Abstract
We scrutinise the ability of the primary QED finalstate resummation tools, combined with electroweak virtual corrections, to reproduce the exact nexttoleading order electroweak calculation in the fourchargedlepton final state. We further examine the dependence of the findings on the leptonphoton dressingcone size as well as the resonance identification strategy. Overall we find excellent agreement with the fixedorder result, but partial differences not directly connected with resummationinduced higherorder effects at the fewpercent level are observed in some cases, which are relevant for precision measurements.
Introduction
The production of four charged leptons in proton–proton collisions offers a rich gamut of processes contributing to the same final state, bound through higherorder electroweak effects, in an experimentally clean environment. Precise measurements of this diverse spectrum are crucial for our understanding of irreducible backgrounds in Higgs boson production as well as vector boson scattering topologies, where chargeparityviolating effects could reveal compelling signs of physics beyond the Standard Model [1]. As such, a detailed study of the fourlepton invariant mass, the azimuthal decorrelation and other similar observables in \(pp\rightarrow \ell \ell \ell ^\prime \ell ^\prime \) production constitutes a vital probe of the gauge structure of the Standard Model whilst providing the ideal test bed to validate stateoftheart theoretical calculations that feed into the experimental analyses. Both ATLAS and CMS and have produced fiducial differential crosssection measurements of fourlepton production in an inclusive phase space [2] as well as onshell regions consistent with \(ZZ \rightarrow 4\ell \) production [3, 4] and \(H\rightarrow ZZ^*\rightarrow 4\ell \) production [5, 6]. Differential crosssection measurements of the fourlepton final state have already been used to set limits on both chargeparity violation [7] as well as the Higgs selfcouplings [8].
Of course, precision measurements necessitate precise calculations to be able to extract as much information as possible. To this end, the nexttoleading order (NLO) QCD corrections to onshell ZZ production are known for almost three decades [9, 10]. The offshell fourlepton production then followed no ten years later [11, 12]. Recently, the nexttonexttoleading order (NNLO) QCD corrections were added [13,14,15], stabilising the cross section predictions on the percent level with respect to the usual QCD scale uncertainties. Although gluoninitiated four lepton production, being a loopinduced process, formally contributes only at NNLO QCD and beyond, its contribution is phenomenologically relevant. Therefore, it was calculated early on [16,17,18,19], and even the NLO QCD corrections are known by now [20,21,22]. In terms of experimentally usable particlelevel predictions, at the moment only the NLO QCD calculations are matched to parton showers in various schemes [23,24,25,26,27,28], benefiting also from the respective event generators’ higherorder QED corrections which is especially important for observables sensitive to energy loss through photon radiation.
The electroweak (EW) correction to fourlepton production, on the other hand, were first calculated in the EW Sudakov approximation [29,30,31,32,33,34], tailored to describe observables sensitive to momentum transfers much larger than the electroweak scale. Photonic corrections, which are of particular importance to observables that contain resonance peaks or thresholds, were analytically calculated in [35]. The complete NLO EW corrections were only calculated in the last ten years [36,37,38,39] and were found to be important ingredients in precision phenomenology in four lepton final states. They have recently also been combined with the NNLO QCD corrections to form the highestprecision fixedorder calculation available [40]. During the completion of the present paper, also a first calculation matching the combined NLO QCD and NLO EW corrections to the parton shower has been presented in [41].
In the MonteCarlo event generators currently used by the LHC experiments, NLO QCD matrix elements are matched to parton showers, possibly merging in highermultiplicity process [42]. Therein, QED corrections are provided by universal QED parton showers [43,44,45,46,47] or other QEDspecific resummations [48,49,50,51]. Processspecific EW corrections are either applied a posteriori on the level of measured observables by extracting correction factors from the fixedorder calculations or they are applied in either the Sudakov [52, 53] or the recently formulated EW virtual approximation [54] on an eventbyevent basis.
Therefore, the aim of this paper is to quantify in a tuned comparison the inherent differences of the two commonly used tools for higherorder QED corrections, PHOTOS [48] and SHERPA’s YennieFrautschiSuura (YFS) [55] based softphoton resummation [51], combined with the EW virtual approximation, in order to ascertain their ability to reproduce the exact NLO EW results and to be able to quantify the algorithmic uncertainties associated with these corrections. This paper is thus organised as follows: in Sect. 2 we summarise the calculational methods and tools that are used in this paper. In Sect. 3 we then present a detailed comparison and analysis of the quality of the different approximations compared to the fixedorder NLO EW calculation. Finally, we offer our conclusions in Sect. 4.
Computational methods
In this paper, we compare the results obtained combining a calculation of LO accuracy in the electroweak sector with both a dedicated QED finalstate photon radiation resummation and approximate virtual EW corrections in the scheme of [54], for the production of four charged leptons to the exact NLO EW result.
The exact fixedorder NLO EW results have been obtained with the SHERPA+OPENLOOPS[47, 56,57,58] framework, allowing for a fully automated calculation of cross sections and observables at nexttoleading order in the electroweak coupling. In this framework, renormalised virtual amplitudes are provided by OPENLOOPS[57, 58], which uses the COLLIERtensor reduction library [59] as well as CUTTOOLS [60] together with the ONELOOP library [61]. All remaining tasks, i.e. the bookkeeping of partonic subprocesses, phasespace integration, and the subtraction of all QED infrared singularities, are provided by SHERPAusing the AMEGICmatrix element generator [62,63,64]. SHERPAin combination with OPENLOOPS(and other providers of renormalised oneloop corrections) has been employed successfully in a range of different calculations [54, 65,66,67,68,69,70,71,72,73,74] and has been validated against other tools in [75].
