∎

11institutetext: K. Ocalan 22institutetext: Necmettin Erbakan University,
Faculty of Aviation and Space Sciences, Konya, Turkey
ORCID ID: http://orcid.org/0000-0002-8419-1400
22email<EMAIL_ADDRESS>
# NNLO QCD predictions of the electron charge asymmetry for the inclusive
$pp\rightarrow W+X$ production in the forward region at 13 TeV and 14 TeV

Kadir Ocalan

(Received: date / Accepted: date)

###### Abstract

This paper presents perturbative QCD predictions of the electron charge
asymmetry for the inclusive $W^{\pm}+X\rightarrow e^{\pm}\nu+X$ production in
proton-proton (pp) collisions. Perturbative QCD calculations are performed at
next-to-next-to-leading order (NNLO) accuracy by using different parton
distribution function (PDF) models at 8 TeV, 13 TeV, and 14 TeV center-of-mass
energies of the CERN LHC pp collisions. The NNLO calculations are performed
for the electrons to have transverse momentum more than 20 GeV in the forward
electron pseudorapidity region $2.0\leq\eta_{e}\leq 4.25$. The NNLO
predictions are first compared at 8 TeV with the measurement of the LHCb
experiment at the LHC for the W+/W- cross section ratio and charge asymmetry
distributions. The 8 TeV predictions using NNPDF3.1, CT14, and MMHT2014 PDF
sets are reported to be in good agreement with the LHCb data for the entire
$\eta_{e}$ region justifying the extension of the calculations to 13 TeV and
14 TeV energies. The charge asymmetry predictions at NNLO accuracy are also
reported in the forward $\eta_{e}$ bins at 13 TeV and 14 TeV and compared
among NNPDF3.1, CT14, and MMHT2014 PDF sets. Overall the predicted W±
differential cross section and charge asymmetry distributions based on
different PDF sets are found to be consistent with each other for the entire
$\eta_{e}$ region. The charge asymmetry distributions are shown to be more
sensitive to discriminate among different PDF models in the 14 TeV
predictions.

###### Keywords:

High Energy Physics phenomenology Perturbative QCD calculations W bosons W
boson charge asymmetry

###### pacs:

12.38.-t 14.70.Fm 13.85.Qk

##  1 Introduction

Production of W and Z bosons has been an important benchmark process to
measure at past colliders and in proton-proton (pp) collisions at the CERN
Large Hadron Collider (LHC). Precise measurements of their production cross
sections provide important tests for the quantum chromodynamic (QCD) and
electroweak (EW) sectors of the Standard Model (SM) and valuable inputs to
constrain parton distribution functions (PDFs) in the proton. Their
measurements enable improvements in background modeling for several rarer SM
and beyond the SM processes as W and Z boson productions constitute a major
background for those processes. In addition, W and Z boson processes are
extensively used for calibrating detector response to improve reconstruction
of leptons, jet, and missing energy signature with better performances. W and
Z bosons are produced in abundance in their leptonic decay channels with
larger cross sections and clean experimental signatures in pp collisions.

Experimentally, W bosons are reconstructed through their leptonic decay
channels $pp\rightarrow W^{\pm}\rightarrow l^{\pm}\nu$. A typical W boson
event is characterized by one isolated charged lepton with high transverse
momentum $p_{T}$ and large missing transverse energy due to the neutrino. W
bosons are produced primarily through the annihilation of a valence quark from
one of the colliding protons with a sea antiquark from the other:
$u\bar{d}\rightarrow W^{+}$, $d\bar{u}\rightarrow W^{-}$. W+ bosons are
produced more often than W- bosons due to the presence of two valence $u$
quarks in the proton. This leads to a production asymmetry between W+ and W-
bosons, which is referred to as W boson charge asymmetry and is usually
defined in terms of cross sections $\sigma(W^{+})$ and $\sigma(W^{-})$
differential in the W boson rapidity $y_{W}$ as

$A(y_{W})=\frac{d\sigma(W^{+})/dy_{W}-d\sigma(W^{-})/dy_{W}}{d\sigma(W^{+})/dy_{W}+d\sigma(W^{-})/dy_{W}}.$
(1)

The important capability of this charge asymmetry variable $A(y_{W})$ is to
discriminate between PDF models as $y_{W}$ is strongly correlated with the
initial-state parton momentum fractions $x$ (Bjorken-$x$ values). However, the
neutrino from the W boson decay leaves the detector unobserved and its
longitudinal momentum cannot be measured directly. For this reason, the full
momentum of the W boson and its rapidity $y_{W}$ cannot be directly
reconstructed. In spite of this experimental complication, the same
information can be accessed by measuring the charge asymmetry from the W boson
decay products. A commonly used approach is to measure the charge asymmetry as
a function of the decay lepton (electron111”Electron” refers to both $e^{+}$
and $e^{-}$ generically throughout the entire paper., in this case)
pseudorapidity222Pseudorapidity is defined as $\eta=-ln[(\theta/2)]$, where
$\theta$ is a polar angle relative to the beam axis. $\eta_{e}$, which is
strongly correlated with the $y_{W}$, defined as

$A_{e}=\frac{d\sigma(W^{+}\rightarrow
e^{+}\nu)/d\eta_{e}-d\sigma(W^{-}\rightarrow
e^{-}\nu)/d\eta_{e}}{d\sigma(W^{+}\rightarrow
e^{+}\nu)/d\eta_{e}+d\sigma(W^{-}\rightarrow e^{-}\nu)/d\eta_{e}}.$ (2)

The electron charge asymmetry $A_{e}$ variable provides significant
constraints on the ratio of $u$ and $d$ quark distributions in the proton as a
function of Bjorken-$x$ values of the partons. It can be used to discriminate
among PDF models that predict different shapes of valence and sea quark
distributions. In addition, this variable can be measured with high accuracy
as because of many sources of systematic uncertainties cancel in the ratio,
and therefore offers a unique opportunity for precision tests of the SM
physics.

