Abstract
The hints for a new resonance at 750 GeV from ATLAS and CMS have triggered a significant amount of attention. Since the simplest extensions of the standard model cannot accommodate the observation, many alternatives have been considered to explain the excess. Here we focus on several proposed renormalisable weakly-coupled models and revisit results given in the literature. We point out that physically important subtleties are often missed or neglected. To facilitate the study of the excess we have created a collection of 40 model files, selected from recent literature, for the Mathematica package SARAH. With SARAH one can generate files to perform numerical studies using the tailor-made spectrum generators FlexibleSUSY and SPheno. These have been extended to automatically include crucial higher order corrections to the diphoton and digluon decay rates for both CP-even and CP-odd scalars. Additionally, we have extended the UFO and CalcHep interfaces of SARAH, to pass the precise information about the effective vertices from the spectrum generator to a Monte-Carlo tool. Finally, as an example to demonstrate the power of the entire setup, we present a new supersymmetric model that accommodates the diphoton excess, explicitly demonstrating how a large width can be obtained. We explicitly show several steps in detail to elucidate the use of these public tools in the precision study of this model.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
The first data from the 13 TeV run of the large hadron collider (LHC) contained a surprise: ATLAS and CMS reported a resonance at about 750 GeV in the diphoton channel with local significances of \(3.9 \sigma \) and \(2.6 \sigma \), respectively [1, 2]. When including the look-elsewhere-effect, the deviations from standard model (SM) expectations drop to \(2.3 \sigma \) and \(1.2 \sigma \).Footnote 1
This possible signal caused a lot of excitement, as it is the largest deviation from the SM which has been seen by both experiments. This in turn led to an avalanche of papers, released very quickly, which analysed the excess from various perspectives [5–359].
It is hard to explain the excess within the most commonly considered frameworks for physics beyond the standard model (BSM), like two-Higgs-doublet models (THDM) or the minimal supersymmetric standard model (MSSM) [360], to mention a couple of well-known examples. Thus, many alternative ideas for BSM models have been considered, some of which lack a deep theoretical motivation and are rather aimed at just providing a decent fit to the diphoton bump. Most of the papers in the avalanche were written quickly, some in a few hours, many in a few days, so the analyses of the new models are likely to have shortcomings. Some effects could be missed in the first attempt and some statements might not hold at a second glance. Indeed we have found a wide range of mistakes or unjustified assumptions, which represented the main motivation that prompted this work.
Now that the dust has settled following the stampede caused by the presentation of the ATLAS and CMS data, the time has come for a more detailed and careful study of the proposed ideas. In the past few years several tools have been developed which can be very helpful in this respect. In the context of renormalisable models, the Mathematica package SARAH [361–366] offers all features for the precise study of a new model: it calculates all tree-level properties of the model (mass, tadpoles, vertices), the one-loop corrections to all masses as well as the two-loop renormalisation group equations, and it can be interfaced with the spectrum generators SPheno [367, 368] and FlexibleSUSY [369]. These codes, in turn, can be used for a numerical analysis of any model, which can compete with the precision of state-of-the-art spectrum generators dedicated just to the MSSM and NMSSM [370]. The RGEs are solved numerically at the two-loop level and the mass spectrum is calculated at one loop. Both codes have the option to include the known two-loop corrections [371–376] to the Higgs masses in the MSSM and NMSSM, which may, depending on the model, provide a good approximation of the dominant corrections. SPheno can also calculate the full two-loop corrections to the Higgs masses in the gaugeless limit at zero external momentum [377, 378]. FlexibleSUSY has an extension to calculate the two-loop Higgs mass corrections using the complementary effective field theory approach, which is to be released very soon. SPheno makes predictions for important flavour observables, which have been not yet implemented in FlexibleSUSY. Of particular importance for the current study is that SPheno and FlexibleSUSY calculate the effective vertices for the diphoton and digluon couplings of the scalars, which can then be used by Monte-Carlo (MC) tools like CalcHep [379, 380] or MadGraph [381, 382]. Other numerical tools like MicrOmegas [383], HiggsBounds [384, 385], HiggsSignals [386] or Vevacious [387] can easily be included in the framework.
These powerful packages provide a way to get a thorough understanding of the new models. The main goal of this work is to support the model builders and encourage them to use these tools. We provide several details about the features of the packages in the spirit of making this paper self-contained and bringing the reader unfamiliar with the tools to the level of knowledge necessary to use them. More information can be found in the manuals of each package. We have created a database of diphoton models in SARAH, by implementing 40 among those proposed in recent literature, which is now available to all interested researchers. For each model we have written model files to interface SARAH with SPheno and FlexibleSUSY.
Although in each case we have tried to check very carefully that we implement the model which has been proposed in the literature, it is of course possible that some details have been missed. The original authors of these models are encouraged to check the implementation themselves to satisfy that what we have implemented really does correspond to the model they proposed. In the description of some of the models we state cases where the model has problems or where we find difficulties for the proposed solution. This helps inform potential users about what they may see when running these models through the tools we are discussing here. However especially in these cases we encourage the original authors to check what we have written and let us know if they disagree with any claim we make.
The aim of this paper is to give a self-consistent picture of how – and why – the diphoton excess can be studied with the above mentioned public tools and the provided model files. For this purpose, we do not only summarise the implementation and validation of the models, but we also give a short introduction to the tools and explain their basic usage. In addition, we present the example of an U(1) extended model and how this can be studied in all detail. This should enable the interested reader to directly make use of these powerful packages without the need to consult other references or manuals. However, before we start we also summarise common shortcomings of too simplified analyses and emphasise how they are easily avoided by using the tools. This provides the main motivation of this paper and we hope that other model builders will also see the necessity of using these packages. Of course, we do not intend to present a thorough study of all the models which we have implemented. However, we comment on some observations concerning the motivation or validity of a model regarding the diphoton excess which came to our mind during the implementation.
This paper is long but modular, and each section is to a large extent self contained, so the reader can easily jump to the section of greater interest. We have structured it as follows:
-
In Sect. 2, we give a list of common mistakes we have found in the literature.
-
In Sect. 3, we discuss at some length the implementation of the diphoton and digluon effective vertices.
-
In Sect. 4, we give an overview of the models which we have implemented in SARAH.
-
In Sect. 5, we provide an explicit example of how to quickly work out the details of a model, analytically with SARAH and numerically with the other tools. For this purpose we extended a natural supersymmetric (SUSY) model to accommodate the 750 GeV resonance.
-
We conclude in Sect. 6.
2 Motivation
Precision studies in high energy physics have reached a high level of automation. There are publicly available tools to perform Monte-Carlo studies at LO or NLO [388–393], many spectrum generators [367, 368, 394–405] for the calculation of pole masses including important higher order corrections, codes dedicated only to Higgs [406–409] or sparticle decays [410–412], and codes to check flavour [413–417] or other precision observables [418]. This machinery has been used in the past mainly for detailed studies of some promising BSM candidates, like the MSSM, NMSSM or variants of THDMs. There are two main reasons why these tools are usually the preferred method to study these models: (i) it has been shown that there can be large differences between the exact numerical results and the analytic approximations; (ii) writing private routines for specific calculations is not only time consuming but also error prone. On the other hand, the number of tools available to study the new ideas proposed to explain the diphoton excess is still limited. Of these tools, many are not yet widely used largely due to the community’s reluctance in adopting new codes. However, we think it is beneficial to adopt this new generation of generic tools like SARAH.
We noticed that several studies done in the context of the 750 GeV excess have overlooked important subtleties in some models, neglected important higher order corrections, or made many simplifying assumptions which are difficult to justify. Using generic software tools in this context can help address these issues: many simplifications will no longer be necessary and important higher order corrections can be taken into account in a consistent manner. In order to illustrate this we comment, in the following subsections, on several issues we became aware of when revisiting some of the results in the literature.
2.1 Calculation of the diphoton and digluon widths
2.1.1 The diphoton and digluon rates beyond leading order
A precise calculation of the diphoton rate is of crucial importance. In the validation process of this work, we identified several results in the literature that deviate, often by an order of magnitude or more, in comparison to our results [106, 304, 336]. Additionally we observed that in many cases there are important subtleties which we think are highly relevant.
First of all, the choice of the renormalisation scale of the running couplings appearing in the calculation. The majority of recent studies use the electromagnetic coupling at the scale of the decaying particle. However, one should rather use \(\alpha _{em} (0)\), i.e. the Thompson limit (see for instance Refs. [419, 420]), in order to keep the NLO corrections under control. Taking this into account already amounts to an \(\mathcal O (10\,\%)\) change of the diphoton rate compared to many studies in the literature. In addition, as we will discuss in Sect. 2.2.3, an important prediction of a model is the ratio \(\text {Br}(S\rightarrow gg)/\text {Br}(S\rightarrow \gamma \gamma )\), where S is the 750 GeV scalar resonance. It is well known that the digluon channel receives large QCD corrections. If one neglects these corrections the ratio will be severely underestimated.
To demonstrate these effects we show in Fig. 1 the total decay widthFootnote 2 of the singlet S as a function of the mass \(M_{F_1}\) and coupling \(Y_{F_1}\) for a simple toy model containing only the vector-like fermions \(\Psi _{F_1}\) as presented in Ref. [276]. Table 1 contains benchmark points for the partial widths of the digluon and diphoton channels as well as the ratio of these two channels for both CP-even and CP-odd scalar resonances. This table contains the LO calculations performed using SPheno as a comparison to results previously shown in the literature [276]. We also show the partial widths including NLO corrections for the diphoton channelFootnote 3 and N\({}^3\)LO QCD corrections to the gluon fusion production as implemented in Sect. 3.5. The discrepancies between the LO calculations arise purely through the choice of the renormalisation scale for the gauge couplings. However, the NLO results clearly emphasise that loop corrections at the considered mass scales are the dominant source of errors. To our knowledge, these uncertainties have thus far not received a sufficiently careful treatment in the literature; we give further discussion of this (and the remaining uncertainty in the SARAH calculation) in Sect. 3.7.
2.1.2 Constraints on a large diphoton width
In order to explain the measured signal, one needs a large diphoton rate of \(\Gamma (S\rightarrow \gamma \gamma )/M_S \simeq 10^{-6}\) assuming a narrow width for S, while for a large width one requires \(\Gamma (S\rightarrow \gamma \gamma )/M_S \simeq 10^{-4}\) [192]. In weakly-coupled models there are three different possibilities to obtain such a large width:
-
1.
Assuming a large Yukawa-like coupling between the resonance and charged fermions
-
2.
Assuming a large cubic coupling between the resonance and charged scalars
-
3.
Using a large multiplicity and/or a large electric charge for the scalars and/or fermions in the loop
However, all three possibilities are also constrained by very fundamental considerations, which we briefly summarise in the following.
2.1.2.1 Large couplings to fermions A common idea to explain the diphoton excess is the presence of vector-like states which enhance the loop-induced coupling of a neutral scalar to two photons or two gluons. This led some authors to consider Yukawa-like couplings of the scalar to the vector-like fermions larger than \(\sqrt{4 \pi }\), which is clearly beyond the perturbative regime.Footnote 4 Nevertheless, a one-loop calculation is used in these analyses to obtain predictions for the partial widths [353], despite being in a non-perturbative region of parameter space.
Moreover, even if the couplings are chosen to be within the perturbative regime at the scale \(Q=750\,{\text {GeV}}\), they can quickly grow at higher energies. This issue of a Landau pole has been already discussed to some extent in the literature [40, 68, 192, 212, 343, 346], and one should ensure that the model does not break down at unrealistic small scales.
2.1.2.2 Large couplings to scalars One possibility to circumvent large Yukawa couplings is to introduce charged scalars, which give large loop contributions to the diphoton/digluon decay. A large cubic coupling between the charged scalar and the 750 GeV one does not lead to a Landau pole for the dimensionless couplings because of dimensional reasons. However, it is known that large cubic couplings can destabilise the scalar potential: if they are too large, the electroweak vacuum could tunnel into a deeper vacuum where \(U(1)_{\text {em}}\) gauge invariance is spontaneously broken, depending on the considered scenario. The simplest example with such a scenario is the SM, extended by a real scalar S and a complex scalar X with hypercharge Y. The scalar potential of this example is
In Fig. 2 the dependence of the diphoton partial width as a function of \(\kappa \) and \(M_X\) is shown, and the stability of the electroweak potential as well as the life-time of its ground state is checked with Vevacious and CosmoTransitions. For more details about vacuum stability in the presence of large scalar cubic terms, we refer to Ref. [343]. The overall conclusion of [343] is that the maximal possible diphoton width, even when allowing for a meta-stable but sufficiently long-lived electroweak vacuum, is not much larger than in the case of vector-like fermions when requiring that the model is perturbative up to the Planck scale. It is therefore essential to perform these checks when studying a model that predicts large cubic scalar couplings. An example of the importance of these checks is demonstrated in Ref. [176]. Here it is shown that vacuum stability demands rule out an explanation in the constrained MSSM, proposed in Ref. [120], where the diphoton signal is produced through stop bound states. Moreover, vacuum stability also places stringent constraints on the pMSSM explanation of the excess [171]. Thus far no valid parameter point has been found which is in agreement with both the diphoton rate and vacuum stability constraints. Similar issues were observed in models with trilinear R-parity violation [12, 165] which are disfavoured by these constraints and might work only in very fine-tuned parameter regions.
2.1.2.3 Large multiplicities To circumvent large Yukawa or cubic couplings, other models require a large number of generations of new BSM fields and/or large electric charges. As a consequence the running of the \(U(1)_Y\) gauge coupling, \(g_1\), gets strongly enhanced well below the Planck scale. Moreover, even before reaching the Landau pole, the model develops large (eventually non-perturbative) gauge couplings. This implies an enhancement of Drell–Yan processes at the LHC, with current data already setting stringent constraints and potentially excluding some of the models proposed to explain the diphoton excess [216, 421]. For general studies on running effects in the context of the diphoton excess see [40, 68, 192, 212, 346]. We briefly discuss dramatic examples of this class of models proposed in Refs. [264] and [325], which feature approximately \({\sim }100\) and 6000–9000 generations of doubly-charged scalar fields respectively. In the model of Ref. [325] the SM particle content is enlarged by a vector-like doubly-charged fermion E, a Majorana fermion \(N_R\), a singlet scalar S, a singly-charged scalar \(h^+\) and \(N_k\) generations of the doubly-charged scalar field \(k^{++}\). At the one-loop level the running of \(g_1\) is governed by the renormalisation group equation (RGE)
where \(t = \log \mu \), \(\mu \) being the renormalisation scale, and
is the one-loop \(\beta \) function. It is clear from Eq. 2.3 that a large value of \(N_k\) necessarily leads to a very steep increase of \(g_1\) with the renormalisation scale, soon reaching a Landau pole. This is shown in Fig. 3, obtained by setting the masses of all the charged BSM states to \(\mu _\mathrm{NP} = 2.5\) TeV, which is already the largest mass considered in this analysis. The running up to \(\mu _\mathrm{NP}\) is governed by the SM RGEs, and the result for \(g_1\) is displayed with a black dashed line. For scales above \(\mu _\mathrm{NP} = 2.5\) TeV, the contributions from BSM fields become effective. Figure 3 shows that a Landau pole can be reached at relatively low energies once we allow for such large values of \(N_k\). In fact, for \(N_k = 9000\), we find that a Landau pole appears already at \(\mu \simeq 2.6\) TeV. In this specific example the appearance of a Landau pole below \(10^{16}\) GeV is unavoidable as soon as \(N_k > 10\), as shown in Table 2.
2.1.3 How do the tools help?
The tools which we describe in more detail in the following sections can help to address all the above issues:
-
1.
FlexibleSUSY and SPheno can calculate the diphoton and digluon rate including important higher order corrections.
-
2.
Using the effective vertices calculated by FlexibleSUSY/SPheno and the interface to CalcHep or MadGraph, the gluon-fusion production cross-section of the 750 GeV mediator can be calculated numerically and one does not have to rely on analytical (and sometimes erroneousFootnote 5) approximations.
-
3.
SARAH calculates the RGEs for a model which can be used to check for the presence of Landau pole.
-
4.
Vevacious can be used to check the stability of the scalar potential.
2.2 Properties of the 750 GeV scalar
2.2.1 Mixing with the SM Higgs
It is often assumed that S, although it is a CP-even scalar, does not mix with the SM-like Higgs h. However, if this is done in a very ad hoc way and not motivated by any symmetry, this assumption will not hold when radiative corrections are taken into account. To see this, one can consider, for example, the scalar potential
where H is the SM Higgs \(SU(2)_L\) doublet, which contains the SM Higgs h. This potential in principle has all ingredients to get a large diphoton decay of S via a loop involving the charged scalar X. Note, however, that the potentially dangerous term \(\kappa _H \, S |H|^2\) has been omitted. One can see immediately that this term would be generated radiatively by the diagram below.
Note that it is also not possible to circumvent this decay by forbidding the \(\lambda _{HX}\) term: since H and X are charged under \(SU(2)_L \times U(1)_Y\), also the \(\lambda _{HX} |H|^2 |X|^2\) term would be generated radiatively via diagrams like
A mixing between the singlet and SM-like Higgs state has important consequences, since the mass eigenstate state would have additional tree level couplings to W and Z from the SM Higgs component. For a singlet dominated mass eigenstate, s, this would open up the decay channels \(s\rightarrow h h\) and \(s\rightarrow ZZ\), \(s\rightarrow W^+ W^-\), which are tightly constrained.
Another possibility is that all terms allowed by symmetries are taken into account, but very special relations among them are imposed like in Ref. [313]. When these relations hold, the above-mentioned tree-level decays in SM particles would cancel. However, as long as there is no symmetry behind these relations, they will not be invariant under RGE running. Therefore, immediately the question arises how large the tuning among the parameters must be to have a point that fulfils all constraints. To illustrate this issue, we make small variations in the couplings \(\lambda _{H3}\) and \(\lambda _{36}\),Footnote 6 which cause non-vanishing tree-level couplings between S and the massive vector bosons, and check for which size of the deviations the condition \(\text {Br}(S\rightarrow W^+ W^-)/\text {Br}(S\rightarrow \gamma \gamma ) < 20\) holds. The result is shown in Fig. 4. Here, the diphoton rate was maximised by setting the masses of the vector-like fields to 375 GeV and using a Yukawa coupling of O(1). In principle, one could try to check what this means for the scale dependence of these ratios by calculating the RGEs. However, this cannot really be done for this setup since one obtains the following condition from the relations which have been imposed: \(\lambda _{H3} = f_Y^2 \frac{M_S^2}{M_F^2}\), i.e. \(\lambda _{H3} = 4 f_Y^2\) is needed to maximise the diphoton branching ratio. Thus, \(f_Y\) of \(\mathcal {O}(1)\) immediately leads to a huge quartic coupling.
Thus, in general, it is very difficult to justify the assumption that the 750 GeV particles do not mix with the SM-like Higgs if there is no fundamental symmetry to forbid this mixing. However, this can already be forbidden using the CP symmetry: the mentioned problems can be circumvented in models where the diphoton excess stems from a CP-odd particle. In the case of a CP-even particle, it is crucial to include the mixing effects and to check at least how large the tuning in parameters must be.
2.2.2 To VEV or not to VEV?
The possibility that the new scalar receives a vacuum expectation value (VEV) is also often neglected. However, as we have just discussed, it often occurs that a H–S mixing will be induced, at least radiatively, in many models. Such radiative effects would immediately lead to a non-zero VEV for the new scalar. Even in cases where there is a symmetry which prevents a mixing with the SM Higgs, the \({750}\,{\text {GeV}}\) particle will still receive a VEV if it is a CP-even scalar. This arises due to the introduced couplings to charged particles which are necessary to allow diphoton and digluon decays. More specifically, these introduced couplings will generate one-loop tadpole diagrams for S as shown in Fig. 5. Thus, the tadpole equation reads at the one-loop level
where \(T^{(T)}\) is the tree-level tadpole, given by
Here, we have parametrised the tree-level expression so that the general form has the solution \(v_S=0\). One finds in general that the one-loop corrections are
with \(A(x^2) = \frac{1}{16 \pi ^2} x^2 [1 + \log (Q^2/x^2)]\). Taking \(M_\Psi \), \(\kappa \), \(M_X\) of the order 1 TeV, results in a VEV which is naturally of order \(1\,\text {TeV}^3/(16 \pi ^2 c_1)\). As a result, the simplifying assumption that \(v_S\) vanishes is in general hard to justify – apart from the rare case in which S is a complete singlet under all local and global symmetries; under this circumstance the VEV can always be rotated away. Therefore, it is important to check how the conclusions made about the model depend on this assumption. Here, the tools discussed in the following sections can really help, as including the non-vanishing \(v_S\) is no more difficult than assuming the VEV vanishes.
2.2.3 Additional decay channels
Many analyses concentrate only on the decay \(S\rightarrow \gamma \gamma \) and completely neglect other potential decay channels. However, there are stringent constraints on the branching ratios of S into other SM final states, which are summarised in Table 3.
Thus, any model which tries to explain the excess via additional coloured states in the loop must necessarily worry about limits from dijet searches [423]. Therefore, an accurate calculation of the digluon decay rate is a necessity. As an example that illustrates why both additional channels and the diphoton/digluon width calculation are important we consider the model presented in Refs. [89, 166] and considered in more detail here in Sect. 4.2.1.
This model extends the SM with a singlet and a scalar SU(2)-doublet colour octet. As an approximation the ratio of the singlet decays to gluons and to photons is
In [89] this is quoted as \({\simeq } 750\); before any NLO corrections are applied, we find 700. However, once we include all of the N\(^3\)LO corrections this is enhanced to 1150, near the bound for constraints on dijet production at 8 TeV and significantly squeezing the parameter space of the model.
Additionally in many works we observed that potential decay channels of the resonance were missed. For instance in Ref. [184], the authors, who considered the Georgi-Machacek model [424], missed the decay of the scalar into \(W^\pm H^\mp \), which can be the dominant mode when kinematically allowed.
2.2.4 How do the tools help?
Many of the assumptions which we criticised were made to keep the study simple. However, when using the public tools presented in the next two section, there is no need for these simplifying assumptions:
-
1.
SARAH automatically calculates all expressions for the masses and vertices in any renormalisable model, no matter how complicated they are.
-
2.
FlexibleSUSY and SPheno give numerical predictions for the mass spectrum and the mixing among all states including higher order corrections.
Thus, for users the study becomes no more difficult when they drop all simplifying assumptions, and instead consider the model in full generality. Moreover, there is no chance to miss important effects in the decays of the new scalar:
-
1.
As outlined above, FlexibleSUSY and SPheno calculate the diphoton and digluon rate very accurately
-
2.
SPheno calculates all other two body decaysFootnote 7 of the scalar. This makes it impossible to miss any channel.
2.3 Considering a full model
2.3.1 Additional constraints in a full model
There are several studies which extend an already existing model by vector-like states and then assume that this part of the model is decoupled from the rest. When this assumption is made it is clear that the results from toy models, with the minimal particle content will be reproduced. However, it is often not clear if this decoupling can be done without invoking specific structures in the choice of parameters, and if these assumptions hold at the loop level.
On the other hand, if model-specific features are used to explain the diphoton excess, it is likely that there will be important constraints on the model coming from other sectors. For instance, there might be bounds from flavour observables, dark matter, Higgs searches, neutrino mixing, electroweak precision observables, searches for BSM particles at colliders, and so on. All of that has to be checked to be sure that any benchmark point presented is indeed a valid explanation for all observations. Such a wide range of constraints is much easier to address by making exhaustive use of tools which provide a high level of automation.
2.3.2 Theoretical uncertainties of other predictions
Even if attempts are made to include the effects of the new states on other sectors of the model, it is important to be aware that there are large uncertainties involved in certain calculations. If the level of uncertainty is underestimated, this can have an impact on what is inferred from the calculation. The large uncertainty in a LO calculation of the diphoton and digluon rate has already been addressed in Sect. 2.1.1. However, there are also other important loop corrections especially in SUSY models: the accurate calculation of the Higgs mass is a long lasting endeavour where for the simplest SUSY models even the dominant three-loop corrections are partially tackled [425]. The current ball-park of the remaining uncertainty is estimated to be 3 GeV.
However, most likely the MSSM cannot explain the excess, hence it would have to be extended. A common choice is to add additional pairs of vector-like superfields together with a gauge singlet, see Sect. 5. These new fields can also be used to increase the SM-like Higgs mass. However, this will in general also increase the theoretical uncertainty in the Higgs mass prediction, because these new corrections are not calculated with the same precision as the MSSM corrections. For instance, Ref. [180] has taken into account the effect of the new states on the SM-like Higgs. There, they use a one-loop effective potential approach considering the new Yukawa couplings to be \(\mathcal {O}(1)\) or below, while also including the dominant two-loop corrections from the stop quark. They assumed that including these corrections is sufficient in order to achieve an uncertainty of 2 GeV in the Higgs mass prediction. One can compare their results encoded in Fig. 7 of Ref. [180] with a calculation including, in addition to the corrections taken into account in the paper, momentum dependence and electroweak corrections at the one-loop level, as well as the additional two-loop corrections arising from all the newly introduced states. These corrections can be important, as was shown for instance in Ref. [426]. The result of the comparison is shown in Fig. 6. We find a similar behaviour, but still there are several GeV difference between both calculations. For \(\kappa _{10}=0.8\) and \(X_t=4\), the point would be within the interesting range for \(m_h=[123,127]\) GeV, while the more sophisticated calculation predicts a mass below 120 GeV. Thus, the assumed uncertainty of 2 GeV in Ref. [180], which would even be optimistic in the MSSM, is completely unrealistic without including all the aforementioned higher order corrections.