The NLO EW corrections to \(\mathrm {pp}\rightarrow 4\ell \) are dominated by either EW Sudakov logarithms of virtual origin or QED logarithms stemming from photon radiation off leptons, depending on the kinematic regime. While EW Sudakov logarithms dominate the large \(p_\mathrm {T}\) or large invariant mass regions, radiative energy loss through photon emission dominates invariant mass distributions below the Zpair threshold or around the resonant Z Breit–Wigner peak in two and fourlepton invariant masses. This observation allows to construct a simple yet effective highprecision standin for a full nexttoleading order matched event generator combining:

(i)
The virtual EW approximation. In [54] it was shown that, for observables that are sufficiently inclusive with respect to photon radiation and where all kinematic invariants are large with respect to the electroweak scale, the full NLO EW results can be reproduced with good accuracy by an approximation consisting only of the exact virtual EW corrections, whose infrared divergences have been suitably subtracted. Thus, this approximation, is defined through
$$\begin{aligned} \begin{aligned} \mathrm{d}\sigma _{\text {NLO }\text {EW}_\text {approx}}\!&=\; \mathrm{d}\sigma _\text {LO} + \mathrm{d}\sigma _\text {EW}^\text {V} + \mathrm{d}\sigma _{\text {EW,approx}}^\text {R}\\&= \mathrm{d}\sigma _\text {LO}\;(1 + \delta _{\text {EW}_\text {approx}})\;. \end{aligned} \end{aligned}$$(2.1)Therein, \(\mathrm{d}\sigma _\text {LO}\) is the leading order differential cross section, while \(\mathrm{d}\sigma _\text {EW}^\text {V}\) and \(\mathrm{d}\sigma _{\text {EW,approx}}^\text {R}\) are the exact NLO EW virtual correction and the endpoint part of the emittedphotonintegrated approximate real emission amplitude.^{Footnote 1} Hence, by construction, \(\mathrm{d}\sigma _{\text {EW,approx}}^\text {R}\) does not only ensure a finite result but also supplies real emission QED logarithms to the approximation. This approach captures all Sudakov effects at NLO EW and is also very suitable for a combination of QCD and EW higherorder effects through a simplified multijet merging approach at NLO QCD+EW [54, 71, 74].

(ii)
QED final state radiation. The inherent approximation of the above virtual EW approximation is partially unfolded again by employing dedicated finalstate photon emission resummations. Specifically, we consider a softphoton resummation in the Yennie–Frautschi–Suura (YFS) scheme [55] as implemented in SHERPA[51] and, alternatively, PHOTOS [48, 76,77,78].^{Footnote 2} Both are limited to final state radiation (FSR) and \(1\rightarrow n\) processes, but are currently the tools of choice to calculate QED FSR corrections for the LHC experiments. To understand their FSR resummation properties we sketch here their defining approximation of the allorders decay rate \(\mathrm{d}\varGamma \) in terms of a given LO decay rate \(\mathrm{d}\varGamma _0\). PHOTOS calculates it as
$$\begin{aligned} \begin{aligned}&\mathrm{d}\varGamma ^{\textsc {PHOTOS}}\\&\quad = \mathrm{d}\varGamma _0 \left\{ 1+ \sum \limits _{c=1}^{n_\text {ch}}\sum \limits _{n_\gamma } \frac{\left( \alpha \,L_c\right) ^{n_\gamma }}{n_\gamma !}\! \left[ \prod \limits _{i=1}^{n_\gamma }\mathrm{d}x_c^i\right] \! \right. \\&\qquad \left. \times \left( P_{\epsilon _{\text {cut}}}(x_c^1)\!\otimes \!.\;\!\!.\;\!\!.\!\otimes \! P_{\epsilon _{\text {cut}}}(x_c^{n_\gamma })\right) \right\} \end{aligned} \end{aligned}$$(2.2)where the radiative part is summed over all \(n_\text {ch}\) charged particles. \(L_c\) is the logarithm of the ratio of the decaying particle’s mass over the mass of the charged particle c, and \(x_c=\prod x_c^i\) is its retained energy fraction after the radiation of \(n_\gamma \) photons. The phase space distribution of these photons is described by the AltarelliParisi splitting functions \(P_{\epsilon _{\text {cut}}}(x)\) in the presence of the infrared cutoff \(\epsilon _{\text {cut}}\), modified by suitable weights to recover the correct softphoton limit and implement exact higherorder corrections, and iterated over all \(n_\gamma \) emitted photons. Their precise definitions can be found in [76]. The implementation of the YFS softphoton resummation in SHERPA, on the other hand, calculates the allorders resummed decay rate using
$$\begin{aligned}&\mathrm{d}\varGamma ^\text {YFS} = \mathrm{d}\varGamma _0\cdot e^{\alpha Y(\omega _\text {cut})}\nonumber \\&\quad \times \sum \limits _{n_\gamma }\frac{1}{n_\gamma !} \left[ \prod \limits _{i=1}^{n_\gamma }\mathrm{d}\varPhi _{k_i}\cdot \alpha \, {\tilde{S}}(k_i)\,\varTheta (k_i^0\omega _\text {cut}) \cdot {\mathcal {C}} \right] \,.\nonumber \\ \end{aligned}$$(2.3)Here, \(Y(\omega _\text {cut})\) is the YFS form factor resumming unresolved real and virtual softphoton corrections. The individual resolved photon \(k_i\)’s phase space, \(\varPhi _{k_i}\), is distributed according to the eikonal \({\tilde{S}}(k_i)\), which is built up by the coherent sum of dipoles formed by all pairs of charged particles in the decay. \(\omega _\text {cut}\) separates the explicitlygenerated resolved from the integratedover unresolved real photon emission phase space. The correction factor \({\mathcal {C}}\) restores the correct spindependent collinear limit and contains decayspecific exact higherorder correction, cf. [51] for details.
