EW radiative corrections to theory predictions of charge asymmetry for W-boson hadroproduction

Productions of weak gauge bosons (W and Z bosons) through charged-current and neutral-current Drell-Yan mechanisms [1] have long been important benchmark processes at past and current hadron colliders. Hadroproductions of W and Z bosons provide prominent tests of quantum chromodynamics (QCD) and electroweak (EW) domains of the standard model (SM) of high-energy particle physics. These bosons have sizable production cross sections and are measured very precisely owing to small background contributions in their subsequent leptonic decay channels. These precise measurements enable substantial inputs for determination of parton distribution functions (PDFs) in the proton. Their hadroproductions feature other experimental aspects such as improving modeling of rarer SM background processes and in new physics searches, as well as improving calibration of detector response for better lepton, missing transverse energy, and jet resolutions. In addition, both W- and Z-boson hadroproductions constitute a broad testing ground for Monte Carlo generators and higher-order theoretical tools.

W-boson hadroproduction in its leptonic decay channel is proceeded by one high transverse momentum p_T -isolated lepton and significant amount of missing transverse energy due to detector-escaping neutrino, particularly in proton-proton (pp) collisions at the CERN Large Hadron Collider (LHC) as pp → W^± → l^± ν+X. Here l is either an electron e or a muon μ and X stands for accompanying final state. W^± bosons are predominantly produced via annihilation mechanism of a valence quark with a sea antiquark from interacting protons as $u\bar{d}\to {W}^{+}$ and $d\bar{u}\to {W}^{-}$. W⁺ bosons have higher production rates than W⁻ bosons as required by the excess of two valence u quarks over one valence d quark in the proton. This asymmetry as a consequence of difference in W⁺ and W⁻ boson production rates is called as W-boson charge asymmetry, which is expressed by their cross sections σ(W⁺ → l⁺ ν) and $\sigma ({W}^{-}\to {l}^{-}\bar{\nu })$ in differential form as a function of W boson rapidity y_W as

$$\begin{eqnarray}&&{A}_{{y}_{W}}=\displaystyle \frac{d\sigma ({W}^{+}\to {l}^{+}\nu )/{{dy}}_{W}-d\sigma ({W}^{-}\to {l}^{-}\bar{\nu })/{{dy}}_{W}}{d\sigma ({W}^{+}\to {l}^{+}\nu )/{{dy}}_{W}+d\sigma ({W}^{-}\to {l}^{-}\bar{\nu })/{{dy}}_{W}}.\end{eqnarray} \tag{ 1 }$$

W-boson charge asymmetry ${A}_{{y}_{W}}$ is mainly sensitive to valence u- and d-quark distributions in terms of incoming parton momentum fractions (Bjorken x values), because y_W depends strongly on x values as from the expression ${x}_{\mathrm{1,2}}=({m}_{W}/\sqrt{s}){e}^{\pm y}$, where m_W stands for the mass of the W-boson and $\sqrt{s}$ does for the center-of-mass energy. The sensitivity to the valence quark distributions can be shown by rewriting the equation (1) in terms of the valence (sea) quark distributions u(x) ($\bar{u}(x)$) and d(x) ($\bar{d}(x)$) as

$$\begin{eqnarray}&&{A}_{{y}_{W}}\approx \displaystyle \frac{u(x)\bar{d}(x)-d(x)\bar{u}(x)}{u(x)\bar{d}(x)+d(x)\bar{u}(x)}\approx \displaystyle \frac{u(x)-d(x)}{u(x)+d(x)},\end{eqnarray} \tag{ 2 }$$

where the expression is simplified with the approximation $\bar{u}(x)\sim \bar{d}(x)\sim \bar{q}(x)$ by treating the sea quark contributions to the proton momentum to be equal such as for small x values in the central region y_W ∼ 0. Moreover, ${A}_{{y}_{W}}$ exhibits symmetric characteristic with respect to y_W in pp collisions, whereas it is antisymmetric in $p\bar{p}$ collisions as because of strongly reduced sensitivity to sea-quark contributions. Although ${A}_{{y}_{W}}$ can be predicted theoretically, it cannot be directly determined in experiments due to absence of information about longitudinal component of neutrino's momentum from W-boson decay. Despite this experimental restriction, the same information as ${A}_{{y}_{W}}$ can still be attained by measurements of charge asymmetry using decay lepton from W boson. This lepton charge asymmetry can readily be measured in terms of decay lepton pseudorapidity η_l (where it is literally linked to y_W ), by exploiting a definition analogously from equation (1) as

$$\begin{eqnarray}&&{A}_{{\eta }_{l}}=\displaystyle \frac{d\sigma ({W}^{+}\to {l}^{+}\nu )/d{\eta }_{l}-d\sigma ({W}^{-}\to {l}^{-}\bar{\nu })/d{\eta }_{l}}{d\sigma ({W}^{+}\to {l}^{+}\nu )/d{\eta }_{l}+d\sigma ({W}^{-}\to {l}^{-}\bar{\nu })/d{\eta }_{l}}.\end{eqnarray} \tag{ 3 }$$

Lepton charge asymmetry ${A}_{{\eta }_{l}}$ stands for both the original ${A}_{{y}_{W}}$ observable and the V–A (vector–axial vector) structure of W boson which are convoluted, where the V–A structure signifies anisotropic decay of W-boson in leptonic mode. By the same token as ${A}_{{y}_{W}}$, precise measurement of ${A}_{{\eta }_{l}}$ variable provide considerable constraints for the relative densities of u and d quarks in the proton as functions of parton x values. ${A}_{{\eta }_{l}}$ variable can also be exploited to extract additional information for sea-quark distributions and to distinguish among various PDF models relying on different parametrization. In addition, ${A}_{{\eta }_{l}}$ is favored by actual experiments as experimental uncertainties partially cancel in its measurements resulting in greater precision in tests of the SM.