2.3.3 How do the tools help?
The tools help to ensure that one really considers all aspects of a full model:
-
1.
All masses of the model are calculated with high accuracy: FlexibleSUSY and SPheno include the full one-loop contribution to all pole masses in a model, while SPheno covers even the dominant two-loop corrections introduced by adding new states.
-
2.
SPheno makes predictions for all important flavour observables in the model.
-
3.
A link to MicrOmegas provides the possibility to calculate the dark matter relic density.
-
4.
The interface to HiggsBounds and HiggsSignals offers the possibility to check all constraints from Higgs searches and to check if the results for the SM-like Higgs can be reproduced.
3 The SARAH framework and its diphoton extension
3.1 SARAH
One of the reasons that makes high energy particle physics an exciting field is the vast amount of experimental data available. When proposing a model one first has to check its self consistency, checking for instance the particle mass spectrum and vacuum stability requirements. Then it has to be tested against data related to collider searches, flavour observables, dark matter observations and Higgs measurements. A lot of effort has been devoted to developing an arsenal of specific tools to explore these quantities with high precision for specific classes of models, such as the MSSM, the THDM and the NMSSM to some extent. However, it is often very difficult – if not impossible – to explain the excess in the simplest versions of these models.Footnote 8 For the time being there is no specific model which is clearly preferred over others as an explanation of the excess, as reflected by the large variety of models that different authors have proposed, and it would be impractical to repeat the process of developing a code for each one of them. In the absence of a dedicated tool, the alternative is often to resort to approximations or just to leading order expressions, as described in the previous section, in which case the analysis (in particular for more complicated models) is of limited value.
Luckily, a dedicated powerful tool already exists. It is the Mathematica package SARAH [361–366], which can perform the most advanced quantum field theory computations and apply them generically to any given model. SARAH has been optimised for an easy, fast, and exhaustive study of renormalisable BSM models. Within a given model SARAH analytically calculates the following:
-
All tree-level masses, vertices and tadpole equations
-
The two-loop RGEs for a general quantum field theory and a softly broken SUSY theory using generic results of [427–436]
-
The one-loop corrections to all one- and two-point functions
In addition SARAH also provides routines to export the derived information in order to use it for numerical calculations with dedicated tools. We give in the following a brief overview about the different possibilities.
3.2 SPheno
SARAH writes Fortran source code for SPheno [367, 368] using the derived information about the mass matrices, tadpole equations, vertices, loop corrections and RGEs for the given model. With this code the user gets a fully functional spectrum generator for the model of their choice. The features of a spectrum generator created in this way are
-
Full two-loop running of all parameters
-
One-loop corrections to all masses
-
Two-loop corrections to Higgs masses
-
Complete one-loop thresholds at \(M_Z\)
-
Calculation of the \(h\gamma \gamma \) and hgg effective couplings at \(\text {N}^3\)LO, see Sect. 3.5
-
Calculation of flavour and precision observables at full one-loop level
-
Calculation of decay widths and branching ratios for two– and three body decays
-
Interface to HiggsBounds and HiggsSignals
-
Estimate of electroweak fine-tuning
-
Prediction for LHC cross sections for all neutral scalars
3.3 FlexibleSUSY
FlexibleSUSY is a Mathematica package which uses the SARAH-generated expressions for the mass matrices, self-energies, tadpole equations, vertices and RGEs to create a C++ spectrum generator for both SUSY and non-SUSY models. The spectrum generators created with FlexibleSUSY have the following features:
-
Full two-loop running of all parameters
-
Three-loop running of all parameters in the SM and MSSM, except for the VEVs
-
Calculation of the pole mass spectrum at the full one-loop level
-
Partial two-loop corrections to the Higgs masses in the SM, Split-MSSM, MSSM, NMSSM, UMSSM and E\(_6\)SSM and partial three-loop corrections to the Higgs mass in the Split-MSSM
-
Complete one-loop and partial two-loop and three-loop threshold corrections to the Standard Model at the scale \(Q = M_Z\) or \(Q = M_t\)
-
Calculation of the \(h\gamma \gamma \) and hgg effective couplings at \(\text {N}^3\)LO, see Sect. 3.5
-
An interface to GM2Calc [418] in the MSSM without flavour violation
FlexibleSUSY aims to generate spectrum generators which are modular such that components can be easily reused or replaced. This means that it is quite easy to re-use the precision calculations in FlexibleSUSY spectrum generators for other purposes or add additional routines.
3.4 Mass spectrum calculation: SUSY vs. non-SUSY
We have outlined that FlexibleSUSY and SPheno can include the radiative corrections to all particles up to the two-loop level in the \({\overline{\mathrm {DR}}}'\) scheme. These corrections are included by default for supersymmetric models. It is known that loop corrections, in particular to the Higgs mass, are crucial. Typically the \({\overline{\mathrm {DR}}}'\) and on-shell calculations are in good agreement. Consequently, the remaining difference between both calculations is often a good estimate for the theoretical uncertainty.
The treatment of non-supersymmetric models in FlexibleSUSY and SPheno is very similar to the treatment of supersymmetric models. The main difference is, that in non-supersymmetric models the parameters are defined in the \(\overline{\mathrm {MS}}\) scheme, while in supersymmetric ones the parameters are defined in the \({\overline{\mathrm {DR}}}'\) scheme. In this paper we perform only tree-level mass calculations (if not stated otherwise), in which the definition of the renormalisation scheme is irrelevant. Thus, in the mass spectrum calculations performed in the following, one is allowed to use input parameters which are defined in the on-shell scheme. This is for instance the standard approach in the large majority of studies of the THDM: there are in general enough free parameters to perform a full on-shell renormalisation keeping all masses and mixing angles fixed. We find that the one-loop corrections in the \({\overline{\mathrm {MS}}}\) scheme can give huge corrections to the tree-level masses in nearly all models presented in the following. Therefore large fine-tuning of the parameters is necessary once the loop corrections are taken into account. A detailed analysis using a full on-shell renormalisation scheme is possible for each model, but is beyond the scope of this work. Of course, for models where it turns out to be unavoidable that shifts in the masses and mixings appear at the loop-level, the user can simply turn on the loop corrections in FlexibleSUSY and SPheno via a flag in the Les Houches input file.
3.5 Calculation of the effective diphoton and digluon vertices in SPheno and FlexibleSUSY
For the calculation of the partial width of a neutral scalar \(\Phi \) decaying into two gluons or two photons we follow closely [419] for the LO and NLO contributions. The partial widths at LO are given by
Here, the sums are over all fermions f, scalars s and vector bosons v which are charged or coloured and which couple to the scalar \(\Phi \). Q is the electromagnetic charges of the fields, \(N_c\) are the colour factors and \(D_2\) is the quadratic Dynkin index of the colour representation which is normalised to \(\frac{1}{2}\) for the fundamental representation. We note that the electromagnetic fine structure constant \(\alpha \) must be taken at the scale \(\mu = 0\), since the final state photons are real [437]. In contrast, \(\alpha _s\) is evaluated at \(\mu = m_\Phi \) which, for the case of interest here, is \(\mu = 750\) GeV. \(r^\Phi _i\) are the so-called reduced couplings, the ratios of the couplings of the scalar \(\Phi \) to the particle i normalised to SM values. These are calculated as
Here, v is the electroweak VEV and C are the couplings between the scalar and the different fields with mass \(M_i\) (\(i=f,s,v\)). Furthermore,
holds and the loop functions are given by
with
For a pure pseudo-scalar state only fermions contribute, i.e. the LO widths read
where
and \(r^A_f\) takes the same form as \(r^\Phi _f\) in Eq. (3.3), simply replacing \(C^{L,R}_{\bar{f} f \Phi }\) by \(C^{L,R}_{\bar{f} f A}\).
These formulae are used by SPheno and FlexibleSUSY to calculate the full LO contributions of any CP-even or odd scalar present in a model including all possible loop contributions at the scale \(\mu = m_\Phi \). However, it is well known, that higher order corrections are important. Therefore, NLO, NNLO and even \(\text {N}^3\)LO corrections from the SM are adapted and used for any model under study. In case of heavy colour fermionic triplets, the included corrections for the diphoton decay are
These expressions are obtained in the limit \(\tau _f \rightarrow 0\) and thus applied only when \(m_\Phi < m_f\). \(r^A_f\) does not receive any corrections in this limit. For the case \(m_\Phi > 100 m_f\), we have included the NLO corrections in the light quark limit given by [419]
for \(X=\Phi ,A\). \(\mu _{\text {NLO}}\) is the renormalisation scale used for these NLO corrections, chosen to be \(\mu _{\text {NLO}}= m_\Phi /2\). In the intermediate range of \(100 m_f> m_\Phi > 2 m_f\), no closed expressions for the NLO correction exist. Our approach in this range was to extract the numerical values of the corrections from HDECAY [406] and to fit them. For the digluon decay rate, the corrections up to \(\mathrm{N}^3\)LO are included and parametrised by
and
For pseudo-scalars we include only corrections up to NNLO as the \(\text {N}^3LO\) are not known for CP-odd scalars.
One has to keep in mind that the NLO up to N\({}^3\)LO corrections are calculated in the SM under the assumption that only a (fermionic) colour triplet and the gluons play any role in the loops. Of course, in BSM models this must not necessarily be the case. For instance, in SUSY models gluinos would also contribute at NLO. The impact of these additional corrections is estimated in Sect. 3.7.2. Another possible effect is the presence of a scalar triplet, such as the SUSY top partners. However, it was found that the higher-order corrections for this case can be well approximated by the SM results, see Ref. [443]. Finally, other colour representations beyond triplets can induce an effective digluon coupling in BSM models. To our knowledge, NLO and higher order corrections for these cases have not yet been discussed in the literature. However, motivated by the observation of Ref. [443] that the K-factor for the higher-order corrections in the MSSM is nearly identical to the SM, because the largest contributions by-far come from final state gluons, we consider the SM corrections to also give the dominant effect at NLO and beyond for this case. However, we also provide a flag in SPheno that allows users to turn-off these corrections, if they think that such corrections are not appropriate for the case at hand. Similarly, in FlexibleSUSY these corrections may be turned off by means of a flag in the generated code.
In order to check the accuracy of our implementation, we compared the results obtained with SARAH–SPheno for the SM Higgs boson decays with the ones given in the CERN yellow pages [444]. In Fig. 7 we show the results for the Higgs branching ratios into two photons and two gluons with and without the inclusion of higher order corrections. One sees that good agreement is generally found when including higher order corrections. Figure 8 shows the relative difference of the partial widths \(\Gamma (h \rightarrow \gamma \gamma )\) and \(\Gamma (h \rightarrow gg)\) as calculated by SPheno and FlexibleSUSY compared to the benchmark values provided by the Higgs cross section working group. While the results obtained from the two codes are not identical, there is good agreement between them for both partial widths. The differences between SPheno and FlexibleSUSY originate mainly from a different treatment of unknown higher-order corrections to the pole mass spectrum. In Fig. 9 we show the ratio \(\text {Br}(h\rightarrow gg)/\text {Br}(h\rightarrow \gamma \gamma )\) and compare it again with the recommended numbers by the Higgs cross section working group [444]. Allowing for a 10 % uncertainty, we find that our calculation including higher order corrections agrees with the expectations, while the LO calculation predicts a ratio that is over a wide range much too small. The important range to look at is actually not the one with \(m_h \sim \) 750 GeV because this corresponds to a large ratio of the scalar mass compared to the top mass. Important for most diphoton models is the range where the scalar mass is smaller than twice the quark mass. In this mass range we find that the NLO corrections are crucial and can change the ratio of the diphoton and digluon rate up to a factor of 2. We also note that if we had used \(\alpha (m_h)\) instead of \(\alpha (0)\) in the LO calculation, the difference would have been even larger, with a diphoton rate overestimated by a factor \((\alpha (m_h)/\alpha (0))^2\simeq (137/124)^2 \simeq 1.22\).
3.6 Monte-Carlo studies
3.6.1 Interplay SARAH-spectrum-generator-MC-tool
The tool chains SARAH–SPheno/FlexibleSUSY–MC-Tools have one very appealing feature: the implementation of a model in the spectrum generator (SPheno or FlexibleSUSY) as well as in a MC tool is based on just one single implementation of the model in SARAH. Thus, the user does not need to worry that the codes might use different conventions to define the model. In addition, SPheno also provides all widths for the particles so that this information can be used by the MC-Tool to save time.
3.6.2 Effective diphoton and digluon vertices for neutral scalars
The effective diphoton and digluon vertices calculated by SPheno or FlexibleSUSY are directly available in the UFO model files and the CalcHep model files: SARAH includes the effective vertices for all neutral scalars to two photons and two gluons, and the numerical values for these vertices are read from the spectrum file generated with SPheno or FlexibleSUSY. For this purpose, a new block EFFHIGGSCOUPLINGS is included in these files, which contains the values for the effective couplings including all corrections outlined in Sect. 3.5.
It is important to mention that these effective couplings correspond to the decay of the scalar; if we use CalcHep or MadGraph to compute the decay \(\Phi \rightarrow gg\) then the value matches (as closely as possible) the \(\text {N}^3\)LO value, which includes real emission processes such as \(\Phi \rightarrow g g g\). Therefore, the corrections at NLO and beyond for \(\Phi \rightarrow gg\) are not the same as \(pp \rightarrow \Phi \) via gluon fusion [437]; the full \(\text {N}^3\)LO production cross-section includes all processes \(g g \rightarrow \Phi + \mathrm {jet}\) and is therefore described by a different k-factor to the decay. This k-factor can for instance be obtained via
where \(c_{\Phi gg}\) is the ratio squared of the effective coupling between \(\Phi \) and two gluons at LO in the considered model and the SM. These values can for instance be read off by the block HiggsBoundsInputHiggsCouplingsBosons in the SPheno spectrum file. \(\sigma _\mathrm{SM}(pp \rightarrow H(M_{\Phi }))\) is the cross section for a SM-like Higgs with mass \(M_\Phi \). This value can be calculated for instance with Higlu [445] or Sushi [446] for the considered center-of-mass energy. SPheno also provides values for \(c_{\Phi gg} \cdot \sigma _\mathrm{SM}(pp \rightarrow H(M_{\Phi }))\) for the most common energies in the blocks HiggsLHCX (X=7,8,13,14) and HiggsFCC.
On the other hand, this approach is not entirely appropriate for more refined collider analyses where the user would like to actually include, for example, a hard jet in the final state (without the full loop corrections to the effective vertex this is not an infra-red safe quantity). In this case, we note that around 750 GeV the effective vertex output by SARAH gives a fairly accurate result – to within 30 % – of the total production cross-section, at least in the Standard Model, when we compute \(\sigma _{SM} ( g g \rightarrow \Phi + \mathrm {jet})\) using MadGraph and the standard cuts on momenta. This is illustrated in Fig. 10.
3.7 Accuracy of the diphoton calculation
Before concluding this section, we should draw the reader’s attention to the question of how accurate the results are from SARAH in combination with SPheno and FlexibleSUSY. While every possible correction has been included, there are still some irreducible sources of uncertainty, as we shall discuss below.
3.7.1 Loop corrections to \(ZZ, WW, Z\gamma \) production
So far in SARAH, loop-level decays are only computed for processes where the tree-level process is absent. This is to avoid the technical issues of infra-red divergences. If the particle that explains the 750 GeV excess is a scalar, then it must mix with the Higgs and acquire tree-level couplings to the Z and W bosons. The respective decays are fully taken into account at tree level. However, due to the existence of such terms, the loop corrections to the decays into Zs and Ws are more complicated and are therefore not yet available in SARAH. Even if hese technical issues do not apply for pseudo-scalar bosons, for which the decays into vector bosons are only possible at the loop level, these decays are also not yet available at the loop level. However, it should be mentioned that there are on-going efforts to close this gap in the near future and to provide the full one-loop corrections to any two-body decay of CP-even and -odd scalars.
That these loop induced decays are missing at the moment in SARAH can trigger two issues the user has to keep in mind. First, there are limits on the decays \(S\rightarrow WW\) and \(S\rightarrow ZZ\) which could be violated if the loop induced couplings between S and two massive vector bosons are too large. Therefore, one has to be careful when studying models with large additional SU(2) representations. The second issue is that the prediction for the BR into two photons suffers from an additional uncertainty because of the missing contribution of the ZZ and WW decays to the total width.
To estimate the uncertainty incurred by their absence, let us assume that our 750 GeV resonance S couples to the \(U(1)_Y\) and \(SU(2)_L\) gauge bosons via the effective operators \(S B_{\mu \nu } B^{\mu \nu }\) and \(S W_{\mu \nu } W^{\mu \nu }\). If we can neglect the tree-level contributions to the decays and assume that the dominant contribution originates from a set of particles in the loops, which have the hypercharge Y and the Dynkin index \(D_2(W)\) and dimension of the SU(2) representation \(d_2\), then the decay widths are approximately given by
where we abbreviated \(t_W\) for \(\tan \theta _W\). Put together, the uncertainty that we find for the decay \(S\rightarrow \gamma \gamma \) reads
The factor in square brackets is therefore largest for fields that only couple to \(SU(2)_L\) gauge bosons, giving a factor of \(\sim 55\), and for SU(2) doublets with hypercharge 1 / 2 it is 13, although the former case yields too many W bosons (the limit from run 1 searches is \(\frac{\Gamma (S \rightarrow WW)}{\Gamma (S \rightarrow \gamma \gamma )} \lesssim 20 \)). Thus, provided that \(\text {Br} ( S \rightarrow \gamma \gamma ) \lesssim 10^{-3}\), the relative uncertainty is guaranteed to be less than 10 %. In such cases, the proportional error in the total width transfers directly into the proportional error in the total cross-section:
On the other hand, for models where the dominant decay channel of the singlet is into gluons, it is not possible to have \(\text {Br} ( S \rightarrow \gamma \gamma ) \lesssim 10^{-3}\) without violating constraints from dijet production, and the reader should be careful about the possible errors incurred. Fortunately, provided that the loop particles have a hypercharge the error is much smaller, for example in the case that \(D_2 = 0\) the coefficient above is less than one, thus giving an error of \(\sim 10^{-3}\) for \(\text {Br} ( S \rightarrow \gamma \gamma ) = 10^{-3}\) .
3.7.2 BSM NLO corrections
As discussed above, SARAH includes the leading-order computation of the diphoton and digluon decay amplitudes including the effects of all Standard Model and Beyond-the-Standard-Model particles in the loops. Furthermore, it also includes the leading-log corrections to the digluon rate at NLO, NNLO and \(\text {N}^3\)LO order in \(\alpha _s\) in the Standard Model, and some NLO corrections due to diagrams with an extra gluon to both the digluon and diphoton rates. However, the NLO corrections are absent for all other particles, which in the case of large Yukawa couplings or hierarchies could be sizeable. Two examples of such a diagrams are given in Fig. 11; in the context of supersymmetric theories, particularly important are diagrams involving the gluino, which (if it is a Majorana particle) would not couple to a singlet at leading order – naively their contribution is
although as we shall discuss below this can be (potentially significantly) an underestimate.
3.7.3 Presence of light fermions
The higher order corrections to the Higgs production and decay via the effective digluon coupling is calculated in the SM using an effective-field-theory (EFT) approach. This is possible because the top mass is sufficiently heavy compared to the Higgs boson. Also the presence of vector-like quarks with masses below 750 GeV is already tightly constrained by direct searches at the LHC [447]. Therefore, for realistic scenarios the EFT approximation is also typically valid. Even so, one might wonder how large the additional uncertainty is due to the presence of light quarks. For a detailed discussion of this, we refer to Ref. [448]. The overall result is that the additional uncertainty is larger than the one stemming from the choice of the QCD scale. Nevertheless, it was found that the EFT computation still gives a good estimate for the overall K-factor.
3.7.4 Tree vs pole masses in loops
For consistency of the perturbative series and technical expediency, the masses inside loops (to calculate pole masses and loop decay amplitudes) are \(\overline{\text {MS}}\) or \(\overline{\text {DR}}^\prime \) parameters, not the pole masses of observed particles. The difference between calculations performed in this scheme and the on-shell scheme are at two-loop order, and so is generally small. However, in particular when there are large hierarchies or Yukawa couplings in a model, there can be a large difference between the Lagrangian parameters and the pole masses, and therefore a large discrepancy between the loop amplitudes calculated from these. In principle, this should be accounted for by including higher-order corrections such as the right-hand diagram in Fig. 11 – but applying such a correction to each propagator in the loop would actually correspond to a four-loop diagram. The effect of using the pole mass instead is to essentially resum part of these diagrams, which as is well known is relevant in the case of large hierarchies of masses – and so should give a more accurate result in that case.
If we define
then for one dominant particle p in the loop, we can estimate the uncertainty as
where the factor of 1 or 1 / 2 assumes that the couplings \(C_{\overline{p}p\Phi }\) do not depend upon the mass \(m_p\) (but the prefactor in \(r_p^\Phi \) therefore does). For most values of \(m_p\) the loop functions are slowly changing (only peaking around \(\tau _p =1\)) so we will have a proportional uncertainty in the result of \( \frac{2\delta m_{s,v}^2}{m_{s,v}^2}\) or \( \frac{2\delta m_f}{m_f}\). As an example, in supersymmetric theories the soft masses of coloured scalars \(\tilde{S}^\prime \) acquire a significant decrease from gluino loops:
If the scalar is a colour triplet with a pole mass of 800 GeV, then for 2 TeV gluinos and the \(\overline{\text {DR}}^\prime \) mass is \({\sim } 1100\) GeV; but \(\frac{\delta m_{\tilde{S}^\prime }^2}{m_{\tilde{S}^\prime }^2} \sim 1!\) This corresponds to a shift of a factor of two in the amplitude, and, if the scalar dominates the total amplitude, a factor of four in \( \Gamma ( S \rightarrow gg/\gamma \gamma ) \); in fact in this case SARAH would be potentially underestimating the diphoton rate. This is a relatively mild example regarding this excess: given that the vast majority of models proposed to explain the diphoton signature contain large Yukawa couplings and many new particles, there is a significant potential for large values of \(\frac{\delta m^2}{m^2} \), about which the user should be careful. It is worth noting that this is an effect that would not be significant in the (N)MSSM, where the Higgs couplings to photons/gluons are dominated by the top quark (and, for photons, the W bosons) whose masses are protected by chiral symmetry from large shifts: this issue is a novelty for the 750 GeV excess.
For non-supersymmetric models, due to the fact that (almost) every parameter point is essentially fine-tuned, we have not calculated loop corrections to the masses by default, and this issue does not arise in the same way. The user is then free to regard the result as involving the pole masses of particles instead, if they so desire – the issue then becomes one of tuning the potentially large corrections to the other input parameters.
4 Models
A large variety of models have been proposed to explain the diphoton excess at \({750}\,{\text {GeV}}\). We have selected and implemented several possible models in SARAH. Our selection is not exhaustive, but we have tried to implement a sufficient cross-section which are representative of many of the ideas put forward in the context of renormalisable models. These are the ones that SARAH can handle. Their description is organised in the subsections that follow. Before we turn to this discussion we first want to mention other proposals which we do not deal with in this paper.
Many authors [16, 35, 50, 59, 67, 72, 99, 144, 150, 190, 192, 207, 249, 251, 265, 298, 301, 319, 331, 341] have studied the excess with effective (non-renormalisable) models, which is sensible given that there are thus far no other striking hints of new physics at the LHC. As more data becomes available and the evidence for new physics becomes more substantial, one might want to UV complete these models, at which point the tools we are advertising become relevant and necessary. Other authors [42, 52, 60, 73, 114, 122, 131, 233, 235, 273, 293, 297, 308, 315, 320, 323, 346] considered strongly coupled models, in which the resonance is a composite state. This possibility would be favoured by a large width of the resonance, as first indications seem to suggest. Another possibility is to interpret the signal in the context of extra-dimensional models [5, 9, 29, 30, 48, 84, 125, 141, 203, 225, 312], with the resonance being a scalar, a graviton, a dilaton, or a radion, depending on the scenario. However, some of these interpretations are in tension with the non-observation of this resonance in other channels. In supersymmetry, the scalar partner of the goldstino could provide an explanation to the diphoton signal [97, 147, 167, 335]. Other ideas, slightly more exotic, include: a model with a space-time varying electromagnetic coupling constant [135], Gluinonia [337], Squarkonium/Diquarkonium [299], flavons [244], axions in various incarnations [8, 24, 63, 246, 274, 336], a natural Coleman–Weinberg theory [22, 307], radiative neutrino mass models [264, 325, 327], and string-inspired models [19, 132, 188, 240, 254].
