We investigate a possibility for explaining the recently announced 750 GeV diphoton excess by the ATLAS and CMS experiments at the CERN LHC in a model with multiple doubly charged particles, that was originally suggested for explaining tiny neutrino masses through a three-loop effect in a natural way. The enhanced radiatively generated effective coupling of a new singlet scalar S with diphoton with multiple charged particles in the loop enlarges the production rate of S in ppS+X via a photon fusion process and also the decay width Γ(Sγγ) even without assuming a tree-level production mechanism. We provide detailed analysis on the cases with or without allowing mixing between S and the standard model Higgs doublet.

1. Introduction

In mid-December 2015, both the ATLAS and CMS experiments announced the observation of a new resonance around 750 GeV as a bump in the diphoton invariant mass spectrum from the run-II data in s=13TeV [1,2]. Their results are based on the accumulated data of 3.2fb1 (ATLAS) and 2.6fb1 (CMS), and local/global significances are 3.9σ/2.3σ (ATLAS) [1] and 2.6σ/1.2σ (CMS) [2], respectively. The best-fit values of the invariant mass are 750GeV by ATLAS and 760GeV by CMS, where ATLAS also reported the best-fit value of the total width as 45GeV.

During/after Moriond EW in March 2016, updated results were reported with the new analysis with different hypotheses on spin (spin-0 or spin-2) and the width to mass ratio (Γ/m<1% “narrow width” or Γ/m~610% “wide width”) [3,4]. Based on the 3.2fb1 data set, the ATLAS group claimed that the largest deviation from the background-only hypothesis was observed near a mass of 750GeV, which corresponds to a local excess of 3.9σ for the spin-0 case of Γ45GeV (Γ/m6%). However, we note that the preference for wide width compared with narrow width is only minor by ~0.3σ significance so that we would take it with caution. In our analysis below, we simply allow both cases with narrow and wide widths. The global significance is still low ~2.0σ.

On the other hand, based on the upgraded amount of the data of 3.3fb1, the CMS group reported a modest excess of events at 760GeV with a local significance of 2.82.9σ depending on the spin hypothesis. The narrow width (Γ/m=1.4×102) maximizes the local excess. In addition, CMS reported the result of a combined analysis of 8TeV and 13TeV data, where the largest excess (3.4σ) was observed at 750GeV for the narrow width (Γ/m=1.4×104). The global significances are <1σ (1.6σ) in the 13TeV (8TeV+13TeV) analyses, respectively. No official combined (ATLAS & CMS) result has been made so far.

Just after the advent of the first announcement, various ways to explain the 750 GeV excess were proposed, even within December 2015, in Refs. [5125]. The first unofficial interpretation of the excess in terms of the signal strength of a scalar (or a pseudoscalar) resonance S, ppS+Xγγ+X, was made immediately after the first announcement in Ref. [11] based on the expected and observed exclusion limits in both of the experiments. The authors claimed  

(1.1)
μ13TeVATLAS=σ(ppS+X)13TeV×B(Sγγ)=(103+4)fb,
 
(1.2)
μ13TeVCMS=σ(ppS+X)13TeV×B(Sγγ)=(5.6±2.4)fb,
with a Poissonian likelihood function (for the ATLAS measurement) and a Gaussian approximation (for the CMS measurement), respectively.

On the other hand, both the ATLAS and CMS groups reported that no significant excess over the standard model (SM) background was observed in their analyses based on the run-I data at s=8TeV [126,127], while a mild upward bump was found in their data around 750GeV. In Ref. [11], the signal strengths at s=8TeV were extracted by use of the corresponding expected and observed exclusion limits given by the experiments, in the Gaussian approximation, for a narrow-width scalar resonance as  

(1.3)
μ8TeVATLAS=σ(ppS+X)8TeV×B(Sγγ)=(0.46±0.85)fb,
 
(1.4)
μ8TeVCMS=σ(ppS+X)8TeV×B(Sγγ)=(0.63±0.35)fb.
It is mentioned that when we upgrade the collider energy from 8TeV to 13TeV, a factor 4.7 enhancement is expected~[11,128], when the resonant particle is produced via gluon fusion, and then the data at s=8TeV and 13TeV are compatible at around the 2σ confidence level (C.L.). Indeed, in the second announcement [3], the ATLAS group discussed this point based on the reanalyzed 8TeV data corresponding to an integrated luminosity of 20fb1 with the latest photon energy calibration in the run-I, which is close to the calibration used for the 13TeV data. When m=750GeV and Γ/m=6%, the difference between the 8TeV and 13TeV results corresponds to statistical significances of 1.2σ (2.1σ) if gluon–gluon (quark–antiquark) productions are assumed. These observations would give us a stimulating hint for surveying the structure of physics beyond the SM above the electroweak scale even though the accumulated amount of data would not be enough for detailed discussions and the errors are large at the present stage.

A key point to understand the resonance is the fact that no bump around 750GeV has been found in the other final states in either the 8TeV or 13TeV data. If B(Sγγ) is the same as the 750GeV Higgs one, B(hγγ)|750 GeV SM=1.79×107 [129], we can immediately recognize that such a possibility is inconsistent with the observed results, e.g., in the ZZ final state, at s=8TeV, where the significant experimental 95% C.L. upper bound on the ZZ channel is 12fb by ATLAS [130] and the branching ratio B(hZZ)|750 GeV SM=0.290 [129]. In general, the process Sγγ should be loop induced since S has zero electromagnetic charge and then the value of B(Sγγ) tends to be suppressed because tree-level decay branches generate primary components of the total width of S. Then, a reasonable setup for explaining the resonance consistently is that all of the decay channels of S are one loop induced, where S would be a gauge singlet under SU(3)C and SU(2)L since a nonsinglet gauge assignment leads to tree-level gauge interactions, which are not desirable in our case.

An example of this direction is that S is a singlet scalar and it couples to vector-like quarks, which contribute to both ppS+X and Sγγ via gluon fusion and photon fusion, respectively. The possibility of diphoton production solely due to photon fusion is also an open possibility as discussed in Refs. [34,40] in the context of the 750GeV excess. The basic idea is simple: when a model contains multiple SU(2)L singlet particles with large U(1)Y hypercharges, the magnitude of the photon fusions in the production and decay sequences is largely enhanced.

In this paper, we focus on the radiative seesaw models [131135], especially where neutrino masses are generated at the three-loop level [136153]. In such scenarios, multiple charged scalars are introduced for realizing the three-loop origin of the neutrino mass, (distinct from the models with one or two loops). We show that when these charged scalars couple to the singlet S strongly enough, we can achieve a reasonable amount of the production cross section in ppS+Xγγ+X through photon fusion. Concretely, we start from the three-loop model [150], and extend the model with additional charged scalars to explain the data.1

This paper is organized as follows. In Sect. 2, we introduce our model based on the model for three-loop induced neutrino masses. In Sect. 3, we show detail of the analysis and numerical results. Section 4 is devoted to summary and discussions.

2. Model

Multiple (doubly) charged particles would induce a large radiative coupling with a singlet scalar S with γγ via one-loop diagrams. We may find the source from multi-Higgs models or extra dimensions [160177] but here we focus on a model for radiative neutrino masses, recently suggested by some of the authors [150] as a benchmark model, which can be extended with a singlet scalar S for the 750 GeV resonance.

2.1. Review: A model for three-loop induced neutrino mass

Our strategy is based on the three-loop induced radiative neutrino model with a U(1) global symmetry [150], where we introduce three Majorana fermions NR1,2,3 and new bosons: one gauge-singlet neutral boson Σ0, two singly charged singlet scalars (h1±,h2±), and one gauge-singlet doubly charged boson k±± to the SM. The particle contents and their charges are shown in Table 1.

Table 1.

Contents of lepton and scalar fields and their charge assignment under SU(3)C×SU(2)L×U(1)Y×U(1), where U(1) is an additional global symmetry and x0 (bold letters emphasize that these numbers correspond to representations of the Lie groups of the non-Abelian gauge interactions)

 Lepton fieldsa Scalar fields New scalar fieldsa 
Characters LLi eRi NRi Φ Σ0 h1+ h2+ k++ ja++ S 
SU(3)C 1 1 1 1 1 1 1 1 1 1 
SU(2)L 2 1 1 2 1 1 1 1 1 1 
U(1)Y 1/2 1 1/2 
U(1) x 2x x 2x 2x 
 Lepton fieldsa Scalar fields New scalar fieldsa 
Characters LLi eRi NRi Φ Σ0 h1+ h2+ k++ ja++ S 
SU(3)C 1 1 1 1 1 1 1 1 1 1 
SU(2)L 2 1 1 2 1 1 1 1 1 1 
U(1)Y 1/2 1 1/2 
U(1) x 2x x 2x 2x 

aThe subscripts found in the lepton fields i(=1,2,3) indicate generations of the fields.

bThe scalar particles shown in this category are added to the original model proposed in Ref. [150] to explain the 750 GeV excess.

We assume that only the SM-like Higgs Φ and the additional neutral scalar Σ0 have VEVs, which are symbolized by Φv/2 and Σ0v/2, respectively.

Here, x(0) is an arbitrary number of the charge of the hidden U(1) symmetry, and under the assignments, neutrino mass matrix is generated at the three-loop level, with a schematic picture shown in Fig. 1. A remnant Z2 symmetry remains after the hidden U(1) symmetry breaking and the particles NR1,2,3 and h2± have negative parities. Then, when a Majorana neutrino is the lightest among them, it becomes a dark matter (DM) candidate and the stability is accidentally ensured.

Fig. 1.

A schematic description for the radiative generation of neutrino masses.

Fig. 1.

A schematic description for the radiative generation of neutrino masses.

In the original model, the Lagrangian of the Yukawa sector LY and the scalar potential V, allowed under the gauge and global symmetries, are given as  

(2.1)
LY=(y)ijL¯LiΦeRj+12(yL)ijL¯LicLLjh1++(yR)ijN¯RieRjch2+12(yN)ijΣ0N¯RicNRj+h.c.,
 
(2.2)
V=mΦ2|Φ|2+mΣ2|Σ0|2+mh12|h1+|2+mh22|h2+|2+mk2|k++|2  +[λ11Σ0*h1h1k+++μ22h2+h2+k+h.c.]+λΦ|Φ|4+λΦΣ|Φ|2|Σ0|2+λΦh1|Φ|2|h1+|2  +λΦh2|Φ|2|h2+|2+λΦk|Φ|2|k++|2+λΣ|Σ0|4+λΣh1|Σ0|2|h1+|2+λΣh2|Σ0|2|h2+|2  +λΣk|Σ0|2|k++|2+λh1|h1+|4+λh1h2|h1+|2|h2+|2+λh1k|h1+|2|k++|2  +λh2|h2+|4+λh2k|h2|2|k++|2+λk|k++|4,
where the indices i,j indicate matter generations and the superscript “c” means charge conjugation (with the SU(2)L rotation by iσ2 for SU(2)L doublets). We assume that yN is diagonal, where the right-handed neutrino masses are calculated as MNi=v2(yN)ii with the assumed ordering MN1(=DM mass)<MN2<MN3. The neutral scalar fields are shown in the unitary gauge as  
(2.3)
Φ=[0v+ϕ2],  Σ0=v+σ2exp(iG/v),
with v246GeV and an associated Nambu–Goldstone~(NG) boson G via the global U(1) breaking due to the occurrence of nonzero v. Requiring the tadpole conditions, V/ϕ|ϕ=v=V/σ|σ=v=0, the resultant mass matrix squared of the CP even components (ϕ,σ) is given by  
(2.4)
m2(ϕ,σ)=[2λΦv2λΦΣvvλΦΣvv2λΣv2]=[cosαsinαsinαcosα][mh200mH2][cosαsinαsinαcosα],
where h is the SM-like Higgs (mh=125GeV) and H is an additional CP even Higgs mass eigenstate. The mixing angle α is determined as  
(2.5)
sin2α=2λΦΣvvmH2mh2.
The neutral bosons ϕ and σ are represented in terms of the mass eigenstates h and H as  
(2.6)
ϕ=hcosα+Hsinα,σ=hsinα+Hcosα.
The two CP even scalars h and H could work as DM-portal scalars and participate in the DM pair annihilation. The mass eigenvalues for the singly charged bosons h1±, h2± and the doubly charged boson k±± are given as  
(2.7)
mh1±2=mh12+12(λΦh1v2+λΣh1v2),mh2±2=mh22+12(λΦh2v2+λΣh2v2),mk±±2=mk2+12(λΦkv2+λΣkv2).

