Precision tools and models to narrow in on the 750 GeV diphoton resonance
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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 weaklycoupled 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 tailormade 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 CPeven and CPodd 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 MonteCarlo 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.
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 lookelsewhereeffect, the deviations from standard model (SM) expectations drop to \(2.3 \sigma \) and \(1.2 \sigma \).^{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, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359].
It is hard to explain the excess within the most commonly considered frameworks for physics beyond the standard model (BSM), like twoHiggsdoublet models (THDM) or the minimal supersymmetric standard model (MSSM) [360], to mention a couple of wellknown 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, 362, 363, 364, 365, 366] offers all features for the precise study of a new model: it calculates all treelevel properties of the model (mass, tadpoles, vertices), the oneloop corrections to all masses as well as the twoloop 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 stateoftheart spectrum generators dedicated just to the MSSM and NMSSM [370]. The RGEs are solved numerically at the twoloop level and the mass spectrum is calculated at one loop. Both codes have the option to include the known twoloop corrections [371, 372, 373, 374, 375, 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 twoloop corrections to the Higgs masses in the gaugeless limit at zero external momentum [377, 378]. FlexibleSUSY has an extension to calculate the twoloop 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 MonteCarlo (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 selfcontained 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 selfconsistent 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.

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 MonteCarlo studies at LO or NLO [388, 389, 390, 391, 392, 393], many spectrum generators [367, 368, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405] for the calculation of pole masses including important higher order corrections, codes dedicated only to Higgs [406, 407, 408, 409] or sparticle decays [410, 411, 412], and codes to check flavour [413, 414, 415, 416, 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.
Branching fraction ratio, as well as the partial decay widths for the digluon and diphoton channels for a toy model containing only the relevant vectorlike fermion pair \(\Psi _{F_i}\). The above values are for the benchmark points \(Y_{F_{i}}=1\) and \(m_{F_{i}}=1\,{\text {TeV}}\), where the values are for a CPeven/CPodd scalar resonance, respectively. The SPheno NLO calculation includes \(\text {N}^{3}\)LO corrections for the digluon channel, while the diphoton decay width is calculated at NLO and LO for a CPeven and odd scalar respectively
Model  Br (\(gg/\gamma \gamma \))  \(\Gamma _{S\rightarrow gg}\) (\(\mathrm{MeV}\))  \(\Gamma _{S\rightarrow \gamma \gamma }\) (\(\mathrm{MeV}\)) 

\(\Psi _{F_1}\)  
Ref. [276] LO  11.62/−  6.74/−  0.58/− 
SPheno LO  13.47/12.22  6.78/14.27  0.50/1.17 
SPheno NLO  23.27/20.27  11.04/23.71  0.47/1.17 
\(\Psi _{F_2}\)  
Ref. [276] LO  24.42/−  15.14/−  0.62/− 
SPheno LO  28.32/25.70  15.26/32.12  0.54/1.25 
SPheno NLO  48.93/42.67  24.85/52.34  0.51/1.25 
\(\Psi _{F_3}\)  
Ref. [276] LO  33.80/−  6.76/−  0.20/− 
SPheno LO  39.20/35.56  6.78/14.27  0.17/0.40 
SPheno NLO  67.72/59.06  11.04/23.71  0.16/0.40 
\(\Psi _{F_4}\)  
Ref. [276] LO  49.84/−  14.95/−  0.30/− 
SPheno LO  57.80/52.44  15.26/32.12  0.26/0.61 
SPheno NLO  99.85/87.09  24.85/53.34  0.25/0.61 
\(\Psi _{F_5}\)  
Ref. [276] LO  150.0/−  1.50/−  10.0 \(\times \) 10\(^{3}\)/− 
SPheno LO  177.0/160.6  1.70/3.57  9.58 \(\times \) 10\(^{3}\)/22.22 \(\times \) 10\(^{3}\) 
SPheno NLO  305.8/266.7  2.76/5.93  9.03 \(\times \) 10\(^{3}\)/22.22 \(\times ~\)10\(^{3}\) 
\(\Psi _{F_6}\)  
Ref. [276] LO  390.0/−  7.80/−  2.00 \(\times \) 10\(^{2}\)/− 
SPheno LO  453.2/411.1  6.78/14.27  1.50 \(\times \) 10\(^{2}\)/3.47 \(\times \) 10\(^{2}\) 
SPheno NLO  782.8/682.8  11.04/23.71  1.41 \(\times \) 10\(^{2}\)/3.47 \(\times \) 10\(^{2}\) 
To demonstrate these effects we show in Fig. 1 the total decay width^{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 vectorlike 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 CPeven and CPodd 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 channel^{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
 1.
Assuming a large Yukawalike 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
2.1.2.1 Large couplings to fermions A common idea to explain the diphoton excess is the presence of vectorlike states which enhance the loopinduced coupling of a neutral scalar to two photons or two gluons. This led some authors to consider Yukawalike couplings of the scalar to the vectorlike fermions larger than \(\sqrt{4 \pi }\), which is clearly beyond the perturbative regime.^{4} Nevertheless, a oneloop calculation is used in these analyses to obtain predictions for the partial widths [353], despite being in a nonperturbative region of parameter space.
Energy scale at which a Landau pole in \(g_1\) appears as a function of \(N_k\) in the model of Ref. [325]
\(N_k\)  \(\mu _{\text {Landau}}\) 

