Accurate simulation of W, Z, and Higgs boson decays in Sherpa
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Abstract
We discuss the inclusion of nextto–nextto leading order electromagnetic and of nextto leading order electroweak corrections to the leptonic decays of weak gauge and Higgs bosons in the Sherpa event generator. To this end, we modify the Yennie– Frautschi–Suura scheme for the resummation of soft photon corrections and its systematic improvement with fixedorder calculations, to also include the effect of virtual corrections due to the exchange of weak gauge bosons. We detail relevant technical aspects of our implementation and present numerical results for observables relevant for highprecision Drell–Yan and Higgs boson production and decay simulations at the LHC.
1 Introduction
The experiments at the LHC are stresstesting the Standard Model (SM) of particle physics at unprecedented levels of precision. In particular, leptonic standardcandle signatures like charged and neutralcurrent Drell–Yan production offer large cross sections together with very small experimental uncertainties, often at or even below the percent level. This allows to extract fundamental parameters in the electroweak (EW) sector of the SM at levels of precision surpassing the LEP heritage. Measurements of the Wboson mass, a key EW precision observable, are already reaching the 20 MeV level [1] based on 7 TeV data alone, with theory uncertainties being one of the leading systematics. Another example for the impressive achievements on the experimental side, challenging currently available theoretical precision, is the recent measurement of the triple differential cross section in neutral current Drell–Yan production based on 8 TeV data [2], the first of its kind at a hadron collider. Furthermore, precision measurements of the Z transverse momentum spectrum [3, 4] have been used to constrain parton distribution functions (PDFs) [5]. In order to fully harness available and future experimental datasets excellent theoretical control of various very subtle effects of higherorder QCD and EW origin is required. For recent reviews and studies on these issues, see e.g. [6, 7, 8]. With this paper we contribute to this effort by investigating higherorder QED/EW effects in the modelling of softphoton radiation off vectorboson decays.
The demand for (sub)percent precision in Drell–Yan production has led to formidable achievements in the theoretical description of corresponding collider observables, often pushing boundaries of technical limitations. The pioneering nexttonexttoleading (NNLO) QCD corrections for differential Drell–Yan production [9, 10, 11] are available as public computer codes [12, 13, 14] and have recently been matched to QCD parton showers, using the \(\mathrm {UN}^2\mathrm {LOPS}\) framework within Sherpa [15, 16], and via a reweighting of a MiNLO improved calculation in Dynnlops [17]. Since recently also NNLO corrections to Drell–Yan production at finite transverse momentum are available [18, 19, 20, 21, 22, 23, 24, 25]. Higherorder EW corrections at the NLO level for inclusive Drell–Yan production are available for quite some time [26, 27] and are implemented in a large number of public codes, including Wzgrad [28, 29, 30], Sanc [31], Rady [32, 33], Horace [34, 35] and Fewz [36]. At finite transverse momentum they have been calculated in [37, 38, 39, 40]. The combination of higherorder QCD and EW effects is available within the Powheg framework [8, 41, 42, 43, 44] matched to partonshowers, and also in [45]. Efforts to calculate the fixedorder NNLO mixed QCD and EW corrections explicitly are underway [46, 47, 48, 49]. Their effect has been studied in the pole approximation [50, 51].
At the desired level of precision QED effects impacting in particular the leptonic final state have to be considered and understood in detail. In this case, soft and collinear photon radiation provides the major contributions. These can be resummed to all orders, and also improved order by order in perturbation theory. Implementations of such calculations have been performed via a QED parton shower matching in Horace [52, 53] and in the Powheg framework, in the structure function approach in Rady, and through a YFStype exponentiation for particle decays in Photos [54], Winhac [55], the Herwig module Sophty [56] and the Sherpa module Photons [57]. In this paper, we present an extension of the Sherpa module Photons, which provides a simulation of QED radiation in (uncoloured) particle decays. Photons implements the approach of Yennie, Frautschi and Suura (YFS) [58] for the calculation of higher order QED corrections. In the YFS approach, leading soft logarithms, which are largely independent of the actual hard process involved, are resummed to all orders. Beyond this, the method also allows for the systematic improvement of the description through the inclusion of full fixedorder matrix elements. The present implementation allows for the inclusion of a collinear approximation to the real matrix element using dipole splitting kernels [59, 60]. Furthermore, for several relevant processes, including the decays of electroweak bosons, \(\tau \) decays as well as generic decays of uncharged scalars, fermionic and vector hadrons, the full real and virtual NLO QED matrix elements are included. This module has also been used for the description of electroweak corrections in the semileptonic decays of B mesons [61]. The aim of this publication is to further enhance the level of precision in the case of the decay of electroweak gauge and Higgsbosons by implementing the full oneloop EW corrections, as well as NNLO QED corrections in the case of Z and Higgsdecays. The electroweak virtual corrections to particle decays are known for a long time [62, 63] and our implementation will be based on these results. In the case of Zboson decays, the double virtual corrections in the limit of small lepton masses are known for about 30 years [64], which we will rely on.
