Precision studies of observables in \(p p \rightarrow W \rightarrow l\nu _l\) and \( pp \rightarrow \gamma ,Z \rightarrow l^+ l^\) processes at the LHC
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Abstract
This report was prepared in the context of the LPCC Electroweak Precision Measurements at the LHC WG (https://lpcc.web.cern.ch/lpcc/index.php?page=electroweak_wg) and summarizes the activity of a subgroup dedicated to the systematic comparison of public Monte Carlo codes, which describe the Drell–Yan processes at hadron colliders, in particular at the CERN Large Hadron Collider (LHC). This work represents an important step towards the definition of an accurate simulation framework necessary for very highprecision measurements of electroweak (EW) observables such as the W boson mass and the weak mixing angle. All the codes considered in this report share at least nexttoleadingorder (NLO) accuracy in the prediction of the total cross sections in an expansion either in the strong or in the EW coupling constant. The NLO fixedorder predictions have been scrutinized at the technical level, using exactly the same inputs, setup and perturbative accuracy, in order to quantify the level of agreement of different implementations of the same calculation. A dedicated comparison, again at the technical level, of three codes that reach nexttonexttoleadingorder (NNLO) accuracy in quantum chromodynamics (QCD) for the total cross section has also been performed. These fixedorder results are a welldefined reference that allows a classification of the impact of higherorder sets of radiative corrections. Several examples of higherorder effects due to the strong or the EW interaction are discussed in this common framework. Also the combination of QCD and EW corrections is discussed, together with the ambiguities that affect the final result, due to the choice of a specific combination recipe. All the codes considered in this report have been run by the respective authors, and the results presented here constitute a benchmark that should be always checked/reproduced before any highprecision analysis is conducted based on these codes. In order to simplify these benchmarking procedures, the codes used in this report, together with the relevant input files and running instructions, can be found in a repository at https://twiki.cern.ch/twiki/bin/view/Main/DrellYanComparison.
1 Introduction
Precision electroweak (EW) measurements in Drell–Yanlike processes at the Fermilab Tevatron and CERN Large Hadron Collider (LHC), \(p p (p \bar{p}) \rightarrow W^\pm \rightarrow l^\pm \nu _l\) and \( pp (p\bar{p})\rightarrow \gamma ,Z \rightarrow l^+ l^\) (\(l=e,\mu \)), require the development of sophisticated simulation tools that should include the best theoretical knowledge available (for recent reviews see, e.g., [1, 2, 3]). Several different theoretical effects enter in the accurate evaluation of total cross sections and kinematic distributions: higherorder QCD corrections, higherorder EW corrections, the interplay between EW and QCD effects, matching of fixedorder results with QCD/QED Parton Showers (PS), tuning of QCD PS to reproduce nonperturbative lowenergy effects, and effects of Parton Distribution Functions (PDF) and their uncertainties. The usage of different Monte Carlo (MC) programs that implement some or all of the above mentioned effects is not trivial.
As an explicit example of the need for the best theoretical predictions, we can consider for instance the measurement of the W boson mass (\(M_W\)), which is extracted from the transverse mass distribution of the \(l\nu \) pair in \(p p (p \bar{p}) \rightarrow W^\pm \rightarrow l^\pm \nu _l\) by means of a template fit to the experimental data. The inclusion of different subsets of radiative corrections in the preparation of the templates modifies the final result of the fit. Having in mind an accuracy target of \(\mathcal{O}(10\) MeV), it is important to include the \(\mathcal{O}(\alpha )\) QED finalstate radiation effects which yield a shift of \(M_W\) of about 100–200 MeV (depending on the precise definition of the final state), but also finalstate multiple photon radiation to all orders, which induces an additional shift of up to \(\mathcal{O}(10\%)\) of the \(\mathcal{O}(\alpha )\) [4]. One may thus also wonder about the size of the shift in \(M_W\) induced by weak or mixed QCDEW corrections. Different subsets of corrections became available separately in the past years in codes that simulate purely QCD or purely EW effects. The combination of QCD and EW corrections is an important step in the development of the MC programs that will be used in highprecision measurements and is one of the main topics of the present report.
 1.
Two codes that have the same perturbative approximation, the same input parameters (couplings, masses, PDFs), the same setup (choice of scales, acceptance cuts), should yield exactly the same results, within the accuracy of the numerical integration.
 2.
The results of different codes can be meaningfully combined only if they satisfy the previous point.
 1.
to verify at any time that a given code works properly according to what its authors have foreseen,
 2.
to demonstrate explicitly the level of agreement of different codes which include identical subsets of radiative corrections, and
 3.
to expose the impact of different subsets of higherorder corrections and of differences in their implementations.
The report is structured as follows: In Sect. 2.1 we describe the common setup for the tuned comparison and the observables under study in this report. The choice of observables was guided by the relevance to the study of Drell–Yan processes at the LHC, in particular to a precise measurement of the W boson mass. In Sects. 2.2 and 2.3 we present the results of the tuned comparison at NLO: in Sect. 2.2 we show the predictions of NLOEW and NLOQCD total cross sections, and in Sect. 2.3 we show the results at NLO EW and NLO QCD for a sample of kinematic distributions listed in Sect. 2.1.
In Sect. 3 we discuss the impact of higherorder QCD and EW corrections, i.e. corrections beyond NLO accuracy, on a selected set of W and Z boson observables. For each code used in this study we consider all the subsets of available corrections which are beyond NLO. To compute the results presented in this section, we adopted an EW input scheme, described in Sect. 3.1, which absorbs known higherorder corrections already in the (N)LO predictions, thus minimizing the impact of neglected orders in perturbation theory. All results obtained in this benchmark setup can serve as a benchmark for future studies. For completeness we provide the results for the total cross sections at NLO EW and NLO QCD obtained in this benchmark setup in Sect. 3.2. In Sect. 3.3 we discuss the effects of purely QCD corrections: after a short introduction in Sect. 3.3.1 on the impact of the \(\mathcal{O}(\alpha _s)\) corrections on the observables under study, we consider in Sects. 3.3.2 and 3.3.3 exact results at \(\mathcal{O}(\alpha _s^2)\) respectively for the total cross sections and for some differential distributions; in Sect. 3.3.4 we briefly introduce the problem of matching fixed and allorder results in perturbation theory; we present results of (NLO+PS)QCD matching in Sect. 3.3.5 and of (NNLO+PS)QCD matching in Sect. 3.3.6. In Sect. 3.4 we discuss the effects of purely EW corrections: after a short introduction in Sect. 3.4.1 on the role of the \(\mathcal{O}(\alpha )\) corrections on the observables under study, we compare in Sect. 3.4.2 the predictions for the partonic subprocesses induced by photons, which are naturally part of the NLO EW results. We discuss different EW input scheme choices in Sect. 3.4.3 and the impact of different gauge boson mass definitions in Sect. 3.4.4. In Sects. 3.4.5–3.4.7, we describe respectively the impact of higherorder corrections introduced via the \(\rho \) parameter or via the definition of effective couplings or due to multiple photon radiation described with a QED PS properly matched to the NLO EW calculation. The effect of light fermionpair emission is discussed in Sect. 3.4.8.
In Sect. 4 we consider the combination of QCD and EW corrections and discuss some possibilities which are allowed by our presently incomplete knowledge of the \(\mathcal{O}(\alpha \alpha _s)\) corrections to the DY processes. In Sect. 4.1 we compare the results that can be obtained with the codes presently available and discuss the origin of the observed differences. In Sect. 4.2 the results of a first calculation of \(\mathcal{O}(\alpha \alpha _s)\) corrections in the pole approximation are used to assess the validity of simple prescriptions for the combination of EW and QCD corrections.
In Appendix A we provide a short description of the MC codes used in this study. In Appendix B we present a tuned comparison of the total cross sections at NLO EW and NLO QCD for \(W^\pm \) and Z production with LHCb cuts.
1.1 Reproducibility of the results: a repository of the codes used in this report
The goal of this report is to provide a quantitative assessment of the technical level of agreement of different codes, but also a classification of the size of higherorder radiative corrections.
The usage of modern MC programs is quite complex and it is not trivial to judge whether the numerical results “outofthebox” of a code are correct. The numbers presented here, computed by the respective authors, should be considered as benchmarks of the codes; every user should thus be able to reproduce them, provided that he/she uses the same inputs and setup and runs with the appropriate amount of statistics.
In order to guarantee the reproducibility of the results presented in this report, we prepared a repository that contains a copy of all the MC codes used in this study, together with the necessary input files and the relevant instructions to run them. The repository can be found at the following URL: https://twiki.cern.ch/twiki/bin/view/Main/DrellYanComparison It should be stressed that simulation codes may evolve in time, because of improvements but also of bug fixes.
2 Tuned comparison of the codes
2.1 Setup for the tuned comparison
To compute the hadronic cross section we use the MSTW2008 [30] set of parton distribution functions, and take the renormalization scale, \(\mu _r\), and the QCD factorization scale, \(\mu _\mathrm{QCD}\), to be the invariant mass of the finalstate lepton pair, i.e. \(\mu _r=\mu _\mathrm{QCD}=M_{l\nu }\) in the W boson case and \(\mu _r=\mu _\mathrm{QCD}=M_{l^+l^}\) in the Z boson case.
All numerical evaluations of EW corrections require the subtraction of QED initialstate collinear divergences, which is performed using the QED DIS scheme. It is defined analogously to the usual DIS [31] scheme used in QCD calculations, i.e. by requiring the same expression for the leading and nexttoleading order structure function \(F_2\) in deep inelastic scattering, which is given by the sum of the quark distributions. Since \(F_2\) data are an important ingredient in extracting PDFs, the effect of the \(\mathcal{O}(\alpha )\) QED corrections on the PDFs should be reduced in the QED DIS scheme. The QED factorization scale is chosen to be equal to the QCD factorization scale, \(\mu _{QED}=\mu _{QCD}\). The QCD factorization is performed in the \(\overline{\mathrm{MS}}\) scheme. The subtraction of the QED initial state collinear divergences is a necessary step to obtain a finite partonic cross section. The absence of a QED evolution in the PDF set MSTW2008 has little phenomenological impact on the kinematic distributions as discussed in Sect. 3.4.2. However, to be consistent in the order of higher order corrections in a best EW prediction, modern PDFs which include QED corrections, such as NNPDF2.3QED [32] and CT14QED [33], should be used.
For the sake of simplicity and to avoid additional sources of discrepancies in the tuned comparison we use the finestructure constant \(\alpha (0)\) throughout in both the calculation of CC and NC cross sections. We will discuss different EW input schemes in Sect. 3.4.3.
In the course of the calculation of radiative corrections to W boson observables the Kobayashi–Maskawa mixing has been neglected, but the final result for each parton level process has been multiplied with the square of the corresponding physical matrix element \(V_{ij}\). From a numerical point of view, this procedure does not significantly differ from a consideration of the Kobayashi–Maskawa matrix in the renormalisation procedure as it has been pointed out in [34].
Twoloop and threeloop running of \(\alpha _s(\mu _r^2)\)
\(\mu _r\) [GeV]  \(\alpha _s\)(NLO)  \(\alpha _s\)(NNLO) 