10  \(2 \times 10^{13}\) GeV 
100  \(1.2 \times 10^5\) GeV 
1000  3.8 TeV 
6000  2.7 TeV 
9000  2.6 TeV 
2.1.3 How do the tools help?
 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 gluonfusion production crosssection of the 750 GeV mediator can be calculated numerically and one does not have to rely on analytical (and sometimes erroneous^{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
2.2.2 To VEV or not to VEV?
2.2.3 Additional decay channels
Limits on \(\Gamma (S\rightarrow X)/\Gamma (S\rightarrow \gamma \gamma )\) assuming a production of S via gluon fusion or heavy quarks. Values are taken from Ref. [192]
\(e^+e^ + \mu ^+ \mu ^\)  \(\tau ^+ \tau ^\)  \(Z\gamma \)  ZZ  Zh  hh  \(W^+W^\)  \(t\bar{t}\)  \(b \bar{b}\)  jj  Inv. 

0.6  6  6  6  10  20  20  300  500  1300  400 
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.
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 GeorgiMachacek 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?
 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.
 1.
As outlined above, FlexibleSUSY and SPheno calculate the diphoton and digluon rate very accurately
 2.
SPheno calculates all other two body decays^{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 vectorlike 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.
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 threeloop corrections are partially tackled [425]. The current ballpark 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 vectorlike superfields together with a gauge singlet, see Sect. 5. These new fields can also be used to increase the SMlike 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 SMlike Higgs. There, they use a oneloop effective potential approach considering the new Yukawa couplings to be \(\mathcal {O}(1)\) or below, while also including the dominant twoloop 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 oneloop level, as well as the additional twoloop 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?
 1.
All masses of the model are calculated with high accuracy: FlexibleSUSY and SPheno include the full oneloop contribution to all pole masses in a model, while SPheno covers even the dominant twoloop 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 SMlike 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.^{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.
3.2 SPheno

Full twoloop running of all parameters

Oneloop corrections to all masses

Twoloop corrections to Higgs masses

Complete oneloop 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 oneloop level

Calculation of decay widths and branching ratios for two– and three body decays

Interface to HiggsBounds and HiggsSignals

Estimate of electroweak finetuning

Prediction for LHC cross sections for all neutral scalars
3.3 FlexibleSUSY

Full twoloop running of all parameters

Threeloop running of all parameters in the SM and MSSM, except for the VEVs

Calculation of the pole mass spectrum at the full oneloop level

Partial twoloop corrections to the Higgs masses in the SM, SplitMSSM, MSSM, NMSSM, UMSSM and E\(_6\)SSM and partial threeloop corrections to the Higgs mass in the SplitMSSM

Complete oneloop and partial twoloop and threeloop 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
3.4 Mass spectrum calculation: SUSY vs. nonSUSY
We have outlined that FlexibleSUSY and SPheno can include the radiative corrections to all particles up to the twoloop 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 onshell calculations are in good agreement. Consequently, the remaining difference between both calculations is often a good estimate for the theoretical uncertainty.
The treatment of nonsupersymmetric models in FlexibleSUSY and SPheno is very similar to the treatment of supersymmetric models. The main difference is, that in nonsupersymmetric 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 treelevel 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 onshell 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 onshell renormalisation keeping all masses and mixing angles fixed. We find that the oneloop corrections in the \({\overline{\mathrm {MS}}}\) scheme can give huge corrections to the treelevel masses in nearly all models presented in the following. Therefore large finetuning of the parameters is necessary once the loop corrections are taken into account. A detailed analysis using a full onshell 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 looplevel, 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
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 higherorder 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 MonteCarlo studies
3.6.1 Interplay SARAHspectrumgeneratorMCtool
The tool chains SARAH–SPheno/FlexibleSUSY–MCTools 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 MCTool 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.
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, looplevel decays are only computed for processes where the treelevel process is absent. This is to avoid the technical issues of infrared divergences. If the particle that explains the 750 GeV excess is a scalar, then it must mix with the Higgs and acquire treelevel 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 pseudoscalar 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 ongoing efforts to close this gap in the near future and to provide the full oneloop corrections to any twobody decay of CPeven 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.
3.7.2 BSM NLO corrections
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 effectivefieldtheory (EFT) approach. This is possible because the top mass is sufficiently heavy compared to the Higgs boson. Also the presence of vectorlike 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 Kfactor.
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 onshell scheme are at twoloop 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 higherorder corrections such as the righthand diagram in Fig. 11 – but applying such a correction to each propagator in the loop would actually correspond to a fourloop 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.
For nonsupersymmetric models, due to the fact that (almost) every parameter point is essentially finetuned, 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 crosssection 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 (nonrenormalisable) 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 extradimensional 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 nonobservation 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 spacetime 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 stringinspired 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 vectorlike 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 nonsupersymmetric models with an enlarged gauge group such as gauged U(1) extensions or leftrightsymmetric 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 twoHiggsdoublet 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 leftrightsymmetric 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 331 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.
Part I of the overview of proposed models to explain the diphoton excess which are now available in SARAH. Special characters are added in the last column if we found serious problems with the model during the implementation. The respective problem is described in the above text
Model  Name  Refs.  

