# Flavor Changing Heavy Higgs Interactions with Leptons at Hadron Colliders

###### Abstract

In a general two Higgs doublet model, we study flavor changing neutral Higgs (FCNH) decays into leptons at hadron colliders, , where could be a CP-even scalar (, ) or a CP-odd pseudoscalar (). The light Higgs boson is found to resemble closely the Standard Model Higgs boson at the Large Hadron Collider. In the alignment limit of for – mixing, FCNH couplings of are naturally suppressed, but such couplings of the heavier are sustained by . We evaluate physics backgrounds from dominant processes with realistic acceptance cuts and tagging efficiencies. We find promising results for TeV, which we extend further to TeV and 100 TeV future pp colliders.

###### pacs:

12.60.Fr, 12.15Mm, 14.80.Ec, 14.65.Ha^{†}

^{†}preprint: The University of Oklahoma arXiv: [hep-ph] OU-HEP-181225 January 2019

## I Introduction

Recent 13 TeV studies at the LHC by ATLAS and CMS experiments confirm that the properties of the 125 GeV Higgs boson are in good agreement with the expectations from the Standard Model (SM) Higgs boson ATLAS:2018doi (); CMS:2018lkl (). This is in sharp contrast to the persistent signs of significant deviation from SM in the flavor sector. The 3–4 hints of lepton universality violation, in simple tree-level semi-leptonic B decays as well as in flavor-changing loop processes, have been very much in the news flavor_anoms_2018 (). Moreover, for over a decade now the muon anomalous magnetic moment measurement at BNL Bennett:2006fi () also seem to show about 3.5 deviation from SM. While so far none of these constitutes compelling evidence against the SM, but even if just one of them pans out, it would be physics beyond SM. In particular, it is important to recall that lepton universality is purely an accidental symmetry of SM. These underpinnings prompt us to question lepton universality and lepton flavor violation in the Higgs sector itself.

Our investigation was motivated by the experimental 2 hint for from CMS Khachatryan:2015kon (). While it has subsequently disappeared Sirunyan:2017xzt (), it in fact motivates further the search for , involving heavy exotic Higgs bosons, as we shall explain. Note in particular that, in face of the current semileptonic anomalies in B decays, a general two Higgs doublet model (g2HDM) had been invoked Crivellin:2012ye () over the disfavored conventional Type II of two Higgs doublet models (2HDM-II). While the situation with the anomalies are as yet inconclusive, we adopt the g2HDM set up in this work, i.e. without the usual symmetry to forbid flavor changing neutral Higgs (FCNH) couplings. Another mechanism may be at work instead of or Natural Flavor Conservation Glashow:1976nt (): alignment Hou:2017hiw (). Removing interactions of the extra scalars with vector boson pairs ( and ), other than the SM-Higgs, is known as the alignment limit Gunion:2002zf (); Carena:2013ooa (); Dev:2014yca (). Influenced by the LHC results on the 125 GeV boson ATLAS:2018doi (); CMS:2018lkl (), we will assume that one must work close to this limit.

We seek the discovery of the leptonic flavor changing decay, specifically , where . In SM, is highly suppressed at loop level by the extremely tiny neutrino masses, but in g2HDM without any symmetry, this decay is in principle possible at tree level. We adopt the following interaction Lagrangian Davidson:2005cw (); Mahmoudi:2009zx (),

(1) |

where , , , , and is the mixing angle between neutral Higgs scalars, in the notation Gunion:1989we () of 2HDM-II. The matrices are diagonal and fixed by fermion masses, with GeV, while matrices are in general not diagonal. The off diagonal elements of are tree level FCNH couplings. However, in the exact alignment limit of , the boson approaches SM Higgs and couples diagonally, but and can still lead to at tree level.