The W boson production asymmetry was previously measured in p$\rm{\bar{p}}$
collisions by the CDF and D0 Collaborations at the Tevatron Abe:1998rv ;
Abazov:2007pm ; Abazov:2008qv ; Aaltonen:2009ta ; Abazov:2013rja . W boson
lepton charge asymmetry measurements at the LHC were performed in the central
lepton pseudorapidity region $|\eta_{l}|\leq$ 2.5 by the ATLAS and CMS
Collaborations at different center-of-mass energies up to 8 TeV Aad:2011dm ;
Chatrchyan:2012xt ; Chatrchyan:2013mza ; Khachatryan:2016pev ; Aaboud:2016btc
; Aad:2019bdc ; Aaboud:2018nic ; Aad:2019rou . The LHCb Collaboration at the
LHC has also reported W boson lepton charge asymmetry measurements at 7 TeV
Aaij:2012vn ; Aaij:2014wba  and 8 TeV Aaij:2015zlq ; Aaij:2016qqz  in the
forward lepton pseudorapidity region 2.0 $\leq\eta_{l}\leq$ 4.5 that go beyond
the ATLAS and CMS acceptance. In all these complementary results, measurements
are compared with the theoretical predictions up to next-to-next-to-leading
order (NNLO) accuracy in perturbative QCD convolved with different PDF models.
NNLO predictions were obtained either by using FEWZ Li:2012wna  or DYNNLO
Catani:2009sm  program for the theoretical comparisons in these measurements.
Among all the experimental results available, the measurements of the
differential cross sections and thus production charge asymmetries are of
particular importance in the forward region of the detector acceptance. The
PDFs exhibit significantly large uncertainties at very low and large $x$
values of the interacting leptons, where $x$ values depend also on the
acceptance defined by means of the lepton $\eta_{l}$. Therefore, measurements
and theoretical predictions of the W boson production asymmetry in the forward
detector acceptance 2.0 $\leq\eta_{l}\leq$ 4.5 offer a unique situation to
provide valuable inputs on determining accurate PDFs at small and large $x$
values between $10^{-4}\leq x\leq 10^{-1}$.

In this work we present precise predictions of the electron charge asymmetry
between the processes $pp\rightarrow W^{+}+X\rightarrow e^{+}\nu_{e}+X$ and
$pp\rightarrow W^{-}+X\rightarrow e^{-}\bar{\nu_{e}}+X$ in the forward
$\eta_{e}$ region 2.0 $\leq\eta_{e}\leq$ 4.25. The calculations are performed
at NNLO accuracy in the perturbative QCD expansion by using different PDF
models at 8 TeV, 13 TeV, and 14 TeV LHC pp collision energies. The predicted
W+/W- cross section ratio and charge asymmetry distributions as a function of
the $\eta_{e}$ are compared with the 8 TeV measurement by the LHCb
Collaboration Aaij:2016qqz . The 8 TeV predictions are validated with the data
in the fiducial phase space of the LHCb measurement. Further, the NNLO
calculations are extended to the 13 TeV and 14 TeV LHC pp energies for the
precise predictions of the charge asymmetry in bins of the $\eta_{e}$. The W+
and W- boson differential cross distributions along with their W+/W- ratios
and the charge asymmetry distributions are predicted and compared among
different PDF sets. The charge asymmetry results are then reported and
compared among different PDF sets. The results reported in this paper
represent the first phenomenological study by means of the NNLO predictions
for the electron charge asymmetry in the forward $\eta_{e}$ region 2.0
$\leq\eta_{e}\leq$ 4.25 of 13 TeV and 14 TeV pp collisions.

##  2 Methodology

###  2.1 Computational setup

The calculations of the differential cross sections and charge asymmetries are
performed by using the _MATRIX_ (v1.0.3) computational framework
Grazzini:2017mhc ; Catani:2009sm . The framework enables calculations of
differential cross sections in the form of binned distributions up to NNLO
accuracy in perturbative QCD. The so-called transverse momentum _$q_{T}$_
-subtraction method Catani:2007vq ; Catani:2012qa  is employed within the
_MATRIX_ computations for the cancellations of infrared divergences that arise
at the intermediate stages of the calculations. These divergences are
regulated by introducing a fixed cut-off value $r_{cut}=$ 0.0015 (0.15%) for
the residual dependence parameter which is defined as $r=p_{T}/m$ in terms of
the $p_{T}$ distribution and invariant mass $m$ for a system of colorless
particles. Moreover, all the spin- and color-correlated tree-level and one-
loop scattering amplitudes are acquired with the use of the _OpenLoops_ tool
Matsuura:1988sm ; Cascioli:2011va ; Denner:2016kdg  along with the
computations. Further to perform theoretical calculations of the differential
cross sections in pp collisions, knowledge of the PDFs is required. The
_LHAPDF 6.2.0_ framework Buckley:2014ana  is used for the evaluation of PDFs
from data files in the computations. Different PDF sets are used in the
calculations, all based on a constant strong coupling $\alpha_{s}=$ 0.118. The
_NNPDF3.1_ Ball:2014uwa , _CT14_ Dulat:2015mca , and _MMHT2014_ Harland-
Lang:2014zoa  PDF sets are used each at NNLO accuracy. To this end, all
predicted results are obtained by treating leptons and light quarks massless
in the Fermi constant $G_{F}$ input scheme of the computational setup. The
default setup of the _MATRIX_ framework is used for the input scheme which
encompasses the choices for the relevant SM parameters that are based on
$G_{F}$ = 1.16639 x 10-5 GeV-2 value.