Toy models with vectorlike fermions  
CPeven singlet  SM+VL/CPevenS  
CPodd singlet  SM+VL/CPoddS  
Complex singlet  SM+VL/complexS  
Models based on the SM gaugegroup  
Portal dark matter  SM+VL/PortalDM  
Scalar octet  SMSOctet  \(\clubsuit \)  
SU(2) triplet quark model  SM+VL/TripletQuarks  [62]  
Single scalar leptoquark  LeptoQuarks/ScalarLeptoquarks  [53]  
Two scalar leptoquarks  LeptoQuarks/TwoScalarLeptoquarks  [106]  \(\spadesuit \) 
GeorgiMachacek model  GeorgiMachacek  
THDM w. colour triplet  THDM+VL/min3  [74]  
THDM w. colour octet  THDM+VL/min8  [74]  
THDMI w. exotic fermions  THDM+VL/TypeIVL  
THDMII w. exotic fermions  THDM+VL/TypeIIVL  
THDMI w. SMlike fermions  THDM+VL/TypeISMlikeVL  [36]  
THDMII w. SMlike fermions  THDM+VL/TypeIISMlikeVL  [36]  
THDM w. scalar septuplet  THDM/ScalarSeptuplet 
Part II of the overview of proposed models to explain the diphoton excess which are now available in SARAH. Special characters are added in the last column if we found serious problems with the model during the implementation. The respective problem is described in the above text
Model  Name  Refs.  

U(1) extensions  
Dark \(U(1)'\)  U1Extensions/darkU1  [280]  
Hidden U(1)  U1Extensions/hiddenU1  [136]  
Simple U(1)  U1Extensions/simpleU1  [104]  
Scotogenic U(1)  U1Extensions/scotoU1  [355]  † 
Unconventional \(U(1)_{BL}\)  U1Extensions/BLVL  [313]  
Sample of \(U(1)'\)  U1Extensions/VLsample  [107]  
Flavournonuniversal charges  U1Extensions/nonUniversalU1  [304]  
Leptophobic U(1)  U1Extensions/U1Leptophobic  [277]  ‡ 
\(Z'\) mimicking a scalar resonance  U1Extensions/trickingLY  [102]  
Nonabelian gaugegroup extensions of the SM  
LR without bidoublets  LRmodels/LRVL  §  
LR with \(U(1)_L \times U(1)_R\)  LRmodels/LRLR  [90]  ¶ 
LR with triplets  LRmodels/tripletLR  [64]  
Dark LR  LRmodels/darkLR  [154]  
331 model without exotic charges  331/v1  [80]  
331 model with exotic charges  331/v2  [88]  
Gauged THDM  GTHDM  [250]  
Supersymmetric models  
NMSSM with vectorlike top  NMSSM+VL/VLtop  [353]  \(\triangle \) 
NMSSM with 5’s  NMSSM+VL/5plets  
NMSSM with 10’s  NMSSM+VL/10plets  
NMSSM with 5’s & 10’s  NMSSM+VL/10plets  [220]  
NMSSM with 5’s and RpV  NMSSM+VL/5plets+RpV  [180]  
Broken MRSSM  brokenMRSSM  [100]  
\(U(1)^\prime \)extended MSSM  MSSM+U1primeVL  
\(E_6\) with extra U(1)  E6MSSMalt  [110] 
4.1 Validation
 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 tailormade 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.

\(\clubsuit \) As explained in more detail in the following subsection, after the inclusion of higherorder 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 gaugeinvariant 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 gaugeinvariant 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
Extra scalar field content of the octet extended SM
Field  Gen.  \(SU(3)_C\)  \(SU(2)_L\)  \(U(1)_Y\) 

S  1  \(\mathbf {1}\)  \(\mathbf {1}\)  0 
O  1  \(\mathbf {8}\)  \(\mathbf {2}\)  \(\frac{1}{2}\) 
Octet masses
4.2.2 331 models
Models based on the \(\mathrm {SU(3)_c} \times \mathrm {SU(3)_L} \times \mathrm {U(1)_\mathcal {X}}\) gauge symmetry [451, 452, 453, 454, 455, 456, 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 \(\mathrm {SU(3)_c} \times \mathrm {SU(2)_L} \times \mathrm {U(1)_Y}\) vectorlike 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 \(+\) vectorlike fermions framework proposed to explain the diphoton excess.
4.2.2.1 On the SU(3) generators in SARAH

Reference: [80]

Model name: 331/v1
Fermionic and scalar particle content of the 331v1 model. The scalar and fermion fields are shown in the top and bottom of the table respectively
Field  Gen.  \(SU(3)_C\)  \(SU(2)_L\)  \(U(1)_{\mathcal {X}}\)  \(U(1)_{\mathcal {L}}\)  \(\mathbb {Z}_2\) 