Charge asymmetry measurements for inclusive W-boson hadroproduction were mostly carried out using ${A}_{{\eta }_{l}}$ observable by the CDF and D0 Collaborations at the Tevatron [2–9] based on $p\bar{p}$ collisions data. ${A}_{{\eta }_{l}}$ measurements were carried out in the ATLAS and CMS experiments at the LHC using pp collisions data in the central detector acceptance ∣η_l ∣ ≤ 2.5 at various center-of-mass energies up to 8 TeV [10–17]. ${A}_{{\eta }_{l}}$ was complementarily measured in the LHCb experiment at the LHC up to 8 TeV [18–21] for the forward region 2.0 ≤ η_l ≤ 4.5 going beyond the central acceptance region. The LHC measurements have already investigated into ${A}_{{\eta }_{l}}$ variable for a broad range of x values 10⁻⁴ < x < 1, enabling improvements for existing PDF constraints at very small and large x values. Furthermore, precise measurements of ${A}_{{\eta }_{l}}$ pave the way for stringent tests of higher-order theoretical calculations beyond leading order (LO) which are convoluted with different PDFs. Ensuing the precision level attained by the LHC measurements, this in turn necessitates theoretical computations to include next-to-LO (NLO) and next-to-NLO (NNLO) radiative corrections in the strong coupling α_S and as well as in the EW coupling α of underlying perturbation theory. Calculations for radiative corrections to Drell-Yan processes in terms of inclusive cross section have been carried out since for some time at NLO [22] and NNLO [23, 24] in QCD perturbation theory. NNLO QCD corrections to W boson differential cross section in leptonic decay modes have also been calculated [25–29]. The dominant EW radiative corrections are induced by quantum electrodynamics (QED) effects in distributions of leptons in final states, and by substantial Sudakov logarithms [30] regarding kinematic distributions at very high energies. In accordance with this information, calculations of EW corrections up to NLO were also reported thoroughly for W-boson leptonic decay in the related literature [31–35]. Higher-order calculations continue to advance by also the inclusion of mixed QCD-EW effects of ${ \mathcal O }({\alpha }_{S}\alpha )$ which are based on dedicated approaches including such as the pole approximation [36–38].

In this phenomenological work, the state-of-the-art predictions, including both NNLO QCD and NLO EW radiative corrections, are reported for the charge asymmetry observable ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) of the W-boson hadroproduction at the LHC. The study aims to fill gaps in the related literature by explicitly carrying out the computations of the combined predictions NNLO QCD+NLO EW and NNLO QCD × NLO EW to have explicit accounting for the NLO EW effects for the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) distribution in the fiducial phase space spanning from central to forward region as η_l ≤ 4.5 (y_W ≤ 4.5). The combined NNLO QCD and NLO EW predictions are compared with the fixed-order NNLO QCD predictions, and are more necessarily justified by the comparisons with the LHC pp collisions data at 8 TeV energy. The impact of the NLO EW effects on the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) distribution at 13 TeV and 14 TeV energies are assessed by performing detailed studies of the NLO EW correction factors with regards to the (N)NLO QCD, and to the LO based on a detailed K-factor analysis. Moreover, the relative sizes of the NLO EW effects in the combined predictions are considered for the W^±-boson differential cross sections for a comprehensive assessment of the NLO EW effects for the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$). The presented results clearly represent an important input for both theoretical predictions and experimental measurements to consider proper inclusion of the NLO EW corrections for a thorough description of the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) distributions.

Higher-order differential cross section calculations, from which the charge asymmetry ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) predictions are obtained, are performed by using the MATRIX framework (version 2.0.0) [28, 39, 40]. The working instructions for setting up and running the MATRIX framework can be found in the [39], where they are readily available to use in a practical manner. The computations of cross sections are performed by relying on the Catani-Seymour dipole-subtraction approach at NLO QCD [41, 42] and NLO EW [43–47]. Besides that the computations of cross sections at NNLO QCD are achieved relying on the q_T-subtraction formalism [48, 49]. The q_T-subtraction method facilitates regularization for soft and collinear divergences of the real radiation contributions in the calculations by means of a cut-off value for the slicing parameter r_cut , which is fixed at r_cut = 0.0015(0.15%). The MATRIX framework is incorporated with the OpenLoops program through an interface to compute the required tree-level and one-loop scattering amplitudes [50–54], including spin- and color-correlated amplitudes for the subtraction of divergences. Leptons (both e and μ) as well as light quarks are considered to be massless except for top quark with mass m_t = 173.2 GeV from the default model-related settings of the framework. The W boson is treated off-shell with leptonic final states based on the complex-mass scheme [55]. In this scheme, the EW mixing angle as ${\sin }^{2}{\theta }_{W}$ (and α) is defined using complex W- and Z-boson mass terms Moreover, the G_μ input scheme is employed for the parametrization of the EW sector, implying the W-boson interactions with fermion currents, which encompasses the Fermi constant G_F and the weak boson masses m_W and m_Z . The related input values in the computations are used in line with the PDG [56] values for m_W , m_Z , decay widths Γ_W and Γ_Z , and G_F as the followings

$$\begin{eqnarray}\begin{array}{rcl}{m}_{W} & = & 80.385\quad \mathrm{GeV},{\text{}}{m}_{Z}=91.1876\quad \mathrm{GeV},\\ {{\rm{\Gamma }}}_{{\text{}}W} & = & 2.0854\quad \mathrm{GeV},{{\rm{\Gamma }}}_{{\text{}}Z}=2.4952\quad \mathrm{GeV},\\ \mathrm{and}\quad {\text{}}{G}_{F} & = & 1.16639\times {10}^{-5}\quad {\mathrm{GeV}}^{-2}.\end{array}\end{eqnarray} \tag{ 4 }$$

The values for the CKM matrix elements are fully adopted from the PDG [56] as per the default setting. Exceptionally, the one-loop diagrams with CKM in the EW contributions to off-shell W^± process are not available within the OpenLoops program and their EW contributions are approximated using a trivial CKM matrix.

