<mH2<500GeV) and the one of H_{1}.
We have verified that the LHC constraints on the heavy scalars (H_{2}, H_{3} and A) are all satisfied. As an example, the reader can convince her/himself by looking at figure 11 that H_{2} easily passes the restriction stemming from σ(ggH2)BR(H2→ττ) (top left) and/or σ(bbH2)BR(H2→ττ) (top right). The black continuous lines on those plots represent the experimental results from Run 1 of the CMS experiment [30]. We also found that the square of the reduced coupling of H_{2} to the Standard Model states is C22≲0.1 for mH2=[150,500]GeV. Then, one finds that the experimental upper bounds set by the search for a heavy Higgs in the H→WW and H→ZZ decay channels in [3, 29] are automatically fulfilled. However, improved sensitivities expected from Run 2 may provide a meaningful probe of the theoretically consistent region, depicted in green.
Analysis (ii). On the top right (left) σ(ggH2)BR(H2→ττ) (σ(bbH2)BR(H2→ττ)) versus the mass of H_{2}.
Also in this case, H_{3} is decoupled, so the restrictions on H_{3} and the massive pseudoscalar A are automatically fulfilled.
Conclusions
In this paper we have presented the main features of the electroweak symmetry breaking sector of the simplest type-II seesaw model with spontaneous violation of lepton number. The Higgs sector has two characteristic features: (a) the existence of a (nearly) massless Nambu–Goldstone boson and (b) all neutral CP-even and CP-odd, as well as singly and doubly-charged scalar bosons coming mainly from the triplet are very close in mass, as illustrated in figure 12 of appendix A. However, one extra CP-even state, namely H_{2} coming from a doublet-singlet mixture can be light. After reviewing the ‘theoretical’ and experimental restrictions which apply on the Higgs sector, we have studied the sensitivities of the searches for Higgs Bosons at the ongoing ATLAS/CMS experiments, including not only the new contributions to Standard Modeldecay channels, but also the novel Higgs decays to majorons. For these we have considered two cases, when the 125 GeV state found at CERN is either (i) the second-to-lightest or (ii) the lightest CP-even scalar boson. For case (i), we have enforced the constraints coming from LEP-II data on the lightest CP-even scalar coupling to the Standard Modelstates and those coming from the LHC Run-1 on the heavier scalars. In case (ii), only the constraints coming from the LHC must be taken into account. Such ‘invisible’ Higgs Boson decays give rise to missing momentum events. We have found that the experimental results from Run 1 on the search for a heavy Higgs in the H→WW and H→ZZ decay channels are automatically fulfilled. However, improved sensitivities expected from Run 2 may provide a meaningful probe of this scenario. In short we have discussed how the neutrino mass generation scenario not only suggests the need to reconsider the physics of electroweak symmetry breaking from a new perspective, but also provides a new theoretically consistent and experimentally viable paradigm.
Type-II seesaw Higgs Boson mass spectrum.
Acknowledgments
Work supported by the Spanish grants FPA2014-58183-P, Multidark CSD2009-00064 and SEV-2014-0398 (MINECO), and PROMETEOII/2014/084 (Generalitat Valenciana). CB thanks Departamento de Física and CFTP, Instituto Superior Técnico, Universidade de Lisboa, for its hospitality while part of this work was carried out. JCR is also support in part by the Portuguese Fundação para a Ciência e Tecnologia (FCT) under contracts UID/FIS/00777/2013 and CERN/FIS-NUC/0010/2015, which are partially funded through POCTI (FEDER), COMPETE, QREN and the EU.
Higgs Boson mass spectrum
Loop functions
The one-loop function f(τ) used in equation (37) is given by,f(τ)=arcsin2(1/τ)ifτ≥1−14log1+1−τ1−1−τ−iπ2ifτ<1
The functions C^{b}_{0} and ΔB0b are given in terms of the Passarino–Veltman functions [54],C0b=C0(mZ2,0,mHa2,mb2,mb2,mb2)=−1mb2I2(τb,λb),ΔB0b=B0(mHa2,mb2,mb2)−B0(mZ2,mb2,mb2)=−mHa2−mZ2mZ2−(mHa2−mZ2)22ma2mZ2I1(τb,λb)+2mHa2−mZ2mZ2I2(τb,λb)where λb=4mb2/mHa2,I1(τ,λ)=τλ2(τ−λ)+τ2λ22(τ−λ)2[f(τ)−f(λ)]+τ2λ(τ−λ)2[g(τ)−g(λ)],I2(τ,λ)=−τλ2(τ−λ)[f(τ)−f(λ)],andg(τ)=τ−1arcsinτforτ≥1121−τlog1+1−τ1−1−τ−iπifτ<1
Higgs Boson couplings
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