\(\Phi _1\)  1  \(\mathbf {1}\)  \({\bar{\mathbf {3}}}\)  \(\frac{2}{3}\)  \(\frac{2}{3}\)  \(+\) 
\(\Phi _2\)  1  \(\mathbf {1}\)  \({\bar{\mathbf {3}}}\)  \(\frac{1}{3}\)  \(\frac{4}{3}\)  \(+\) 
\(\Phi _3\)  1  \(\mathbf {1}\)  \({\bar{\mathbf {3}}}\)  \(\frac{1}{3}\)  \(\frac{2}{3}\)  − 
\(\Phi _X\)  1  \(\mathbf {1}\)  \({\bar{\mathbf {3}}}\)  \(\frac{1}{3}\)  \(\frac{4}{3}\)  \(+\) 
\(\psi _L\)  3  \(\mathbf {1}\)  \({\bar{\mathbf {3}}}\)  \(\frac{1}{3}\)  \(\frac{1}{3}\)  \(+\) 
\(e_R\)  3  \(\mathbf {1}\)  \(\mathbf {1}\)  \(1\)  \(1\)  \(+\) 
s  3  \(\mathbf {1}\)  \(\mathbf {1}\)  0  1  \(+\) 
\(Q_L^{1,2}\)  2  \(\mathbf {3}\)  \(\mathbf {3}\)  0  \(\frac{2}{3}\)  \(+\) 
\(Q_L^3\)  1  \(\mathbf {3}\)  \({\bar{\mathbf {3}}}\)  \(\frac{1}{3}\)  \(\frac{2}{3}\)  − 
\(u_R\)  3  \(\mathbf {3}\)  \(\mathbf {1}\)  \(\frac{2}{3}\)  0  \(+\) 
\(T_R\)  1  \(\mathbf {3}\)  \(\mathbf {1}\)  \(\frac{2}{3}\)  0  − 
\(d_R\)  3  \(\mathbf {3}\)  \(\mathbf {1}\)  \(\frac{1}{3}\)  0  − 
\(D_R,S_R\)  2  \(\mathbf {3}\)  \(\mathbf {1}\)  \(\frac{1}{3}\)  0  \(+\) 
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 \(U(1)_\mathcal {X}\).
Fermionic and scalar particle content of the 331v2 model. The scalar and fermion fields are shown in the top and bottom of the table respectively
Field  Gen.  \(SU(3)_C\)  \(SU(2)_L\)  \(U(1)_{\mathcal {X}}\) 

\(\rho \)  1  \(\mathbf {1}\)  \(\mathbf {3}\)  1 
\(\eta \)  1  \(\mathbf {1}\)  \(\mathbf {3}\)  0 
\(\chi \)  1  \(\mathbf {1}\)  \(\mathbf {3}\)  \(1\) 
\(\psi _L\)  3  \(\mathbf {1}\)  \({\bar{\mathbf {3}}}\)  \(1\) 
\(e_R\)  3  \(\mathbf {1}\)  \(\mathbf {1}\)  \(1\) 
\(E_R\)  3  \(\mathbf {3}\)  \(\mathbf {3}\)  \(2\) 
\(Q_L^{1,2}\)  2  \(\mathbf {3}\)  \(\mathbf {3}\)  \(\frac{2}{3}\) 
\(Q_L^3\)  1  \(\mathbf {3}\)  \({\bar{\mathbf {3}}}\)  \(\frac{1}{3}\) 
\(u_R\)  3  \(\mathbf {3}\)  \(\mathbf {1}\)  \(\frac{2}{3}\) 
\(T_R\)  1  \(\mathbf {3}\)  \(\mathbf {1}\)  \(\frac{4}{3}\) 
\(d_R\)  3  \(\mathbf {3}\)  \(\mathbf {1}\)  \(\frac{1}{3}\) 
\(D_R,S_R\)  2  \(\mathbf {3}\)  \(\mathbf {1}\)  \(\frac{5}{3}\) 
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 \(U(1)_\mathcal {X}\).
4.2.3 \(E_6\)inspired SUSY model with extra U(1)

Reference: [110]

Model name: SUSYmodels/E6SSMalt
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 loopinduced decay through the exotic states.
The representations of the chiral superfields under the \(SU(3)_C\) and \(SU(2)_L\) gauge groups, and their \(U(1)_Y\) and \(U(1)_N\) charges without the \(E_6\) normalisation. The GUT normalisations are \(\sqrt{\frac{5}{3}}\) for \(U(1)_Y\) and \(\sqrt{40}\) for \(U(1)_N\). The transformation properties under the discrete symmetries \(\mathbb {Z}_2^H\), \(\mathbb {Z}_2^L\) are also shown, where ‘\(+\)’ indicates the superfield is even under the symmetry and ‘−’ indicates that it is odd under the symmetry
Field  Gen  \(SU(3)_C\)  \(SU(2)_L\)  \(U(1)_Y\)  \(U(1)_N\)  \(\mathbb {Z}_2^H\)  \(\mathbb {Z}_2^L\) 