With Eqs. (2.2) and (2.3) at hand, we observe that through the inclusion of exact NLO QED matrix element corrections^{Footnote 3} to their initial photon distributions (collinear splitting functions in PHOTOS, soft eikonal in YFS), both resummations should produce very similar results in \(Z\rightarrow \ell ^+\ell ^\) decays. As both approaches, however, resum different quantities, the logarithm \(L_c\) in PHOTOS and the YFS form factor Y in the softphoton resummation, differences are expected when resummation effects become important.
Finally, conversions of photons into lepton pairs is not accounted for in either program. It needs to be noted that both resummations are unitary and do not alter the event weight.
Consequently, the combination of either QED FSR resummation with the virtual EW approximation are dubbed NLO \(\text {EW}_\text {approx}\times {\textsc {YFS}}\) and NLO \(\text {EW}_\text {approx}\times {\textsc {PHOTOS}}\) approximations in the following. Its validity was further tested for other classes of processes, among them the production of \(2\ell 2\nu \) final states, [66, 74]. While this construction is of course not formally NLO accurate, it provides an accurate description of both logarithmically enhanced regions. Its performance will be assessed in detail in Sect. 3. One crucial input, however, is the treatment of resonances in the QED FSR tools. It is described in the following.
Resonance identification The implementation of resummed final state photon emission corrections in SHERPAincludes a generic resonance identification, ensuring that collective multipole radiation off the chargedlepton ensemble preserves all resonance structures present in the event. This is more relevant in softphoton resummations than in collinear ones, since soft wideangle emissions have a stronger effect on the lepton direction than collinear ones and are not recombined into a physical dressed lepton momentum. To this end, first the final state of a scattering process is analysed and possible resonances decaying into lepton pairs are identified on the basis of event kinematics and existing vertices in the model. For the process studied in this paper, \({\mathrm {pp}}\rightarrow \ell ^+\ell ^\ell ^{\prime +}\ell ^{\prime }\) (\(\ell ,\ell ^\prime \in e,\mu \)), multiple resonance structures are possible. They are disentangled on the basis of the distance measure \(\varDelta _{\ell \ell }^Z= m_{\ell ^+\ell ^}  m_Z/\varGamma _Z\), where of course only sameflavour pairs are taken into account. A lepton pair is then considered to be produced by a resonance if \(\varDelta _{\ell \ell }^Z<\varDelta _\text {thr}\), with \(\varDelta _\text {thr}\) being a free parameter of order 1. Subsequently, identified resonantproduction subprocesses are separated from the rest of the event, and the emerging decay is dressed with photon radiation respecting the Breit–Wigner distribution of the resonance, i.e. preserving the original virtuality of the offshell leptonic system. Finally, all leftover nonresonantly produced leptons are grouped in a fictitious process, \(X\rightarrow \ell ^+\ell ^\) or \(X\rightarrow \ell ^+\ell ^\ell ^{\prime +}\ell ^{\prime }\), with suitably adjusted masses for X.
Thus, depending on the fourlepton kinematics, three cases can be distinguished, cf. Fig. 1:

(a)
Double resonant. Two pairs of opposite sign and same flavour leptons whose respective \(\varDelta _{\ell \ell }^Z\) is smaller than \(\varDelta _\text {thr}\) are identified by the above algorithm. Hence, both \(Z\rightarrow \ell ^+\ell ^\) decays are reconstructed (setting the Z mass equal to \(m_{\ell \ell }\)) and passed separately to the QED FSR resummation.

(b)
Single resonant. Only one pair of opposite sign and same flavour leptons with \(\varDelta _{\ell \ell }^Z\) is smaller than \(\varDelta _\text {thr}\) is found. Only for this pair a \(Z\rightarrow \ell ^+\ell ^\) decay is reconstructed, and passed on as such to the QED FSR resummation. The remaining leptons are treated as nonresonantly produced and passed to the QED FSR resummation as such. In consequence, no specific \(Z\rightarrow \ell ^+\ell ^\) higherorder corrections are applied.

(c)
Nonresonant. No opposite sign and same flavour lepton pair with \(\varDelta _{\ell \ell }^Z<\varDelta _\text {thr}\) is found. Consequently, the complete four lepton final state is passed to the QED FSR resummation as is and no specific \(Z\rightarrow \ell ^+\ell ^\) higherorder corrections are applied.
