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 $pp\u2192S+X$ via a photon fusion process and also the decay width $\Gamma (S\u2192\gamma \gamma )$ 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 $s=13\u2002TeV$ [1,2]. Their results are based on the accumulated data of $3.2\u2009fb\u22121$ (ATLAS) and $2.6\u2009fb\u22121$ (CMS), and local/global significances are $3.9\sigma /2.3\sigma $ (ATLAS) [1] and $2.6\sigma /\u22721.2\sigma $ (CMS) [2], respectively. The best-fit values of the invariant mass are $750\u2002GeV$ by ATLAS and $760\u2002GeV$ by CMS, where ATLAS also reported the best-fit value of the total width as $45\u2002GeV$.

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 ($\Gamma /m<1%$ “narrow width” or $\Gamma /m~6\u221210%$ “wide width”) [3,4]. Based on the $3.2\u2002fb\u22121$ data set, the ATLAS group claimed that the largest deviation from the background-only hypothesis was observed near a mass of $750\u2002GeV$, which corresponds to a local excess of $3.9\sigma $ for the spin-0 case of $\Gamma \u224845\u2002GeV\u2002(\Gamma /m\u22486%)$. However, we note that the preference for wide width compared with narrow width is only minor by $~0.3\sigma $ 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 $~2.0\sigma $.

On the other hand, based on the upgraded amount of the data of $3.3\u2002fb\u22121$, the CMS group reported a modest excess of events at $760\u2002GeV$ with a local significance of $2.8\u22122.9\sigma $ depending on the spin hypothesis. The narrow width ($\Gamma /m=1.4\xd710\u22122$) maximizes the local excess. In addition, CMS reported the result of a combined analysis of $8\u2002TeV$ and $13\u2002TeV$ data, where the largest excess ($3.4\sigma $) was observed at $750\u2002GeV$ for the narrow width ($\Gamma /m=1.4\xd710\u22124$). The global significances are $<1\sigma $ ($1.6\sigma $) in the $13\u2002TeV$ ($8\u2002TeV+13\u2002TeV$) 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. [5–125]. The first unofficial interpretation of the excess in terms of the signal strength of a scalar (or a pseudoscalar) resonance $S$, $pp\u2192S+X\u2192\gamma \gamma +X$, 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

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 $s=8\u2002TeV$ [126,127], while a mild upward bump was found in their data around $750\u2002GeV$. In Ref. [11], the signal strengths at $s=8\u2002TeV$ 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

A key point to understand the resonance is the fact that no bump around $750\u2002GeV$ has been found in the other final states in either the $8\u2002TeV$ or $13\u2002TeV$ data. If $B(S\u2192\gamma \gamma )$ is the same as the $750\u2002GeV$ Higgs one, $B(h\u2192\gamma \gamma )|750\u2002GeV\u2002SM=1.79\xd710\u22127$ [129], we can immediately recognize that such a possibility is inconsistent with the observed results, e.g., in the $ZZ$ final state, at $s=8\u2002TeV$, where the significant experimental $95%$ C.L. upper bound on the $ZZ$ channel is $12\u2002fb$ by ATLAS [130] and the branching ratio $B(h\u2192ZZ)|750\u2002GeV\u2002SM=0.290$ [129]. In general, the process $S\u2192\gamma \gamma $ should be loop induced since $S$ has zero electromagnetic charge and then the value of $B(S\u2192\gamma \gamma )$ 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 $pp\u2192S+X$ and $S\u2192\gamma \gamma $ 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 $750\u2002GeV$ 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 [131–135], especially where neutrino masses are generated at the three-loop level [136–153]. 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 $pp\u2192S+X\u2192\gamma \gamma +X$ 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 $\gamma \gamma $ via one-loop diagrams. We may find the source from multi-Higgs models or extra dimensions [160–177] 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 $\Sigma 0$, two singly charged singlet scalars ($h1\xb1,h2\xb1$), and one gauge-singlet doubly charged boson $k\xb1\xb1$ to the SM. The particle contents and their charges are shown in Table 1.