The paper is organized as follows. In Sect. 2, we review the YFS algorithm, motivating and investigating the procedure to include higher order corrections at a given perturbative order. In Sect. 3, we summarize the numerical results for the decays \(Z\rightarrow \ell ^+\ell ^\), \(W\rightarrow \ell \nu \) in Drell–Yan production. There we also present results for \(H\rightarrow \ell ^+\ell ^\) decays in hadronic Higgs production. The measurement of the latter is highly challenging due to small leptonic Higgs couplings but potentially achievable at the HLLHC. We discuss and conclude in Sect. 4.
2 Implementation
2.1 The YFS formalism
 the YFS form factorwith the sum running over all pairs of charged particles and the soft factors given by$$\begin{aligned} Y(\Omega ) = \sum _{i<j} Y_{ij}(\Omega ) = 2\alpha \left( B_{ij}+\tilde{B}_{ij}(\Omega )\right) , \end{aligned}$$(2.14)$$\begin{aligned} B_{ij} =&\frac{i}{8\pi ^3}Z_iZ_j\theta _i\theta _j\int \mathrm {d}^4k \frac{1}{k^2}\nonumber \\&\times \left( \frac{2q_i\theta _ik}{k^22\left( k\cdot q_i\right) \theta _i} {+} \frac{2q_j\theta _j+k}{k^2+2\left( k\cdot q_j\right) \theta _j}\right) ^2, \end{aligned}$$(2.15)These two terms contain all infrared virtual and real divergences which cancel due to the KinoshitaLeeNauenberg theorem [65, 66], guaranteeing the finiteness of \(Y(\Omega )\) and of the decay width. \(Z_i\) and \(Z_j\) are the charges of the particles i and j, and the factors \(\theta = \pm 1\) for particles in the final or initial state, respectively. We provide expressions for \(B_{ij}\) in finalfinal and initialfinal dipoles in terms of scalar master integrals in Appendix C. The calculation of the full form factor can be found in [57];$$\begin{aligned} \tilde{B}_{ij}\left( \Omega \right) =&\, \frac{1}{4\pi ^2}Z_iZ_j\theta _i\theta _j\int \mathrm {d}^4k\ \delta \left( k^2\right) \nonumber \\&\times \left( \phantom {\frac{1}{2}}1\Theta \left( k,\Omega \right) \right) \left( \frac{q_i}{q_i\cdot k}  \frac{q_j}{q_j\cdot k}\right) ^2. \end{aligned}$$(2.16)
 the eikonal factor \(\tilde{S}\left( k\right) \)describing the soft emission of a photon off a collection of charged particles;$$\begin{aligned} \tilde{S}\left( k\right)= & {} \sum _{i<j} \tilde{S}_{ij}\left( k\right) \nonumber \\= & {} \frac{\alpha }{4\pi ^2}\sum _{i<j}Z_iZ_j\theta _i\theta _j \left( \frac{q_i}{q_i\cdot k}  \frac{q_j}{q_j\cdot k}\right) ^2 \end{aligned}$$(2.17)

the lowest order matrix element \(\tilde{\beta _0^0}\);
 a correction factor \(\mathcal {C}\) to the full matrix element, given byThe terms in the first bracket describe the nexttoleading order (NLO), i.e. the \(\mathcal {O}\left( \alpha \right) \) term of the expansion, and the terms in the second bracket describe the nextto–nexttoleading order (NNLO), i.e. the \(\mathcal {O}\left( \alpha ^2\right) \) term of the expansion. In this publication, we will primarily be concerned with this correction factor, in particular with the virtual corrections at NLO, the term \(\tilde{\beta }_0^1\), which we extend to an expression at full NLO in the electroweak theory for the decays of the weak bosons, as well as the complete NNLO bracket which we will be calculating for the neutral weak bosons;$$\begin{aligned} \mathcal {C}= & {} 1 + \frac{1}{\tilde{\beta }_0^0} \left( \tilde{\beta }_0^1 + \sum _{i=1}^{n_\gamma } \frac{\tilde{\beta _1^1}\left( k_i\right) }{\tilde{S}\left( k_i\right) }\right) \nonumber \\&+ \frac{1}{\tilde{\beta }_0^0}\left( \tilde{\beta }_0^2 + \sum _{i=1}^{n_\gamma } \frac{\tilde{\beta _1^2}\left( k_i\right) }{\tilde{S}\left( k_i\right) } + \sum _{\begin{array}{c} i,j=1 \\ i\ne j \end{array}}^{n_\gamma } \frac{\tilde{\beta }_2^2\left( k_i,k_j\right) }{\tilde{S}\left( k_i\right) \tilde{S}\left( k_j\right) }\right) \nonumber \\&+ \frac{1}{\tilde{\beta }_0^0} \mathcal {O}\left( \alpha ^3\right) . \end{aligned}$$(2.18)

and the Jacobean \(\mathcal {J}\) capturing the effect of the momenta mapping.
