On a possible large width 750 GeV diphoton resonance at ATLAS and CMS
Abstract
The ATLAS and CMS experiments at the LHC have reported an excess of diphoton events with invariant mass around 750 GeV, with local significance of about and , respectively. We entertain the possibility that this excess is due to new physics, in which case the data suggest a new particle with 13 TeV LHC production cross section times diphoton branching ratio of about 5 fb. Interestingly, ATLAS reports a mild preference for a sizeable width for the signal of about 45 GeV; this result appears consistent with CMS, and is further supported by improving the compatibility of the 8 TeV and 13 TeV analyses. We focus on the possibility that the new state is a scalar. First, we show that, in addition to the new state that is needed directly to produce the diphoton bump, yet more new particles beyond the Standard Model are needed to induce diphoton decay rate of the right size. Second, we note that if the excess is attributed to the BreitWigner peak of a single new state, then the signal strength and width – taken together – suggest a total LHC production cross section of order fb. Restricting to perturbative models without adhoc introduction of many new states or exotic charges, we reach the following conclusions: (i) Gluonfusion cannot explain the required large production cross section. (ii) Tree level production from initial state quarks cannot explain the required branching ratio to two photons. (iii) Tree level production is constrained by flavor data as well as LHC RunI and Tevatron dijet analyses. Insisting on a large width we are led to suggest that more than one scalar states, nearly degenerate in mass, could conspire to produce an observed wide bump.
Contents
I Introduction
The Large Hadron Collider (LHC) experiments, ATLAS and CMS, reported an excess of diphoton events in their RunII 13 TeV analyses, using 3.2fb and 2.6fb of data, respectively ATL (2015); CMS (2015a). The local and global significance of the excess are estimated by and , respectively, at ATLAS (CMS). The occurrence of an excess in both ATLAS and CMS, compatible with RunI results, makes an exciting case for new physics. Here we assume that this is indeed the case. The reported excess is centered around GeV, and requires a 13 TeV cross section around fb, where is either a scalar or a spintwo particle produced in schannel; in what follows we focus on the scalar possibility.
Interestingly, ATLAS reports a pull in the 13 TeV data towards a sizeable signal width^{1}^{1}1See Sec. 10 in ATL (2015). The effect is reported as a systematic pull in the nuisance parameter associated with the photon energy resolution uncertainty., with best fit GeV. A finite width is also somewhat preferred, by about 0.8 compared to the narrow width alternative, for compatibility with RunI results. In terms of the theoretical interpretation, there is crucial distinction between the large and narrow width possibilities, suggesting very different modelbuilding avenues. It is therefore interesting to clarify the theoretical perspective on this issue, even while the observational evidence is still inconclusive. This is our main objective in the current paper. We combine complimentary constraints from collider and precision flavor physics, showing that a large intrinsic width for the new particle would pose a generic theoretical challenge for perturbative model interpretations. We show how relaxing the large width assumption alleviates these difficulties, making the reported signal width a key experimental observable for theory interpretation. In addition to the issue of signal width, we show that new states beyond the SM, in addition to the particle itself, are required to explain the coupling of to photons.
Our reasoning is as follows. The partial production cross section in the diphoton channel is related to the total cross section by^{2}^{2}2The expression we use is applicable in the narrow width approximation, that is valid to about ten percent accuracy even if we consider total width larger by a factor of two or so than the ATLAS best fit value. Thus even though we refer to as “large width” in the text, what we mean by large width – as will become clear in discussing the physics implications – is essentially or so. . This gives the constraint
(1) 
at the 13 TeV LHC. On the right, we put the observational data scaled to the best fit values. On the left, we put the implied theory requirement. Here we scale the total production cross section of by 1 pb, comparable to the 0.74 pb NNLO ggF production cross section LHC Higgs Cross Section Working Group et al. (CERN, Geneva, 2013) for a heavy SMlike Higgs boson at 750 GeV. Concerning the diphoton decay width, we note that perturbative models, without baroque model building, predict^{3}^{3}3We will get back to this point with explicit examples later on, e.g. in App. B. The more explicit dependence on parameters is , where is the mass of some other new particle running in the loop amplitude, with EM charge and multiplicity . We are assuming mundane , as well as , since both and would suppress the amplitude, requiring an even larger total production cross section in Eq. (1). . Note that in Eq. (1) we do not assume schannel exchange, or indeed any particular partonic topology, for the production process of at the LHC.
