The Virtual Diphoton Excess
Abstract
Interpreting the excesses around 750 GeV in the diphoton spectra to be the signal of a new heavy scalar decaying to photons, we point out the possibility of looking for correlated signals with virtual photons. In particular, we emphasize that the effective operator that generates the decay will also generate decays of () and () independently of the couplings to and . Depending on the relative sizes of these effective couplings, we show that the virtual diphoton component can make up a sizable, and sometimes dominant, contribution to the total and partial widths. We also discuss modifications to current experimental cuts in order to maximize the sensitivity to these virtual photon effects. Finally, we briefly comment on prospects for channels involving other Standard Model fermions as well as more exotic decay possibilities of the putative resonance.
I Introduction
There has been tremendous interest in the excesses recently reported by both ATLAS collaboration (2015) and CMS Collaboration (2015) in the diphoton spectrum around 750 GeV. If this is a sign of a new resonance, the simplest explanation for the decay is through the photon field strength or dual field strength tensor. For concreteness we will consider the dual field strength case via the dimension five operator
(1) 
where and . We take to be some new mass scale associated with this operator that will cancel in all the ratios we will consider. Our choice of operator in Eq. (1) implies the new resonance is a parity odd scalar, but our considerations largely apply if it turns out to be a parity even or CP violating scalar as well as a spin 2 resonance.
Assuming electroweak gauge symmetry holds in the UV, the operator in Eq. (1) must descend from a linear combination of the operators Low and Lykken (2015):
(2) 
As has already been pointed out many times Cai et al. (2015); Harigaya and Nomura (2015); Mambrini et al. (2015); Nakai et al. (2015); Knapen et al. (2015); Buttazzo et al. (2015); Franceschini et al. (2015); Higaki et al. (2015); McDermott et al. (2015); Ellis et al. (2015); Bellazzini et al. (2015); Low et al. (2015); Gupta et al. (2015); Molinaro et al. (2015); Cao et al. (2015a); Matsuzaki and Yamawaki (2015); Kobakhidze et al. (2015); Cox et al. (2015); Curtin and Verhaaren (2015); Bian et al. (2015); Ahmed et al. (2015); Falkowski et al. (2015); Bai et al. (2015); Benbrik et al. (2015); Alves et al. (2015); Cao et al. (2015b); Liao and Zheng (2015); de Blas et al. (2015); Belyaev et al. (2015); Altmannshofer et al. (2015); Craig et al. (2015); Cao et al. (2015c); Dev et al. (2015), these operators will lead to correlated signals in decays to and , as well as if is nonzero. Searches for diboson resonances have been performed by ATLAS Aad et al. (2014, 2015a, 2015b) and CMS Khachatryan et al. (2014) placing constraints on models which can explain the diphoton resonance.
In this letter, we emphasize that the operator in Eq. (1) alone is enough to produce and decays of the resonance through virtual photons, irrespective of its UV origin. We examine under which circumstances the virtual photon component makes up a sizable contribution, or even dominates over the and components, to these three and four body decays, with particular emphasis on the leptonic and channels.
We also examine what effects cuts on the lepton invariant masses have on the relative composition of the and partial widths. Should the diphoton excess persist, then knowing the mass of will allow a search for and decays imposing only minimal constraints on any subset of the final states. We take advantage of this to motivate modifying current experimental searches in the and channels in order to maximize the sensitivity to the virtual diphoton effects. We also briefly discuss possibilities in the less experimentally clean decays to other SM fermions.
Ii Decay of to
If there is indeed a new particle decaying to , then it will also decay to via a virtual photon. The rate of this decay is strongly sensitive to the phase space cuts, particularly on the invariant mass of the lepton pair. In particular, if an experimental analysis allows lepton pairs with an invariant mass between and , then the ratio of partial widths gives
(3) 
The factor of comes from the additional photon coupling, while the log comes from integrating the photon propagator over the phase space. From this formula we see that if a search has a narrow invariant mass window around the pole, as in the ATLAS search Aad et al. (2014) which requires GeV, then the effects from virtual photons will be tiny. On the other hand, making a search as inclusive as possible will raise the rate from virtual photons even in the absence of contributions from ’s.