###  2.2 Fiducial acceptance

The 8 TeV LHCb measurement that was performed in pp collisions at 8 TeV
Aaij:2016qqz  is chosen to be the reference for the justification of the
calculations of this paper. This reference measurement focuses on the
differential cross sections and charge asymmetries in the electron decay
channel of the W± bosons in the forward detector acceptance as well. The
fiducial phase space requirements in the calculations are employed to be in
line with the reference LHCb measurement. The electrons are required to have
transverse momentum $p_{T}>$ 20 GeV and to lie in the forward pseudorapidity
region of $2.0\leq\eta_{e}\leq 4.25$. There is no need to impose any
requirements for the final state hadronic jets as the calculations are
considered for the inclusive W boson production. The fiducial acceptance is
also relaxed to have no explicit requirement for the missing transverse
momentum of the final state (anti)neutrino. Furthermore, no requirement for
the W boson transverse mass is imposed unlike to the ATLAS and CMS W boson
charge asymmetry measurements performed in the central $\eta_{l}$ detector
coverage.

###  2.3 Theoretical uncertainties

Cross section calculations in perturbative QCD acquire dependence on the
renormalization $\mu_{R}$ and factorization $\mu_{F}$ scales and the numerical
results will depend on the choice of these scales. In this paper, the central
value for the scales is chosen to be the W boson mass $\mu_{R}$ = $\mu_{F}$ =
$m(W)$ = 80.385 GeV in the usual way. The theoretical uncertainties due to the
choice of the scales, which are simply called as scale uncertainties,
referring to missing higher-order contributions in the calculations are
estimated by varying independently the $\mu_{R}$ and $\mu_{F}$ up and down by
a factor of 2 around the central value. All possible combinations are
considered in the variations while imposing the constraint
$0.5\leq\mu_{R}/\mu_{F}\leq 2.0$. In the cross section ratio and charge
asymmetry calculations, this constraint is generalized to an uncorrelated
scale variations while restricting to $0.5\leq\mu/\mu^{{}^{\prime}}\leq 2.0$
between all pairs of scales. Since the scale uncertainties arise from the
missing terms in the calculations, the higher the calculated perturbative
order, the smaller the theoretical uncertainties associated with the
renormalization and factorization procedure will be. Moreover, PDF and
$\alpha_{s}$ uncertainties are also considered. The PDF uncertainties for
NNPDF3.1, CT14, and MMHT2014 PDF sets are estimated by using the prescription
of the PDF4LHC working group Butterworth:2015oua ; Buckley:2014ana . The
$\alpha_{s}$ uncertainty is estimated by varying the $\alpha_{s}$ value by
$\pm 0.001$ around 0.118. Estimated theoretical uncertainties for the 8 TeV
inclusive cross sections of the W+ and W- boson processes at NNLO accuracy are
tabulated in Table 1 as an example to compare relative size of each type of
theoretical uncertainty considered in this paper. Total theoretical
uncertainties are obtained by summing scale, PDF, and $\alpha_{s}$
uncertainties in quadrature. Then, total uncertainties are symmetrized by
taking the bigger values from estimated up and down uncertainties in a
conservative approach.

Table 1: The estimated sizes of the scale, PDF, and $\alpha_{s}$ uncertainties for the 8 TeV inclusive cross section predictions of the W+ and W- bosons at NNLO accuracy. Estimated uncertainties are given as percentage (%) of the central values of the predictions from NNPDF3.1, CT14, and MMHT2014 PDF sets. Uncertainty | NNPDF3.1 | CT14 | MMHT2014
---|---|---|---
Values for $W^{+}\rightarrow e^{+}\nu$ process
Scale (%) | 0.74 | 0.76 | 0.78
PDF (%) | 1.96 | 2.40 | 1.64
$\alpha_{s}$ (%) | 1.06 | 1.04 | 1.10
Total (%) | 2.35 | 2.72 | 2.12
Values for $W^{-}\rightarrow e^{-}\nu$ process
Scale (%) | 0.72 | 0.64 | 0.80
PDF (%) | 2.22 | 2.90 | 1.50
$\alpha_{s}$ (%) | 1.16 | 1.00 | 1.14
Total (%) | 2.61 | 3.13 | 2.05

##  3 Phenomenological results at 8 TeV

The NNLO calculations and their comparisons with the reference LHCb results at
8 TeV Aaij:2016qqz  are reported in this section. The calculations are
performed in the fiducial phase space as discussed in Sec. 2.2. W+/W-
differential cross section ratio $R_{W^{\pm}}$, where

$R_{W^{\pm}}=\frac{d\sigma(W^{+}\rightarrow
e^{+}\nu)/d\eta_{e}}{d\sigma(W^{-}\rightarrow e^{-}\nu)/d\eta_{e}},$ (3)

and electron charge asymmetry $A_{e}$ (Eq. 2) variables as a function of
$\eta_{e}$ are used in justification of the calculations. The $R_{W^{\pm}}$
and $A_{e}$ variables are calculated in the $\eta_{e}$ bins of (2.0, 2.25),
(2.25, 2.5), (2.5, 2.75), (2.75, 3.0), (3.0, 3.25), (3.25, 3.5), (3.5, 3.75),
(3.75, 4.25) in line with the 8 TeV LHCb paper. The NNPDF3.1, CT14, and
MMHT2014 NNLO PDF sets are used in the calculations. Total theoretical
uncertainties that are estimated as discussed in Sec. 2.3 are propagated to
the $R_{W^{\pm}}$ and $A_{e}$ calculations. The LHCb data results that were
obtained at Born level, do not incorporate the effect of quantum
electrodynamics final-state radiation, are used to enable direct comparisons
to the calculations. The data central results in the comparisons are used
together with their total experimental uncertainties that are obtained by
adding statistical, systematic, and the LHC beam energy uncertainties in
quadrature.