91.1876  0.1201789  0.1170699 
50  0.1324396  0.1286845 
100  0.1184991  0.1154741 
200  0.1072627  0.1047716 
500  0.0953625  0.0933828 
Summary of lepton identification requirements in the calo setup
Electrons  Muons 

Combine e and \(\gamma \) momentum four vectors, if \({\varDelta } R(e,\gamma )<0.1\)  Reject events with \(E_\gamma >2\) GeV for \({\varDelta } R(\mu ,\gamma )<0.1\) 
Reject events with \(E_\gamma >0.1~E_e\) for \(0.1<{\varDelta } R(e,\gamma )<0.4\)  Reject events with \(E_\gamma >0.1~E_\mu \) for \(0.1<{\varDelta } R(\mu ,\gamma )<0.4\) 
Since we consider predictions inclusive with respect to QCD radiation, we do not impose any jet definition.
We use the Pythia version 6.4.26, Perugia tune (PYTUNE(320)). When producing NLO QCD+EW results with Pythia, the QED showering effects are switched off by setting MSTJ(41)=MSTP(61)=MSTP(71)=1.
In the following we list the observables considered in this study for charged (CC) and neutral current (NC) processes: \(pp \rightarrow W^\pm \rightarrow l^\pm \nu _l\) and \(pp \rightarrow \gamma ,Z \rightarrow l^+ l^\) with \(l=e,\mu \).
2.1.1 W boson observables

\(\sigma _W\): total inclusive cross section of W boson production.
 \(\frac{d\sigma }{dM_\perp (l\nu )}\): transverse mass distribution of the lepton leptonneutrino pair. The transverse mass is defined aswhere \(p_\perp ^\nu \) is the transverse momentum of the neutrino, and \(\phi ^{\ell \nu }\) is the angle between the charged lepton and the neutrino in the transverse plane.$$\begin{aligned} M_\perp =\sqrt{2p_\perp ^\ell p_\perp ^\nu (1\cos \phi ^{\ell \nu })}, \end{aligned}$$(4)

\(\frac{d\sigma }{d p_\perp ^l}\): charged lepton transverse momentum distribution.

\(\frac{d\sigma }{d p_\perp ^\nu }\): missing transverse momentum distribution.

\(\frac{d\sigma _W}{d p_\perp ^W}\): leptonpair (W) transverse momentum distribution.
2.1.2 Z boson observables

\(\sigma _Z\): total inclusive cross section of Z boson production.

\(\frac{d\sigma }{dM_{l^+l^}}\): invariant mass distribution of the lepton pair.

\(\frac{d\sigma }{dp_\perp ^l}\): transverse lepton momentum distribution (l is the positively charged lepton).

\(\frac{d\sigma _Z}{d p_\perp ^Z}\): leptonpair (Z) transverse momentum distribution.
2.2 Tuned comparison of total cross sections at NLO EW and NLO QCD with ATLAS/CMS cuts
In this section we provide a tuned comparison of the total cross sections computed at fixed order, namely LO, NLO EW and NLO QCD, using the setup of Sect. 2.1 for the choice of input parameters and ATLAS/CMS acceptance cuts.
Tuned comparison of total cross sections (in pb) for \(p p \rightarrow W^+\rightarrow l^+ \nu _l+X\) at the 8 TeV LHC, with ATLAS/CMS cuts and bare leptons. (\(\times \)) indicates that although POWHEG_BW provides NLO EW results also for bare electrons, due to the smallness of the electron mass it would require very highstatistics to obtain permille level precision. Thus, we recommend to use the bare setup in POWHEG_BW only for muons
Code  LO  NLO QCD  NLO EW \(\mu \)  NLO EW e 

HORACE  2897.38(8)  \(\times \)  2988.2(1)  2915.3(1) 
WZGRAD  2897.33(2)  \(\times \)  2987.94(5)  2915.39(6) 
RADY  2897.35(2)  2899.2(4)  2988.01(4)  2915.38(3) 
SANC  2897.30(2)  2899.9(3)  2987.77(3)  2915.00(3) 
DYNNLO  2897.32(5)  2899(1)  \(\times \)  \(\times \) 
FEWZ  2897.2(1)  2899.4(3)  \(\times \)  \(\times \) 
POWHEGw  2897.34(4)  2899.41(9)  \(\times \)  \(\times \) 
POWHEG_BMNNP  2897.36(5)  2899.0(1)  2988.4(2)  2915.7(1) 
POWHEG_BW  2897.4(1)  2899.2(3)  2987.7(4)  (\(\times \)) 
2.3 Tuned comparison of kinematic distributions at NLO EW and NLO QCD with ATLAS/CMS cuts
Tuned comparison of total cross sections (in pb) for \(p p \rightarrow W^+\rightarrow l^+ \nu _l+X\) at the 8 TeV LHC, with ATLAS/CMS cuts and calorimetric leptons
Code  LO  NLO EW \(\mu \) calo  NLO EW e calo 

HORACE  2897.38(8)  2899.0(1)  3003.5(1) 
WZGRAD  2897.33(2)  2898.33(5)  3003.33(6) 
RADY  2897.35(2)  2898.37(4)  3003.36(4) 
SANC  2897.30(2)  2898.18(3)  3003.00(4) 
The corresponding total cross sections can be found in Sect. 2.2.
Tuned comparison of total cross sections (in pb) for \(p p \rightarrow W^\rightarrow l^ \bar{\nu }_l+X\) at the 8 TeV LHC, with ATLAS/CMS cuts and bare leptons. (\(\times \)) indicates that although POWHEG_BW provides NLO EW results also for bare electrons, due to the smallness of the electron mass it would require very highstatistics to obtain permille level precision. Thus, we recommend to use the bare setup in POWHEG_BW only for muons
Code  LO  NLO QCD  NLO EW \(\mu \)  NLO EW e 

HORACE  2008.84(5)  \(\times \)  2076.48(9)  2029.15(8) 
WZGRAD  2008.95(1)  \(\times \)  2076.51(3)  2029.26(3) 
RADY  2008.93(1)  2050.5(2)  2076.62(2)  2029.29(2) 
SANC  2008.926(8)  2050.3(3)  2076.56(2)  2029.19(3) 
DYNNLO  2008.89(3)  2050.2(9)  \(\times \)  \(\times \) 
FEWZ  2008.9(1)  2049.97(8)  \(\times \)  \(\times \) 
POWHEGw  2008.93(3)  2050.14(5)  \(\times \)  \(\times \) 
POWHEG_BMNNP  2008.94(3)  2049.9(1)  2076.9(1)  2029.71(6) 
POWHEG_BW  2009.2(4)  2050.2(4)  2076.0(3)  (\(\times \)) 
Tuned comparison of total cross sections (in pb) for \(p p \rightarrow W^\rightarrow l^ \bar{\nu }_l+X\) at the 8 TeV LHC, with ATLAS/CMS cuts and calorimetric leptons
Code  LO  NLO EW \(\mu \) calo  NLO EW e calo 

HORACE  2008.84(5)  2013.67(7)  2085.42(8) 
WZGRAD  2008.95(1)  2013.42(3)  2085.26(3) 
RADY  2008.93(1)  2013.49(2)  2085.37(2) 
SANC  2008.926(8)  2013.48(2)  2085.24(4) 
Tuned comparison of total cross sections (in pb) for \(p p \rightarrow \gamma ,Z \rightarrow l^ l^++X\) at the 8 TeV LHC, with ATLAS/CMS cuts and bare leptons. (\(\times \)) indicates that FEWZ provides NLO EW results only in the \(G_\mu \) scheme, and thus no results are available for the setup of the tuned comparison (see Sect. 2.1)
Code  LO  NLO QCD  NLO EW \(\mu \)  NLO EW e 

HORACE  431.033(9)  \(\times \)  438.74(2)  422.08(2) 
WZGRAD  431.048(7)  \(\times \)  439.166(6)  422.78(1) 
RADY  431.047(4)  458.16(3)  438.963(4)  422.536(5) 
SANC  431.050(2)  458.27(3)  439.004(5)  422.56(1) 
DYNNLO  431.043(8)  458.2(2)  \(\times \)  \(\times \) 
FEWZ  431.00(1)  458.13(2)  (\(\times \))  (\(\times \)) 
POWHEGz  431.08(4)  458.19(8)  \(\times \)  \(\times \) 
POWHEG_BMNNPV  431.046(9)  458.16(7)  438.9(1)  422.2(2) 
2.3.1 Tuned comparison of \(W^\pm \) boson observables
In the following we present a tuned comparison of results for the \(M_\perp , p_\perp ^W\) and \(p_\perp ^l,p_\perp ^\nu \) distributions for \(W^\pm \) production in \(pp\rightarrow \mu ^\pm \nu _\mu +X\) at the 8 TeV LHC with ATLAS/CMS cuts in the bare setup. To compare the results of different codes at NLO EW we show in Figs. 1, 2, 3 and 4 the ratios R=code/HORACE, where code=HORACE, POWHEG_BMNNP, POWHEG_BW, RADY, SANC, WZGRAD, and at NLO QCD we show in Figs. 5, 6, 7, 8, 9 and 10 the ratios R=code/POWHEG, where code=DYNNLO, FEWZ, POWHEG, RADY, SANC.
Tuned comparison of total cross sections (in pb) for \(p p \rightarrow \gamma , Z\rightarrow l^+ l^+X\) at the 8 TeV LHC, with ATLAS/CMS cuts and calorimetric leptons
Code  LO  NLO EW \(\mu \) calo  NLO EW e calo 