We find that a width of would require either very large total production cross section, or very large partial decay width to photons, or some large combination of both. This modelindependent constraint summarizes the basic modelbuilding challenge in explaining a large intrinsic total width for .
In Sec. II, before going into the signal width issue, we show that existing RunI constraints on and production at the 8 TeV LHC imply that new states, in addition to itself, must be introduced to facilitate a large enough coupling. In Sec. III we show that the cross section given by Eq. (1) is difficult to achieve with a gluon fusion (ggF) loop, unless the true width is smaller than the GeV bestfit ATLAS value by about two orders of magnitude. Forced to consider tree level production of a 750 GeV new state, we are also led to a tight spot in model building with various constraints including flavor and compatibility with RunI. We summarise our results and conclude in Sec. IV. In App. A we recap some loop functions used in the analysis. App. B illustrates the theory difficulty in obtaining within the specific example of two Higgs doublet models.
Ii New physics states in addition to are needed at least for the diphoton decay amplitude
New particles beyond the SM, in addition to the scalar , are needed to explain the effective coupling of to photons. This is exciting news: it means that itself is probably the tip of the iceberg and more new physics states, charged under EM, await discovery. To derive this result, note that the only SM particles that could be relevant to the loop amplitude are the top quark and the boson, as all other charged states in the SM have small masses , resulting with strong suppression from the loop amplitude. Since , utilising and/or in the loop amplitude implies that the tree level decays and/or are open, and are related to the amplitude by one and the same underlying coupling. This point brings about constraints from RunI and top pair production, that exclude the amplitude from being explained entirely by SM contributions.
We bound the top quark contribution to using the RunI 8 TeV ATLAS and CMS analyses Aad et al. (2015a); Khachatryan et al. (2016). For our purpose in this section, these analyses can be summarized by the approximate bound pb at 95%CL. (Here and elsewhere, a subscript on a cross section means that the cross section applies to the LHC with the stated center of mass energy. Cross section without subscript should be understood to apply to the 13 TeV LHC.)
To induce the diphoton effect we need a coupling between and the top,
(2) 
We can take to be an SU(2) doublet, in which case is a dimensionless Yukawa coupling, but we could also allow to be e.g. an SMsinglet, in which case the interaction above would have dimension five at least, namely , where is the SM Higgs VEV, some effective field theory (EFT) cutoff scale, and a dimensionless coupling. Our results apply equally to these possibilities.
Eq. (2) induces both an effective coupling of to the photons from the top loop, and treelevel decay of to . This is illustrated by the Feynmann graphs in Fig. 1. We find,
(3)  
(4) 
We recall the loop amplitude , , in App. A. The phase space factor in the decay is . Note that the total production cross section for at the 8 TeV LHC is related to that at 13 TeV via , where is the corresponding ratio of parton luminosities relevant for the production of . Combining these results we find
(5)  
where the inequality comes from imposing our rough 1 pb limit based on Aad et al. (2015a); Khachatryan et al. (2016). Note that the coupling cancels in the partial decay width ratio . We scaled this ratio of parton luminosities by a number close to the ggF value of 4.7, that would apply for schannel production of , noting that this ratio for quarkgluon fusion and qq is smaller than that for ggF.
From Eq. (5) we conclude that, at least considering direct schannel production of , the decay to two photons cannot be mediated by a top loop alone, as that would lead to violation of the 8 TeV limits of Refs. Aad et al. (2015a); Khachatryan et al. (2016) by about two orders of magnitude.