As mentioned, there has been a lot of interest in this FCNH interaction among leptons at the LHC. There was a excess of above the background in CMS Run 1 data, with the best fit branching fraction Khachatryan:2015kon () , which is consistent with ATLAS Run 1 result Aad:2016blu () of . But the excess was ruled out by 2016 CMS data Sirunyan:2017xzt (), with upper limit , giving the bound on FCNH coupling , where . However, may be small because of alignment, or . The leptonic FCNH Yukawa couplings of the heavy boson, would approach the FCNH coupling in strength in the alignment limit, since . While the recent CMS limit implies must be small, and can still be sizable and should be probed experimentally.

In this paper, we study the discovery potential for the decays , , followed by decays into an electron and neutrinos or into a -jet (, or ) and neutrino. Imposing the current LHC Higgs data, CMS and B physics constraints, we calculate the full tree level matrix elements for both signals and backgrounds. We use realistic acceptance cuts to reduce the backgrounds with current b-tag, tag, and mistag efficiencies. Some promising results are presented for 14 and 27 TeV center of mass (CM) energies for an integrated luminosity = 300 and 3000 fb, in sync with future High Luminosity (HL) and High Energy (HE) LHC Barletta:2013ooa (); Tomas:2016kuo (); Zimmermann:2017bbr (); Shiltsev:2017tjx ().

We discuss experimental limits on relevant parameters from B physics and LHC Higgs data in Sec. II, and give in Sec. III the production cross sections for the Higgs signal and the dominant background with realistic acceptance cuts, as well as our strategy to determine the reconstructed masses for the Higgs bosons. Sec. IV presents the discovery potential at the LHC for TeV, and also for future hadron colliders with and TeV. Optimistic conclusions are drawn in Sec. V.

## Ii Constraints on Relevant Parameters

The most relevant parameters are , for the decay , and for the production via the triangle-top loop. A potentially large induces Altunkaynak:2015twa () , which can dilute the branching ratio, while is subject to tight constraints by physics data. LHC data for the 125 GeV boson ATLAS:2018doi (); CMS:2018lkl () suggest in 2HDM-II. We take for illustration, although larger values are allowed in the general 2HDM Altunkaynak:2015twa (); Hou:2018uvr (). As for other matrix elements, we set for diagonal elements except , and ignore off-diagonal ones except , and . Degenerate extra scalar masses, i.e. , is assumed for simplicity. In this section, we consider phenomenological constraints on , , and under these assumptions. In general is complex and it may contribute to CP violation and Baryogenesis Fuyuto:2017ewj (). For simplicity, we will take it to be real in this work.

The FCNH couplings and induce decay, with branching ratio

(2) |

where GeV, and the and modes are added up. The total width is estimated by the sum of and partial widths obtained by rescaling of SM values deFlorian:2016spz () with added. We impose the 95% C.L. limit % by CMS Sirunyan:2017xzt ().

Constraints on and by various low-energy processes containing tau and muon are discussed in the literature (see, e.g. Ref. Davidson:2010xv (); Sierra:2014nqa (); Dorsner:2015mja (); Omura:2015xcg ()). It is found that is most relevant. Its branching ratio is given by Omura:2015xcg ()

(3) |

where we take % Tanabashi:2018oca (), and gives the strength of the amplitude with different chiral structure. In addition to the one-loop contribution mediated by the neutral and charged scalar bosons, we also include the two-loop Barr-Zee type contribution in , following Ref. Omura:2015xcg (). The latter contribution can be obtained by the obvious translation of the expression for Chang:1993kw (), and we include the dominant contribution from the effective () vertex, which brings in dependence on via the top loop. Current limits on are by Belle Hayasaka:2007vc () and by BABAR Aubert:2009ag (), both at 90% C.L. Belle II may improve the limit by a factor of 100 Kou:2018nap (). We conservatively take to illustrate future sensitivity.

is also constrained by physics, in particular by the () meson mixings and Altunkaynak:2015twa (). We update the results of Ref. Altunkaynak:2015twa () with the latest experimental and theoretical values as summarized below. We adopt the Summer 2018 result by UTfit Bona:2006sa () for values of CKM parameters and constraints on the - mixing amplitude ():