HORACE  431.033(9)  407.67(1)  439.68(2) 
WZGRAD  431.048(7)  407.852(7)  440.29(1) 
RADY  431.047(4)  407.568(6)  440.064(5) 
SANC  431.050(2)  407.687(5)  440.09(1) 
2.3.2 Tuned comparison of Z boson observables
In Figs. 11 and 12 and in Figs. 13 and 14 we present a tuned comparison of results for NLO EW and QCD predictions, respectively, for the \(M_{l^+l^}, p_\perp ^Z\) and \(p_\perp ^l\) distributions in \(pp\rightarrow \gamma ,Z\rightarrow \mu ^+\mu ^ +X\) at the 8 TeV LHC with ATLAS/CMS cuts in the bare setup of Sect. 2.1. The agreement of different codes providing NLO EW predictions for these distributions in the kinematic regions under study are at the five per mill level or better, apart from a difference at the one per cent level in the transverse momentum distribution of the lepton pair for small values of \(p_\perp ^Z\). As it is the case for CC DY, these results should be considered just for technical checks, since \(p_\perp ^Z\) receives large contributions from QCD radiation. The combined effects of EW and QCD corrections in \(p_\perp ^Z\) can be studied for instance by using a calculation of NLO EW corrections to \(Z+j\) production [40] and the implementation of NLO EW corrections in POWHEG [41] as discussed in Sect. 4.
3 Impact of higherorder radiative corrections
On the other hand, the setup of Sect. 2.1 suffers for two reasons, relevant from the phenomenological but also from the theoretical point of view: (1) the choice of the finestructure constant as input parameter in the EW Lagrangian introduces an explicit dependence on the value of the lightquark masses via the electric charge renormalization; these masses are not well defined quantities and introduce a nonnegligible parametric dependence of all the results; (2) the strength of the coupling of the weak currents is best expressed in terms of the Fermi constant, whose definition reabsorbs to all orders various classes of large radiative corrections; when using the Fermi constant, the impact of the remaining, process dependent corrections is thus reduced in size with respect to other input schemes, like, e.g., the one of Sect. 2.1.
3.1 Setup for benchmark predictions
 1.
In the case of W boson production, in addition to the acceptance cuts we apply \(M_\perp (l\nu ) >40\) GeV.
 2.To account for the fact that we are using the constant width approach, we have to adjust the W, Z mass and width input parameters that have been measured in the sdependent width approach accordingly, as follows [18, 42] (\(\gamma _V={\varGamma }_V/M_V\)):Consequently, the input values for the W, Z masses and widths change to$$\begin{aligned} M_V \rightarrow \frac{M_V}{\sqrt{1+\gamma _V^2}};\quad {\varGamma }_V \rightarrow \frac{{\varGamma }_V}{\sqrt{1+\gamma _V^2}}. \end{aligned}$$$$\begin{aligned}&M_Z = 91.1535~\mathrm{{GeV}},\quad {\varGamma }_Z = 2.4943~\mathrm{{GeV}} \nonumber \\&M_W = 80.358~\mathrm{{GeV}},\quad {\varGamma }_W =2.084~\mathrm{{GeV}}. \end{aligned}$$(5)
 3.We use the following EW input scheme: In the calculation of the treelevel couplings we replace \(\alpha (0)\) by the effective coupling \(\alpha _{G_\mu }=\sqrt{2} G_\mu M_W^2 (1M_W^2/M_Z^2)/\pi \). The relative \(\mathcal{O}(\alpha )\) corrections are calculated with the fine structure constant \(\alpha (0)\). At NLO EW this replacement implies an additional contribution of \({\varDelta } r\) to the relative \(\mathcal{O}(\alpha )\) corrections. The oneloop result for \({\varDelta } r\) has been calculated in Refs. [43, 44] and can be decomposed as follows:When using the input values of Eq. (1) and the values for \(M_W\) and \(M_Z\) given in item (2) \({\varDelta } r(\mathrm{1loop})=0.0295633444\) (\({\varDelta } r=0.0296123554\) for the unshifted W / Z masses of Eq. (1)).$$\begin{aligned} {\varDelta } r(1\text {loop})={\varDelta } \alpha \frac{c_w^2}{s_w^2} {\varDelta } \rho +{\varDelta } r_{rem}(M_H). \end{aligned}$$
3.1.1 Setup for the evaluation of photoninduced contributions

In order to have a modern parametrization of the photon density, we used the central NNPDF2.3_lo_as_130 _qed PDF set [32].

We use as input parameters \((\alpha (0), M_W, M_Z)\) for all photoninduced processes.
3.2 Total cross sections in the benchmark setup at NLO EW and NLO QCD with ATLAS/CMS cuts
\(p p \rightarrow W^+\rightarrow l^+ \nu _l\) total cross sections (in pb) at LO, NLO EW and NLO QCD at the 8 TeV LHC with ATLAS/CMS cuts in the benchmark setup
Code  LO  NLO QCD  NLO EW \(\mu \) bare  NLO EW e calo 

HORACE  3109.65(8)  \(\times \)  3022.8(1)  3039.5(2) 
WZGRAD  3109.62(2)  \(\times \)  3022.68(4)  3039.13(5) 
SANC  3109.66(2)  (\(\times \))  3022.53(4)  3038.94(4) 
DYNNLO  3109.5(2)  3092.3(9)  \(\times \)  \(\times \) 
FEWZ  3109.20(8)  3089.1(3)  \(\times \)  \(\times \) 
POWHEGw  (\(\times \))  3090.4(2)  \(\times \)  \(\times \) 
POWHEG_BMNNP  3109.68(7)  3089.6(2)  3022.8(2)  (\(\times \)) 
\(p p \rightarrow W^\rightarrow l^ \bar{\nu }_l\) total cross sections (in pb) at LO, NLO EW and NLO QCD at the 8 TeV LHC with ATLAS/CMS cuts in the benchmark setup
Code  LO  NLO QCD  NLO EW \(\mu \) bare  NLO EW e calo 

HORACE  2156.36(6)  \(\times \)  2101.17(8)  2111.1(2) 
WZGRAD  2156.46(2)  \(\times \)  2101.23(2)  2110.65(4) 
SANC  2156.46(2)  (\(\times \))  2101.31(4)  2110.69(4) 
DYNNLO  2156.38(2)  2189.3(7)  \(\times \)  \(\times \) 
FEWZ  2156.09(4)  2187.1(1)  \(\times \)  \(\times \) 
POWHEGw  (\(\times \))  2187.72(6)  \(\times \)  \(\times \) 
POWHEG_BMNNP  2156.44(4)  2187.5(1)  2101.5(1)  (\(\times \)) 
\(p p \rightarrow \gamma ,Z \rightarrow l^ l^+\) total cross sections (in pb) at LO, NLO EW and NLO QCD at the 8 TeV LHC with ATLAS/CMS cuts in the benchmark setup
Code  LO  NLO QCD  NLO EW \(\mu \) bare  NLO EW e calo 

HORACE  462.663  \(\times \)  443.638  
WZGRAD  462.677(4)  \(\times \)  443.950(6)  445.178(7) 
SANC  462.675(2)  (\(\times \))  443.794(4)  444.963(4) 
DYNNLO  (\(\times \))  491.94(5)  \(\times \)  \(\times \) 
FEWZ  462.631(9)  491.62(4)  443.84(2)  444.67(2) 
POWHEGz  (\(\times \))  491.744(4)  \(\times \)  \(\times \) 
POWHEG_BMNNPV  462.67(1)  491.3(8)  443.4(1)  (\(\times \)) 
3.3 Impact of QCD corrections on W and Z boson observables in the benchmark setup
3.3.1 NLO QCD corrections
At LO the DY processes are described in terms of quark–antiquark annihilation subprocesses.^{3} The NLO QCD corrections are due to real and virtual corrections to the incoming quark–antiquark line, but they receive a contribution also from the (anti)quarkgluon scattering subprocesses.
Some observables, such as the leptonpair transverse momentum, the \(\phi ^*\) variable or the singlelepton transverse momentum, are strongly sensitive to the details of real QCD radiation. The leptonpair transverse momentum or the \(\phi ^*\) distributions are indeed absent at LO (\(p_\perp ^V=0\) and \(\phi ^*=\pi \)), so that for these quantities NLO QCD is the first perturbative nonvanishing order. In the singlelepton transverse momentum case, the distribution receives, on top of the LO value, a large contribution from the recoil of the intermediate gauge boson against initialstate QCD radiation, enhanced by its collinearly divergent behaviour. Even if this is not formally the case, NLO QCD is numerically the lowest perturbative order which can be used to assess the impact of higher order corrections. On the contrary the (pseudo)rapidity distributions and the invariant/transverse mass distributions receive a milder, slowly varying NLO QCD correction, close in size to the value of the total NLO Kfactor.
3.3.2 NNLO QCD corrections: total cross section
We study the predictions for DY processes with the inclusion of QCD nexttonexttoleading order (NNLO) corrections in the strong coupling constant using^{4} the following three MC codes, DYNNLO [5], FEWZ [7, 46], and SHERPANNLOFO [21].
Tuned comparison of NNLO QCD total cross sections (in pb) at the 8 TeV LHC in the benchmark setup with ATLAS/CMS cuts
Process  DYNNLO  FEWZ  SHERPANNLOFO 

\(pp\rightarrow l^+\nu _l+X\)  3191(7)  3207(2)  3204(4) 
\(pp\rightarrow l^\bar{\nu }_l+X\)  2243(6)  2238(1)  2252(3) 
\(pp\rightarrow l^+l^+X\)  502.4(4)  504.6(1)  502.0(6) 
The impact of NNLO QCD corrections on the total cross section of the DY processes depends on the corrections to the lowerorder processes but also on a small contribution from new partonic channels. The second order corrections reduce the renormalization/factorization scale dependence of the final result, with respect to NLO QCD, and bring it down to the 1% level [5, 46].
3.3.3 NNLO QCD corrections: kinematic distributions
The NNLO QCD predictions for kinematic distributions are compared for a subset of observables in Figs. 15 and 16, where the ratio to the SHERPANNLOFO prediction is shown. As it can be seen, the predictions agree within the statistical uncertainties of the MC integration.
The impact of NNLO QCD corrections on the kinematic distributions of the DY processes depends on the observable under study. Since some observables such as the leptonpair transverse momentum, the singlelepton transverse momentum or the \(\phi ^*\) variable are strongly sensitive to the details of real QCD radiation at NLO, they are significantly modified by the second order QCD corrections. On the contrary the (pseudo)rapidity distributions and the invariant/transverse mass distributions receive a milder corrections, closer in size to the value of the total NNLO Kfactor.
To illustrate the impact of the NNLO QCD corrections we compute for a given observable \(\mathcal{O}\) the ratio \(R_{\mathcal{O}}=\left( \frac{d\sigma ^{NNLO}}{d\mathcal{O}}\right) / \left( \frac{d\sigma ^{NLO}}{d\mathcal{O}} \right) \) with the same distribution evaluated respectively with NNLO QCD and NLO QCD accuracy. We consider the distributions at NLO QCD as perfectly tuned and neglect here the differences introduced by the choice in the denominator of one NLO QCD code with respect to another one. We present the results in Figs. 17, 18 and 19.
Figures 17 and 19 show the relative correction to the lepton and to the neutrino transverse momentum distributions. The NNLO QCD corrections, expressed in terms of the NLO QCD result, are quite flat and moderate (smaller than 10%) below the Jacobian peak, they have a sharply peaked behaviour about the Jacobian peak, where fixed order perturbation theory breaks down, while they are of \(\mathcal{O}(20\%)\) and are growing for increasing transverse momentum above the Jacobian peak. Again, the pronounced corrections that appear at the lower end of the distributions can be understood as an effect of the acceptance cuts.
In Figs. 18 and 19 we show the relative corrections to the leptonpair transverse momentum distributions, for the three processes (NC and CC) under consideration, in two ranges of transverse momentum (\(p^{V}_\perp \,\in [0,25]\) GeV and \(p^{V}_\perp \,\in [0,250]\) GeV). In fixedorder perturbation theory the distribution is divergent in the limit of vanishing transverse momentum; the sign of the first bin and the slope of the distributions in this limit depend on the perturbative order, so that a comparison between NLO QCD and NNLO QCD predictions is merely of technical interest. At large leptonpair transverse momentum, where the perturbative regime of QCD allows to study the convergence of the perturbative expansion, the NNLO QCD corrections are large, of \(\mathcal{O}(40\%)\), and quite flat in the range \(50\le p^{V}_\perp \,\le 300\) GeV.
3.3.4 Higherorder QCD corrections to all orders: generalities