The quark-photon channels at NLO EW can result in considerable enhancement of the photon-induced effects and the contributions from these effects need to be accounted for in the computations. Collinear photon and charged lepton pairs are recombined into dressed leptons within a cone, where radius of cone ΔR is defined in terms of separations in azimuthal angle and pseudorapidity between photons and leptons and its value is set up to 0.1. Throughout the computations all phase space cuts, kinematic observables, and scales are considered using dressed lepton definition. Moreover, PDFs which take into account QED effects, are required to include ${ \mathcal O }(\alpha )$ contributions for reliable cross-section calculation of hadronic decay processes. The NNPDF31 [57] PDF models are utilized based on the LUXqed methodology [58] which also include QED evolution for the determination of photon density. More precisely, NNPDF31_nlo_as_0118_luxqed and NNPDF31_nnlo_as_0118_luxqed PDF sets are used at appropriate perturbative orders that are all based on the α_s (m_Z ) = 0.118 value. The LHAPDF (v6.2.0) platform [59] is exploited for the purpose of evaluation of the PDF sets.

The calculations are carried out using realistic fiducial phase spaces regarding leptonic decay of W boson. Fiducial requirements are adopted from the actual charge asymmetry measurements that were based on 8 TeV data by the CMS [13] and LHCb [20] experiments at the LHC. The decay leptons (e or μ) with transverse momentum either ${p}_{T}^{l}\gt$ 25GeV in the central region 0 ≤ η_l ≤ 2.4 or ${p}_{T}^{l}\gt$ 20 GeV in the forward region 2.0 ≤ η_l ≤ 4.5 are required to validate the calculations with the reference 8-TeV CMS and LHCb results, respectively. In the calculations at 13 TeV and 14 TeV, the leptons with ${p}_{T}^{l}\gt$ 25 GeV are required in the entire acceptance spanning over both central and forward regions as 0 ≤ η_l ≤ 4.5. There are no requirements applied for either the transverse mass of the W boson or the missing transverse energy resulted from the decay neutrino, whereas these requirements can intentionally be utilized in pure experimental studies. Despite no explicit criteria are required for final-state jet(s), unresolved hadronic emission is permitted accompanying the leptonic final state of the W boson.

Calculations of hadronic cross sections in the context of perturbative QCD theory have dependance on both the renormalization scale μ_R and the factorization scale μ_F . The μ_R and μ_F are set equal to

$$\begin{eqnarray}&&{\mu }_{R,F}={\zeta }_{R,F}{\text{}}{\mu }_{0},\quad \mathrm{with}\quad {\text{}}{\mu }_{0}={\text{}}{m}_{W}=80.385\quad \mathrm{GeV}\quad \mathrm{and}\quad 1/2\leqslant {\zeta }_{R},{\zeta }_{F}\leqslant 2\end{eqnarray} \tag{ 5 }$$

in the computations. The ζ_R = ζ_F = 1 condition refers to the choice of central scale. Theoretical uncertainties from excluded QCD corrections beyond NNLO are estimated by varying the scales μ_R and μ_F using the usual 7-point variations (ζ_R , ζ_F )=(2,2), (2,1), (1,2), (1,1), (1,1/2), (1/2,1), (1/2,1/2). PDF uncertainties are also estimated by following the steps aas prescribed by the PDF4LHC working group [59, 60] so as to account for imperfect modeling of PDFs. In addition, uncertainties due to α_s value are estimated by shifting α_s by 0.001 up and down with regards to the default value of 0.118, which is the one exactly considered in the PDF sets. Finally, quadratic sum of scale, PDF, and α_s uncertainties is considered in evaluation of total theoretical uncertainties, where they are quoted symmetrically in the predictions by taking the uncertainty only from the largest variation.

To this end, combination procedure of QCD and EW corrections in the computations needs to be reminded briefly (based on the [40]) in the following. Corrections at NNLO in QCD and at NLO in EW are combined using both standard additive and factorised (multiplicative) prescriptions. At higher orders, these corrections to differential cross section d σ_{NNLO  QCD} and d σ_{NLO  EW} can be expressed by making use of correction factors δ_QCD and δ_EW relative to the differential cross section at LO d σ_LO, respectively as

$$\begin{eqnarray}&&d{\sigma }_{\mathrm{NNLO}\quad \ \mathrm{QCD}}=d{\sigma }_{\mathrm{LO}}(1+{\delta }_{\mathrm{QCD}})\quad \mathrm{and}\quad d{\sigma }_{\mathrm{NLO}\quad \ \mathrm{EW}}=d{\sigma }_{\mathrm{LO}}(1+{\delta }_{\mathrm{EW}}).\end{eqnarray} \tag{ 6 }$$

The first prescription corresponds to an additive combination as NNLO QCD+NLO EW, which is to be expressed in the following form

$$\begin{eqnarray}&&d{\sigma }_{\mathrm{NNLO}\quad \ \mathrm{QCD}+\mathrm{NLO}\quad \ \mathrm{EW}}=d{\sigma }_{\mathrm{LO}}(1+{\delta }_{\mathrm{QCD}}+{\delta }_{\mathrm{EW}}).\end{eqnarray} \tag{ 7 }$$

Factorised prescription is considered as an alternate combination NNLO QCD × NLO EW as given below

$$\begin{eqnarray}&&d{\sigma }_{\mathrm{NNLO}\quad \ \mathrm{QCD}\times \mathrm{NLO}\quad \ \mathrm{EW}}=d{\sigma }_{\mathrm{LO}}(1+{\delta }_{\mathrm{QCD}})(1+{\delta }_{\mathrm{EW}}),\end{eqnarray} \tag{ 8 }$$

where it apparently enables an approximation of the additional QCD-EW contributions that are mixed at higher orders in the forms of ${ \mathcal O }({\alpha }_{s}\alpha )$ and ${ \mathcal O }({\alpha }_{s}^{2}\alpha )$. Consequently, the factorised approach can be used to estimate roughly systematic uncertainty due to missing higher-order EW effects from the generated mixed QCD-EW contributions for the combined predictions based on the additive approach. Finally in the computations of the combined predictions from both the prescriptions, individual partonic channels $q\bar{q}$ at all orders of QCD, qg and $\bar{q}g$ at NLO QCD, gg and $q(\bar{q}){q}^{{\prime} }$ at NNLO QCD, and photon-induced channels q(g)γ at NLO EW are all naturally included.