This model can explain the smallness of the observed neutrino masses and the presence of DM without severe parameter tuning. A summary of the features in the model is given in Appendix A.

Here we introduce a real singlet scalar S in the model and assume that it couples with the doubly charged scalar(s). Due to the contributions of the charged particles in the loop, a large branching ratio B(Sγγ) is achievable without assuming tree-level interactions [34,40]. When B(Sγγ) is sizable, the production cross section of the resonance particle, σ(ppS+X), becomes large through photon fusion processes; thus we do not have to rely on gluon fusion processes, which often request additional colored particles that bring in dangerous hadronic activities. Thus we may explain the 750 GeV excess as pointed out in Refs. [34,40].

2.2. Extension with a scalar S for the 750 GeV resonance

In the following part, we consider an extension of the original model with the new interactions as  

(2.8)
ΔV=μ^SkS|k++|2+λ^SkS2|k++|2+V(S)  +a=1Nj{m^ja±±2|ja++|2+μ^SjaS|ja++|2+λ^SjaS2|ja++|2  +[λ11(a)Σ0h1h1ja+++μ22(a)h2+h2+ja+h.c.]},
where S is a real scalar and ja±±(a=1,2,,Nj) are additional SU(2)L singlet doubly charged scalars with hypercharge +2 and a global U(1) charge +2x. Here, V(S) represents the potential of the singlet scalar S. Here, we assume that S has a VEV, and S should be replaced with SS+S. After the replacement, we pick up the relevant terms for our analysis and summarize:  
(2.9)
ΔVeff=μSkS|k++|2+12mS2S2+a=1Nj{mja±±2|ja++|2+μSjaS|ja++|2+[λ11(a)Σ0h1h1ja+++μ22(a)h2+h2+ja+h.c.]},
with  
(2.10)
mja±±2m^ja±±2+μ^SjaS+λ^SjaS2,μSkμ^Sk+2λ^SkS,μSjaμ^Sja+2λ^SjaS.
The squared physical masses of S and ja±± are mS2 and mja±±2, respectively and we set mS to 750GeV for our explanation of the 750GeV excess.2 Here, ja±± has the same charges as k±± and then can contribute to the three-loop induced neutrino masses shown in Fig. 1.3 The trilinear terms in the square brackets are required for evading the stability of ja±±. We also ignore such possible terms as |ja++|2|Φ|2, |ja++|2|Σ0|2 and S|Φ|2, S|Σ0|2 in Eq. (2.8) in our analysis below. This is justified as a large VEV of S generates large effective trilinear couplings μSk and μSja through the original terms S2|k++| and S2|ja++|, respectively, even when the dimensionless coefficients λ^Sk and λ^Sja are not large.

3. Analysis

3.1. Formulation of p(γ)p(γ)S+Xγγ+X

Additional interactions in Eq. (2.9) provide possible decay channels of S to γγ, Zγ, ZZ, and k++k or ja++ja up to the one-loop level. We assume that mk±± and mja±± are greater than mS/2(=375GeV), where the last two decay channels at the tree level are closed kinematically. Here, we show the case when S is a mass eigenstate and there is no mixing through mass terms with other scalars. In the present case that no tree-level decay branch is open and only SU(2)L singlet charged scalars describe the loop-induced partial widths, the relative strengths among ΓSγγ, ΓSZγ, ΓSZZ, and ΓSW+W are governed by quantum numbers at the one-loop level4 as  

(3.1)
ΓSγγ:ΓSZγ:ΓSZZ:ΓSW+W1:2(sW2cW2):(sW4cW4):0.
In the following, we calculate ΓSZZ in a simplified way:  
(3.2)
ΓSZZsW22cW2ΓSZγ0.15ΓSZγ.

Here, we represent a major part of partial decay widths of S with our notation for loop functions with the help of Refs. [179183]. In the following part, for simplicity, we set all the masses of the doubly charged scalars mja±± the same as mk±±, while we ignore the contributions from the two singly charged scalars h1,2± since they should be at least as heavy as around 3TeV and decoupled as mentioned in Appendix A. The concrete forms of ΓSγγ and ΓSZγ are  

(3.3)
ΓSγγ=αEM2mS3256π3v2|12vμmk±±2Qk2A0γγ(τk)|2,
 
(3.4)
ΓSZγ=αEM2mS3512π3(1mZ2mS2)3|μmk±±2(2QkgZkk)A0Zγ(τk,λk)|2,
with  
(3.5)
μ=Σaμa[μSk+a=1NjμSja],gZkk=Qk(sWcW),τk=4mk±±2mS2,λk=4mk±±2mZ2.
Here, Qk(=2) is the electric charge of the doubly charged scalars in units of the positron's one, cW and sW are the cosine and the sine, respectively, of the Weinberg angle θW, and αEM is the electromagnetic fine structure constant. In the following calculation, we use sW2=0.23120 and αEM=1/127.916. The loop factors take the following forms,:  
(3.6)
A0γγ(x)=x2[x1f(x1)],A0Zγ(x,y)=xy2(xy)+x2y22(xy)2[f(x1)f(y1)]+x2y(xy)2[g(x1)g(y1)].
The two functions f(z) and g(z) (zx1 or y1) are formulated as  
(3.7)
f(z)=arcsin2z for z1,
 
(3.8)
g(z)=z11arcsinz for z1,
where the situation mS2mk±±,mZ2mk±± corresponds to z1. For simplicity, we assume the relation  
(3.9)
μSk=μSja,
for all a.

For the production of S corresponding to the 750GeV resonance, we consider the photon fusion process, as first discussed in the context of the 750GeV excess in Refs. [34,40]. We take the photon parton distribution function (PDF) from Ref. [184], which adopted the methods in Ref. [185].5 The inclusive production cross section of a scalar (or pseudoscalar) resonance R is generally formulated as  

(3.10)
dσinc(p(γ)p(γ)R+X)dMR2dyR=dLincdMR2dyRσ^(γγR),
where MR and yR are the mass and the rapidity of the resonance R, and σ^(γγR) shows the parton-level cross section for the process γγR. The inclusive luminosity function can be conveniently written in terms of the photon PDF as  
(3.11)
dLγγincdMR2dyR=1sγ(x1,μ)γ(x2,μ),
where x1,2=MRse±yR represent the momentum fractions of the photons inside the protons and s means the total energy. The value of γ(x,μ) can be evaluated by taking the Dokshitzer–Gribov–Lipatov–Altarelli–Parisi (DGLAP) evolution from the starting scale μ0(=1GeV) to μ after an estimation of coherent and incoherent components of the initial form of γ(x,μ=μ0) at μ=μ0 (see [184] for details).

By adopting the narrow width approximation, which is fine in our case, the parton-level cross section of the particle S of mass mS and rapidity yS is  

(3.12)
σ^(γγS)=8π2Γ(Sγγ)mSδ(MR2mS2)      =8π2Γtot(S)mSB(Sγγ)δ(MR2mS2).
The inclusive differential cross section is obtained in a factorized form:  
(3.13)
dσinc(p(γ)p(γ)S+X)dyS=8π2Γ(Sγγ)mS×dLγγincdMR2dyS|MR=mS.

Now taking the values for γ(x,μ) in Ref. [184], we obtain a convenient form of cross section  

(3.14)
σinc(p(γ)p(γ)S+X)=91fb(Γtot(S)1GeV)B(Sγγ)
or  
(3.15)
σinc(p(γ)p(γ)S+Xγγ+X)=91fb(Γtot(S)1GeV)B2(Sγγ),
for evaluating production cross sections at s=13TeV. The reference magnitude of the cross section, 91 fb, is much greater than that in Ref. [40] obtained under the narrow width approximation and effective photon approximation [208,209], 1.63.6fb (depending on the minimum impact parameter for elastic scattering), while it is smaller than that in Ref. [187] through a similar calculation in Ref. [184], 240fb. We also find at MR=750GeV in Ref. [184],  
(3.16)
Lγγinc(s=13TeV)Lγγinc(s=8TeV)2.9.
Having the above relations in Eqs. (3.14)–(3.16), it is straightforward to evaluate the inclusive production cross section at s=8TeV. We note that the resultant value is greater than the value (2) cited in Ref. [187].

3.2. Results

3.2.1. Case 1: Without mass mixing

In this part, we discuss the case that the field S is a mass eigenstate, where no mixing effect is present through mass terms with other scalars. Under our assumptions, the relevant parameters are (mk±±,μSk,Nj): the universal physical mass of the doubly charged scalars (assuming mk±±=mja±± for all a), the universal effective scalar trilinear coupling (assuming μSk=μSja for all a), and the number of additional doubly charged singlet scalars. We observe the unique relation among the branching ratios of S irrespective of mk±± and μSk, which is suggested by Eq. (3.1), as  

(3.17)
B(Sγγ)0.591,B(SγZ)0.355,B(SZZ)0.0535.

In Ref. [215], reasonable target values for the cross section of σγγσ(ppS+Xγγ+X) at the s=13TeV LHC were discussed as functions of the variable R13/8, which is defined as  

(3.18)
R13/8σ(ppS)|s=13TeVσ(ppS)|s=8TeV,
where the published data after Moriond 2016 are included, and the four categories distinguished by the two features (spin-0 or spin-2; narrow width [ΓS/mS0] or wide width [ΓS/mS=6%]) are individually investigated. As pointed out in Eq. (3.17), the value of B(Sγγ) is uniquely fixed as 60% and S is produced only through photon fusion in the present case. As shown in Eq. (3.16) in our estimation of the photo-production, R13/8 corresponds to 2.9, where the best fit values of σγγ at s=13TeV are extracted from [215] as  
(3.19)
2.0±0.5f b (for ΓS/mS0), 4.25±1.0f b (for ΓS/mS=6%).
The theoretical error in the present formulation of the photo-production was evaluated as ±1520% in [184]. Then, we decide to focus on the 2σ favored regions including the error (20%, fixed) also, concretely speaking,  
(3.20)
[0.8,3.6]f b (for ΓS/mS0),[1.8,7.5]f b (for ΓS/mS=6%).
Here, the 95% C.L. upper bound on σγγ at s=8TeV is 2.4f b [126,127] and the favored regions are still consistent with the 8 TeV result (or just on the edge). It is found that the bounds on the Zγ, ZZ final states are weaker than that of γγ. Relevant information is summarized in Table 2.

Table 2.