Of course most models that explain the diphoton excess via Eq. (1) will also generate the operator
(4) 
Naively, the effects from this operator should be parametrically larger than the the operator since the can be produced on shell. However, the suppression is not nearly so large for two important reasons:

The coupling to leptons is suppressed relative to that of the photon.

Unlike the photon, there is no log enhancement when integrating the region of phase space away from the pole.
Therefore, if the phase space cuts are very inclusive, the offshell photon can be an important effect.
In Fig. 1 we plot the three different contributions to the process as a function of the ratio of couplings
(5) 
We have normalized the three components of to the partial width so the ratio involving the component (blue curve) is flat. We plot these ratios for both ATLASlike phase space cuts (solid) and for much more inclusive ‘Full’ cuts^{1}^{1}1Note that we have only considered cuts on the lepton invariant masses and not on the lepton or rapidity. Since the rate is dominated by the pole structure of the vector boson propogators, this simplifications captures qualitatively the features we wish to emphasize in this study. with (dashed). The lower cutoff of GeV is inspired by studies looking for similar offshell photon effects involving the Higgs boson at 125 GeV GonzalezAlonso and Isidori (2014); Chen et al. (2015a, b). We see that with these relaxed phase space cuts, the component can be a few per cent of the onshell rate because the log in Eq. (3) is large, while with current cuts the virtual photon contribution is an order of magnitude smaller.
We also see in Fig. 1 that for small , the component dominates, while for large the component dominates as expected. Another expected feature is that the contribution from is relatively unaltered by these cuts since they both contain the Zpole. The interference between the two components is always small, but is significantly enhanced by the more inclusive cuts, making this effect potentially observable with a large number of decays. This type of interference also opens up the possibility of observing CP violation in the three body decays as proposed for the Higgs boson Chen et al. (2014a).
From the ATLAS 8 TeV search Aad et al. (2014), one can bound the cross section into , although the bound depends on how the cross section scales going from 8 to 13 TeV. In the case of gluon initiated production, the two body decay is limited to be about twice (see for example Franceschini et al. (2015)) so we place a grey vertical line to indicate this limit. The production mechanism could however be photon Fichet et al. (2015); Csaki et al. (2015) or quark Franceschini et al. (2015); Gao et al. (2015) initiated, or perhaps some more exotic production mechanism Cho et al. (2015); Li et al. (2015); An et al. (2015); Bernon and Smith (2015); Liu et al. (2015). Therefore, we show results for even larger values of due to this uncertainty.
The central observation of this study is that the invariant mass spectrum of the lepton pair (rather than the full system) contains significant information on the couplings of the new resonance to gauge bosons. In Fig. 2 we plot the normalized invariant mass distributions for two extreme values (10 and 0.1) of the ratio of couplings defined in Eq. (5). We also show the two simple cases of (red) and (green) using the operators in Eq. (2). These predict and respectively Low and Lykken (2015), where is the Weinberg angle. Unsurprisingly, we see that larger values of raises the height of the peak around the pole, while lower values raises the value at low . A perhaps more unexpected feature, is that for low values of the ratio there are also more events at high above the peak. This comes from the fact that the distributions are normalized so the peak is not as large.
We can exploit the fact that the virtual photon and have very different distributions in the invariant mass of the lepton pair to make a crude but very simple measurement of . The idea is to simply take the fraction of events that have leptons near the pole:
(6) 
where the total number of events is defined by the inclusive phase space cuts GeV. As can be seen in Fig. 3, is strongly dependent on . We plot various different values of the mass window , and we see that for , the slope of the curve is large and this variable becomes quite sensitive. For larger couplings, the virtual photon contribution to this channel becomes subdominant and this observable becomes less sensitive. In this case, however, the total rate of events will be larger so a more statistically precise measurement will be possible.