The combined predictions NNLO QCD+NLO EW and NNLO QCD × NLO EW, based on both additive and multiplicative prescriptions in turn, are compared with the actual ${A}_{{\eta }_{l}}$ measurements available in the η_l bins at 8 TeV. The combined predictions are compared also with the fixed-order NNLO QCD in addition to the CMS [13] and LHCb [20] measurements in the central η_l ≤ 2.4 and forward 2.0 ≤ η_l ≤ 4.5 regions, respectively. The predictions are acquired for the μ decay channel with the criterion of ${p}_{T}^{\mu }\gt$ 25 (20) GeV in the central (forward) acceptance to allow for direct comparisons with the measurements. The bin ranges for the η_μ are identical to those used in the measurements for the central and forward regions, respectively, as provided below

$$\begin{eqnarray}&&\begin{array}{l}\{{\eta }_{\mu }\leqslant 2.4\}=\{(0.00,0.20),(0.20,0.40),(0.40,0.60),(0.60,0.80),(0.80,1.00),\\ (1.00,1.20),(1.20,1.40),(1.40,1.60),(1.60,1.85),(1.85,2.10),(2.10,2.40)\}\\ \quad \mathrm{and}\quad \\ \{2.0\leqslant {\eta }_{\mu }\leqslant 4.5\}=\{(2.00,2.25),(2.25,2.50),(2.50,2.75),(2.75,3.00),\\ (3.00,3.25),(3.25,3.50),(3.50,4.00),(4.00,4.50)\}.\end{array}\end{eqnarray} \tag{ 9 }$$

The NNLO PDF set NNPDF31 LUXqed is exploited in the predictions. Estimates of total theoretical uncertainty in the form of quadratic sum of scale, PDF, and α_s uncertainties are included for the predicted results, while total experimental uncertainty of both statistical and systematic components, which are quadratically summed as well, is quoted for the data points. The predicted ${A}_{{\eta }_{\mu }}$ distributions at NNLO QCD, NNLO QCD+NLO EW, and NNLO QCD × NLO EW are shown in comparisons with the data distributions in figure 1. The combined predictions are observed to exhibit only slight deviations from the fixed-order NNLO QCD in the entire region η_μ ≤ 4.5. The combined predictions generally reproduce the data distributions in the central η_μ region up to a difference that amounts to less than ∼5% within uncertainties. Nevertheless the combined predictions appear to overestimate the data up to ∼5%–6% in the forward η_μ region 2.75–4.0, the theoretical uncertainty is observed to be generally larger in this region as well. Consequently, the combined predictions including NLO EW corrections are justified without the presence of significant deviations from the pp collision data at 8 TeV. Apart from the presented validation, comparisons of the same kind of NNLO QCD predictions with the data are readily available for the ${A}_{{\eta }_{l}}$ distribution in the [61, 62], exploiting several PDF sets without the QED effects. In these references, it has been relevantly shown that the small residual discrepancy with the data can be attributed to the large power corrections. The power-suppressed terms in these corrections, which survive after the elimination of infrared singularities at finite r_cut value, essentially stem from fiducial phase space cuts applied on the decay products.

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The ${A}_{{\eta }_{l}}$ and ${A}_{{y}_{W}}$ distributions by the combined predictions are compared with the NNLO QCD one to assess the impact of the inclusion of NLO EW effects at 13 and 14 TeV of current and future LHC energies. The predictions are provided for the combined e and μ decay channels with ${p}_{T}^{l}\gt$ 25 GeV requirement for the entire detector acceptance comprising both central and forward acceptances as η_l ≤ 4.5. The bin edges are used as the same for the η_l and y_W , which are taken the same as those given in equation (9) in the entire region 0–4.5 except for the modified bin ranges 2.4–2.7, 2.7–3.0, and 3.0–3.5. The ${A}_{{\eta }_{l}}$ distributions are displayed in figure 2 with the comparisons of the combined predictions to the QCD prediction at NNLO. The contribution of NLO EW corrections are at a few percent level for the η_l range 0–3.5 regardless of the type of prescription employed in the combination with respect to the NNLO QCD. The differences among the predictions with and without NLO EW are relatively larger for the η_l range 3.5–4.5, where also uncertainties grow larger. Alternatively the ${A}_{{y}_{W}}$ distributions are provided in figure 3 by noting larger uncertainties towards the central y_W range 0–0.4. Similarly to the ${A}_{{\eta }_{l}}$, the predictions using both types of combination prescription are consistent with each other regarding their estimated uncertainties for the probed y_W range. The differences among the predictions with and without NLO EW are noted throughout the entire y_W region, but they are relatively larger in the central and intermediate region for the ${A}_{{y}_{W}}$ distributions as opposed to the ${A}_{{\eta }_{l}}$ distributions, where the inclusion of NLO EW effects appear to have larger impact.