95% C.L. upper bounds on decay channels of a 750GeV scalar resonance

Final state Upper bound (in fb, 95% C.L.) Category Ref. 
γγ 2.4/2.4 8 TeV-ATLAS/CMS [126,127
 13/13 13 TeV-ATLAS/CMS [3,4
Zγ 4.0/27 8 TeV/13 TeV-ATLAS [210,211
ZZ 12/99 8 TeV/13 TeV-ATLAS [130,212
WW 35 8 TeV-ATLAS [213
hh 40 8 TeV-ATLAS [214
Final state Upper bound (in fb, 95% C.L.) Category Ref. 
γγ 2.4/2.4 8 TeV-ATLAS/CMS [126,127
 13/13 13 TeV-ATLAS/CMS [3,4
Zγ 4.0/27 8 TeV/13 TeV-ATLAS [210,211
ZZ 12/99 8 TeV/13 TeV-ATLAS [130,212
WW 35 8 TeV-ATLAS [213
hh 40 8 TeV-ATLAS [214

In Fig. 2, situations in our model are summarized. Six cases with different numbers of doubly charged scalars are considered with Nj=0, 1, 10, 100, 200, and 300. Here, we should mention an important issue. As indicated in Fig. 2, when Nj is zero, more than 10~20TeV is required in the effective trilinear coupling μSk. Such a large trilinear coupling would immediately lead to a violation of tree-level unitarity in the scattering amplitudes including μSk, e.g., k++kk++k or SSk++k at around the energy 1TeV, where the physics of our interest is spread. Also, the vacuum is possibly threatened by destabilization via the large trilinear coupling, which calls charge breaking minima. To avoid the problems, naively speaking, the value of μSk is less than 1~5TeV.6

Fig. 2.

Six cases with different numbers of doubly charged scalars are considered with Nj=0, 1, 10, 100, 200, and 300. Inside the green regions, the best-fit value of the production cross section is realized by taking account of ±20% theoretical error discussed in Ref. [184]. The yellow regions indicate the areas where we obtain the 2σ-favored values in the production cross section of p(γ)p(γ)S+Xγγ+X, where we take account of both the theoretical (±20%) and experimental (shown in Eq. (3.19)) errors. Cross-section evaluations are due to Eq. (3.15). The gray shaded region mk±±438GeV in Nj=0 shows the excluded parts in the 95% C.L. via the ATLAS 8 TeV search for doubly charged particles with the assumption of B(k±±μ±μ±)=100% [217]. The vertical black dotted lines represent corresponding bounds on the universal physical mass mk±± when we assume B(ja±±μ±μ±)=100% for all ja±±. Two types of constraints with respect to the “Landau pole” of gY (defined as gY(μ)=4π) are meaningful when Nj is large (Nj=200, 300). The red lines indicate three reference boundaries of the correction factor cδ=1 with c=1, 0.1, 0.01 to the trilinear couplings μSk(=μSja) defined in Eq. (3.22). For each choice of c, the region below the corresponding boundary is favored from the viewpoint of perturbativity.

Fig. 2.

Six cases with different numbers of doubly charged scalars are considered with Nj=0, 1, 10, 100, 200, and 300. Inside the green regions, the best-fit value of the production cross section is realized by taking account of ±20% theoretical error discussed in Ref. [184]. The yellow regions indicate the areas where we obtain the 2σ-favored values in the production cross section of p(γ)p(γ)S+Xγγ+X, where we take account of both the theoretical (±20%) and experimental (shown in Eq. (3.19)) errors. Cross-section evaluations are due to Eq. (3.15). The gray shaded region mk±±438GeV in Nj=0 shows the excluded parts in the 95% C.L. via the ATLAS 8 TeV search for doubly charged particles with the assumption of B(k±±μ±μ±)=100% [217]. The vertical black dotted lines represent corresponding bounds on the universal physical mass mk±± when we assume B(ja±±μ±μ±)=100% for all ja±±. Two types of constraints with respect to the “Landau pole” of gY (defined as gY(μ)=4π) are meaningful when Nj is large (Nj=200, 300). The red lines indicate three reference boundaries of the correction factor cδ=1 with c=1, 0.1, 0.01 to the trilinear couplings μSk(=μSja) defined in Eq. (3.22). For each choice of c, the region below the corresponding boundary is favored from the viewpoint of perturbativity.

Also, we consider the doubly charged singlet scalars produced via ppγ/Z+Xk++k+X. Lower bounds at 95% C.L. on mk±± via the 8TeV LHC data were provided by the ATLAS group in Ref. [217] as 374GeV, 402GeV, 438GeV when assuming a 100% branching ratio to e±e±, e±μ±, μ±μ± pairs, respectively. In our model, the doubly charged scalars can decay through the processes as shown in Fig. 3, where h1+'s are off shell since it should be heavy, at least 3TeV. In the case of k++ in Nj=0, when the values of μ11 and μ22 are the same or similar, from Eq. (2.2), the relative branching ratios between k++μ+μ+νiνj and k++μ+μ+ are roughly proportional to (yL)2i(yL)2j and ((yR)22)2. As concluded in our previous work [150], the absolute value of (yR)22 should be as large as around 8~9 to generate the observed neutrino properties, while a typical magnitude of (yL)2i is 0.5~1. Then, the decay branch k++μ+μ+ is probably as dominant as ~100% and we need to consider the 8 TeV bound seriously. The simplest attitude would be to avoid examining the shaded regions in Fig. 2, which indicate the excluded parts in the 95% C.L. via the ATLAS 8 TeV data with the assumption of B(k±±μ±μ±)=100% [217].

Fig. 3.

A schematic description for the decay patterns of k++ or ja++ with two anti-muons in the final state. Here, h1+'s in the left diagram are off-shell particles.

Fig. 3.

A schematic description for the decay patterns of k++ or ja++ with two anti-muons in the final state. Here, h1+'s in the left diagram are off-shell particles.

When one more doubly charged scalar j1++ (Nj=1) exists, a detailed analysis is needed for precise bounds on k±± and j1±±. Benchmark values are given in Fig. 2 by the vertical black dotted lines, which represent corresponding bounds on the universal physical mass mk±± when we assume B(ja±±μ±μ±)=100% for all ja±±. We obtain the 95% C.L. lower bounds on the universal mass value mk±± as ~500GeV(Nj=1), ~660GeV(Nj=10), ~900GeV(Nj=100), ~980GeV(Nj=200), and ~1030GeV(Nj=300), respectively, through numerical simulations by

MadGraph5_aMC@NLO
[218,219] with the help of
FeynRules
[220222] for model implementation.

The method that we adopt for evaluating the corresponding 95% C.L. bounds with the assumption of B(ja±±μ±μ±)=100% for all ja±±, where more than one doubly charged scalar exists, is as follows. When N doubly charged scalars are present, the expected number of the total signal receives the multiplicative factor N. Following this statement, we can estimate the bound on the universal mass mk±± via the pair production cross section of a doubly charged scalar k±± (in the N=1 case) though the sequence ppγ/Z+Xk++k+X. The bound should correspond to the mass where the production cross section is N times smaller than the benchmark value in mk±±=438GeV, which is the 95% C.L. lower bound on mk±± from the ATLAS 8 TeV data [217]. We obtained the leading-order cross section as 0.327fb, which is fairly close to the ATLAS value, 0.357fb read from Ref. [217, Fig. 4(c)). In calculations, we used the

CTEQ6L
proton PDF [223] and set the renormalization and factorization scales to 2mk±±.

Fig. 4.

Positions of the “Landau pole” defined as gY(μ)=4π.

Fig. 4.

Positions of the “Landau pole” defined as gY(μ)=4π.

Here, we point out an interesting possibility. From Eq. (2.9), if λ11(1)Σ0 is rather larger than μ22(1), the pattern j1++μ+μ+νiνj possibly becomes considerable, where we cannot reconstruct the invariant mass of the doubly charged scalar since missing energy exists in this decay sequence. Then, the significance for exclusion would be dropped and we could relax the bound on mj1±± to some extent. An extreme case is with a nonzero λ11(1)Σ0 and μ22(1)=0, where the branching ratio of j1++μ+μ+ becomes zero at the one-loop level and the significance takes the lowest value, which is the best for avoiding the 8 TeV LHC bound. Also in this situation, no additional contribution to the neutrino mass matrix exists and the original successful structure is not destroyed. Similar discussions are applicable when Nj is more than 1.

When we assume 100% branching fractions in ja++μ+μ+ for all ja++, the common trilinear coupling μSk should be larger than ~10TeV(Nj=0), ~8TeV(Nj=1), ~3TeV(Nj=10), less than 1TeV(Nj=100,200,300), to obtain a reasonable amount of the production cross section taking into account the ±20% theoretical error in cross section as suggested by Fig. 2. As mentioned, large trilinear couplings λ11(a)Σ0 can help us to alleviate the 8 TeV bound.

Another theoretical bound is reasonably expected when, as in the present situation, many new particles with nonzero gauge charges are introduced around 1TeV. The presence of multiple doubly charged SU(2)L singlet scalars deforms the energy evolution of the U(1)Y gauge coupling gY as  

(3.21)
1gY2(μ)=1gY2(minput)bYSM16π2log(μ2minput2)θ(μmthreshold)ΔbY16π2log(μ2mthreshold2),
where bYSM=41/6, ΔbY=4/3(Nj+1), and we implicitly assume the relation minput(=mZ)<mthreshold(=mk±±=mja±±). As a reasonable criterion, we require that the theory is still not drastically strongly coupled within the LHC reach ~10TeV.7 Positions of the “Landau pole” μ, which is defined as gY(μ)=4π, are calculated with ease as functions of Nj and mthreshold(=mk±±=mja±±) as shown in Fig. 4. Now, we recognize that under the criterion, the case with Nj100 is not restricted in the sense that the bound via the “Landau pole” is much weaker than the phenomenological requirement mk±±(=mja±±)375GeV (for preventing the decays Sk++k,ja++ja). On the other hand when Nj is rather larger than 100, meaningful bounds are expected from Fig. 4. For example, when Nj=200 (300), mk±±(=mja±±) should be greater than ~1.1TeV (~2.2TeV).

There also arises a largish loop contribution to the universal trilinear coupling μSk(=μSja) as  

(3.22)
μSkμSk(1+cδ),δ=Nj+116π2×(μSkmk±±)2.
A convenient parameter, c1, encapsulates the effects from all higher-order contributions. Precise determination of c is beyond the scope of this paper thus, instead, we show the cases with c=0.01, 0.1, and 1 as benchmarks (see Fig. 2). It is easily noticed that the loop-induced value could dominate over the tree-level value unless cδ<1, or equivalently μSk/mk±±<4π/[c(Nj+1)]1/2. This may affect the convergence of the multiloop expansion even though the theory is still renormalizable.8

Unfortunately when Nj is only a few, explaining the diphoton excess is not consistent since the value of μSk is too large and tree-level unitarity is violated. This problem is avoided when Nj10, whereas the evolution of gY through the renormalization group effect puts additional bounds on mk±±(=mja±±) when Nj100. The preferred parameter would be further constrained by cδ<1 as in Fig. 2. In conclusion, we can explain the 750 GeV excess consistently even when B(ja±±μ±μ±)=100% for all ja±±.

3.2.2. Case 2: With mass mixing

In this section, we investigate the situation when the mass mixing between S and Φ are allowed. At first, we phenomenologically introduce the mixing angle β as  

(3.23)
(ϕS)=(cβsβsβcβ)(hS),
where we use the shorthand notation cβcosβ, sβsinβ, and express the observed 125GeV and 750GeV scalars (mass eigenstates) by h and S, respectively. We assume the following effective interactions among scalars:  
(3.24)
ΔVeff=12mh2h2+12mS2S2+μSkS|k++|2+μSjaS|ja++|2+μ^SΦS|Φ|2+λ^SΦS2|Φ|2,
where mh and mS represent the mass eigenvalues 125GeV and 750GeV; μSk and μSja are effective trilinear couplings as defined in Eq. (2.10), where the contents of them are not important in this study. We note that we safely ignore the terms ϕ|k++|2 and ϕ|ja++|2 since these terms originate from the gauge-invariant interactions |Φ|2|k++|2 and |Φ|2|ja++|2, where effective trilinear couplings of them are small compared with μSk and μSja. Because of the mixing in Eq. (3.23), the terms h|k++| and h|ja++| are induced and can affect the signal strength of h.