One could imagine varying in an experimental analysis to get more information about this coupling ratio. Taking this to the extreme and using the full phase space information contained in the differential mass distribution event by event would allow for even better measurements. Of course using a socalled matrix element method where the likelihood is constructed from the fully differential decay width using all observables in uses the maximum amount of information. Furthermore, at 750 GeV these kinematic observables may be more discriminating than was found for a 125 GeV Higgs boson Gainer et al. (2012) decaying to . However, we leave a fully differential analysis utilizing all observables in using the framework of Chen et al. (2013); Chen and VegaMorales (2014); Chen et al. (2014a, 2015c) to ongoing work Chen et al. (2016).
Finally, we briefly comment on backgrounds. The dominant background around 750 GeV in the current search Aad et al. (2014) comes from genuine , while a jet faking a photon is the second most important but highly subdominant. The dominant background has been calculated very precisely in both the and initial states Ametller et al. (1985); van der Bij and Glover (1988); Ohnemus (1993); Baur et al. (1998); De Florian and Signer (2000); Adamson et al. (2003); Hollik and Meier (2004); Accomando et al. (2006); Campbell et al. (2011); Hamilton et al. (2012); Grazzini et al. (2014); Barze et al. (2014); Denner et al. (2015); Grazzini et al. (2015). A crude estimate using treelevel Madgraph Alwall et al. (2014) simulation finds that opening the lepton invariant mass cut from being just around the pole to simply requiring GeV roughly doubles the background. This should also give a reasonable estimate for the fake photon background because the underlying process is + jets, so the invariant mass distribution when a photon is replaced with a jet should be similar. Ultimately, the background is smooth and rapidly falling in the center of mass energy, allowing for good background discrimination with a simple sideband analysis. Therefore, we do not expect relaxing the cuts on the lepton pair invariant mass to be an obstruction for enhancing the virtual diphoton signal.
Iii Decays to four leptons
We now turn to four body decays where again . In this case the operator
(7) 
will also contribute and is naively the dominant effect due to the fact that both bosons can be onshell at GeV. There are however, still contributions from the and operators studied in the previous section. If these operators descend only from the invariant operators of Eq. (2), then there are only two unknowns and the system is overconstrained. Therefore, measuring the contribution of all three operators is a nontrivial test of the SM gauge symmetry at the scale of the mass of the new resonance. While the rate alone is not enough to measure all three operators, a fully differential analysis may be able to determine all three in a single channel Chen et al. (2016), but we do not explore this here.
The current best limits for decays to in Run I come from the channel Aad et al. (2015a) from which one can extract that the decay to is at most a factor of six bigger than the rate to Franceschini et al. (2015) assuming that is produced from gluon initial states. This channel has a significantly higher branching ratio than the channel, but suffers from a worse signal to background ratio. Therefore, this search requires that both pairs of objects are roughly on the pole. There is also a search for decays to four leptons Aad et al. (2015b) which has a significantly smaller rate, but is experimentally much cleaner. In this search, there is also a requirement that one lepton pair invariant mass be between 50 and 120 GeV while the second is required to be between 12 and 120 GeV. This not only reduces the total signal rate, but also the relative size of any non contribution to , analogous to the three body case of described above.
Here we will study ratios of partial widths involving in the two dimensional parameter space of defined in Eq. (5) and a second ratio of couplings,
(8) 
The kinematics of are more complicated than and have been studied at length in the context of a heavy Higgs decay (see for example Soni and Xu (1993); Barger et al. (1994); Choi et al. (2003); Buszello et al. (2004); Gao et al. (2010)). Although there are multiple angular observables which contain useful information, in this simplified study we focus on the information contained in the two invariant mass distributions of the lepton pairs. In particular, as with our study of , we examine how the rate as well as its composition in terms of the , and components is affected by phase space cuts on the invariant mass of the lepton pairs.
We label the lepton pair invariant masses and and define following the conventions and definitions in Chen et al. (2013); Chen and VegaMorales (2014). Since we are considering only rates, the difference between the and channels due to identical final state interference is negligible. However, as pointed out in Chen et al. (2015a), these identical final state effects can greatly influence event selection and these channels should be treated separately in a more complete fully differential likelihood analysis Chen and VegaMorales (2014); Chen et al. (2014b, 2015c, 2015a). Since these subtleties are not relevant for current purposes, we simply study the channel and multiply by a factor of two to include and .