We turn now to the weakly coupled models, and list the ones which we have implemented in SARAH.
All model files are available for download at http://sarah.hepforge.org/Diphoton_Models.tar.gz and an overview of all implemented models is given in Tables 4 and 5, where we have divided the models into five different categories. The first three models can be regarded as toy models which simply extend the Standard Model by some basic ingredients for explaining the diphoton excess, namely a singlet scalar and a number of different vector-like fermions. The second category contains models which are also based on the SM gauge group but feature a more complicated structure than the toy models mentioned before. Table 5 contains a variety of non-supersymmetric models with an enlarged gauge group such as gauged U(1) extensions or left-right-symmetric models, as well as some supersymmetric models, both with and without an enlarged gauge sector.
Some of the models which were implemented can be seen as a straightforward extension or a modification of known models like the Standard Model, the NMSSM, a two-Higgs-doublet model, or a \(U(1)'\) model. They are derived from models already available in the SARAH model repository and will not be discussed here in detail. Note however, that some model classes, like left-right-symmetric models, are now for the first time publicly available for SARAH. For all the necessary information regarding the particle content, symmetries and the Lagrangian, we refer the interested reader to the documentation provided alongside the tarball containing the model files. As a selection, we discuss below in some detail the implementation of four rather involved models (one with scalar octets, two 3-3-1 models and one supersymmetric \(E_6\)-inspired model).
It is beyond the scope of this paper to discuss every model with its diphoton phenomenology in detail: many of the original papers for which we created the model files discussed their model in specific limits, e.g. decoupling complete parts of the sector without showing that the respective limit can even be consistently obtained. Therefore, a complete phenomenological study of each model would be necessary for checking all claims. Instead, we regard our model implementations as a starting point for the authors of these models or other researchers to perform a more thorough study themselves. Whenever benchmark points in terms of the model parameters were given in the respective literature, however, we have compared our results, and deviations are noted below.
In case of questions, comments or bug reports concerning these models, please, send an e-mail to diphoton-tools @cern.ch which includes all authors.
4.1 Validation
All SARAH model files which have been created, as well as the numerical codes derived thereof, have been validated by us using the following procedure:
-
1.
First, the SARAH files themselves have been tested for consistency using basic SARAH commands, which are easy to use and we recommend these to readers. First of all, we have checked every model for anomalies as well as for the invariance under all gauge and discrete symmetries which is automatically done when the model is loaded within SARAH. Furthermore, the CheckModel command was executed which in addition checks the sanity of all field and parameter definitions as well as whether all possible particle admixtures have been correctly taken into account.
-
2.
Whenever analytic formulas such as mass matrices were presented in the original studies which propose the model, we have reproduced and checked the respective expressions with SARAH.
-
3.
For each model, we have produced and successfully compiled the tailor-made code for the spectrum generators SPheno and FlexibleSUSY.
-
4.
Whenever the reference proposing the model has presented the necessary information to reproduce their results, we have done so. Differences are noted below.
-
5.
The model files for MadGraph and CalcHep have been produced for all models and checked for consistency using the internal routines of the respective tools. Furthermore, we have computed representative processes like the production and/or decay of the candidate for the diphoton resonance and compared the obtained branching ratios between MadGraph, CalcHep and SPheno/FlexibleSUSY.
-
6.
For each model, we provide a set of input parameters which can be used to produce a valid spectrum which itself can then serve as an input for programs like MadGraph or CalcHep.
During the validation process, we noticed inconsistencies in the definition of some models when using the field content and symmetries as provided in the respective references. We therefore modified the respective model in order to restore consistency. For other models, we obtain results that are different to those quoted in the study proposing the model. We individually marked the affected models in the last column of the tables with a special character. The issues we found are the following:
-
\(\clubsuit \) As explained in more detail in the following subsection, after the inclusion of higher-order corrections, the dijet constraints cut deeply into the allowed parameter space.
-
\(\spadesuit \) We find disagreement with the diphoton rate as calculated in the original reference: we have reproduced the partial widths presented in Fig. 3 of Ref. [106] and find values which are roughly an order of magnitude smaller.
-
†The \(U(1)_D\) charge of the \(H'\) field as defined in Ref. [355] has been changed to \(-1\) in order to make the Yukawa interaction terms gauge invariant.
-
‡The model cannot explain the diphoton excess with Yukawa couplings in the perturbative range, but the authors use values between 5 and 10. As stressed in Sect. 2.1.2.1, this renders the perturbative calculation, and hence the results, to be invalid.
-
§We had to change the Yukawa interactions: in Ref. [138], they are defined as, e.g., \( \overline{q_L} H^\dagger _L U_L\) which contracts to zero because of the implicit left/right projection operators. Moreover, in Refs. [138, 149] the ‘conjugate’ assignments of the fields \(H_{L/R}\) need to be exchanged in order to obtain a gauge-invariant Lagrangian. For more details see the actual model implementation or the notes provided with the model files.
-
¶We had to adapt the scalar potential as Eq. (6) and Eq. (7) in Ref. [90] are not gauge invariant. In the model implementation, we allow for every gauge-invariant term in the Lagrangian.
-
\(\triangle \) Here, couplings of about 5 are needed to explain diphoton excess, rendering the perturbative calculation to be inconsistent.
4.2 Examples of model implementations
4.2.1 Scalar octet extension
A charged scalar colour octet O coupled to a scalar singlet S was proposed in Refs. [89, 166]. Here the singlet is the \({750}\,{\text {GeV}}\) candidate, while the octet enters the loops that contribute to the generation of the couplings of the singlet to the gauge bosons. While Ref. [166] considers a toy model involving only the term \(S\,|O|^2\), Ref. [89] takes the singlet extended Manohar-Wise model [450]. For the SARAH implementation we have used the full model. However, since the cubic and quartic terms in O do not play a significant role, they are turned off by default in the SARAH model file.
The extra particle content with respect to the SM is a real singlet S and a scalar colour octet O which is also charged under \(SU(2)_L \times U(1)_Y\), see Table 6. The isospin components of O are
where \(A=1,\ldots ,8\) is the adjoint colour index. The full scalar
Electroweak symmetry-breaking (EWSB) is driven by the VEV of the neutral component of the SM Higgs doublet, which can be decomposed as
Here \(\phi _H \equiv h\) is the Higgs boson, to be identified with the 125 GeV state discovered at the LHC. Similarly, the singlet S receives a VEV, and the neutral component of the octet is split into its CP-even and CP-odd eigenstates:
We will now briefly discuss the parameter space of the model in order to justify our choice of input parameters. First, we consider the tadpole equations, which can be automatically derived by SARAH. Their solution for \(\mu ^2\) and \(\kappa _1\) is
The tree-level mass matrix for the CP-even neutral scalars in the \((\phi _H, \phi _S)\) basis is given by
We note that, in general, there is singlet-doublet mixing. There are two reasons to consider a small singlet-doublet mixing angle, \(\theta \). First, the stringent constraints derived from Higgs physics measurements, and second, the required suppressed decay widths into Higgses, Ws and Zs in order to fit the diphoton signal – indeed in [89] values of \({\sim } 10^{-2}\) were found to be required. If we have a small mixing angle, then we can write
This implies \(\lambda _S > 0\), but also
However, we also have \(v_S^2 \sim m_{750}^2/8\lambda _S\), and so
We thus require a tachyonic \(m_S^2\) for the SM Higgs mass condition:
where in the last step we have taken \(\lambda _S = 1.2\), a rather large value. If we want \(\kappa _1 \sim -1\) then we require \(m_S^2 \sim - (600\, \mathrm {GeV})^2\). On the other hand, from the second tadpole equation we have
which, if we require \(|\kappa _1| < 2\), gives
so putting these together we find the narrow window
Alternative implementation in SARAH
The above discussion suggests to use a different choice for the input parameters of the model in our SARAH implementation: ideally we would like the particle masses, the mixing and only dimensionless couplings to be the inputs. We shall take the input parameters to be
In terms of these the other parameters are determined to be
The exact version of these equations is implemented in SARAH and can be selected using the InputFile \(\rightarrow \) “SPheno_diphoton.m” option in MakeAll or MakeSPheno.
Octet masses
One further input is taken in [89]: the physical mass of the octet scalars. These are given in terms of the Lagrangian parameters as:
The values of \(\lambda _i\) are taken to be small and equal in order for the octets to have similar masses, but since this is not the general case, we do not impose this choice in SARAH. The choice in that paper does however hide the possibility of tachyonic \(m_O^2\) (and hence possible charge/colour breaking minima) – indeed, if we insist that \(m_O^2 > 0\) we have a lower bound on the masses of
Clearly this is violated for \(m_{O^{0,+}} = 600\) GeV when \(\kappa _2 \sim 1, \lambda _S \ll 1\). On the other hand, this does not guarantee a problem.
The desired vacuum has energy
If we instead concentrate on the potential terms containing the octets, where only one component develops a VEV, we find
Arranging for the additional minimum of the potential to be higher than the colour-breaking one then places a lower bound on the octet self-couplings, but for the phenomenology of the diphoton excess – when we neglect loop corrections to the mass of the octet – they play no other role.
Comments on fitting the excess
In [89] the authors find that the diphoton excess can be easily fit with octets at 600 or 1000 GeV and \(\kappa _2 \sim 1.5\) or 4.5, respectively. The scenario involves merely the simplifying assumption \(\lambda _1 = \lambda _2 = \lambda _3\) so that the octets are of approximately equal mass. The ratio between the digluon and diphoton decay rates is then
In [89] this is quoted as \({\simeq } 715\). In SARAH, before any NLO corrections are applied, the running of the Standard Model gauge couplings yields \(\alpha _s (750\, \mathrm {GeV}) = 0.091 \) and we use \(\alpha (0) \simeq 137^{-1}\), giving a ratio of 700, in good agreement. However, when we include corrections up to N\(^3\)LO, this ratio rises to 1150, putting the model near the boundary of exclusion due to dijet production at 8 TeV. These differences are illustrated in plots produced from SARAH/SPheno in Fig. 12. To produce these plots, all branching ratios/widths are calculated in SPheno, as is the production cross-section of the resonance at 8 TeV. To calculate 13 TeV cross-sections the 8 TeV cross-section was rescaled by the parton luminosity factor for gluons of 4.693.
4.2.2 3-3-1 models
Models based on the \(\mathrm {SU(3)_c} \times \mathrm {SU(3)_L} \times \mathrm {U(1)_\mathcal {X}}\) gauge symmetry [451–457], 331 for short, constitute an extension of the SM that could explain the number of generations of matter fields. This is possible as anomaly cancellation forces the number of generations to be equal to the number of quark colours.
Regarding the diphoton excess, 331 models automatically include all the required ingredients to explain the hint. First, the usual \(SU(2)_L\) Higgs doublet must be promoted to a \(SU(2)_L\) triplet, the new component being a singlet under the standard \(\mathrm {SU(3)_c} \times \mathrm {SU(2)_L} \times \mathrm {U(1)_Y}\) symmetry. Similarly, the group structure requires the introduction of new coloured fermions to complete the \(SU(3)_L\) quark multiplets, these exotic quarks being \(\mathrm {SU(3)_c} \times \mathrm {SU(2)_L} \times \mathrm {U(1)_Y}\) vector-like singlets after the breaking of \(\mathrm {SU(3)_c} \times \mathrm {SU(3)_L} \times \mathrm {U(1)_\mathcal {X}}\). Therefore, \(\mathrm {SU(3)_c} \times \mathrm {SU(3)_L} \times \mathrm {U(1)_\mathcal {X}}\) models naturally embed the simple singlet \(+\) vector-like fermions framework proposed to explain the diphoton excess.
There are several variants of \(\mathrm {SU(3)_c} \times \mathrm {SU(3)_L} \times \mathrm {U(1)_\mathcal {X}}\) models. These are characterized by their \(\beta \) parameter,Footnote 9 which defines the electric charge operator asFootnote 10
First, in Sect. 4.2.2.1 we consider the model in Ref. [80]. This 331 variant has \(\beta = 1/\sqrt{3}\), which fixes the electric charges of all the states contained in the \(SU(2)_L\) triplets and anti-triplets to the usual \(0,\pm 1\) values. In Sect. 4.2.2.2 we consider a 331 model with \(\beta = -\sqrt{3}\), a value leading to exotic electric charges. This 331 variant has been discussed in the context of the diphoton excess in [88, 243, 459]. Although the mechanism to explain the diphoton excess is exactly the same as in [80], the presence of the exotic states leads to slightly different numerical results.
4.2.2.1 On the SU(3) generators in SARAH
The most common choice for the SU(3) generators is \(T_a = \frac{\lambda _a}{2}\), with \(\lambda _a\) (\(a=1,\ldots ,8\)) the Gell-Mann matrices. However, this is just one of the possible representations. In fact, SARAH uses a different set of matrices, \(T_a^{{\mathtt{SARAH}}} = \frac{\Lambda _a}{2}\), following the conventions of Susyno [460]. The relation between the non-diagonal matrices in the two bases is
Concerning the diagonal matrices, the usual \(\lambda _{3,8}\) Gell-Mann matrices,
are replaced by \(\Lambda _{7,8}\),
The electric charge operator can be written, using the conventions in SARAH, as
This in turn implies that the charge assignments in the SU(3) multiplets must be adapted as well. For example, one can easily check that the electric charges of the first and third components of a SU(3) triplet t are exchanged when going from the usual Gell-Mann representation to the basis choice employed in SARAH,
In the following we will use the standard conventions based on the Gell-Mann matrices in order to keep the discussion as close to the original works as possible. However, we emphasize that the implementation of the 331 models in SARAH requires this dictionary between the bases. It should also be noted that in the current implementation in SARAH of the 331 models described below, vertices involving four vector bosons in the generated model files for CalcHep cannot yet be handled correctly. In order to generate model files that will work with CalcHep, one must therefore exclude these vertices from being written out by SARAH by specifying Exclude -> {VVVV} in the options of SARAH ’s MakeCHep.
4.2.2.1 331 model without exotic charges
-
Reference: [80]
-
Model name: 331/v1
The model is based on the \(\mathrm {SU(3)_c} \times \mathrm {SU(3)_L} \times \mathrm {U(1)_\mathcal {X}}\) gauge symmetry, extended with a global \(U(1)_{\mathcal {L}}\) and an auxiliary \(\mathbb {Z}_2\) symmetry to forbid some undesired couplings. The fermionic and scalar particle content of the model is summarized in Table 7. In addition, due to the extended group structure, the model contains 17 gauge bosons: the usual 8 gluons; 8 \(W_i\) bosons associated to \(SU(3)_L\) and the B boson associated to \(U(1)_\mathcal {X}\).
The fermionic \(SU(3)_L\) triplets of the model can be decomposed as
The notation used for the extra quarks that constitute the third components of the \(SU(3)_L\) triplets \(Q^{1,2,3}_L\) is motivated by the fact that their electric charges are \(-1/3\) and 2 / 3 for D/S and T, respectively. The scalar multiplets can be written as
While \(\phi _1^-\), \(\phi _{2,3}^+\) and \(S_1^-\) are electrically charged scalars, the components \(\phi _{1,2,3,X}\), \(S_{2,3}\) and X are neutral.
The Yukawa Lagrangian of the model can be split as
where
and
We defined \(\hat{d}_R \equiv ( D_R, S_R )\). We note that Eq. (4.31) leads to an inverse seesaw mechanism for neutrino masses [461, 462]. Here, \(y^a\) is anti-symmetric while \(m_s\) is symmetric, whereas the rest of Yukawa couplings are generic matrices, including those in Eq. (4.30). An additional term \(y^s_X \bar{\psi }_L s \Phi _X\) could be added to Eq. (4.31), but given that \(\langle \Phi _X \rangle = 0\), it does not contribute to neutrino masses and we will drop it for simplicity. Finally, the scalar potential is given by
where \(i=1,2,3,X\). The \(\mathbb {Z}_2\)-soft-breaking term, \(f \Phi _1 \Phi _2 \Phi _3\), is required to break unwanted accidental symmetries in the scalar potential.
We will assume the following symmetry breaking pattern
4.2.2.2 331 model with exotic charges
Now, we will consider a 331 variant with \(\beta = -\sqrt{3}\), as discussed in the context of the diphoton excess in [88]. The fermionic and scalar particle content of the model is summarized in Table 8. In addition, the model contains 17 gauge bosons: the usual 8 gluons; 8 \(W_i\) bosons associated to \(SU(3)_L\) and the B boson associated to \(U(1)_\mathcal {X}\).
The fermionic \(SU(3)_L\) triplet representations of the model can be decomposed as
Due to the choice \(\beta = - \sqrt{3}\), the electric charges for the extra quarks that constitute the third components of the \(SU(3)_L\) triplets \(Q^{1,2,3}_L\) are 5 / 3, 5 / 3 and \(-4/3\), respectively. The scalar triplets can be written as
Therefore, the particle spectrum of the model contains the exotic quarks in Eq. (4.34), as well as the doubly-charged fermion \(E^{--}\) and the scalars \(\rho ^{++}\) and \(\chi ^{--}\).
The Yukawa Lagrangian of the model can be split as
where
where we have defined \(\hat{d}_R \equiv ( D_R, S_R )\), and
We note that the exotic fermions E, D, S and T only couple to the \(\chi \) scalar triplet, and thus only via its vacuum expectation value (VEV) they will acquire masses. Finally, the scalar potential is given by
We will assume the following symmetry breaking pattern
In this case, the non-zero VEV of \(\chi \) is responsible for the breaking \(SU(3)_L \times U(1)_X \rightarrow SU(2)_L \times U(1)_Y\). The requirement that this occurs at a scale much above the EW scale then imposes a hierarchy amongst the VEVs, namely that \(v_3 \gg v_1, v_2\). Consequently, one of the CP-even scalar states is predominantly from the \(\chi \) triplet and decouples from the SM. This scalar is then identified as the candidate for the 750 GeV resonance in this model. The decays of this state into two photons proceed via loops involving the heavy fermions, as well as those involving the charged scalars and additional charged vector bosons.
4.2.3 \(E_6\)-inspired SUSY model with extra U(1)
-
Reference: [110]
-
Model name: SUSYmodels/E6SSMalt
\(E_6\)-inspired SUSY models predict extra SM-gauge singlets and extra exotic fermions, so they immediately have the ingredients that many authors have tried to use to fit the diphoton excess. These models are often motivated as a solution to the \(\mu \)-problem of the MSSM, because the extra U(1) gauge symmetry forbids the \(\mu \)-term, while when one of the singlet fields develops a VEV at the TeV scale this breaks the extra U(1) giving rise to a massive \(Z^\prime \) vector boson and at the same time generates an effective \(\mu \) term through the singlet interaction with the up- and down-type Higgs fields, \(\lambda \hat{S} \hat{H}_u \hat{H}_d\). The matter content of the model at low energies fills three generations of complete 27-plet representations of \(E_6\), which ensures that anomalies automatically cancel.
A number of models of this nature have been proposed as explanations of the diphoton excess [110, 275, 463]. The example we implement here [110] is a variant of the E\(_6\)SSM [464, 465]. In this version two singlet states develop VEVs and the idea is that the 750 GeV excess is explained by one of these singlet states with a loop-induced decay through the exotic states.
In \(E_6\) models the extra U(1) which extends the SM gauge group is given as a linear combination of \(U(1)_\psi \) and \(U(1)_\chi \) which appear from the breakdown of the \(E_6\) symmetry as \(E_6\rightarrow SO(10)\times U(1)_{\psi }\) followed by SO(10) into SU(5), \(SO(10)\rightarrow SU(5) \times U(1)_{\chi }\). In the E\(_6\)SSM and the variant implemented here the specific combination is,
To allow one-step gauge coupling unification some incomplete multiplets must be included in the low energy matter content. So in addition to the matter filling complete 27 representations of \(E_6\) there are also two SU(2) multiplets \(\hat{H}^\prime \) and \(\hat{\overline{H}}^\prime \), which are the only components from additional \(\varvec{27'}\) and \(\varvec{\overline{27}'}\) representations that survive to low energies. All gauge anomalies cancel between these two states, so they do not introduce any gauge anomalies. Furthermore, the low energy matter content of the model beyond the MSSM includes three generations of exotic diquarks,Footnote 11 \(\hat{D}_i, \hat{\bar{D}}_i\), three generations of SM singlet superfields \(\hat{S}_i\) and extra Higgs-like states \(H^u_{1,2}\) and \(H^d_{1,2}\) that do not get VEVs.
The full set of superfields are given in Table 9 along with their representations under SU(3) and SU(2) and their charges under the two U(1) gauge groups and the discrete symmetries, which we will now discuss.
The \(\mathbb {Z}_2^L\) symmetry plays a role similar to R-parity in the MSSM, being imposed to avoid rapid proton decay in the model. However with this imposed there are still terms in the superpotential that can lead to dangerous flavour changing neutral currents (FCNCs). To forbid these, an approximate \(\mathbb {Z}_2^H\) symmetry is imposed. In the original E\(_6\)SSM model only \(\hat{S}_3\), \(\hat{H}_d\) and \(\hat{H}_u\) were even under the \(\mathbb {Z}_2^H\) symmetry, however in this variant \(S_2\) is also even under this approximate symmetry.
The full superpotential before imposing any discrete symmetries is given by
where
However, with the discrete symmetries imposed and integrating out the heavy right-handed neutrinos, the superpotential in this specific variant reduces to,Footnote 12
One should remember that the \(\mathbb {Z}_2^H\) can only be an approximate symmetry as otherwise the exotic quarks could not decay. In this variant the exotic quarks decay through the \(\mathbb {Z}_2^H\) violating interactions of \(W_1\).
In the paper it is assumed that the singlet mixing can be negligible and the numerical calculation was performed under this assumption, neglecting any mixing between the singlet state which decays to \(\gamma \gamma \) via the exotic states and the other CP-even Higgs states from the standard SU(2) doublets. However it is clear that there must be some mixing from the D-terms, and therefore if that is included one important check would be to test whether other decays are sufficiently suppressed. Moreover, the parameters needed to simultaneously get a 125 GeV SM-like Higgs state and a 750 GeV singlet-dominated state are not given. In this respect we note that the singlet VEVs appear both in the diagonal entries of the mass matrix and in the off-diagonal entries that mix the singlet states with the doublet states.
We finally note that other similar \(E_6\) models have also been proposed in the context of the diphoton excess. These include a model by two authors from the original paper [275], a model with a different U(1) group at low energies [466], and a model that is still \(E_6\)-inspired, but where no extra U(1) survives down to low energies [269].
5 Study of a natural SUSY explanation for the diphoton excess
We show in this section how one can use the described setup to perform easily a detailed study of a new model that aims at explaining the diphoton anomaly. This model was not proposed before in the literature to explain the diphoton excess and offers a very rich phenomenology. We will not only discuss the main phenomenological features of the model, but we will also show the necessary steps to obtain this information with the discussed tools. However, we emphasise once again that we are not aiming at a thorough exploration of the entire phenomenology of the model, something that would be clearly beyond the purpose of this example.
5.1 The model
We are now going to study a SUSY model which enhances the tree-level Higgs mass due to non-decoupling D-terms. The model is based on that proposed in Ref. [467] as a natural SUSY model which allows for light stops compatible with the measured Higgs boson mass, extended by three generations of pairs of vector-like quarks and leptons. We want to achieve a tree-level enhancement of the SM-like Higgs mass and an explanation of the diphoton excess via the loop-induced decay of a CP-odd scalar. In addition, we will also check whether one can get a broad diphoton resonance in this model.
The matter field content is shown in Table 10 and the considered superpotential reads:
We will not make the simplifying assumption that mixings between the MSSM states and the new vector-like fields can be neglected. Of course, such mixing could have been forbidden by choosing different \(U(1)_X\) charges for the new particles. However, in such case there would be a conserved \(\mathbb {Z}_2\) symmetry associated to the vector-like states (under which all vector-like superfields are odd and the rest are even) that would make the lightest of them absolutely stable. This would be a problem unless that state is neutral and colourless, and thus this scenario can only be viable if we also consider additional singlet vector-like states, such as vector-like partners for the right-handed neutrinos, and make them lighter than the other vector-like states. Thus, this setup would predict two stable particles to make the dark matter. Such a scenario could also be studied with the tools presented here. However, we decided not to consider this option in the following.
The other main ingredients of the model are the general soft-SUSY breaking terms, which read
Note that we have included the gaugino mass term \( M_{1X}\) arising from gauge kinetic mixing. All the terms shown in Eq. (5.2) are automatically added by SARAH based on the information provided by the user about the particle content and the superpotential. Several scalar fields acquire VEVs. We decompose them as
We define \(\tan \beta =\frac{v_u}{v_d}\), \(v=\sqrt{v_d^2+v_u^2}\) as well as \(\tan \beta _x = \frac{v_{\eta }}{v_{\bar{\eta }}}\), \(x=\sqrt{v_\eta ^2+v_{\bar{\eta }}^2}\). In addition, the sneutrinos are decomposed with respect to their CP eigenstates,
which in general have different masses due to the Majorana mass-term \(Y_X \langle \bar{\eta }\rangle \) in the superpotential. Since \(H_d^0\) and \(H_u^0\) carry charges under both U(1) gauge groups, there will be non-zero Z–\(Z'\) mixing even in the limit of vanishing gauge kinetic mixing. The list of particle mixings, which go beyond the usual MSSM mixings reads
The model files which implement this model in SARAH are available in the SARAH model repository as U1xMSSM3G. A FlexibleSUSY model file for the model is also available in the current release of FlexibleSUSY. Finally, we provide all files to reproduce the computations that follow at http://sarah.hepforge.org/U1xMSSM_example.tar.gz.