The Shh interaction may be also introduced via the interaction Lagrangian:  

(3.25)
12μShSh2 with μShmSh[cβ32cβsβ2],
where mSh represents a mass scale and the mixing factor could be determined via the gauge-invariant term S|Φ|2.9

A significant distinction from the previous no-mixing case is that the 750GeV scalar can couple to the SM particles through the mixing effect. The inclusive production cross section at the LHC is deformed as  

(3.27)
σ(ppS+X)(σppH750GeVSMggF+σppH750GeVSMVBF)sβ2+σppSpf,
where σppH750GeVSMggF and σppH750GeVSMVBF represent the inclusive production cross section of the SM-like Higgs boson with 750GeV mass through the gluon fusion and vector boson fusion processes, respectively, and σppSpf shows a corresponding value through the photon fusion in Eq. (3.14). We adopt the following digits in [91,129,228,229]:  
(3.28)
σppH750GeVSMggF={156.8fbat s=8TeV,590fbat s=13TeV,σppH750GeVSMVBF={50fbat s=8TeV,220fbat s=13TeV,
 
(3.29)
Γtot(H750 GeVSM)=247GeV,B(H750 GeVSMWW)=58.6%,B(H750 GeVSMZZ)=29.0%.
Part of the relevant partial decay widths are written down as  
(3.30)
ΓSWW=ΓH750GeVSMWWsβ2,
 
(3.31)
ΓShh~(μSh)232πmS(14mh2mS2)1/2,
and the total width takes the form  
(3.32)
Γtot(S)~[Γtot(H750 GeVSM)ΓH750GeVSMZZ]sβ2+ΓSγγ+ΓSZγ+ΓSZZ+ΓShh,
where the minuscule parts B(H750 GeVSMγγ)=1.79×105%, B(H750 GeVSMZγ)=1.69×104%, and B(H750 GeVSMgg)=2.55×102% [129] could be safely neglected. Here, ΓSγγ, ΓSZγ, and ΓSZZ describe decay widths at the one-loop level, where the multiple doubly charged scalars propagate in the loops. When we take the limit sβ0, they are reduced to Eqs. (3.2)–(3.4). Explicit forms of these widths are summarized in Appendix B.

In Fig. 5, prospects are widely discussed in the choice of the mass of the degenerate doubly charged scalars (mk±±[=mja±±]=900GeV) and two different choices of mSh (0.5TeV [left panel] and 1.9TeV [right panel]). First, we emphasize that the 125GeV Higgs h couples to the doubly charged scalars through the mixing in Eq. (3.23) in the present setup. As in Ref. [150], we take the results at s=7 and 8TeV of the five Higgs decay channels reported by the ATLAS and CMS experiments into consideration, which are hγγ, hZZ, hWW, hbb¯, hτ+τ [230235], and calculate a χ2 variable for estimating 2σ allowed ranges of the parameter space, which are depicted in light blue.10 Here, we find two types of allowed regions with and without including sβ=0, which correspond to the cases with and without accidental cancelation between SM contributions and the new contributions through the mixing, respectively.

Fig. 5.

Allowed ranges of the parameters {Σa(μa),sβ} are shown in the choice of the mass of the degenerate doubly charged scalars (mk±±[=mja±±]=900GeV) and two different choices of mSh (0.5TeV [left panel] and 1.9TeV [right panel]). The light blue regions represent 2σ allowed regions of 125GeV Higgs signal strengths, while the orange regions suggest the areas where the 750GeV excess is suitably explained. The gray/cyan regions are excluded in 95% C.L.s by the ATLAS 8TeV results for Sγγ/ZZ. For better understanding, several contours for the total width of S (ΓS), total production cross sections at s=8/13TeV (σtot,8/13), and the percentage of the production through the photon fusion at s=13TeV (σpf,13) are illustrated.

Fig. 5.

Allowed ranges of the parameters {Σa(μa),sβ} are shown in the choice of the mass of the degenerate doubly charged scalars (mk±±[=mja±±]=900GeV) and two different choices of mSh (0.5TeV [left panel] and 1.9TeV [right panel]). The light blue regions represent 2σ allowed regions of 125GeV Higgs signal strengths, while the orange regions suggest the areas where the 750GeV excess is suitably explained. The gray/cyan regions are excluded in 95% C.L.s by the ATLAS 8TeV results for Sγγ/ZZ. For better understanding, several contours for the total width of S (ΓS), total production cross sections at s=8/13TeV (σtot,8/13), and the percentage of the production through the photon fusion at s=13TeV (σpf,13) are illustrated.

The orange regions suggest the 2σ-favored areas with taking account of the 20% theoretical error in the present way for photon-fusion production cross section summarized in Eq. (3.20). Here, for an illustration we use the values in the cases of Γ/m0 and Γ/m=6% for the regions Γ/m<1% and Γ/m1%, respectively. The gray/cyan regions are excluded in 95% C.L.s by the ATLAS 8TeV results for Sγγ/ZZ. For a better understanding, several contours for the total width of S (ΓS), total production cross sections at s=8/13TeV (σtot,8/13), and the percentage of the production through the photon fusion at s=13TeV (σpf,13) are illustrated. Relevant branching ratios of S are shown in Fig. 6 for the two configurations in Fig. 5.

Fig. 6.

Relevant branching ratios of S in the two configurations in Fig. 5 are shown. Here, values of Σa(μa) are suitably fixed as typical digits in the corresponding allowed regions.

Fig. 6.

Relevant branching ratios of S in the two configurations in Fig. 5 are shown. Here, values of Σa(μa) are suitably fixed as typical digits in the corresponding allowed regions.

Now we focus on two types of consistent solutions around sβ0 and sβ0.15. The physics in the situation sβ0 is basically the same as the previous “case 1” without the mass mixing effect, where the total decay width is small, concretely less than 1GeV. On the other hand, when sβ0.15, partial widths of decay branches that are opened by a nonzero value of sβ become sizable and expected values of the total width can become, interestingly, near 10.5GeV or 45GeV, which are the latest 13TeV best-fit values of the CMS and ATLAS groups, respectively.

Finally, we briefly comment on tree-level unitarity. When we consider mk±±[=mja±±]=900GeV, the bound via tree-level unitarity is relaxed in both sβ0 and sβ0.15. However, with a large value of the universal trilinear coupling in the 3 to 6 TeV range, cδ<1 is achieved only if c1 when B(ja±±μ±μ±)=100% for all ja±±, which may require further model-building efforts.

4. Conclusion and discussion

In this paper, we investigated a possibility for explaining the recently announced 750 GeV diphoton excess by the ATLAS and CMS experiments at the CERN LHC in the context of loop-induced singlet production and decay through photon fusion. When a singlet scalar S, which is a candidate of the resonance particle, couples to doubly charged particles, we can obtain a suitable amount of the cross section of ppS+Xγγ+X without introducing a tree-level production of S. In three-loop radiative neutrino models, SU(2)L singlet multiple doubly charged scalars are introduced such that the Sγγ vertex is radiatively generated and enhanced. When we consider such doubly charged scalar(s), the branching ratio B(Sγγ) is uniquely fixed at 60% by quantum numbers when S is a mass eigenstate. Constraints from 8TeV LHC data are all satisfied.

A fascinating feature in the single S production through photon fusion is that the value of B(Sγγ) as well as ΓS determines the production cross section, as shown in Eqs. (3.14) and (3.15). With the branching fraction to diphoton Sγγ60% (see Sect. 3.2.1), when we take the “wide-width” scenario with Γ/m~6%, the expected cross section to diphoton is too large. However, in the “narrow-width” scenario with ΓS=62.9MeV, it fits nicely to the best-fit value for the inclusive cross section of 2fb. We also note that the width is close to the 8+13TeV best-fit value announced by the CMS group (105MeV) (see Appendix C). This is an informative prediction of our present scenario that should be tested in the near future. Also the relative strengths of the one-loop-induced partial decay widths are insensitive to Nj as shown in Eq. (3.1) when the mixing effect between S and the Higgs doublet Φ is negligible. This universality is a remarkable property of our scenario and this relation can be tested when more data is available.

When S and the Higgs doublet Φ can mix, some distinctive and interesting features are found. In the first thought, only a small mixing sinβ1 is allowed to circumvent drastic modifications to 125GeV Higgs signal strengths but we can see another interesting region of parameter space with sinβ0.15, where the 750GeV excess can be explained consistently within the “wide-width” scenario (see Sect. 3.2.2). However a big part of the parameter space, especially in the case with the scalar mixing, would lie outside the cδ<1 region, which requires c1 for a viable model.

Finally, we discuss further extensions of the model and other phenomenological issues.

  • A possible extension of the present direction is to introduce NSSU(2)L singlet scalars, (S=S1,S2,,SNS), without hypercharge in the theory. If the masses of the scalars are almost degenerate to 750GeV, the current experiment may not be able to detect the multiple bumps so that they would look like a single bump as we see it. The total cross section, then, is enhanced by the multiplicative factor NS2 as  

    (4.1)
    σtot(ppγγ+X)NS2σ(ppS+Xγγ+X).

  • Another possible extension is that we also introduce the singly charged scalars h˜1,2± that hold the same quantum numbers as h1,2± and have the same interaction with ja±± as h1,2± do with k±±. In such a possibility, contributions to the neutrino mass matrix are enhanced and we can reduce the value of the large coupling required for a consistent explanation in the original model, especially in (yR)22. See the appendix for details.

  • The triple coupling of the Higgs boson could be enhanced in our case that may activate strong first-order phase transition, which is a necessity for realizing the electroweak baryogenesis scenario [236]. In such a case, radiative seesaw models can explain not only neutrino mass and dark matter but also baryon asymmetry of the universe.

  • The decays k±±±± and ja±±±± provide very clean signatures. The 13 TeV LHC would be expected to replace the current bound on the universal mass, e.g., mk±±>438GeV when B(k±/ja±μ±μ±)=100% for all the doubly charged scalars, from the 8 TeV data [217] soon. An important feature recognized from Fig. 2 is that when Nj is not so large as around 10, only light doubly charged scalars are consistent with the bound from tree-level unitarity. Such possibilities will be exhaustively surveyed and eventually confirmed or excluded in the near future. On the other hand, when Nj is as large as around 10, from Fig. 2, more than ~700GeV doubly charged scalars can exist holding tree-level unitarity. Such heavy particles require a suitable amount of integrated luminosity for being tested in colliders. In other words, such possibilities will be hard to discard in the near future.

  • It might be worth mentioning the distinction between our model discussed here and the other well-known radiative models, namely, the Zee model [131] at the one-loop level, the Zee–Babu model [133,134] at the two-loop level, the Kraus–Nasri–Trodden (KNT) model [136], the Aoki–Kanemura–Seto (AKS) model [137,138], and the Gustafsson–No–Rivera (GNR) model [139] at the three-loop level. Essentially, any model that includes isospin singlet charged bosons potentially explains the 750 GeV diphoton excess along the same lines as discussed in this paper. Among those, three-loop models have natural DM candidates by construction, which we regard as a phenomenologically big advantage. Our model shares this virtue. On the other hand, in view of the charged boson, our model and also the GNR model include doubly charged particles. From the currently available data, it is not possible to distinguish the effect of a singly charged scalar from a doubly charged scalar. However, we still see that a doubly charged boson is in favor of the explanation of the 750 GeV diphoton excess simply because of the enhanced diphoton coupling.

  • As we discussed before, k±± decays to μ±μ± with an almost 100% branching fraction, distinctively from other models, e.g., the Zee–Babu model, due to the large coupling (yR)222π, which is required to realize the observed neutrino data in our setup consistently.

