# Impact of LHCb 13 TeV $\mathbb{W}$ and $\mathbb{Z}$ pseudo-data on the
parton distribution functions

Qilin Deng  Institute of Particle Physics, Central China Normal University,
Wuhan, Hubei 430079 China  Sayipjamal Dulat<EMAIL_ADDRESS>School of
Physics Science and Technology, Xinjiang University, Urumqi, Xinjiang 830046
China  Qundong Han  Institute of Particle Physics, Central China Normal
University,
Wuhan, Hubei 430079 China  Tie-Jiun Hou<EMAIL_ADDRESS>Department
of Physics, College of Sciences, Northeastern University, Shenyang, Liaoning
110819 China  Hang Yin<EMAIL_ADDRESS>Institute of Particle Physics,
Central China Normal University,
Wuhan, Hubei 430079 China  C.–P. Yuan<EMAIL_ADDRESS>Department of Physics
and Astronomy, Michigan State University, East Lansing, MI 48824 U.S.A.

(September 8, 2020)

###### Abstract

We study the potential of the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ single
$W^{\pm}$ and $Z$ boson pseudo-data for constraining the parton distribution
functions (PDFs) of the proton. As an example, we demonstrate the sensitivity
of the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ data, collected with integrated
luminosities of 5$\mbox{\,fb}^{-1}$ and 300$\mbox{\,fb}^{-1}$, to reducing the
PDF uncertainty bands of the CT14HERA2 PDFs, using the error PDF updating
package ePump. The sensitivities of various experimental observables are
compared. Generally, sizable reductions in PDF uncertainties can be observed
in the 300$\mbox{\,fb}^{-1}$ data sample, particularly in the small-$x$
region. The double-differential cross section measurement of $Z$ boson
$p_{\mathrm{T}}$ and rapidity can greatly reduce the uncertainty bands of $u$
and $d$ quarks in almost the whole $x$ range, as compared to various single
observable measurements.

PDF; W/Z boson production;

###### pacs:

12.15.Ji, 12.15.Lk, 13.85.Qk

††preprint: MSUHEP-20-015

###### Contents

  1. I Introduction
  2. II  Updating Error PDFs
  3. III Impact of the LHCb 13 TeV $W^{\pm}$ and $Z$ pseudo-data on CT14HERA2 PDFs
    1. III.1 Update from LHCb 13 TeV $W^{\pm}$ pseudo-data
    2. III.2 Update from LHCb 13 TeV $Z$ pseudo-data
    3. III.3 Update from LHCb 13 TeV $W^{+}\;+\;W^{-}\;+\;Z$ pseudo-data
    4. III.4 Update from LHCb 13 TeV $(W^{+}+W^{-})/Z$ pseudo-data
  4. IV Tolerance criteria in ePump update
  5. V Conclusion
  6. VI Acknowledgments

##  I Introduction

In hadron colliders, most physics analyses rely heavily on the understanding
of the parton picture of hadronic beam particles, including the precision
measurements of the Standard Model (SM) parameters Aaltonen:2012bp ;
Abazov:2012bv ; Aaboud:2017svj , and new physics searches. The parton picture
follows the factorization theorem of quantum chromodynamics (QCD). The parton
distribution functions (PDFs) are nonperturbative, and therefore cannot be
calculated. They are functions of the Bjorken-$x$ values ($x$, momentum
fraction) of partons at a momentum transfer scale ($Q$), which are determined
phenomenologically by a global analysis of experimental data from a wide range
of physics processes, such as Deep Inelastic Scattering (DIS), Drell-Yan (DY),
inclusive jets, and top quark pair production processes. The PDF dependencies
on $Q$ are determined by the renormalization-group based evolution equations,
e.g. DGLAP equation Gribov:1972ri ; Dokshitzer:1977sg ; Altarelli:1977zs .