Matching a resummed and a (N)NLO fixedorder expressions requires a procedure that avoids double countings and possibly allows for the MC simulation of events with a probabilistic interpretation. The solution of this problem at NLO was developed in [48, 49] and more recently in [21, 50, 51] also for the inclusion of NNLO partonic results. Each approach solves the matching problem in a different way, yielding predictions that respect the nominal perturbative accuracy for observable that are stable under the inclusive evaluation of radiative effects, but differ in the treatment of higherorder terms. The matching ambiguity, parametrized in different ways, should be considered as an additional source of theoretical uncertainty, together with the one usually expressed by the choice of the renormalization/factorization scales.

In the MC codes the resummation to all orders of some classes of contributions is done by means of a Parton Shower (PS) approach, with leading logarithmic (LL) accuracy in the log of the gauge boson transverse momentum. There are differences of subleading logarithmic order in the available PS algorithms, which yield a difference in the final predictions.

The PS codes are usually interfaced with models that describe nonperturbative effects of the strong interaction at low energy scales; the parameters of these models are usually tuned to reproduce some relevant distribution, but their choice (and the corresponding quality of the description of the data) represents an additional source of ambiguity in the predictions.
In Figs. 20, 21, 22 and 23 we expose the impact of higherorder corrections, \(\mathcal{O}(\alpha _s^2)\) and higher, in units of the NLO QCD results. In this way we appreciate where the higher orders play a crucial role, how well the NNLO QCD results are approximated by a NLO+PS formulation (Figs. 20, 21), and the impact of matching the NNLO QCD fixedorder calculation and a QCDPS (Figs. 22, 23). The disadvantage of this choice of presenting the results is that for some observables the NLO QCD is not a sensible lowest order approximation.
3.3.5 Comparison of (NLO+PS)QCD vs NNLO QCD results
The POWHEG +PYTHIA and the SHERPA NLO+PS NLO+PS predictions are based on the same exact matrix elements present in all the codes that have NLO QCD accuracy for the total cross section, but they add the higherorder effects due to multiple parton emissions to all orders via a QCDPS, with two different matching procedures. At \(\mathcal{O}(\alpha _S^2)\) they both have a partial overlap with those by the fixedorder NNLO results, because of the inclusion of the LL terms. It should be stressed that the POWHEG +PYTHIA and the SHERPA NLO+PS NLO+PS codes do not have NNLO QCD accuracy for the total cross section nor do they have an accurate description of the large leptonpair transverse momentum region, where exact matrix element effects for the second emission are important. On the other hand, they include the resummation to all orders of multiple parton emissions, which is important to yield a sensible description of the small leptonpair transverse momentum region, of the low\(\phi ^*\) region of the \(\phi ^*\) distribution or of the Jacobian peak of the single lepton transverse momentum distribution.
We observe in Figs. 20 and 21 that the QCDPS corrections in POWHEG +PYTHIA have a small impact on the invariantmass (NC DY) or transversemass (CC DY) distributions (middle plots); the correction is slowly varying over the entire mass range, with the exception of the lower end of the distribution, where the acceptance cuts yield a distinction between oneemission and multipleemissions final states.
In the same figures, we show the corrections to the lepton transverse momentum distribution (upper plots). We observe at the jacobian peak the distortion due to the fact that in this region a fixed order description is not sufficient to describe this observable. Below the jacobian peak the corrections of \(\mathcal{O}(\alpha _S^2)\) and higher become smaller for decreasing values of the transverse momentum, before reaching the acceptance cut. Above the jacobian peak, the QCDPS effects follow those obtained at NNLO QCD. This result can be interpreted by observing that the lepton transverse momentum has two components, one from the gauge boson decay at LO and one due to the gaugeboson recoil against QCD radiation; immediately above the jacobian peak, the recoil component is characterized by a small value of the leptonpair transverse momentum; in this region the collinear approximation on which the PS is based is quite accurate, and thus the second real emission in the PS approximation is close to the exact result. For larger values of the leptonpair transverse momentum the QCDPS becomes inadequate to describe the spectrum; the role of the first and second order exact matrix element corrections is shown in the lower plots of Figs. 20 and 21. The difference between the two approximations vary between zero and 40% in the interval \(p^{V}_\perp \,\in [70,300]\) GeV.
The resummation of multiple parton emissions to all orders via the PS makes the distribution vanish in the limit of vanishing leptonpair transverse momentum, as it is physically expected (Sudakov suppression). The size of the QCDPS correction in units NLO QCD is infinitely negative when \(p^{V}_\perp \,\rightarrow 0\); this peculiar result is a consequence of the choice of the NLO QCD prediction as unit to express the higherorder effects, which is inappropriate in this specific corner of the phasespace. This comment is at variance with respect to the one for the NNLO QCD corrections: also in that case the size of the correction is infinitely large, but only because at each fixed order the distribution diverges, each time with a different coefficient.
3.3.6 Comparison of different (NNLO+PS)QCD matching schemes
The matching of NNLO QCD results with a QCDPS has been achieved first in the MiNLO approach [50, 51, 55]. In the DY case the calculation has been implemented in a code based on POWHEG +MiNLO combined with DYNNLO , and henceforth denoted DYNNLOPS [6]. This method is based on the NLO+PS formulation of the original hard process plus onejet, and supplements it with Sudakov form factors that lead to finite predictions as the additional jet becomes unresolved. The NNLO accuracy is achieved by reweighing via a pretabulated phasespace dependent Kfactors.
Another NNLO+PS matching approach is called UN2LOPS [21, 56] and it is a variant of the UNLOPS [57] method. UNLOPS is one of the unitary merging techniques recently developed to merge multijet NLO calculations while preserving the inclusive cross section of the process with the lowest jet multiplicity. In UN2LOPS, by only keeping events with resolvable QCD emissions, which are available as part of the NNLO calculation, the description of the DY processes at large transverse momentum becomes equivalent to the study of W(Z) plus one additional jet at NLO. The remainder of the phase space is filled by a calculation at NNLO, with a corresponding veto on any QCD activity, forming the zero jet bin. This is essentially the phase space slicing method, and the goal of the UN2LOPS approach is to merge the two parts after the PS is added. Only the part of W(Z) plus one jet at NLO is matched with PS, where any standard methods could be used. Events in the zero jet bin should not be showered to avoid double counting because QCD radiation has already been described by the PS matched W(Z) plus one jet process at NLO.^{6} The merging is done by suppressing the divergence in W(Z) plus one jet via the shower veto algorithm in which the vetoed events are added back to the zero jet bin to preserve the inclusive cross section. In order to generate physically meaningful results, the separation cut scale \(q_\perp \) must be smaller than the terminating scale of the parton shower. In contrast to the MiNLO method, realemission configurations do not receive a contribution from the NNLO calculation because twoloop virtual contributions in the 0jet bin are not showered. The resulting difference is beyond NNLO accuracy for the original hard process. Formally the resummation of UN2LOPS is limited by the accuracy of the parton shower, while in the MiNLO method, a higher logarithmic accuracy of the first emission can be achieved with analytic Sudakov form factor for the corresponding observable.^{7} Nevertheless, for other observables or subsequent emissions, resummation in MiNLO is only as accurate as the parton shower can provide. The calculation of the DY processes in the UN2LOPS approach has been implemented in the code SHERPA NNLO+PS .
Both these two matching approaches should not be considered as a final answer to the problem of matching NNLO fixed order with PS results, but rather as a first step towards more general methods.
We note that results for Drell–Yan production at NNLL’+NNLO matched to a PS in the GENEVA MonteCarlo framework are presented in Ref. [58], but not included in this study.
In Fig. 22 we show the results obtained with the SHERPA NNLO+PS code, in the case of CC DY, and compare them to the corresponding NNLO fixedorder predictions. We present two different uncertainty bands: the first one, in black in the plots, is obtained by varying the renormalization \(\mu _R\) and factorization \(\mu _F\) scales of the underlying fixed order calculation, with \(\mu _R=\mu _F\) and \(1/2\le \mu _R/M_{ll} \le 2\); the second one, in green in the plots, is obtained by varying the shower scale Q of the QCDPS in the interval \(1/2 \le Q/M_{ll} \le 2\).
In Fig. 23 we show the results obtained with the two codes SHERPA NNLO+PS and DYNNLOPS , in the case of NC DY, and compare them with each other and with the corresponding NNLO fixedorder predictions. The SHERPA NNLO+PS uncertainty bands have been computed as described above, while in the DYNNLOPS case the band is obtained by varying by a factor 2 up and down independently all renormalization and factorization scales appearing in the underlying MiNLO procedure (at variance with the report setup, in the MiNLO approach both renormalization and factorization scales are set equal to the gauge boson transverse momentum), keeping their ratio between 1 / 2 and 2. This leads to seven different scale choices. Independently of this we vary by a factor 2 up and down the renormalization and factorization scale in the underlying DYNNLO calculation keeping the two equal. This leads to three different scale choices. As these scale choices are taken to be independent, this leads to \(3\cdot 7=21\) scale choices of which the envelope is taken as the uncertainty band. The procedure is described in more detail in [6]. Since the procedures used to evaluate the uncertainty bands are different for the two codes, we present separately in the two columns: the DYNNLOPS band and the central scales SHERPA NNLO+PS prediction (left plots) and the two SHERPA NNLO+PS bands and the central scales DYNNLOPS prediction (right plots).