5.2 Analytical results with Mathematica
Before we perform a numerically precise study of the model, we show how already with just SARAH and Mathematica one can gain a lot of information about a new model.
5.2.1 Consistency checks
The model is initialised after loading it in SARAH via
SARAH automatically performs some basic consistency checks for the model. For instance, it checks whether the model is free from gauge anomalies:
One can see that SARAH tests all different combinations of gauge anomalies and, given that no warning is printed on the screen, confirms that all of them cancel. Similarly, it also checks that all terms in the superpotential are in agreement with all global and local symmetries. More detailed checks can be carried out by running CheckModel[] when the initialisation is finished.
After a few seconds, a message is printed telling that the model is loaded.
5.2.2 Particles and parameters
An overview of all particles and parameters present in this model is given in Tables 11, 12 and 13. The user has also access to this information by calling
to get all particles present after EWSB and by calling
to see all existing parameters. Moreover, it is possible to get similar tables as the ones shown here in LaTeX-format for each model via the commands
5.2.3 Gauge sector
Before we discuss the matter sector or the scalar potential, we have a brief look at the gauge bosons. We make use of the mass matrices calculated by SARAH during the initialisation of the model. We find a handy expression for the mass matrix of the neutral gauge bosons in the limit of vanishing gauge kinetic mixing (\(g_{X1}=g_{1X}=0\)) via
which reads
Note, that MassMatrix[VP] and MassMatrix[VZ] would have given the same result. We can check the eigenvalues of this matrix to first order in \(\frac{v^2}{x^2}\) using the Series command of Mathematica
and find
As expected, the first two eigenvalues are just the ones of the SM gauge bosons, while the mass of the new gauge boson is given by
We will use this relation in the following to replace x by \(M_{Z'}\) in all equations.
5.2.4 Scalar sector
Solving the tadpole equations We turn now to the scalar sector of the model. First, we make a list with a few simplifying assumptions which we are going to use in the following
Here we assume all parameters to be real, remove any complex conjugation (conj) and use the Landau gauge ( RXi[_]->0), then we turn off again gauge kinetic mixing and take the VEVs of \(\eta \) and \(\bar{\eta }\) to be equal. In the fourth line, we parametrise \(v_d\) and \(v_u\) as usual in terms of v and \(\tan \beta \). Finally, we set the parameters \(\kappa \), \(T_\kappa \), \(\lambda \), \(T_\lambda \) and \(L_\xi \) to zero. We can now solve the tadpole equations, stored by SARAH in TadpoleEquations[Eigenstates], with respect to the parameters \(m_{H_d}^2\), \(m_{H_u}^2\), \(m_{\eta }^2\), \(m_S^2\) and \(\xi \) using the aforementioned assumptions:
We have saved the solution in the variable sol for further usage.
Obtaining a 750 GeV pseudo-scalar We use the solution and our assumptions to get simpler expressions for the mass matrix of the CP-even (called hh) and CP-odd (called Ah) scalars:
These matrices can be expressed as
with
We omit here the analytical expressions for \(m_A^{2,X}\) and \(m_H^{2,X}\) because of their length and since they are not needed for the following brief discussion. The mass matrix for the CP-odd states is block-diagonal since the MSSM part is unchanged, while we have mixing in the CP-even sector among all five components.Footnote 13 The additional D-Terms can be found in the MSSM block, \(m_H^{2,\text {MSSM}}\). This also explains our choice of a pseudo-scalar as the resonance behind the diphoton excess: the tree-level mixing between the scalar singlet and the doublets would cause tree-level decays of a 750 GeV scalar into all kinds of SM particles. In particular, those into WW and ZZ are constrained and could easily spoil our setup as an explanation of the excess in this model. Of course, we have to check whether it is possible to obtain a pseudo-scalar of the correct mass, and get the corresponding scalar sufficiently heavy so as to escape detection. For that purpose, we calculate the eigenvalues of the lower \(3 \times 3\) block of the pseudo-scalar mass matrix, and fix \(B_S\) by demanding to have a pseudo-scalar of the correct mass:
We now make an arbitrary choice for the numerical values of the remaining parameters, except \(m_{\bar{\eta }}^2\) and \(M_{Z'}\),
and calculate all CP-even and CP-odd mass eigenvalues for specific values of \(m_{\bar{\eta }}^2\) and \(M_{Z'}\):
The results are
Thus, as expected, we have two massless (up to numerical errors) states in the CP-odd sector, which are the neutral Goldstone bosons to be eaten by the Z and \(Z^\prime \) gauge bosons, accompanied by a particle with a mass of 750 GeV. In the scalar sector we find the lightest state with a mass very close to \(M_Z\) and another scalar below 1 TeV. However, checking the composition of the 750 and 825 GeV particles via
we see that the CP-odd state is, as expected, mainly a singlet while the CP-even one is mainly a X-Higgs (composed by \(\phi _\eta \) and \(\phi _{\bar{\eta }}\)). That looks already very promising.
Higgs mass enhancement via non-decoupling D -terms Now, we want to confirm that one gets non-decoupling D-terms in this model which cause an enhancement of the tree-level mass of the SM-like scalar. For this purpose, we define a simple function which calculates the lightest CP-even mass for input values of \(m_{\bar{\eta }}\) and \(M_{Z'}\),
and create a contour plot using this function. The result is depicted in Fig. 13, where one sees that for \(m_{\bar{\eta }} \gg M_{Z'}\) it is indeed possible to find a tree-level mass well above 100 GeV, while for \(m_{\bar{\eta }} \ll M_{Z'}\) the tree-level mass approaches \(M_Z\).
Is there a second light scalar? One can now start to play also with the values we have chosen for num to see how the eigenvalues of both matrices change. One finds, for instance, that it is also possible to get a second, relatively light scalar in the model. With the values
we find a tree-level mass of 38 GeV for the lightest CP-even scalar, which is mainly a mixture of \(\eta \) and \(\bar{\eta }\). It will be interesting to see if this scenario is still in agreement with all experimental constraints and how important the loop corrections are.
How to obtain a broad width? So far, we have not considered the total decay width of the 750 GeV scalar. The experimental data shows a slight preference for a rather large width of about 40 GeV, which is not easy to accommodate in weakly coupled models, typically requiring a large branching ratio into invisible states. Therefore, it would be interesting to see if this can be realised in this model. There are three possibilities for invisible decays: (i) neutralinos, (ii) (heavy) neutrinos, (iii) sneutrinos. We are going to consider the third option here. For this purpose, we have to check two ingredients: can the mass of the sneutrinos be sufficiently light and how can the coupling to the 750 GeV scalar be maximised? To get a feeling for that, we first consider the mass matrix of the CP-even and CP-odd sneutrinos. We assume that flavour and left-right mixing effects are negligible. In that case, it is sufficient to take a look only at the (4,4) entry of the mass matrices:
After some simplifications, we get the following expressions from SARAH:
We see that the terms in the second line, \(\propto T_x M_{Z'}\) and \(\propto v_S \lambda _C Y_x\), induce a mass splitting between the CP-even and CP-odd states. Thus, in order to have the decay \(A \rightarrow \tilde{\nu }^I \tilde{\nu }^R\) kinematically allowed, these terms must be individually small or cancel each other. In addition, one has to compensate the large terms \({\sim } M_{Z'}\) in order to get sufficiently light sneutrinos. This could be done by assuming a negative \(m_ {\nu }^{2} = -\frac{1}{4 g_X^2} (4 M_{Z'}^{2} - g_{X}^{2} v^{2} )Y_{x,{1 1}}^{2}\). Of course, we must check whether this leads to spontaneous R-parity breaking via sneutrino VEVs, and for this purpose one can use Vevacious, see below.
We can now check the vertex \(A \tilde{\nu }^I \tilde{\nu }^R\) using the same assumptions:
and we obtain after some simplificationFootnote 14
If the pseudo-scalar is a pure singlet, only the term \(\propto Z_{35}^A\) contributes. This term is independent of \(v_S\) and \(T_x\), i.e. we can reduce the mass splitting between the CP-even and CP-odd sneutrinos by adjusting these parameters without having a negative impact on the coupling strength to the 750 GeV scalar.
5.2.5 Vector-like sector
Before we finish the analytical discussion of the masses, we briefly discuss the extended matter sector. The mass matrices responsible for the mixing between the SM fermions and the vector-like fermions can be obtained from SARAH by calling
which return
Thus, for large \(\lambda _i\) (\(i=u,e\)) and \(v_S\), there are two important sources for the mass of the vector-like states. The full sfermion matrices containing the new scalars are too lengthy to be shown here. We only check the new mass matrix for one generation of the vector-like selectrons which are the 7th and 10th gauge eigenstates. We can pick the values via
and obtain by setting all parameters to be diagonal
where we have defined \(\tilde{D} = (t_\beta ^2-1) v^2 (2 g_1^2+g_X^2)\). There is a potentially dangerous term \(\lambda _e \xi \) which rapidly increases for increasing \(\lambda _e\). To keep all scalar masses positive, it is necessary to choose a rather large \(B_E\) as well. Therefore, we are going to choose always
in our numerical study to circumvent tachyonic scalars.
5.2.6 RGEs and gauge kinetic mixing
We have so far made the simplifying assumption that gauge kinetic mixing vanishes. However, if the two Abelian gauge groups are not orthogonal, kinetic mixing would be generated via RGE running even if it vanishes at some energy scale. Thus, one of the first checks on the RGEs of the model we can make is whether the two U(1) gauge groups are orthogonal. For this purpose, we first calculate the one-loop RGEs with SARAH via
We have chosen one-loop RGEs only to save time. Without the TwoLoop->False flag, the full two-loop RGEs would have been calculated automatically. Other options for CalcRGEs are:
-
ReadLists: If the RGEs have already be calculated, the results are saved in the output directory. The RGEs can be read from these files instead of doing the complete calculation again.
-
VariableGenerations: Some theories contain heavy superfields which should be integrated out above the SUSY scale. Therefore, it is possible to calculate the RGEs assuming the number of generations of specific superfields as free variable to make the dependence on these fields obvious. The new variable is named NumberGenerations[X], where X is the name of the superfield.
-
NoMatrixMultiplication: Normally, the \(\beta \)-functions are simplified by writing the sums over generation indices as matrix multiplication. This can be switched off using this option.
-
IgnoreAt2Loop: The calculation of 2-loop RGEs for models with many new interactions can be very time-consuming. However, often one is only interested in the dominant effects of the new contributions at the 1-loop level. Therefore, IgnoreAt2Loop -> $LIST can be used to neglect parameters at the two-loop level The entries of $LIST can be superpotential or soft SUSY-breaking parameters as well as gauge couplings.
-
WriteFunctionsToRun: Defines if a file should be written to evaluate the RGEs numerically in Mathematica
We can now check the entries in BetaGauge and find
The standard normalisation factor \(\sqrt{5/3}\) for the hypercharge has been included. One can see that the \(\beta \)-functions for \(g_{YX}\) and \(g_{XY}\) are non-zero even in the limit \(g_{XY}, g_{YX} \rightarrow 0\), i.e. these couplings will be induced radiatively. Thus, in general one has not only two couplings \(g_1\) and \(g_X\) in this model, but a gauge coupling matrix G defined as
In the limit of vanishing kinetic mixing, \(g_{YX} = g_{XY} = 0\), the relations \(g_{YY} = g_1\) and \(g_{XX} = g_X\) hold. Even if gauge kinetic mixing is present, one has the freedom to perform a change in basis to bring G into a particular form. The most commonly considered cases are the symmetric basis with \(g_{XY} = g_{YX}\) and the triangle basis with \(g_{YX} = 0\). The triangle basis has the advantage that the new scalars do not contribute to the electroweak VEV, and the entire impact of gauge kinetic mixing is encoded in one new coupling \(\tilde{g}\). The relations between \(g_{ij}\) (\(i,j=X,Y\)) and \(g_1\), \(g_X\), \(\tilde{g}\) are [468]
It is interesting to see how large \(\tilde{g}\) is naturally. With ‘naturally’ we mean under the assumption that the off-diagonal \(g_{YX}\) and \(g_{XY}\) couplings vanish at some high scale \(\Lambda \) and are generated by RGE running down to the SUSY scale. Thus, in this setup, the size of gauge kinetic mixing is a function of \(\Lambda \) and \(g_X\) at this scale. We can write a simple Mathematica function to get a feeling for the off-diagonal gauge couplings:
In the first line, we load the file written by SARAH which provides the RGEs in a form which Mathematica can solve. This file also contains the function RunRGEs that can be used to solve the RGEs numerically. As boundary condition, we used \(g_1 = 0.45\) at the scale 1 TeV. After the running we rotate the couplings to the basis where \(g_{XY}\) vanishes. We can make a contour plot via
and get the result shown in Fig. 14. Thus, we find that at the SUSY scale the gauge mixing coupling \(\tilde{g}\) is negative and not much smaller than an ordinary gauge coupling unless \(\Lambda \) is assumed to be very small.
5.2.7 Boundary conditions and free parameters
For the subsequent numerical analysis we are going to assume some simplified boundary conditions applied at the SUSY scale:
In addition, we can set \(Y_\nu = 0\) since this parameter is highly constrained to be small by the small neutrino masses. In addition, we set the mixing parameters \(m_{eE}^2\), \(m_{uU}^2\), \(\tilde{M}_E\), \(\tilde{M}_U\), \(\tilde{B}_E\), \(\tilde{B}_U\), \(M_{1X}\) to zero and also assume vanishing \(\lambda \), \(\kappa \), and \(T_\kappa \). However, we stress that this is just done to keep the following discussion short and simple. All effects of these parameters can be included without any additional efforts. Thus, the free parameters mainly considered in the following are
The tadpole equations are solved for \(m_{H_d}^2\), \(m_{H_u}^2\), \(m_{\eta }^2\), \(m_S^2\) and \(\xi \), while \(B_E\) and \(B_U\) are fixed via Eq. (5.28).
5.3 Analysis of the important loop corrections to the Higgs mass
We now turn to the numerical analysis of this model. In the first step, we have written a SPheno.m file for the boundary conditions, see Sect. 5.2.7, and generated the SPheno code with the SARAH command
We copy the generated Fortran code to a new sub-directory of SPheno-3.3.8 and compile it via
We now have an executable SPhenoU1xMSSM which expects the input parameters from a file called LesHouches.in.U1xMSSM. The SPheno code provides many important calculations which would be very time-consuming to be performed ‘by hand’ for this model, but could be expected to be relevant. A central point is the calculation of the pole mass spectrum at the full one-loop (and partially two-loop) level. In particular, the loop corrections from the vector-like states are known to be very important. However, the focus in the literature has usually been only on the impact on the SM-like Higgs. We can automatically go beyond that and consider the corrections to the 750 GeV state as well. Moreover, SPheno calculates all additional two-loop corrections in the gaugeless limit including all new matter interactions. Thus, we can check the impact of the vector-like states even at two-loop level. These effects have not been studied in any of the SUSY models proposed so far to explain the diphoton excess. Of course, SPheno also makes a very precise prediction for the diphoton and digluon decay rate of all neutral scalars as described in Sect. 3.5, and it checks for any potential decay mode. Thus, it is impossible to miss any important decay as sometimes has happened in the literature when discussing the diphoton excess. Finally, there are also other important constraints for this model like those from flavour observables or Higgs coupling measurements. As will be shown in the next sections, all of this can be checked automatically with SPheno and tools interacting with it.
If not mentioned otherwise, we make the following choice for the input parameters
5.3.1 New loop corrections to the SM-like Higgs
In this model we have two new important loop corrections to the SM-like Higgs: (i) the corrections from vector-like states, proportional to \(Y'_u\) and \(Y'_e\), and (ii) the new corrections from the extended gauge sector. The corrections from vector-like (s)tops up to two-loop have been discussed in detail in Ref. [426] using the SARAH/SPheno framework. There are several important effects which are often neglected in studies of vector-like states which only make use of the one-loop effective potential: the momentum effects at one-loop, the two-loop corrections, and the shift of the top-Yukawa coupling. In general, the user does not need to worry about these details because SARAH/SPheno take care of them automatically. However, it might be interesting to have an intuitive feeling about the size of the different effects. Since it demands some ‘hacking’ of the code to disentangle the calculation in that way, we are not making this analysis here, but we briefly summarise the main results of Ref. [426] in Fig. 15. We see that all these effects can alter the Higgs mass by several GeV. Thus, an estimated uncertainty of about 2–3 GeV when using only the one-loop effective potential approximation is usually over optimistic.
Furthermore, in models with non-decoupling D-terms the new loop corrections are usually neglected in the literature. Therefore, we are going to check whether this is a good approximation or not. For this purpose we show the SM-like Higgs pole mass at tree and one-loop level as a function of \(g_X\) for two different values of \(M_{Z'}\). Since SPheno performs the two-loop corrections in the gaugeless limit, additional corrections from the extended gauge sector are not included at two-loop, and we concentrate on the one-loop effects here. For this purpose, we use the different flags in the Les Houches input file from SPheno to turn the corrections at the different loop levels on or off:
Here, A and B are either 1 or 0. With flag 55 the entire loop-corrections to all masses can be turned on (1) or off (0), while flag 7 only skips (1) or includes (0) the two-loop corrections in the Higgs sector. The results are shown in Fig. 16. All scans have been performed using the Mathematica package SSP [469] for which SARAH already writes a template input when generating the SPheno code (SSP_Template.m.U1xMSSM) for a given model. We see that the tree-level mass rises quickly with increasing \(g_X\). However, for both \(M_{Z'}\) values this shift is compensated to some extent when one-loop corrections are included. Thus, the inclusion of non-decoupling D-terms only at tree-level would overestimate the positive effect on the SM-like Higgs mass by 20–30 %. In addition, we also see that off-diagonal gauge couplings of a realistic size arising from gauge kinetic mixing reduce the positive effect from the non-decoupled D-terms on the Higgs mass by a few GeV.
5.3.2 Loop corrections to the 750 GeV scalar
There are also important loop corrections to all other scalars in the model if large Yukawa-like couplings are present. We discuss this briefly for the 750 GeV pseudo-scalar: in Fig. 17, the mass at tree, one- and two-loop level for varying \(\lambda _V \equiv \lambda _e=\lambda _u\) for two different values of \(m_{SUSY}\), 1.5 and 2.5 TeV, is given. For \(m_{SUSY}=1.5\) TeV there is only a moderate difference between tree-level, one- and two-loop for \(\lambda _V\rightarrow 0\), but for \(\lambda _V\) of O(1) the one-loop corrections cause a shift by 100 GeV and more, which is compensated to some extent by the two-loop corrections. For larger \(m_{SUSY}\) we see already a large positive shift for small \(\lambda _V\), which quickly increases and reaches 300–400 GeV for \(\lambda _V \sim 0.8\). For even larger values of \(\lambda _V\), the difference between tree-level and the loop corrected mass becomes smaller. Still, the overall shift is more than 100 GeV, and this would be highly overestimated by only including one-loop corrections. As we will see in the next section, one needs \(\lambda _V \sim O(1)\) to explain the diphoton signal. For this value, a naive tree-level analysis gives a mass for the lightest CP-odd state which is far off the correct value. Thus, one has to be much more careful with the choice for \(B_S\).
5.4 Diphoton and digluon rate
We now discuss the diphoton and digluon decay rate of the pseudo-scalar, and its dependence on the new Yukawa-like couplings. As we have just seen, large couplings induce a non-negligible mass shift. Therefore, it is necessary to adjust \(B_S\) carefully to get the correct mass, 750 GeV, after including all loop corrections. This can be done by SSP, which can adjust \(B_S\) for each point to obtain the correct mass within 5 GeV uncertainty. The results for the calculated diphoton and digluon rate at LO and with the higher order corrections discussed in Sect. 3.5 are shown in Fig. 18. In order to see the size of the higher order corrections, one can use the flag 521 in SPheno to turn them on and off
This can also be achieved in FlexibleSUSY by setting a flag in the generated C++ code as desired. Note that, as discussed in Sect. 3.5, there is close agreement for the diphoton and digluon effective vertices computed using SPheno and FlexibleSUSY. Therefore we present here only the results obtained using SPheno. Nevertheless, it might be often helpful to compare the results between both codes since they use a slightly different matching and renormalisation procedure which results in some differences in the mass spectrum and consequently also in the calculated decays and branching ratios. Therefore, these differences can be used as a rough estimate of the theoretical uncertainty of the different calculations.
One finds the expected behaviour: the partial widths rise quadratically with the coupling. For about \(\lambda _V \simeq 1.0\) one has \(\Gamma (S\rightarrow \gamma \gamma )/M_S \sim 10^{-6}\), which is necessary to explain the observed excess. In Fig. 18 we also show a comparison between a purely LO calculation and the one including the higher order QCD corrections described in Sect. 3.5. There is no change for the decay into two photons, because its NLO corrections for a pseudo-scalar are non vanishing only for \(m_A > 2 M_F\). Instead, the digluon width is enhanced by a factor of 2 when including NLO and NNLO QCD corrections. This also changes the ratio of the digluon-to-diphoton width from about 10 (LO only) to 20 (including higher orders).
5.5 Constraints on choice of parameters
5.5.1 Singlet-doublet mixing
So far, we made some strong assumptions about some parameters in this model. In particular, we set the coupling between the singlet and the two Higgs doublets \(\lambda =0\). This raises the question how sensitive the results are to this choice. For this purpose, we can test what happens if we slightly deviate from it. The branching ratios of the CP-odd scalar of 750 GeV mass, which is nearly a pure singlet, as a function of \(\lambda \) are shown in Fig. 19. For comparison we also show the branching ratios for the CP-even scalar with a mass around 800 GeV. This particle is mainly a mixture of \(\eta \) and \(\bar{\eta }\) with a small singlet component. For both particles we depict the branching ratios when calculating only tree-level masses and when including loop-corrections. At the tree level we find that the impact of \(\lambda \) on the branching ratios of A is very small. This does not change much when including the loop corrections to the pseudo-scalar rotation matrix. On the other hand, for vanishing \(\lambda \) we already have a large branching ratio of the CP-even scalar into hh even at tree level. Moreover, the decay modes into two massive vector bosons or \(t\bar{t}\) at tree level increase very quickly with \(\lambda \) and for \(\lambda > 0.01\) they already dominate. At one-loop level, the large dependence on \(\lambda \) is no longer visible, because for very small \(\lambda \) the branching ratios into massive SM vector bosons and fermions are already large. This can be seen in Fig. 20 where we compare the doublet fraction of the two states at tree level and one loop. In general, the behaviour shows that a CP-odd scalar might be a much less fine-tuned candidate for the observed excess.
5.5.2 Constraints from Higgs coupling measurements
We have seen in the Mathematica session that it is possible to obtain two light scalars at tree-level. One question is: is this also possible when including all loop contributions? In order to address this question we change some input parameters to
The pole masses and the doublet fraction of two lightest CP-even states as a function of \(\tan \beta _X\) is shown in Fig. 21. We find the very strong dependence on \(\tan \beta _X\), known in many U(1) extensions [468, 470, 471]. One difference here is that, due to the mixing with the singlet, the light state in the extended sector does not become massless for \(\tan \beta _X = 1\), but for small deviations from it. We see in Fig. 21 that the SM-like Higgs gets a positive mass shift after the level crossing, while the mass of the lightest state drops very quickly. Of course, it is important to know if such light Higgs-like particles are compatible with all limits from Higgs searches at LEP, Tevatron and the LHC. For this purpose, we can make use of HiggsBounds, which checks whether the decay rates of a scalar into SM states are compatible with the observations at all experiments performed so far. If any of these rates is above 1 (normalised to the SM expectation), such a parameter point would be ruled out. Similarly, we can use HiggsSignals to obtain a \(\chi ^2\) estimator for each parameter point, based on how well the measured Higgs properties are reproduced. In order to use HiggsBounds and HiggsSignals, we set the flag
in the input file for SPheno. In this way, SPheno writes out all files which are necessary to run HiggsBounds and HiggsSignals via the effective couplings input mode (effC). However, there is one caveat: SPheno does not automatically write the file MHall_uncertainties which gives an estimate for the theoretical uncertainty in the mass prediction of all scalars. The reason is that SPheno cannot do such an estimate automatically. However, if this file is missing, HiggsBounds and HiggsSignals would assume that the uncertainty is zero. Therefore, we add this file by hand and assume a 3 GeV uncertainty for all masses. We can now use this setup to make a scan in the (\(\tan \beta _X\),\(\lambda \)) plane, for instance by using the SSP option to automatically call HiggsBounds and HiggsSignals during a parameter scan. The results are shown in Fig. 22. One can see that both, the discovery potential and the \(\chi ^2\) value, are very sensitive on small changes in these two parameters. The reason is mainly the large dependence of the masses of the two lightest scalars and their mixing. One sees that the best \(\chi ^2\) value is found close to the \(\tan \beta _X\) range where the SM-like particle is the second lightest CP-even state, and the lightest one is about 80 GeV. In addition, for a very small stripe close to \(\lambda =0\) also points with very light scalars with masses below 40 GeV pass all constraints, but for slightly larger values of \(\lambda \) the mixing already becomes too large and the points are excluded by \(e^+ e^-\rightarrow (h_1)Z \rightarrow (b \bar{b})Z\) from LEP searches.
