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Comments on the diphoton excess:

critical reappraisal of effective field theory interpretations

###### Abstract

We consider the diphoton excess observed by ATLAS and CMS using the most up-to-date data and estimate the preferred enhancement in the production rate between 8 TeV and 13 TeV. Within the framework of effective field theory (EFT), we then show that for both spin- and spin- Standard Model (SM) gauge-singlet resonances, two of the three processes , , and must occur with a non-zero rate. Moreover, we demonstrate that these branching ratios are highly correlated in the EFT. Couplings of to additional SM states may be constrained and differentiated by comparing the production rates with and without the vector-boson fusion (VBF) cuts. We find that for a given VBF to inclusive production ratio there is maximum rate of to gauge bosons, , and lighter quark anti-quark pairs. Simultaneous measurements of the width and the VBF ratio may be able to point towards the existence of hidden decays.

^{†}

^{†}preprint: MIT-CTP/4786

## I Introduction

First data at the 13 TeV LHC hint to the existence of a diphoton resonance, , with a mass 750 GeV Aad et al. (2015); on behalf of the ATLAS Collaboration (2016); Collaboration (2015); on behalf of the CMS Collaboration (2016); Collaboration (2016). Because decays to two photons, , one generically expects that also decays to the other gauge boson pairs: , , and . The aim of this manuscript is to make this generic expectation more precise and to consider implications of couplings to other Standard Model (SM) states.

We begin our discussion by performing an up-to-date fit of to the available data (after Moriond EW 2016). We quantify the enhancement of the production cross section between the 8 TeV and 13 TeV LHC runs implied by the current data. The large preferred enhancement can be interpreted as a preference for heavy quark annihilation or gluon fusion dominated production.^{1}^{1}1For alternative production mechanisms leading to large ratios of 13 TeV and 8 TeV production cross-sections c.f. Franceschini et al. (2015); Knapen et al. (2015); Liu et al. (2015); Huang et al. (2015). Accordingly, we extract the preferred diphoton signal strengths for various possible production modes.

One of our central results is that, within our working assumptions, at least two out of three branching ratios, , , and , need to be nonzero if the resonance is a SM gauge singlet.^{2}^{2}2We also show that the branching ratios are constrained if is part of an doublet or triplet representation; in general, we find that in the EFT framework at least one of the additional branching ratios to electroweak gauge bosons needs to be non vanishing.
Our working assumptions are: (1) that the resonance is either a spin-0 or spin-2 particle, and (2) that effective field theory (EFT) may be used to describe the interactions of with the SM. Implicitly, this means that we can truncate the EFT after the first few lowest dimension operators. In our case, we keep systematically all the terms up to and including operators of dimension 5 (dimension 6 for that is an electroweak doublet).

If the mixing of with the SM-Higgs and its coupling with the Higgs kinetic mixing can be neglected, two branching ratios (e.g., and ) are predicted in terms of the third one (in this case, ). If mixes with the Higgs, then measuring two branching ratios out of three, , , or , predicts the third. Below, we derive the sum rules relating these branching ratios. While the importance of these decay modes has already been stressed in the literature Franceschini et al. (2015); Low et al. (2015); Low and Lykken (2015); Alves et al. (2015); Altmannshofer et al. (2015); Di Chiara et al. (2016); Ellis et al. (2015); Petersson and Torre (2015); Cao et al. (2015); Kobakhidze et al. (2015); Fichet et al. (2015); Han et al. (2015); Feng et al. (2015); Heckman (2016); Berthier et al. (2015); Craig et al. (2015); Dev et al. (2015); Stolarski and Vega-Morales (2016); Fichet et al. (2016); Buttazzo et al. (2016); Gupta et al. (2015); Agrawal et al. (2015), we phrase the discussion directly in terms of observables, making contact with the experiments very explicit.

Determining the dominant production channel(s) of the resonance is paramount as more data accumulates. Towards that end, we consider the simultaneous measurements of the rate after applying vector boson fusion (VBF) cuts along with the total width. This can distinguish between different production channels and help resolve whether hidden decays are required. For example, we show that the ratio of the rates with and without applying VBF cuts is an efficient discriminator between gluon fusion and heavy quark production. Furthermore, for a given VBF ratio there is a maximum allowed rate to electroweak gauge bosons and quark pairs (excluding ); measuring a rate beyond this value may indicate decays to a hidden sector or to currently relatively unconstrained final states, such as .