Note Added: In the recent update in ICHEP 2016 (on 5th August 2016) after we submitted this manuscript to PTEP, which includes the analyzed data accumulated in 2016 (ATLAS: 15.4f b1, CMS: 12.9f b1), the 750 GeV diphoton signal now turns out to be statistically disfavored [237,238]. Nevertheless, we are still motivated to study the diboson resonance which may show up in a higher energy domain11 and the generic results in this paper would be useful in the future in any case.

Acknowledgements

S.K., K.N., Y.O., and S.C.P. thank the workshop, Yangpyung School 2015, for providing us with an opportunity to initiate this collaboration. We are grateful to Eung Jin Chun, Satoshi Iso, Takaaki Nomura, and Hiroshi Yokoya for fruitful discussions. K.N. thanks Koichi Hamaguchi for useful comments when the first revision had been prepared. S.K. was supported in part by Grant-in-Aid for Scientific Research, Ministry of Education, Culture, Sports, Science and Technology (MEXT), No. 23104006, and Grant H2020-MSCA-RISE- 2014 No. 645722 (Non-minimal Higgs). This work is supported in part by the National Research Foundation of Korea (NRF) Research No. 2009-0083526 (Y.O.) of the Republic of Korea. S.C.P. is supported by an NRF grant funded by the Korean government (MSIP) (Nos. 2016R1A2B2016112 and 2013R1A1A2064120). This work was supported by IBS under the project code IBS-R018-D1 for R.W.

Funding

Open Access funding: SCOAP3.

Appendix A. Brief review of the original model

Here, we briefly summarize features in the model discussed in Ref. [150].

  • (a) In this model, the sub-eV neutrino masses are radiatively generated at the three-loop level with the loop suppression factor 1/(4π)6. In such a situation, a part of couplings, including scalar trilinear couplings, contributing to the neutrino matrix tends to be close to unity.

  • (b) When a scalar trilinear coupling is large, it can put a negative effect on scalar quartic couplings at the one-loop level, which threatens the stability of the vacuum.

  • (c) The doubly charged scalar k±± is isolated from the charged lepton at the leading order under the assignment of the global U(1) charges summarized in Table 1. Then, the charged particle does not contribute to lepton-flavor-violating processes significantly and a few hundred GeV mass is possible.

  • (d) The two singly charged scalars h1± and h2± have couplings to the charged leptons at the tree level. Since in our model a part of couplings are sizable, constraints from lepton flavor violations and vacuum stability do not allow a few hundred GeV masses, especially when k±± is around a few hundred GeV. The result of the global analysis in our previous paper [150] says that when k±± is 250GeV (which is around the minimum value of mk±±), mh1± and mh2± should be greater than 3TeV.

  • (e) In the allowed parameter configurations, we found that the absolute value of the coupling (yR)22 (in front of N¯R2eR2ch2) tends to be 8~9, while the peak of the distribution of the scalar trilinear couplings μ11λ11v/2 (in front of h1h1k++) and μ22 (in front of h2+h2+k) is around 14~15TeV. We assumed that values of μ11 and μ22 are the same and real in the analysis.

  • (f) The two CP even components are mixed each other as shown in Eq. (2.4). By the (simplified) global analysis in Ref. [150] based on the data in Refs. [231235], the sine of the mixing angle α should be  

    (A.1)
    |sinα|0.3,
    within 2σ allowed regions.

  • (g) On the other hand, the observed relic density requires a specific range of sinα. In our model, the Majorana DM NR1 communicates with the SM particles and the U(1) NG boson G through the two CP even scalars h and H. When v is O(1)TeV, DM – DM – h/H couplings are significantly suppressed as (MN1/v) and then we should rely on the two scalar resonant regions. When we consider the situation mDM/2mh(125GeV), a reasonable amount of the mixing angle α is required as  

    (A.2)
    |sinα|0.3,
    where a tense situation with Eq. (A.1) is observed. The allowed range of v is a function of sinα and the maximum value is  
    (A.3)
    v|max~9TeV when |sinα|~0.3.
    When the other resonant point is selected as mDM/2mH, the requirement on the angle is  
    (A.4)
    |sinα|0.3
    when mH=250GeV or a bit more. We find that the heavy H as mH=500GeV cannot explain the relic density because of the suppression in the resonant propagator of H. The maximum of v is found as  
    (A.5)
    v|max~6TeV when 0|sinα|0.05,
    where the couplings of H to the SM particles become so weak and hard to be excluded from the 8 TeV LHC results.

Appendix B. Decay widths at one loop

Here, we summarize the forms of relevant decay widths at the one-loop level in the presence of the scalar mixing in Eq. (3.23). We mention that we ignore ΓSgg since this value is tiny because of the fact B(H750 GeVSMgg)=2.55×102%. The widths of the 125GeV Higgs boson are used for global fits of signal strengths of the observed Higgs:  

(B.1)
Γhgg=αs2mh372π3v2|34(A1/2γγ(τtSM))cβ|2,
 
(B.2)
Γhγγ=αEM2mh3256π3v2|(A1γγ(τWSM)+NCQt2A1/2γγ(τtSM))cβ+12v[Σaμa]mk±±2Qk2A0γγ(τkSM)(sβ)|2,
 
(B.3)
ΓhZγ=αEM2mh3512π3(1mZ2mh2)3|ASMZγcβ[Σaμa]mk±±2(2QkgZkk)A0Zγ(τkSM,λk)(sβ)|2,
 
(B.4)
ΓSγγ=αEM2mS3256π3v2|(A1γγ(τW)+NCQt2A1/2γγ(τt))sβ+12v[Σaμa]mk±±2Qk2A0γγ(τk)cβ|2,
 
(B.5)
ΓSZγ=αEM2mS3512π3(1mZ2mS2)3|ASMZγ(τW,tSMτW,t)sβ[Σaμa]mk±±2(2QkgZkk)A0Zγ(τk,λk)cβ|2,
 
(B.6)
ΓSZZ=|(Γtot(H750 GeVSM)B(H750 GeVSMZZ))1/2sβ+MSZZcβ|2,
with the factors  
(B.7)
ASMZγ=2v[cotθWA1Zγ(τWSM,λW)+NC(2Qt)(T3(t)2QtsW2)sWcWA1/2Zγ(τtSM,λt)],
 
(B.8)
MSZZ={(sW22cW2)αEM2mS3512π3(1mZ2mS2)3}1/2[[Σaμa]mk±±2(2QkgZkk)A0Zγ(τk,λk)],
 
(B.9)
A1γγ(x)=x2[2x2+3x1+3(2x11)f(x1)],
 
(B.10)
A1/2γγ(x)=2x2[x1+(x11)f(x1)],
 
(B.11)
A1Zγ(x,y)=4(3tan2θW)I2(x,y)+[(1+2x1)tan2θW(5+2x1)]I1(x,y),
 
(B.12)
A1/2Zγ(x,y)=I1(x,y)I2(x,y).

Here, the ratios and the two functions are defined for convenience:  

(B.13)
τiSM=4mi2mh2,τi=4mi2mS2(i=t,W,k),
 
(B.14)
I1(x,y)=xy2(xy)+x2y22(xy)2[f(x1)f(y1)]+x2y(xy)2[g(x1)g(y1)]=A0Zγ(x,y),
 
(B.15)
I2(x,y)=xy2(xy)[f(x1)f(y1)].
Here, αs, NC(=3), Qt(=2/3), and T3(t)(=1/2) are the fine structure constants of the QCD coupling, the QCD color factor for quarks, the electric charges of the top quark in units of the positron's one, and the weak isospin of the top quark, respectively. Other variables have already been defined around Eqs. (3.5)–(3.8). When we take the limit sβ0, ΓSZZ is reduced to the form in Eq. (3.2).

Appendix C. Additional plots

In this appendix, we provide plots for discussing the case of the mixing of two fields S and Φ through mass terms under the assumption S=0. Here, the mass parameter mSh in the Shh interaction is automatically determined by the two mass eigenvalues and the mixing angle β as shown in Eq. (3.26). We note that the two choices in the universal mass of doubly charged scalars (660GeV and 900GeV) are from the expected 95% C.L. lower bounds under the assumption B(ja±±μ±μ±)=100% when Nj=10 and Nj=100, respectively.

Fig. C.1.

Allowed ranges of the parameters {Σa(μa),sβ} are shown in the two choices of the mass of the degenerate doubly charged scalars (mk±±[=mja±±]=660/900GeV [left panel/right panel]). Under the assumption S=0, the value of mSh is fixed as shown in Eq. (3.26). The light blue regions represent 2σ allowed regions of 125GeV Higgs signal strengths, while the orange regions suggest the areas where the 750GeV excess is suitably explained. The gray/cyan regions are excluded in 95% C.L.s by the ATLAS 8TeV results for Sγγ/ZZ. For a better understanding, several contours for the total width of S (ΓS), total production cross sections at s=8/13TeV (σtot,8/13), and the percentage of the production through the photon fusion at s=13TeV (σpf,13) are illustrated.

Fig. C.1.

Allowed ranges of the parameters {Σa(μa),sβ} are shown in the two choices of the mass of the degenerate doubly charged scalars (mk±±[=mja±±]=660/900GeV [left panel/right panel]). Under the assumption S=0, the value of mSh is fixed as shown in Eq. (3.26). The light blue regions represent 2σ allowed regions of 125GeV Higgs signal strengths, while the orange regions suggest the areas where the 750GeV excess is suitably explained. The gray/cyan regions are excluded in 95% C.L.s by the ATLAS 8TeV results for Sγγ/ZZ. For a better understanding, several contours for the total width of S (ΓS), total production cross sections at s=8/13TeV (σtot,8/13), and the percentage of the production through the photon fusion at s=13TeV (σpf,13) are illustrated.

Fig. C.2.

Relevant branching ratios of S in the two configurations in Fig. C.1 are shown. Here, values of Σa(μa) are suitably fixed as typical digits in the corresponding allowed regions.

Fig. C.2.

Relevant branching ratios of S in the two configurations in Fig. C.1 are shown. Here, values of Σa(μa) are suitably fixed as typical digits in the corresponding allowed regions.