In essence, due to the inclusivity of the cuts employed for the analysis in Sect. 3, the bulk of the cross section is classified as doubly resonant. The precise fraction, however, depends on the free parameter \(\varDelta _\text {thr}\), or the answer to the question when is a lepton pair considered to be produced resonantly or not.
Results
For the numerical results presented in this section we use the tools and methods summarised in Sect. 2. Both the NLO EW calculation as well as the approximate NLO \(\text {EW}_\text {approx}\times {\textsc {YFS}}\) and NLO \(\text {EW}_\text {approx}\times {\textsc {PHOTOS}}\) are calculated (and renormalised) in the \(G_\mu \)scheme with the following input parameters
All other particles are considered massless. The electromagnetic coupling is thus defined as
with the complex masses and mixing angles,
The additional power of \(\alpha \) occuring at NLO is set to its value in the Thomson limit,
in order to facilitate the comparison to the FSR resummation tools. Higherorder EW corrections are estimated by changing the renormalisation scheme to the \(\alpha (m_Z)\) scheme,^{Footnote 4} still keeping the additional power in the EW coupling at NLO at \(\alpha (0)\). As this delivers only a discrete twopoint variation, an estimate of the renormalisation scheme uncertainty would be obtained by symmetrising the difference between the two predictions around our chosen central value.
Furthermore, we use the NNPDF30_nnlo_as_0118 PDFs [80], SHERPA’s default PDF also used by the LHC experiments, interfaced through LHAPDF 6.2.1 [81]. This choice removes \(\gamma \)induced contributions, which both facilitates the comparisons against the QED finalstate resummations and has been found to be phenomenologically unimportant [38, 39]. It also makes our findings directly transferable to current LHC applications which all use this PDF set. However, as we nonetheless include QED initialstate mass factorisation terms to render the NLO EW calculation finite [63], we incur a slight mismatch in the initialstate evolution between the PDF and NLO EW calculation, which again does not impact the comparison presented in the following.
Our results are independent of the QCD renormalisation scale \(\mu _R\) throughout, and only weakly depend on the factorisation scale \(\mu _F\). To avoid having to resolve ambiguities in the sameflavour channel, we simply set it to
where the sum includes all four dressed lepton momenta defined below. In addition, both the YFS softphoton resummation and PHOTOS use the electromagnetic coupling in the Thomson limit, cf. Eq. (3.3). As infrared cutoffs we use \(\omega _\text {cut}=1\,\text {MeV}\) for the YFS softphoton resummation, applied to the photon energy in the restframe of the radiating multipole after radiation, and \(\epsilon _\text {cut}=1\times 10^{5}\) for PHOTOS, which translates into \(\omega _\text {cut}=\epsilon _\text {cut}\cdot m\) where m is the invariant mass of the reconstructed decaying particle in its rest frame, as detailed in Sect. 2. In both cases, we investigate the impact of a conservative and a relaxed choice of clustering threshold, setting \(\varDelta _\text {thr}=1\) and \(\varDelta _\text {thr}=10\) respectively.
We analyse the events with RIVET[82] using an event selection based on a recent ATLAS measurement of the inclusive fourlepton lineshape at 13 TeV [2]. Electrons and muons are defined at the dressed level, meaning the lepton fourmomentum is combined with the fourmomenta of nearby prompt photons for different dressingcone sizes. The dressingcone size itself is varied between \(\varDelta R_\text {dress}=0.005,0.02,0.1,0.2\).^{Footnote 5} Prompt photons used in the dressing procedure are subsequently removed from the final state. Exactly four muons are selected in the sameflavour case or exactly two electrons and two muons in the differentflavour case. All leptons are required to be within a pseudorapidity of \(\left \eta _\ell \right < 2.47\) and to have a minimum transverse momentum of \(20\,\text {GeV}\) for the leading lepton, \(15\,\text {GeV}\) for the subleading lepton, and \(10\,\text {GeV}\) and \(7\,\text {GeV}\) for the third and fourth lepton, respectively. All sameflavour lepton pairs have to be separated by at least \(\varDelta R = \sqrt{(\varDelta \eta )^2 + (\varDelta \phi )^2} > 0.1\), while a stricter separation of \(\varDelta R > 0.2\) is required for differentflavour leptons. In case the dressing cone size is larger than half of the pairwise lepton separation, photons are combined with the closest lepton.
Exactly two oppositecharge dilepton pairs are required in the event, where the leading lepton pair is chosen to be the one whose invariant dilepton mass is closest to the Zboson resonance. A dilepton invariant mass window of \(50\,\text {GeV}< {m_{\ell \ell }}< 106\,\text {GeV}\) is used for the leading lepton pair, while a dynamic invariant mass cut is employed for the subleading lepton pair, depending on the overall fourlepton invariant mass, \({m_{4\ell }}\) using the following slidingwindow algorithm:

for \({m_{4\ell }}< 100\,\text {GeV}\), require \({m_{\ell \ell }}> 5\,\text {GeV}\) for the subleading pair;

for \(100\,\text {GeV}\le {m_{4\ell }}< 110\,\text {GeV}\), require \({m_{\ell \ell }}> 5\,\text {GeV}+ 0.7\times \left( m_{4\ell }  100\,\text {GeV}\right) \) for the subleading pair;

for \(110\,\text {GeV}\le {m_{4\ell }}< 140\,\text {GeV}\), require \({m_{\ell \ell }}> 12\,\text {GeV}\) for the subleading pair;

for \(140\,\text {GeV}\le {m_{4\ell }}< 190\,\text {GeV}\), require \({m_{\ell \ell }}> 5\,\text {GeV}+ 0.76\times \left( m_{4\ell }  140\,\text {GeV}\right) \) for the subleading pair;

for \(190\,\text {GeV}\le {m_{4\ell }}\), require \({m_{\ell \ell }}> 50\,\text {GeV}\) for the subleading pair.