The predicted $R_{W^{\pm}}$ distributions based on different PDF sets are
compared with the data in Fig. 1. The $R_{W^{\pm}}$ predictions at NNLO
accuracy using different PDF sets are generally in good agreement with each
other and with the data within uncertainties through the entire $\eta_{e}$
bins. The prediction using MMHT2014 PDF set provides the best agreement with
the data among other predictions except for the 3.0–3.25 bin where it
overestimates the data by up to 5%. The predictions using NNPDF3.1 and CT14
overestimate the data in 3.5–3.75 bin by up to 10%, while the prediction using
MMHT2014 is in quite good agreement with the data in this bin. The predictions
using different PDF sets are able to describe the data well in the very
forward $\eta_{e}$ bin 3.75–4.25. It is worth to notice that the FEWZ
predictions provided in the LHCb paper tend to underestimate the $R_{W^{\pm}}$
data distribution in the far forward bins 3.5–3.75 and 3.75–4.25. The
discrepancy between the _MATRIX_ and FEWZ predictions is checked and
attributed mainly to the residual dependence of the $r_{cut}$ = 0.15% value of
the _$q_{T}$_ -subtraction method in the far forward region. The _MATRIX_
framework differs by the _$q_{T}$_ -subtraction method from the FEWZ which is
based on sector decomposition Binoth:2000ps . This residual dependence is due
to power-suppressed terms that remain after the subtraction of infrared
singular contribution at finite $r_{cut}$ value. The impact of the choice of
$r_{cut}$ = 0.15% is checked with respect to a lower cut-off value 0.05%,
which amounts to $\sim$2–7% difference. This difference shows that the power
corrections due to finite $r_{cut}$ = 0.15% value are large in this region
which can only vanish in the limit $r_{cut}\rightarrow$ 0\. A systematic
uncertainty up to 4% is propagated to the $R_{W^{\pm}}$ predictions in the
last two bins from the extrapolation $r_{cut}\rightarrow$ 0 to account for
this dependency to the $r_{cut}$ value by means of the large power
corrections. In addition, the impact of the difference in the input parameter
choices of the _MATRIX_ and the FEWZ default setup Li:2012wna  is checked to
be 1–2%. Together with these checks, the NNLO calculations in terms of the
predicted $R_{W^{\pm}}$ distributions are therefore justified quite well by
using the LHCb results.

Figure 1: The differential cross section ratio $R_{W^{\pm}}$ distributions as
a function of the $\eta_{e}$ which are predicted at NNLO accuracy by using
NNPDF3.1, CT14, and MMHT2014 PDF sets and their comparisons with the LHCb data
at 8 TeV. The predictions include total theoretical uncertainties obtained by
adding scale, PDF, and $\alpha_{s}$ uncertainties in quadrature. The data
includes total experimental uncertainty obtained by adding statistical,
systematic, and the LHC beam energy uncertainties in quadrature. In the lower
inset, the ratios of the predictions to the data for the $R_{W^{\pm}}$
variable are provided.

The predictions from different PDF sets and their comparisons with the data at
NNLO accuracy for the $A_{e}$ variable are shown in Fig. 2 and also tabulated
in Table 2. The predictions using different PDF sets generally reproduce the
data well within uncertainties throughout the entire $\eta_{e}$ ranges. The
predictions are also in good agreement with each other and with the data
except for the bins 3.0–3.25 and 3.5–3.75, where the predictions using
NNPDF3.1 and CT14 exhibit lower values as compared to the data and the
prediction using MMHT2014. The discrepancy between the data and the
predictions using NNPDF3.1 and CT14 are up to $\sim$15% in the bins 3.0–3.25
and 3.5–3.75. The best description of the data is achieved with the prediction
using MMHT2014 over the other PDF sets for the entire $\eta_{e}$ ranges. The
predictions using different PDF sets are able to describe the data in the far
forward bins 3.5–3.75 and 3.75–4.25 in comparison to the FEWZ predictions
provided in the LHCb paper, where they slightly tend to underestimate the
data. More generally the trend of increasing discrepancy between the data and
predictions towards the very forward $\eta_{e}$ region as presented in the
LHCb paper is not observed in Fig. 2 for the central values of the
predictions. This discrepancy between the _MATRIX_ and FEWZ predictions for
the description of the data in the far forward bins is also mainly due to the
large power corrections that appear at finite cut-off $r_{cut}$ = 0.15% value
in the MATRIX calculation. The difference of the choice of $r_{cut}$ = 0.15%
relative to a lower cut-off value $r_{cut}$ = 0.05% has been checked to be
$\sim$14–19%. Therefore, a systematic uncertainty up to 5% from the
extrapolation in the limit $r_{cut}\rightarrow$ 0 is also propagated to the
predictions in the last two bins of the $A_{e}$ variable to account for the
effect from the large power corrections at the finite $r_{cut}$ value in the
subtraction method. To this end, the NNLO calculations using different PDF
sets are successfully justified with the LHCb results for also the $A_{e}$
variable at 8 TeV.