Lepton fields^{a} | Scalar fields | New scalar fields^{a} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Characters | $LLi$ | $eRi$ | $NRi$ | $\Phi $ | $\Sigma 0$ | $h1+$ | $h2+$ | $k++$ | $ja++$ | $S$ |

$SU(3)C$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |

$SU(2)L$ | $2$ | $1$ | $1$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |

$U(1)Y$ | $\u22121/2$ | $\u22121$ | 0 | $1/2$ | 0 | 1 | 1 | 2 | 2 | 0 |

$U(1)$ | 0 | 0 | $\u2212x$ | 0 | $2x$ | 0 | $x$ | $2x$ | $2x$ | 0 |

Lepton fields^{a} | Scalar fields | New scalar fields^{a} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

Characters | $LLi$ | $eRi$ | $NRi$ | $\Phi $ | $\Sigma 0$ | $h1+$ | $h2+$ | $k++$ | $ja++$ | $S$ |

$SU(3)C$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |

$SU(2)L$ | $2$ | $1$ | $1$ | $2$ | $1$ | $1$ | $1$ | $1$ | $1$ | $1$ |

$U(1)Y$ | $\u22121/2$ | $\u22121$ | 0 | $1/2$ | 0 | 1 | 1 | 2 | 2 | 0 |

$U(1)$ | 0 | 0 | $\u2212x$ | 0 | $2x$ | 0 | $x$ | $2x$ | $2x$ | 0 |

^{a}The subscripts found in the lepton fields $i(=1,2,3)$ indicate generations of the fields.

^{b}The 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 $\Phi $ and the additional neutral scalar $\Sigma 0$ have VEVs, which are symbolized by $\u2329\Phi \u232a\u2261v/2$ and $\u2329\Sigma 0\u232a\u2261v\u2032/2$, respectively.

Here, $x(\u22600)$ 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 $h2\xb1$ 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.

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

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 $B(S\u2192\gamma \gamma )$ is achievable without assuming tree-level interactions [34,40]. When $B(S\u2192\gamma \gamma )$ is sizable, the production cross section of the resonance particle, $\sigma (pp\u2192S+X)$, 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}Here, $ja\xb1\xb1$ has the same charges as $k\xb1\xb1$ 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 $ja\xb1\xb1$. We also ignore such possible terms as $|ja++|2|\Phi |2$, $|ja++|2|\Sigma 0|2$ and $S|\Phi |2$, $S|\Sigma 0|2$ in Eq. (2.8) in our analysis below. This is justified as a large VEV of $S$ generates large effective trilinear couplings $\mu Sk$ and $\mu Sja$ through the original terms $S2|k++|$ and $S2|ja++|$, respectively, even when the dimensionless coefficients $\lambda ^Sk$ and $\lambda ^Sja$ are not large.

## 3. Analysis

### 3.1. Formulation of $p(\gamma )p(\gamma )\u2192S+X\u2192\gamma \gamma +X$

Additional interactions in Eq. (2.9) provide possible decay channels of $S$ to $\gamma \gamma $, $Z\gamma $, $ZZ$, and $k++k\u2212\u2212$ or $ja++ja\u2212\u2212$ up to the one-loop level. We assume that $mk\xb1\xb1$ and $mja\xb1\xb1$ are greater than $mS/2(=375\u2002GeV)$, 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 $\Gamma S\u2192\gamma \gamma $, $\Gamma S\u2192Z\gamma $, $\Gamma S\u2192ZZ$, and $\Gamma S\u2192W+W\u2212$ are governed by quantum numbers at the one-loop level^{4} as

Here, we represent a major part of partial decay widths of $S$ with our notation for loop functions with the help of Refs. [179–183]. In the following part, for simplicity, we set all the masses of the doubly charged scalars $mja\xb1\xb1$ the same as $mk\xb1\xb1$, while we ignore the contributions from the two singly charged scalars $h1,2\xb1$ since they should be at least as heavy as around $3\u2002TeV$ and decoupled as mentioned in Appendix A. The concrete forms of $\Gamma S\u2192\gamma \gamma $ and $\Gamma S\u2192Z\gamma $ are

For the production of $S$ corresponding to the $750\u2002GeV$ resonance, we consider the photon fusion process, as first discussed in the context of the $750\u2002GeV$ 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

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

Now taking the values for $\gamma (x,\mu )$ in Ref. [184], we obtain a convenient form of cross section

### 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 $(mk\xb1\xb1,\mu Sk,Nj)$: the universal physical mass of the doubly charged scalars (assuming $mk\xb1\xb1=mja\xb1\xb1$ for all $a$), the universal effective scalar trilinear coupling (assuming $\mu Sk=\mu Sja$ 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 $mk\xb1\xb1$ and $\mu Sk$, which is suggested by Eq. (3.1), as