As expected, for the invariant mass distribution of the lepton pair, in Fig. 23, all predictions agree very well. In particular in the central region, closer to the peak, the large statistics allow us to appreciate that also uncertainty bands are very similar among the two NNLO+PS results, and that the central line of one lies well within the (very narrow) uncertainty band of the other tool. For smaller and larger invariant masses, the conclusions are similar, although the limited statistics do not allow such a precise comparison.
Turning to the lepton transverse momentum, \(p^{l}_\perp \,\), spectrum, in Fig. 23 one observes that in the range where this distribution is NNLO accurate (i.e. where \(p^{l}_\perp \,\)is less than half the mass of the Z boson), the results of the two NNLO+PS codes are again in good agreement with each other and with the NNLO QCD reference line. The uncertainty band is very thin, as expected, until one approaches the Jacobian peak region. As explained in the previous section, in this region resummation effects are important. Although the two NNLO+PS results are obtained with very different approaches, the mutual agreement is very good. One should notice however, that to the left of the Jacobian peak, the NNLO+PS result from DYNNLOPS seems to depart from the pure fixedorder results a few bins earlier than the one from SHERPA NNLO+PS . These differences are likely to be due to the differences in how events are generated close to the Sudakov peak in \(p^{Z}_\perp \,\), which is a phasespace region where resummation is crucial, and the two NNLO+PS calculations perform it using very different approaches. Therefore differences at the few percent level are not unexpected. The differences between the NNLO+PS and the fixedorder results at the lower end of the \(p^{l}_\perp \,\)spectrum have already been noticed and commented on earlier in this chapter. For transverse momenta larger than \(M_Z/2\), the two NNLO+PS results rapidly start to reapproach the fixedorder line, which in this region is NLO QCD accurate. However, towards the end of the plotted range, some differences among the results can be observed: firstly, the DYNNLOPS result exhibits a moderately harder spectrum, which would probably be more evident at higher \(p^{l}_\perp \,\)values. Secondly, the uncertainty band of the two NNLO+PS results (the one due to the \(\mu _R,\mu _F\) scale variation only) is larger in the DYNNLOPS result than in the SHERPA NNLO+PS one. Both these differences can be understood by looking at the differences amongst the results for the vectorboson transverse momentum in the medium to low range ([0, 50] GeV), which is the phase space region where the bulk of the events with \(p^{l}_\perp \,\)approximately equal to [55, 60] GeV are generated.
The transverse momentum spectrum \(p^{Z}_\perp \,\)of the lepton pair is the observable that exposes most clearly the differences between the two results. For the purpose of this comparison, the more relevant difference to explain is the difference in shape (and absolute value) for \(p^{Z}_\perp \,\in [20,100]\) GeV, that we will address in the next paragraph. At very high \(p^{Z}_\perp \,\), differences are also fairly large, but in that region they can be mostly attributed to the MiNLO scale choice: when \(p^{Z}_\perp \,\)is large (above \(M_Z\)), the MiNLO Sudakov form factor switches off, but the strong coupling is evaluated at \(p^{Z}_\perp \,\), whereas in SHERPA NNLO+PS and in the fixedorder calculation it is evaluated at the dilepton invariant mass \(m_{ll}\).
The range \(p^{Z}_\perp \,\in [20,50]\) GeV is a “transition” region, since it is the region where higherorder corrections (of fixedorder origin as well as from resummation) play a role, but none of them is dominant. Due to Sudakov suppression, in DYNNLOPS the first two bins of the \(p^{Z}_\perp \,\)distribution are suppressed compared to the fixedorder results; in turn, the unitarity fulfilled by the matching procedure, in order to respect the total cross section normalization, spreads part of the cross section close to the singular region across several bins in \(p^{Z}_\perp \,\), including those to the right of the Sudakov peak.
The SHERPA NNLO+PS results instead are closer to the fixedorder prediction in the first bins, which is may be a consequence of the PS not being applied to the events of the 0jet bin.
Since the first bins are the region where most of the crosssection is sitting, a relatively small difference among the two NNLO+PS results in the peak region will show up, greatly amplified, in the transition region (to preserve the total cross section). At, say, 50 GeV, both the NNLO+PS results have a cross section larger than the pure fixedorder, with DYNNLOPS larger than SHERPA NNLO+PS . Moreover, although at large \(p^{Z}_\perp \,\)the cross section is small, the DYNNLOPS result is, by construction, below the others, as explained previously. This difference must also be compensated, and this takes place in the transition region too.
For the DYNNLOPS results, the scale choice in the transition region is inherited from the underlying MiNLO simulation. This means that the conventional factor 1/2 or 2 is applied to a dynamical scale choice (\(\mu = p^{Z}_\perp \,\)), and this fact helps in explaining why not only the result is larger than the fixed order and the SHERPA NNLO+PS distributions, but it also exhibits a different shape and uncertainty band. In the SHERPA NNLO+PS approach, effects similar to the latter in the transition region are mainly taken into account by the variation of the resummation scale, as the corresponding plot supports. In fact, this is the dominant uncertainty of the SHERPA NNLO+PS result in the transition region.
In spite of all the aforementioned details, one should also notice that for \(p^{Z}_\perp \,\), the two NNLO+PS results are mutually compatible over almost all the entire spectrum, once the uncertainty bands are considered.
3.4 Impact of EW corrections on W and Z boson observables in the benchmark setup
In Sect. 3.3 we presented the impact of higherorder QCD corrections, using the fixedorder NLO QCD results (which have been demonstrated to be fully under control) as unit to express the relative effect of different subsets. We follow the same approach now to discuss the EW corrections.
We discuss in Sect. 3.4.1 the main features of the NLO EW corrections, with special emphasis on the observables that are relevant to EW precision measurements. In Sects. 3.4.2–3.4.8, we present the impact of different subsets and combinations of higherorder corrections and if not stated otherwise express their effect using as a unit the results computed at NLO EW.
3.4.1 NLO EW corrections
Among the observables which are sensitive to the absolute normalization of the process, we have the single lepton pseudorapidity and the leptonpair rapidity distributions, and also the largemass tail of the leptonpair invariant mass distribution. The former receive a correction which is very close in size to the one of the total cross section, and which is quite flat along the whole (pseudo)rapidity range (the FSR corrections and the redefinition of the couplings via renormalization do not modify the LO kinematics, yielding, in first approximation, a global rescaling of the distributions).
The NLO EW virtual corrections become large and negative in the tails of the singlelepton transverse momentum, leptonpair invariant and transversemass distributions, when at least one kinematical invariant becomes large, because of the contribution of the purely weak vertex and box corrections. This effect of the socalled EW Sudakov logarithms can not be reabsorbed in a redefinition of the couplings and is process dependent. A recent discussion of the DY processes in the Sudakov regime can be found, e.g., in Refs. [59, 60].
The size of the effects due to the emission of real photons depends on the experimental definition of the lepton, i.e. on the recombination procedure of the momenta of the lepton with those of the surrounding photons. The radiation of photons collinear to the emitting lepton has a logarithmic enhancement, with a natural cutoff provided by the lepton mass. These mass logarithms cancel completely in the total inclusive cross section (Kinoshita–Lee–Nauenberg theorem), but leave an effect on the differential distributions. The recombination of the photons and lepton momenta effectively acts like the integration over the collinear corner of the photon phase space, yielding a cancellation of the singular contribution from that region; as a consequence, the logarithmic enhancement of the corrections is reduced, as if the lepton had acquired a heavier effective mass.
3.4.2 Photoninduced processes
In Fig. 25 we present the evaluation at hadron level of these contributions in the case of the NC DY process, done with the proton PDF set NNPDF2.3_lo_as_0130_qed, using the codes HORACE and SANC . We show the ratios \(R=1+d\sigma (\gamma \gamma , \gamma q)/d\sigma (q\bar{q})\) to illustrate the relative effect of including the photoninduced processes in the LO prediction. The reason for the contribution of the \(\gamma \mathop {q}\limits ^{{}()}\rightarrow \mu ^+\mu ^\mathop {q}\limits ^{{}()}\) subprocess to be negative, i.e. values smaller than 1 in the plots, can be understood as being due to the presence of subtraction terms for the collinear divergences, which are necessary in a NLO calculation.
3.4.3 EW input scheme choices
The calculation of the NLO EW set of corrections to the DY processes, requires the renormalization of EW couplings and masses, which is typically done by imposing onshell conditions on the relevant Green’s functions. The choice of the set of physical observables necessary to evaluate the parameters \((g,g',v)\) of the gauge sector of the Lagrangian is done following two main criteria: (1) the quantities which are best determined from the experimental point of view minimize the parametric uncertainties affecting all the predictions; (2) some observables automatically include in their definition important classes of radiative corrections, so that their use reduces the impact of the radiative corrections to the scattering process under study.
A convenient set of parameters that describes EW processes at hadron colliders is \((G_\mu ,M_W,M_Z)\), the so called \(G_\mu \) scheme. The Fermi constant \(G_\mu \) measured from muon decay naturally parameterize the CC interaction, while the W and Z masses fix the scale of EW phenomena and the mixing with the hypercharge field. A drawback of this choice is the fact that the coupling of real photons to charged particles is computed from the inputs and in lowest order is equal to \(\alpha _{G_\mu }=G_\mu \sqrt{2} M_W^2 (1M_W^2/M_Z^2)/\pi \sim 1/132\) much larger than the fine structure constant \(\alpha (0)\sim 1/137\), which would be the natural value for an onshell photon.