5.5.3 Large decay width and constraints from vacuum stability
We have already considered the possibility to enhance the total decay width of the CP-odd scalar via decays in pairs of right-sneutrinos. In our tree-level analysis with SARAH we found that one can reduce the mass splitting between the two mass eigenstates by demanding
In addition, as already discussed above, one has to use a negative soft-mass for the sneutrinos,
to get the states light enough. This immediately raises two questions: (i) how large can the total width be for large values of \(Y_x\)? (ii) Is the electroweak vacuum stable or not? First of all, we notice that a negative \(m_\nu ^2\) does not necessarily imply spontaneous R-parity violation, as shown in Ref. [472], in contrast to some claims in this direction in the previous literature. However, the danger of disastrous vacuum decays increases, of course, with decreasing \(m_\nu ^2\). Therefore, we use Vevacious to check the stability of the potential. For this purpose, we have written a second SARAH model file where we include the possibility of VEVs for the right sneutrinos. We also added in this new implementation those mixings among states which were forbidden by R-parity conservation. This is actually necessary because Vevacious calculates the one-loop corrections to the effective potential and the full mass matrices are required. The Vevacious model file is generated via
We can now run a point with SPheno. If we turn on
we can pass the SPheno spectrum file in a second step to Vevacious, which finds all minima of the potential with the additional VEV. If the global minimum is not the local one found by SPheno with correct EWSB, Vevacious uses CosmoTransitions [473] to get the life-time of ‘our’ vacuum. If our survival probability is found to be below 10 %, we label the points as short-lived. Metastable points with a longer life-time are called long-lived. We choose the following set of input parametersFootnote 15
The final result is summarised in Fig. 23.Footnote 16 To maximise the effect on the total width, we take all sneutrinos to be degenerate and with the same coupling to the 750 GeV scalar.
We see that we can get a large total width of the pseudo-scalar for large diagonal entries in \(Y_x\). Up to values of \(Y_x\) of 0.25, which corresponds to a total width of 15 GeV, the vacuum is absolutely stable. One can even reach \(Y_x \sim 0.36\) (\(\Gamma \sim 30\) GeV) before the life-time of the correct vacuum becomes too short. The dependence of the tunnelling time on the value of \(m^2_\nu \) is shown in the middle of Fig. 23. One might wonder how dangerous this vacuum decay is, since spontaneous R-parity violation is not a problem per se. However, we show also in the right plot in Fig. 23 that the electroweak VEV v changes dramatically in the global minimum. Therefore, these points are clearly ruled out.
Even if we cannot reach a width of 45 GeV with the chosen point, we see that the principle idea to enhance the width is working very well. Thus, with a bit more tuning of the parameters, one might even be able to accommodate this value. Furthermore, as the diphoton rate decreases with increasing total width, the relevant couplings or masses would need to be adjusted in order to maintain the explanation of the \(\gamma \gamma \) excess. However, this is beyond the scope of this example. We emphasise that, since the large coupling responsible for the large width is a dimensionful parameter, it will not generate a Landau pole. Thus, the large width hypothesis does not necessarily point to a strongly coupled sector close to the observed resonance.
5.5.4 Dark matter relic density
We have seen in the last section that light sneutrinos are a good possibility in this model to enhance the width of the 750 GeV particle. Of course, it would be interesting to see if they can also be a dark matter candidate. For this purpose, we can implement the model in MicrOmegas to calculate the relic density and to check current limits from direct and indirect detection experiments. In order to implement the model in MicrOmegas, it is sufficient to generate the model files for CalcHep with SARAH via
and copy the generated files into the work/models directory of a new MicrOmegas project. SARAH also writes main files which can be used to run MicrOmegas. For instance, the file CalcOmega.cpp calculates the dark matter relic density and writes the result as well as all important annihilation channels to an external file. This information can then be stored when running a parameter scan. The parameters are easily exchanged between MicrOmegas and a SARAH-based spectrum generator by copying the spectrum file into the main directory of the current MicrOmegas project directory.Footnote 17 However, it is important to remember that MicrOmegas cannot handle complex parameters. Therefore, one has to make sure, even in the case without CP violation, that all rotation matrices of Majorana fermions are real. This can be done by using the following flag for SPheno:
The results from a small scanFootnote 18 are shown in Fig. 24. Here, we have used again the condition of Eq. (5.44) as well as very small deviations from it. One can see that the impact of this small variation on the total width is marginal, but the relic density is clearly affected. Thus, with some tuning of the parameters one can expect that it is possible to explain the dark matter relic density and the total width by light right-handed sneutrinos. However, also finding such a point is again beyond the scope of the example here.
Moreover, there are plenty of other dark matter candidates which mainly correspond to the gauge eigenstates \(\tilde{S}\), \(\tilde{X}\), \(\tilde{\eta }\), \(\tilde{\bar{\eta }}\) beyond the ones from the MSSM. The properties of all of them could be checked with MicrOmegas as well. A detailed discussion of neutralino and sneutrino dark matter in U(1) extensions of the MSSM and different mechanism to obtain the correct abundance was given for instance in Ref. [477].
5.5.5 Flavour constraints
As mentioned above, we decided to include in this model mixing terms between the extra vector-like fermions and the MSSM particles in order to let the new states decay. In this way, we have a safe solution to circumvent any potential cosmological problem. If one assumes the new coupling matrices to have a generic form, i.e. large entries of O(1), including off-diagonal ones as well, they can trigger flavour violation effects. For instance, let us assume that \(Y'_e\) has the following form
with degenerate diagonal entries X, and flavour violating entries \(\alpha \), \(\beta \), \(\gamma \). We can now check how strong the constraints on \(\alpha \), \(\beta \), \(\gamma \) would be for given X. For this purpose, we use the results from SPheno for Br(\(\mu \rightarrow 3e\)), Br(\(\tau \rightarrow 3\mu \)), Br(\(\tau \rightarrow 3e\)), and \(\mu \)–e conversion in Ti and Au, and compare the results with the current experimental limits, see Fig. 25. We find, for instance for \(X=0.1\), that \(\alpha \) must be smaller than \({\sim } 10^{-9}\), while the limits on \(\beta \), \(\gamma \), obtained from \(\tau \) decays, can still be as large as \(O(10^{-6})\).
If other vector-like states mixing with the left-handed quarks or the right-handed down-like quarks are present – as would be the case for instance when assuming 5 or 10-plets of SU(5) – there would also be stringent constraints on their couplings: they would cause tree-level contributions to \(\Delta M_{B_s}\). Since these observables are also calculated by SPheno, one can easily check the limits on models featuring those states.
5.6 \(Z'\) mass limits
So far, we have picked a \(Z'\) mass of at least 2.5 TeV. Of course, we have to check that this is consistent with current exclusion limits. Recent exclusion limits for \(pp \rightarrow Z' \rightarrow e^+ e^-\) have been released by ATLAS using 13 TeV data and 3.2 \(\text {fb}^{-1}\) [482]. To compare the prediction for our model with these numbers, we can use the UFO model files generated by SARAH via
and add them to MadGraph. For this purpose, we copy the SARAH generated files to a subdirectory models/U1xMSSM of the MadGraph installation. Afterwards, we generate all necessary files to calculate the cross section for the process under consideration by running in MadGraph
Note the option -modelname when loading the model. This ensures that MadGraph is using the names for the particles as defined in our model implementation. Using the default names of MadGraph causes naming conflicts because of the extended Higgs sector. One can give the spectrum files written by SPheno as input (param_card.dat) for MadGraph. One just has to make sure that the blocks written for HiggsBounds and HiggsSignals are turned off because the SLHA parser of MadGraph is not able to handle them. This can be done by setting the following flag in the Les Houches input file:
In principle, one could also change the mass directly in the param_card without re-running SPheno for each point. However, the advantage of SPheno is that it calculates the width of the \(Z'\) gauge boson including SUSY and non-SUSY states. This usually has some impact on the obtained limits [471, 483, 484]. We can now scan over \(M_{Z'}\) for fixed values of \(g_X\) and compare the predicted cross section with the exclusion limits. In addition, we can also check the impact of gauge-kinetic mixing: as we have seen, these couplings are negative and can be sizeable. Therefore, we compare the results without gauge kinetic mixing and when setting \(g_{1X} = -\frac{1}{5} g_X\) at the SUSY scale. The results are summarised in Fig. 26. We see that for \(g_X = 0.5\) the limit is about 2.8 TeV without gauge-kinetic mixing. Including kinetic mixing, it gets reduced by about 200 GeV. Thus, one sees that kinetic mixing is not necessarily a small effect. This contradicts some claims that sometimes appear in the literature, where it is often argued that kinetic mixing can be ignored. In particular, we emphasise that this is very relevant when discussing a GUT theory with RGE running over many orders of magnitude in energy scale.
6 Summary
We have given an overview on weakly-coupled renormalisable models proposed to explain the excess observed by ATLAS and CMS around 750 GeV in the diphoton channel. We have pointed out that many of the papers quickly written after the announcement of the excess are based on assumptions and simplifications which are often unjustified and can lead to wrong conclusions. A very common mistake is the lack of inclusion of higher order corrections to the digluon and diphoton decay rates, which results in underestimating the ratio typically by a factor of 2. Several authors assume that the new 750 GeV scalar does not mix with the SM Higgs, which is often not justified. Including such a mixing can give large constraints. These and other problems can be easily avoided by using SARAH and related tools which were created with the purpose of facilitating precision studies of high energy physics models. In particular, the link between SARAH and the spectrum generators FlexibleSUSY and SPheno is a powerful approach to obtain the mass spectrum and all the rotation matrices for any given model without neglecting flavour mixing, complex phases or 1st and 2nd generation Yukawa couplings. Optionally, one can also include all the important radiative corrections up to two loops. In addition, we have improved the functionality of FlexibleSUSY and SPheno to calculate the diphoton and digluon decay widths of neutral scalars, including the higher order QCD corrections up to N\({}^3\)LO. One can now pass on this information directly to Monte-Carlo tools, like CalcHep and MadGraph, by using the appropriate model files generated with SARAH.
In order to study as many models in as much detail as possible, we have created a database of SARAH model files for many of the ideas proposed so far in the literature. The database is also meant to provide many examples in the context of the diphoton excess with which the novel user can try out to familiarise with SARAH, in order to build up the level of expertise needed to implement their own models in the future.
Finally, we have introduced an attractive SUSY model which combines the idea of non-decoupling D-terms with the explanation of the diphoton excess. We have used this as a new example to show how to use SARAH to first understand the model analytically at leading order. As a second step, we have performed a numerical analysis of the important loop corrections to the different masses, checked limits from Higgs searches, neutral gauge bosons searches, and from lepton flavour violation. We have demonstrated that this model could explain a large width of the 750 GeV scalar, but in this context limits from spontaneous R-parity violation become important. These limits can be checked by using the interface to Vevacious.
Notes
Here, the total width is simply the sum of the diphoton and digluon channels ignoring small contributions from other sub-dominant channels.
NLO corrections in the case of a CP-odd vanish in the limit \(m_f \gg m_S\), see Sect. 3.5 for more detail.
This diphoton excess could be triggered by strong interactions. Of course, in this case one cannot use perturbative methods to understand it.
It is straightforward to see that the analytical estimate of the production cross section in Eq. (10) of Ref. [336] is wrong by orders of magnitude: consider the production of a SM-like scalar H with \(m_H={750}\,{\text {GeV}}\) via top-loops. Then, the factor \(h_F^2 m_t^2/m_F^2\) drops out and one obtains \(\sigma =1440\) pb which is too large by roughly three orders of magnitude [422]. The authors of Ref. [304] (which originally made use of this analytic estimate) have revised their results in an updated version of their paper.
This model assumes the 750 GeV boson to be a linear combination of two scalar fields \(\chi _3\) and \(\chi _6\). The quoted couplings arise in the scalar potential as \(\lambda _{H3} |\chi _3|^2 |H|^2 + \lambda _{36} |\chi _3|^2 |\chi _6|^2\), with H being the usual Higgs doublet. The couplings of \(\chi _3\) to vector-like quarks are given by the Yukawa couplings \(f_Y\) and \(f_X\).
Even three-body decays into another scalar and two fermions can be calculated with SPheno.
The MSSM with and without trilinear R-parity violation is disfavoured by the constrains from vacuum stability, see Sect. 2.1.2. In the NMSSM one can explain this excess by assuming the presence of additional final states via four-body decays like \(\Phi \rightarrow (\phi _a)(\phi _a) \rightarrow 4 \gamma \gamma \) [173, 183]. This explanation requires the mass of \(\phi _a\) to be tiny and very close to the pion mass. In the absence of a specific approximate U(1) symmetry (either Peccei-Quinn or R-symmetry), a very delicate fine-tuning would be needed, rendering this possibility less attractive. Another possibility was presented in Ref. [37] where the diphoton signal originates from a parent resonance decay in very finely-tuned parameter regions with a low UV cutoff.
See [458] for a complete discussion of 331 models with generic \(\beta \).
Equation (4.21) assumes that the SU(3) generators are \(T_a = \frac{\lambda _a}{2}\), with \(\lambda _a\) (\(a=1,\ldots ,8\)) the Gell-Mann matrices. However, this is not the convention used in SARAH, see below.
In the original E\(_6\)SSM these states could be either diquark or leptoquark in nature, depending on the choice of a discrete symmetry, but in the model considered here the allowed superpotential terms for the decay of these exotic quarks imply they are diquark.
In the paper proposing this variant to explain the excess [110], the terms involving the surviving Higgs states on the second line are omitted from the superpotential.
The mixing between the MSSM scalars and S is vanishing here only because of our simplifying assumption \(\lambda =0\) but is non-zero in general.
We give for simplicity the results in LaTeXformat. The SARAH internal conventions for the vertices are the following: the results for each vertex are returned as arrays in the format {Particles,{Coeff1,Lor1},{Coeff2,Lor2}}: first, the involved particles with the names for their generation and colour indices are shown and then the coefficients for the different Lorentz structures are given. For the example of a triple scalar vertex, {Coeff2,Lor2} is absent, and Lor1 is just 1. For vertices involving fermions, PL and PR are used for the polarisation operators.
This choice might be a bit unlucky but shows the dangers of the two-loop effective potential calculation: in the gauge-less limit, one of the pseudo-scalars has a tree-level mass close to 0. This causes divergences (‘Goldstone catastrophe’) [474, 475] and makes it necessary to turn off the 2L corrections in SPheno via the flag 7 set to 1.
For this example we had to turn off the thermal corrections to the tunnelling by inserting vcs.ShouldTunnelThermally = False in Vevacious.py because CosmoTransitions failed otherwise to calculate the tunnelling time in the six-dimensional potential.
If the spectrum file is not called SPheno.spc.$MODEL, one can change the file-name by editing the fourth line in func1.mdl written by SARAH.
The relic density calculation for this model can be very time-consuming, especially for the sneutrinos where a large number of co-annihilation channels have to be calculated: the first parameter point might take several hours, all following points should take no longer than seconds, if no new channels are needed.
References
ATLAS Collaboration, M. Aaboud et al., Search for resonances in diphoton events at \(\sqrt{s}\) = 13 TeV with the ATLAS detector. arXiv:1606.03833
CMS Collaboration, V. Khachatryan et al., Search for resonant production of high-mass photon pairs in proton–proton collisions at sqrt(s) = 8 and 13 TeV. arXiv:1606.04093
ATLAS Collaboration, T.A. collaboration, Search for scalar diphoton resonances with 15.4 fb\(^{-1}\) of data collected at \(\sqrt{s}\) = 13 TeV in 2015 and 2016 with the ATLAS detector
CMS Collaboration, C. Collaboration, Search for resonant production of high mass photon pairs using \(12.9\,{\rm fb}^{-1}\) of proton–proton collisions at \(\sqrt{s} = 13\,{\rm TeV}\) and combined interpretation of searches at 8 and 13 TeV
S. Abel, V.V. Khoze, Photo-production of a 750 GeV di-photon resonance mediated by Kaluza–Klein leptons in the loop. arXiv:1601.07167
F. Abu-Ajamieh, R. Houtz, R. Zheng, Interpretation of 750 GeV diphoton resonance within the RS model and the associated radion phenomenology. arXiv:1607.01464
B. Agarwal, J. Isaacson, K.A. Mohan, Minimal dilaton model and the diphoton excess. arXiv:1604.05328
P. Agrawal, J. Fan, B. Heidenreich, M. Reece, M. Strassler, Experimental considerations motivated by the diphoton excess at the LHC. arXiv:1512.05775
A. Ahmed, B.M. Dillon, B. Grzadkowski, J.F. Gunion, Y. Jiang, Higgs-radion interpretation of 750 GeV di-photon excess at the LHC. arXiv:1512.05771
A. Ahriche, G. Faisel, S. Nasri, J. Tandean, Addressing the LHC 750 GeV diphoton excess without new colored states. arXiv:1603.01606
S. Alexander, L. Smolin, Enhanced color gauge invariance and a new di-photon state at the LHC. arXiv:1601.03091
B.C. Allanach, P.S.B. Dev, S.A. Renner, K. Sakurai, Di-photon excess explained by a resonant sneutrino in R-parity violating supersymmetry. arXiv:1512.07645
D. Aloni, K. Blum, A. Dery, A. Efrati, Y. Nir, On a possible large width 750 GeV diphoton resonance at ATLAS and CMS. arXiv:1512.05778
W. Altmannshofer, J. Galloway, S. Gori, A.L. Kagan, A. Martin, J. Zupan, On the 750 GeV di-photon excess. arXiv:1512.07616
E. Alvarez, L. Da Rold, J. Mazzitelli, A. Szynkman, A 750 GeV graviton and the Higgs as a pNGB. arXiv:1606.05326
A. Alves, A.G. Dias, K. Sinha, The 750 GeV \(S\)-cion: where else should we look for it? arXiv:1512.06091
A. Alves, A.G. Dias, K. Sinha, Diphotons at the \(Z\)-pole in models of the 750 GeV resonance decaying to axion-like particles. arXiv:1606.06375
H. An, C. Cheung, Y. Zhang, Broad diphotons from narrow states. arXiv:1512.08378
L.A. Anchordoqui, I. Antoniadis, H. Goldberg, X. Huang, D. Lust, T.R. Taylor, 750 GeV diphotons from closed string states. arXiv:1512.08502
L.A. Anchordoqui, H. Goldberg, X. Huang, Constraints on 750 GeV colorless Q-onia from running couplings. arXiv:1605.01937
L.A. Anchordoqui, I. Antoniadis, H. Goldberg, X. Huang, D. Lust, T.R. Taylor, Update on 750 GeV diphotons from closed string states. Phys. Lett. B 759, 223–228 (2016). arXiv:1603.08294
O. Antipin, M. Mojaza, F. Sannino, A natural Coleman–Weinberg theory explains the diphoton excess. arXiv:1512.06708
O. Antipin, P. Culjak, K. Kumericki, I. Picek, Radiative neutrino models in light of diphoton signals. arXiv:1606.05163
L. Aparicio, A. Azatov, E. Hardy, A. Romanino, Diphotons from diaxions. arXiv:1602.00949
T. Appelquist, J. Ingoldby, M. Piai, J. Thompson, A spartan model for the LHC diphoton excess. arXiv:1606.00865