Figure 2: The electron charge asymmetry $A_{e}$ distributions as a function of the $\eta_{e}$ which are predicted at NNLO accuracy by using NNPDF3.1, CT14, and MMHT2014 PDF sets and their comparisons with the LHCb data at 8 TeV. The predictions include total theoretical uncertainties while the data include total experimental uncertainty. In the lower inset, the ratios of the predictions to the data for the $A_{e}$ variable are provided. Table 2: The 8 TeV electron charge asymmetry (in percent) $A_{e}$(%) predictions by using NNPDF3.1, CT14, and MMHT2014 PDF sets at NNLO accuracy and their comparisons with the LHCb data in bins of the $\eta_{e}$. The predictions include total theoretical uncertainties while the data include total experimental uncertainty. $\eta_{e}$ | Data | NNPDF | CT | MMHT
---|---|---|---|---
2.00–2.25 |  26.78$\pm$0.8 |  28.37$\pm$0.7 |  29.24$\pm$0.8 |  27.95$\pm$0.6
2.25–2.50 |  26.98$\pm$0.7 |  25.06$\pm$0.6 |  26.14$\pm$0.7 |  27.66$\pm$0.7
2.50–2.75 |  25.84$\pm$0.7 |  25.30$\pm$1.3 |  27.78$\pm$1.2 |  26.53$\pm$1.0
2.75–3.00 |  20.39$\pm$0.8 |  21.85$\pm$0.8 |  22.37$\pm$0.9 |  22.31$\pm$0.9
3.00–3.25 |  14.15$\pm$0.8 |  11.62$\pm$1.1 |  13.40$\pm$1.2 |  15.50$\pm$0.8
3.25–3.50 |  5.25$\pm$1.2 |  6.28$\pm$0.8 |  6.12$\pm$0.9 |  5.53$\pm$1.0
3.50–3.75 |  -6.25$\pm$1.4 |  -4.55$\pm$1.5 |  -4.26$\pm$1.5 |  -5.51$\pm$1.8
3.75–4.25 |  -23.85$\pm$1.9 |  -21.70$\pm$2.8 |  -22.61$\pm$3.6 |  -23.30$\pm$3.2

##  4 Predictions at 13 TeV and 14 TeV

The predictions from the NNLO calculations are validated by using the LHCb
data at 8 TeV in Sec. 3. The 8 TeV predictions using all the PDF sets, in
particular the MMHT2014 PDF set, described the data well in a consistent
manner throughout the entire $\eta_{e}$ region and already motivated
extensions of the calculations to 13 TeV and 14 TeV, which are the current and
the near-future planned center-of-mass energies of the LHC, respectively. The
13 TeV and 14 TeV NNLO calculations are performed by employing the same
fiducial phase space requirements as discussed in Sec. 2.2. Total theoretical
uncertainties are estimated using the procedure as detailed in Sec. 2.3.
Additionally, a systematic uncertainty is included for the predictions in the
last two forward bins to account for the dependency to the finite cut-off
$r_{cut}$ = 0.15% value, where the large power corrections can have an impact
as already discussed in Sec. 3. The PDF sets NNPDF3.1, CT14, and MMHT2014 are
used in the calculations.

The 13 TeV and 14 TeV calculations are first performed for the predictions of
the differential cross sections of the W± bosons $d\sigma(W^{\pm})$ and their
ratios $R_{W^{\pm}}$ in bins of the $\eta_{e}$. The differential distributions
are predicted and compared among different PDF models in Fig. 3. The 13 TeV
differential distributions are predicted consistently within uncertainties for
the W± boson processes only with a few exceptions where the predictions using
different PDF sets exhibited slight deviations from each other such as in bins
2.75–3.0 and 3.0–3.25 for the W- boson process. The prediction using NNPDF3.1
slightly underestimates the W- boson cross section for the lower $\eta_{e}$
bins with respect to the predictions using other PDF sets. However, the
predicted $R_{W^{\pm}}$ distributions are in very good agreement among
different PDF models. The $R_{W^{\pm}}$ distributions tend to decrease towards
forward $\eta_{e}$ bins as expected. The 14 TeV differential cross sections
are also predicted consistently within uncertainties for the W± boson
processes regardless of the PDF model. The predictions only exhibit small
discrepancies that are more pronounced in bins 3.25–3.5 and 3.5–3.75. The
predicted 14 TeV $R_{W^{\pm}}$ distributions are also in good agreement among
PDF sets, however become more sensitive to distinguish among different PDF
models as compared to 13 TeV distributions. This is clearly seen in the
intermediate to higher bins of $\eta_{e}$ that the predictions tend to deviate
from each other by up to $\sim$12%. Therefore, the NNLO differential cross
section distributions for the W± boson processes can be further used to
calculate $A_{e}$ distributions at 13 TeV and 14 TeV.

Figure 3: The 13 TeV (top) and 14 TeV (bottom) differential cross section
$d\sigma$ distributions for the W+ and W- boson processes as a function of the
$\eta_{e}$ which are predicted at NNLO accuracy by using NNPDF3.1, CT14, and
MMHT2014 PDF sets. The predictions include total theoretical uncertainties. In
the lower inset, the predictions for the differential cross section ratios
$R_{W^{\pm}}$ are provided.