This somewhat intricate definition of the fiducial volume increases the number of experimentally cleanly measurable events in particular in the region below the ZZ continuum where at most one of the ZZ bosons can be onshell. In particular, the \(Z\rightarrow 4\ell \) resonance is strongly enhanced when compared to uniform acceptance criteria for all leptons. For our comparison this has the advantage that the performance of both approximations can be extensively tested in various regimes, each comprising very different resonant structures.
In the following, we compare the Bornlevel prediction (black) with the exact NLO EW prediction (green) and the approximate NLO \(\text {EW}_\text {approx}\) approximation, augmented with PHOTOS (dotted) or YFS(solid) using either a conservative (red) or relaxed (blue) clustering threshold. We also study the effect of using a range of different dressingcone sizes, where we expect the dependence of the respective cross sections on the dressingcone size to be better described by the QED FSR tools than the fixedorder calculations. In particular, we expect both the fixedorder calculations and the QED FSR resummations to agree well for the most inclusive dressingcone size of \(\varDelta R_\text {dress}=0.2\), while the largest dressingconesize induced deviations are to be expected for the smallest size of \(\varDelta R_\text {dress}=0.005\).
Inclusive cross sections
Before we turn to discuss several classes of differential distribution we briefly scrutinise the inclusive cross section in the fiducial phase space described above. Table 1 summarises these inclusive fiducial cross section for both the sameflavour and differentflavour channel and the representative lepton dressing cone of \(\varDelta R_\text {dress}=0.1\). Most notable, the fixedorder cross section displays a marked dependence on the EW input and renormalisation scheme as it is proportional to \(\alpha ^4\) at the leading order. To estimate the uncertainty due to missing higherorder EW corrections, we vary the renormalisation scheme from our default, the \(G_\mu \) scheme, to the \(\alpha (m_Z)\) scheme. Both schemes are generally considered suitable for the processes under consideration. Indeed, the NLO corrections in the \(G_\mu \) and \(\alpha (m_Z)\) schemes are both at the fewpercent level, albeit of opposite sign: \(4.9\%(4.8\%)\) vs. \(+2.6\%(+2.7\%)\) in the differentflavour (sameflavour) channel, respectively. In any case, in line with our expectation, the EW schemeuncertainty decreases from \(9.8\%\) at LO to \(2.7\%\) at NLO. It is to be expected though that in regions of phase space with larger EW corrections this uncertainty rises as well. Finally, given this higherorder uncertainty, the NLO \(\text {EW}_\text {approx}\times {\textsc {YFS}}\) and NLO \(\text {EW}_\text {approx}\times {\textsc {PHOTOS}}\) approximations very well reproduce the exact result to within less than 0.5%. By their construction, including the exact renormalised virtual contributions, they also well reproduce the exact renormalisation scheme dependence. The agreement for the other, somewhat less standard, dressing cones can be gauged from Fig. 6. Disagreements for both stay well below 1% for \(\varDelta R_\text {dress}=0.2\) and 0.02, only rising to slightly above 1% for \(\varDelta R_\text {dress}=0.005\), in line with our earlier expectation. At this point it is again imperative to stress that this excellent level of agreement is to some degree accidental: despite the wellmotivated construction of the approximation it is formally not NLO EW accurate. As an example, this level of agreement for inclusive cross sections was not observed in, e.g., \(\mu ^+\nu _\mu e^{\bar{\nu }}_e\) production [74].
Lepton transverse momentum distributions
The first class of observables we are examining are the transverse momentum distributions of the four leptons. They are shown in Figs. 2, 3, 4 and 5, respectively. Looking at the fixedorder result first, its renormalisation scheme uncertainty increases as the size of the NLO EW correction gets larger, rising from slightly over \(2\%\) in the peak of each distribution to quickly to more than 5% as the transverse momenta increase.
The dominant effect of the electroweak corrections in the lepton transverse momentum distributions is a depletion of the crosssection in the high \(p_\mathrm {T}\) tails through the EW Sudakov logarithms, which is well reproduced by the NLO \(\text {EW}_\text {approx}\times {\textsc {YFS}}\) and NLO \(\text {EW}_\text {approx}\times {\textsc {PHOTOS}}\) approximations in all distributions. Deviations are typically much smaller than the EW renormalisation scheme uncertainty. When comparing the two approximations to the fixedorder calculation, it can be seen that for both the differentflavour and sameflavour channel both PHOTOS and YFSbehave similarly across the spectrum, except for the low\(p_\mathrm {T}\) end of the leading and secondleading lepton \(p_\mathrm {T}\) distribution. Here, depending on the dressingcone size, YFS slightly undershoots the fixedorder calculation. The effect is most pronounced just below the peak of the respective distribution. This behaviour can be attributed to the fact that the YFS softphoton resummation has more wideangle radiation than PHOTOS that will not be recombined into the dressed lepton object. In turn, this causes more events to fail the minimum \(p_\mathrm {T}\) requirements of both leptons, leading to the correspondingly slightly reduced inclusive cross section already reported in Table 1. A similar effect is not present in the third and fourth lepton in the \(p_\mathrm {T}\) region under consideration.