2.2 Motivation for higher order corrections
The previous subsection introduced the YFS procedure for dressing the lowest order matrix element with soft radiation to all orders. This basic procedure, in which \(\mathcal {C} = 1\), yields photon distributions that are correct in the limit of soft radiation. For the remainder of this paper, we will call this the soft approximation. Away from the strict soft limit, exact matrix elements are necessary to describe observables at the required accuracy, and we described the procedure for their systematic incorporation. Hard photon radiation occurs predominantly collinear to the emitter and more frequently in processes with large energytomass ratios of the involved particles. With this in mind, generic collinear corrections for the real matrix element, based on the splitting functions developed in [59, 60], were employed in [57] to account for hard QED radiation in the softcollinear approximation. While this approximation correctly describes radiation in the limits of soft and collinear radiation, it does neither account for interference effects nor hard wideangle radiation. In order to capture these effects correctly, full matrix elements for real and virtual photon radiation have to be added, some of which have already been included in [57].
For illustration, in Fig. 1 (left) we compare the softcollinear, the NLO QEDcorrect and the NNLO QEDcorrect results for the invariant mass \(m_{\ell \ell }\) of the electrons produced in Zboson decays. To guide the eye we also show the leadingorder result. The NLO QED result represents the maximal accuracy of the implementation in Photons as described in [57]. Figure 1 (left) clearly shows the necessity to include photon radiation in the first place. Photon radiation causes a significant shape difference, shifting events from large to lower \(m_{\ell \ell }\). This effect is a lot more striking in the decay into the lighter leptons, such as for the electrons exhibited here, which are much more likely to radiate photons. We can also appreciate that while the softcollinear approximation does a good job of describing the distribution near the peak, it predicts a harder spectrum at lower values of \(m_{\ell \ell }\). The peak region corresponds to the limit of soft photon radiation, while the latter region corresponds to hard photon radiation. This observation thus suggests that in order to capture the behaviour of the distribution over its entirety, we need to employ full matrix elements. It is then natural to ask whether higher order corrections beyond the NLO in QED are required as well. The impact of NNLO QED corrections is already illustrated in Fig. 1 (left) and the description of these and of full NLO EW corrections will be the focus of the next two sections.
2.3 NLO electroweak corrections
The discussion in Sect. 2.1 was restricted to QED corrections only. Since the exponentiation relies on the universal behaviour of the amplitudes in the soft limit only, additional fixedorder corrections can easily be added, as long as they are not divergent in the soft limit and thus do not spoil the softphoton exponentiation. This is, in fact, the case for the weak part of the corrections in the full electroweak theory, where the masses of the weak bosons regulate the soft divergence that is plaguing the massless photon. In this work, we are concerned with the decays of weak bosons; consequently, there is no phase space available for the emission of a real, massive weak boson, and the additional electroweak corrections contribute only to the virtual corrections \(\tilde{\beta }_0^{n_V}\).
The known oneloop virtual corrections for the decays of the electroweak bosons [62, 67] have been implemented in a number of programs dedicated to electroweak precision calculations already mentioned in the introduction. They can be calculated analytically with programs such as FeynCalc [68, 69], FormCalc [70] or PackageX [71], and numerically with programs such as GoSam [72, 73], MadGraph5 [74, 75], OpenLoops [76, 77] or Recola [78, 79]. The twoloop virtual electroweak corrections are not fully known yet, with only partial results for particular observables available, see for example [80, 81].