References

[1]
ATLAS Collaboration
, Technical Report ATLAS-CONF-2015-081.
[2]
CMS Collaboration
, Technical Report CMS-PAS-EXO-15-004.
[3]
ATLAS Collaboration
, Technical Report ATLAS-CONF-2016-018.
[4]
CMS Collaboration
, Technical Report CMS-PAS-EXO-16-018.
[5]
Harigaya
K.
Nomura
Y.
arXiv:1512.04850 [Search inSPIRE].
[6]
Mambrini
Y.
Arcadi
G.
Djouadi
A.
arXiv:1512.04913 [Search inSPIRE].
[7]
Backovic
M.
Mariotti
A.
Redigolo
D.
arXiv:1512.04917 [Search inSPIRE].
[8]
Angelescu
A.
Djouadi
A.
Moreau
G.
arXiv:1512.04921 [Search inSPIRE].
[9]
Nakai
Y.
Sato
R.
Tobioka
K.
arXiv:1512.04924 [Search inSPIRE].
[10]
Knapen
S.
Melia
T.
Papucci
M.
Zurek
K.
arXiv:1512.04928 [Search inSPIRE].
[11]
Buttazzo
D.
Greljo
A.
Marzocca
D.
arXiv:1512.04929 [Search inSPIRE].
[12]
[13]
Franceschini
R.
Giudice
G. F.
Kamenik
J. F.
McCullough
M.
Pomarol
A.
Rattazzi
R.
Redi
M.
Riva
F.
Strumia
A.
Torre
R.
arXiv:1512.04933 [Search inSPIRE].
[14]
Di Chiara
S.
Marzola
L.
Raidal
M.
arXiv:1512.04939 [Search inSPIRE].
[15]
Higaki
T.
Jeong
K. S.
Kitajima
N.
Takahashi
F.
arXiv:1512.05295 [Search inSPIRE].
[16]
McDermott
S. D.
Meade
P.
Ramani
H.
arXiv:1512.05326 [Search inSPIRE].
[17]
Ellis
J.
Ellis
S. A. R.
Quevillon
J.
Sanz
V.
You
T.
arXiv:1512.05327 [Search inSPIRE].
[18]
Low
M.
Tesi
A.
Wang
L.-T.
arXiv:1512.05328 [Search inSPIRE].
[19]
Bellazzini
B.
Franceschini
R.
Sala
F.
Serra
J.
arXiv:1512.05330 [Search inSPIRE].
[20]
Gupta
R. S.
Jager
S.
Kats
Y.
Perez
G.
Stamou
E.
arXiv:1512.05332 [Search inSPIRE].
[21]
Petersson
C.
Torre
R.
arXiv:1512.05333 [Search inSPIRE].
[22]
Molinaro
E.
Sannino
F.
Vignaroli
N.
arXiv:1512.05334 [Search inSPIRE].
[23]
Falkowski
A.
Slone
O.
Volansky
T.
arXiv:1512.05777 [Search inSPIRE].
[24]
Dutta
B.
Gao
Y.
Ghosh
T.
Gogoladze
I.
Li
T.
arXiv:1512.05439 [Search inSPIRE].
[25]
Cao
Q.-H.
Liu
Y.
Xie
K.-P.
Yan
B.
Zhang
D.-M.
arXiv:1512.05542 [Search inSPIRE].
[26]
Matsuzaki
S.
Yamawaki
K.
arXiv:1512.05564 [Search inSPIRE].
[27]
Kobakhidze
A.
Wang
F.
Wu
L.
Yang
J. M.
Zhang
M.
arXiv:1512.05585 [Search inSPIRE].
[28]
Martinez
R.
Ochoa
F.
Sierra
C. F.
arXiv:1512.05617 [Search inSPIRE].
[29]
Cox
P.
Medina
A. D.
Ray
T. S.
Spray
A.
arXiv:1512.05618 [Search inSPIRE].
[30]
Becirevic
D.
Bertuzzo
E.
Sumensari
O.
Funchal
R. Z.
arXiv:1512.05623 [Search inSPIRE].
[31]
No
J. M.
Sanz
V.
Setford
J.
arXiv:1512.05700 [Search inSPIRE].
[32]
Demidov
S. V.
Gorbunov
D. S.
arXiv:1512.05723 [Search inSPIRE].
[33]
Chao
W.
Huo
R.
Yu
J.-H.
arXiv:1512.05738 [Search inSPIRE].
[34]
Fichet
S.
von Gersdorff
G.
Royon
C.
arXiv:1512.05751 [Search inSPIRE].
[35]
Curtin
D.
Verhaaren
C. B.
arXiv:1512.05753 [Search inSPIRE].
[36]
Bian
L.
Chen
N.
Liu
D.
Shu
J.
arXiv:1512.05759 [Search inSPIRE].
[37]
Chakrabortty
J.
Choudhury
A.
Ghosh
P.
Mondal
S.
Srivastava
T.
arXiv:1512.05767 [Search inSPIRE].
[38]
Ahmed
A.
Dillon
B. M.
Grzadkowski
B.
Gunion
J. F.
Jiang
Y.
arXiv:1512.05771 [Search inSPIRE].
[39]
Agrawal
P.
Fan
J.
Heidenreich
B.
Reece
M.
Strassler
M.
arXiv:1512.05775 [Search inSPIRE].
[40]
Csaki
C.
Hubisz
J.
Terning
J.
arXiv:1512.05776 [Search inSPIRE].
[41]
Aloni
D.
Blum
K.
Dery
A.
Efrati
A.
Nir
Y.
arXiv:1512.05778 [Search inSPIRE].
[42]
Bai
Y.
Berger
J.
Lu
R.
arXiv:1512.05779 [Search inSPIRE].
[43]
Gabrielli
E.
Kannike
K.
Mele
B.
Raidal
M.
Spethmann
C.
Veermae
H.
arXiv:1512.05961 [Search inSPIRE].
[44]
Benbrik
R.
Chen
C.-H.
Nomura
T.
arXiv:1512.06028 [Search inSPIRE].
[45]
Kim
J. S.
Reuter
J.
Rolbiecki
K.
Ruiz de Austri
R.
arXiv:1512.06083 [Search inSPIRE].
[46]
Alves
A.
Dias
A. G.
Sinha
K.
arXiv:1512.06091 [Search inSPIRE].
[47]
Megias
E.
Pujolas
O.
Quiros
M.
arXiv:1512.06106 [Search inSPIRE].
[48]
Carpenter
L. M.
Colburn
R.
Goodman
J.
arXiv:1512.06107 [Search inSPIRE].
[49]
Bernon
J.
Smith
C.
arXiv:1512.06113 [Search inSPIRE].
[51]
Arun
M. T.
Saha
P.
arXiv:1512.06335 [Search inSPIRE].
[52]
Han
C.
Lee
H. M.
Park
M.
Sanz
V.
arXiv:1512.06376 [Search inSPIRE].
[54]
Chakraborty
I.
Kundu
A.
arXiv:1512.06508 [Search inSPIRE].
[55]
Ding
R.
Huang
L.
Li
T.
Zhu
B.
arXiv:1512.06560 [Search inSPIRE].
[56]
Han
H.
Wang
S.
Zheng
S.
arXiv:1512.06562 [Search inSPIRE].
[57]
Han
X.-F.
Wang
L.
arXiv:1512.06587 [Search inSPIRE].
[58]
Luo
M.-x.
Wang
K.
Xu
T.
Zhang
L.
Zhu
G.
arXiv:1512.06670 [Search inSPIRE].
[59]
Chang
J.
Cheung
K.
Lu
C.-T.
arXiv:1512.06671 [Search inSPIRE].
[60]
Bardhan
D.
Bhatia
D.
Chakraborty
A.
Maitra
U.
Raychaudhuri
S.
Samui
T.
arXiv:1512.06674 [Search inSPIRE].
[61]
Feng
T.-F.
Li
X.-Q.
Zhang
H.-B.
Zhao
S.-M.
arXiv:1512.06696 [Search inSPIRE].
[62]
Antipin
O.
Mojaza
M.
Sannino
F.
arXiv:1512.06708 [Search inSPIRE].
[63]
Wang
F.
Wu
L.
Yang
J. M.
Zhang
M.
arXiv:1512.06715 [Search inSPIRE].
[64]
Cao
J.
Han
C.
Shang
L.
Su
W.
Yang
J. M.
Zhang
Y.
arXiv:1512.06728 [Search inSPIRE].
[65]
Huang
F. P.
Li
C. S.
Liu
Z. L.
Wang
Y.
arXiv:1512.06732 [Search inSPIRE].
[66]
Liao
W.
Zheng
H.-q.
arXiv:1512.06741 [Search inSPIRE].
[67]
[68]
Dhuria
M.
Goswami
G.
arXiv:1512.06782 [Search inSPIRE].
[69]
Bi
X.-J.
Xiang
Q.-F.
Yin
P.-F.
Yu
Z.-H.
arXiv:1512.06787 [Search inSPIRE].
[70]
Kim
J. S.
Rolbiecki
K.
de Austri
R. R.
arXiv:1512.06797 [Search inSPIRE].
[71]
Berthier
L.
Cline
J. M.
Shepherd
W.
Trott
M.
arXiv:1512.06799 [Search inSPIRE].
[72]
Cho
W. S.
Kim
D.
Kong
K.
Lim
S. H.
Matchev
K. T.
Park
J.-C.
Park
M.
arXiv:1512.06824 [Search inSPIRE].
[73]
Cline
J. M.
Liu
Z.
arXiv:1512.06827 [Search inSPIRE].
[74]
Bauer
M.
Neubert
M.
arXiv:1512.06828 [Search inSPIRE].
[75]
Chala
M.
Duerr
M.
Kahlhoefer
F.
Schmidt-Hoberg
K.
arXiv:1512.06833 [Search inSPIRE].
[76]
Barducci
D.
Goudelis
A.
Kulkarni
S.
Sengupta
D.
arXiv:1512.06842 [Search inSPIRE].
[77]
Pelaggi
G. M.
Strumia
A.
Vigiani
E.
arXiv:1512.07225 [Search inSPIRE].
[78]
Boucenna
S. M.
Morisi
S.
Vicente
A.
arXiv:1512.06878 [Search inSPIRE].
[79]
[80]
Cárcamo Hernández
A. E.
Nisandzic
I.
arXiv:1512.07165 [Search inSPIRE].
[81]
Dey
U. K.
Mohanty
S.
Tomar
G.
arXiv:1512.07212 [Search inSPIRE].
[82]
de Blas
J.
Santiago
J.
Vega-Morales
R.
arXiv:1512.07229 [Search inSPIRE].
[83]
Belyaev
A.
Cacciapaglia
G.
Cai
H.
Flacke
T.
Parolini
A.
Serôdio
H.
arXiv:1512.07242 [Search inSPIRE].
[84]
Dev
P. S. B.
Teresi
D.
arXiv:1512.07243 [Search inSPIRE].
[85]
Huang
W.-C.
Tsai
Y.-L. S.
Yuan
T.-C.
arXiv:1512.07268 [Search inSPIRE].
[86]
Moretti
S.
Yagyu
K.
arXiv:1512.07462 [Search inSPIRE].
[87]
Patel
K. M.
Sharma
P.
arXiv:1512.07468 [Search inSPIRE].
[89]
Chakraborty
S.
Chakraborty
A.
Raychaudhuri
S.
arXiv:1512.07527 [Search inSPIRE].
[90]
Cao
Q.-H.
Chen
S.-L.
Gu
P.-H.
arXiv:1512.07541 [Search inSPIRE].
[91]
Altmannshofer
W.
Galloway
J.
Gori
S.
Kagan
A. L.
Martin
A.
Zupan
J.
arXiv:1512.07616 [Search inSPIRE].
[92]
Cvetič
M.
Halverson
J.
Langacker
P.
arXiv:1512.07622 [Search inSPIRE].
[93]
[94]
Allanach
B. C.
Dev
P. S. B.
Renner
S. A.
Sakurai
K.
arXiv:1512.07645 [Search inSPIRE].
[95]
Davoudiasl
H.
Zhang
C.
arXiv:1512.07672 [Search inSPIRE].
[96]
Craig
N.
Draper
P.
Kilic
C.
Thomas
S.
arXiv:1512.07733 [Search inSPIRE].
[97]
Das
K.
Rai
S. K.
arXiv:1512.07789 [Search inSPIRE].
[98]
Cheung
K.
Ko
P.
Lee
J. S.
Park
J.
Tseng
P.-Y.
arXiv:1512.07853 [Search inSPIRE].
[99]
Liu
J.
Wang
X.-P.
Xue
W.
arXiv:1512.07885 [Search inSPIRE].
[100]
Zhang
J.
Zhou
S.
arXiv:1512.07889 [Search inSPIRE].
[101]
Casas
J. A.
Espinosa
J. R.
Moreno
J. M.
arXiv:1512.07895 [Search inSPIRE].
[102]
Hall
L. J.
Harigaya
K.
Nomura
Y.
arXiv:1512.07904 [Search inSPIRE].
[103]
Han
H.
Wang
S.
Zheng
S.
arXiv:1512.07992 [Search inSPIRE].
[104]
Park
J.-C.
Park
S. C.
arXiv:1512.08117 [Search inSPIRE].
[105]
Salvio
A.
Mazumdar
A.
arXiv:1512.08184 [Search inSPIRE].
[106]
Chway
D.
Dermíšek
R.
Jung
T. H.
Kim
H. D.
arXiv:1512.08221 [Search inSPIRE].
[107]
Li
G.
Mao
Y.-n.
Tang
Y.-L.
Zhang
C.
Zhou
Y.
Zhu
S.-h.
arXiv:1512.08255 [Search inSPIRE].
[108]
Son
M.
Urbano
A.
arXiv:1512.08307 [Search inSPIRE].
[109]
Tang
Y.-L.
Zhu
S.-h.
arXiv:1512.08323 [Search inSPIRE].
[110]
An
H.
Cheung
C.
Zhang
Y.
arXiv:1512.08378 [Search inSPIRE].
[111]
Cao
J.
Wang
F.
Zhang
Y.
arXiv:1512.08392 [Search inSPIRE].
[112]
Wang
F.
Wang
W.
Wu
L.
Yang
J. M.
Zhang
M.
arXiv:1512.08434 [Search inSPIRE].
[113]
Cai
C.
Yu
Z.-H.
Zhang
H.-H.
arXiv:1512.08440 [Search inSPIRE].
[114]
Cao
Q.-H.
Liu
Y.
Xie
K.-P.
Yan
B.
Zhang
D.-M.
arXiv:1512.08441 [Search inSPIRE].
[116]
Gao
J.
Zhang
H.
Zhu
H. X.
arXiv:1512.08478 [Search inSPIRE].
[118]
Bi
X.-J.
Ding
R.
Fan
Y.
Huang
L.
Li
C.
Li
T.
Raza
S.
Wang
X.-C.
Zhu
B.
arXiv:1512.08497 [Search inSPIRE].
[119]
Goertz
F.
Kamenik
J. F.
Katz
A.
Nardecchia
M.
arXiv:1512.08500 [Search inSPIRE].
[120]
Anchordoqui
L. A.
Antoniadis
I.
Goldberg
H.
Huang
X.
Lust
D.
Taylor
T. R.
arXiv:1512.08502 [Search inSPIRE].
[121]
Dev
P. S. B.
Mohapatra
R. N.
Zhang
Y.
arXiv:1512.08507 [Search inSPIRE].
[122]
Bizot
N.
Davidson
S.
Frigerio
M.
Kneur
J. L.
arXiv:1512.08508 [Search inSPIRE].
[123]
Hamada
Y.
Noumi
T.
Sun
S.
Shiu
G.
arXiv:1512.08984 [Search inSPIRE].
[124]
Kanemura
S.
Machida
N.
Odori
S.
Shindou
T.
arXiv:1512.09053 [Search inSPIRE].
[125]
Jiang
Y.
Li
Y.-Y.
Liu
T.
arXiv:1512.09127 [Search inSPIRE].
[126]
Aad
G.
et al
Phys. Rev. D
 