While the lepton \(p_\mathrm {T}\) distributions are generally insensitive to the choice of clustering threshold \(\varDelta _\text {thr}\), a small dependence on the size of the dressingcone size can be seen, which can be expected since the amount of FSR radiation off the leptons captured by the dressing algorithm determines whether or not the event will pass the fiducial selection. The two larger dressingcone sizes are more inclusive and so generally better reproduce the fixedorder calculation, which in turn is not expected to reasonably describe the energy profile within the cone. This is where the resummation employed by the two approximations becomes relevant in order to describe the dressingcone dependence accurately.
Fourlepton observables
Similar to the individual lepton \(p_\mathrm {T}\) spectra, both PHOTOS and the YFSbased resummation agree well with the fixedorder calculation also for multilepton observables in the different and sameflavour channels. In almost all regions their deviation from the exact result is much smaller than the renormalisation scheme uncertainty, which can be seen in the fourlepton rapidity distribution in Fig. 6 but also in the fourlepton invariant mass spectrum in Fig. 7. As before, the fixedorder scheme uncertainty increases as the overall size of the electroweak correction increases. However, this uncertainty is estimated only by a discrete twopoint variation, producing pinchpoints whenever the two schemes switch their roles as the one predicting the larger cross section. The thus assessed uncertainty, even after symmetrisation, is underestimated in these regions and should be compared with nearby regions away from the pinch points.
The fourlepton invariant mass distribution covers a wide range of topologies: the ZZ continuum sharply turns on around 180 \(\text {GeV}\), just before the horizontal axis transitions from a linear to a logarithmic scale at 200 \(\text {GeV}\). Below the continuum threshold, one of the bosons has to be increasingly offshell and the crosssection drops accordingly. The cross section then experiences a small rise caused by the virtuality of the offshell \(\gamma ^*\) to move towards zero until such topologies are disallowed by the otherwise comparably inclusive cuts on the subleading leptons. For \(m_{4\ell }\approx m_Z\) the \(Z\rightarrow 4\ell \) peak is well developed, again due to the loose cuts on the subleading leptons which allow for a large number of the preferred hierarchical structure in \(Z\rightarrow \ell \ell \gamma [\rightarrow \ell \ell ]\) decays. With the leptons of the subleading pair allowed to become soft, a Drell–Yanlike topology is picked out where a primary lepton pair radiates a photon that subsequently splits into a secondary lepton pair with a typically much smaller invariant mass. Since this topology is described with fixedorder matrix elements, all possible combinations and interferences between primary and secondary lepton pair are accounted for.
QED finalstate radiation that is not captured by the dressing algorithm will cause the fourlepton system to lose energy and hence migrate from higher to lower invariant mass values. The effect will be largest, with corrections reaching \({\mathcal {O}}(1)\), just below the Z resonance and the ZZ continuum threshold due events migrating from these regions of enhanced crosssection through radiative energy loss. The precise size of this effect, however, strongly depends on the size of the dressing cone, as it determines how much photon radiation is recombined. These effects are seen in the NLO EW fixedorder prediction and are well reproduced by both approximations for large dressingcone sizes. As expected, the differences increase the smaller \(\varDelta R_\text {dress}\), with the resummations again being expected to yield more reliable results for very small dressingcone sizes.
In the offshell regions below the resonances, the impact of the different clustering thresholds, which determine when a leptonpair is considered to be produced resonantly, also becomes visible. Not unexpectedly, the effect is larger in the \(4\mu \)channel than in the \(2e2\mu \)channel, as the number of potential pairings is larger. Generally, it can be observed that the tighter clustering threshold is somewhat too strict, whereas the looser threshold typically reproduces the full fixedorder calculation better in this region of phase space. Overall, due to its construction around the collinear limit, PHOTOS shows a smaller clustering threshold dependence than the softphoton resummation of YFS, except for extremely low fourlepton invariant masses.
The large invariantmass tails are dominated by virtual EW Sudakov logarithms, but a residual dressingconesize dependence remains. In all cases, PHOTOS and YFSgive almost identical results in both the differentflavour and sameflavour channels. For the most inclusive \(\varDelta R_\text {dress}\) they also excellently agree with the fixedorder calculation, as expected.
Leptonpair observables
Turning now to leptonpair observables, Fig. 8 shows the invariant mass of the muon pair in the differentflavour process in the top row and the oppositesign lepton pair whose invariant is closest to the nominal Z mass for the sameflavour process in the bottom row. In both cases the expected resonance around 91 \(\text {GeV}\) is accompanied by a smaller enhancement at lower invariant mass values, the shape of which is induced by the fiducial selection criteria. The region below 50 GeV and above 106 GeV is only filled in the differentflavour case where the identification of the two leptonpairs, and Z candidates, is unambiguous and therefore, the muonpair may be very far offshell. Whereas in the sameflavour case the leptons, and corresponding Z candidates are identified by choosing the one out of four possible pairings which has the closest invariant mass to the nominal Z mass, and is thus limited by the event selection to a minimal and maximal value of 50 and 106 GeV, respectively. The biggest effect of the electroweak corrections is then again seen just below the Z resonance and the selectioninduced enhancement below.