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The NLO EW effects in the combined (N)NLO QCD+NLO EW and (N)NLO QCD × NLO EW predictions are investigated relative to the (N)NLO QCD predictions at both 13 TeV and 14 TeV. This is targeted to better figure out impact of the included EW corrections for achieving adequate description of the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) distribution. Relative sizes of the (N)NLO QCD+NLO EW and (N)NLO QCD × NLO EW corrections with regards to either NLO or NNLO QCD corrections, denoted by δ(λ), are defined respectively based on the following expressions

$$\begin{eqnarray}&&\begin{array}{l}\delta (\lambda )=\displaystyle \frac{{\lambda }_{\mathrm{NLO}\quad \mathrm{QCD}+\mathrm{NLO}\quad \mathrm{EW}}}{{\lambda }_{\mathrm{NLO}\quad \mathrm{QCD}}}-1\quad \mathrm{and}\quad \delta (\lambda )=\displaystyle \frac{{\lambda }_{{NLO}\quad {QCD}\times {NLO}\quad {EW}}}{{\lambda }_{{NLO}\quad {QCD}}}-1\\ \quad \mathrm{or}\quad \\ \delta (\lambda )=\displaystyle \frac{{\lambda }_{\mathrm{NNLO}\quad \mathrm{QCD}+\mathrm{NLO}\quad \mathrm{EW}}}{{\lambda }_{\mathrm{NNLO}\quad \mathrm{QCD}}}-1\quad \mathrm{and}\quad \delta (\lambda )=\displaystyle \frac{{\lambda }_{{NNLO}\quad {QCD}\times {NLO}\quad {EW}}}{{\lambda }_{{NNLO}\quad {QCD}}}-1,\end{array}\end{eqnarray} \tag{ 10 }$$

where the quantity λ stands for differential distributions of the d σ(W⁺), d σ(W⁻), and ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) in terms of the η_l (y_W ). The relative correction factors in the differential cross sections δ(d σ(W⁺)/d η_l ) and δ(d σ(W⁻)/d η_l ) for the entire η_l region 0–4.5 are displayed in figure 4. The included NLO EW corrections in the combined predictions are negative for both the d σ(W⁺) and d σ(W⁻) distributions throughout the entire η_l region. The sizes of the NLO EW corrections are down to −2% in the η_l range 0–3.0 and noticeably down to −6% in the η_l range 3.0–4.5. The NLO EW effects are generally similar in size between W⁺ and W⁻ boson leptonic decay channels, nevertheless they are not predicted exactly the same for the entire η_l region. The differences emerge in both central and forward η_l regions and amount up to ∼0.5% (∼1.0%) in the η_l range 0–3.0 (3.0–4.5). Apparently, the differences of the EW effects between W⁺ and W⁻ processes are more sizable towards the high-η_l region which should be taken into account in a reliable higher-order prediction. Next up, the relative correction factor $\delta ({A}_{{\eta }_{l}})$ for the ${A}_{{\eta }_{l}}$ distribution is given in figure 5. The NLO EW corrections appear to be minimized by cancellation in some of the bins in the central η_l region, whereas they can exhibit sizable impact up to 2%–3% in the η_l range 0–3.5. They grow larger than ∼4% onwards the very last two η_l bins between 3.5–4.5. Therefore the NLO EW effects should not be disregarded for a thorough description of the ${A}_{{\eta }_{l}}$ distribution as the they can amount to a few percent in the central acceptance region and grow larger in the forward region.

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Additionally, the impact of the NLO EW corrections are also investigated for the d σ(W⁺), d σ(W⁻), and ${A}_{{y}_{W}}$ in terms of the y_W at 14 TeV, as shown in figure 6. The EW effects appear to be negative for the d σ(W⁺), d σ(W⁻) as a function of the y_W . The NLO EW corrections amount to ∼−1.5% in the most central region and decrease continuously up to ∼−5% towards the forward y_W for the d σ(W⁺). Similarly for the d σ(W⁻), the NLO EW corrections have a decreasing trend in the y_W range 0–3.5 from ∼−1.5% to ∼−5%, but they can amount to ∼−3% in the very last two y_W bins. The EW effects are sizable for the ${A}_{{y}_{W}}$ as the correction factor $\delta ({A}_{{y}_{W}})$ can be as large as 4%–5% depending on the y_W range. The EW effects turn out to be minimized by cancellation in the last two y_W bins between 3.5–4.5. By the same token as ${A}_{{\eta }_{l}}$, the NLO EW corrections should rather be accounted for a complete and reliable description of the ${A}_{{y}_{W}}$. As a final remark on this part, the NLO EW corrections for the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) distribution are observed to be comparable between the combined predictions relying on either additive or factorised prescription.

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The sizes of the NLO EW corrections that are included together with the NNLO QCD corrections in the combined predictions of NNLO QCD+NLO EW and NNLO QCD × NLO EW are investigated relative to the LO accuracy for the observables d σ(W⁺), d σ(W⁻), and ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) in bins of the η_l (y_W ). For this purpose the so-called K-factor analysis is carried out to obtain the relative sizes of the NLO EW corrections with regards to the LO at 13 TeV and 14 TeV energies. The K-factor analysis is also considered for the sizes of the NLO QCD corrections with respect to the LO as well as of the NNLO QCD corrections to NLO QCD to enable comparisons with the relative sizes of the NLO EW corrections over the LO. It is worth to state that the (N)NLO predictions are obtained using the (N)NLO NNPDF31 LUXqed PDF set as before, while the LO predictions are obtained using the NLO NNPDF31 LUXqed PDF set as the corresponding PDF set at LO is not available. The K-factors in this regard are defined suitably as

$$\begin{eqnarray}&&{K}_{\mathrm{NLO}\,\mathrm{EW}}=\displaystyle \frac{{\lambda }_{\mathrm{NLO}\,\mathrm{EW}}}{{\lambda }_{\mathrm{LO}}}\quad \mathrm{and}\quad {\text{}}{K}_{({\rm{N}})\mathrm{NLO}\,\mathrm{QCD}}=\displaystyle \frac{{\lambda }_{({\rm{N}})\mathrm{NLO}\,\mathrm{QCD}}}{{\lambda }_{({\rm{N}})\mathrm{LO}\,\mathrm{QCD}}},\end{eqnarray} \tag{ 11 }$$