We first consider the ratio of the rate to the decay rate as shown Fig. 4. As with the Higgs boson at 125 GeV, this ratio will not be very large, but this is compensated by the very high precision with which it can be measured Djouadi et al. (2015). Depending on the coupling ratios, the rate will not be bigger than of the rate for coupling ratios which are still allowed by and direct searches Franceschini et al. (2015). This happens to be roughly the same as for the 125 GeV Higgs boson where this ratio is Dittmaier et al. (2011); Andersen et al. (2013). As the Higgs boson was discovered in both and Chatrchyan et al. (2012); Aad et al. (2012), this gives some hope that if the GeV diphoton excess persists, a signal in may also be observable soon.
From Fig. 4, we also see that the rate can be enhanced by going to more inclusive phase space cuts: GeV, compared to those used by the ATLAS search Aad et al. (2015a) which requires GeV and GeV. The effect is largest when since in this case the and components make up a larger fraction of . Thus phase space cuts have a larger effect compared to when dominates, since in that case both bosons can be onshell in either the more inclusive or the ATLASlike cuts. We again show values of and larger than allowed by and searches Franceschini et al. (2015) due to the various assumptions which go into these limits as discussed above.
In Fig. 5 we show the relative contribution of the naively subdominant components to , namely those arising from (blue) and (orange). Again we see that expanding the phase space cuts gives significantly more sensitivity to these components than current ATLAS cuts. The absolute size of the component is relatively unaffected when and as can also be inferred from Fig. 4 because the inclusive and ATLAS cut contours become very similar in that region. We also see in Fig. 5 that the component dominates when , and that the dominates when . Finally, we note the sharply rising slope for the size of the component when and , indicating a strong sensitivity in this regime.
We again propose a simple way to measure and analogous to the one from the previous section for . Namely we define a similar ratio
(9) 
where again the total number of events is defined by the inclusive phase space with GeV. We show contours of in Fig. 6 where we see that it is very sensitive to for while less sensitive to . The stronger sensitivity to can be understood from the pole structure of the two bosons which can both be onshell at 750 GeV. Again we also see the benefits of using more inclusive cuts to enhance the nonZZ components.
We have not discussed interference between the different intermediate states since it has a negligible effect on the rates. However, in a fully differential analysis where shape information is used, these interference effects can potentially be important. In particular, as has been shown in many studies of the Higgs boson, these interference effects would give us access to the CP properties of and to potential CP violating effects. An investigation of these interesting possibilities using the framework of Chen et al. (2013); Chen and VegaMorales (2014); Chen et al. (2014a, 2015c) is ongoing Chen et al. (2016).
In an experimental analysis, backgrounds must of course be taken into account, but, as with Higgs decays to four leptons, the background is very small. The dominant source of background is quark initiated production, with the oneloop gluon process also contributing, but again is very subdominant Aad et al. (2015b). As far as we know, there are no higher order calculations of these backgrounds, but that is partially because they are quite small. As with the case, enlarging the mass window from the current searches will increase the background, but it will still be small and smooth, so a sideband analyses can again be used. Of course using a fully differential likelihood analysis would increase the ability to discriminate signal from background further Gainer et al. (2011), but we do not investigate this possibility here. For present purposes we have simply used naive estimates to ensure that the dominant background can easily be controlled.
Iv Nonleptonic and Exotic Decays
If the diphoton excess proves to be more than a statistical fluctuation and indeed due to a new scalar , we will want to search for decays in as many channels as possible, not just the experimentally clean ones with leptons. Furthermore, our considerations of the virtual diphoton contributions to and also apply when one considers other charged fermions in the SM, though of course experimentally these channels are much less cleanly measured. While the branching ratios and couplings of the and photon are well measured, looking for decays in and is an important test to see if there is other new physics or couplings of to SM fermions.
In Fig. 7 we consider the (where ) partial width normalized to for the various light SM fermions. We see that for small the leptons (solid red) dominate. By comparing the solid red curve, which is the full decay width, to the dashed red curve which is only the onshell mediated width, we see that the low behavior is dominated by photon contributions. This explains why the leptons are the largest contribution at small , since they have larger electric charge than SM quarks. At larger , decays with quarks and neutrinos become more important. While these are experimentally more difficult, kinematic shape information can perhaps be used to uncover the signal from the background, though we do not explore this issue here. We also note that if one imposes the limit (vertical line) derived from Franceschini et al. (2015), this implies a limit on () and () of and of the rate, respectively. Of course, all the caveats discussed above about the production mechanism still apply.