Again, there is good agreement between the FSR resummations and the fixedorder calculation for inclusive dressingcone sizes, in particular compared to the fixedorder resummation scheme uncertainty, though as before, differences grow larger for smaller \(\varDelta R_\text {dress}\). The dependence on the clustering threshold \(\varDelta _\text {thr}\) is also larger for the YFSsoftphoton resummation than for PHOTOS, with the conservative \(\varDelta _\text {thr}=1\) being too restrictive.
The corresponding transverse momentum spectra are shown in Fig. 9, which also features a cutinduced enhancement around 20–30 GeV as well as the usual electroweak Sudakov suppression in the tail of the distribution. Variations of the dressingcone size result in a global shift of the two approximations compared to the fixedorder calculation where the latter tends to be better reproduced by the larger dressingcone sizes. A notable exception here is the aforementioned cutinduced hump around 25 GeV where the EW corrections display a stronger dressingconesize dependence. Both effects are not surprising as every cut in the fiducial selection adds sensitivity to the modelling of QED finalstate radiation, which is required to accurately describe the fraction of events predicted to pass the selection cuts.
Although the transverse momentum observables display hardly any dependence on \(\varDelta _\text {thr}\), the YFSsoftphoton resummation and PHOTOS predict noticeably different results on the 1% level below \(\approx 30\) GeV, with PHOTOS being consistently larger for every considered dressing cone size in both the sameflavour as well as the differentflavour channel.
Azimuthal correlations
Figure 10 shows a few possible phasespace configurations of the fourlepton final state in the \(p_\mathrm {T} \)–\(\phi \) plane. In the Born configuration, the leading two leptons are typically in opposite hemispheres resulting in a large azimuthal difference between them. Here, either the leading lepton \(\ell _1\) balances all three subleading leptons \(\ell _2\), \(\ell _3\) and \(\ell _4\) (a), or either the third or fourth lepton may cross over to the leading lepton’s hemisphere (b). In order for the azimuthal opening angle \(\varDelta \phi \) between the leading and the subleading lepton to become small, and in particular for the subleading lepton to cross over into the leading lepton’s hemisphere, both the relative transverse momenta of all four leptons have to become almost degenerate and the opening angle between the third and fourth lepton has to be smaller than that of the leading and subleading one (c). All of these restrictions are lifted once an additional object to recoil against is present (d), greatly enhancing the available phase space for configurations with small \(\varDelta \phi (\ell _1,\ell _2)\).
Figure 11 now displays the azimuthal separation of the two leading leptons, showing exactly the aforementioned suppression for small \(\varDelta \phi \) at leading order. For \(\varDelta \phi (\ell _1,\ell _2)>\tfrac{\pi }{2}\), where the leading and subleading leptons reside in opposite hemispheres, the NLO EW corrections and their uncertainties are roughly constant and reproduce the total NLO EW corrections to the inclusive cross section. Here, both YFSand PHOTOS agree well with the fixedorder calculation, with deviations in the permille range being much smaller than the renormalisation scheme uncertainty of 2.5–3%, for the most inclusive dressingcone sizes. The smaller dressing cones again induce shape and rate differences between the resummations and the fixedorder result. Only minute \(\varDelta _\text {thr}\)dependences can be observed.
In the region \(\varDelta \phi (\ell _1,\ell _2)<\tfrac{\pi }{2}\) now, the NLO EW corrections, through the presence of the additional real emission photon, lifts the abovediscussed kinematic restrictions and induce strongly increasing positive EW corrections, although the absolute cross section in this region remains tiny. Correspondingly, as this correction is driven by the real emission corrections only, the scheme uncertainty becomes leadingorderlike and increases to over 10%. Nonetheless, as the nature of the large corrections indicates, \({\mathcal {O}}(\alpha ^2)\) corrections are expected to be large. This is confirmed by the large deviation the resummations exhibit throughout all dressingcone sizes, being in rather good agreement between themselves. Also in this region, \(\varDelta _\text {thr}\)dependences are small.
Since the first and second lepton are typically in opposite hemispheres, there is a lot of freedom for the orientation of the third lepton. In fact, all \(\varDelta \phi \) between 0 and \(\tfrac{2\pi }{3}\) are well populated, with exception of the dilepton \(\varDelta R\) imposed by the selection cut, cf. Fig. 12. The fact that this drop happens at \(\varDelta R(\ell _2,\ell _3)<\tfrac{\pi }{15}\approx 0.2\) suggests that both leptons are not coming from the same Z boson in the differentflavour channel in this region at Born level. In the sameflavour channel, likely due to the presence of a photonpole between four out of the six leptonpair combinations, the cross section slightly rises as \(\varDelta \phi \) tends to zero, until the selection criteria regulate the pole. In turn, the NLO EW corrections and their uncertainties show no shape in this region and reproduce the inclusive corrections. They are, independent of the clustering threshold, also well reproduced by both the NLO \(\text {EW}_\text {approx}\times {\textsc {YFS}}\) and NLO \(\text {EW}_\text {approx}\times {\textsc {PHOTOS}}\) approximations, notwithstanding small differences at the level of 1% in both the same and differentflavour channel as \(\varDelta \phi \rightarrow 0\). As before, the agreement with the fixedorder result for large \(\varDelta R_\text {dress}\) is much better than the renormalisation scheme uncertainty, but is worsened for smaller dressingcone sizes, in line with observations made for earlier observables.