\(\hat{Q}_i\)  3  \(\mathbf {3}\)  \(\mathbf {2}\)  \(\frac{1}{6}\)  1  −  \(+\) 
\(\hat{u}_i^c\)  3  \(\mathbf {\overline{3}}\)  \(\mathbf {1}\)  \(\frac{2}{3}\)  1  −  \(+\) 
\(\hat{d}_i^c\)  3  \(\mathbf {\overline{3}}\)  \(\mathbf {1}\)  \(\frac{1}{3}\)  2  −  \(+\) 
\(\hat{L}_i\)  3  \(\mathbf {1}\)  \(\mathbf {2}\)  \(\frac{1}{2}\)  2  −  − 
\(\hat{e}_i^c\)  3  \(\mathbf {1}\)  \(\mathbf {1}\)  1  1  −  − 
\(\hat{N}_i^c\)  3  \(\mathbf {1}\)  \(\mathbf {1}\)  0  0  −  − 
\(\hat{S}_i\)  2  \(\mathbf {1}\)  \(\mathbf {1}\)  0  5  \(+\)  \(+\) 
\(\hat{S}_1\)  1  \(\mathbf {1}\)  \(\mathbf {1}\)  0  5  −  \(+\) 
\(\hat{H}_u\)  1  \(\mathbf {1}\)  \(\mathbf {2}\)  \(\frac{1}{2}\)  \(2\)  \(+\)  \(+\) 
\(\hat{H}_d\)  1  \(\mathbf {1}\)  \(\mathbf {2}\)  \(\frac{1}{2}\)  \(3\)  \(+\)  \(+\) 
\(\hat{H}_{\alpha }^u\)  2  \(\mathbf {1}\)  \(\mathbf {2}\)  \(\frac{1}{2}\)  \(2\)  −  \(+\) 
\(\hat{H}_{\alpha }^d\)  2  \(\mathbf {1}\)  \(\mathbf {2}\)  \(\frac{1}{2}\)  \(3\)  −  \(+\) 
\(\hat{D}_i\)  3  \(\mathbf {3}\)  \(\mathbf {1}\)  \(\frac{1}{3}\)  \(2\)  −  \(+\) 
\(\hat{\overline{D}}\)  3  \(\mathbf {\overline{3}}\)  \(\mathbf {1}\)  \(\frac{1}{3}\)  \(3\)  −  \(+\) 
\(\hat{L}_4\)  1  \(\mathbf {1}\)  \(\mathbf {2}\)  \(\frac{1}{2}\)  2  −  \(+\) 
\(\hat{\overline{L}}_4\)  1  \(\mathbf {1}\)  \(\mathbf {\overline{2}}\)  \(\frac{1}{2}\)  \(2\)  −  \(+\) 
The \(\mathbb {Z}_2^L\) symmetry plays a role similar to Rparity 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.
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 CPeven Higgs states from the standard SU(2) doublets. However it is clear that there must be some mixing from the Dterms, 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 SMlike Higgs state and a 750 GeV singletdominated 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 offdiagonal 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
Scalars and fermions in the \(U(1)_X\)extended MSSM
SF  Spin 0  Spin \(\frac{1}{2}\)  Generations  \(U(1)_Y\)  \(SU(2)_L\)  \(SU(3)_C\)  \(U(1)_X\) 

\(\hat{q}\)  \(\tilde{q}\)  q  3  \(\frac{1}{6}\)  \(\mathbf{2}\)  \(\mathbf{3}\)  0 
\(\hat{l}\)  \(\tilde{l}\)  l  3  \(\frac{1}{2}\)  \(\mathbf{2}\)  \(\mathbf{1}\)  0 
\(\hat{d}\)  \(\tilde{d}_R^{*}\)  \(d^*_R\)  3  \(\frac{1}{3}\)  \(\mathbf{1}\)  \(\mathbf{\overline{3}}\)  \(\frac{1}{2} \) 
\(\hat{u}\)  \(\tilde{u}_R^{*}\)  \(u^*_R\)  3  \(\frac{2}{3}\)  \(\mathbf{1}\)  \(\mathbf{\overline{3}}\)  \(\frac{1}{2} \) 
\(\hat{e}\)  \(\tilde{e}_R^*\)  \(e^*_R\)  3  1  \(\mathbf{1}\)  \(\mathbf{1}\)  \(\frac{1}{2} \) 
\(\hat{\nu }\)  \(\tilde{\nu }_R^*\)  \(\nu ^*_R\)  3  0  \(\mathbf{1}\)  \(\mathbf{1}\)  \(\frac{1}{2} \) 
\(\hat{U}\)  \(\tilde{U}^*\)  \(U^*\)  3  \(\frac{2}{3}\)  \(\mathbf{1}\)  \(\mathbf{\overline{3}}\)  \(\frac{1}{2} \) 
\(\hat{\bar{U}}\)  \(\tilde{\bar{U}}\)  \(\bar{U}\)  3  \(\frac{2}{3}\)  \(\mathbf{1}\)  \(\mathbf{3}\)  \(\frac{1}{2} \) 
\(\hat{E}\)  \(\tilde{E}^*\)  \(E^*\)  3  1  \(\mathbf{1}\)  \(\mathbf{1}\)  \(\frac{1}{2} \) 
\(\hat{\bar{E}}\)  \(\tilde{\bar{E}}\)  \(\bar{E}\)  3  \(1\)  \(\mathbf{1}\)  \(\mathbf{1}\)  \(\frac{1}{2} \) 
\(\hat{H}_d\)  \(H_d\)  \(\tilde{H}_d\)  1  \(\frac{1}{2}\)  \(\mathbf{2}\)  \(\mathbf{1}\)  \(\frac{1}{2} \) 
\(\hat{H}_u\)  \(H_u\)  \(\tilde{H}_u\)  1  \(\frac{1}{2}\)  \(\mathbf{2}\)  \(\mathbf{1}\)  \(\frac{1}{2} \) 
\(\hat{\eta }\)  \(\eta \)  \(\tilde{\eta }\)  1  0  \(\mathbf{1}\)  \(\mathbf{1}\)  \(1 \) 
\(\hat{\bar{\eta }}\)  \(\bar{\eta }\)  \(\tilde{\bar{\eta }}\)  1  0  \(\mathbf{1}\)  \(\mathbf{1}\)  1 
\(\hat{S}\)  S  \(\tilde{S}\)  1  0  \(\mathbf{1}\)  \(\mathbf{1}\)  0 
Fermions in the considered model. We show here the names used by SARAH during the Mathematica session as well as the names in the output files for MonteCarlo tools. Here, g denotes a generation index and c a colour index
LaTeX  SARAH  Output 