Interference effects between the SM and new physics contributions to production could be significant for new physics contributions to the differential cross section of the same order of magnitude as the SM one. Eq. (5), however, relates a topmediated decay to factor of 10100 excess in cross section. This result is therefore insensitive to the interference effects. To clarify this point, in Fig. 2 we show the 8 TeV LHC differential cross section, calculated for the SM alone and for (left) an additional scalar or (right) pseudoscalar particle with parameters chosen to account for the diphoton excess. We use an effective gluon coupling ( for pseudoscalar ) to induce ggF production of (interference effects are maximized if the production mechanism is ggF), and the interaction of Eq. (2) to induce diphoton decay (with real for scalar or for pseudoscalar). Our matrix element is computed at LO with FeynCalc Mertig et al. (1991); Shtabovenko et al. (2016) and convolved with MSTW2008 NNLO pdf Martin et al. (2009). We use a flat Kfactor of to normalize the SM total cross section to NNLO calculations Czakon et al. (2013).
The set of three bumps corresponds to adjusting the parameters to fb, at the lowest end of the cross section required for the diphoton excess. The difference between these three curves corresponds to different choices for the sign of the product of couplings , with the middle curve corresponding to artificially removing interference by hand. Clearly interference, being a ten percent effect in the relevant parameter space, is unimportant to our basic conclusion that topinduced diphoton decay would imply a major discrepancy in the distribution. In Fig. 3 we further compare the distributions to data, using for concreteness the measurements in Ref. Aad et al. (2015a) presenting the binned distribution divided by the expected SM prediction in each bin. The plot assumes fb, again at the lowest end to address the diphoton excess. Topmediated diphoton decay is unambiguously excluded by data.
Finally, while would apply for schannel direct production of , we can contemplate the possibility that is produced in association with some other massive new physics particle with mass , that then decays into . In this case, the ratio , that we scale in Eq. (5) by a value relevant for 750 GeV center of mass energy, could be larger than 5, potentially relaxing the constraint. Assuming for definiteness that the parent state is pairproduced^{4}^{4}4The argument follows similarly if the heavy parent state is singly produced, though one would then need to require TeV to obtain the required increase in parton luminosity ratio from 8 to 13 TeV. We do not discuss here additional collider signatures of such scenario, including the expectation that would be born significantly boosted in the partonic center of mass frame, or the possibility of missing transverse energy or additional hadronic activity – none of which where reported in Ref.ATL (2015) – following the heavy state decay., we note that to obtain fb, namely above the lower range required for the signal, one would need . This translates to TeV. At such large , however, we expect that the LHC production cross section for the heavy states is already quite small even at tree level QCD. For example, for this mass, gluino pair production has in the ballpark of 1 fb. This small cross section leaves no room for the additional parametric suppression of . We conclude that it is highly unlikely for the SM top quark to be the sole mediator of decay.
Analogously to the top quark discussion above, the contribution to can be bounded using the RunI 8 TeV ATLAS analysis Aad et al. (2015b), from which we have pb at 95%CL. Adopting the simple model , we find^{5}^{5}5The SM Higgs would have, in this notation, , with the electroweak coupling ; while for a heavy we expect a further suppression factor of in . We note that the loop amplitude contributing to the effective decay rate is suppressed, relative to the corresponding contribution, by a factor , due to the smaller ratio . Also note that, with our assumed coupling, the decay is dominated by longitudinal polarization.,
(6)  
(7) 
We recall the loop amplitude in App. A. Combining these results we find
(8)  
Repeating the arguments of the constraints we see that the loop cannot induce a large enough amplitude, falling short by about three orders of magnitude given current bounds from direct measurements. Thus, new particles in addition to need to be added in consistent new physics explanations of the ATLAS and CMS diphoton excess.