Next the 13 TeV and 14 TeV electron charge asymmetry $A_{e}$ calculations are
performed as a function of the $\eta_{e}$ based on the differential cross
section predictions of the W± boson processes. The differential $A_{e}$
predictions are compared among the PDF sets as shown in Fig. 4 and tabulated
in Tables 3 and 4. Overall the 13 TeV predictions for the $A_{e}$ variable
from different PDF sets are in good agreement with each other within
uncertainties almost for the entire $\eta_{e}$ ranges. The $A_{e}$ predictions
in the most forward $\eta_{e}$ bin 3.75–4.25 distinguish MMHT2014 PDF set from
other PDF sets, where the prediction using MMHT2014 provides much higher
$A_{e}$ result. The 14 TeV predictions for the $A_{e}$ variable are mostly in
good agreement among the PDF sets within uncertainties, however more sensitive
to distinguish among them as compared to the 13 TeV predictions. The
predictions using NNPDF3.1 and CT14 agree with each other more than the
prediction using MMHT2014. The prediction using CT14 exhibits some deviations
that are more pronounced in intermediate region. No single PDF set is able to
provide consistent agreement with the other throughout the entire $\eta_{e}$
ranges. The sensitivity to the PDF choice regarding the $A_{e}$ distributions
is clearly increased in going from 13 TeV results to 14 TeV results.

Figure 4: The 13 TeV (top) and 14 TeV (bottom) electron charge asymmetry $A_{e}$ distributions as a function of the $\eta_{e}$ which are predicted at NNLO accuracy by using NNPDF3.1, CT14, and MMHT2014 PDF sets. The predictions include total theoretical uncertainties. Table 3: The 13 TeV electron charge asymmetry (in percent) $A_{e}$(%) predictions by using NNPDF3.1, CT14, and MMHT2014 PDF sets at NNLO accuracy in bins of the $\eta_{e}$. The predictions include total theoretical uncertainties. $\eta_{e}$ | NNPDF3.1 | CT14 | MMHT2014
---|---|---|---
2.00–2.25 | 23.78$\pm$0.6 | 22.84$\pm$0.9 | 21.11$\pm$1.3
2.25–2.50 | 26.40$\pm$0.7 | 23.13$\pm$0.7 | 24.77$\pm$0.6
2.50–2.75 | 24.27$\pm$1.0 | 23.47$\pm$0.7 | 26.00$\pm$1.6
2.75–3.00 | 25.75$\pm$1.2 | 22.51$\pm$1.1 | 24.85$\pm$1.1
3.00–3.25 | 23.18$\pm$1.2 | 22.38$\pm$1.9 | 19.50$\pm$1.0
3.25–3.50 | 20.57$\pm$1.1 | 23.23$\pm$0.8 | 22.07$\pm$1.1
3.50–3.75 | 13.86$\pm$1.7 | 13.57$\pm$1.9 | 13.41$\pm$1.9
3.75–4.25 | 0.15$\pm$2.0 | 0.91$\pm$2.2 | 6.67$\pm$2.3
Table 4: The 14 TeV electron charge asymmetry (in percent) $A_{e}$(%) predictions by using NNPDF3.1, CT14, and MMHT2014 PDF sets at NNLO accuracy in bins of the $\eta_{e}$. The predictions include total theoretical uncertainties. $\eta_{e}$ | NNPDF3.1 | CT14 | MMHT2014
---|---|---|---
2.00–2.25 | 25.15$\pm$0.9 | 20.60$\pm$1.3 | 19.90$\pm$1.6
2.25–2.50 | 25.26$\pm$0.7 | 26.05$\pm$0.8 | 23.09$\pm$0.5
2.50–2.75 | 25.38$\pm$0.7 | 21.91$\pm$0.7 | 28.24$\pm$0.6
2.75–3.00 | 27.90$\pm$1.0 | 29.22$\pm$0.8 | 25.04$\pm$0.8
3.00–3.25 | 21.24$\pm$1.5 | 21.93$\pm$1.1 | 27.04$\pm$1.1
3.25–3.50 | 25.74$\pm$1.1 | 25.13$\pm$1.6 | 21.76$\pm$1.5
3.50–3.75 | 16.06$\pm$1.8 | 22.46$\pm$1.4 | 13.87$\pm$1.7
3.75–4.25 | 5.08$\pm$2.4 | 6.97$\pm$2.6 | 7.53$\pm$2.0

The $A_{e}$ distributions are also compared at different center-of-mass
energies 8 TeV, 13 TeV, and 14 TeV for the prediction using MMHT2014 as a
baseline PDF set in Fig. 5. It has been discussed in Sec. 3 that the
prediction using MMHT2014 provides the best description of 8 TeV LHCb data and
also provides $A_{e}$ distributions reasonably well at 13 TeV and 14 TeV. The
13 TeV and 14 TeV predictions provided lower distributions in the lower
$\eta_{e}$ bins 2.0–2.5 and higher distributions in the intermediate to higher
bins 2.75–4.25 relative to the 8 TeV prediction. The distributions at
different energies almost overlap within uncertainties in the bins 2.5–3.0.
Right after the bin 2.75–3.0, the 13 TeV and 14 TeV predictions separate from
the 8 TeV prediction by exhibiting higher distributions towards the very
forward $\eta_{e}$ region. In the most forward bins 3.25–4.25, 13 TeV and 14
TeV predictions overlap within uncertainties, however 14 TeV distribution is
predicted slightly higher than 13 TeV distribution. The charge asymmetry
increases by going from 8 TeV to higher energies in the very forward
$\eta_{e}$ bins 3.0–4.25. Another outstanding observation is that $\eta_{e}$
bin where $A_{e}$ distribution reaches a peak shifts towards higher bins in
going from 8 TeV to higher energies.