For the case of four fermion decays, there are many more possibilities including and which are also experimentally challenging. They are expected however to have much larger rates than in much of the parameter space, particularly when the coupling is not parametrically larger than for and . One can also consider decays to or other channels. While the computations utilized in this work can be extended to these cases as well, the experimental analyses become more difficult and backgrounds have to be treated more carefully. If the resonance at 750 GeV turns out to be genuine new physics, fully understanding all these channels will be crucial to characterizing the new state and any theory it might be associated with.
Finally, we note that the simplified analysis presented here is also useful if the new physics is not one simple resonance decaying to diphoton but is instead multiple resonances Franceschini et al. (2015); Potter (2016), not a resonance Cho et al. (2015); Li et al. (2015); An et al. (2015); Bernon and Smith (2015); Liu et al. (2015), or a resonance that decays through a cascade Knapen et al. (2015); Falkowski et al. (2015); Agrawal et al. (2015); Chang et al. (2015); Chala et al. (2015). Each of these kinds of models has different predictions for both the correlated searches via and as well as with virtual photons. Furthermore, the improved signal to background ratio, particularly in the case of four leptons, will allow a more precise measurement of the lineshape allowing discrimination of many possibilities. Should the excess persist, an exploration of these cases would also be interesting.
V Conclusions and Outlook
In this work, we interpret the excess observed by ATLAS and CMS in the diphoton spectra around 750 GeV to be indicative of a new scalar resonance decaying to photons. We show in particular that the effective operator responsible for the decay will also lead to a signal in and (where is a SM fermion) decays independently of the effective couplings of to and . We have focused in particular on the leptonic and channels (). Depending on the relative sizes of these effective couplings, we show that the virtual diphoton component can make up a sizable, and sometimes dominant, contribution to the total and partial widths.
We have also explored the effects that phase space cuts on the invariant mass of the lepton pairs have on the total rates and composition of and . We have emphasized the contribution from virtual photons and pointed out that current experimental searches should be modified in order to enhance the sensitivity to these virtual photon effects. We find that a more inclusive phase space cut (while still requiring the full system to be at the resonance mass) would allow an increased signal rate and larger contributions from all components of and . The virtual photon contributions in particular can be increased by an order of magnitude. This allows us to study in more detail while still keeping the backgrounds under control.
Finally, we have used a simple cut and count method with ratios of partial widths to assess the potential sensitivity of and to ratios of effective couplings between and , , and . We find particularly strong sensitivity when the effective coupling of to is larger than to or . A full analysis taking advantage of all the final state kinematics can reveal more about the nature of the resonance, but we have left this to ongoing work. We have also briefly discussed nonleptonic channels and potential applications of our analysis methods to more exotic possibilities for explaining the diphoton excess. Should the excess persist, the methods utilized and discussed here will prove useful for ascertaining the nature of the putative new resonance.
Acknowledgments: We would like to thank Yi Chen, Jose Santiago, and Jorge de Blas for useful conversations. R.V.M. is supported by MINECO, under grant number FPA201347836C32P.
References
 collaboration (2015) T. A. collaboration (ATLAS) (2015), eprint ATLASCONF2015081.
 Collaboration (2015) C. Collaboration (CMS) (2015), eprint CMSPASEXO15004.
 Low and Lykken (2015) I. Low and J. Lykken (2015), eprint 1512.09089.
 Cai et al. (2015) H. Cai, T. Flacke, and M. Lespinasse (2015), eprint 1512.04508.
 Harigaya and Nomura (2015) K. Harigaya and Y. Nomura (2015), eprint 1512.04850.
 Mambrini et al. (2015) Y. Mambrini, G. Arcadi, and A. Djouadi (2015), eprint 1512.04913.