Iii The modelbuilding challenge of a large width
A large decay width, as possibly hinted to by ATLAS data, would be difficult to account for with ggF loop production. In Sec. III.1 we substantiate this point with a toy model example, illustrating the generic problem. We then turn in Sec. III.2 to examine the possibility of treelevel production of the 750 GeV state.
iii.1 Large total width poses a challenge for ggF loop production of
Consider a toy model with a real singlet scalar^{6}^{6}6We denote the quantum numbers of a field w.r.t. the SM gauge groups by , where and are the representations under SU(3) and SU(2) and is the hypercharge. together with colortriplet, EMcharge , vectorlike fermions , such that
(9) 
This model mimics the SM Higgstop coupling. The effective Lagrangian, integrating out and at one loop, is (see e.g. Kniehl and Spira (1995))
(10) 
where we allowed for identical copies of fermions.
The scalar decay width to two photons is
(11) 
using the fact that the factor , limiting to GeV (meaning ), is bounded by^{7}^{7}7The absolute maximum value of the factor is , reached around , or . But here GeV would be excluded by current collider limits. .
The toy model at fixed GeV has two free parameters for ggF: and the coupling . (The SM, of course, has fixed by , where is the top quark mass.) The ggF production cross section for is identical to that of the SM Higgs , up to the replacement and , with GeV being the SM Higgs vacuum expectation value (VEV). Comparing to the SM ggF cross section for a heavy Higgs, we therefore have
(12) 
The LHC 13 TeV SM NNLO+NNLL result at GeV is pb (the LO result is smaller by a Kfactor of 2.6). Limiting ourselves to perturbative and to GeV, as a conservative estimate to the lowest mass allowed by existing searches for new colored fermions, we find pb for . Comparing with Eq. (1), and allowing GeV, the requirement of the toy model to explain the ATLAS and CMS excess translates into
(13) 
To accommodate Eq. (13) at the best fit values of and , without resorting to exotic charges , we would need, e.g., 8 copies of the vectorlike fermions, all with large Yukawa couplings and rather light mass 500 GeV. It is unlikely that such multitude of subTeV colored fermions would have evaded RunI constraints.
Somewhat larger cross section could be obtained with more complex modelbuilding, namely, adding more new colored states with large couplings to to increase the ggF production further, or adding several or exoticallycharged uncolored EMcharged vectorlike fermions to boost the diphoton decay. However, the main message we see in this result is that Eq. (1), the generic impact of which is illustrated in our toy model constraint of Eq. (13), highlights the width as a key observable in the theoretical interpretation of the excess: allowing to decrease by 23 orders of magnitude from its best fit value would make it rather easy to attribute the excess to ggF production in relatively minimal perturbative extensions of the SM.
On the other hand, without appealing to multicomponent models and if we insist on believing that the large width is correct, then the ATLAS signal appears to call for tree level production of the new 750 GeV state. In the next section we study this direction, showing that it runs into difficulties of its own.
iii.2 Constraints on a large width with tree level production of
To couple a scalar field at tree level to the initial state in pp collisions – guessing in advance that sizable couplings are required, suggesting renormalizable models – we would need a Higgslike electroweak doublet with Yukawa couplings to the quarks, notably of the first generation. We therefore consider a twoHiggs doublet model (2HDM): in addition to the usual light Higgs of the SM, we add the new field .
We first consider the possibility of Yukawa coupling of to first generation up quarks. Here, production at the LHC is dominated by
(14)  
written in the quark mass basis, with being the CKM matrix. We assume for simplicity that the couplings are real.
Avoiding finetuning for the first generation quark masses requires
(15) 
where GeV and are the SMlike Higgs and the new scalar vacuum expectation values, respectively. While some mass mixing of with the SM Higgs is unavoidable, the details are modeldependent and we ignore this point here for simplicity. We note that quartic terms of the form could split the states at order GeV (or perhaps more likely GeV if we judge based on the very perturbative SM Higgs selfquartic ). This is a parametrically small splitting compared to the scale and we ignore this complication in what follows in kinematics and loop functions, though we comment about other implications later on.