Figure 5: The comparison of the electron charge asymmetry $A_{e}$
distributions at 8 TeV, 13 TeV, and 14 TeV as a function of the $\eta_{e}$
which are predicted by using MMHT2014 PDF set. The prediction includes total
theoretical uncertainties.

##  5 Summary and conclusion

In this paper a detailed study of the theoretical predictions of the electron
charge asymmetry for the inclusive $W^{\pm}+X\rightarrow e^{\pm}\nu+X$
production in the forward region of the pp collisions is presented. The charge
asymmetry calculations are performed with the inclusion of QCD NNLO
corrections in the perturbation expansion at 8 TeV, 13 TeV, and 14 TeV LHC
energies. The calculations are considered for the fiducial acceptance of the
decay electron with transverse momentum $p_{T}>$ 20 GeV in the forward
electron pseudorapidity region $2.0\leq\eta_{e}\leq 4.25$. The charge
asymmetry in the forward $\eta_{e}$ region is sensitive to the ratio of $u$
and $d$ quark distributions in the proton as a function of Bjorken-$x$ values
of the partons, particularly providing valuable inputs for the accurate
determination of PDFs at small and large $x$ values between $10^{-4}\leq x\leq
10^{-1}$.

The NNLO predictions that are based on the PDF sets NNPDF3.1, CT14, and
MMHT2014 at 8 TeV are first presented and compared with the reference LHCb
measurement Aaij:2016qqz . The 8 TeV predictions are obtained for the W+/W-
cross section ratio and the charge asymmetry differential in bins of the
$\eta_{e}$. The predicted results from different PDF sets are found to be in
good agreement with each other and with the LHCb data for most of the
$\eta_{e}$ bins. The results showed that the best description of the 8 TeV
data is achieved by the prediction using MMHT2014 for both the cross section
ratio and charge asymmetry. Furthermore, the predictions from different PDF
sets are able to describe the data quite well in the far forward $\eta_{e}$
bins 3.5–3.75 and 3.75–4.25 in comparison to the predictions provided with the
LHCb measurement, where they tend to underestimate the data within theoretical
and experimental uncertainties. The NNLO predictions using different PDF sets
are therefore successfully justified with the LHCb data at 8 TeV.

The 13 TeV and 14 TeV NNLO results that are presented in this paper represent
the first NNLO predictions of the charge asymmetry for the W± boson processes
in the forward region $2.0\leq\eta_{e}\leq 4.25$. The 13 TeV and 14 TeV
predictions are reported for the differential cross sections of the W± bosons
and the charge asymmetry as a function of the $\eta_{e}$. The predicted
results from NNPDF3.1, CT14, and MMHT2014 are compared to test the sensitivity
to discriminate among different PDF models. The W+ and W- boson differential
cross sections are predicted consistently among the PDF sets apart from some
$\eta_{e}$ bins where the predictions from different PDF models exhibited
slight discrepancies with respect to each other. The 14 TeV predictions for
the differential cross sections and their ratios are found to be more
sensitive to distinguish among the PDF models in comparison to the 13 TeV
predictions. The NNLO charge asymmetry results are reported to be in good
agreement within total theoretical uncertainties among the predictions using
different PDF sets for almost the entire $\eta_{e}$ ranges at 13 TeV and 14
TeV. In addition, the sensitivity of the charge asymmetry distributions to the
choice of PDF set is shown to be increased with the increasing collision
energy in going from 13 TeV to 14 TeV. Moreover, The predicted charge
asymmetry results are compared at 8 TeV, 13 TeV, and 14 TeV collision energies
by using MMHT2014 as a baseline PDF set. The 13 TeV and 14 TeV distributions
are shown to separate from 8 TeV distribution in the $\eta_{e}$ bin 3.0–3.25
towards the very forward $\eta_{e}$ region, where the charge asymmetry is
increased in going from 8 TeV to 13 TeV and 14 TeV distributions. In the
forward bins $\eta_{e}$ 3.25–4.25, the 13 TeV and 14 TeV distributions are
observed to overlap within uncertainties, however 14 TeV charge asymmetry
distribution is predicted to be slightly higher than 13 TeV distribution. The
results clearly show that the very forward $\eta_{e}$ bins become more of
interest to probe precisely the $u$ and $d$ quark distributions in the proton
by shifting towards 13 TeV and 14 TeV collision energies.

## Conflict of interest

The author declares that he has no conflict of interest.