We implemented the electroweak corrections for the decays \(Z\rightarrow \ell ^+\ell ^\), \(H\rightarrow \ell ^+\ell ^\) and \(W\rightarrow \ell \nu \) in the YFS correction factor \(\mathcal {C}\). In doing so, we also reimplemented and revalidated the QED corrections in a more straightforward way. In our calculation we retain the full dependence on the lepton masses in the decay \(H\rightarrow \ell ^+\ell ^\). In the two other decays we keep them only in the QED corrections, where they are required to regularize the collinear singularities, while we neglect them in the other contributions. To this end we used the vertex form factors found in [63] to describe the virtual corrections to the vertices. We renormalize the theory using the onshell renormalization scheme, following the treatment described in [62]. We have validated the amplitudes on a pointbypoint level against an implementation in OpenLoops [76, 77], all in the case of massless leptons for Z and Wboson decays, and for the case of a Higgs decay into massive fermions. In addition, we also validated the values of the scalar integrals including masses against Collier [82, 83, 84, 85] and QCDLoop [86], as well as individual renormalization constants for massive leptons against OpenLoops. Real corrections due to the emission of an additional photon are calculated in the helicity formalism [87, 88, 89] using building blocks available within Sherpa [90]. We validated these corrections explicitly, against Wzgrad [28, 29, 30] and by internal comparison with Amegic [90] and Comix [91]. These comparisons have yielded maximal relative deviations between our implementation and these references of at most \(10^{10}\) in \(\mathcal {O}(100)\) randomly generated phase space points. We also validated full cross sections against Wzgrad [28, 29, 30] (containing only FSR corrections) and found very good agreement.
For the decays of Z and Higgsbosons, we further implement an option including only QED corrections. In the decay of neutral bosons, this choice forms a gauge–invariant subset of the full electroweak corrections and can thus be considered independently. Practically, this amounts to turning off the purely weak vertex form factors as well as turning off those parts of the renormalization constants that are of weak origin. This option is not available in the case of a Wboson decay as the W itself couples to the photon. We list the relevant form factors, renormalization constants and the necessary modifications in the pure QED case in Appendix B.
As an illustration in Fig. 1 (right) we compare the LO, the softcollinear and the full NLOcorrect results for the invariant mass of the charged electron and the neutrino in Wboson decays. As for the Z decay, the inclusion of the exact fixedorder corrections is mandatory for a reliable prescription below the resonance peak.
2.4 NNLO QED corrections
We will now turn to the discussion of NNLO QED corrections to Z and Higgsboson decays. They comprise doublevirtual, realvirtual and realreal contributions. The NNLO QED corrections can be combined with the full NLO EW corrections, and we will label this combination as “NNLO QED \(\oplus \) NLO EW”. As illustrated in Fig. 1 (left) the NNLO corrections yield very small corrections beyond NLO  at least in observables defined at LO. However, their inclusion ensures precision at the subpercent level required for future Drell–Yan measurements.
2.4.1 Double virtual corrections
The twoloop pure QED corrections to the form factor for the Zboson decay have been known in the limit of small lepton masses since the LEP era [64, 92]. Including full mass dependence, currently the twoloop QED form factor is only known for the decay of a virtual photon [93].
To the best of our knowledge, no QED twoloop formfactors are available for the decay of Higgs bosons. In principle they could be obtained from corresponding QCD results [94, 95, 96, 97, 98] However, for simplicity here we rely on the leading logarithmic behaviour only, \(\tilde{\beta }_0^2 = \frac{1}{2} \log ^2\left( \frac{s}{m^2}\right) \). We find that for the decays into bare muons, this is a sufficient approximation. Appreciable effects due to this approximation might only be noticeable in Higgs decays into \(\tau \)leptons.
For the decay of Zbosons, we use the results in Eqs. (2.15) and (2.22) of [64], together with the subtraction term B expanded in the limit \(s \gg m^2\). The results for the form factors given in [64] are sufficient as we only require the squared contribution \(\mathrm {Re}({M_0^2M_0^{0 *}})\). In fact, here the twoloop amplitude \(M_0^2\) factorizes into a simple factor multiplying the leading order matrix element.