(4) |

where . As for , we adopt a recent world average Amhis:2016xyh (), which includes the recent Belle result Belle:2016ufb (), and the updated SM prediction Misiak:2015xwa (); Czakon:2015exa () for the photon energy GeV. We then use the ratio Crivellin:2013wna () to constrain based on our LO calculation, allowing the 2 experimental uncertainty of with the theoretical uncertainty linearly added. Note that the new experimental and theoretical values result in rather strong limits on in 2HDM-II Misiak:2017bgg (): 570–800 GeV at 95% CL, depending on the method used to extract the limit. Our method gives GeV in 2HDM-II at large

We ignore effect of on mixings and as it enters via the charm loop, making its impact minor Crivellin:2013wna () compared with and entering via the top loop. But induces decay Hou:1991un (); Chen:2013qta (), and the recent ATLAS limit Aaboud:2018oqm () of

(5) |

directly constrains if is nonzero. The width is given by

(6) | |||||

where , , , and we define as a convenient FCNH coupling Altunkaynak:2015twa (); Jain:2019ebq (). Combining with the LO width to obtain the total top width, we recast the ATLAS limit Aaboud:2018oqm () of Eq. (5) to obtain for .

In our numerical calculations of this section, we take the latest PDG values Tanabashi:2018oca () for particle masses, in particular the top quark pole mass GeV and bottom quark mass GeV as input. Fig. 1 summarizes the constraints on the plane with for (a) 150 GeV and (b) 300 GeV: exclusions are shown by the blue-hatched regions for by CMS, gray-shaded regions for by BABAR, pink-shaded regions for the mixing () and green-hatched regions for . The other three observables in Eqs. (4) give weaker limit than and are not shown in the figures. The dashed contours with are shown as future Belle II sensitivity. We note that the constraints by and are highly sensitive to the choice of parameters: the constraint gets weaker for a smaller and eventually loses sensitivity if ; the constraint is relaxed for a smaller , and becomes weaker than the mixing constraint if . In passing, the effect Omura:2015xcg () on the muon is insignificant ( in the shown parameter regions) due to small and values, which suppress the one-loop contribution, and the – mass degeneracy, which leads to cancellation of the one-loop and contributions.

Combining experimental limits from LHC Higgs data and physics, we consider , and Aaboud:2018oqm (). To be consistent with physics constraints, we choose

(7) |

for or , which always satisfies the constraint for the heavy Higgs scalar mass considered in our study.

## Iii Higgs Signal and Physics Background

In this section, we discuss the prospect of discovering FCNH interactions from heavy Higgs bosons and decaying into . There are several parameters that can affect the signal cross section in the 2HDM. We use the experimental results and constraints to optimize the parameter range. Recent data from LHC point toward a Higgs sector in which the light CP even Higgs state is the SM-like Higgs ATLAS:2018doi (); CMS:2018lkl (). This constraint suggests that is very small. For case studies in our analysis we set .

### iii.1 The Higgs Potential and Decay Final States

For the heavy CP-even boson, the most important SM decay channels are , and . In addition, and channels might become dominant in some regions of parameter space. The CP-odd pseudoscalar boson has significant decays into , as well as possible dominant contributions from and channels.

To study heavy boson or decays involving the light Higgs boson , let us consider a general CP-conserving Higgs potential Gunion:2002zf ()

(8) | |||||

Applying minimization conditions, we can express the triple Higgs coupling in terms of physical masses and mixing angles Gunion:2002zf (); Craig:2013hca ()

(9) | |||||

which vanishes in the alignment limit, as the self coupling is proportional to .

For simplicity, we take the heavy Higgs states and to be degenerate and we set . As a sample study, we choose three values of and 0, to maintain tree level unitarity. For Yukawa couplings, except (Eq. 7), we set , which is in good agreement with the current constraints from B Physics and LHC. For off-diagonal elements , we perform case studies for = 0.1 and 0.5, and set all the remaining off-diagonal terms to be 0 except .