In Ref. [215], reasonable target values for the cross section of $\sigma \gamma \gamma \u2261\sigma (pp\u2192S+X\u2192\gamma \gamma +X)$ at the $s=13\u2002TeV$ LHC were discussed as functions of the variable $R13/8$, which is defined as

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 $10~20\u2002TeV$ is required in the effective trilinear coupling $\mu Sk$. Such a large trilinear coupling would immediately lead to a violation of tree-level unitarity in the scattering amplitudes including $\mu Sk$, e.g., $k++k\u2212\u2212\u2192k++k\u2212\u2212$ or $SS\u2192k++k\u2212\u2212$ at around the energy $1\u2002TeV$, 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 $\mu Sk$ is less than $1~5\u2002TeV$.^{6}

Also, we consider the doubly charged singlet scalars produced via $pp\u2192\gamma \u2217/Z+X\u2192k++k\u2212\u2212+X$. Lower bounds at $95%$ C.L. on $mk\xb1\xb1$ via the $8\u2002TeV$ LHC data were provided by the ATLAS group in Ref. [217] as $374\u2002GeV$, $402\u2002GeV$, $438\u2002GeV$ when assuming a $100%$ branching ratio to $e\xb1e\xb1$, $e\xb1\mu \xb1$, $\mu \xb1\mu \xb1$ 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 $3\u2002TeV.$ In the case of $k++$ in $Nj=0$, when the values of $\mu 11$ and $\mu 22$ are the same or similar, from Eq. (2.2), the relative branching ratios between $k++\u2192\mu +\mu +\nu i\nu j$ and $k++\u2192\mu +\mu +$ 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 $k++\u2192\mu +\mu +$ 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 $B(k\xb1\xb1\u2192\mu \xb1\mu \xb1)=100%$ [217].

When one more doubly charged scalar $j1++$ ($Nj=1$) exists, a detailed analysis is needed for precise bounds on $k\xb1\xb1$ and $j1\xb1\xb1$. Benchmark values are given in Fig. 2 by the vertical black dotted lines, which represent corresponding bounds on the universal physical mass $mk\xb1\xb1$ when we assume $B(ja\xb1\xb1\u2192\mu \xb1\mu \xb1)=100%$ for all $ja\xb1\xb1$. We obtain the $95%$ C.L. lower bounds on the universal mass value $mk\xb1\xb1$ as $~500\u2002GeV(Nj=1)$, $~660\u2002GeV(Nj=10)$, $~900\u2002GeV(Nj=100)$, $~980\u2002GeV(Nj=200)$, and $~1030\u2002GeV(Nj=300)$, respectively, through numerical simulations by

The method that we adopt for evaluating the corresponding $95%$ C.L. bounds with the assumption of $B(ja\xb1\xb1\u2192\mu \xb1\mu \xb1)=100%$ for all $ja\xb1\xb1$, 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 $mk\xb1\xb1$ via the pair production cross section of a doubly charged scalar $k\xb1\xb1$ (in the $N=1$ case) though the sequence $pp\u2192\gamma \u2217/Z+X\u2192k++k\u2212\u2212+X$. The bound should correspond to the mass where the production cross section is $N$ times smaller than the benchmark value in $mk\xb1\xb1=438\u2002GeV$, which is the $95%$ C.L. lower bound on $mk\xb1\xb1$ from the ATLAS 8 TeV data [217]. We obtained the leading-order cross section as $0.327\u2002fb$, which is fairly close to the ATLAS value, $0.357\u2002fb$ read from Ref. [217, Fig. 4(c)). In calculations, we used the

Here, we point out an interesting possibility. From Eq. (2.9), if $\lambda 11(1)\u2329\Sigma 0\u2217\u232a$ is rather larger than $\mu 22(1)$, the pattern $j1++\u2192\mu +\mu +\nu i\nu j$ 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 $mj1\xb1\xb1$ to some extent. An extreme case is with a nonzero $\lambda 11(1)\u2329\Sigma 0\u2217\u232a$ and $\mu 22(1)=0$, where the branching ratio of $j1++\u2192\mu +\mu +$ 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 $ja++\u2192\mu +\mu +$ for all $ja++$, the common trilinear coupling $\mu Sk$ should be larger than $~10\u2002TeV(Nj=0)$, $~8\u2002TeV(Nj=1)$, $~3\u2002TeV(Nj=10)$, less than $1\u2002TeV(Nj=100,200,300)$, to obtain a reasonable amount of the production cross section taking into account the $\xb120%$ theoretical error in cross section as suggested by Fig. 2. As mentioned, large trilinear couplings $\lambda 11(a)\u2329\Sigma 0\u2217\u232a$ 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 $1\u2002TeV$. The presence of multiple doubly charged $SU(2)L$ singlet scalars deforms the energy evolution of the $U(1)Y$ gauge coupling $gY$ as