2.4.2 Realvirtual corrections

The YFS formalism relies on fermion masses to regularize the collinear singularities, which in the case of small fermion masses may amount to the evaluation of expressions very close to logarithmic singularities, of the type \(\log (s_{ij}/m^2)\), where \(s_{ij} = (p_i+p_j)^2\) is the invariant mass of two momenta. We find that in our implementation the amplitudes for the decays into electrons and to some extent also into muons are affected by numerical instabilities while the amplitudes for the decays into \(\tau \)’s are wellbehaved.

In addition, the employed PassarinoVeltmann reduction may lead to the appearance of small Gram determinants in denominators. One way to circumvent this issue is by employing an expansion in the Gram determinant for the problematic tensor integrals rather than the full reduction, as implemented for example in Collier [85]. Since this requires the implementation of a significant number of expressions for different combinations of arguments in the tensor integrals and thus amounts to a large overhead, this is not pursued in this work.
2.4.3 Realreal corrections
3 Results
3.1 Setup
In this section we present the numerical effects induced by the NLO EW and NNLO QED corrections presented in the previous section, focussing on the decays \(Z\rightarrow \ell ^+\ell ^\), \(W\rightarrow \ell \nu \) and \(H\rightarrow \ell ^+\ell ^\) with \(\ell =\{e,\mu ,\tau \}\) following hadronic neutralcurrent and chargedcurrent Drell–Yan and Higgs production respectively.
The results presented here are based on an implementation in the Photons module [57] of the Sherpa Monte Carlo framework (release version 2.2.4) [15]. We consider hadronic collisions at the LHC at 13 TeV for the production of Z, W and Higgsbosons and their subsequent decays. In the neutralcurrent Drell–Yan case we require \(65\,\mathrm {GeV}< m_{\ell \ell } < 115\,\mathrm {GeV}\), while for the other modes no generation cuts are applied. Since we aim to purely focus on the effects of photon radiation in the decays, we turn off the QCD shower, fragmentation and underlying event simulation. We use Rivet 2.5.4 [101] for the analysis. For the case of electrons in the final state, we perform the analysis either using bare leptons or using dressed leptons with a radius parameter \(\Delta R = 0.1\) or \(\Delta R = 0.2\). For the case of muon and \(\tau \) final states only bare results are shown. We focus our results on a few key distributions and always normalize to the respective inclusive cross section. Overall, we choose to focus on ratios between different predictions, in order to highlight small subtle differences relevant for precision Drell–Yan and Higgs physics.
Input parameters for the numerical results are chosen as listed in Table 1. The weak coupling \(\alpha \) is defined in the onshell \(\alpha (0)\) scheme. This choice is sensible as we are explicitly also investigating distributions in resolved finalstate photons. At the same time, the YFS formalism is strictly only defined in the limit of soft photon emissions. In this input scheme, the sine of the weak mixing angle is a derived quantity \(s_W^2 =1 \frac{M_W^2}{M_Z^2}\). Gauge and Higgsboson widths are taken into account in a fixedwidth scheme.
Electroweak input parameters: gauge and Higgs boson masses and widths, lepton masses and the EW coupling in the \(\alpha (0)\) scheme
Mass (GeV)  Width (GeV)  

Z  91.1876  2.4952 
W  80.385  2.085 
H  125  0.00407 
e  \(0.511 \,\mathrm {MeV}\)  – 
\(\mu \)  \(0.105 \,\mathrm {GeV}\)  – 
\(\tau \)  \(1.777 \,\mathrm {GeV}\)  – 
\(1/\alpha \left( 0\right) \)  137.03599976 
3.2 Neutralcurrent Drell–Yan production
In Fig. 2, we present the distributions of the invariant mass of the two leptons (left) and of the invariant mass of the system made up of the decay leptons and the photon closest to either of them (right). Already from the plots in Sect. 2.2, it is clear that the inclusion of photon radiation is crucial for a reliable description of the dilepton invariant mass. All higherorder corrections significantly differ from the LO prediction, which fails to describe the lineshape below the peak. At the NLO QED level corrections beyond the soft and softcollinear approximations induce distortions up to the 1% level. In fact, the soft approximation does not generate enough hard radiation, while the softcollinear approximation produces about 1% too many events at low \(m_{\ell \ell }\), i.e. it seems to generate too much hard photon radiation. In this observable both the NLO EW and NNLO QED corrections provide only a marginal effect on the order of permille, and neither of these corrections provides a significant shift of the peak of the distribution. Clearly, the dressing of the electrons has a significant effect on this distribution, reflecting the sensitivity to QED radiation. Bare electrons show a significant shape difference compared to dressed electrons. The results based on different dressing parameters however differ by at most a few %, suggesting that much of the photon radiation occurs close to the electron. Comparing different lepton species, we see that muons, in comparison to the dressed electrons, radiate significantly more, yielding up to 25% more events at low \(m_{\ell \ell }\). In contrast, the heavier \(\tau \)’s radiate less in comparison, resulting in differences with respect to dressed electrons of only a few %.