Figure 2 shows all major two body decays for the heavy and , with and . Note that, in keeping as in Fig. 1 and fixing the value to 0.01, , dominates over , which is interesting by itself. For the boson, , and play crucial roles in affecting the branching ratio. We use 2HDMC Eriksson:2009ws () to scan over 150 GeV 500 GeV and for . For , , and channels might become predominant. This suggests values close to 1 TeV may not be visible in the channels, so we limit our case study to GeV.

The pseudoscalar decays mostly into fermions, as shown in Figs. 2(c) and 2(d). Its decay is independent of and in a general 2HDM. Only has significant impact on the branching fractions. For , becomes dominant. Furthermore, for GeV, also makes significant contribution. For , the channel starts to dominate, hence we limit our study to GeV to ensure significance.

### iii.2 Higgs Signal

Our main signal channel is the production and FCNH decay of a heavy Higgs boson () via gluon fusion, Han:2000jz (); Assamagan:2002kf (); Harnik:2012pb (); Buschmann:2016uzg (); Sher:2016rhh (); Primulando:2016eod (); Bednyakov:2018hfq (). With the decaying leptonically, we are looking for a final state of two opposite sign, different flavor leptons and missing energy. With a hadronically decaying , a final state with a -jet (), a muon, and missing energy is needed. We have evaluated the FCNH signal cross sections with analytic matrix element and leading order CT14 parton distribution functions Dulat:2015mca (); Gao:2013xoa (). To include higher order corrections we calculate K-factors with Higlu Spira () for .

### iii.3 Standard Model Backgrounds

The dominant background for leptonic final states comes from , and . For hadronic channel, we have considered as the most dominant background along with the channel. For hadronic channel, contribution is highly suppressed, when we veto any event with more than one b jet, with GeV and . We have used MADGRAPH Alwall:2011uj () and HELAS Hagiwara:2008jb () to generate matrix elements for the backgrounds. To include the NLO corrections, we have employed MCFM Campbell:2010ff (); Campbell:2015qma () to evaluate higher order cross sections.

Kinematic Variables | ||

### iii.4 Realistic acceptance cuts

To study the discovery potential for the FCNH signal, we apply realistic acceptance cuts proposed by CMS Khachatryan:2015kon (); Sirunyan:2017xzt () at TeV as shown in Table I. In addition, we apply Gaussian smearing for particle momenta ATLAS:2013-004 (); Khachatryan:2016kdb () to simulate detector effects based on ATLAS Aad:2009wy () and CMS Colaleo:2015vsq () specifications.

(10) |

CM Energy () | 14 TeV | 27 TeV | 100 TeV |

Backgrounds for | |||

31.96 | 58.7 | 195.1 | |

12.27 | 23.73 | 86.29 | |

1.92 | 5.06 | 27.9 | |

0.95 | 2.51 | 13.9 | |

Total | 47.1 | 90.0 | 323.2 |

Backgrounds for | |||

6139 | 14074 | 61285 | |

109.8 | 202.3 | 676.9 | |

6.4 | 16.9 | 93.3 | |

Total | 6255 | 14293 | 62055 |

We present in Table II the cross sections for physics backgrounds with acceptance cuts as well as tagging efficiency for -jets, = 0.7 Friis:2011zz (); Lumb:2010btk (), and mistag efficiency = 0.01 ATLAS:2018bpl (); Sirunyan:2017ezt ().

We note that, as the Higgs boson mass increases, cut becomes more effective, and for , , are almost completely vetoed. For leptonic channel, becomes more dominant than .

## Iv Discovery Potential

To estimate the discovery potential, we require that the lower limit on the signal plus background should be larger than the corresponding upper limit on the background with statistical fluctuations, which leads to HGG ()

(11) |

where and are the signal and background cross sections, respectively, and is the integrated luminosity. Choosing , we obtain a significance. For a large number of background events, it simplifies to the statistical significance

(12) |

where and are the number of signal and background events.

### iv.1 Discovery Reach for Pseudoscalar

The pseudoscalar has higher production cross section, and with no suppression coming from , which is forbidden, it is more promising than the heavy scalar . Fig. 3 shows the discovery region for in the ( plane, for = 0.1 and 0.5, including both the leptonic channel (upper panels) and the hadronic channel (lower panels). Because of high QCD backgrounds, performance for hadronic decay is worse than leptonic decay, despite its higher branching ratios.