 Nakai et al. (2015) Y. Nakai, R. Sato, and K. Tobioka (2015), eprint 1512.04924.
 Knapen et al. (2015) S. Knapen, T. Melia, M. Papucci, and K. Zurek (2015), eprint 1512.04928.
 Buttazzo et al. (2015) D. Buttazzo, A. Greljo, and D. Marzocca (2015), eprint 1512.04929.
 Franceschini et al. (2015) R. Franceschini, G. F. Giudice, J. F. Kamenik, M. McCullough, A. Pomarol, R. Rattazzi, M. Redi, F. Riva, A. Strumia, and R. Torre (2015), eprint 1512.04933.
 Higaki et al. (2015) T. Higaki, K. S. Jeong, N. Kitajima, and F. Takahashi (2015), eprint 1512.05295.
 McDermott et al. (2015) S. D. McDermott, P. Meade, and H. Ramani (2015), eprint 1512.05326.
 Ellis et al. (2015) J. Ellis, S. A. R. Ellis, J. Quevillon, V. Sanz, and T. You (2015), eprint 1512.05327.
 Bellazzini et al. (2015) B. Bellazzini, R. Franceschini, F. Sala, and J. Serra (2015), eprint 1512.05330.
 Low et al. (2015) M. Low, A. Tesi, and L.T. Wang (2015), eprint 1512.05328.
 Gupta et al. (2015) R. S. Gupta, S. Jager, Y. Kats, G. Perez, and E. Stamou (2015), eprint 1512.05332.
 Molinaro et al. (2015) E. Molinaro, F. Sannino, and N. Vignaroli (2015), eprint 1512.05334.
 Cao et al. (2015a) Q.H. Cao, Y. Liu, K.P. Xie, B. Yan, and D.M. Zhang (2015a), eprint 1512.05542.
 Matsuzaki and Yamawaki (2015) S. Matsuzaki and K. Yamawaki (2015), eprint 1512.05564.
 Kobakhidze et al. (2015) A. Kobakhidze, F. Wang, L. Wu, J. M. Yang, and M. Zhang (2015), eprint 1512.05585.
 Cox et al. (2015) P. Cox, A. D. Medina, T. S. Ray, and A. Spray (2015), eprint 1512.05618.
 Curtin and Verhaaren (2015) D. Curtin and C. B. Verhaaren (2015), eprint 1512.05753.
 Bian et al. (2015) L. Bian, N. Chen, D. Liu, and J. Shu (2015), eprint 1512.05759.
 Ahmed et al. (2015) A. Ahmed, B. M. Dillon, B. Grzadkowski, J. F. Gunion, and Y. Jiang (2015), eprint 1512.05771.
 Falkowski et al. (2015) A. Falkowski, O. Slone, and T. Volansky (2015), eprint 1512.05777.
 Bai et al. (2015) Y. Bai, J. Berger, and R. Lu (2015), eprint 1512.05779.
 Benbrik et al. (2015) R. Benbrik, C.H. Chen, and T. Nomura (2015), eprint 1512.06028.
 Alves et al. (2015) A. Alves, A. G. Dias, and K. Sinha (2015), eprint 1512.06091.
 Cao et al. (2015b) J. Cao, C. Han, L. Shang, W. Su, J. M. Yang, and Y. Zhang (2015b), eprint 1512.06728.
 Liao and Zheng (2015) W. Liao and H.q. Zheng (2015), eprint 1512.06741.
 de Blas et al. (2015) J. de Blas, J. Santiago, and R. VegaMorales (2015), eprint 1512.07229.
 Belyaev et al. (2015) A. Belyaev, G. Cacciapaglia, H. Cai, T. Flacke, A. Parolini, and H. Serodio (2015), eprint 1512.07242.
 Altmannshofer et al. (2015) W. Altmannshofer, J. Galloway, S. Gori, A. L. Kagan, A. Martin, and J. Zupan (2015), eprint 1512.07616.
 Craig et al. (2015) N. Craig, P. Draper, C. Kilic, and S. Thomas (2015), eprint 1512.07733.
 Cao et al. (2015c) Q.H. Cao, Y. Liu, K.P. Xie, B. Yan, and D.M. Zhang (2015c), eprint 1512.08441.