A very similar behaviour can be found in the invariant mass of the dilepton system combined with the closest photon. As this observable requires the emission of at least one photon, the NLO QED curve corresponds effectively to a LO prediction. However, also the soft and softcollinear approximations describe this observable reasonably well and higher order NNLO QED or NLO EW corrections are negligible. Comparing the dressing parameters, we find much smaller differences here: bare electrons only differing by about 15% from the dressed versions. There is barely a difference between the two dressings. In the same manner, the difference between lepton species is subdued as well: muons differing up to 2% at most from dressed electrons.
In Fig. 3, we present the distribution of the transverse momentum of the lepton, \(p_{\perp ,\ell }\), alongside the transverse momentum of the system of the two leptons, \(p_{\perp ,\ell \ell }\). The transverse momentum of the leptons, \(p_{\perp ,\ell }\), receives small corrections from the inclusion of higher order corrections beyond NLO QED into the YFS formalism. Only the phenomenologically irrelevant region of very low \(p_{\perp ,\ell }\) receives corrections at the permille level at NLO EW. Both the soft and softcollinear approximations agree at the permill level with NLO QED for \(p_{\perp ,\ell }>20\) GeV. Correspondingly, also the dressing of the electrons has a small effect on this distribution, with bare electrons carrying significantly less transverse momentum than the dressed versions. The difference between lepton species is marginal, up to about 5% at very low \(p_{\perp ,\ell }\) and above the Jacobi peak.
In contrast, the transverse momentum of the system of leptons, \(p_{\perp ,\ell \ell }\), shows significantly larger effects. Of course this distribution is not defined at LO and correspondingly it is very sensitive to the modelling of photon radiation. This can be appreciated when comparing the NLO QED prediction with the soft and softcollinear approximations. Only at small \(p_{\perp ,\ell \ell }\) the approximations agree. In this observable also the inclusion of NLO EW effects shows a significant impact, with differences reaching up to 5%. The NNLO QED effects provide a competing effect to the NLO EW corrections, lifting the distributions by about 2% across the entire distribution. The effects of the dressing on the distribution is unsurprisingly very large as well. Bare electrons show significantly more events at nonvanishing values of \(p_{\perp ,\ell \ell }\), while a different dressing parameter leads to an almost flat decrease across the spectrum. The comparison of the different lepton species shows that the muons again radiate a lot more, with up to 75% more events at medium \(p_{\perp ,\ell \ell }\). \(\tau \)’s in comparison show a reduction in the number of events at large \(p_{\perp ,\ell \ell }\) of up to 50%.
Finally, in Fig. 4, we show the distribution of the sum of the photon energies in the decay rest frame, \(\sum _{n_{\gamma }} E_{\gamma }\), and the distribution of the socalled \(\phi _{\eta }^{*}\)variable. The sum of the photon energies is largely correlated with the \(p_{\perp ,\ell \ell }\), as discussed before. This distribution shows a distinct edge at about half the Zboson mass, which is being washed out by multiple radiation. The kinematics of the decay restrict the energy of a single radiated photon to be smaller than \(E_{\gamma , \mathrm {max}}^1 = \frac{\hat{s}4m_\ell ^2}{2\sqrt{\hat{s}}}\), which is roughly equal to half the boson mass near the resonance. For an event to have a total photon energy beyond this edge, two hard photons need to recoil at least partly against each other. The region above the kinematical edge is then only described approximately, as long as no NNLO corrections are considered. The NLO EW prediction mildly increases the number of events without photon radiation, leading to a decrease at the kinematic edge of about 3%. The NNLO QED corrections again provide a competing effect, leading to a difference of about 1% to the NLO QED predictions near the edge. Beyond it, the NNLO QED corrections show a significant departure from the shape of the previous predictions as this region is for the first time described correctly at fixedorder. The behaviour of different dressings and lepton species is very similar to the case of the \(p_{\perp ,\ell \ell }\). The bare electrons show a significantly larger number of hard photons, while another dressing only leads to an approximately flat decrease. Muonic decays contain a larger number of events with hard photons, while \(\tau \)’s radiate significantly less.