^{7}Positions of the “Landau pole” $\mu $, which is defined as $gY(\mu )=4\pi $, are calculated with ease as functions of $Nj$ and $mthreshold(=mk\xb1\xb1=mja\xb1\xb1)$ as shown in Fig. 4. Now, we recognize that under the criterion, the case with $Nj\u2272100$ is not restricted in the sense that the bound via the “Landau pole” is much weaker than the phenomenological requirement $mk\xb1\xb1(=mja\xb1\xb1)\u2273375\u2002GeV$ (for preventing the decays $S\u2192k++k\u2212\u2212,ja++ja\u2212\u2212$). On the other hand when $Nj$ is rather larger than 100, meaningful bounds are expected from Fig. 4. For example, when $Nj=200$ (300), $mk\xb1\xb1(=mja\xb1\xb1)$ should be greater than $~1.1\u2002TeV$ ($~2.2\u2002TeV$).

There also arises a largish loop contribution to the universal trilinear coupling $\mu Sk(=\mu Sja)$ as

^{8}

Unfortunately when $Nj$ is only a few, explaining the diphoton excess is not consistent since the value of $\mu Sk$ is too large and tree-level unitarity is violated. This problem is avoided when $Nj\u227310$, whereas the evolution of $gY$ through the renormalization group effect puts additional bounds on $mk\xb1\xb1(=mja\xb1\xb1)$ when $Nj\u2273100$. The preferred parameter would be further constrained by $c\delta <1$ as in Fig. 2. In conclusion, we can explain the 750 GeV excess consistently even when $B(ja\xb1\xb1\u2192\mu \xb1\mu \xb1)=100%$ for all $ja\xb1\xb1$.

#### 3.2.2. Case 2: With mass mixing

In this section, we investigate the situation when the mass mixing between $S$ and $\Phi $ are allowed. At first, we phenomenologically introduce the mixing angle $\beta $ as

The $S\u2032\u2212h\u2212h$ interaction may be also introduced via the interaction Lagrangian:

^{9}

A significant distinction from the previous no-mixing case is that the $750\u2002GeV$ scalar can couple to the SM particles through the mixing effect. The inclusive production cross section at the LHC is deformed as

In Fig. 5, prospects are widely discussed in the choice of the mass of the degenerate doubly charged scalars ($mk\xb1\xb1[=mja\xb1\xb1]=900\u2002GeV$) and two different choices of $mS\u2032h$ ($0.5\u2002TeV$ [left panel] and $1.9\u2002TeV$ [right panel]). First, we emphasize that the $125\u2002GeV$ 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 $s=7\u2002and\u20028\u2002TeV$ of the five Higgs decay channels reported by the ATLAS and CMS experiments into consideration, which are $h\u2192\gamma \gamma $, $h\u2192ZZ$, $h\u2192WW$, $h\u2192bb\xaf$, $h\u2192\tau +\tau \u2212$ [230–235], and calculate a $\chi 2$ variable for estimating $2\sigma $ 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 $s\beta =0$, which correspond to the cases with and without accidental cancelation between SM contributions and the new contributions through the mixing, respectively.

The orange regions suggest the $2\sigma $-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 $\Gamma /m\u21920$ and $\Gamma /m=6%$ for the regions $\Gamma /m<1%$ and $\Gamma /m\u22651%$, respectively. The gray/cyan regions are excluded in $95%$ C.L.s by the ATLAS $8\u2002TeV$ results for $S\u2032\u2192\gamma \gamma /ZZ$. For a better understanding, several contours for the total width of $S\u2032$ ($\Gamma S\u2032$), total production cross sections at $s=8/13\u2002TeV$ ($\sigma tot,8/13$), and the percentage of the production through the photon fusion at $s=13\u2002TeV$ ($\sigma pf,13$) are illustrated. Relevant branching ratios of $S\u2032$ are shown in Fig. 6 for the two configurations in Fig. 5.