where λ is again used for the d σ(W⁺), d σ(W⁻), and ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$), which are predicted as a function of the η_l (y_W ). The predicted factors for K_{NLO EW}, K _{NLO QCD}, and K_{NNLO QCD} are provided in bins of the η_l at 13 (14) TeV in table 1 (table 2). K_{NLO EW} factors differ by up to 1%–2% between the d σ(W⁺) and d σ(W⁻) at both 13 TeV and 14 TeV. Although they are generally around unity for the ${A}_{{\eta }_{l}}$ within 1%–2% in the η_l range 0–3.0, the NLO EW corrections have impact equal to or larger than 3% for the η_l range between 3.0–4.5. The NLO corrections in EW are predicted to impact the ${A}_{{\eta }_{l}}$ distribution more in the very forward η_l region, where they require proper inclusion. As anticipated K_{NLO  QCD} factors are quite more sizable than K_{NLO EW} ones for the d σ(W⁺) and d σ(W⁻) in the entire η_l region. The NLO corrections in QCD appear to impact the ${A}_{{\eta }_{l}}$ in the range up to 12%, depending on aparticular η_l bin. Moreover, K_{NNLO QCD} factors differ by up to ∼5%–6% between the d σ(W⁺) and d σ(W⁻), which cause considerably large impact on the ${A}_{{\eta }_{l}}$ throughout the entire η_l region. Conclusively, the higher-order predictions are unavoidably required to account for the NLO EW effects for the ${A}_{{\eta }_{l}}$, more importantly in the very forward η_l region, even though the impact induced by the NLO EW corrections over LO is less sizable as opposed to the ones by the (N)NLO QCD effects. The performed K-factor analysis also reveals that K_{NLO EW} factors exhibit a stable pattern for the ${A}_{{\eta }_{l}}$, i.e., do not vary much for the entire η_l region as opposed to K_{NLO QCD} and K_{NNLO QCD} factors. Therefore, implementation of the NLO EW effects for the ${A}_{{\eta }_{l}}$ distribution in Monte Carlo-based or higher-order theoretical predictions could be more straightforwardly handled.

Table 1.  K-factors for the NLO EW corrections relative to LO (K_{NLO EW}), NLO QCD corrections relative to LO (K_{NLO QCD}), and NNLO QCD corrections relative to NLO QCD (K_{NNLO QCD}), predicted using equation (11). K-factors are provided at 13 TeV for the observables d σ(W⁺), d σ(W⁻), and ${A}_{{\eta }_{l}}$ in bins of the η_l .

K-factor	KNLO EW			KNLO QCD			KNNLO QCD
ηl bin	d σ(W+)	d σ(W−)	${A}_{{\eta }_{l}}$	d σ(W+)	d σ(W−)	${A}_{{\eta }_{l}}$	d σ(W+)	d σ(W−)	${A}_{{\eta }_{l}}$
0.00–0.20	0.98	0.98	1.00	1.18	1.16	1.10	1.00	0.95	1.31
0.20–0.40	0.98	0.98	1.00	1.17	1.17	1.00	0.99	0.99	0.98
0.40–0.60	0.98	0.98	0.98	1.17	1.17	0.97	1.00	0.96	1.19
0.60–0.80	0.98	0.98	0.99	1.17	1.16	1.02	1.04	1.02	1.10
0.80–1.00	0.98	0.99	0.99	1.16	1.18	0.92	0.97	1.03	0.98
1.00–1.20	0.98	0.98	0.99	1.19	1.16	1.12	1.00	0.99	1.04
1.20–1.40	0.98	0.98	0.99	1.16	1.17	0.97	0.94	1.00	0.93
1.40–1.60	0.98	0.99	0.99	1.16	1.18	0.93	1.00	0.96	1.13
1.60–1.85	0.98	0.98	1.00	1.16	1.17	0.98	0.96	0.97	0.98
1.85–2.10	0.98	0.98	1.00	1.18	1.18	1.00	0.96	0.96	1.01
2.10–2.40	0.98	0.98	1.00	1.15	1.17	0.97	0.95	1.00	0.88
2.40–2.70	0.99	0.98	1.00	1.18	1.15	1.04	1.00	1.01	0.97
2.70–3.00	0.99	0.99	1.00	1.14	1.18	0.94	0.97	1.02	0.90
3.00–3.50	0.95	0.94	1.03	1.16	1.16	0.99	0.96	0.98	0.96
3.50–4.00	0.95	0.94	1.05	1.17	1.14	1.12	0.95	0.94	1.02
4.00–4.50	0.96	0.94	0.93	1.17	1.15	0.92	0.98	0.98	0.98

Table 2.  K-factors at 14 TeV, predicted using equation (11), following the same description as table 1.

K-factor	KNLO EW			KNLO QCD			KNNLO QCD
ηl bin	d σ(W+)	d σ(W−)	${A}_{{\eta }_{l}}$	d σ(W+)	d σ(W−)	${A}_{{\eta }_{l}}$	d σ(W+)	d σ(W−)	${A}_{{\eta }_{l}}$
0.00–0.20	0.98	0.98	1.01	1.17	1.17	1.04	1.00	0.93	1.34
0.20–0.40	0.98	0.99	0.98	1.17	1.17	0.99	0.98	0.97	1.10
0.40–0.60	0.98	0.98	1.03	1.16	1.17	0.95	1.01	0.98	1.16
0.60–0.80	0.98	0.99	0.99	1.17	1.17	1.02	1.03	1.02	1.04
0.80–1.00	0.98	0.99	0.98	1.16	1.16	1.02	1.03	1.02	1.08
1.00–1.20	0.98	0.99	1.00	1.17	1.17	0.99	0.93	0.99	0.91
1.20–1.40	0.98	0.98	1.01	1.18	1.18	0.99	0.97	1.00	0.88
1.40–1.60	0.98	0.99	0.99	1.18	1.18	1.00	0.95	0.95	1.10
1.60–1.85	0.99	0.98	1.01	1.16	1.16	0.98	0.96	1.00	0.87
1.85–2.10	0.98	0.99	0.98	1.17	1.17	1.01	0.97	0.93	1.12
2.10–2.40	0.98	0.98	1.00	1.17	1.17	1.00	0.96	1.00	0.90
2.40–2.70	0.98	0.98	1.00	1.15	1.16	0.98	0.98	1.01	0.92
2.70–3.00	0.98	0.98	1.00	1.16	1.17	0.97	1.00	0.97	1.05
3.00–3.50	0.95	0.94	1.03	1.16	1.17	0.98	1.00	1.00	1.00
3.50–4.00	0.95	0.94	1.07	1.15	1.14	1.07	0.94	0.97	0.90
4.00–4.50	0.96	0.94	0.90	1.16	1.13	0.86	0.97	0.93	0.84