 Dev et al. (2015) P. S. B. Dev, R. N. Mohapatra, and Y. Zhang (2015), eprint 1512.08507.
 Aad et al. (2014) G. Aad et al. (ATLAS), Phys. Lett. B738, 428 (2014), eprint 1407.8150.
 Aad et al. (2015a) G. Aad et al. (ATLAS) (2015a), eprint 1512.05099.
 Aad et al. (2015b) G. Aad et al. (ATLAS) (2015b), eprint 1509.07844.
 Khachatryan et al. (2014) V. Khachatryan et al. (CMS), JHEP 08, 174 (2014), eprint 1405.3447.
 Chen et al. (2013) Y. Chen, N. Tran, and R. VegaMorales, JHEP 1301, 182 (2013), eprint 1211.1959.
 Chen and VegaMorales (2014) Y. Chen and R. VegaMorales, JHEP 1404, 057 (2014), eprint 1310.2893.
 Chen et al. (2014a) Y. Chen, A. Falkowski, I. Low, and R. VegaMorales, Phys.Rev. D90, 113006 (2014a), eprint 1405.6723.
 GonzalezAlonso and Isidori (2014) M. GonzalezAlonso and G. Isidori, Phys. Lett. B733, 359 (2014), eprint 1403.2648.
 Chen et al. (2015a) Y. Chen, R. Harnik, and R. VegaMorales (2015a), eprint 1503.05855.
 Chen et al. (2015b) Y. Chen, D. Stolarski, and R. VegaMorales, Phys. Rev. D92, 053003 (2015b), eprint 1505.01168.
 Fichet et al. (2015) S. Fichet, G. von Gersdorff, and C. Royon (2015), eprint 1512.05751.
 Csaki et al. (2015) C. Csaki, J. Hubisz, and J. Terning (2015), eprint 1512.05776.
 Gao et al. (2015) J. Gao, H. Zhang, and H. X. Zhu (2015), eprint 1512.08478.
 Cho et al. (2015) W. S. Cho, D. Kim, K. Kong, S. H. Lim, K. T. Matchev, J.C. Park, and M. Park (2015), eprint 1512.06824.
 Li et al. (2015) G. Li, Y.n. Mao, Y.L. Tang, C. Zhang, Y. Zhou, and S.h. Zhu (2015), eprint 1512.08255.
 An et al. (2015) H. An, C. Cheung, and Y. Zhang (2015), eprint 1512.08378.
 Bernon and Smith (2015) J. Bernon and C. Smith (2015), eprint 1512.06113.
 Liu et al. (2015) J. Liu, X.P. Wang, and W. Xue (2015), eprint 1512.07885.
 Gainer et al. (2012) J. S. Gainer, W.Y. Keung, I. Low, and P. Schwaller, Phys. Rev. D86, 033010 (2012), eprint 1112.1405.
 Chen et al. (2015c) Y. Chen, E. Di Marco, J. Lykken, M. Spiropulu, R. VegaMorales, et al., JHEP 1501, 125 (2015c), eprint 1401.2077.
 Chen et al. (2016) Y. Chen, D. Stolarski, R. VegaMorales, et al. (2016), eprint Work in progress.
 Ametller et al. (1985) L. Ametller, E. Gava, N. Paver, and D. Treleani, Phys. Rev. D32, 1699 (1985).
 van der Bij and Glover (1988) J. J. van der Bij and E. W. N. Glover, Phys. Lett. B206, 701 (1988).
 Ohnemus (1993) J. Ohnemus, Phys. Rev. D47, 940 (1993).
 Baur et al. (1998) U. Baur, T. Han, and J. Ohnemus, Phys. Rev. D57, 2823 (1998), eprint hepph/9710416.
 De Florian and Signer (2000) D. De Florian and A. Signer, Eur. Phys. J. C16, 105 (2000), eprint hepph/0002138.
 Adamson et al. (2003) K. L. Adamson, D. de Florian, and A. Signer, Phys. Rev. D67, 034016 (2003), eprint hepph/0211295.
 Hollik and Meier (2004) W. Hollik and C. Meier, Phys. Lett. B590, 69 (2004), eprint hepph/0402281.