Now we focus on two types of consistent solutions around $s\beta \u22430$ and $s\beta \u2243\u22120.15$. The physics in the situation $s\beta \u22430$ is basically the same as the previous “case 1” without the mass mixing effect, where the total decay width is small, concretely less than $1\u2002GeV$. On the other hand, when $s\beta \u2243\u22120.15$, partial widths of decay branches that are opened by a nonzero value of $s\beta $ become sizable and expected values of the total width can become, interestingly, near $10.5\u2002GeV$ or $45\u2002GeV$, which are the latest $13\u2002TeV$ best-fit values of the CMS and ATLAS groups, respectively.

Finally, we briefly comment on tree-level unitarity. When we consider $mk\xb1\xb1[=mja\xb1\xb1]=900\u2002GeV$, the bound via tree-level unitarity is relaxed in both $s\beta \u22430$ and $s\beta \u2243\u22120.15$. However, with a large value of the universal trilinear coupling in the 3 to 6 TeV range, $c\delta <1$ is achieved only if $c\u226a1$ when $B(ja\xb1\xb1\u2192\mu \xb1\mu \xb1)=100%$ for all $ja\xb1\xb1$, 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 $pp\u2192S+X\u2192\gamma \gamma +X$ 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 $S\u2212\gamma \u2212\gamma $ vertex is radiatively generated and enhanced. When we consider such doubly charged scalar(s), the branching ratio $B(S\u2192\gamma \gamma )$ is uniquely fixed at $\u224360%$ by quantum numbers when $S$ is a mass eigenstate. Constraints from $8\u2002TeV$ LHC data are all satisfied.

A fascinating feature in the single $S$ production through photon fusion is that the value of $B(S\u2192\gamma \gamma )$ as well as $\Gamma S$ determines the production cross section, as shown in Eqs. (3.14) and (3.15). With the branching fraction to diphoton $S\u2192\gamma \gamma \u224360%$ (see Sect. 3.2.1), when we take the “wide-width” scenario with $\Gamma /m~6%$, the expected cross section to diphoton is too large. However, in the “narrow-width” scenario with $\Gamma S=62.9\u2002MeV$, it fits nicely to the best-fit value for the inclusive cross section of $2\u2002fb$. We also note that the width is close to the $8+13\u2002TeV$ best-fit value announced by the CMS group ($105\u2002MeV$) (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 $\Phi $ 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 $\Phi $ can mix, some distinctive and interesting features are found. In the first thought, only a small mixing $sin\beta \u226a1$ is allowed to circumvent drastic modifications to $125\u2002GeV$ Higgs signal strengths but we can see another interesting region of parameter space with $sin\beta \u2243\u22120.15$, where the $750\u2002GeV$ 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 $c\delta <1$ region, which requires $c\u226a1$ 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 $NS$$SU(2)L$ singlet scalars, $(S=S1,S2,\u2026,SNS)$, without hypercharge in the theory. If the masses of the scalars are almost degenerate to $750\u2002GeV$, 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)$\sigma tot(pp\u2192\gamma \gamma +X)\u2248NS2\sigma (pp\u2192S+X\u2192\gamma \gamma +X).$Another possible extension is that we also introduce the singly charged scalars $h\u02dc1,2\xb1$ that hold the same quantum numbers as $h1,2\xb1$ and have the same interaction with $ja\xb1\xb1$ as $h1,2\xb1$ do with $k\xb1\xb1$. 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 $k\xb1\xb1\u2192\u2113\xb1\u2113\xb1$ and $ja\xb1\xb1\u2192\u2113\xb1\u2113\xb1$ provide very clean signatures. The 13 TeV LHC would be expected to replace the current bound on the universal mass, e.g., $mk\xb1\xb1>438\u2002GeV$ when $B(k\xb1/ja\xb1\u2192\mu \xb1\mu \xb1)=100%$ 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 $~700\u2002GeV$ 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, $k\xb1\xb1$ decays to $\mu \xb1\mu \xb1$ with an almost $100%$ branching fraction, distinctively from other models, e.g., the Zee–Babu model, due to the large coupling $(yR)22\u22732\pi $, 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: $15.4\u2002f\u2009b\u22121$, CMS: $12.9\u2002f\u2009b\u22121$), 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 domain^{11} 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: SCOAP^{3}.

#### 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 $1/(4\pi )6$. 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 $k\xb1\xb1$ 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 $h1\xb1$ and $h2\xb1$ 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 $k\xb1\xb1$ is around a few hundred GeV. The result of the global analysis in our previous paper [150] says that when $k\xb1\xb1$ is $250\u2002GeV$ (which is around the minimum value of $mk\xb1\xb1$), $mh1\xb1$ and $mh2\xb1$ should be greater than $3\u2002TeV$.