The paper is organized as follows. In Section II we present an updated fit of the resonance to the most recent ATLAS and CMS 13 TeV and 8 TeV analyses. In Sec. III we introduce the EFT interactions of to the SM particles, assuming is either spin- and spin-. In Sec. IV we derive the correlations between branching ratios of to different EW gauge boson pairs. Section V discusses the importance of searching for the potential VBF production of , while Sec. VI considers the implications of simultaneous measurements of the VBF production rate and the total width. We conclude in Sec. VII.

## Ii Fit to the current data

We start our discussion of the diphoton excess by performing a analysis of the current publicly available ATLAS and CMS data. Our analysis addresses the following questions: (i) What is the significance of the excess after combining all publicly available ATLAS and CMS data? (ii) What is the preferred production channel for ? (iii) What is the compatibility between the 8 TeV and 13 TeV data sets? (iv) What is the preferred diphoton production cross-section, , at 13 TeV? In the combination below we resort to several approximations. Most importantly, we assume constant efficiencies for different channels. The results should thus be taken as a rough guide only.

The production mechanism of is presently unknown. Some handle on it can be obtained already now, though, by comparing the LHC data collected at the 8 TeV and 13 TeV center of mass energies. For this purpose, we examine the dependence of the current experimental results on the ratio between the 8 TeV and the 13 TeV production rates

(1) |

Examining the compatibility between the two sets of measurements gives valuable information on the production mechanism because different parton luminosities scale differently with collider energy.

In our analysis we include the 8 TeV and 13 TeV diphoton searches, including the Moriond EW 2016 updates, by ATLAS on behalf of the ATLAS Collaboration (2016) and CMS on behalf of the CMS Collaboration (2016); Collaboration (2016) and distinguish between the narrow and wide decay width hypotheses. In combining the results we assume uncorrelated measurements and construct a as function of . For CMS and narrow-width approximation we use the reported functions, (see Fig. 10 of Collaboration (2016)),

(2) |

where was chosen as the reference value by CMS for the spin-0 (spin-2) hypothesis. Here is the diphoton signal rate at 13 TeV. For the wide resonance hypothesis, , CMS does not provide the functions directly. However, we can construct a quadratic functions based on the public -value distributions for 8 TeV and 13 TeV. These are described by two parameters each, the two minima and the two curvatures.

Similarly, for ATLAS results the is defined analogously to (2), but assuming quadratic functions for . In this case, the parameters are fixed using the quoted significance of the excess for the 13 TeV and 8 TeV analyses, the compatibility of the two, and the global minimum . The first three inputs are provided in on behalf of the ATLAS Collaboration (2016) except for the narrow width case where the significance of the TeV data (re)analysis is not provided. We estimate this by fitting the signal to a single bin. Such a narrow resonance fit to a binned distribution might not faithfully represent the maximal significance of the constraint. In order to be conservative, we also consider the three bins nearest to and employ the weakest constraint in the combination fit.

The global minimum we obtained from

(3) |

where are the 8 TeV (13 TeV) integrated luminosities and runs over the relevant data points presented in on behalf of the ATLAS Collaboration (2016). The are the observed (estimated background) number of events in the -th bin and the corresponding estimated uncertainty. Signal is modeled using a normalized Breit-Wigner resonance function centered at with a width , whose integral over the -th bin is given by . Finally, is the corresponding signal efficiency for the 8 (13) TeV analysis. ATLAS presented two analyses with different cuts on the transverse energies of the photons. In the following, we employ the “spin-0” analysis which requires . While it contains only a subset of data passing the cuts of the “spin-2” analysis, it exhibits a slightly more significant excess and reduced tension with the 8 TeV results. Using MadGraph 5 Alwall et al. (2014) simulations we estimate the signal efficiency to be close to for a spin-0 (spin-2) resonance . The obtained spin-0 efficiency is consistent with the range quoted in on behalf of the ATLAS Collaboration (2016).