We show our results for = 14, 27 and 100 TeV. At low masses, GeV or roughly the threshold, the entire range of is detectable at 3000 fb, independent of the center-of-mass energy. For an intermediate range (200 GeV 300 GeV), our discovery region starts shrinking because of predominance (plus a milder effect from turn-on), which is more striking for the larger value as shown in the right panel plots of Fig. 3. For higher mass range ( GeV), we see a slight increase in the 5 region before and around , owing to the rise in production cross section for , before the turn-on of decay further suppresses our signal towards higher masses beyond GeV. Note that is actually larger than for our mass range (see Eq. (7)), which is constrained by B physics.

### iv.2 Discovery Reach for Heavy CP-even Scalar

For the heavy CP-even boson , the situation is quite different. The branching fraction for is affected by , tan and . The latter Higgs sector parameter affects the decay, where in Fig. 2 we illustrated with . In order to understand the effect of , we perform a case study for with GeV, , and scan over for tan = 1. The results are shown in Fig. 4 for , 27 and 100 TeV for the leptonic channel and . The hadronic channel is similar except it will have higher QCD background.

We observe that for a fixed value of , increasing from to 0 lowers the cross section of while increasing the trilinear Higgs coupling, , which enhances the branching fraction of ,

(13) |

with . As a case study, let us choose the values of , 0, with to preserve tree-level unitarity and stability for a general 2HDM, which resembles the generic case more closely, and perform a scan for and 150 GeV 500 GeV. The results are shown in Fig. 5. There is a large discoverable region in the low mass regime ( GeV). However, as we start increasing , first , then , then become dominant. The discovery potential is improved somewhat around the threshold because of rise in production cross section, appearing as ‘dips’ of contours in Fig. 3 and Fig. 5. Beyond that region, we still have some parameter space that can be probed, and a 100 TeV high energy collider can probe to lower couplings. The likelihood of detection increases as we reduce the value of , from 0 to . The situation for would be worse than , the right panels of Fig. 3.

## V Conclusion

The general two Higgs doublet model offers a very rich phenomenology for flavor changing neutral Higgs interactions with fermions, because of the absence of any symmetry to suppress them. Strong experimental constraints exist for these FCNH interactions, but third generation fermions might offer promising signatures for new physics at the LHC and future hadron colliders. Experimental data from LHC Run 1 had shown some hints for the light CP-even Higgs boson , but became insignificant with 2016 CMS data at Run 2. However, in the general 2HDM, the coupling probed is , which is expected to be small in the alignment limit of , where the light CP-even Higgs boson approaches the standard Higgs boson.

For heavy Higgs states, the pseudoscalar boson has FCNH coupling that is independent of , while the heavy CP-even scalar has FCNH coupling , where is expected to be close to unity. Thus, they offer great promise to discover FCNH signals with lepton flavor violating production of at the LHC and future hadron colliders.

We have investigated the prospects of discovering for the high luminosity (HL) and high energy (HE) LHC and future high energy colliders. With gluon fusion being the dominant mode of production for both heavy scalars because of finite , we find promising results for LHC with , , when is not yet overwhelming for up to 300 GeV. The choice of - mixing parameter and values are meant as illustrative. Having taken degenerate , and , GeV can still evade constraint and should be taken seriously. It should be noted that is more promising than because of its higher production cross section and fewer decay channels affecting its decay to , but decay depends also on Higgs potential due to mode. If is considerably larger than 0.1, , decay to would suppress observability, and a higher energy collider would be needed.

Our study has focused on discovering , in final state, but companion final states such as (for ), (for ) and , are worthy pursuits in their own right, some of which have been studied elsewhere.

Acknowledgments. MK thanks E. Senaha for useful discussion on the constraint. C.K. thanks the High Energy Physics Group at National Taiwan University for hospitality, where part of the research was completed. This research was supported in part by the U.S. Department of Energy (RJ, CK, BM, and AS) as well as by grants MOST 106-2112-M-002-015-MY3, 107-2811-M-002-039, and 107-2811-M-002-3069 (WSH and MK).