C. Arbelaez, A.E. Cárcamo Hernández, S. Kovalenko, I. Schmidt, Linking radiative seesaw-type mechanism of fermion masses and non-trivial quark mixing with the 750 GeV diphoton excess. arXiv:1602.03607
B.A. Arbuzov, I.V. Zaitsev, On a possibility of a consistent interpretation of diboson excesses at LHC. arXiv:1602.08293
G. Arcadi, P. Ghosh, Y. Mambrini, M. Pierre, Re-opening dark matter windows compatible with a diphoton excess. arXiv:1603.05601
M.T. Arun, P. Saha, Gravitons in multiply warped scenarios—at 750 GeV and beyond. arXiv:1512.06335
M.T. Arun, D. Choudhury, Bulk gauge and matter fields in nested warping: II. Symmetry breaking and phenomenological consequences. arXiv:1601.02321
J. Ashfaque, L. Delle Rose, A.E. Faraggi, C. Marzo, The LHC di-photon excess and gauge coupling unification in extra \(Z^{\prime }\) heterotic-string derived models. arXiv:1606.01052
J.M. Ashfaque, The observed diphoton excess in F-theory inspired heterotic string-derived model. arXiv:1607.03076
U. Aydemir, T. Mandal, Interpretation of the 750 GeV diphoton excess with colored scalars in \({\bf SO(10)}\) grand unification. arXiv:1601.06761
U. Aydemir, D. Minic, C. Sun, T. Takeuchi, The 750 GeV diphoton excess in unified \(SU(2)_L\times SU(2)_R\times SU(4)\) models from noncommutative geometry. Mod. Phys. Lett. A 31(18), 1650101 (2016). arXiv:1603.01756
M. Backovic, A. Mariotti, D. Redigolo, Di-photon excess illuminates dark matter. arXiv:1512.04917
M. Badziak, Interpreting the 750 GeV diphoton excess in minimal extensions of two-Higgs-doublet models. arXiv:1512.07497
M. Badziak, M. Olechowski, S. Pokorski, K. Sakurai, Interpreting 750 GeV diphoton excess in plain NMSSM. arXiv:1603.02203
K.J. Bae, C.-R. Chen, K. Hamaguchi, I. Low, From the 750 GeV diphoton resonance to multilepton excesses. arXiv:1604.07941
K.J. Bae, K. Hamaguchi, T. Moroi, K. Yanagi, Probing the origin of 750 GeV diphoton excess with the precision measurements at the ILC. Phys. Lett. B 759, 575–582 (2016). arXiv:1604.08307
K.J. Bae, M. Endo, K. Hamaguchi, T. Moroi, Diphoton excess and running couplings. arXiv:1602.03653
S. Baek, J.-H. Park, LHC 750 GeV diphoton excess and muon \((g-2)\). Phys. Lett. B 758, 416–422 (2016). arXiv:1602.05588
Y. Bai, J. Berger, R. Lu, A 750 GeV dark pion: cousin of a dark G-parity-odd WIMP. arXiv:1512.05779
Y. Bai, V. Barger, J. Berger, Color-octet companions of a 750 GeV heavy pion. arXiv:1604.07835
Y. Bai, J. Berger, J. Osborne, B.A. Stefanek, A chiral composite model for the 750 GeV diphoton resonance. arXiv:1605.07183
S. Banerjee, D. Barducci, G. Bélanger, C. Delaunay, Implications of a high-mass diphoton resonance for heavy quark searches. arXiv:1606.09013
P. Baratella, J. Elias-Miro, J. Penedo, A. Romanino, A closer look to the sgoldstino interpretation of the diphoton excess. JHEP 06, 086 (2016). arXiv:1603.05682
R. Barbieri, D. Buttazzo, L.J. Hall, D. Marzocca, Higgs mass and unified gauge coupling in the NMSSM with vector matter. arXiv:1603.00718
D. Bardhan, D. Bhatia, A. Chakraborty, U. Maitra, S. Raychaudhuri, T. Samui, Radion candidate for the LHC diphoton resonance. arXiv:1512.06674
D. Bardhan, P. Byakti, D. Ghosh, T. Sharma, The 750 GeV diphoton resonance as an sgoldstino: a reappraisal. JHEP 06, 129 (2016). arXiv:1603.05251
D. Barducci, A. Goudelis, S. Kulkarni, D. Sengupta, One jet to rule them all: monojet constraints and invisible decays of a 750 GeV diphoton resonance. arXiv:1512.06842
N.D. Barrie, A. Kobakhidze, S. Liang, M. Talia, L. Wu, Heavy leptonium as the origin of the 750 GeV diphoton excess. arXiv:1604.02803
N.D. Barrie, A. Kobakhidze, M. Talia, L. Wu, 750 GeV composite axion as the LHC diphoton resonance. arXiv:1602.00475
M. Bauer, M. Neubert, Flavor anomalies, the diphoton excess and a dark matter candidate. arXiv:1512.06828
M. Bauer, C. Hoerner, M. Neubert, Diphoton resonance from a warped extra dimension. arXiv:1603.05978
M. Bauer, M. Neubert, A. Thamm, The “forgotten” decay \(S\rightarrow Zh\) as a CP analyzer. arXiv:1606.01016
D. Becirevic, E. Bertuzzo, O. Sumensari, R.Z. Funchal, Can the new resonance at LHC be a CP-odd Higgs boson? arXiv:1512.05623
G. Bélanger, C. Delaunay, A dark aector for \(g_\mu -2\), \(R_K\) and a diphoton resonance. arXiv:1603.03333
A. Belhaj, S.-E. Ennadifi, On 750 GeV diphoton resonance in stringy standard-like models. arXiv:1606.02956
B. Bellazzini, R. Franceschini, F. Sala, J. Serra, Goldstones in diphotons. arXiv:1512.05330
A. Belyaev, G. Cacciapaglia, H. Cai, T. Flacke, A. Parolini, H. Serôdio, Singlets in composite Higgs models in light of the LHC di-photon searches. arXiv:1512.07242
K. Benakli, L. Darmé, M.D. Goodsell, J. Harz, The di-photon excess in a perturbative SUSY model. arXiv:1605.05313
R. Benbrik, C.-H. Chen, T. Nomura, Higgs singlet as a diphoton resonance in a vector-like quark model. arXiv:1512.06028
I. Ben-Dayan, R. Brustein, Hypercharge axion and the diphoton \(750\) GeV resonance. arXiv:1601.07564
A. Berlin, The diphoton and diboson excesses in a left-right symmetric theory of dark matter. arXiv:1601.01381
J. Bernon, C. Smith, Could the width of the diphoton anomaly signal a three-body decay? arXiv:1512.06113
J. Bernon, A. Goudelis, S. Kraml, K. Mawatari, D. Sengupta, Characterising the 750 GeV diphoton excess. JHEP 05, 128 (2016). arXiv:1603.03421
L. Berthier, J.M. Cline, W. Shepherd, M. Trott, Effective interpretations of a diphoton excess. arXiv:1512.06799
E. Bertuzzo, P.A.N. Machado, M. Taoso, Di-photon excess in the 2HDM: hasting towards the instability and the non-perturbative regime. arXiv:1601.07508
A. Bharucha, A. Djouadi, A. Goudelis, Threshold enhancement of diphoton resonances. arXiv:1603.04464
S. Bhattacharya, S. Patra, N. Sahoo, N. Sahu, 750 GeV di-photon excess at CERN LHC from a dark sector assisted scalar decay. arXiv:1601.01569
X.-J. Bi, R. Ding, Y. Fan, L. Huang, C. Li, T. Li, S. Raza, X.-C. Wang, B. Zhu, A promising interpretation of diphoton resonance at 750 GeV. arXiv:1512.08497
X.-J. Bi, Q.-F. Xiang, P.-F. Yin, Z.-H. Yu, The 750 GeV diphoton excess at the LHC and dark matter constraints. arXiv:1512.06787
L. Bian, N. Chen, D. Liu, J. Shu, A hidden confining world on the 750 GeV diphoton excess. arXiv:1512.05759
N. Bizot, S. Davidson, M. Frigerio, J.L. Kneur, Two Higgs doublets to explain the excesses \(pp\rightarrow \gamma \gamma (750\ {\rm GeV})\) and \(h \rightarrow \tau ^\pm \mu ^\mp \). arXiv:1512.08508
A. Bolaños, J.L. Diaz-Cruz, G. Hernández-Tomé, G. Tavares-Velasco, Has a Higgs-flavon with a \(750\) GeV mass been detected at the LHC13? arXiv:1604.04822
K. Bondarenko, A. Boyarsky, O. Ruchayskiy, M. Shaposhnikov, Features in the standard model diphoton background. arXiv:1606.09592
C. Bonilla, M. Nebot, R. Srivastava, J.W.F. Valle, Flavor physics scenario for the 750 GeV diphoton anomaly. Phys. Rev. D 93(7), 073009 (2016). arXiv:1602.08092
E.E. Boos, V.E. Bunichev, I.P. Volobuev, Can the 750 GeV diphoton LHC excess be due to a radion-dominated state? arXiv:1603.04495
D. Borah, S. Patra, S. Sahoo, Subdominant left-right scalar dark matter as origin of the 750 GeV di-photon excess at LHC. arXiv:1601.01828
S.M. Boucenna, S. Morisi, A. Vicente, The LHC diphoton resonance from gauge symmetry. arXiv:1512.06878
M.R. Buckley, Wide or narrow? The phenomenology of 750 GeV diphotons. arXiv:1601.04751
D. Buttazzo, A. Greljo, D. Marzocca, Knocking on new physics’ door with a scalar resonance. arXiv:1512.04929
D. Buttazzo, A. Greljo, G. Isidori, D. Marzocca, Toward a coherent solution of diphoton and flavor anomalies. arXiv:1604.03940
C. Cai, Z.-H. Yu, H.-H. Zhang, The 750 GeV diphoton resonance as a singlet scalar in an extra dimensional model. arXiv:1512.08440
X. Calmet, The dual standard model and the 750 GeV events at the LHC. arXiv:1604.06185
J. Cao, F. Wang, Y. Zhang, Interpreting the 750 GeV diphton excess within TopFlavor seesaw model. arXiv:1512.08392
Q.-H. Cao, Y. Liu, K.-P. Xie, B. Yan, D.-M. Zhang, A boost test of anomalous diphoton resonance at the LHC. arXiv:1512.05542
Q.-H. Cao, Y. Liu, K.-P. Xie, B. Yan, D.-M. Zhang, The diphoton excess, low energy theorem and the 331 model. arXiv:1512.08441
J. Cao, C. Han, L. Shang, W. Su, J. M. Yang, Y. Zhang, Interpreting the 750 GeV diphoton excess by the singlet extension of the Manohar–Wise model. arXiv:1512.06728
Q.-H. Cao, S.-L. Chen, P.-H. Gu, Strong CP problem, neutrino masses and the 750 GeV diphoton resonance. arXiv:1512.07541
Q.-H. Cao, Y.-Q. Gong, X. Wang, B. Yan, L.L. Yang, One bump or two peaks? The 750 GeV diphoton excess and dark matter with a complex mediator. arXiv:1601.06374
J. Cao, L. Shang, W. Su, Y. Zhang, J. Zhu, Interpreting the 750 GeV diphoton excess in the minimal dilaton model. arXiv:1601.02570
M. Carena, P. Huang, A. Ismail, I. Low, N.R. Shah, C.E.M. Wagner, A second peak in diphoton (or diboson) resonances. arXiv:1606.06733
A. Carmona, A 750 GeV graviton from holographic composite dark sectors. arXiv:1603.08913
A. Carmona, F. Goertz, A. Papaefstathiou, Uncovering the relation of a di-photon scalar resonance to the Higgs boson. arXiv:1606.02716
L.M. Carpenter, R. Colburn, J. Goodman, Supersoft SUSY models and the 750 GeV diphoton excess, beyond effective operators. arXiv:1512.06107
J.A. Casas, J.R. Espinosa, J.M. Moreno, The 750 GeV diphoton excess as a first light on supersymmetry breaking. arXiv:1512.07895
N. Chakrabarty, B. Mukhopadhyaya, S. SenGupta, Diphoton excess via Chern–Simons interaction in a warped geometry scenario. arXiv:1604.00885
J. Chakrabortty, A. Choudhury, P. Ghosh, S. Mondal, T. Srivastava, Di-photon resonance around 750 GeV: shedding light on the theory underneath. arXiv:1512.05767
S. Chakraborty, A. Chakraborty, S. Raychaudhuri, Diphoton resonance at 750 GeV in the broken MRSSM. arXiv:1512.07527
I. Chakraborty, A. Kundu, Diphoton excess at 750 GeV: singlet scalars confront naturalness. arXiv:1512.06508
M. Chala, M. Duerr, F. Kahlhoefer, K. Schmidt-Hoberg, Tricking Landau–Yang: how to obtain the diphoton excess from a vector resonance. arXiv:1512.06833
M. Chala, C. Grojean, M. Riembau, T. Vantalon, Deciphering the CP nature of the 750 GeV resonance. arXiv:1604.02029
S. Chang, A simple \(U(1)\) gauge rheory explanation of the diphoton excess. arXiv:1512.06426
J. Chang, K. Cheung, C.-T. Lu, Interpreting the 750 GeV di-photon resonance using photon-jets in hidden-valley-like models. arXiv:1512.06671
W. Chao, Neutrino catalyzed diphoton excess. arXiv:1512.08484
W. Chao, Symmetries behind the 750 GeV diphoton excess. arXiv:1512.06297
W. Chao, R. Huo, J.-H. Yu, The minimal scalar-stealth top interpretation of the diphoton excess. arXiv:1512.05738
W. Chao, The diphoton excess inspired electroweak baryogenesis. arXiv:1601.04678
W. Chao, The diphoton excess from an exceptional supersymmetric standard model. arXiv:1601.00633
C.-Y. Chen, M. Lefebvre, M. Pospelov, Y.-M. Zhong, Diphoton excess through dark mediators. arXiv:1603.01256
C.-H. Chen, T. Nomura, 750 GeV diphoton resonance and the implications in an \(SU(2)_1 \times SU(2)_2 \times U(1)_Y\) model. arXiv:1606.03804
K. Cheung, P. Ko, J.S. Lee, J. Park, P.-Y. Tseng, A Higgcision study on the 750 GeV di-photon resonance and 125 GeV SM Higgs boson with the Higgs-singlet mixing. arXiv:1512.07853
C.-W. Chiang, M. Ibe, T.T. Yanagida, Revisiting scalar quark hidden sector in light of 750-GeV diphoton resonance. arXiv:1512.08895
C.-W. Chiang, H. Fukuda, M. Ibe, T.T. Yanagida, 750 GeV diphoton resonance in a visible heavy QCD axion model. Phys. Rev. D 93(9), 095016 (2016). arXiv:1602.07909
C.-W. Chiang, A.-L. Kuo, Can the 750-GeV diphoton resonance be the singlet Higgs boson of custodial Higgs triplet model? arXiv:1601.06394
R.S. Chivukula, A. Farzinnia, K. Mohan, E.H. Simmons, Diphoton resonances in the renormalizable coloron model. arXiv:1604.02157
W.S. Cho, D. Kim, K. Kong, S.H. Lim, K.T. Matchev, J.-C. Park, M. Park, The 750 GeV diphoton excess may not imply a 750 GeV resonance. arXiv:1512.06824
K. Choi, S.H. Im, H. Kim, D.Y. Mo, 750 GeV diphoton resonance and electric dipole moments. arXiv:1605.00206
D. Choudhury, K. Ghosh, The LHC diphoton excess at 750 GeV in the framework of the constrained minimal supersymmetric standard model. arXiv:1605.00013
D. Chway, R. Dermíšek, T.H. Jung, H.D. Kim, Glue to light signal of a new particle. arXiv:1512.08221
J.M. Cline, Z. Liu, LHC diphotons from electroweakly pair-produced composite pseudoscalars. arXiv:1512.06827
T. Cohen, G.D. Kribs, A.E. Nelson, B. Ostdiek, 750 GeV diphotons from supersymmetry with Dirac gauginos. arXiv:1605.04308
J.H. Collins, C. Csaki, J.A. Dror, S. Lombardo, Novel kinematics from a custodially protected diphoton resonance. Phys. Rev. D 93(11), 115001 (2016). arXiv:1603.09350
P. Cox, A.D. Medina, T.S. Ray, A. Spray, Diphoton excess at 750 GeV from a radion in the bulk-Higgs scenario. arXiv:1512.05618
N. Craig, P. Draper, C. Kilic, S. Thomas, How the \(\gamma \gamma \) resonance stole Christmas. arXiv:1512.07733
C. Csaki, J. Hubisz, J. Terning, The minimal model of a diphoton resonance: production without gluon couplings. arXiv:1512.05776
C. Csaki, L. Randall, A diphoton resonance from bulk RS. arXiv:1603.07303
C. Csaki, J. Hubisz, S. Lombardo, J. Terning, Gluon vs. photon production of a 750 GeV diphoton resonance. arXiv:1601.00638
Y. Cui, T. Okui, A. Yunesi, The 750 GeV diphoton casting the first light on baryogenesis. arXiv:1605.08736
D. Curtin, C.B. Verhaaren, Quirky explanations for the diphoton excess. arXiv:1512.05753
M. Cvetič, J. Halverson, P. Langacker, String consistency, heavy exotics, and the \(750\) GeV diphoton excess at the LHC. arXiv:1512.07622
G. Cynolter, J. Kovács, E. Lendvai, Diphoton excess and VV-scattering. Mod. Phys. Lett. A 31 (22), 1650133 (2016). arXiv:1604.01008
M. Dalchenko, B. Dutta, Y. Gao, T. Ghosh, T. Kamon, Exploring the jet multiplicity in the 750 GeV diphoton excess. arXiv:1606.03067
U. Danielsson, R. Enberg, G. Ingelman, T. Mandal, The force awakens—the 750 GeV diphoton excess at the LHC from a varying electromagnetic coupling. arXiv:1601.00624
K. Das, S. K. Rai, The 750 GeV diphoton excess in a \(U(1)\) hidden symmetry model. arXiv:1512.07789
K. Das, T. Li, S. Nandi, S.K. Rai, A new proposal for diphoton resonance from \(E_6\) motivated extra \(U(1)\). arXiv:1607.00810
A. Dasgupta, M. Mitra, D. Borah, Minimal left-right symmetry confronted with the 750 GeV di-photon excess at LHC. arXiv:1512.09202
B. Dasgupta, J. Kopp, P. Schwaller, Photons, photon jets and dark photons at 750 GeV and beyond. arXiv:1602.04692
J.H. Davis, M. Fairbairn, J. Heal, P. Tunney, The significance of the 750 GeV fluctuation in the ATLAS Run 2 diphoton data. arXiv:1601.03153
H. Davoudiasl, C. Zhang, A 750 GeV messenger of dark conformal symmetry breaking. arXiv:1512.07672
H. Davoudiasl, P.P. Giardino, C. Zhang, Exo-Higgs at 750 GeV and genesis of baryons. arXiv:1605.00037
S. Dawson, I.M. Lewis, Singlet model interference effects and a 750 GeV diphoton resonance. arXiv:1605.04944
J. de Blas, J. Santiago, R. Vega-Morales, New vector bosons and the diphoton excess. arXiv:1512.07229
C. Delaunay, Y. Soreq, Probing new physics with isotope shift spectroscopy. arXiv:1602.04838
A. Delgado, M. Garcia-Pepin, M. Quiros, J. Santiago, R. Vega-Morales, Diphoton and diboson probes of fermiophobic Higgs bosons at the LHC. JHEP 06, 042 (2016). arXiv:1603.00962
S.V. Demidov, D.S. Gorbunov, On sgoldstino interpretation of the diphoton excess. arXiv:1512.05723
F.F. Deppisch, S. Kulkarni, H. Päs, E. Schumacher, Leptoquark patterns unifying neutrino masses, flavor anomalies and the diphoton excess. arXiv:1603.07672
F.F. Deppisch, C. Hati, S. Patra, P. Pritimita, U. Sarkar, Implications of the diphoton excess on left-right models and gauge unification. arXiv:1601.00952
F. D’Eramo, J. de Vries, P. Panci, A 750 GeV portal: LHC phenomenology and dark matter candidates. arXiv:1601.01571
V. De Romeri, J.S. Kim, V. Martín-Lozano, K. Rolbiecki, R. Ruiz de Austri, Confronting dark matter with the diphoton excess from a parent resonance decay. Eur. Phys. J. C 76(5), 262 (2016). arXiv:1603.04479
P.S.B. Dev, D. Teresi, Asymmetric dark matter in the Sun and the diphoton excess at the LHC. arXiv:1512.07243
P.S.B. Dev, R.N. Mohapatra, Y. Zhang, Quark seesaw vectorlike fermions and diphoton excess. arXiv:1512.08507
U.K. Dey, S. Mohanty, G. Tomar, 750 GeV resonance in the dark left-right model. arXiv:1512.07212
U.K. Dey, S. Mohanty, G. Tomar, Leptoquarks: 750 GeV diphoton resonance and IceCube events. arXiv:1606.07903
M. Dhuria, G. Goswami, Perturbativity, vacuum stability and inflation in the light of 750 GeV diphoton excess. arXiv:1512.06782
S. Di Chiara, L. Marzola, M. Raidal, First interpretation of the 750 GeV di-photon resonance at the LHC. arXiv:1512.04939
S. Di Chiara, A. Hektor, K. Kannike, L. Marzola, M. Raidal, Large loop-coupling enhancement of a 750 GeV pseudoscalar from a light dark sector. arXiv:1603.07263
A. Di Iura, J. Herrero-Garcia, D. Meloni, Phenomenology of SU(5) low-energy realizations: the diphoton excess and Higgs flavor violation. arXiv:1606.08785
B.M. Dillon, V. Sanz, A little KK graviton at 750 GeV. arXiv:1603.09550
B.M. Dillon, C. Han, H.M. Lee, M. Park, Confronting diphoton resonance with cascade decays in warped gravity. arXiv:1606.07171
L. Di Luzio, J.F. Kamenik, M. Nardecchia, Implications of perturbative unitarity for the \(\gamma \gamma \) resonance at 750 GeV. arXiv:1604.05746
S. Dimopoulos, A. Hook, J. Huang, G. Marques-Tavares, A 750 GeV QCD axion. arXiv:1606.03097
K. Dimopoulos, C. Owen, How the diphoton excess may save minimal hybrid inflation in supergravity. arXiv:1606.06677
R. Ding, L. Huang, T. Li, B. Zhu, Interpreting \(750\) GeV diphoton excess with R-parity violation supersymmetry. arXiv:1512.06560
R. Ding, Z.-L. Han, Y. Liao, X.-D. Ma, Interpretation of 750 GeV diphoton excess at LHC in singlet extension of color-octet neutrino mass model. arXiv:1601.02714
R. Ding, Y. Fan, L. Huang, C. Li, T. Li, S. Raza, B. Zhu, Systematic study of diphoton resonance at 750 GeV from sgoldstino. arXiv:1602.00977
D.N. Dinh, N.A. Ky, P.Q. Vǎn, N.T.H. Vân, A see-saw scenario of an \(A_4\) flavour symmetric standard model. arXiv:1602.07437
A. Djouadi, J. Ellis, J. Quevillon, Interference effects in the decays of 750 GeV states into \(\gamma \gamma \) and \(t\bar{t}\). arXiv:1605.00542
A. Djouadi, J. Ellis, R. Godbole, J. Quevillon, Future collider signatures of the possible 750 GeV state. arXiv:1601.03696
A. Djouadi, A. Pilaftsis, The 750 GeV diphoton resonance in the MSSM. arXiv:1605.01040
B.A. Dobrescu, P.J. Fox, J. Kearney, Multi-step production of a diphoton resonance. arXiv:1605.08772
F. Domingo, S. Heinemeyer, J.S. Kim, K. Rolbiecki, The NMSSM lives: with the 750 GeV diphoton excess. Eur. Phys. J. C 76(5), 249 (2016). arXiv:1602.07691
I. Dorsner, S. Fajfer, N. Kosnik, Is symmetry breaking of SU(5) theory responsible for the diphoton excess? arXiv:1601.03267
P. Draper, D. McKeen, Diphotons, new vacuum angles, and strong CP. arXiv:1602.03604
H.K. Dreiner, M.E. Krauss, B. O’Leary, T. Opferkuch, F. Staub, Validity of the CMSSM interpretation of the diphoton excess. Phys. Rev. D 94 (5), 055013 (2016). arXiv:1606.08811
M. Duerr, P. Fileviez Perez, J. Smirnov, New forces and the 750 GeV resonance. arXiv:1604.05319
B. Dutta, Y. Gao, T. Ghosh, I. Gogoladze, T. Li, Interpretation of the diphoton excess at CMS and ATLAS. arXiv:1512.05439
B. Dutta, Y. Gao, T. Ghosh, I. Gogoladze, T. Li, J.W. Walker, An SU(6) GUT origin of the TeV-scale vector-like particles associated with the 750 GeV diphoton resonance. arXiv:1604.07838
B. Dutta, Y. Gao, T. Ghosh, I. Gogoladze, T. Li, Q. Shafi, J. W. Walker, Diphoton excess in consistent supersymmetric SU(5) models with vector-like particles. arXiv:1601.00866
A. Efrati, J.F. Kamenik, Y. Nir, The phenomenology of the di-photon excess and \(h\rightarrow \tau \mu \) within 2HDM. arXiv:1606.07082
J. Ellis, S.A.R. Ellis, J. Quevillon, V. Sanz, T. You, On the interpretation of a possible \(\sim 750\) GeV particle decaying into \(\gamma \gamma \). arXiv:1512.05327
U. Ellwanger, C. Hugonie, A 750 GeV diphoton signal from a very light pseudoscalar in the NMSSM. arXiv:1602.03344
M. Fabbrichesi, A. Urbano, The breaking of the \(SU(2)_L\times U(1)_Y\) symmetry: the 750 GeV resonance at the LHC and perturbative unitarity. arXiv:1601.02447
M. Fabbrichesi, M. Pinamonti, A. Urbano, Telling the spin of the di-photon resonance. arXiv:1604.06948
A. Falkowski, O. Slone, T. Volansky, Phenomenology of a 750 GeV singlet. arXiv:1512.05777
A. Falkowski, J.F. Kamenik, Di-photon portal to warped gravity. arXiv:1603.06980
A.E. Faraggi, J. Rizos, The 750 GeV diphoton LHC excess and Extra Z’s in heterotic-string derived models. arXiv:1601.03604
T.-F. Feng, X.-Q. Li, H.-B. Zhang, S.-M. Zhao, The LHC 750 GeV diphoton excess in supersymmetry with gauged baryon and lepton numbers. arXiv:1512.06696
S. Fichet, G. von Gersdorff, C. Royon, Scattering light by light at 750 GeV at the LHC. arXiv:1512.05751
S. Fichet, G. von Gersdorff, C. Royon, Measuring the diphoton coupling of a 750 GeV resonance. arXiv:1601.01712
R. Franceschini, G.F. Giudice, J.F. Kamenik, M. McCullough, A. Pomarol, R. Rattazzi, M. Redi, F. Riva, A. Strumia, R. Torre, What is the gamma gamma resonance at 750 GeV? arXiv:1512.04933