92
(
3
),
032004
[arXiv:1504.05511] [Search inSPIRE].
[127]
CMS Collaboration
, Technical Report CMS-PAS-HIG-14-006.
[129]
Andersen
J. R.
et al
arXiv:1307.1347 [Search inSPIRE].
[130]
Aad
G.
et al
Eur. Phys. J. C
 
76
(
1
),
45
(
2016
) [arXiv:1507.05930] [Search inSPIRE].
[131]
Zee
A.
Phys. Lett. B
 
93
,
389
, (
1980
);
95
,
461
(
1980
) [erratum].
[132]
Cheng
T. P.
Li
L.-F.
Phys. Rev. D
 
22
,
2860
(
1980
).
[133]
Zee
A.
Nucl. Phys. B
 
264
,
99
(
1986
).
[134]
Babu
K. S.
Phys. Lett. B
 
203
,
132
(
1988
).
[135]
[136]
Krauss
L. M.
Nasri
S.
Trodden
M.
Phys. Rev. D
 
67
,
085002
(
2003
) [arXiv:hep-ph/0210389] [Search inSPIRE].
[137]
Aoki
M.
Kanemura
S.
Seto
O.
Phys. Rev. Lett.
 
102
,
051805
(
2009
) [arXiv:0807.0361] [Search inSPIRE].
[138]
Aoki
M.
Kanemura
S.
Yagyu
K.
Phys. Rev. D
 
83
,
075016
(
2011
) [arXiv:1102.3412] [Search inSPIRE].
[139]
Gustafsson
M.
No
J. M.
Rivera
M. A.
Phys. Rev. Lett.
 
110
,
211802
(
2013
);
112
,
259902
(
2014
) [erratum]; [arXiv:1212.4806] [Search inSPIRE].
[140]
Kajiyama
Y.
Okada
H.
Yagyu
K.
J. High Energy Phys.
 
10
,
196
(
2013
) [arXiv:1307.0480] [Search inSPIRE].
[141]
Ahriche
A.
Chen
C.-S.
McDonald
K. L.
Nasri
S.
Phys. Rev. D
 
90
,
015024
(
2014
) [arXiv:1404.2696] [Search inSPIRE].
[142]
Ahriche
A.
McDonald
K. L.
Nasri
S.
J. High Energy Phys.
 
10
,
167
(
2014
) [arXiv:1404.5917] [Search inSPIRE].
[143]
Chen
C.-S.
McDonald
K. L.
Nasri
S.
Phys. Lett. B
 
734
,
388
(
2014
) [arXiv:1404.6033] [Search inSPIRE].
[144]
Okada
H.
Orikasa
Y.
Phys. Rev. D
 
90
,
075023
(
2014
) [arXiv:1407.2543] [Search inSPIRE].
[145]
Hatanaka
H.
Nishiwaki
K.
Okada
H.
Orikasa
Y.
Nucl. Phys. B
 
894
,
268
(
2015
) [arXiv:1412.8664] [Search inSPIRE].
[146]
Jin
L.-G.
Tang
R.
Zhang
F.
Phys. Lett. B
 
741
,
163
(
2015
) [arXiv:1501.02020] [Search inSPIRE].
[147]
Culjak
P.
Kumericki
K.
Picek
I.
Phys. Lett. B
 
744
,
237
(
2015
) [arXiv:1502.07887] [Search inSPIRE].
[148]
Geng
C.-Q.
Huang
D.
Tsai
L.-H.
Phys. Lett. B
 
745
,
56
(
2015
) [arXiv:1504.05468] [Search inSPIRE].
[149]
Ahriche
A.
McDonald
K. L.
Nasri
S.
Toma
T.
Phys. Lett. B
 
746
,
430
(
2015
) [arXiv:1504.05755] [Search inSPIRE].
[150]
Nishiwaki
K.
Okada
H.
Orikasa
Y.
Phys. Rev. D
 
92
,
093013
(
2015
) [arXiv:1507.02412] [Search inSPIRE].
[151]
Okada
H.
Yagyu
K.
arXiv:1508.01046 [Search inSPIRE].
[152]
Ahriche
A.
McDonald
K. L.
Nasri
S.
arXiv:1508.02607 [Search inSPIRE].
[153]
Ahriche
A.
McDonald
K. L.
Nasri
S.
Phys. Rev. D
 
92
,
095020
(
2015
) [arXiv:1508.05881 [Search inSPIRE].
[154]
Nomura
T.
Okada
H.
Phys. Lett. B
 
755
,
306
(
2016
) [arXiv:1601.00386] [Search inSPIRE].
[155]
Nomura
T.
Okada
H.
arXiv:1601.04516 [Search inSPIRE].
[156]
Okada
H.
Yagyu
K.
Phys. Lett. B
 
756
,
337
(
2016
) [arXiv:1601.05038] [Search inSPIRE].
[157]
Nomura
T.
Okada
H.
Phys. Lett. B
 
756
,
295
(
2016
) [arXiv:1601.07339] [Search inSPIRE].
[158]
Arbeláez
C.
Cárcamo Hernández
A. E.
Kovalenko
S.
Schmidt
I.
arXiv:1602.03607 [Search inSPIRE].
[159]
Ko
P.
Nomura
T.
Okada
H.
Orikasa
Y.
arXiv:1602.07214 [Search inSPIRE].
[160]
Park
S. C.
Shu
Ji.
Phys. Rev. D
 
79
,
091702
(
2009
) [arXiv:0901.0720] [Search inSPIRE].
[161]
Chen
C.-R.
Nojiri
M. M.
Park
S. C.
Shu
J.
Takeuchi
M.
J. High Energy Phys.
 
09
,
078
(
2009
) [arXiv:0903.1971] [Search inSPIRE].
[162]
Chen
C.-R.
Nojiri
M. M.
Park
S. C.
Shu
J.
arXiv:0908.4317 [Search inSPIRE].
[163]
Park
S. C.
Shu
J.
AIP Conf. Proc. 1200, 587 (2010); 1200, 1051 (2010) [arXiv:0910.0931] [Search inSPIRE].
[164]
Kong
K.
Park
S. C.
Rizzo
T. G.
J. High Energy Phys.
 
04
,
081
(
2010
) [arXiv:1002.0602] [Search inSPIRE].
[165]
Kong
K.
Park
S. C.
Rizzo
T. G.
J. High Energy Phys.
 
07
,
059
(
2010
) [arXiv:1004.4635] [Search inSPIRE].
[166]
Csaki
C.
Heinonen
J.
Hubisz
J.
Park
S. C.
Shu
J.
J. High Energy Phys.
 
01
,
089
(
2011
) [arXiv:1007.0025] [Search inSPIRE].
[167]
Nishiwaki
K.
J. High Energy Phys.
 
05
,
111
(
2012
) [arXiv:1101.0649] [Search inSPIRE].
[168]
Nishiwaki
K.
Oda
K.-y.
Okuda
N.
Watanabe
R.
Phys. Lett. B
 
707
,
506
(
2012
) [arXiv:1108.1764] [Search inSPIRE].
[169]
Nishiwaki
K.
Oda
K.-y.
Okuda
N.
Watanabe
R.
Phys. Rev. D
 
85
,
035026
(
2012
) [arXiv:1108.1765] [Search inSPIRE].
[170]
Kim
D.
Oh
Y.
Park
S. C.
J. Korean Phys. Soc.
 
67
,
1137
(
2015
) [arXiv:1109.1870] [Search inSPIRE].
[171]
Datta
A. K.
Nishiwaki
K.
Niyogi
S.
J. High Energy Phys.
 
11
,
154
(
2012
) [arXiv:1206.3987] [Search inSPIRE].
[172]
Flacke
T.
Kong
K.
Park
S. C.
J. High Energy Phys.
 
05
,
111
(
2013
) [arXiv:1303.0872] [Search inSPIRE].
[173]
Kakuda
T.
Nishiwaki
K.
Oda
K.-y.
Watanabe
R.
Phys. Rev. D
 
88
,
035007
(
2013
) [arXiv:1305.1686] [Search inSPIRE].
[174]
Flacke
T.
Kong
K.
Park
S. C.
Phys. Lett. B
 
728
,
262
(
2014
) [arXiv:1309.7077] [Search inSPIRE].
[175]
Datta
A. K.
Nishiwaki
K.
Niyogi
S.
J. High Energy Phys.
 
01
,
104
(
2014
) [arXiv:1310.6994] [Search inSPIRE].
[176]
Dohi
H.
Kakuda
T.
Nishiwaki
K.
Oda
K.-y.
Okuda
N.
Afr. Rev. Phys.
 
9
,
0069
(
2014
) [arXiv:1406.1954] [Search inSPIRE].
[177]
Flacke
T.
Kong
K.
Park
S. C.
Mod. Phys. Lett.
 
A30
,
1530003
(
2015
) [arXiv:1408.4024] [Search inSPIRE].
[178]
Lee
H. M.
Kim
D.
Kong
K.
Park
S. C.
J. High Energy Phys.
 