## References

  * (1) F. Abe et al., Measurement of the lepton charge asymmetry in W boson decays produced in p$\rm{\bar{p}}$ collisions, Phys. Rev. Lett., 81, 5754-5759 (1998)
  * (2) V. M. Abazov et al., Measurement of the muon charge asymmetry from W boson decays, Phys. Rev. D, 77, 011106 (2008)
  * (3) V. M. Abazov et al., Measurement of the electron charge asymmetry in $\rm{p\rm{\bar{p}}\rightarrow W+X\rightarrow e\nu+X}$ events at $\sqrt{s}$ = 1.96 TeV, Phys. Rev. Lett., 101, 211801 (2008)
  * (4) T. Aaltonen et al., Direct measurement of the W production charge asymmetry in p$\rm{\bar{p}}$ collisions at $\sqrt{s}$ = 1.96 TeV, Phys. Rev. Lett., 102, 181801 (2009)
  * (5) V. M. Abazov et al., Measurement of the muon charge asymmetry in $\rm{p\rm{\bar{p}}\rightarrow W+X\rightarrow\mu\nu+X}$ events at $\sqrt{s}$ = 1.96 TeV, Phys. Rev. D, 88, 091102 (2013)
  * (6) G. Aad et al., Measurement of the inclusive $W^{\pm}$ and $Z/\gamma$ cross sections in the electron and muon decay channels in pp collisions at $\sqrt{s}$ = 7 TeV with the ATLAS detector, Phys. Rev. D, 85, 072004 (2012)
  * (7) S. Chatrchyan et al., Measurement of the electron charge asymmetry in Inclusive W production in pp collisions at $\sqrt{s}$ = 7 TeV, Phys. Rev. Lett., 109, 111806 (2012)
  * (8) S. Chatrchyan et al., Measurement of the muon charge asymmetry in Inclusive $pp\rightarrow W+X$ production at $\sqrt{s}$ = 7 TeV and an improved determination of light parton distribution functions, Phys. Rev. D, 90, 032004 (2014)
  * (9) V. Khachatryan et al., Measurement of the differential cross section and charge asymmetry for inclusive $pp\rightarrow W^{\pm}+X$ production at $\sqrt{s}$ = 8 TeV, Eur. Phys. J. C, 76, 469 (2016)
  * (10) M. Aaboud et al., Precision measurement and interpretation of inclusive $W^{+}$, $W^{-}$, and $Z/\gamma^{\star}$ production cross sections with the ATLAS detector, Eur. Phys. J. C, 77, 367 (2017)
  * (11) G. Aad et al., Measurement of $W^{\pm}$-boson and $Z$-boson production cross-sections in pp collisions at $\sqrt{s}$ = 2.76 TeV with the ATLAS detector, Eur. Phys. J. C, 79, 901 (2019)
  * (12) M. Aaboud et al., Measurements of $W$ and $Z$ boson production in pp collisions at $\sqrt{s}$ = 5.02 TeV with the ATLAS detector, Eur. Phys. J. C, 79, 128 (2019)
  * (13) G. Aad et al., Measurement of the cross-section and charge asymmetry of $W$ bosons produced in proton-proton collisions at $\sqrt{s}$ = 8 TeV with the ATLAS detector, Eur. Phys. J. C, 79, 760 (2019)
  * (14) R. Aaij et al., Inclusive W and Z production in the forward region at $\sqrt{s}$ = 7 TeV, JHEP, 06, 058 (2012)
  * (15) R. Aaij et al., Measurement of the forward W boson cross-section in pp collisions at $\sqrt{s}$ = 7 TeV, JHEP, 12, 079 (2014)
  * (16) R. Aaij et al., Measurement of forward W and Z boson production in pp collisions at $\sqrt{s}$ = 8 TeV, JHEP, 01, 155 (2016)
  * (17) R. Aaij et al., Measurement of forward $W\rightarrow e\nu$ production in pp collisions at $\sqrt{s}$ = 8 TeV, JHEP, 10, 030 (2016)
  * (18) Y. Li and F. Petriello, Combining QCD and electroweak corrections to dilepton production in FEWZ, Phys. Rev. D, 86, 094034 (2012)
  * (19) S. Catani, L. Cieri, G. Ferrera, D. de Florian, M. Grazzini, Vector boson production at hadron colliders: a fully exclusive QCD calculation at NNLO, Phys. Rev. Lett., 103, 082001 (2009)
  * (20) M. Grazzini, S. Kallweit, and M. Wiesemann, Fully differential NNLO computations with MATRIX, Eur. Phys. J. C, 78, 537 (2018)
  * (21) S. Catani and M. Grazzini, An NNLO subtraction formalism in hadron collisions and its application to Higgs boson production at the LHC, Phys. Rev. Lett., 98, 222002 (2007)
  * (22) S. Catani, L. Cieri, D. de Florian, G. Ferrera, and M. Grazzini, Vector boson production at hadron colliders: hard-collinear coefficients at the NNLO, Eur. Phys. J. C, 72, 2195 (2012)
  * (23) T. Matsuura, S. C. van der Marck, and W. L. van Neerven, The calculation of the second order soft and virtual contributions to the drell-yan cross-section, Nucl. Phys. B, 319, 570-622 (1989)
  * (24) F. Cascioli, P. Maierhofer, and S. Pozzorini, Scattering amplitudes with Open Loops, Phys. Rev. Lett., 108, 111601 (2012)
  * (25) A. Denner, S. Dittmaier, L. Hofer, Collier: a fortran-based complex one-loop lIbrary in extended regularizations, Comput. Phys. Commun., 212, 220-238 (2017)
  * (26) A. Buckley, J. Ferrando, S. Lloyd, K. Nordstrom, and B. Page, LHAPDF6: parton density access in the LHC precision era, Eur. Phys. J. C, 75, 132 (2015)
  * (27) R. D. Ball et al., Parton distributions for the LHC Run II, JHEP, 04, 040 (2015)
  * (28) S. Dulat, T. Hou, J. Gao, M. Guzzi, and J. Huston, New parton distribution functions from a global analysis of quantum chromodynamics, Phys. Rev. D, 93, 033006 (2016)
  * (29) L. A. Harland-Lang, A. D. Martin, P. Motylinski, and R. S. Thorne, Parton distributions in the LHC era: MMHT 2014 PDFs, Eur. Phys. J. C, 75, 204 (2015)
  * (30) J. Butterworth et al., PDF4LHC recommendations for LHC Run II, J. Phys. G, 43, 023001 (2016)
  * (31) T. Binoth and G. Heinrich, An automatized algorithm to compute infrared divergent multiloop integrals, Nucl. Phys. B, 585, 741-759 (2000)