In addition to the K-factors as a function of the η_l , K_{NLO EW}, K_{NLO QCD}, and K_{NNLO QCD} are investigated for the d σ(W⁺), d σ(W⁻), and ${A}_{{y}_{W}}$ as a function of the y_W at 13 (14) TeV in table 3 (table 4). The analysis of K-factors in bins of the y_W is used for enabling comparisons with the presented K-factors in bins of the η_l , which are more relevant for actual experimental measurements. The K_{NLO EW} factors for the d σ(W⁺) and d σ(W⁻) in bins of the y_W differ only by up to 1%–2% at both 13 TeV and 14 TeV, as also observed for the ones in bins of the η_l . The impact of the NLO EW effects over the LO on the ${A}_{{y}_{W}}$ is as large as 3%–4% for the entire y_W region. More significantly, the ${A}_{{y}_{W}}$ is observed to be less sensitive to the NLO EW effects than the ${A}_{{\eta }_{l}}$ in the very forward y_W region 3.0–4.5, in particular K_{NLO EW} factors turn out to be unity in the last two y_W bins. The impact of the inclusion of the (N)NLO corrections in QCD in bins of the y_W follow a similar pattern for the d σ(W⁺), d σ(W⁻), and ${A}_{{y}_{W}}$ with the ones in bins of the η_l apart from a few exceptions. Such as, the K_{NNLO QCD} factors vary in a wider range for the ${A}_{{y}_{W}}$ than the ${A}_{{\eta }_{l}}$. Moreover, K_{NLO QCD} and K_{NNLO QCD} factors do not reveal large sensitivity to (N)NLO QCD effects for the ${A}_{{y}_{W}}$ in the forward y_W region 3.0–4.5 in addition to K_{NLO EW} factors, which is in contrast with the observation of the corresponding K-factors as discussed for the ${A}_{{\eta }_{l}}$ in the forward η_l region.

Table 3.  K-factors for the NLO EW corrections relative to LO (K_{NLO EW}), NLO QCD corrections relative to LO (K_{NLO QCD}), and NNLO QCD corrections relative to NLO QCD (K_{NNLO QCD}), predicted using equation (11). K-factors are provided at 13 TeV for the observables d σ(W⁺), d σ(W⁻), and ${A}_{{y}_{W}}$ in bins of the y_W .

K-factor	KNLO EW			KNLO QCD			KNNLO QCD
yW bin	d σ(W+)	d σ(W−)	${A}_{{y}_{W}}$	d σ(W+)	d σ(W−)	${A}_{{y}_{W}}$	d σ(W+)	d σ(W−)	${A}_{{y}_{W}}$
0.00–0.20	0.98	0.98	0.99	1.17	1.16	1.17	0.93	1.06	1.21
0.20–0.40	0.98	0.98	1.01	1.16	1.16	1.01	0.97	0.97	1.00
0.40–0.60	0.98	0.99	0.97	1.18	1.17	1.07	0.99	0.99	1.00
0.60–0.80	0.98	0.98	0.98	1.17	1.16	1.14	1.04	0.96	1.79
0.80–1.00	0.98	0.98	1.01	1.17	1.16	1.08	0.96	0.94	1.19
1.00–1.20	0.98	0.98	1.01	1.17	1.15	1.18	1.04	0.95	2.01
1.20–1.40	0.98	0.98	1.00	1.18	1.18	0.99	0.99	1.03	0.84
1.40–1.60	0.98	0.98	1.02	1.17	1.16	1.09	1.01	1.00	1.08
1.60–1.85	0.98	0.99	0.99	1.17	1.18	0.92	0.98	1.03	0.87
1.85–2.10	0.98	0.98	1.00	1.18	1.16	1.05	0.97	0.93	1.18
2.10–2.40	0.98	0.97	1.03	1.17	1.17	1.01	0.95	0.99	0.84
2.40–2.70	0.98	0.97	1.03	1.17	1.16	1.01	0.93	0.98	0.83
2.70–3.00	0.97	0.96	1.03	1.18	1.18	1.01	0.97	0.99	0.93
3.00–3.50	0.97	0.95	1.04	1.15	1.16	0.98	0.99	0.96	1.05
3.50–4.00	0.96	0.96	1.00	1.14	1.16	0.99	0.97	0.97	1.00
4.00–4.50	0.96	0.97	1.00	1.13	1.17	0.99	0.91	0.96	0.98

Table 4.  K-factors at 14 TeV, predicted using equation (11), following the same description as table 3.