Next, the ATLAS and CMS functions are marginalized over

(4) |

Comparing the marginalized with the zero-signal hypothesis,

(5) |

gives the significance of the excess. We plot in Fig. 1 assuming a spin-0 with either a narrow (left panel) or wide decay width (right panel). At present the difference between spin-0 and spin-2 hypotheses is negligible, so that we do not plot the corresponding results for spin-2. The vertical lines in Fig. 1 indicate ratios expected for different production mechanisms, computed using the NLO NNPDF 2.3 Ball et al. (2015) pdf set. The excess becomes more significant for larger values of , i.e., for larger ratios of 13 TeV to 8 TeV production cross sections.

This feature can be seen also from the compatibility of the 8 TeV and 13 TeV datasets, which can be assessed through

(6) |

Here is the minimum of when varying . The dependence of on is shown in Fig. 2. Since the significance of the excess in the 8 TeV data is much smaller than for the 13 TeV measurements, the fit prefers a large enhancement of the 13 TeV production rates compared to 8 TeV. The above may be interpreted as a preference for having sea partons (gluons or heavy quarks) as initial states. These predict higher ratios, . In contrast, if is produced through valence quark annihilation or photon fusion Fichet et al. (2015); Csaki et al. (2016); Fichet et al. (2016); Harland-Lang et al. (2016a); Ababekri et al. (2016); Harland-Lang et al. (2016b) which is disfavored by more than for a narrow (wide) . In the following we will consider these results as indicative but keep the possibility of valence quark annihilation and photon (or more generally EW vector boson) fusion dominated production open. The results for spin-0 and spin-2 are, again, very similar.

Finally, we combine the ATLAS and CMS data to estimate the preferred value for the cross section . We use the combined (2), (II) to find the band as function of . The results are presented in Fig. 3. Note that the best-fit cross section grows larger as the ratio increases.
In Tab. 1 we summarize the best-fit 13 TeV diphoton rates, , for a number of assumed production mechanisms. In Tab. 1 we take the efficiencies, , to be independent of the production mechanism.
The errors due to this approximation are expected to be subleading compared to the current sizable experimental uncertainties and the limitations of our fitting procedure. For the spin-0 case one can understand the smallness of these effects most easily by noting that the photon distributions are boost invariant and thus at parton level independent of the particular parton luminosity integration.^{3}^{3}3We thank Gilad Perez for insightful discussions on this point.

In the subsequent sections we will also make use of experimental constraints on the branching ratios of to different final states, such as , , , and . These constraints are taken from Tab. 1 of Franceschini et al. (2015) (see also Gupta et al. (2015)). Using the 8 TeV data Aad et al. (2016a); collaboration (2014); Khachatryan et al. (2015); Aad et al. (2016b) one has at 95 % CL

(7) | ||||

(8) | ||||

(9) | ||||

(10) |

where we have defined

(11) |

and in the second step conservatively assumed (as in the case of ) for maximal enhancement of the prompt production between 8 TeV and 13 TeV. The corresponding 13 TeV searches are less sensitive CMS (2016). For the final state we use the recent 13 TeV bound presented by ATLAS collaboration (2016):

(12) |

to be compared with the rescaled 8 TeV bound Aad et al. (2014). The 13 TeV bound in (12) is , using the central value fb for spin-0 with wide decay width produced from the initial state, cf. Table 1.

narrow width () | wide width (%) | |||
---|---|---|---|---|

production mech. | spin-0 | spin-2 | spin-0 | spin-2 |

## Iii Effective Field Theory framework

We setup an EFT description of the interactions between and the SM fields. Our discussion partially overlaps with and extends previous results presented in Franceschini et al. (2015); Low et al. (2015); Low and Lykken (2015); Alves et al. (2015); Altmannshofer et al. (2015); Ellis et al. (2015); Petersson and Torre (2015); Cao et al. (2015); Kobakhidze et al. (2015); Fichet et al. (2015); Han et al. (2015); Feng et al. (2015); Heckman (2016); Berthier et al. (2015); Craig et al. (2015); Stolarski and Vega-Morales (2016); Fichet et al. (2016); Buttazzo et al. (2016). We make two choices for the spin of . We start with the spin- scenario and assume that is either an singlet or triplet, commenting also on the possibility that is an electroweak doublet. The resulting phenomenology is quite similar also in the case where has spin-, which we cover next. In all cases we consistently keep all the terms up to and including dimension 5 (dimension 6 for the doublet) and comment on effects at higher powers.