3.3 Charged Drell–Yan leptonneutrino pair production
In Fig. 5, we start with the transverse mass of the lepton neutrino system, \(M^{\perp }_{\ell \nu }\), and the invariant mass of the charged lepton and the nearest photon, \(m_{\ell \gamma }\). The \(M^{\perp }_{\ell \nu }\) observable is barely affected by the NLO EW corrections. In fact the softcollinear approximation agrees with NLO EW at the permill level. The dressing of the electrons has a rather large impact, with differences with respect to a bare treatment reaching up to 10% at the edge. A slight shift of the edge is observed when comparing different lepton species with one another, affecting the distribution to up to a few %.
The invariant mass of the charged lepton and the closest photon, \(m_{\ell \gamma }\) shows significantly larger corrections. Compared to the NLO EW corrections, the soft approximation predicts a spectrum that is too soft, while the softcollinear approximation produces up to 5% more events with large \(m_{\ell \gamma }\). Bare electrons have a lot more events at low \(m_{\ell \gamma }\) coming from those photons that have not been clustered in comparison to the dressed cases. On the other hand, those electrons dressed with \(\Delta R = 0.2\) have a reduced number of events at low \(m_{\ell \gamma }\). The comparison between lepton species shows significant differences close to low \(m_{\ell \gamma }\), illustrating the differing size of the dead cone.
In Fig. 6, we show the transverse momentum of the charged lepton, \(p_{\perp ,\ell }\), alongside the missing transverse energy, \(E_{\perp }^{\mathrm {miss}}\). The latter corresponds in our simple setup to the transverse energy that the neutrino carries away. Both distributions are related and indeed they behave very similarly. As in the neutralcurrent case, the transverse momentum of the charged lepton is barely affected by NLO EW corrections, with corrections only becoming appreciable for very low values of \(p_{\perp ,\ell }\). The dressing affects the distributions by up to about 10% in the peak region, while different lepton species differ by up to 4% in the peak region and at low \(p_{\perp ,\ell }\).
In Fig. 7, we present the sum of photon energies in the decay rest frame, \(\sum _{n_{\gamma }} E_{\gamma }\), and the number of photons with energy \(E_{\gamma } > 0.1 \,\mathrm {GeV}\), \(n_{\gamma }\). The sum of photon energies shows a kinematic edge just as in the neutral current case. While the soft approximation predicts too soft a spectrum of photon energies, the softcollinear approximation does a much better job in Wdecays as the effects coming from NLO EW corrections reach at most 5% at the kinematic edge. The reason for this behaviour can be read from the distribution of the \(n_{\gamma }\). The soft approximation is shown to produce too few photons, while the softcollinear approximation predicts more events with 1–3 photons. Analyses using bare electrons show a significantly larger number of photons, with already 4 times more events with 1 photon. At the same time, for \(\Delta R = 0.2\) electrons, the number of photons is suppressed significantly. A similar picture presents itself when comparing lepton species. Muonic decays contain significantly more photons, while decays into \(\tau \)’s end up with a lot less events with at least one photon.
As a noteworthy observation we want to point out a difference between neutralcurrent and chargedcurrent processes: the softcollinear approximation is more reliable in the chargedcurrent case. This can be understood from the fact that here collinear radiation predominantly originates from just one particle, the lepton, rather than two competing particles as in the Zboson case. In the latter case the effect of the error due to missing interference contributions in the softcollinear approximation is thus enhanced.
3.4 Leptonic Higgsboson decays
Finally we highlight the effect of higherorder corrections in photon radiation off leptonic Higgs decays. Numerical results are shown in Fig. 8, where the nominal distribution corresponds to \(H\rightarrow \mu ^+ \mu ^\) with bare muons. Here we focus on the dilepton invariant mass \(m_{\ell \ell }\) and \(p_{\perp ,\ell \ell }\) recoil. As for neutralcurrent Drell–Yan we consider higherorder corrections at the level of soft and softcollinear approximations, full NLO QED, NLO EW and also combining NLO EW with NNLO QED. The LO prediction clearly fails to describe the invariant mass distribution. Yet, the soft and softcollinear approximations provide a quite reliable description with corrections smaller than 1–2% with respect to full NLO QED. The weak corrections are slightly larger compared to the neutralcurrent Drell–Yan case, still they alter the invariant mass distribution only at the permille level and are overcompensated by NNLO QED effects of the same order. As mentioned in Sect. 3.1 we are unable to resolve the sharp mass peak of the Higgsboson with the lowest energy photons we generate. However, investigating the low energy tail of the invariant mass distribution, we observe that the NLO QED corrections provide a mostly flat contribution in the peak region. Comparing decays into bare muons with decays into bare \(\tau \)’s, we can appreciate a significantly smaller sensitivity of the \(\tau \) distribution to QED radiation.