11
,
150
(
2015
) [arXiv:1507.06312] [Search inSPIRE].
[179]
Ellis
J. R.
Gaillard
M. K.
Nanopoulos
D. V.
Nucl. Phys. B
 
106
,
292
(
1976
).
[180]
Shifman
M. A.
Vainshtein
A. I.
Voloshin
M. B.
Zakharov
V. I.
Sov. J. Nucl. Phys.
 
30
,
711
(
1979
); Yad. Fiz.
30
,
1368
(
1979
).
[182]
Carena
M.
Low
I.
Wagner
C. E. M.
J. High Energy Phys.
 
08
,
060
(
2012
) [arXiv:1206.1082] [Search inSPIRE].
[183]
Chen
C.-S.
Geng
C.-Q.
Huang
D.
Tsai
L.-H.
Phys. Rev. D
 
87
,
075019
(
2013
) [arXiv:1301.4694] [Search inSPIRE].
[184]
Harland-Lang
L. A.
Khoze
V. A.
Ryskin
M. G.
J. High Energy Phys.
 
03
,
182
(
2016
) [arXiv:1601.07187] [Search inSPIRE].
[185]
Martin
A. D.
Ryskin
M. G.
Eur. Phys. J. C
 
74
,
3040
(
2014
) [arXiv:1406.2118] [Search inSPIRE].
[186]
Danielsson
U.
Enberg
R.
Ingelman
G.
Mandal
T.
(
2016
) [arXiv:1601.00624] [Search inSPIRE].
[187]
Csaki
C.
Hubisz
J.
Lombardo
S.
Terning
J.
(
2016
) [arXiv:1601.00638] [Search inSPIRE].
[188]
Ito
H.
Moroi
T.
Takaesu
Y.
Phys. Lett. B
 
756
,
147
(
2016
) [arXiv:1601.01144] [Search inSPIRE].
[189]
D'Eramo
F.
de Vries
J.
Panci
P.
arXiv:1601.01571 [Search inSPIRE].
[191]
Fichet
S.
von Gersdorff
G.
Royon
C.
arXiv:1601.01712 [Search inSPIRE].
[192]
Harland-Lang
L. A.
Khoze
V. A.
Ryskin
M. G.
arXiv:1601.03772 [Search inSPIRE].
[193]
Franzosi
D. B.
Frandsen
M. T.
arXiv:1601.05357 [Search inSPIRE].
[194]
Abel
S.
Khoze
V. V.
arXiv:1601.07167 [Search inSPIRE].
[195]
Ben-Dayan
I.
Brustein
R.
arXiv:1601.07564 [Search inSPIRE].
[196]
Martin
A. D.
Ryskin
M. G.
J. Phys. G
 
43
,
04LT02
(
2016
) [arXiv:1601.07774] [Search inSPIRE].
[197]
Barrie
N. D.
Kobakhidze
A.
Talia
M.
Wu
L.
Phys. Lett. B
 
755
,
343
(
2016
) [arXiv:1602.00475] [Search inSPIRE].
[198]
Ito
H.
Moroi
T.
Takaesu
Y.
arXiv:1602.01231 [Search inSPIRE].
[199]
Gross
C.
Lebedev
O.
No
J. M.
arXiv:1602.03877 [Search inSPIRE].
[200]
Baek
S.
Park
J.-h.
arXiv:1602.05588 [Search inSPIRE].
[201]
Molinaro
E.
Sannino
F.
Vignaroli
N.
arXiv:1602.07574 [Search inSPIRE].
[202]
Panico
G.
Vecchi
L.
Wulzer
A.
arXiv:1603.04248 [Search inSPIRE].
[203]
Bharucha
A.
Djouadi
A.
Goudelis
A.
arXiv:1603.04464 [Search inSPIRE].
[204]
Ababekri
M.
Dulat
S.
Isaacson
J.
Schmidt
C.
Yuan
C. P.
arXiv:1603.04874 [Search inSPIRE].
[205]
Anchordoqui
L. A.
Antoniadis
I.
Goldberg
H.
Huang
X.
Lust
D.
Taylor
T. R.
arXiv:1603.08294 [Search inSPIRE].
[206]
Howe
K.
Knapen
S.
Robinson
D. J.
arXiv:1603.08932 [Search inSPIRE].
[207]
Frandsen
M. T.
Shoemaker
I. M.
arXiv:1603.09354 [Search inSPIRE].
[208]
Von Weizsacker
C. F.
Z. Phys.
 
88
,
612
(
1934
).
[209]
Williams
E. J.
Phys. Rev.
 
45
,
729
(
1934
).
[210]
Aad
G.
et al
Phys. Lett. B
 
738
,
428
(
2014
) [arXiv:1407.8150] [Search inSPIRE].
[211]
ATLAS Collaboration
, Technical Report ATLAS-CONF-2016-010.
[212]
ATLAS Collaboration
, Technical Report ATLAS-CONF-2016-016.
[214]
ATLAS Collaboration
, Technical Report ATLAS-CONF-2014-005.
[215]
Kamenik
J. F.
Safdi
B. R.
Soreq
Y.
Zupan
J.
arXiv:1603.06566 [Search inSPIRE].
[216]
Schuessler
A.
Zeppenfeld
D.
SUSY 2007 Proc., 15th Int. Conf. Supersymmetry and Unification of Fundamental Interactions, July 26–August 1, 2007, Karlsruhe, Germany (
2007
) [arXiv:0710.5175] [Search inSPIRE].
[217]
Aad
G.
et al
J. High Energy Phys.
 
03
,
041
(
2015
) [arXiv:1412.0237] [Search inSPIRE].
[218]
Alwall
J.
Herquet
M.
Maltoni
F.
Mattelaer
O.
Stelzer
T.
J. High Energy Phys.
 
06
,
128
(
2011
) [arXiv:1106.0522] [Search inSPIRE].
[219]
Alwall
J.
Frederix
R.
Frixione
S.
Hirschi
V.
Maltoni
F.
Mattelaer
O.
Shao
H. S.
Stelzer
T.
Torrielli
P.
Zaro
M.
J. High Energy Phys.
 
07
,
079
(
2014
) [arXiv:1405.0301] [Search inSPIRE].
[220]
Christensen
N. D.
Duhr
C.
Comput. Phys. Commun.
 
180
,
1614
(
2009
) [arXiv:0806.4194] [Search inSPIRE].
[221]
Alloul
A.
Christensen
N. D.
Degrande
C.
Duhr
C.
Fuks
B.
Comput. Phys. Commun.
 
185
,
2250
(
2014
) [arXiv:1310.1921] [Search inSPIRE].
[222]
Degrande
C.
Duhr
C.
Fuks
B.
Grellscheid
D.
Mattelaer
O.
Reiter
T.
Comput. Phys. Commun.
 
183
,
1201
(
2012
) [arXiv:1108.2040] [Search inSPIRE].
[223]
Pumplin
J.
Stump
D. R.
Huston
J.
Lai
H. L.
Nadolsky
P. M.
Tung
W. K.
J. High Energy Phys.
 
07
,
012
(
2002
) [arXiv:hep-ph/0201195] [Search inSPIRE].
[224]
Alves
D. S. M.
Galloway
J.
Ruderman
J. T.
Walsh
J. R.
J. High Energy Phys.
 
02
,
007
(
2015
) [arXiv:1410.6810] [Search inSPIRE].
[225]
Becciolini
D.
Gillioz
M.
Nardecchia
M.
Sannino
F.
Spannowsky
M.
Phys. Rev. D
 
91
,
015010
(
2015
); 92, 079905 (2015) [erratum] [arXiv:1403.7411] [Search inSPIRE].
[226]
Bae
K. J.
Endo
M.
Hamaguchi
K.
Moroi
T.
arXiv:1602.03653 [Search inSPIRE].
[227]
Kanemura
S.
Okada
Y.
Senaha
E.
Yuan
C. P.
Phys. Rev. D
 
70
,
115002
(
2004
) [arXiv:hep-ph/0408364] [Search inSPIRE].
[228]
Dittmaier
S.
et al
arXiv:1101.0593 [Search inSPIRE].
[229]
Dittmaier
S.
et al
arXiv:1201.3084 [Search inSPIRE].
[230]
Aad
G.
et al
Phys. Rev. D
 
91
,
012006
(
2015
) [arXiv:1408.5191] [Search inSPIRE].
[231]
Aad
G.
et al
Phys. Rev. D
 
90
,
112015
(
2014
) [arXiv:1408.7084] [Search inSPIRE].
[232]
Aad
G.
et al
J. High Energy Phys.
 
01
,
069
(
2015
) [arXiv:1409.6212] [Search inSPIRE].
[233]
ATLAS Collaboration
, Technical Report ATLAS-CONF-2014-061.
[234]
Aad
G.
et al
Phys. Rev. D
 
92
,
012006
(
2015
) [arXiv:1412.2641] [Search inSPIRE].
[235]
Khachatryan
V.
et al
Eur. Phys. J. C
 
75
,
212
(
2015
) [arXiv:1412.8662] [Search inSPIRE].
[236]
Kanemura
S.
Okada
Y.
Senaha
E.
Phys. Lett. B
 
606
,
361
(
2005
) [arXiv:hep-ph/0411354] [Search inSPIRE].
[237]
ATLAS Collaboration
, Technical Report ATLAS-CONF-2016-059.
[238]
CMS Collaboration
, Technical Report CMS-PAS-EXO-16-027.
[239]
Fuks
B.
Kang
D. W.
Park
S. C.
Seo
M.-S.
Phys. Lett. B
 
761
,
344
(
2016
) [arXiv:1608.00084] [Search inSPIRE].
1 Recently, several other works have emerged in this direction [154159].
2 In a later part of Sect. 3.2.2, we have discussions on the situation when S and Φ are mixed.
3 In general, mixing between k±± and ja±± could be allowed but the induced value via the renormalization group running at the scale of our interest is expected to be small with heavy masses of h1± and h2±, thus is neglected.
4The branching fractions are easily understood in an effective theory with the standard model gauge symmetries. See, e.g., [178] with s2=0 in the paper.
5 See also [13,120,154,157,159,186207] for related issues.
6In the case of MSSM with a light t˜1 (100GeV), A=At=Ab, tanβ1, mAMZ, |μ|MQ˜ and Mb˜, the bound on the trilinear coupling |A|5TeV was reported in Ref. [216].
7We note that measurements of running electroweak couplings put bounds on additional contributions to the beta functions of the SU(2)L and U(1)Y gauge couplings [224] even though the work [224] did not survey the parameter range that is relevant to our discussion. Similar discussions have been had in the QCD coupling [225,226], which is basically irrelevant in our case.
8One should note, however, that cδ<1 is not an absolute requirement for a consistent theory. See, e.g., Ref. [227] where a loop-induced value overwhelms the tree-level counterpart in the context of the two Higgs doublet model.
9 When S=0, the scale of mSh is determined through the two mass eigenvalues and the mixing angle β as  
(3.26)
mSh=(mS2mh2v)sin(2β),
since the mass mixing term Sϕ and the three point vertex Sϕ2 have the unique common origin S|Φ|2. Plots in this situation are provided in Appendix C.
10 The original model contains invisible channels in the 125GeV Higgs boson due to the existence of a dark matter candidate and a Nambu–Goldstone boson from the spontaneous breaking of a global U(1). We ignore the invisible widths in the global fit for simplicity.
11It is suggested that additional jet activity could provide a useful handle to understand the underlying physics of heavy resonance in Ref. [239].

Author notes

These authors contributed equally to this work
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted reuse, distribution, and reproduction in any medium, provided the original work is properly cited.Funded by SCOAP3