 Accomando et al. (2006) E. Accomando, A. Denner, and C. Meier, Eur. Phys. J. C47, 125 (2006), eprint hepph/0509234.
 Campbell et al. (2011) J. M. Campbell, R. K. Ellis, and C. Williams, JHEP 07, 018 (2011), eprint 1105.0020.
 Hamilton et al. (2012) K. Hamilton, P. Nason, and G. Zanderighi, JHEP 10, 155 (2012), eprint 1206.3572.
 Grazzini et al. (2014) M. Grazzini, S. Kallweit, D. Rathlev, and A. Torre, Phys. Lett. B731, 204 (2014), eprint 1309.7000.
 Barze et al. (2014) L. Barze, M. Chiesa, G. Montagna, P. Nason, O. Nicrosini, F. Piccinini, and V. Prosperi, JHEP 12, 039 (2014), eprint 1408.5766.
 Denner et al. (2015) A. Denner, S. Dittmaier, M. Hecht, and C. Pasold, JHEP 04, 018 (2015), eprint 1412.7421.
 Grazzini et al. (2015) M. Grazzini, S. Kallweit, and D. Rathlev, JHEP 07, 085 (2015), eprint 1504.01330.
 Alwall et al. (2014) J. Alwall, R. Frederix, S. Frixione, V. Hirschi, F. Maltoni, O. Mattelaer, H. S. Shao, T. Stelzer, P. Torrielli, and M. Zaro, JHEP 07, 079 (2014), eprint 1405.0301.
 Soni and Xu (1993) A. Soni and R. M. Xu, Phys. Rev. D48, 5259 (1993), eprint hepph/9301225.
 Barger et al. (1994) V. D. Barger, K.m. Cheung, A. Djouadi, B. A. Kniehl, and P. M. Zerwas, Phys. Rev. D49, 79 (1994), eprint hepph/9306270.
 Choi et al. (2003) S. Y. Choi, D. J. Miller, M. M. Muhlleitner, and P. M. Zerwas, Phys. Lett. B553, 61 (2003), eprint hepph/0210077.
 Buszello et al. (2004) C. P. Buszello, I. Fleck, P. Marquard, and J. J. van der Bij, Eur. Phys. J. C32, 209 (2004), eprint hepph/0212396.
 Gao et al. (2010) Y. Gao, A. V. Gritsan, Z. Guo, K. Melnikov, M. Schulze, and N. V. Tran, Phys. Rev. D81, 075022 (2010), eprint 1001.3396.
 Chen et al. (2014b) Y. Chen, R. Harnik, and R. VegaMorales, Phys.Rev.Lett. 113, 191801 (2014b), eprint 1404.1336.
 Djouadi et al. (2015) A. Djouadi, J. Quevillon, and R. VegaMorales (2015), eprint 1509.03913.
 Dittmaier et al. (2011) S. Dittmaier et al. (LHC Higgs Cross Section Working Group) (2011), eprint 1101.0593.
 Andersen et al. (2013) J. R. Andersen et al. (LHC Higgs Cross Section Working Group) (2013), eprint 1307.1347.
 Chatrchyan et al. (2012) S. Chatrchyan et al. (CMS), Phys. Lett. B716, 30 (2012), eprint 1207.7235.
 Aad et al. (2012) G. Aad et al. (ATLAS), Phys. Lett. B716, 1 (2012), eprint 1207.7214.
 Gainer et al. (2011) J. S. Gainer, K. Kumar, I. Low, and R. VegaMorales, JHEP 1111, 027 (2011), eprint 1108.2274.
 Potter (2016) C. T. Potter (2016), eprint 1601.00240.
 Agrawal et al. (2015) P. Agrawal, J. Fan, B. Heidenreich, M. Reece, and M. Strassler (2015), eprint 1512.05775.
 Chang et al. (2015) J. Chang, K. Cheung, and C.T. Lu (2015), eprint 1512.06671.
 Chala et al. (2015) M. Chala, M. Duerr, F. Kahlhoefer, and K. SchmidtHoberg (2015), eprint 1512.06833.