(e) In the allowed parameter configurations, we found that the absolute value of the coupling $(yR)22$ (in front of $N\xafR2eR2ch2\u2212$) tends to be $8~9$, while the peak of the distribution of the scalar trilinear couplings $\mu 11\u2261\lambda 11v\u2032/2$ (in front of $h1\u2212h1\u2212k++$) and $\mu 22$ (in front of $h2+h2+k\u2212\u2212$) is around $14~15\u2002TeV$. We assumed that values of $\mu 11$ and $\mu 22$ 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. [231–235], the sine of the mixing angle $\alpha $ should be

within $2\sigma $ allowed regions.(A.1)$|sin\alpha |\u22720.3,$(g) On the other hand, the observed relic density requires a specific range of $sin\alpha $. 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 $v\u2032$ is $O(1)\u2002TeV$, DM – DM – $h/H$ couplings are significantly suppressed as $(MN1/v\u2032)$ and then we should rely on the two scalar resonant regions. When we consider the situation $mDM/2\u2243mh(\u2243125\u2002GeV)$, a reasonable amount of the mixing angle $\alpha $ is required as

where a tense situation with Eq. (A.1) is observed. The allowed range of $v\u2032$ is a function of $sin\alpha $ and the maximum value is(A.2)$|sin\alpha |\u22730.3,$When the other resonant point is selected as $mDM/2\u2243mH$, the requirement on the angle is(A.3)$v\u2032|max~9\u2002TeV\u2002when\u2002|sin\alpha |~0.3.$when $mH=250\u2002GeV$ or a bit more. We find that the heavy $H$ as $mH=500\u2002GeV$ cannot explain the relic density because of the suppression in the resonant propagator of $H$. The maximum of $v\u2032$ is found as(A.4)$|sin\alpha |\u22720.3$where the couplings of $H$ to the SM particles become so weak and hard to be excluded from the 8 TeV LHC results.(A.5)$v\u2032|max~6\u2002TeV\u2002when\u20020\u2272|sin\alpha |\u22720.05,$

#### 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 $\Gamma S\u2032\u2192gg$ since this value is tiny because of the fact $B(H750\u200aGeVSM\u2192gg)=2.55\xd710\u22122%$. The widths of the $125\u2002GeV$ Higgs boson are used for global fits of signal strengths of the observed Higgs:

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

#### Appendix C. Additional plots

In this appendix, we provide plots for discussing the case of the mixing of two fields $S$ and $\Phi $ through mass terms under the assumption $\u2329S\u232a=0$. Here, the mass parameter $mS\u2032h$ in the $S\u2032\u2212h\u2212h$ interaction is automatically determined by the two mass eigenvalues and the mixing angle $\beta $ as shown in Eq. (3.26). We note that the two choices in the universal mass of doubly charged scalars ($660\u2002GeV$ and $900\u2002GeV$) are from the expected $95%$ C.L. lower bounds under the assumption $B(ja\xb1\xb1\u2192\mu \xb1\mu \xb1)=100%$ when $Nj=10$ and $Nj=100$, respectively.

## References

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**1200**, 1051 (2010) [arXiv:0910.0931] [Search inSPIRE].

**92**, 079905 (2015) [erratum] [arXiv:1403.7411] [Search inSPIRE].

^{2}In a later part of Sect. 3.2.2, we have discussions on the situation when $S$ and $\Phi $ are mixed.

^{3}In general, mixing between $k\xb1\xb1$ and $ja\xb1\xb1$ 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 $h1\xb1$ and $h2\xb1$, thus is neglected.

^{4}The 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.

^{6}In the case of MSSM with a light $t\u02dc1$ ($100\u2002GeV$), $A=At=Ab$, $tan\beta \u226b1$, $mA\u226bMZ$, $|\mu |\u226aMQ\u02dc$ and $Mb\u02dc$, the bound on the trilinear coupling $|A|\u22725\u2002TeV$ was reported in Ref. [216].

^{7}We 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.

^{8}One should note, however, that $c\delta <1$ 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 $\u2329S\u232a=0$, the scale of $mS\u2032h$ is determined through the two mass eigenvalues and the mixing angle $\beta $ as

^{10}The original model contains invisible channels in the $125\u2002GeV$ 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.

^{11}It is suggested that additional jet activity could provide a useful handle to understand the underlying physics of heavy resonance in Ref. [239].

## Author notes

^{3}