### iii.1 Spin-, singlet

We first consider the case where is a gauge singlet spin- particle. Assuming CP invariance, the remaining choice is whether is a scalar or a pseudo-scalar. At the level of observables the differences between these two scenarios are small. We begin with the scalar case and later mention how the pseudo-scalar case differs.

The only renormalizable interactions of the scalar with the SM are through the Higgs doublet field ,

(13) |

where in the unitary gauge , with GeV. We will see below that the dimensionful parameter is required to be small, , in order not to induce too large of a mixing between and the Higgs, . The scalar potential for contains, in addition to terms in (13), the terms involving only , . For simplicity, we assume that the scalar potential for does not introduce a vacuum expectation value for . If this is not the case, one can simply shift , and then appropriately redefine the coefficients in the SM and the interaction Lagrangians.

The dimension 5 operators induce couplings to all the SM fields,

(14) |

where is the sine (cosine) of the weak mixing angle, , , , are the QCD, hypercharge and weak isospin field strengths, respectively, and are the quark and lepton left-handed doublets, respectively, and , , are the right-handed fields for down-type quarks, up-type quarks, and leptons.
In general, the coefficients are complex matrices, where we do not display the dependence on the generational indices. However, since we are considering the CP conserving case, are assumed real.
The normalization of the operators in the first line of (14) reflects the expectation that they are induced at 1-loop level. The and terms in the denominators ensure that the parameters and do not contain the loop factors and . For simplicity we take as the operator normalization scale. ^{4}^{4}4Note that this does not mean that the EFT expansion is in , but rather in with the scale of new states that is parametrically larger then .

After electroweak symmetry breaking (EWSB), the term in (13) and the term in (14) lead to the mixing between and . In terms of the mass eigenstates, , we have , , where the mixing angle is

(15) |

The searches for heavy Higgses decaying to exclude Falkowski et al. (2015).^{5}^{5}5The bound on from WW resonance searches does not necessarily apply in this case, however, since the scalars can decay to SM fermions un-suppressed, reducing the branching fraction.
In the following, we work to leading non-trivial order in the small parameter .

After EWSB the couplings of with the vector bosons are given by^{6}^{6}6From now on we drop the prime on the notation for the mass eigenstate (i.e., and ).

(16) |

where we have defined

(17) |

Note that this parameter controls the decay rate to the longitudinal and bosons. The coefficients in the first line in (16) may be written in terms of and :

(18) | ||||

(19) | ||||

(20) |

The coupling of to a pair of Higgses is, to leading order in ,

(21) |

where GeV is the Higgs mass. These couplings mediate decays, thus, non-observation of this decay channel may then be used to put constraints on , , and .

The couplings of to the SM fermions arise from the mixing with the Higgs and from dimension-5 operators in Eq. (14). The couplings to up quarks are thus given by

(22) |

and similarly for the down quarks and charged leptons. The first term in the parenthesis is due to the mixing with the SM Higgs and is flavor diagonal. The couplings of dimension-5 operators can in principle be flavor violating, though such terms are tightly constrained Goertz et al. (2015).

In the case of pseudoscalar there is a smaller number of dimension-5 operators that we may write:

(23) |

As for the scalar case, we assume CP conservation so that the are real.

### iii.2 Spin-0, doublet

Another interesting possibility is that is one of the neutral components of a doublet with hypercharge. The scalar can in general mix with the SM Higgs doublet and the setup is captured by a general two Higgs doublet model. First, one is free to rotate the two Higgs doublets () into a basis where only one obtains a vev

(24) |

The renormalizable scalar potential of the theory has the form

(25) |

subject to the condition . In the CP conserving limit, i.e., assuming all parameters in the scalar potential to be real, the CP-odd pseudoscalar does not mix with the other neutral states and thus forms a mass eigenstate . The two CP even scalars , on the other hand, do mix to form the mass eigenstates and

(26) |

Close to the decoupling limit , one can identify with the observed Higgs boson at GeV and and/or with . In this limit,

(27) |

up to corrections of order . The renormalizable interactions in the scalar potential lead to coupling to pairs of Higgs bosons ( coupling is forbidden by CP invariance). To leading order in it can be written as

(28) |

where are in general linear combinations of the . It is very important that this coupling of to two Higgses is tuned to be small in order not to violate constraints on the branching ratio of the resonance to .