\(\tilde{\chi }^_{{i}} = \left( \begin{array}{c} \lambda ^_{{i}}\\ \lambda ^{+,*}_{{i}}\end{array} \right) \)  \( \texttt {Cha[\{g\}]} = \left( \begin{array}{c} \texttt {Lm[\{g\}]} \\ \texttt {conj[Lp[\{g\}]]}\end{array} \right) \)  C 
\(\tilde{\chi }^0_{{i}} = \left( \begin{array}{c} \lambda ^0_{{i}}\\ \lambda ^{0,*}_{{i}}\end{array} \right) \)  \( \texttt {Chi[\{g\}]} = \left( \begin{array}{c} \texttt {L0[\{g\}]} \\ \texttt {conj[L0[\{g\}]]}\end{array} \right) \)  N 
\(d_{{i \alpha }} = \left( \begin{array}{c} D_{L,{i \alpha }}\\ D^*_{R,{i \alpha }}\end{array} \right) \)  \( \texttt {Fd[\{g, c\}]} = \left( \begin{array}{c} \texttt {FDL[\{g, c\}]} \\ \texttt {conj[FDR[\{g, c\}]]}\end{array} \right) \)  d 
\(e_{{i}} = \left( \begin{array}{c} E_{L,{i}}\\ E^*_{R,{i}}\end{array} \right) \)  \( \texttt {Fe[\{g\}]} = \left( \begin{array}{c} \texttt {FEL[\{g\}]} \\ \texttt {conj[FER[\{g\}]]}\end{array} \right) \)  e 
\(u_{{i \alpha }} = \left( \begin{array}{c} U_{L,{i \alpha }}\\ U^*_{R,{i \alpha }}\end{array} \right) \)  \( \texttt {Fu[\{g, c\}]} = \left( \begin{array}{c} \texttt {FUL[\{g, c\}]} \\ \texttt {conj[FUR[\{g, c\}]]}\end{array} \right) \)  u 
\(\nu _{{i}} = \left( \begin{array}{c} \lambda _{\nu ,{i}}\\ \lambda ^*_{\nu ,{i}}\end{array} \right) \)  \( \texttt {Fv[\{g\}]} = \left( \begin{array}{c} \texttt {Fvm[\{g\}]} \\ \texttt {conj[Fvm[\{g\}]]}\end{array} \right) \)  nu 
\(\tilde{g}_{{\alpha }} = \left( \begin{array}{c} \lambda _{{\tilde{g}},{\alpha }}\\ \lambda ^*_{{\tilde{g}},{\alpha }}\end{array} \right) \)  \( \texttt {Glu[\{c\}]} = \left( \begin{array}{c} \texttt {fG[\{c\}]} \\ \texttt {conj[fG[\{c\}]]}\end{array} \right) \)  go 
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 LaTeXformat for each model via the commands
Scalars, vector bosons and ghosts in the considered model. We show here the names used by SARAH during the Mathematica session as well as the names in the output files for MonteCarlo tools. Here, t denotes a generation index and c a colour index
LaTeX  SARAH  Output  LaTeX  SARAH  Output 

\(\tilde{d}_{{i \alpha }}\)  Sd[{g, c}]  sd  \(\tilde{u}_{{i \alpha }}\)  Su[{g, c}]  su 
\(\tilde{e}_{{i}}\)  Se[{g}]  se  \(\nu ^i_{{i}}\)  SvIm[{g}]  nI 
\(\nu ^R_{{i}}\)  SvRe[{g}]  nR  \(h_{{i}}\)  hh[{g}]  h 
\(A^0_{{i}}\)  Ah[{g}]  Ah  \(H^_{{i}}\)  Hpm[{g}]  {Hm, Hp} 
\(g_{{\alpha \rho }}\)  VG[{c, lorentz}]  g  \(\gamma _{{\rho }}\)  VP[{lorentz}]  A 
\(Z_{{\rho }}\)  VZ[{lorentz}]  Z  \({Z'}_{{\rho }}\)  VZp[{lorentz}]  Zp 
\(W^_{{\rho }}\)  VWm[{lorentz}]  {Wm, Wp}  
\(\eta ^G_{{\alpha }}\)  gG[{c}]  gG  \(\eta ^{\gamma }\)  gP  gA 
\(\eta ^Z\)  gZ  gZ  \(\eta ^{Z'}\)  gZp  gZp 
\(\eta ^\)  gWm  gWm  \(\eta ^+\)  gWmC  gWpC 
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
Names of parameters in the considered model used by SARAH within Mathematica and in the output for other codes
LaTeX  SARAH  Output  LaTeX  SARAH  Output  LaTeX  SARAH  Output 