The decay width of to two quarks is
(16) 
with a similar result for . We use MG5@NLO Alwall et al. (2014) to estimate the schannel total production cross section of at the 13 TeV LHC, finding
(17) 
at GeV, where is an order unity factor that we introduce here by hand. The result for is similar. We get a rough idea of the magnitude of the factor by computing the cross section , for which we get pb. This suggests . We use this estimate below, though our results are not very sensitive to the precise value of as long as it is .
In the next two subsections we examine flavor and collider constraints, finding that, because of these constraints on the production vertex, large partial width to diphotons would be needed for the treelevel coupling 2HDM to explain the 750 GeV resonance. Before going into the detailed analyses we summarise the results in Fig. 4, that we now explain.
Fig. 4 shows contours of constant in the diphoton cross section (yaxis)signal width (xaxis) plane. The value of indicated by the contours is the lower limit that is needed to match the corresponding values of cross section and total width.
The left (right) panels in Fig. 4 show the results when considering diagonal Yukawa couplings of to the first generation up (down) type quarks^{8}^{8}8We comment in the next subsections about the implications of coupling to both first and second generation quarks in a flavor U(2)symmetric way; as we show, this possibility, while it ameliorates the flavor constraints associated with breaking the approximate light flavor U(2) symmetry of the SM, makes the dijet constraints correspondingly stronger and so does not affect our conclusions.. The grey band marks the cross section required to explain the ATLAS and CMS excess. Solid lines show the constraint arising from flavor [Eq. (22) and discussion around it], while dashed lines show the constraint due to dijet searches [Eq. (24) and discussion around it].
To substantiate the theory difficulty in obtaining such large diphoton partial width, , in App. B we explore the associated auxiliary modelbuilding gymnastics in two concrete examples. In App. B.1 we show that the charged Higgs loop of the state contained in can make only a negligible contribution to the diphoton partial width of and , compared to the required . In App. B.2 we add vectorlike fermions to boost the diphoton decay, and show that extreme limits of the parameters would be needed to achieve the required effect.
iii.2.1 Flavor constraints
Any model with extra Higgs doublets, and in particular the model in Eq. (14), has flavor changing couplings for either the neutral scalar or the charged scalar or both. Consequently, our model is constrained by measurements of and mixing Blum et al. (2011), where box diagrams containing the charged and/or neutral states in lead to contributions that we summarize by:

mixing:
(18) 
mixing:
(19)
We took the zero mass limit for internal quarks in the loop function.
There are two limiting cases where the contributions to flavor changing neutral current processes appear in only one of the two quark sectors. (In a more generic case, the constraints are similar, typically stronger.) First, consider . In this case, the charged scalar couplings to are given by . Box diagrams, involving the exchange of contribute to mixing. We obtain:
(20) 
Second, consider . In this case, the neutral scalar couplings to are given by . Box diagrams, involving the exchange of the neutral scalars contribute to mixing, and give us again a limit quantitatively similar to Eq. (20).
Using Eq. (20) in Eq. (17) we have
(21) 
with a similar bound for . Using now Eq. (21) in Eq. (1), summing the contributions of and and denoting the sum collectively by , etc., we find
(22) 
Eq. (22) is depicted in the left panel of Fig. 4 as solid contours of constant in the plane.
We find results comparable to Eq. (22) when considering a dominant coupling of to down quarks, obtained by replacing in the first line of Eq. (14) with . The main difference is that the constraint analogous to Eq. (22) becomes somewhat tighter, because of the smaller PDF for compared to at the LHC. This effect for the scenario can be captured by replacing in Eq. (22). The result is shown in the right panel of Fig. 4. We note that constraints similar to Eq. (20) are obtained when coupling to any one particular case of the eight pairs of first two generations quarks, such as or . However, going to, e.g., or for the initial state introduces a significant PDF suppression in the production vertex, making the corresponding version of Eq. (22) much more constraining.