The alternative choice \((\alpha (0),M_W,M_Z)\), the socalled \(\alpha (0)\) scheme, does not suffer of the problem with real photon radiation, but introduces: (i) a dependence on the unphysical quantities, lightquark masses, via the electric charge renormalization, and (ii) it leaves large radiative corrections at NLO and in higher orders.
Further modifications may be considered. For NC DY the gauge invariant separation of complete EW radiative corrections into pure weak (PW) and QED corrections (involving virtual or real photons) is possible. Therefore, these two contributions may be considered at different scales, PW at \(\mathcal{O}(G_\mu ^3)\), and QED still at \(\mathcal{O}(\alpha G_\mu ^2)\). These different scales seem to be most natural for PW and QED contributions correspondingly. For CC DY PW and QED corrections are not separately gauge invariant, so that usually the complete NLO EW contribution (PW+QED) is considered using the same overall scale, either \(\mathcal{O}(G_\mu ^3)\) or \(\mathcal{O}(\alpha G_\mu ^2)\). More refined modifications may be considered, for instance based on defining gauge invariant subsets by using the Yennie–Frautschi–Suura approach [61]. The spread of predictions with different modifications of the \(G_\mu \) scheme may be considered as an estimate for the uncertainty due to missing higherorder EW effects.
3.4.4 Impact of different gauge boson mass definitions
3.4.5 Universal higherorder corrections in NC DY
 Following Ref. [18], the leading \(G_\mu m_t^2\) universal higher order corrections are taken into account via the replacements:in the LO expression for the NC DY cross section. As was argued in Refs. [62, 63], this approach correctly reproduces terms up to \(\mathcal{O}({\varDelta }\rho ^2)\). The quantity \({\varDelta }\rho \)$$\begin{aligned}&s^2_{\scriptscriptstyle W}\rightarrow \bar{s}^2_{\scriptscriptstyle W}\equiv s^2_{\scriptscriptstyle W}+{\varDelta }\rho \,c^2_{\scriptscriptstyle W},\nonumber \\&\quad c^2_{\scriptscriptstyle W}\rightarrow \bar{c}^2_{\scriptscriptstyle W}\equiv 1\bar{s}^2_{\scriptscriptstyle W}=(1{\varDelta }\rho )\,\bar{c}^2_{\scriptscriptstyle W}\end{aligned}$$(8)contains two contributions:$$\begin{aligned} {\varDelta }\rho =3x_t\,[1+\rho ^{(2)}\,(m_{\scriptscriptstyle H}^2/m_t^2)\,x_t]\, \left[ 1\frac{2\alpha _s(m_t^2)}{9\pi }(\pi ^2+3)\right] \nonumber \\ \end{aligned}$$(9)
 (i)
the twoloop EW part at \(\mathcal{O}(G_\mu ^2)\), second term in the first square brackets [64, 65, 66, 67], with \(\rho ^{(2)}\) given in Eq. (12) of Refs. [66, 67] (actually, after the discovery of the Higgs boson and the determination of its mass it became sufficient to use the low Higgs mass asymptotic, Eq. (15) of Refs. [66, 67]);
 (ii)the mixed EW\(\otimes \)QCD at \(\mathcal{O}(G_\mu \alpha _s)\), second term in the second square brackets [68, 69]. The quantity \({\varDelta }\rho ^{(1)}\)represents the leading NLO EW correction to \({\varDelta }\rho \) at \(\mathcal{O}(G_\mu )\) and should be subtracted from higherorder effects. Therefore, the contribution of higherorder effects has the following generic form:$$\begin{aligned} {\varDelta }\rho ^{(1)}\Big ^{G_\mu }=3x_t=\frac{3\sqrt{2}G_\mu m_t^2}{16\pi ^2} \end{aligned}$$(10)where \(c_i\) and \(R_{1i,2i}\) are combinations of \(Z(\gamma )f\bar{f}\) couplings and the ratio \(c^2_{\scriptscriptstyle W}/s^2_{\scriptscriptstyle W}\), and their explicit form depends on the parametrization of the LO cross section where the replacements (8) are performed (cf. Eq. (3.49) of [18]). This approach is implemented in RADY and SANC .$$\begin{aligned} \sum _i\,c_i[2({\varDelta }\rho {\varDelta }\rho ^{(1)})\,R_{1i}+{\varDelta }\rho ^2\,R_{2i}], \end{aligned}$$(11)
 As described in Ref. [26], the implementation of the NC DY in WZGRAD closely follows Refs. [70, 71] for a careful treatment of higherorder corrections, which is important for a precise description of the Z resonance. The NLO differential parton cross section including weak \(\mathcal{O}(\alpha )\) and leading \(\mathcal{O}(\alpha ^2)\) has the following form\(\mathrm{d} \hat{\sigma }_{\mathrm{box}}\) describes the contribution of the box diagrams and the matrix elements \(A_{\gamma ,Z}^{(0+1)}\) comprise the Born matrix elements, \(A_{\gamma ,Z}^0\), the \(\gamma ,Z, \gamma Z\) self energy insertions, including a leadinglog resummation of the terms involving the light fermions, and the oneloop vertex corrections. \(A_{\gamma ,Z}^{(0+1)}\) can be expressed in terms of effective vector and axialvector couplings \(g_{V,A}^{(\gamma ,Z),f}, f=l,q\), including vertex corrections and self energy insertions. Moreover, the \(M_Z\) renormalization constant \(\delta M_Z^2=\mathcal{R}e({\varSigma }^Z(M_Z^2))\) is replaced by \(\delta M_Z^2=\mathcal{R}e({\varSigma }^Z(M_Z^2) \frac{(\hat{\varSigma }^{\gamma Z}(M_Z^2))^2}{M_Z^2+\hat{\varSigma }^\gamma (M_Z^2)})\) where \({\varSigma }^V\, (\hat{\varSigma }^V)\) denotes the transverse part of the unrenormalized (renormalized) gauge boson self energy corrections. Higherorder (irreducible) corrections connected to the \(\rho \) parameter are taken into account by performing the replacement$$\begin{aligned} \mathrm{d} \hat{\sigma }^{(0+1)}= & {} \mathrm{dP_{2f}} \, \frac{1}{12} \, \sum A_{\gamma }^{(0+1)}+ A_Z^{(0+1)}^2(\hat{s},\hat{t},\hat{u})\nonumber \\&+\,\mathrm{d} \hat{\sigma }_{\mathrm{box}}(\hat{s},\hat{t},\hat{u}). \end{aligned}$$(12)where \({\varDelta }\rho ^{h.o.}={\varDelta } \rho {\varDelta }\rho ^{(1)}^{G_\mu }\) with \({\varDelta }\rho \) of Eq. (9) and \({\varDelta } \rho ^{(1)}^{G_\mu }\) of Eq. (10).$$\begin{aligned} \frac{\delta M_Z^2}{M_Z^2} \frac{\delta M_W^2}{M_W^2} \rightarrow \frac{\delta M_Z^2}{M_Z^2} \frac{\delta M_W^2}{M_W^2} {\varDelta }\rho ^{h.o.} \end{aligned}$$(13)
3.4.6 Higherorder effects to all orders via running couplings in NC DY
The purely EW fixedorder results, in the case of the NC DY process, can be improved with the systematic inclusion of some classes of universal higherorder corrections. The strategy to achieve this result is given by the matching of an Improved Born Approximation (IBA) of the LO description of the process, together with the full \(\mathcal{O}(\alpha )\) calculation, avoiding any double counting.
The use of the amplitudes in Eqs. (14)–(15) to compute the cross section represents an approximation of the exact NLO EW calculation for the non radiative part of the cross section; since they contain terms beyond NLO EW, one can also read a partial improvement over pure NLO. Their matching with the exact NLO EW expressions allows to recover this perturbative accuracy, but also to have a systematic inclusion of universal higherorder terms. Double counting is avoided by subtracting the \(\mathcal{O}(\alpha )\) part of the effective couplings in Eq. (15), in that part of the virtual corrections where the UV counterterms are introduced.
In the invariant mass region below the Z resonance the QED corrections increase the cross section by up to 100% of the fixedcoupling LO result. The introduction of the effective couplings yields a net effect at the few per cent level of the LO result. The impact of this redefinition of the LO couplings is demonstrated in Fig. 29, where we take the ratio of these improved predictions with those computed at NLO EW in the best setup of Sect. 3.1; the deviation from 1 is entirely due to terms of \(\mathcal{O}(\alpha ^2)\) or higher, present in the effective couplings.
3.4.7 QED shower matched to NLO EW matrix elements
The inclusion of multiple photon radiation in the presence of NLO EW matrix elements requires a matching procedure to avoid double counting. Several examples have been proposed in the literature following different algorithms, which have been implemented in the codes HORACE , POWHEG , and WINHAC , for instance. In Fig. 30 we use HORACE to illustrate the effect of all photon emissions beyond the first one in the NC (upper plots) and CC (lower plots) processes in the benchmark setup of Sect. 3.1 for the case of bare muons. The ratio shows the impact of the improved NLO EW prediction, when the NLO EW correction is matched to multiple photon radiation, over the NLO EW prediction; thus a deviation from 1 is entirely due to terms of \(\mathcal{O}(\alpha ^2)\) or higher. The impact of \(\mathcal{O}(\alpha )\) corrections on the LO distributions shown in Fig. 24 is largely due to photon radiation and thus we also observe a nonnegligible effect on the shape from higherorder multiple photon radiation in Fig. 30; the size of these effects, as expected, is in the 1% per cent ballpark, and depends on the shape of the observable. For example, while the \(\mathcal{O}(\alpha )\) corrections to the leptonpair transverse mass distribution can be as large as \(8\%\) of the LO prediction around the Jacobian peak, the \(\mathcal{O}(\alpha ^2)\) corrections of multiple photon radiation are \({<}0.5\%\) of the NLO EW prediction. The leptonpair invariant mass is the only observable that significantly changes because of multiple photon radiation: in fact the \(\mathcal{O}(\alpha )\) radiative effect is of \(\mathcal{O}(85\%)\) below the Z resonance, while at \(\mathcal{O}(\alpha ^2)\) the effects are a fraction of the previous order correction and can be as large as \(5\%\).
The best results of WINHAC for the CC DY process are obtained when interfaced with a parton shower MC (here: PYTHIA ), which also handles the initialstate photon radiation, and when including multiplephoton radiation in the YFS scheme. The impact of the YFS exponentiation is shown in Fig. 32 on the example of the \(p_T\) distribution of the charged lepton and the transverse mass distribution of the \(l\nu \) pair with and without taking into account the PYTHIA shower for initialstate photon and parton radiation.
The impact of YFS exponentiation observed in Fig. 32 is very similar to the multiplephoton radiation effects obtained with HORACE as shown in Fig. 30, i.e. also in the YFS exponentiation scheme of WINHAC the \(\mathcal{O}(\alpha ^2)\) corrections (and higher) amount to at most \(0.5\%\) of the \(\mathtt{NLO \, EW_\mathrm{sub}}\) prediction. As expected, in the presence of the QCD PS the multiplephoton radiation effects are less pronounced in the lepton \(p_T\) distribution but are unchanged in the leptonpair transverse mass distribution (see also Sect. 4 for a discussion of the interplay of QCD and QED effects in these observables).
3.4.8 Additional lightfermionpair emission
We used the MC codes SANC and HORACE to study the impact of the emission of an additional lightfermion pair in the NC DY process. In Fig. 33 the relative effect with respect to the NLO EW result is shown for the lepton transverse mass and leptonpair invariant mass distributions. The effect of additional lightfermion pair emission in the CC DY process has also been studied with the SANC code and was found to be less numerically important compared to the NC DY case.
4 Interplay of QCD and EW corrections
 1.