\(g_1\)  g1  g1  \(g_2\)  g2  g2  \(g_3\)  g3  g3 
\(g_{X}\)  gX  gX  \(g_{Y X}\)  g1X  gYX  \(g_{X Y}\)  gX1  gXY 
lw  lw  lw  \(L_{lw}\)  L[lw]  Llw  \(\tilde{M}_E\)  MtE  MtE 
\(\tilde{B}_E\)  B[MtE]  BtE  \(M_E\)  MVE  mve  \(B_{E}\)  B[MVE]  Bmve 
\(\mu \)  \(\backslash \) [Mu]  Mu  \(B_{\mu }\)  B[ \(\backslash \) [Mu]]  Bmu  \(M_S\)  MS  ms 
\(B_{S}\)  B[MS]  Bms  \(\tilde{M}_U\)  MtU  MtU  \(\tilde{B}_U\)  B[MtU]  BtU 
\(M_U\)  MVU  mvu  \(B_{U}\)  B[MVU]  Bmvu  \(Y_d\)  Yd  Yd 
\(T_d\)  T[Yd]  Td  \(Y_e\)  Ye  Ye  \(T_e\)  T[Ye]  Te 
\({Y'_e}\)  Yep  yep  \({T'_{e}}\)  T[Yep]  Tyep  \({\lambda }_{C}\)  lambdaC  lamc 
\(T_{{\lambda }_{C}}\)  T[lambdaC]  Tlc  \({\lambda }_{E}\)  lambdaE  lame  \(T_{{\lambda }_{E}}\)  T[lambdaE]  Tle 
\({\lambda }_{H}\)  lambdaH  lamh  \(T_{{\lambda }_{H}}\)  T[lambdaH]  Tlh  \(\kappa \)  kappa  kap 
\(T_{\kappa }\)  T[kappa]  Tkap  \({\lambda }_{U}\)  lambdaU  lamu  \(T_{{\lambda }_{U}}\)  T[lambdaU]  Tlu 
\(Y_u\)  Yu  Yu  \(T_u\)  T[Yu]  Tu  \({Y'_u}\)  Yup  yup 
\({T'_{u}}\)  T[Yup]  Tyup  \(Y_x\)  Yn  Yx  \(T_x\)  T[Yn]  Tx 
\(Y_\nu \)  Yv  Yv  \(T_\nu \)  T[Yv]  Tv  \(m_q^2\)  mq2  mq2 
\(m_l^2\)  ml2  ml2  \(m_{H_d}^2\)  mHd2  mHd2  \(m_{H_u}^2\)  mHu2  mHu2 
\(m_d^2\)  md2  md2  \(m_u^2\)  mu2  mu2  \(m^2_{uUX}\)  muUX2  muux2 
\(m_e^2\)  me2  me2  \(m^2_{eEX}\)  meEX2  meex2  \(m_{\nu }^2\)  mvR2  mv2 
\(m_{\eta }^2\)  mC12  mC12  \(m_{\bar{\eta }}^2\)  mC22  mC22  \(m^2_{S}\)  mS2  ms2 
\(m^2_{UX}\)  mUX2  mux2  \(m^2_{UXp}\)  mUXp2  muxp2  \(m^2_{EX}\)  mEX2  mex2 
\(m^2_{EXp}\)  mEXp2  mexp2  \(M_1\)  MassB  M1  \(M_2\)  MassWB  M2 
\(M_3\)  MassG  M3  \({M}_{BL}\)  MassBX  MBp  \({M}_{B B'}\)  MassBBX  MBBp 
\(v_d\)  vd  vd  \(v_u\)  vu  vu  \(v_{\eta }\)  x1  x1 
\(v_{\bar{\eta }}\)  x2  x2  xS  xS  xS  \(Z^{\gamma Z Z'}\)  ZZ  ZZ 
\(Z^{W}\)  ZW  ZW  \(Z^{\tilde{W}}\)  ZfW  ZfW  \(\phi _{\tilde{g}}\)  PhaseGlu  pG 
\(Z^D\)  ZD  ZD  \(Z^U\)  ZU  ZU  \(Z^E\)  ZE  ZE 
\(Z^i\)  ZVI  ZVI  \(Z^R\)  ZVR  ZVR  \(Z^H\)  ZH  ZH 
\(Z^A\)  ZA  ZA  \(Z^+\)  ZP  ZP  N  ZN  ZN 
U  UM  UM  V  UP  UP  \(U^V\)  UV  UV 
\(U^e_L\)  ZEL  ZEL  \(U^e_R\)  ZER  ZER  \(U^d_L\)  ZDL  ZDL 
\(U^d_R\)  ZDR  ZDR  \(U^u_L\)  ZUL  ZUL  \(U^u_R\)  ZUR  ZUR 
e  e  el  \(\Theta _W\)  ThetaW  TW  \(\beta \)  \(\backslash \) [Beta]  betaH 
\({\Theta '}_W\)  ThetaWp  TWp  \(\alpha ^{1}\)  v  v  v 
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 pseudoscalar We use the solution and our assumptions to get simpler expressions for the mass matrix of the CPeven (called hh) and CPodd (called Ah) scalars:
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 CPeven and CPodd 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 CPodd 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 CPodd state is, as expected, mainly a singlet while the CPeven one is mainly a XHiggs (composed by \(\phi _\eta \) and \(\phi _{\bar{\eta }}\)). That looks already very promising.
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 treelevel mass well above 100 GeV, while for \(m_{\bar{\eta }} \ll M_{Z'}\) the treelevel 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 treelevel mass of 38 GeV for the lightest CPeven 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 CPeven and CPodd sneutrinos. We assume that flavour and leftright mixing effects are negligible. In that case, it is sufficient to take a look only at the (4,4) entry of the mass matrices:
We can now check the vertex \(A \tilde{\nu }^I \tilde{\nu }^R\) using the same assumptions:
5.2.5 Vectorlike 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 vectorlike fermions can be obtained from SARAH by calling
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 oneloop RGEs with SARAH via

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 2loop RGEs for models with many new interactions can be very timeconsuming. However, often one is only interested in the dominant effects of the new contributions at the 1loop level. Therefore, IgnoreAt2Loop > $LIST can be used to neglect parameters at the twoloop level The entries of $LIST can be superpotential or soft SUSYbreaking parameters as well as gauge couplings.