Up to here we focused on dominant coupling of to one particular pair of quarks. The flavor constraint thus derived manifests the breaking that such a coupling introduces to the approximate U(2) flavor symmetry of the first two generations in the SM. This suggests a way out, by assuming or in Eq. (14) (or similarly for the case)^{9}^{9}9We are grateful to Gilad Perez for a discussion of this point.. Adopting this form of , both Eqs. (1819) vanish up to CKM and GIM suppression factors that make flavor violation unobvservably small. We conclude that it is possible in principle to avoid the flavor constraint on by imposing flavor U(2) involving the first two generation quarks. The cost of the U(2) limit, however, is a factor enhancement in the dijet decay width of . This is shown in the next section to lead to strong constraints on its own.
Finally, we emphasize that in Eq. (22) we took the contributions of and to the signal to be equal and, moreover, assumed that and are equal up to a splitting smaller than or comparable to the resolution of the CMS and ATLAS analysis algorithms. In slight more generality, there is no reason apriori for the Yukawa coupling in Eq. (14) to be real, in which case the contributions of and would be different and include interference effects. Moreover, a mass splitting comparable to the resolution of ATLAS and CMS would lead to the two states contributing to separate, and individually less significant, resonances. Generically, we expect that this will result in a stronger constraint than Eq. (22), by about a factor of 2.
Along the same line, we note that nearlydegenerate scalar and pseudoscalar components of a neutral complex scalar field, despite having individually narrow widths, could perhaps mimic a wider bump as hinted to by the ATLAS analysis. This is an interesting way to potentially get away from the tight spot associated with an experimentally deduced large width, though the precise implementation would require knowledge of the effects in the ATLAS and CMS bump hunt algorithms. While we do not pursue this possibility further, we note that splitting the two states by more than about 10 GeV should allow ATLAS and CMS, once better statistics become available, to resolve the two adjacent bumps with their current electromagnetic energy resolution.
iii.2.2 Dijet constraints
The treelevel couplings of the 2HDM are further constrained by the LHC dijet resonance search at 8 TeV, which can be directly translated to the production crosssection at 13 TeV. We use MSTW2008 pdf set at NNLO Martin et al. (2009) to find
(23) 
The 8 TeV dijet results are reported as C.L. limit on , where is an acceptance factor reflecting the different cuts used by ATLAS and CMS. Estimating the acceptance as using MG5, we approximate the dijet bound, at 750 GeV, as pb Aad et al. (2015c); CMS (2015b). Summing again the contributions of and we thus find
(24) 
The resulting constraint is depicted in dashed lines in Fig. 4.
A comparable bound is provided by the TeVatron dijet resonance search at 1.96 TeV. When coupled to first generation quarks, the production at the TeVatron exhibits a valencePDF enhancement compared to the LHC, that partially compensates for the smaller CoM energy. The luminusity ratios are in this case
(25) 
The Tevatron dijet results are reported as C.L. limit on , for central jets. We again estimate the acceptance using MG5 and find . The Tevatron dijet bound, at 750 GeV, then reads pb Aaltonen et al. (2009), resulting in
(26) 
Iv Conclusions
The ATLAS and CMS experiments have reported a hint for a resonant excess of diphoton events around 750 GeV. We study the possibility that this excess, if not a statistical fluctuation, is generated by a spin0 resonance . In this case, the data imply a cross section of a few fb.
We argued, in a modelindependent way, that new charged particles, beyond the Standard Model (SM) ones, are necessary in order to give a large enough decay rate into two photons. Concretely, we use data from RunI of the LHC, showing that neither of the two heavy charged SM particles – the top quark and the boson – can generate that is large enough. (The light charged SM particles give negligible contributions.)
Intriguingly, the ATLAS data show preference for a large width, of order . Our focus in this work was on the question of whether the combination of the three observables, , and, in particular, large , can be accounted for in reasonable theoretical scenarios.
We first studied the scenario where the dominant production mechanism of is gluon fusion (ggF). We showed that if (i) the number of new particles is of order one, (ii) their electromagnetic charge is of order one, and (iii) their coupling to is of order one, then the rate of diphoton events would be 23 orders of magnitude too small. Thus, only rather exotic models, employing large multiplicity of new states, or large charges, or large couplings, or some combination of these unusual features, can explain the data.