## References

- (1) The ATLAS Collaboration, ATLAS-CONF-2018-031.
- (2) The CMS Collaboration, CMS-PAS-HIG-17-031.
- (3) See e.g. the talk by T. Blake at CKM 2018, Heidelberg, Sept 2018.
- (4) G.W. Bennett et al. [Muon g-2 Collaboration], Phys. Rev. D 73, 072003 (2006).
- (5) V. Khachatryan et al. [CMS Collaboration], Phys. Lett. B 749 (2015) 337.
- (6) A.M. Sirunyan et al. [CMS Collaboration], JHEP 1806, 001 (2018).
- (7) A. Crivellin, C. Greub and A. Kokulu, Phys. Rev. D 86, 054014 (2012).
- (8) S.L. Glashow and S. Weinberg, Phys. Rev. D 15, 1958 (1977).
- (9) W.-S. Hou and M. Kikuchi, EPL 123, 11001 (2018).
- (10) J.F. Gunion and H.E. Haber, Phys. Rev. D 67, 075019 (2003).
- (11) M. Carena, I. Low, N.R. Shah and C.E.M. Wagner, JHEP 1404, 015 (2014).
- (12) P.S. Bhupal Dev and A. Pilaftsis, JHEP 1412, 024 (2014).
- (13) S. Davidson and H.E. Haber, Phys. Rev. D 72, 035004 (2005).
- (14) F. Mahmoudi and O. Stål, Phys. Rev. D 81, 035016 (2010).
- (15) J.F. Gunion, H.E. Haber, G.L. Kane and S. Dawson, Front. Phys. 80, 1 (2000).
- (16) G. Aad et al. [ATLAS Collaboration], Eur. Phys. J. C 77, 70 (2017).
- (17) W. Barletta, M. Battaglia, M. Klute, M. Mangano, S. Prestemon, L. Rossi and P. Skands, arXiv:1310.0290 [physics.acc-ph].
- (18) R. Tomas et al., Nucl. Part. Phys. Proc. 273-275, 149 (2016).
- (19) F. Zimmermann, ICFA Beam Dyn. Newslett. 72, 138 (2017).
- (20) V. Shiltsev, arXiv:1705.02011 [physics.acc-ph].
- (21) B. Altunkaynak, W.-S. Hou, C. Kao, M. Kohda and B. McCoy, Phys. Lett. B 751, 135 (2015).
- (22) W.-S. Hou, M. Kohda and T. Modak, Phys. Rev. D 98, 075007 (2018).
- (23) K. Fuyuto, W.-S. Hou and E. Senaha, Phys. Lett. B 776, 402 (2018).
- (24) D. de Florian et al. [LHC Higgs Cross Section Working Group], arXiv:1610.07922 [hep-ph].
- (25) S. Davidson and G.J. Grenier, Phys. Rev. D 81, 095016 (2010).
- (26) D. Aristizabal Sierra and A. Vicente, Phys. Rev. D 90, 115004 (2014).
- (27) I. Doršner, S. Fajfer, A. Greljo, J.F. Kamenik, N. Košnik and I. Nišandžić, JHEP 1506, 108 (2015).
- (28) Y. Omura, E. Senaha and K. Tobe, Phys. Rev. D 94, 055019 (2016).
- (29) M. Tanabashi et al. [Particle Data Group], Phys. Rev. D 98, 030001 (2018).
- (30) D. Chang, W.-S. Hou and W.-Y. Keung, Phys. Rev. D 48, 217 (1993).
- (31) K. Hayasaka et al. [Belle Collaboration], Phys. Lett. B 666, 16 (2008).
- (32) B. Aubert et al. [BABAR Collaboration], Phys. Rev. Lett. 104, 021802 (2010).
- (33) E. Kou et al. [Belle II Collaboration], arXiv:1808.10567 [hep-ex].
- (34) M. Bona et al. [UTfit Collaboration], http://www.utfit.org/.