As already discussed in the previous sections, many observables relevant for precision EW measurements require a formulation that goes beyond fixedorder perturbation theory and includes the resummation to all orders of some logarithmically enhanced terms, preserving with a matching procedure the (N)NLO accuracy on the total cross section. This problem, which was discussed separately for QCD and for EW corrections, is present also once we consider the effect of mixed QCDxEW terms: in other words we need a matching procedure that preserves the NLO(QCD+EW) accuracy on the total cross section and that describes the emission of the hardest parton (gluon/quark/photon) with exact matrix elements, leaving the remaining emissions to a Parton Shower algorithm.
 2.
As long as the exact \(\mathcal{O}(\alpha \alpha _s)\) corrections to the fourfermion process are not fully known, we need to assess the accuracy of the recipes that combine QCD and EW effects available from independent calculations, e.g., the validity of an ansatz which factorizes QCD and EW terms.
In Sect. 4.3 we additionally show a comparison of different ways to simultaneously include QCD and QED/EW corrections to all orders on top of a LO description of the observables (with LO accuracy for the total cross section) and compare these results with the fixed order NLO predictions, in the case of calorimetric electrons in the final state.
4.1 Combination of QED/EW with QCD results in the POWHEG framework
The study of the DY observables that are relevant for highprecision measurements requires the inclusion of QEDFSR effects to all orders and of QCDISR effects to all orders, in order to obtain a description stable upon inclusion of further higherorder corrections.
The impact of multiple parton radiation has been discussed in Sects. 3.3 and 3.4, separately in the QCD and QED cases, in codes that match the PS algorithm with NLO fixedorder results.
PS codes are often used as standalone tools, since they provide a good approximation of the shape of the differential distributions. When QCDPS and QEDPS are combined together, the resulting description has an exact treatment of the kinematics of each individual QCD/QED parton emission, but lacks the exact matrix element corrections and the normalization which are instead available in a fixedorder NLOaccurate calculation.
We first start from the LO distributions of these two quantities, which show the sharply peaked behavior due to the jacobian factor. The QEDFSR emissions are simulated with the PHOTOS code and yield effects which are similar for the two observables, with a negative correction of \(\mathcal{O}(8\%)\) at the jacobian peak, as shown in the middle plots by the blue points.
We then consider the role of NLOQCD corrections and of a QCDPS in the POWHEG +PYTHIA code and remark (cfr. the upper plots) that, while the shape of the transverse mass distribution is preserved, to a large extend, by QCD corrections, the lepton transverse momentum distribution is instead strongly smeared, with a much broader shape around the jacobian peak. The inclusion of the PHOTOS corrections on top of the POWHEG +PYTHIA simulation has now a different fate, compared to the LO case (cfr. middle plots, red points): the shape and the size of the QED corrections are similar to the LO case for the transverse mass; in the lepton transverse momentum case instead the QED correction is reduced in size and flatter in shape, with respect to the LO case. The comparison of the percentage corrections due to QEDFSR in the two examples discussed above (blue and red points in the middle plots) shows a difference which is due to mixed QCDxQED corrections, since the set of pure QED corrections is common to the two simulations.
The impact on the \(M_W\) determination of the interplay between QCD and EW corrections in the POWHEG(QCD+EW) framework has been presented in [75].
4.2 Towards exact \(\mathcal{O}(\alpha \alpha _s)\): assessment of the accuracy of current approximations
As mentioned earlier, the question how to properly combine QCD and EW corrections in predictions will only be settled by a full NNLO calculation of the \(\mathcal{O}(\alpha \alpha _s)\) corrections that is not yet available, although first steps in this direction have been taken by calculating twoloop contributions [77, 78, 79, 80, 81], the full \(\mathcal{O}(\alpha \alpha _s)\) correction to the W/Zdecay widths [82, 83], and the full \(\mathcal{O}(\alpha )\) EW corrections to W/Z+jet production including the W/Z decays [39, 40, 84].
In the numerical results shown below, all terms of Eq. (18) are consistently evaluated using the NNPDF2.3QED NLO set [32], which includes \(\mathcal {O}(\alpha )\) corrections. We consider the case of “bare muons” without any photon recombination. Results obtained assuming a recombination of leptons with collinear photons can be found in Ref. [87] and show the same overall features, with corrections that typically reduced by a factor of two.
In the case of the \(M_{\mathrm {T},\nu l}\) distribution (left plot in Fig. 36), which is rather insensitive to Wboson recoil due to jet emission, both versions of the naive product ansatz approximate the PA prediction quite well near the Jacobian peak and below. Above the peak, the product \(\delta _{\alpha _s}' \delta _\alpha \) based on the full NLO EW correction factor deviates from the other curves, which signals the growing importance of effects beyond the PA. In contrast, the product ansatz fails to provide a good description for the lepton \(p_{\mathrm {T},l}\) distributions (right plots in Figs. 36, 37), which are sensitive to the interplay of QCD and photonic realemission effects. In this case one also observes a larger discrepancy of the two different implementations of the naive product, which indicates a larger impact of the missing \(\mathcal{O}(\alpha \alpha _s)\) initialinitial corrections of Fig. 35a, and in particular the realemission counterparts. For the \(M_{l^+l^}\) distribution for Z production (left plot in Fig. 37), the naive products approximate the full initial–final corrections reasonably well for \(M_{l^+l^}\ge M_{\mathrm {Z}}\), but completely fail already a little below the resonance where they do not even reproduce the sign of the full correction \(\delta ^{{\mathrm {prod}}\times {\mathrm {dec}}}_{\alpha _s\alpha }\). This failure can be understood from the fact that the naive product ansatz multiplies the corrections locally on a binbybin basis, while a more appropriate treatment would apply the QCD correction factor at the resonance, \(\delta _{\alpha _s}'(M_{l^+l^}=M_{\mathrm {Z}})\approx 6.5\%\), for the events that are shifted below the resonance by photonic FSR. The observed mismatch is further enhanced by a sign change in the QCD correction \(\delta _{\alpha _s}'\) at \(M_{l^+l^}\approx 83\,\mathrm {GeV}\).
These examples show that a naive product approximation has to be used with care and does not hold for all distributions. The results are also sensitive to the precise definition of the correction factors \(\delta _\alpha \) and \(\delta _{\alpha _s}\) [86]. As shown in Ref. [87], a more suitable factorized approximation of the dominant \(\mathcal {O}(\alpha \alpha _s)\) effects can be obtained by combining the full NLO QCD corrections to vectorboson production with the leadinglogarithmic approximation for FSR through a structurefunction or a parton shower approach such as used in PHOTOS [12]. In this way the interplay of the recoil effects from jet and photon emission is properly taken into account, while certain nonuniversal, subleading, effects are neglected.
4.3 Comparing different ansatzes of higherorder QED/EW corrections combined with QCD parton showers
In this section we compare the higherorder QED corrections predicted by SHERPA ’s Yennie–Frautschi–Suura (YFS) softphoton resummation [61, 88], the standard DGLAP collinear higherorder QED corrections as implemented in PYTHIA8 [89], and the exact NLO EW calculation performed by SHERPA using oneloop matrix elements from OPENLOOPS [90, 91, 92]. In Ref. [38], for the case of the NC DY process, the quality of the YFS implementation of SHERPA has been checked against the exact NLO EW \(\mathcal {O}(\alpha )\) calculation and the NNLO QCDEW mixed \(\mathcal {O}(\alpha _s\alpha )\) calculation in the pole approximation of [85, 87]; we point to this reference for the quantitative results. In the following, the calculations including YFS exponentiation, standard DGLAP QED and fixedorder NLOEW corrections have been performed also for the CC DY process and shall be compared among each other in a realistic scenario. We consider electrons dressed with the surrounding \({\varDelta } R=0.1\), which are required to have \(p_\mathrm{T}>25\,\mathrm{GeV}\) and \(y<2.4\), and a missing transverse momentum of at least \(25\,\mathrm{GeV}\).
Figure 38 (left) shows the comparison of the different calculations for the reconstructed transverse mass of the W boson. Besides the leading QCD higherorder corrections, the higherorder EW corrections between either the YFS resummation or the partonshower approach agree well with the fixedorder result (see the central inset), only PYTHIA8 ’s QED parton shower predicts a stronger correction around the peak and near the threshold. The differences with respect to the NLO EW correction can be traced to multiphoton emissions present in the allorder results and to genuine weak effects only present in the NLO EW calculation. The same findings were reported for the case of lepton pair production in Ref. [38]. Applying the YFS resummation in addition to higherorder QCD corrections, the implementation corresponds to a multiplicative combination of both effects and preserves these findings for the leptonpair transverse mass distribution (lower inset), as already observed in Sect. 4.1. Again, subpercent level agreement is found with the fixedorder calculation in the peak region. At low transverse masses the resummation of QCD corrections is important and drives the difference to the fixedorder result.
Figure 38 (right) details the comparison of the different calculations for the transverse momentum of the dressed electron. Again, the exact \(\mathcal {O}(\alpha )\) calculation is in subpercent level agreement with the YFS resummation, and again, the general offset can be attributed to both multiple photon emission corrections and genuine weak corrections (central inset). The PYTHIA8 QED parton shower shows a different behavior in the peak region. Once NLO QCD effects are also taken into account (lower inset), the importance of their resummation with respect to their simple fixedorder treatment, as already observed in Sect. 3.3.4, overwhelms the comparison between the YFS soft photon resummation and the fixedorder NLO EW calculation for this observable.
The investigation of the observed difference in the behavior of the QED parton shower in PYTHIA8 and the YFS softphoton resummation is left to a future study.
5 Conclusions