We then considered tree level production, . We showed that constraints from flavor changing neutral current processes and from collider dijet analyses put an upper bound on the production cross section. Again, the new charged particles that lead to the decay should be rather exotic, as they need to generate if one aims to accommodate diphoton production rate and bump width in the ballpark of the best fit values. We demonstrated in specific models the difficulty in achieving such a high rate.
Our final conclusion is that a large total width is very difficult to accommodate in perturbative extensions of the Standard Model. It remains to be seen whether the diphoton excess will survive larger statistics and, if it does, whether the large width will still be implied by the measurements.
Note added in proof. The presentation of a 750 GeV excess by the ATLAS and CMS collaborations ATL (2015); CMS (2015a) triggered many analyses by the particle physics community. A few selected published examples, with results corroborating and partially overlapping to ours, include Buttazzo et al. (2016); Low et al. (2016); Franceschini et al. (2016); Ellis et al. (2016); Falkowski et al. (2016); many more analyses can be found in the references to ATL (2015); CMS (2015a). To the best of our knowledge, none of the works includes our flavor analysis, and none analyzed the implications of a large total width in the detailed and general way we attempted here.
Acknowledgements.
We thank Gilad Perez for useful discussions on v2 of this manuscript. YN is the Amos deShalit chair of theoretical physics. This research is supported by the ICORE program of the Planning and Budgeting Committee and the Israel Science Foundation (grant number 1937/12). YN is supported by a grant from the United StatesIsrael Binational Science Foundation (BSF), Jerusalem, Israel.Appendix A Fermion and vector boson loop amplitudes in
Appendix B 2HDM diphoton decays
In Sec. III.2 we showed that treelevel production of , that could be achieved in principle by associating with the scalar and/or pseudoscalar components of the neutral part of a heavy Higgs doublet, would require the model to satisfy the constraint in order to explain the ATLAS and CMS 750 GeV diphoton excess. In the next two subsections we illustrate the parametric difficulty in satisfying this constraint.
b.1 Diphoton decay via Higgs mixing and charged Higgs loop
The first point we clarify is that the loop diagram involving the state cannot (by far) accommodate .
For the charged Higgs loop to contribute to , and given that we must not allow a large VEV for , we note that mixing with the SM state is needed. We can obtain this by turning on
(30)  
where is the SM Higgs in Unitary gauge (with some abuse of notation we denote by both the full SM doublet and its fluctuating real scalar component).
Neglecting the VEV of , the fielddependent tree level charged Higgs mass is
(31) 
We can use the Higgs low energy theorem (LET, see e.g. Kniehl and Spira (1995)) to compute the leadinglog diphoton amplitude,
(32) 
with
(33) 
where . This gives the diphoton width
(34) 
as expected, with a similar result for .
b.2 Diphoton decay via vectorlike fermions
Let us introduce vectorlike leptons , , , , with
(35) 
For simplicity we take all masses and couplings to be real. We will use the Higgs LET to estimate the induced diphoton decay width of . With real couplings, the LET yields an effective vertex, but no analogous vertex. Instead, the vertex can be obtained from the ABJ anomaly as we utilise below. For simplicity we assume , setting , where and are the positive charged Dirac fermion mass eigenvalues. Allowing for identical copies of (35) with vectorlike fermion masses larger than , we then have
(36) 
where
(37) 
with . This gives the diphoton partial width:
(38) 
For the pseudoscalar we have Kniehl and Spira (1995)
(39) 
with , giving
(40) 
Once again tuning to within the resolution of the ATLAS and CMS analyses, the effective decay width [assuming doubled total production cross section as in Eq. (22)] is
(41)  
collectively denoting . In the last line, to obtain a conservative estimate, we set , letting at least one of the charged vectorlike fermions be lighter than 350 GeV.
With three identical copies, , Yukawa couplings , and tuned parameters so that despite large mixing, we can just barely obtain the required diphoton decay width.
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