- (35) Y. Amhis et al. [HFLAV Collaboration], Eur. Phys. J. C 77, 895 (2017); see https://hflav.web.cern.ch for updates.
- (36) A. Abdesselam et al. [Belle Collaboration], arXiv:1608.02344 [hep-ex].
- (37) M. Misiak et al., Phys. Rev. Lett. 114, 221801 (2015).
- (38) M. Czakon, P. Fiedler, T. Huber, M. Misiak, T. Schutzmeier and M. Steinhauser, JHEP 1504, 168 (2015).
- (39) A. Crivellin, A. Kokulu and C. Greub, Phys. Rev. D 87, 094031 (2013).
- (40) M. Misiak and M. Steinhauser, Eur. Phys. J. C 77, 201 (2017).
- (41) W.-S. Hou, Phys. Lett. B 296, 179 (1992).
- (42) K.-F. Chen, W.-S. Hou, C. Kao and M. Kohda, Phys. Lett. B 725, 378 (2013).
- (43) M. Aaboud et al. [ATLAS Collaboration], arXiv:1812.11568 [hep-ex].
- (44) R. Jain and C. Kao, arXiv:1901.00157 [hep-ph].
- (45) N. Craig, J. Galloway and S. Thomas, arXiv:1305.2424 [hep-ph].
- (46) D. Eriksson, J. Rathsman and O. Stål, Comput. Phys. Commun. 181, 189 (2010).
- (47) T. Han and D. Marfatia, Phys. Rev. Lett. 86, 1442 (2001).
- (48) K.A. Assamagan, A. Deandrea and P.A. Delsart, Phys. Rev. D 67, 035001 (2003).
- (49) R. Harnik, J. Kopp and J. Zupan, JHEP 1303, 026 (2013).
- (50) M. Buschmann, J. Kopp, J. Liu and X.-P. Wang, JHEP 1606, 149 (2016).
- (51) M. Sher and K. Thrasher, Phys. Rev. D 93, 055021 (2016).
- (52) R. Primulando and P. Uttayarat, JHEP 1705, 055 (2017).
- (53) A. Bednyakov and V. Rutberg, Mod. Phys. Lett. A 33, 1850152 (2018).
- (54) J. Gao et al., Phys. Rev. D 89, 033009 (2014).
- (55) S. Dulat et al., Phys. Rev. D 93, 033006 (2016).
- (56) M. Spira, hep-ph/9510347; Nucl. Instrum. Meth. A 389, 357 (1997).
- (57) J. Alwall, M. Herquet, F. Maltoni, O. Mattelaer and T. Stelzer, JHEP 1106, 128 (2011).
- (58) K. Hagiwara, J. Kanzaki, Q. Li and K. Mawatari, Eur. Phys. J. C 56, 435 (2008).
- (59) J.M. Campbell and R.K. Ellis, Nucl. Phys. Proc. Suppl. 205-206, 10 (2010).
- (60) J.M. Campbell, R.K. Ellis and W.T. Giele, Eur. Phys. J. C 75, 246 (2015).
- (61) The ATLAS Collaboration, ATLAS-PHYS-PUB-2013-004
- (62) V. Khachatryan et al. [CMS Collaboration], JINST 12, P02014 (2017).
- (63) G. Aad et al. [ATLAS Collaboration], arXiv:0901.0512 [hep-ex].
- (64) A. Colaleo, A. Safonov, A. Sharma and M. Tytgat, CERN-LHCC-2015-012, CMS-TDR-013.
- (65) E.K. Friis [CMS Collaboration], Nucl. Phys. Proc. Suppl. 218, 256 (2011).
- (66) D. Lumb, CERN-THESIS-2010-350.
- (67) The ATLAS Collaboration, ATLAS-CONF-2018-001.
- (68) A.M. Sirunyan et al. [CMS Collaboration], JINST 13, P05011 (2018).
- (69) H. Baer, M. Bisset, C. Kao and X. Tata, Phys. Rev. D 46, 1067 (1992).