The distribution of the transverse momentum of the dilepton system shows similar effects as in the case of the Zboson decay. The soft approximation predicts a distribution that is far too soft, while the softcollinear approximation predicts too many events with large \(p_{\perp ,\ell \ell }\). The NLO EW corrections increase the number of events by about a permille at low \(p_{\perp ,\ell \ell }\), and decrease them at high values up to about 5%. The NNLO QED corrections in this case do not provide a large competing effect, and the NNLO QED \(\oplus \) NLO EW prediction agrees with the NLO EW one at the permille level. Decays into \(\tau \)’s show about 40% less events with nonvanishing \(p_{\perp ,\ell \ell }\), the effect being close to constant across the entire distribution.
4 Conclusions and outlook
In this paper, we have presented an implementation of NLO EW and NNLO QED corrections to the decays of weak gauge and Higgs bosons within the YFS formalism. For this purpose, we extended Sherpa ’s module Photons to include the relevant matrix elements, renormalized in the onshell scheme, and subtractions needed within this formalism. In our numerical results we find that observables relating only to the leptons in the process are only marginally affected by the corrections, up to the level of a few permil. In particular, the peak of the invariant mass distributions is practically not affected. Distributions that relate to the energies of the generated photons themselves, or can be related to them, such as the transverse momentum of the pair of the leptons \(p_{\perp ,\ell \ell }\), naturally receive larger corrections. The electroweak corrections increase the likelihood of hard photon radiation by up to 23% for very hard radiation. The NNLO QED corrections compete with these corrections by reducing the likelihood of hard radiation, albeit to a smaller extent. At the same time, some regions of phase space are only described at leading order in \(\alpha \) upon the inclusion of the double real radiation, such that in these regions the corrections can be significantly larger. Examples for such regions are those where the sum of the photon energies exceeds half the boson mass or regions of large \(\phi ^{*}\). Angular distributions of the photons are not affected by higher order contributions confirming the general radiation pattern of QED radiation. The results give us confidence that the inclusion of the full EW corrections to particle decays within the YFS formalism in Sherpa are sufficient to achieve precise results for most leptonic observables. Beyond the corrections investigated in this work, it will be interesting to consider the YFS formalism also including initial state effects and the matching to NLO EW corrections to the hard production process, see [103].
The implemented NNLO QED and NLO EW corrections provide high precision also in extreme phase space regions and can be seamlessly added to standard precision QCD simulations. This provides an important theoretical input to future precision determinations of fundamental parameters of the EW theory at hadron colliders and beyond.
Footnotes
 1.
For an agreement correct up to order \(\mathcal {O}\left( \alpha \right) \), we would need to remove \(\tilde{\beta }_0^2\), \(\tilde{\beta }_1^2\) and \(\tilde{\beta }_2^2\). By far and large this has already been implemented in [57].
 2.
It should of course be noted that the SM Higgs has a resonance width of only \(\sim 4 \,\mathrm {MeV}\), which is smaller than this photon cut, suggesting that we still do not resolve the resonance well with this cut. However, we find that a cut of the order \(10 \,\mathrm {MeV}\) is necessary in order to guarantee a good performance of the method in both decay channels. In any case, this smaller choice of the cutoff still allows a closer investigation of the regions close to the resonance in plots generated from the lepton momenta, as long as the binning is not chosen too fine. In particular the regions that will be populated through the radiation of photons from leptons in the resonance region will be included in this description.
Notes
Acknowledgements
We would like to thank our colleagues from the Sherpa and OpenLoops collaborations for discussions. We are indebted to D. Wackeroth and A. Vicini for support with Wzgrad and for clarification of sometimes subtle issues in the calculational framework. This work was financially supported by the European Commission under Grant Agreements PITNGA2012315877 (MCnet) and PITNGA2012316704 (HiggsTools), and by the ERC Advanced Grant 340983 (MC@NNLO).
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