The renormalizable couplings of the scalars to SM fermions can be described as

(29) |

where denotes the SM quark and lepton left-handed doublets, while stands for the corresponding right-handed singlet fields of up-, down-quarks and charged leptons. The are in general complex matrices. In the mass basis of SM fermions after EWSB, . The couplings to fermions are given by

(30) |

where and are components of given in the SM fermion mass basis.

At operator dimension five, only couple to lepton doublets

(31) |

The term contributes to Majorana neutrino masses leading to a severe constraint . The , on the other hand, lead to couplings to neutrinos

(32) |

Finally, direct couplings of to pairs of transverse gauge bosons are induced at dimension six (in the limit these can be the leading contributions also for ). The field strengths may couple to four independent combinations of scalar bilinears,

(33) |

along with four additional variants, , that have generators inserted in-between the fields (e.g., ). Assuming CP conservation the dimension-six couplings of , to gauge bosons are then given by

(34) |

Hermiticity ensures that the coefficients are real.
The leading operators that may generate decays of () to electroweak gauge bosons are and ( and ).
In principle, either or may be ^{7}^{7}7Because of their mass degeneracy it is also possible that they both contribute to the diphoton signal leading to apparently wide resonant feature.. Since the calculations are similar for both scenarios, we assume in the following that the 750 GeV resonance is the scalar . Then, the couplings of to transverse electroweak bosons takes the same form as in (16), except that now

(35) | ||||

(36) | ||||

(37) |

where

(38) |

The parameter controlling the decays to longitudinal and , , is unrelated to the above parameters, mirroring the singlet discussion. Note that, in this case, the couplings of to electroweak gauge bosons have one additional parameter compared to the singlet scenario. This means that given two of the above couplings, the other two may be determined.

### iii.3 Spin-0, triplet

In this sub-section we introduce the effective Lagrangian assuming that the 750 GeV resonance is the charge-neutral component, , of an triplet, . For simplicity we set the hypercharge to , so that is accompanied by two charge-1 components, . The leading interactions with the SM are of dimension three and four,

(39) | ||||

(40) |

The dimensionful parameter, , induces a VEV for and is tightly constrained by EWPTs, Olive et al. (2014); Yagyu (2013). Consequently, its contribution to the decay rate is negligible. The terms lead, after EWSB, to a universal mass shift of all components. The charged state is thus almost degenerate in mass with . The – mass splitting comes from the small mixing of with the higgs, (39), and from dimension six operator .

The couplings of to SM gauge bosons and fermions start at dimension five,

(41) |

where . Note that at dimension 5, does not couple to gluons. At dimension 7 we find

(42) |

It can be easily verified that the number of independent parameters is sufficient to completely de-correlate (and ) decay rates to various EW gauge boson final states.

For the triplet with there is one renormalizable operator, . First nonrenromalizable interaction occur at dimension 7 where there are three operators of the form , , . The analysis is thus similar to the case of the being part of the electroweak doublet.

### iii.4 Spin-2

Next we consider the spin-2 case. At leading dimension five operator level the most general interactions of a massive spin-2 field satisfying the mass-shell conditions (c.f. Buchbinder et al. (2000)) with the SM can be described in terms of the traceless components of the energy momentum tensor^{8}^{8}8
Since is traceless, it couples only to the traceless components of the SM stress tensors in (III.4). For each of the SM scalars, vectors, and fermions, there is only a single dimension traceless, symmetric operator that may couple to . In (III.4) we chose these tensors to be conserved. However, there are no extra terms in the dimension effective Lagrangian for , beyond that written in Eq. (III.4), even if we assume that does not necessary couple to conserved stress tensors.

(43) |

After EWSB the gauge part becomes

(44) |

with