R. Franceschini, G.F. Giudice, J.F. Kamenik, M. McCullough, F. Riva, A. Strumia, R. Torre, Digamma, what next? arXiv:1604.06446
M.T. Frandsen, I.M. Shoemaker, Asymmetric dark matter models and the LHC diphoton excess. JCAP 1605(05), 064 (2016). arXiv:1603.09354
D.B. Franzosi, M.T. Frandsen, Symmetries and composite dynamics for the 750 GeV diphoton excess. arXiv:1601.05357
C.D. Froggatt, H.B. Nielsen, Production and decay of 750 Gev state of 6 top and 6 anti top quarks. arXiv:1605.03909
C. Frugiuele, E. Fuchs, G. Perez, M. Schlaffer, Atomic probes of new physics. arXiv:1602.04822
H. Fukuda, M. Ibe, O. Jinnouchi, M. Nojiri, Cracking down on fake photons—a case of 750 GeV diphoton resonance. arXiv:1607.01936
E. Gabrielli, K. Kannike, B. Mele, M. Raidal, C. Spethmann, H. Veermäe, A SUSY inspired simplified model for the 750 GeV diphoton excess. arXiv:1512.05961
J. Gao, H. Zhang, H.X. Zhu, Diphoton excess at 750 GeV: gluon–gluon fusion or quark-antiquark annihilation? arXiv:1512.08478
C. Garcia-Cely, J. Heeck, Polynomial spectral features from dark matter and connection to the diphoton resonance. arXiv:1605.08049
S.-F. Ge, H.-J. He, J. Ren, Z.-Z. Xianyu, Realizing dark matter and Higgs inflation in light of LHC diphoton excess. arXiv:1602.01801
C.-Q. Geng, D. Huang, Note on Spin-2 particle interpretation of the 750 GeV diphoton excess. arXiv:1601.07385
H. Georgi, Y. Nakai, Diboson excess from a new strong force. arXiv:1606.05865
D.M. Ghilencea, H.M. Lee, Diphoton resonance at 750 GeV in supersymmetry. arXiv:1606.04131
T. Gherghetta, N. Nagata, M. Shifman, A visible QCD axion from an enlarged color group. Phys. Rev. D 93(11), 115010 (2016). arXiv:1604.01127
K. Ghorbani, H. Ghorbani, The 750 GeV diphoton excess from a pseudoscalar in fermionic dark matter scenario. arXiv:1601.00602
K. Ghosh, S. Jana, S. Nandi, Neutrino mass generation and 750 GeV diphoton excess via photon–photon fusion at the large hadron collider. arXiv:1607.01910
A. Ghoshal, On electroweak phase transition and di-photon excess with a 750 GeV scalar resonance. arXiv:1601.04291
S.B. Giddings, H. Zhang, Kaluza–Klein graviton phenomenology for warped compactifications, and the 750 GeV diphoton excess. arXiv:1602.02793
S.I. Godunov, A.N. Rozanov, M.I. Vysotsky, E.V. Zhemchugov, New physics at 1 TeV? arXiv:1602.02380
F. Goertz, J.F. Kamenik, A. Katz, M. Nardecchia, Indirect constraints on the scalar di-photon resonance at the LHC. arXiv:1512.08500
S. Gopalakrishna, T.S. Mukherjee, The 750 GeV diphoton excess in a two Higgs doublet model and a singlet scalar model, with vector-like fermions, unitarity constraints, and dark matter implications. arXiv:1604.05774
B. Gripaios, M. Nardecchia, T. You, An anomalous composite Higgs and the 750 GeV di-gamma resonance. arXiv:1605.09647
B. Gripaios, D. Sutherland, An operator basis for the standard model with an added scalar singlet. arXiv:1604.07365
C. Gross, O. Lebedev, J.M. No, Drell–Yan constraints on new electroweak states and the di-photon anomaly. arXiv:1602.03877
J. Gu, Z. Liu, Running after diphoton. arXiv:1512.07624
Y.-C. Guo, C.-X. Yue, Z.-H. Zhao, 750 GeV diphoton excess confronted with a top-pion in the TTM model. Int. J. Mod. Phys. A 31(16), 1650086 (2016). arXiv:1605.08183
R.S. Gupta, S. Jaeger, Y. Kats, G. Perez, E. Stamou, Interpreting a 750 GeV diphoton resonance. arXiv:1512.05332
L.J. Hall, K. Harigaya, Y. Nomura, 750 GeV diphotons: implications for supersymmetric unification. arXiv:1512.07904
L.J. Hall, K. Harigaya, Y. Nomura, 750 GeV diphotons: implications for supersymmetric unification II. arXiv:1605.03585
Y. Hamada, T. Noumi, S. Sun, G. Shiu, An O(750) GeV resonance and inflation. arXiv:1512.08984
Y. Hamada, H. Kawai, K. Kawana, K. Tsumura, Models of LHC diphoton excesses valid up to the Planck scale. arXiv:1602.04170
K. Hamaguchi, S.P. Liew, Models of 750 GeV quarkonium and the LHC excesses. arXiv:1604.07828
C. Han, H.M. Lee, M. Park, V. Sanz, The diphoton resonance as a gravity mediator of dark matter. arXiv:1512.06376
H. Han, S. Wang, S. Zheng, Scalar explanation of diphoton excess at LHC. arXiv:1512.06562
X.-F. Han, L. Wang, Implication of the 750 GeV diphoton resonance on two-Higgs-doublet model and its extensions with Higgs field. arXiv:1512.06587
H. Han, S. Wang, S. Zheng, Dark matter theories in the light of diphoton excess. arXiv:1512.07992
X.-F. Han, L. Wang, L. Wu, J.M. Yang, M. Zhang, Explaining 750 GeV diphoton excess from top/bottom partner cascade decay in two-Higgs-doublet model extension. arXiv:1601.00534
X.-F. Han, L. Wang, J.M. Yang, An extension of two-Higgs-doublet model and the excesses of 750 GeV diphoton, muon g-2 and \(h\rightarrow \mu \tau \). arXiv:1601.04954
C. Han, T.T. Yanagida, N. Yokozaki, Implications of the 750 GeV diphoton excess in gaugino mediation. arXiv:1602.04204
C. Han, K. Ichikawa, S. Matsumoto, M.M. Nojiri, M. Takeuchi, Heavy fermion bound states for diphoton excess at 750 GeV—collider and cosmological constraints. JHEP 04, 159 (2016). arXiv:1602.08100
K. Harigaya, Y. Nomura, Composite models for the 750 GeV diphoton excess. arXiv:1512.04850
K. Harigaya, Y. Nomura, Hidden pion varieties in composite models for diphoton resonances. arXiv:1603.05774
K. Harigaya, Y. Nomura, A composite model for the 750 GeV diphoton excess. arXiv:1602.01092
L.A. Harland-Lang, V.A. Khoze, M.G. Ryskin, The photon PDF in events with rapidity gaps. arXiv:1601.03772
L.A. Harland-Lang, V.A. Khoze, M.G. Ryskin, The production of a diphoton resonance via photon–photon fusion. arXiv:1601.07187
C. Hati, Explaining the diphoton excess in alternative left-right symmetric model. arXiv:1601.02457
M. He, X.-G. He, Y. Tang, A \(\gamma \gamma \) collider for the 750 GeV resonant state. Phys. Lett. B 759, 166–170 (2016). arXiv:1603.00287
J.J. Heckman, 750 GeV diphotons from a D3-brane. arXiv:1512.06773
A. Hektor, L. Marzola, Di-photon excess at LHC and the gamma ray excess at the Galactic Centre. arXiv:1602.00004
A.E. Cárcamo Hernández, The 750 GeV diphoton resonance can cause the SM fermion mass and mixing pattern. arXiv:1512.09092
A.E. Cárcamo Hernández, I. Nisandzic, LHC diphoton 750 GeV resonance as an indication of \(SU(3)_c\times SU(3)_L\times U(1)_X\) gauge symmetry. arXiv:1512.07165
A.E. Cárcamo Hernández, I. de Medeiros Varzielas, E. Schumacher, The \(750\,\text{GeV}\) diphoton resonance in the light of a 2HDM with \(S_3\) flavour symmetry. arXiv:1601.00661
J.L. Hewett, T.G. Rizzo, 750 GeV diphoton resonance in warped geometries. arXiv:1603.08250
T. Higaki, K.S. Jeong, N. Kitajima, F. Takahashi, The QCD axion from aligned axions and diphoton excess. arXiv:1512.05295
B. Holdom, M. Ratzlaff, Heavy Higgs decay to \(t\bar{t} Z\) and constraints on a \(750\) GeV pseudoscalar. arXiv:1605.08411
K. Howe, S. Knapen, D.J. Robinson, Diphotons from an electroweak triplet-singlet. arXiv:1603.08932
F.P. Huang, C.S. Li, Z.L. Liu, Y. Wang, 750 GeV diphoton excess from cascade decay. arXiv:1512.06732
W.-C. Huang, Y.-L.S. Tsai, T.-C. Yuan, Gauged two Higgs doublet model confronts the LHC 750 GeV di-photon anomaly. arXiv:1512.07268
X.-J. Huang, W.-H. Zhang, Y.-F. Zhou, A 750 GeV dark matter messenger at the Galactic Center. arXiv:1512.08992
X.-J. Huang, W.-H. Zhang, Y.-F. Zhou, Connecting the LHC diphoton excess to the Galatic center gamma-ray excess. arXiv:1605.09018
D.T. Huong, P.V. Dong, Left-right asymmetry and 750 GeV diphoton excess. Phys. Rev. D 93(9), 095019 (2016). arXiv:1603.05146
L.E. Ibanez, V. Martin-Lozano, A megaxion at 750 GeV as a first hint of low scale string theory. arXiv:1512.08777
H. Ito, T. Moroi, Y. Takaesu, Studying 750 GeV di-photon resonance at photon–photon collider. arXiv:1601.01144
S. Iwamoto, G. Lee, Y. Shadmi, R. Ziegler, Diphoton signals from colorless hidden quarkonia. arXiv:1604.07776
S. Jain, F. Margaroli, S. Moretti, L. Panizzi, The 750 GeV threshold to a new particle world. arXiv:1605.08741
Y. Jiang, Y.-Y. Li, T. Liu, 750 GeV resonance in the gauged \(U(1)^{\prime }\)-extended MSSM. arXiv:1512.09127
Y. Jiang, L. Li, R. Zheng, Boosted scalar confronting 750 GeV di-photon excess. arXiv:1605.01898
S. Jung, J. Song, Y.W. Yoon, How resonance-continuum interference changes 750 GeV diphoton excess: signal enhancement and peak shift. arXiv:1601.00006
J.F. Kamenik, M. Redi, Back to 1974: the \({\cal Q}\)-onium. arXiv:1603.07719
J.F. Kamenik, B.R. Safdi, Y. Soreq, J. Zupan, Comments on the diphoton excess: critical reappraisal of effective field theory interpretations. arXiv:1603.06566
A.Y. Kamenshchik, A.A. Starobinsky, A. Tronconi, G.P. Vacca, G. Venturi, Vacuum energy, standard model physics and the \(750\; {\rm GeV}\) diphoton excess at the LHC. arXiv:1604.02371
S. Kanemura, K. Nishiwaki, H. Okada, Y. Orikasa, S.C. Park, R. Watanabe, LHC 750 GeV diphoton excess in a radiative seesaw model. arXiv:1512.09048
S. Kanemura, N. Machida, S. Odori, T. Shindou, Diphoton excess at 750 GeV in an extended scalar sector. arXiv:1512.09053
K. Kaneta, S. Kang, H.-S. Lee, Diphoton excess at the LHC Run 2 and its implications for a new heavy gauge boson. arXiv:1512.09129
S.K. Kang, J. Song, Top-phobic heavy Higgs boson as the 750 GeV diphoton resonance. arXiv:1512.08963
K. Kannike, G.M. Pelaggi, A. Salvio, A. Strumia, The diphoton as the Higgs of the Higgs. arXiv:1605.08681
A. Karozas, S.F. King, G.K. Leontaris, A.K. Meadowcroft, Diphoton excess from \(E_6\) in F-theory GUTs. arXiv:1601.00640
B.J. Kavanagh, Re-examining the significance of the 750 GeV diphoton excess at ATLAS. arXiv:1601.07330
J. Kawamura, Y. Omura, Diphoton excess at 750 GeV and LHC constraints in models with vector-like particles. arXiv:1601.07396
J.S. Kim, K. Rolbiecki, R.R. de Austri, Model-independent combination of diphoton constraints at 750 GeV. arXiv:1512.06797
J.S. Kim, J. Reuter, K. Rolbiecki, R.R. de Austri, A resonance without resonance: scrutinizing the diphoton excess at 750 GeV. arXiv:1512.06083
J.E. Kim, Is an axizilla possible for di-photon resonance? arXiv:1512.08467
S.F. King, R. Nevzorov, 750 GeV diphoton resonance from singlets in an exceptional supersymmetric standard model. arXiv:1601.07242
S. Knapen, T. Melia, M. Papucci, K. Zurek, Rays of light from the LHC. arXiv:1512.04928
P. Ko, Y. Omura, C. Yu, Diphoton excess at 750 GeV in leptophobic U(1)\(^\prime \) model inspired by \(E_6\) GUT. arXiv:1601.00586
P. Ko, C. Yu, T.-C. Yuan, 750 GeV diphoton excess as a composite (pseudo)scalar boson from new strong interaction. arXiv:1603.08802
P. Ko, T. Nomura, H. Okada, Y. Orikasa, Confronting a new three-loop seesaw model with the 750 GeV diphoton excess. arXiv:1602.07214
P. Ko, T. Nomura, Dark sector shining through 750 GeV dark Higgs boson at the LHC. arXiv:1601.02490
A. Kobakhidze, F. Wang, L. Wu, J.M. Yang, M. Zhang, LHC 750 GeV diphoton resonance explained as a heavy scalar in top-seesaw model. arXiv:1512.05585
M.E. Krauss, T. Opferkuch, F. Staub, M.W. Winkler, Soft Gamma Rays from Heavy WIMPs. Phys. Dark Univ. 14, 29–34 (2016). arXiv:1605.05327
K. Kulkarni, Extension of \(\nu \) MSM model and possible explanations of recent astronomical and collider observations. arXiv:1512.06836
A. Kusenko, L. Pearce, L. Yang, Leptogenesis via the 750 GeV pseudoscalar. Phys. Rev. D 93(11), 115005 (2016). arXiv:1604.02382
L.V. Laperashvili, H.B. Nielsen, C.R. Das, New results at LHC confirming the vacuum stability and multiple point principle. arXiv:1601.03231
G. Lazarides, Q. Shafi, Diphoton resonances in a U(1)\(_{B-L}\) extension of the minimal supersymmetric standard model. Phys. Rev. D 93(11), 111702 (2016). arXiv:1602.07866
P. Lebiedowicz, M. Luszczak, R. Pasechnik, A. Szczurek, Can the diphoton enhancement at 750 GeV be due to a neutral technipion? arXiv:1604.02037
G.K. Leontaris, Q. Shafi, Diphoton resonance in F-theory inspired flipped SO(10). arXiv:1603.06962
G. Li, Y.-N. Mao, Y.-L. Tang, C. Zhang, Y. Zhou, S.-H. Zhu, A Pseudoscalar Decaying Only via Loops as an Explanation for the 750 GeV Diphoton Excess. Phys. Rev. Lett. 116(15), 151803 (2016). arXiv:1512.08255
W. Li, J.N. Ng, Doubly charged vector leptons and the 750 GeV diphoton resonance. arXiv:1605.04885
T. Li, J.A. Maxin, V.E. Mayes, D.V. Nanopoulos, The \(750\) GeV diphoton excesses in a realistic D-brane model. arXiv:1602.09099
T. Li, J.A. Maxin, V.E. Mayes, D.V. Nanopoulos, A flippon related singlet at the LHC II. arXiv:1602.01377
W. Liao, H.-Q. Zheng, Scalar resonance at 750 GeV as composite of heavy vector-like fermions. Commun. Theor. Phys. 66(2), 219–223 (2016). arXiv:1512.06741
J. Liu, X.-P. Wang, W. Xue, LHC diphoton excess from colorful resonances. arXiv:1512.07885
N. Liu, W. Wang, M. Zhang, R. Zheng, 750 GeV diphoton resonance in a vector-like extension of Hill model. arXiv:1604.00728
X. Liu, H. Zhang, RG-improved prediction for 750 GeV resonance production at the LHC. arXiv:1603.07190
M. Low, A. Tesi, L.-T. Wang, A pseudoscalar decaying to photon pairs in the early LHC run 2 data. arXiv:1512.05328
I. Low, J. Lykken, Implications of gauge invariance on a heavy diphoton resonance. arXiv:1512.09089
M.-X. Luo, K. Wang, T. Xu, L. Zhang, G. Zhu, Squarkonium, diquarkonium and octetonium at the LHC and their di-photon decays. Phys. Rev. D 93(5), 055042 (2016). arXiv:1512.06670
E. Ma, N. Pollard, O. Popov, M. Zakeri, Gauge \(B-L\) model of radiative neutrino mass with multipartite dark matter. arXiv:1605.00991
Y. Mambrini, G. Arcadi, A. Djouadi, The LHC diphoton resonance and dark matter. arXiv:1512.04913
S.F. Mantilla, R. Martinez, F. Ochoa, C.F. Sierra, Diphoton decay for a 750 GeV scalar boson in a \(SU(6)\otimes U(1)_{X}\) model. arXiv:1602.05216
A.D. Martin, M.G. Ryskin, Advantages of exclusive \(\gamma \gamma \) production to probe high mass systems. arXiv:1601.07774
R. Martinez, F. Ochoa, C.F. Sierra, Diphoton decay for a \(750\) GeV scalar boson in an \(U(1)^{\prime }\) model. arXiv:1512.05617
R. Martinez, F. Ochoa, C.F. Sierra, \(\varvec {SU(3)_{C}}\otimes \varvec {SU(3)_{L}}\otimes \varvec {U(1)_{X}}\) models in view of the \(750\) GeV diphoton signal. arXiv:1606.03415
A. Martini, K. Mawatari, D. Sengupta, Diphoton excess in phenomenological spin-2 resonance scenarios. arXiv:1601.05729
L. Marzola, A. Racioppi, M. Raidal, F.R. Urban, H. Veermäe, Non-minimal CW inflation, electroweak symmetry breaking and the 750 GeV anomaly. arXiv:1512.09136
S. Matsuzaki, K. Yamawaki, 750 GeV diphoton signal from one-family walking technipion. arXiv:1512.05564
S. Matsuzaki, K. Yamawaki, Walking from 750 GeV to 950 GeV in the technipion zoo. Phys. Rev. D 93(11), 115027 (2016). arXiv:1605.04667
S.D. McDermott, P. Meade, H. Ramani, Singlet scalar resonances and the diphoton excess. arXiv:1512.05326
J. McDonald, The 750 GeV resonance as non-minimally coupled inflaton: unitarity violation and why the resonance is a real singlet scalar. arXiv:1604.01711
E. Megias, O. Pujolas, M. Quiros, On dilatons and the LHC diphoton excess. arXiv:1512.06106
T. Modak, S. Sadhukhan, R. Srivastava, 750 GeV diphoton excess from gauged \(B-L\) symmetry. arXiv:1601.00836
J.W. Moffat, A composite model of quarks and bosons and the 750 GeV resonance. arXiv:1605.09404
E. Molinaro, F. Sannino, N. Vignaroli, Minimal composite dynamics versus axion origin of the diphoton excess. arXiv:1512.05334
E. Molinaro, F. Sannino, N. Vignaroli, Collider tests of (composite) diphoton resonances. arXiv:1602.07574
S. Moretti, K. Yagyu, The 750 GeV diphoton excess and its explanation in 2-Higgs doublet models with a real inert scalar multiplet. arXiv:1512.07462
E. Morgante, D. Racco, M. Rameez, A. Riotto, The 750 GeV diphoton excess, dark matter and constraints from the IceCube experiment. arXiv:1603.05592
C.W. Murphy, Vector leptoquarks and the 750 GeV diphoton resonance at the LHC. arXiv:1512.06976
Y. Nakai, R. Sato, K. Tobioka, Footprints of new strong dynamics via anomaly. arXiv:1512.04924
R. Nevzorov, A.W. Thomas, Diphoton signature of neutral pseudo-Goldstone boson in the E6CHM at the CERN LHC. arXiv:1605.07313
H.P. Nilles, M.W. Winkler, 750 GeV diphotons and supersymmetric grand unification. JHEP 05, 182 (2016). arXiv:1604.03598
J.M. No, V. Sanz, J. Setford, See-saw composite Higgses at the LHC: linking Naturalness to the \(750\) GeV di-photon resonance. arXiv:1512.05700
J.M. No, Is it \(SU(2)_{\rm L}\) or just \(U(1)_{\rm Y}\)? \(750\) GeV di-photon probes of the electroweak nature of new states. arXiv:1605.05900
T. Nomura, H. Okada, Four-loop neutrino model Inspired by diphoton excess at 750 GeV. arXiv:1601.00386
T. Nomura, H. Okada, Generalized Zee-Babu model with 750 GeV diphoton resonance. arXiv:1601.07339
T. Nomura, H. Okada, Four-loop radiative seesaw model with 750 GeV diphoton resonance. arXiv:1601.04516
H. Okada, K. Yagyu, Renormalizable model for neutrino mass, dark matter, muon \(g-2\) and 750 GeV diphoton excess. arXiv:1601.05038
D. Palle, On the possible new 750 GeV heavy boson resonance at the LHC. arXiv:1601.00618
E. Palti, Vector-like exotics in F-theory and 750 GeV diphotons. arXiv:1601.00285
J.-C. Park, S. C. Park, Indirect signature of dark matter with the diphoton resonance at 750 GeV. arXiv:1512.08117
K.M. Patel, P. Sharma, Interpreting 750 GeV diphoton excess in SU(5) grand unified theory. arXiv:1512.07468
G.M. Pelaggi, A. Strumia, E. Vigiani, Trinification can explain the di-photon and di-boson LHC anomalies. arXiv:1512.07225
M. Perelstein, Y.-D. Tsai, 750 GeV di-photon excess and strongly first-order electroweak phase transition. arXiv:1603.04488
C. Petersson, R. Torre, The 750 GeV diphoton excess from the goldstino superpartner. arXiv:1512.05333
A. Pilaftsis, Diphoton signatures from heavy axion decays at LHC. arXiv:1512.04931
C.T. Potter, Pseudoscalar gluinonia to diphotons at the LHC. arXiv:1601.00240
M. Redi, A. Strumia, A. Tesi, E. Vigiani, Di-photon resonance and dark matter as heavy pions. JHEP 05, 078 (2016). arXiv:1602.07297
J. Ren, J.-H. Yu, SU(2) x SU(2) x U(1) interpretation on the 750 GeV diphoton excess. arXiv:1602.07708
P. Roig, J.J. Sanz-Cillero, A broad 750 GeV diphoton resonance? Not alone. arXiv:1605.03831
I. Sahin, Semi-elastic cross section for a scalar resonance of mass 750 GeV. arXiv:1601.01676
A. Salvio, A. Mazumdar, Higgs stability and the 750 GeV diphoton excess. arXiv:1512.08184
A. Salvio, F. Staub, A. Strumia, A. Urbano, On the maximal diphoton width. arXiv:1602.01460
V. Sanz, Theoretical interpretation of a spin-two diphoton excess. arXiv:1603.05574
R. Sato, K. Tobioka, LHC future prospects of the 750 GeV resonance. arXiv:1605.05366
M. Son, A. Urbano, A new scalar resonance at 750 GeV: towards a proof of concept in favor of strongly interacting theories. arXiv:1512.08307
D. Stolarski, R. Vega-Morales, The virtual diphoton excess. arXiv:1601.02004
A. Strumia, Interpreting the 750 GeV digamma excess: a review (2016). arXiv:1605.09401
F. Takahashi, M. Yamada, N. Yokozaki, Diphoton excess from hidden U(1) gauge symmetry with large kinetic mixing. arXiv:1604.07145
Y.-L. Tang, S.-H. Zhu, NMSSM extended with vector-like particles and the diphoton excess on the LHC. arXiv:1512.08323
Y. Tsai, L.-T. Wang, Y. Zhao, Faking the diphoton excess by displaced dark photon decays. arXiv:1603.00024
F. Wang, L. Wu, J.M. Yang, M. Zhang, 750 GeV diphoton resonance, 125 GeV Higgs and muon g-2 anomaly in deflected anomaly mediation SUSY breaking scenario. arXiv:1512.06715
F. Wang, W. Wang, L. Wu, J.M. Yang, M. Zhang, Interpreting 750 GeV diphoton resonance in the NMSSM with vector-like particles. arXiv:1512.08434
M. Yamada, T.T. Yanagida, K. Yonekura, Diphoton excess as a hidden monopole. Phys. Lett. B 759, 459–463 (2016). arXiv:1604.07203
J.-H. Yu, Hidden gauged U(1) model: unifying scotogenic neutrino and flavor dark matter. arXiv:1601.02609
J. Zhang, S. Zhou, Electroweak vacuum stability and diphoton excess at 750 GeV. arXiv:1512.07889
H. Zhang, The 750 GeV diphoton excess: who introduces it? arXiv:1601.01355
Y.-J. Zhang, B.-B. Zhou, J.-J. Sun, The fourth generation quark and the 750 GeV diphoton excess. arXiv:1602.05539
C. Zhang, M.-Y. Cui, L. Feng, Y.-Z. Fan, Z.-Z. Ren, \(\gamma \)-ray line signal in the 750 GeV-diphoton-excess-motivated gravity mediator dark matter model. arXiv:1606.00557
A. Angelescu, A. Djouadi, G. Moreau, Scenarii for interpretations of the LHC diphoton excess: two Higgs doublets and vector-like quarks and leptons. arXiv:1512.04921
F. Staub, SARAH. arXiv:0806.0538
F. Staub, From superpotential to model files for FeynArts and CalcHep/CompHep. Comput. Phys. Commun. 181, 1077–1086 (2010). arXiv:0909.2863
F. Staub, Automatic calculation of supersymmetric renormalization group equations and self energies. Comput. Phys. Commun. 182, 808–833 (2011). arXiv:1002.0840
F. Staub, SARAH 3.2: Dirac gauginos, UFO output, and more. Comput. Phys. Commun. 184, 1792–1809 (2013). arXiv:1207.0906
F. Staub, SARAH 4: a tool for (not only SUSY) model builders. Comput. Phys. Commun. 185, 1773–1790 (2014). arXiv:1309.7223
F. Staub, Exploring new models in all detail with SARAH. Adv. High Energy Phys. 2015, 840780 (2015). arXiv:1503.04200
W. Porod, SPheno, a program for calculating supersymmetric spectra, SUSY particle decays and SUSY particle production at e+ e\(-\) colliders. Comput. Phys. Commun. 153, 275–315 (2003). arXiv:hep-ph/0301101