In this report we compared several public codes which simulate the Drell–Yan processes in different perturbative approximations. All these codes are at least NLO accurate in the description of inclusive observables in either the EW or strong interaction, or possibly with respect to both.

This common level of accuracy allowed to consistently compare the codes, testing their respective numerical implementations and the resulting level of agreement (see Sect. 2).

Relying on this NLOaccurate framework, it has been possible to define a way to quantify the impact of higherorder corrections, i.e. beyond NLO, which may differ from code to code (see Sect. 3). The study of the impact of different sets of corrections has been performed separately for the EW and strong interactions.

Some codes provide, in the same implementation, QCD and EW corrections, which have been separately tested in Sects. 2 and 3. The interplay of both sets of corrections is discussed in Sect. 4.

The impact of all the higherorder corrections, which are available in some but not in all codes, is expressed as a percentage effect, using a common unit, namely the distribution obtained in the calculation which has NLO accuracy for the total cross section and uses the inputs of the benchmark setup.

The distribution used as common unit may not be the most suitable choice for all the observables: in fact in some phasespace corners perturbation theory breaks down and the fixedorder distribution provides only a technical reference rather than a sensible estimate of the physical observable.

The problem of a consistent matching of fixed and allorders results emerges in several cases discussed in Sect. 3, both in the EW and in the QCD sectors. Different matching procedures may agree on the accuracy on the observables inclusive over radiation (NLO or NNLO) but differ by the inclusion of higherorder subleading terms; the latter, despite their subleading classification, might nevertheless have a sizable impact on some differential distribution, sensitive to radiation effects.

The analytical expression of the terms by which two matching procedures differ is not always available, leaving open only the possibility of a numerical comparison.

In a tuned comparison at NLO, where all the input parameters and the simulation setup are identical and the matrix elements have the same accuracy for all the codes, we observe that the total cross sections agree at the 0.03% level both in the NLO EW and in the NLO QCD calculations; the differential distributions differ at most at the 0.5% level.

The spread of the predictions at differential level reflects the impact of different choices in the numerical implementation of exactly the same calculation, in particular the handling of the subtraction of infrared and collinear divergences.

In a tuned comparison of codes that share NNLO QCD accuracy for the observables inclusive over radiation (cfr. Sect. 3.3.2), the level of agreement for the total cross sections is at the 0.4% level and for the differential distributions is at the \(\mathcal{O}(1\%)\) level, depending on the observable and on the range considered, but always with compatibility within the statistical error bands.

All the EW higherorder effects are of \(\mathcal{O}(\alpha ^2)\) or higher. Their size is in general at the few per mill level, with some exceptions like the leptonpair invariant mass distribution, which receives corrections up to 5%. This particularly large size is due to the combination of two elements: on the one side to the steeply falling shape of the Z boson resonance; on the other side, to the fact that most of the events are produced at the Z peak, but final state radiation reduces the eventual invariant mass of the lepton pair, so that the lowermass bins are populated. At \(\mathcal{O}(\alpha )\)the effect is of \(\mathcal{O}(100\%)\) and multiple photon radiation still yields an additional corrections of several per cent.

In the absence of a full NNLO EW calculation, all the higherorder EW effects are necessarily subsets of the full result. They thus may not be representative of the full result, and care should be taken in using these partial results to estimate the effects of missing higherorder corrections.

The size of the QCD radiative corrections strongly depends on the observable: the differential distributions which require a resummation to all orders in some phasespace corners should be discussed separately from those that are stable upon inclusion of radiative effects. Given our reference results obtained with codes that have NLO QCD accuracy for the total cross section, we studied higherorder effects due to NNLO QCD corrections, NLO QCD corrections matched with a QCD PS, and NNLO QCD corrections matched with a QCD PS. In case of the matched calculations we compared two different matching formulations.

The NNLO QCD corrections to the invariant (transverse) mass distribution of the lepton pair are small in size, at the few per cent level over the whole spectrum. The same codes predict a large positive correction of \(\mathcal{O}(4050\%)\) of the lowerorder result for the leptonpair transverse momentum distribution,^{13} as the effect of having the exact description of two hard real parton emissions. The latter show to play an important role also in the description of the hard tail, above the Jacobian peak, of the singlelepton transverse momentum distribution, with effects again at the \(\mathcal{O}(30{}40\%)\) level.

Matching fixed and allorder results is necessary to obtain a sensible description of the Jacobian peak in the single lepton transverse momentum distribution or the lowmomentum tail of the leptonpair transverse momentum distribution. Even if this goal is achieved, nevertheless two codes that share the same accuracy for the total cross section (in the absence of acceptance cuts), i.e. NLO QCD or NNLO QCD, still exhibit sizable differences in the prediction of these same observables, in the intermediate ranges of the spectra. It should be stressed that these differences can be, in the NLO+PS matching, as large as few percent at the Jacobian peak or even several tens of percent for the leptonpair transverse momentum distribution. The size of these differences is reduced, at the several per cent level, with the NNLO+PS matching. This kind of matching ambiguities should be added to the usual renormalization/factorization scale variations and deserves further investigation. An example of such a study of matching uncertainties can be found in Ref. [93], for the Higgs transverse momentum distribution in gluon fusion.

QCD and EW effects are separately available at first perturbative order and have been extensively tested in Sect. 2. The possibility of combining the differential Kfactors in a factorized ansatz has been shown to be accurate, compared to the \(\mathcal{O}(\alpha \alpha _s)\) results available in pole approximation at the W (Z) resonance, for observables that are insensitive to a redistribution of events by QCD radiation, such as in the transversemass distribution of the W or Z bosons. Naive products fail to capture the dominant QCDxEW corrections in distributions such as in the transverse momentum of the lepton, which is sensitive to QCD initialstate radiation and photonic finalstate radiation. For the invariantmass distribution of the neutralcurrent process the naive product approach is insufficient as well because of large photonic finalstate corrections and initialstate QCD corrections which depend on the reconstructed invariant mass in a nontrivial way.

The POWHEG implementation of QCD+EW corrections shares with the other codes of the present report the NLO(QCD+EW) accuracy for the total cross section. On the other hand, it offers one possible solution to the matching of fixed and allorders results, both in QCD and in the EW sectors, and in turn it introduces mixed QCDxEW factorizable corrections to all orders.

The interplay between QCD and QED corrections is not trivial, as it can be checked in observables like the chargedlepton transverse momentum distribution, where one can appreciate the large size of mixed \(\mathcal{O}(\alpha \alpha _s)\)and higher corrections. The impact, in the same QCD framework, of subleading effects due to weak radiative corrections and to the exact treatment of real radiation matrix elements is not negligible in view of precision EW measurements, e.g. being the correction at the several per mill level in the case of the leptonpair transverse mass distribution.

The estimate of the accuracy available in the prediction of DY observables requires the distinction between: (1) higherorder corrections which have been computed and are available in at least one code and (2) missing higherorder terms which are unknown, whose effect can only be estimated.

The present report provides, for item (1), guidance to assess the size of the corrections which are missing in one code, thanks to the analysis of Sect. 3, so that they can be treated as a theoretical systematic error, when they are not included in the simulation.

On the other hand, item (2) requires a detailed, systematic discussion, which can start from the results of the present report, but goes beyond its scope. The estimate of the actual size of missing higher orders is an observabledependent statement. In some specific cases the available fixedorder perturbative results may offer a handle to estimate the remaining missing corrections. On the other hand, the quantities which require matching of fixed and allorder results are simultaneously affected by several sources of uncertainty whose systematic evaluation will require a dedicated effort (see, e.g., the discussion in Sect. 3.3.6).
Footnotes
 1.
For recent W / Z physics results from the LHC see: ATLAS: https://twiki.cern.ch/twiki/bin/view/AtlasPublic/StandardModelPublicResults CMS: https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsSMP LHCb: https://lhcb.web.cern.ch/lhcb/PhysicsResults/LHCbPhysicsResults.html.
 2.
See also a recent study in Ref. [38].
 3.
We discuss separately the NC DY LO contribution given by \(\gamma \gamma \rightarrow l^+l^\) scattering, which receives a nontrivial QCD correction only starting from third perturbative order.
 4.
Recently, an implementation of NNLO QCD corrections to \(pp \rightarrow Z\) and \(pp \rightarrow W\) including the decays of the unstable gauge bosons became also available in MCFM [45].
 5.
 6.
Except for the pure twoloop virtual contribution, which contributes to W(Z) plus one jet at NNLO if showered.
 7.
The analytic Sudakov form factor is generally observabledependent (not fully differential); in the application to DY here, the relevant observable used by MiNLO is the W(Z) transverse momentum \(p^{V}_\perp \) ).
 8.
For a discussion on the definition of an effective electromagnetic coupling at the 2loop level see Ref. [74].
 9.
In the case of a radiative event, an effective Born configuration is computed to evaluate \(K_{IBA}\).
 10.
We understand that the phasespace factors are properly included in the definition of the various \(d\sigma \) coefficients.
 11.
The treatment of FSR QED radiation present in POWHEG(QCD+EW) , up to svn version 3358, generates artificially enhanced \(\mathcal{O}(\alpha \alpha _s)\)corrections, as pointed out in Refs. [75, 76], published after the completion of the present report. Concerning the POWHEG(QCD+EW) code, an improved treatment which overcomes this problem is described in Ref. [75]. An alternative implementation is described in Ref. [76].
 12.
Note that this correction factor differs from that in the standard QCD K factor \( K_{{\mathrm {NLO}}_{\mathrm {s}}} =\sigma _{{\mathrm {NLO}}_{\mathrm {s}}}/\sigma _{\mathrm {LO}}\equiv 1 +\delta _{\alpha _s}\) due to the use of different PDF sets in the Born contributions. See Ref. [86] for further discussion.
 13.
We remind the reader that the codes that have NNLO QCD accuracy for the total cross section are only NLO QCD accurate in the prediction of the large momentum tail of the leptonpair transverse momentum distribution. For the same reason our reference results, which are NLO QCD accurate for the total cross section, are only LO accurate for this observable.
 14.
In the present version the “photoninduced” processes are not considered.
Notes
Acknowledgements
Work on this report was partially carried out at the LHC Run II and the Precision Frontier workshop at the Kavli Institute for Theoretical Physics (KITP), supported by NSF PHY1125915, at the MIAPP workshop Challenges, Innovations and Developments in Precision Calculations for the LHC, supported by the Munich Institute for Astro and Particle Physics (MIAPP) of the DFG cluster of excellence “Origin and Structure of the Universe”, and the Prospects and precision at the Large Hadron Collider at 14 TeV workshop at the Galileo Galilei Institute for Theoretical Physics (GGI). We are grateful to the CERN LPCC and Michelangelo Mangano for hosting workshops at the LPCC, which allowed for close interactions of members of this working group with members of the experimental collaborations involved in EW precision analysis. We thank the Department of Physics at the University of Milan and INFN for financial support for the kickoff meeting W mass workshop, and John Campbell and Fermilab for hosting the second W mass workshop.
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