WriteFunctionsToRun: Defines if a file should be written to evaluate the RGEs numerically in Mathematica
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
5.2.7 Boundary conditions and free parameters
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 subdirectory of SPheno3.3.8 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 timeconsuming 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 oneloop (and partially twoloop) level. In particular, the loop corrections from the vectorlike states are known to be very important. However, the focus in the literature has usually been only on the impact on the SMlike Higgs. We can automatically go beyond that and consider the corrections to the 750 GeV state as well. Moreover, SPheno calculates all additional twoloop corrections in the gaugeless limit including all new matter interactions. Thus, we can check the impact of the vectorlike states even at twoloop 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.
5.3.1 New loop corrections to the SMlike Higgs
Furthermore, in models with nondecoupling Dterms 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 SMlike Higgs pole mass at tree and oneloop level as a function of \(g_X\) for two different values of \(M_{Z'}\). Since SPheno performs the twoloop corrections in the gaugeless limit, additional corrections from the extended gauge sector are not included at twoloop, and we concentrate on the oneloop 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:
5.3.2 Loop corrections to the 750 GeV scalar
5.4 Diphoton and digluon rate
We now discuss the diphoton and digluon decay rate of the pseudoscalar, and its dependence on the new Yukawalike couplings. As we have just seen, large couplings induce a nonnegligible 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
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 pseudoscalar 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 digluontodiphoton width from about 10 (LO only) to 20 (including higher orders).
5.5 Constraints on choice of parameters
5.5.1 Singletdoublet mixing
5.5.2 Constraints from Higgs coupling measurements
5.5.3 Large decay width and constraints from vacuum stability
We can now run a point with SPheno. If we turn on
We see that we can get a large total width of the pseudoscalar 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 lifetime 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 Rparity 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 SARAHbased spectrum generator by copying the spectrum file into the main directory of the current MicrOmegas project directory.^{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 scan^{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 righthanded 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
If other vectorlike states mixing with the lefthanded quarks or the righthanded downlike quarks are present – as would be the case for instance when assuming 5 or 10plets of SU(5) – there would also be stringent constraints on their couplings: they would cause treelevel 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
In principle, one could also change the mass directly in the param_card without rerunning SPheno for each point. However, the advantage of SPheno is that it calculates the width of the \(Z'\) gauge boson including SUSY and nonSUSY 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 gaugekinetic 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 gaugekinetic 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 weaklycoupled 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 MonteCarlo 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 nondecoupling Dterms 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 Rparity violation become important. These limits can be checked by using the interface to Vevacious.
Footnotes
 1.
 2.
Here, the total width is simply the sum of the diphoton and digluon channels ignoring small contributions from other subdominant channels.
 3.
NLO corrections in the case of a CPodd vanish in the limit \(m_f \gg m_S\), see Sect. 3.5 for more detail.
 4.
This diphoton excess could be triggered by strong interactions. Of course, in this case one cannot use perturbative methods to understand it.
 5.
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 SMlike scalar H with \(m_H={750}\,{\text {GeV}}\) via toploops. 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.
 6.
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 vectorlike quarks are given by the Yukawa couplings \(f_Y\) and \(f_X\).
 7.
Even threebody decays into another scalar and two fermions can be calculated with SPheno.
 8.
The MSSM with and without trilinear Rparity 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 fourbody 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 PecceiQuinn or Rsymmetry), a very delicate finetuning 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 finelytuned parameter regions with a low UV cutoff.
 9.
See [458] for a complete discussion of 331 models with generic \(\beta \).
 10.
Equation (4.21) assumes that the SU(3) generators are \(T_a = \frac{\lambda _a}{2}\), with \(\lambda _a\) (\(a=1,\ldots ,8\)) the GellMann matrices. However, this is not the convention used in SARAH, see below.
 11.
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.
 12.
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.
 13.
The mixing between the MSSM scalars and S is vanishing here only because of our simplifying assumption \(\lambda =0\) but is nonzero in general.
 14.
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.
 15.
This choice might be a bit unlucky but shows the dangers of the twoloop effective potential calculation: in the gaugeless limit, one of the pseudoscalars has a treelevel 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.
 16.
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 sixdimensional potential.
 17.
If the spectrum file is not called SPheno.spc.$MODEL, one can change the filename by editing the fourth line in func1.mdl written by SARAH.
 18.
The relic density calculation for this model can be very timeconsuming, especially for the sneutrinos where a large number of coannihilation 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.
Notes
Acknowledgments
We thank Martin Winkler for betatesting and valuable impact, and Michael Spira for helpful discussions. Mark Goodsell acknowledges support from Agence Nationale de Recherche Grant ANR15CE310002 “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 (2713463B731) funded by the Spanish MINECO as well as from the Spanish Grants FPA201458183P, Multidark CSD200900064 and SEV20140398 (MINECO), FPA201122975 and PROMETEOII/2014/084 (Generalitat Valenciana), and is grateful to WeiChih 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 leftright models. Lorenzo Basso acknowledges support by the OCEVU Labex (ANR11LABX 0060) and the A*MIDEX project (ANR11IDEX000102), 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 Terascale. Dylan Harries is supported by the University of Adelaide and the ARC Centre of Excellence for Particle Physics at the Terascale. Florian Staub thanks Alfredo Urbano for discussions about the GeorgiMachacek model and for testing the model file.
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