W. Porod, F. Staub, SPheno 3.1: extensions including flavour, CP-phases and models beyond the MSSM. Comput. Phys. Commun. 183, 2458–2469 (2012). arXiv:1104.1573
P. Athron, J.-H. Park, D. Stöckinger, A. Voigt, FlexibleSUSY—a spectrum generator generator for supersymmetric models. Comput. Phys. Commun. 190, 139–172 (2015). arXiv:1406.2319
F. Staub, P. Athron, U. Ellwanger, R. Grober, M. Muhlleitner, P. Slavich, A. Voigt, Higgs mass predictions of public NMSSM spectrum generators. arXiv:1507.05093
G. Degrassi, P. Slavich, F. Zwirner, On the neutral Higgs boson masses in the MSSM for arbitrary stop mixing. Nucl. Phys. B 611, 403–422 (2001). arXiv:hep-ph/0105096
A. Brignole, G. Degrassi, P. Slavich, F. Zwirner, On the O(alpha(t)**2) two loop corrections to the neutral Higgs boson masses in the MSSM. Nucl. Phys. B 631, 195–218 (2002). arXiv:hep-ph/0112177
A. Dedes, P. Slavich, Two loop corrections to radiative electroweak symmetry breaking in the MSSM. Nucl. Phys. B 657, 333–354 (2003). arXiv:hep-ph/0212132
A. Brignole, G. Degrassi, P. Slavich, F. Zwirner, On the two loop sbottom corrections to the neutral Higgs boson masses in the MSSM. Nucl. Phys. B 643, 79–92 (2002). arXiv:hep-ph/0206101
A. Dedes, G. Degrassi, P. Slavich, On the two loop Yukawa corrections to the MSSM Higgs boson masses at large tan beta. Nucl. Phys. B 672, 144–162 (2003). arXiv:hep-ph/0305127
G. Degrassi, P. Slavich, On the radiative corrections to the neutral Higgs boson masses in the NMSSM. Nucl. Phys. B 825, 119–150 (2010). arXiv:0907.4682
M.D. Goodsell, K. Nickel, F. Staub, Two-loop Higgs mass calculations in supersymmetric models beyond the MSSM with SARAH and SPheno. Eur. Phys. J. C 75(1), 32 (2015). arXiv:1411.0675
M. Goodsell, K. Nickel, F. Staub, Generic two-loop Higgs mass calculation from a diagrammatic approach. Eur. Phys. J. C 75(6), 290 (2015). arXiv:1503.03098
A. Pukhov, CalcHEP 2.3: MSSM, structure functions, event generation, batchs, and generation of matrix elements for other packages. arXiv:hep-ph/0412191
E.E. Boos, M.N. Dubinin, V.A. Ilyin, A.E. Pukhov, V.I. Savrin, CompHEP: specialized package for automatic calculations of elementary particle decays and collisions (1994). arXiv:hep-ph/9503280
J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer, T. Stelzer, MadGraph 5: going beyond. JHEP 06, 128 (2011). arXiv:1106.0522
J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H.S. Shao, T. Stelzer, P. Torrielli, M. Zaro, The automated computation of tree-level and next-to-leading order differential cross sections, and their matching to parton shower simulations. JHEP 07, 079 (2014). arXiv:1405.0301
G. Belanger, F. Boudjema, A. Pukhov, micrOMEGAs: a code for the calculation of dark matter properties in generic models of particle interaction, in The Dark Secrets of the Terascale, pp. 739–790 (2013). arXiv:1402.0787
P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, K.E. Williams, HiggsBounds: confronting arbitrary Higgs sectors with exclusion bounds from LEP and the Tevatron. Comput. Phys. Commun. 181, 138–167 (2010). arXiv:0811.4169
P. Bechtle, O. Brein, S. Heinemeyer, G. Weiglein, K.E. Williams, HiggsBounds 2.0.0: confronting neutral and charged Higgs sector predictions with exclusion bounds from LEP and the Tevatron. Comput. Phys. Commun. 182, 2605–2631 (2011). arXiv:1102.1898
P. Bechtle, S. Heinemeyer, O. Stål, T. Stefaniak, G. Weiglein, HiggsSignals: confronting arbitrary Higgs sectors with measurements at the Tevatron and the LHC. Eur. Phys. J. C 74(2), 2711 (2014). arXiv:1305.1933
J.E. Camargo-Molina, B. O’Leary, W. Porod, F. Staub, \({\bf Vevacious}\): a tool for finding the global minima of one-loop effective potentials with many scalars. Eur. Phys. J. C 73(10), 2588 (2013). arXiv:1307.1477
M. Bahr et al., Herwig++ physics and manual. Eur. Phys. J. C 58, 639–707 (2008). arXiv:0803.0883
J. Bellm et al., Herwig 7.0/Herwig++ 3.0 release note. arXiv:1512.01178
T. Sjostrand, S. Mrenna, P.Z. Skands, PYTHIA 6.4 physics and manual. JHEP 05, 026 (2006). arXiv:hep-ph/0603175
T. Sjöstrand, S. Ask, J.R. Christiansen, R. Corke, N. Desai, P. Ilten, S. Mrenna, S. Prestel, C.O. Rasmussen, P.Z. Skands, An introduction to PYTHIA 8.2. Comput. Phys. Commun. 191, 159–177 (2015). arXiv:1410.3012
T. Gleisberg, S. Hoeche, F. Krauss, M. Schonherr, S. Schumann, F. Siegert, J. Winter, Event generation with SHERPA 1.1. JHEP 02, 007 (2009). arXiv:0811.4622
S. Höche, F. Krauss, M. Schönherr, F. Siegert, W+ \(n\)-jet predictions at the large hadron collider at next-to-leading order matched with a parton shower. Phys. Rev. Lett. 110(5), 052001 (2013). arXiv:1201.5882
B.C. Allanach, SOFTSUSY: a program for calculating supersymmetric spectra. Comput. Phys. Commun. 143, 305–331 (2002). arXiv:hep-ph/0104145
B.C. Allanach, P. Athron, L.C. Tunstall, A. Voigt, A.G. Williams, Next-to-minimal SOFTSUSY. Comput. Phys. Commun. 185, 2322–2339 (2014). arXiv:1311.7659
B.C. Allanach, M.A. Bernhardt, Including R-parity violation in the numerical computation of the spectrum of the minimal supersymmetric standard model: SOFTSUSY. Comput. Phys. Commun. 181, 232–245 (2010). arXiv:0903.1805
B.C. Allanach, C.H. Kom, M. Hanussek, Computation of neutrino masses in R-parity violating supersymmetry: SOFTSUSY3.2. Comput. Phys. Commun. 183, 785–793 (2012). arXiv:1109.3735
B.C. Allanach, A. Bednyakov, R.R. de Austri, Higher order corrections and unification in the minimal supersymmetric standard model: SOFTSUSY3.5. Comput. Phys. Commun. 189, 192–206 (2015). arXiv:1407.6130
B.C. Allanach, A. Bednyakov, R. Ruiz de Austri, The inclusion of two-loop SUSYQCD corrections to gluino and squark pole masses in the minimal and next-to-minimal supersymmetric standard model: SOFTSUSY3.7. arXiv:1601.06657
A. Djouadi, J.-L. Kneur, G. Moultaka, SuSpect: a Fortran code for the supersymmetric and Higgs particle spectrum in the MSSM. Comput. Phys. Commun. 176, 426–455 (2007). arXiv:hep-ph/0211331
S. Heinemeyer, W. Hollik, G. Weiglein, FeynHiggs: a program for the calculation of the masses of the neutral CP even Higgs bosons in the MSSM. Comput. Phys. Commun. 124, 76–89 (2000). arXiv:hep-ph/9812320
P. Drechsel, L. Galeta, S. Heinemeyer, G. Weiglein, Precise predictions for the Higgs-boson masses in the NMSSM. arXiv:1601.08100
U. Ellwanger, C. Hugonie, NMSPEC: a Fortran code for the sparticle and Higgs masses in the NMSSM with GUT scale boundary conditions. Comput. Phys. Commun. 177, 399–407 (2007). arXiv:hep-ph/0612134
J. Baglio, R. Gröber, M. Mühlleitner, D.T. Nhung, H. Rzehak, M. Spira, J. Streicher, K. Walz, NMSSMCALC: a program package for the calculation of loop-corrected Higgs boson masses and decay Widths in the (complex) NMSSM. Comput. Phys. Commun. 185(12), 3372–3391 (2014). arXiv:1312.4788
J.P. Vega, G. Villadoro, SusyHD: Higgs mass determination in supersymmetry. JHEP 07, 159 (2015). arXiv:1504.05200
A. Djouadi, J. Kalinowski, M. Spira, HDECAY: a program for Higgs boson decays in the standard model and its supersymmetric extension. Comput. Phys. Commun. 108, 56–74 (1998). arXiv:hep-ph/9704448
W. Frisch, H. Eberl, H. Hlucha, HFOLD—a program package for calculating two-body MSSM Higgs decays at full one-loop level. Comput. Phys. Commun. 182, 2219–2226 (2011). arXiv:1012.5025
U. Ellwanger, J.F. Gunion, C. Hugonie, NMHDECAY: a Fortran code for the Higgs masses, couplings and decay widths in the NMSSM. JHEP 02, 066 (2005). arXiv:hep-ph/0406215
U. Ellwanger, C. Hugonie, NMHDECAY 2.0: an updated program for sparticle masses, Higgs masses, couplings and decay widths in the NMSSM. Comput. Phys. Commun. 175, 290–303 (2006). arXiv:hep-ph/0508022
M. Muhlleitner, A. Djouadi, Y. Mambrini, SDECAY: a Fortran code for the decays of the supersymmetric particles in the MSSM. Comput. Phys. Commun. 168, 46–70 (2005). arXiv:hep-ph/0311167
H. Hlucha, H. Eberl, W. Frisch, SFOLD—a program package for calculating two-body sfermion decays at full one-loop level in the MSSM. Comput. Phys. Commun. 183, 2307–2312 (2012). arXiv:1104.2151
D. Das, U. Ellwanger, A.M. Teixeira, NMSDECAY: a Fortran code for supersymmetric particle decays in the next-to-minimal supersymmetric standard model. Comput. Phys. Commun. 183, 774–779 (2012). arXiv:1106.5633
G. Degrassi, P. Gambino, P. Slavich, SusyBSG: a Fortran code for BR[B –> X(s) gamma] in the MSSM with minimal flavor violation. Comput. Phys. Commun. 179, 759–771 (2008). arXiv:0712.3265
F. Mahmoudi, SuperIso v2.3: a program for calculating flavor physics observables in supersymmetry. Comput. Phys. Commun. 180, 1579–1613 (2009). arXiv:0808.3144
J. Rosiek, P. Chankowski, A. Dedes, S. Jager, P. Tanedo, SUSY_FLAVOR: a computational tool for FCNC and CP-violating processes in the MSSM. Comput. Phys. Commun. 181, 2180–2205 (2010). arXiv:1003.4260
A. Crivellin, J. Rosiek, P.H. Chankowski, A. Dedes, S. Jaeger, P. Tanedo, SUSY_FLAVOR v2: a computational tool for FCNC and CP-violating processes in the MSSM. Comput. Phys. Commun. 184, 1004–1032 (2013). arXiv:1203.5023
J.S. Lee, M. Carena, J. Ellis, A. Pilaftsis, C.E.M. Wagner, CPsuperH2.3: an updated tool for phenomenology in the MSSM with explicit CP violation. Comput. Phys. Commun. 184, 1220–1233 (2013). arXiv:1208.2212
P. Athron, M. Bach, H.G. Fargnoli, C. Gnendiger, R. Greifenhagen, J.-H. Park, S. Paßehr, D. Stöckinger, H. Stöckinger-Kim, A. Voigt, GM2Calc: precise MSSM prediction for \((g - 2)\) of the muon. arXiv:1510.08071
M. Spira, A. Djouadi, D. Graudenz, P.M. Zerwas, Higgs boson production at the LHC. Nucl. Phys. B 453, 17–82 (1995). arXiv:hep-ph/9504378
C. Hartmann, M. Trott, On one-loop corrections in the standard model effective field theory; the \(\Gamma (h \rightarrow \gamma \, \gamma )\) case. JHEP 07, 151 (2015). arXiv:1505.02646
F. Goertz, A. Katz, M. Son, A. Urbano, Precision Drell–Yan measurements at the LHC and implications for the diphoton excess. arXiv:1602.04801
LHC Higgs Cross Section Working Group Collaboration, S. Dittmaier et al., Handbook of LHC Higgs cross sections: 1. Inclusive observables. arXiv:1101.0593
ATLAS Collaboration, G. Aad et al., Search for new phenomena in the dijet mass distribution using \(p-p\) collision data at \(\sqrt{s}=8\) TeV with the ATLAS detector. Phys. Rev. D 91(5), 052007 (2015). arXiv:1407.1376
H. Georgi, M. Machacek, Doubly charged Higgs bosons. Nucl. Phys. B 262, 463 (1985)
P. Kant, R.V. Harlander, L. Mihaila, M. Steinhauser, Light MSSM Higgs boson mass to three-loop accuracy. JHEP 08, 104 (2010). arXiv:1005.5709
K. Nickel, F. Staub, Precise determination of the Higgs mass in supersymmetric models with vectorlike tops and the impact on naturalness in minimal GMSB. JHEP 07, 139 (2015). arXiv:1505.06077
M.E. Machacek, M.T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 1. Wave function renormalization. Nucl. Phys. B 222, 83 (1983)
M.E. Machacek, M.T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 2. Yukawa couplings. Nucl. Phys. B 236, 221 (1984)
M.E. Machacek, M.T. Vaughn, Two loop renormalization group equations in a general quantum field theory. 3. Scalar quartic couplings. Nucl. Phys. B 249, 70 (1985)
M. Luo, H. Wang, Y. Xiao, Two loop renormalization group equations in general gauge field theories. Phys. Rev. D 67, 065019 (2003). arXiv:hep-ph/0211440
S.P. Martin, M.T. Vaughn, Two loop renormalization group equations for soft supersymmetry breaking couplings. Phys. Rev. D 50, 2282 (1994). arXiv:hep-ph/9311340. [Erratum: Phys. Rev.D78,039903(2008)]
R.M. Fonseca, M. Malinsky, W. Porod, F. Staub, Running soft parameters in SUSY models with multiple U(1) gauge factors. Nucl. Phys. B 854, 28–53 (2012). arXiv:1107.2670
M.D. Goodsell, Two-loop RGEs with Dirac gaugino masses. JHEP 01, 066 (2013). arXiv:1206.6697
R.M. Fonseca, M. Malinský, F. Staub, Renormalization group equations and matching in a general quantum field theory with kinetic mixing. Phys. Lett. B 726, 882–886 (2013). arXiv:1308.1674
M. Sperling, D. Stöckinger, A. Voigt, Renormalization of vacuum expectation values in spontaneously broken gauge theories. JHEP 07, 132 (2013). arXiv:1305.1548
M. Sperling, D. Stöckinger, A. Voigt, Renormalization of vacuum expectation values in spontaneously broken gauge theories: two-loop results. JHEP 01, 068 (2014). arXiv:1310.7629
A. Djouadi, The anatomy of electro-weak symmetry breaking. I: the Higgs boson in the standard model. Phys. Rep. 457, 1–216 (2008). arXiv:hep-ph/0503172
M. Kramer, E. Laenen, M. Spira, Soft gluon radiation in Higgs boson production at the LHC. Nucl. Phys. B 511, 523–549 (1998). arXiv:hep-ph/9611272
K.G. Chetyrkin, B.A. Kniehl, M. Steinhauser, Hadronic Higgs decay to order alpha-s**4. Phys. Rev. Lett. 79, 353–356 (1997). arXiv:hep-ph/9705240
K.G. Chetyrkin, J.H. Kuhn, C. Sturm, QCD decoupling at four loops. Nucl. Phys. B 744, 121–135 (2006). arXiv:hep-ph/0512060
Y. Schroder, M. Steinhauser, Four-loop decoupling relations for the strong coupling. JHEP 01, 051 (2006). arXiv:hep-ph/0512058
P.A. Baikov, K.G. Chetyrkin, Top quark mediated Higgs boson decay into hadrons to order \(\alpha _s^5\). Phys. Rev. Lett. 97, 061803 (2006). arXiv:hep-ph/0604194
S. Dawson, A. Djouadi, M. Spira, QCD corrections to SUSY Higgs production: the role of squark loops. Phys. Rev. Lett. 77, 16–19 (1996). arXiv:hep-ph/9603423
LHC Higgs Cross Section Working Group Collaboration, J.R. Andersen et al., Handbook of LHC Higgs cross sections: 3. Higgs properties. arXiv:1307.1347
M. Spira, HIGLU: a program for the calculation of the total Higgs production cross-section at hadron colliders via gluon fusion including QCD corrections. arXiv:hep-ph/9510347
R.V. Harlander, S. Liebler, H. Mantler, SusHi: a program for the calculation of Higgs production in gluon fusion and bottom-quark annihilation in the standard model and the MSSM. Comput. Phys. Commun. 184, 1605–1617 (2013). arXiv:1212.3249
ATLAS collaboration, Search for production of vector-like quark pairs and of four top quarks in the lepton plus jets final state in \(pp\) collisions at \(\sqrt{s}=8\) TeV with the ATLAS detector. Tech. Rep. ATLAS-CONF-2015-012, CERN, Geneva, Mar, 2015
C. Anastasiou, C. Duhr, F. Dulat, E. Furlan, T. Gehrmann, F. Herzog, A. Lazopoulos, B. Mistlberger, CP-even scalar boson production via gluon fusion at the LHC. arXiv:1605.05761
H. An, T. Liu, L.-T. Wang, 125 GeV Higgs boson, enhanced di-photon rate, and gauged \(U(1)_PQ\)-extended MSSM. Phys. Rev. D 86, 075030 (2012). arXiv:1207.2473
A.V. Manohar, M.B. Wise, Flavor changing neutral currents, an extended scalar sector, and the Higgs production rate at the CERN LHC. Phys. Rev. D 74, 035009 (2006). arXiv:hep-ph/0606172
M. Singer, J.W.F. Valle, J. Schechter, Canonical neutral current predictions from the weak electromagnetic gauge group SU(3) X \(u\)(1). Phys. Rev. D 22, 738 (1980)
J.W.F. Valle, M. Singer, Lepton number violation with quasi Dirac neutrinos. Phys. Rev. D 28, 540 (1983)
F. Pisano, V. Pleitez, An SU(3) x U(1) model for electroweak interactions. Phys. Rev. D 46, 410–417 (1992). arXiv:hep-ph/9206242
P.H. Frampton, Chiral dilepton model and the flavor question. Phys. Rev. Lett. 69, 2889–2891 (1992)
R. Foot, O.F. Hernandez, F. Pisano, V. Pleitez, Lepton masses in an SU(3)-L x U(1)-N gauge model. Phys. Rev. D 47, 4158–4161 (1993). arXiv:hep-ph/9207264
J.C. Montero, F. Pisano, V. Pleitez, Neutral currents and GIM mechanism in SU(3)-L x U(1)-N models for electroweak interactions. Phys. Rev. D 47, 2918–2929 (1993). arXiv:hep-ph/9212271
V. Pleitez, M.D. Tonasse, Heavy charged leptons in an SU(3)-L x U(1)-N model. Phys. Rev. D 48, 2353–2355 (1993). arXiv:hep-ph/9301232
A.J. Buras, F. De Fazio, J. Girrbach, M.V. Carlucci, The anatomy of quark flavour observables in 331 models in the flavour precision era. JHEP 02, 023 (2013). arXiv:1211.1237
P.V. Dong, N.T.K. Ngan, Phenomenology of the simple 3-3-1 model with inert scalars. arXiv:1512.09073
R.M. Fonseca, Calculating the renormalisation group equations of a SUSY model with Susyno. Comput. Phys. Commun. 183, 2298–2306 (2012). arXiv:1106.5016
R.N. Mohapatra, J.W.F. Valle, Neutrino mass and baryon number nonconservation in superstring models. Phys. Rev. D 34, 1642 (1986)
S.M. Boucenna, J.W.F. Valle, A. Vicente, Predicting charged lepton flavor violation from 3-3-1 gauge symmetry. Phys. Rev. D 92(5), 053001 (2015). arXiv:1502.07546
E. Ma, Diphoton revelation of the utilitarian supersymmetric standard model. arXiv:1512.09159
S.F. King, S. Moretti, R. Nevzorov, Theory and phenomenology of an exceptional supersymmetric standard model. Phys. Rev. D 73, 035009 (2006). arXiv:hep-ph/0510419
S.F. King, S. Moretti, R. Nevzorov, Exceptional supersymmetric standard model. Phys. Lett. B 634, 278–284 (2006). arXiv:hep-ph/0511256
E. Ma, Compendium of models from a gauge U(1) framework. arXiv:1601.01400
R.M. Capdevilla, A. Delgado, A. Martin, Light stops in a minimal U(1)x extension of the MSSM. Phys. Rev. D 92(11), 115020 (2015). arXiv:1509.02472
B. O’Leary, W. Porod, F. Staub, Mass spectrum of the minimal SUSY B-L model. JHEP 05, 042 (2012). arXiv:1112.4600
F. Staub, T. Ohl, W. Porod, C. Speckner, A tool box for implementing supersymmetric models. Comput. Phys. Commun. 183, 2165–2206 (2012). arXiv:1109.5147
M. Hirsch, M. Malinsky, W. Porod, L. Reichert, F. Staub, Hefty MSSM-like light Higgs in extended gauge models. JHEP 02, 084 (2012). arXiv:1110.3037
M. Hirsch, W. Porod, L. Reichert, F. Staub, Phenomenology of the minimal supersymmetric \(U(1)_{B-L}\times U(1)_R\) extension of the standard model. Phys. Rev. D 86, 093018 (2012). arXiv:1206.3516
J.E. Camargo-Molina, B. O’Leary, W. Porod, F. Staub, The stability Of R-parity in supersymmetric models extended By \(U(1)_{B-L}\). Phys. Rev. D 88, 015033 (2013). arXiv:1212.4146
C.L. Wainwright, CosmoTransitions: computing cosmological phase transition temperatures and bubble profiles with multiple fields. Comput. Phys. Commun. 183, 2006–2013 (2012). arXiv:1109.4189
S.P. Martin, Taming the Goldstone contributions to the effective potential. Phys. Rev. D 90(1), 016013 (2014). arXiv:1406.2355
J. Elias-Miro, J.R. Espinosa, T. Konstandin, Taming infrared divergences in the effective potential. JHEP 08, 034 (2014). arXiv:1406.2652
Planck Collaboration, P.A.R. Ade et al., Planck 2013 results. XVI. Cosmological parameters. Astron. Astrophys. 571, A16 (2014). arXiv:1303.5076
L. Basso, B. O’Leary, W. Porod, F. Staub, Dark matter scenarios in the minimal SUSY B-L model. JHEP 09, 054 (2012). arXiv:1207.0507
SINDRUM Collaboration, U. Bellgardt et al., Search for the decay mu+ —> e+ e+ e\(-\). Nucl. Phys. B 299, 1 (1988)
K. Hayasaka et al., Search for lepton flavor violating Tau decays into three leptons with 719 million produced Tau+Tau\(-\) pairs. Phys. Lett. B 687, 139–143 (2010). arXiv:1001.3221
SINDRUM II Collaboration, C. Dohmen et al., Test of lepton flavor conservation in mu –> e conversion on titanium. Phys. Lett. B 317, 631–636 (1993)
SINDRUM II Collaboration, W.H. Bertl et al., A search for muon to electron conversion in muonic gold. Eur. Phys. J. C 47, 337–346 (2006)
ATLAS collaboration, Search for new phenomena in the dilepton final state using proton–proton collisions at \(\sqrt{s}\) = 13 TeV with the ATLAS detector
M.E. Krauss, B. O’Leary, W. Porod, F. Staub, Implications of gauge kinetic mixing on Z’ and slepton production at the LHC. Phys. Rev. D 86, 055017 (2012). arXiv:1206.3513
L. Basso, B. Fuks, M.E. Krauss, W. Porod, Doubly-charged Higgs and vacuum stability in left-right supersymmetry. JHEP 07, 147 (2015). arXiv:1503.08211
Acknowledgments
We thank Martin Winkler for beta-testing and valuable impact, and Michael Spira for helpful discussions. Mark Goodsell acknowledges support from Agence Nationale de Recherche Grant ANR-15-CE31-0002 “HiggsAutomator,” and would like to thank Pietro Slavich and Luc Darmé for interesting discussions. Avelino Vicente acknowledges financial support from the “Juan de la Cierva” program (27-13-463B-731) funded by the Spanish MINECO as well as from the Spanish Grants FPA2014-58183-P, Multidark CSD2009-00064 and SEV-2014-0398 (MINECO), FPA2011-22975 and PROMETEOII/2014/084 (Generalitat Valenciana), and is grateful to Wei-Chih Huang for discussions about the GTHDM model. Manuel E. Krauss is supported by the BMBF Grant 00160287 and thanks Cesar Bonilla for useful discussions on left-right models. Lorenzo Basso acknowledges support by the OCEVU Labex (ANR-11-LABX- 0060) and the A*MIDEX project (ANR-11-IDEX-0001-02), funded by the “Investissements d’Avenir” French government program managed by the ANR. The work by Peter Athron is in part supported by the ARC Centre of Excellence for Particle Physics at the Tera-scale. Dylan Harries is supported by the University of Adelaide and the ARC Centre of Excellence for Particle Physics at the Tera-scale. Florian Staub thanks Alfredo Urbano for discussions about the Georgi-Machacek model and for testing the model file.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Funded by SCOAP3.
About this article
Cite this article
Staub, F., Athron, P., Basso, L. et al. Precision tools and models to narrow in on the 750 GeV diphoton resonance. Eur. Phys. J. C 76, 516 (2016). https://doi.org/10.1140/epjc/s10052-016-4349-5
Received:
Accepted:
Published:
DOI: https://doi.org/10.1140/epjc/s10052-016-4349-5