Conversely, the azimuthal difference between the second and the third lepton is suppressed in the backtoback configuration at \(\varDelta \phi \approx \pi \). This is again a result of the kinematic suppression of the configurations depicted in Fig. 10d. Photon emissions lift the kinematic restrictions also in this case and allow the second and third lepton closer together, thereby opening up phase space for the backtoback topology. This is once more manifested as an electroweak enhancement, this time in the region around \(\pi \). Both PHOTOS and YFSagree well with oneanother, and their difference with fixedorder calculation indicates large \({\mathcal {O}}(\alpha ^2)\) corrections.
For the third and the fourth lepton, the azimuthal difference would be enhanced towards backtoback or closeby values of \(\varDelta \phi \). However, the isolation requirements on the leptons suppress the configurations where two of the leptons are very close to each other, giving rise to a kink towards very low values of the azimuthal difference, as can be seen in Fig. 13. No part of the distribution is kinematically suppressed at leading order, hence no region receives large positive radiative corrections. On the contrary, the NLO EW corrections are flat and featureless throughout, and, apart from a 1% difference between YFSand PHOTOS in both the sameflavour and the differentflavour channel for small \(\varDelta \phi \), are well reproduced by both approximations for inclusive dressingcone sizes. Virtually no \(\varDelta _\text {thr}\) dependence is observed.
Conclusions
In this paper we presented a study of kinematic distributions in the fourchargedleptons final state including Born and oneloop EW corrections using the SHERPAand OPENLOOPSframeworks. In addition to the exact NLO EW calculation, we incorparated EW corrections in an approximation, based on exact virtual NLO contributions supplemented with a softphoton resummation using both PHOTOS as well as SHERPA’s softphoton resummation in the Yennie–Frautschi–Suura scheme. We showed that this approximation is able to reproduce the full NLO EW result for \(pp\rightarrow \ell \ell \ell ^\prime \ell ^\prime \) production to within a few percent, which we studied separately for the sameflavour and the differentflavour configuration. We observed that the setup which uses PHOTOS to model the softphoton emissions consistently predicts a larger crosssection than the setup using the YFSscheme, with the largest differences seen in the differentflavour case, while the YFSscheme is generally closer to the fixedorder NLO EW calculation.
We also studied the dependence on the dressingcone size and find that a cone size of \(\varDelta R_\text {dress}=0.1\) gives the best overall agreement between the two approximations and the fixedorder calculation. Further, while both resummation calculations are expected to give a more reliable dependence on the dressingcone size \(\varDelta R_\text {dress}\), an adoption of the smallest dressingcone radius of 0.005 induces both shape and ratechanges in most distributions. This emphasises the need for a properly matched calculation to combine the resummed description with the formal accuracy of the exact NLO EW calculation.^{Footnote 6}
Finally, we also investigated the effect of the clustering threshold used by SHERPAto preserve resonance structures and observed that, compared to the default value \(\varDelta _\text {thr}=1\), a more relaxed threshold tends to improve the agreement with the fixedorder result in most regions of phase space. This indicates that the QED corrections to the fourlepton final state behave as if the leptons were produced resonantly in a larger region of phase space than a naïve interpretation of the Breit–Wigner width suggests.
Data Availability Statement
This manuscript has no associated data or the data will not be deposited. [Authors’ comment: This a theory calculation. All numbers are contained in the manuscript.]
Notes
 1.
In practice, the CataniSeymour Ioperator is used.
 2.
We use the native implementation of the softphoton resummation of SHERPA2.2.8, and use the C++ interface to PHOTOS 3.6.4 to directly call PHOTOS from within SHERPA. Both tools are handed the exact same reconstructed \(1\rightarrow n\) subprocesses. Each interface and parameter setup is independent of the process (or reconstructed resonant subprocess) under consideration.
 3.
While NNLO QED + NLO EW corrections are available for the YFS implementation in SHERPA[79] it is currently not the default in the experiments, and thus not employed here.
 4.
The \(\alpha (m_Z)\) scheme is defined by the W and Z masses and widths detailed above in addition to \(\alpha (m_Z)=1/128.802\).
 5.
We have studied all of the following dressing cone sizes \(\varDelta R_\text {dress}=0.001,0.002,0.005,0.01,0.02,0.05,0.1,0.2,0.5\). We have chosen the above selection to combine readability with instructiveness, bearing in mind practical relevance.
 6.
One such matched calculation has appeared recently in [41] using the PYTHIA8 QED shower.
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Acknowledgements
We thank Max GoblirschKolb for many fruitful discussions during early stages of the project. This work has received funding from the European Union’s Horizon 2020 research and innovation programme as part of the Marie SkłodowskaCurie Innovative Training Network MCnetITN3 (Grant agreement no. 722104). MS acknowledges the support of the Royal Society (URF\R1\180549) through the award of a University Research Fellowship.
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Gütschow, C., Schönherr, M. Four lepton production and the accuracy of QED FSR. Eur. Phys. J. C 81, 48 (2021). https://doi.org/10.1140/epjc/s10052020088169
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