K-factor	KNLO EW			KNLO QCD			KNNLO QCD
yW bin	d σ(W+)	d σ(W−)	${A}_{{y}_{W}}$	d σ(W+)	d σ(W−)	${A}_{{y}_{W}}$	d σ(W+)	d σ(W−)	${A}_{{y}_{W}}$
0.00–0.20	0.98	0.99	0.97	1.17	1.16	1.08	0.94	0.99	0.77
0.20–0.40	0.99	0.98	1.03	1.17	1.17	1.01	0.93	0.98	0.79
0.40–0.60	0.99	0.98	1.02	1.17	1.16	1.18	1.00	0.97	1.39
0.60–0.80	0.98	0.99	0.98	1.16	1.15	1.07	0.99	0.98	1.07
0.80–1.00	0.98	0.98	1.01	1.17	1.17	1.09	1.03	0.96	1.80
1.00–1.20	0.98	0.98	1.00	1.17	1.16	1.02	1.03	0.93	2.05
1.20–1.40	0.98	0.98	1.03	1.19	1.16	1.16	1.00	1.02	0.85
1.40–1.60	0.98	0.98	1.01	1.17	1.17	1.02	1.03	0.98	1.37
1.60–1.85	0.98	0.98	1.00	1.17	1.18	0.97	1.00	1.02	0.90
1.85–2.10	0.98	0.98	1.02	1.17	1.16	1.03	0.97	0.92	1.27
2.10–2.40	0.98	0.98	1.01	1.18	1.17	1.05	0.98	1.00	0.88
2.40–2.70	0.98	0.97	1.04	1.17	1.18	0.98	0.93	1.04	0.73
2.70–3.00	0.98	0.96	1.04	1.16	1.17	0.98	0.96	0.99	0.91
3.00–3.50	0.97	0.95	1.04	1.16	1.15	1.01	1.01	0.93	1.14
3.50–4.00	0.96	0.96	1.00	1.13	1.15	0.98	0.95	1.00	0.95
4.00–4.50	0.95	0.93	1.01	1.13	1.19	0.98	0.93	0.84	1.03

In the reported work, the charge asymmetry ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) predictions are presented in terms of the η_l (y_W ) for the leptonic final state of the W-boson hadroproduction in pp collisions. The predictions are presented from the computations of both NNLO QCD and NLO EW radiative corrections to achieve adequate theoretical description. The combined predictions NNLO QCD+NLO EW and NNLO QCD × NLO EW, ensuing standard additive and factorised combination prescriptions in turn, for the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) distributions are of importance in the domain of higher-precision studies which aim to reach the precision at or even less than percent level as required by present and future experimental results. The combined predictions for the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) provide significant inputs which can primarily be transferred to the knowledge of precise determination of the relative u- and d-quark density functions in the proton. In addition, the presented combined predictions are substantially necessary for the relevant field of collider phenomenology to have explicit accounting for the NLO EW effects for the observable ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$).

The combined NNLO QCD and NLO EW predictions are first compared with the genuine NNLO QCD predictions along with the actual LHC measurements at 8 TeV pp collisions energy. The combined predictions are thoroughly justified with the measurements in both the central η_l ≤ 2.4 and forward 2.0 ≤ η_l ≤ 4.5 acceptance regions apart from a few deviations which generally amount to less than ∼5%. The combined predictions are also compared with the NNLO QCD predictions at both 13 and 14 TeV energies for the entire acceptance η_l ≤ 4.5 to assess the impact of the included EW corrections at NLO. The NLO EW effects are shown to result in sizable impact for the ${A}_{{\eta }_{l}}$ distribution in the very forward η_l region ∼3.0–4.5 and for the ${A}_{{y}_{W}}$ distribution in almost through the entire low- and intermediate-y_W region. The differences between the combined predictions NNLO QCD+NLO EW and NNLO QCD × NLO EW are found to be small, i.e., at the level of a few per mille. Nevertheless, comparisons of the predictions based on additive and factorised approaches are essentially useful to anticipate the impact of mixed QCD-EW effects, which are literally known to be missing in the additive approach. Owing to the small difference from the checks on the comparisons, no systematic uncertainty due to missing mixed QCD-EW effects is included for the NNLO QCD+NLO EW predictions of the charge asymmetry.

The impact of the EW corrections at NLO in the combined predictions is extensively studied for the differential cross sections d σ(W⁺) and d σ(W⁻), in addition to the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) by means of the relative NLO EW correction factors with regards to the NNLO predictions in QCD. The relative correction factors are shown to be −2% (−6%) in the η_l range 0–3.0 (3.0–4.5) for the d σ(W⁺) and d σ(W⁻), where their differences are observed to be up to ∼1.0% in the high-η_l region 3.0–4.5 between W⁺- and W⁻-boson processes. The NLO EW effects exhibit sizable impact on the ${A}_{{\eta }_{l}}$ distribution as the relative correction factors turn out to be up to 2%–3% in the η_l range 0–3.5 and more than ∼4% in the highest two η_l bins 3.5–4.5. The relative correction factors are also investigated for the d σ(W⁺), d σ(W⁻), and ${A}_{{y}_{W}}$ as a function of the y_W . They are explicitly shown to be ∼-1.5% in the most central y_W acceptance, while decreasing up to ∼−5% in the forward y_W acceptance for the d σ(W⁺) and d σ(W⁻). The impact of the NLO EW effects on the ${A}_{{y}_{W}}$ is found to amount up to 4%–5% depending on the y_W range. These observations are in support of requirement of proper accounting for the NLO corrections in EW in theoretical predictions to achieve a complete and reliable description of the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) distribution. In addition, a comprehensive K-factor analysis is performed to further assess the relative sizes of the NLO corrections in EW with regards to the LO accuracy in comparisons with those of the (N)NLO QCD corrections. The NLO EW K-factors for the ${A}_{{\eta }_{l}}$ are predicted to vary in the range 1%–2% in the η_l region 0–3.0, whereas they are 3% or even more in the η_l region 3.0–4.5. The K-factor analysis conclusively revealed that the forward η_l region 3.0–4.5 is more prone to the requirement of the inclusion of the NLO corrections in EW in ${A}_{{\eta }_{l}}$ predictions. The NLO EW K-factors over the LO for the ${A}_{{y}_{W}}$ are investigated to be as large as 3%–4% in some of the y_W ranges. More significantly, ${A}_{{y}_{W}}$ is shown to be less sensitive to the NLO EW effects in the very forward region 3.0–4.5 contrary to the ${A}_{{\eta }_{l}}$, by the K-factor analysis over the LO. The presented NLO EW K-factors can be considered in the related phenomenological studies of the ${A}_{{\eta }_{l}}$ (${A}_{{y}_{W}}$) for properly correcting for the NLO EW effects.

The data generated and/or analysed during the current study are not publicly available for legal/ethical reasons but are available from the corresponding author on reasonable request.

The author declares no conflict of interest.