Precision measurements of the single $W^{\pm}$ and $Z$ gauge boson production
cross section111Throughout this paper, $Z$ includes both the $Z$ boson and the
virtual photon contributions. at the CERN Large Hadron Collider (LHC) provide
important tests of the QCD and the electroweak (EW) sectors of the SM.
Theoretical predictions for these cross sections are available up to next-to-
next-to-leading order (NNLO) in perturbative QCD Rijken:1994sh ;
Hamberg:1990np ; Harlander:2002wh ; vanNeerven:1991gh ; Anastasiou:2003ds ,
where one of the dominated systematic uncertainties comes from the PDFs. The
CT14 PDFs Dulat:2015mca  are the first CTEQ-TEA PDFs that include published
results from the ATLAS, CMS, and LHCb collaborations at
7$\mathrm{\,Te\kern-1.00006ptV}$, including the $W^{\pm}$ and $Z$ gauge boson
production cross sections and the lepton charge asymmetry measurements from
the ATLAS Collaboration Aad:2011dm , the lepton charge asymmetry in the
electron Chatrchyan:2012xt  and muon channels Chatrchyan:2013mza  from the CMS
Collaboration, the lepton charge asymmetry in the decay of $W^{\pm}$-bosons to
an electron or a muon, and the $Z$ boson rapidity distribution from the LHCb
Collaboration Aaij:2012vn . The ATLAS and CMS measurements primarily impose
constraints on the light quark and antiquark PDFs at $x\gtrsim 0.01$. As
studied in Refs. Alekhin:2013nda ; Gao:2017yyd , the LHCb
7$\mathrm{\,Te\kern-1.00006ptV}$ and 8$\mathrm{\,Te\kern-1.00006ptV}$
$W^{\pm}$ and $Z$ boson measurements, though with larger statistical
uncertainties as compared to the corresponding results from the ATLAS and CMS
experiments, could also impose significant constraints on $u$ and $d$ PDFs.

In the past few decades, a large number of experimental results have been used
in the PDF global analysis, but we still have limited knowledge of the PDFs in
very small- and very large-$x$ ranges. In the single $W^{\pm}$ and $Z$ boson
production, the $x$ value of interacting partons ($x_{1}$ and $x_{2}$) are
correlated with the boson production, through its rapidity ($y$), as
$y=\frac{1}{2}\ln\frac{x_{1}}{x_{2}}$. Therefore, the single $W^{\pm}$ and $Z$
data in the forward detector region are valuable in the PDF global analysis,
as events with larger boson rapidity are produced by partons with small or
large $x$. Correlations between the predicted LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ $Z$ boson production cross section and the
$u$ and $d$ quark PDFs as a function of Bjorken-$x$ are shown in Fig. 1. As
shown in the figure, the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ data is
expected to have strong correlations with $u$ and $d$ quarks in the small-$x$
region, indicating the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ $W^{\pm}$ and
$Z$ data can be used to constrain the corresponding PDFs.

Figure 1: Correlation $\cos\phi$ Nadolsky:2008zw  between (left) $u(x,Q)$ PDF
or (right) $d(x,Q)$ PDF, and the DY differential cross-section in 18 bins of
$Z$ boson rapidity, as predicted by ResBos with CT14HERA2 PDFs, at
$Q=100\,{\rm GeV}$. The same binning scheme as the LHCb paper Aaij:2012vn  is
used.

The LHCb detector Alves:2008zz ; Aaij:2014jba  is a single-arm forward
spectrometer designed for the study of particles containing $b$ or $c$ quarks,
covering the pseudorapidity range $2<\eta<5$. With a high-performance tracking
system and a muon sub-detector, the LHCb data can also be extended to
precision EW measurements. Using $pp$ collision data collected at
${\sqrt{s}=7\mathrm{\,Te\kern-1.00006ptV}}$, the LHCb has measured the single
$W^{\pm}$ and $Z$ boson production cross sections using both muon and electron
channel Aaij:2012vn ; Aaij:2012mda ; Aaij:2014wba , and the same measurements
had been performed using ${\sqrt{s}=8\mathrm{\,Te\kern-1.00006ptV}}$ data
Aaij:2015zlq ; Aaij:2015vua ; Aaij:2016qqz . These results have been used to
constrain the PDFs Harland-Lang:2014zoa ; Ball:2014uwa ; Dulat:2015mca , bring
valuable information to the PDF analysis. The
${\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}}$ $pp$ collision data sample has
been collected with a larger center-of-mass energy than previous publications,
with more $W^{\pm}$ and $Z$ boson events boosted to the forward region, and
therefore allows access to even smaller (or larger) values of $x$ than the
previous 7 and 8$\mathrm{\,Te\kern-1.00006ptV}$ results.

The LHCb detector runs at reduced luminosity compared to the ATLAS and CMS
detectors Aaij:2014ida , because the detector occupancy is extremely high in
the forward region. In the LHC Run 2 period (2015–2018), the LHCb detector
collected more than 5$\mbox{\,fb}^{-1}$ $pp$ collision data Aaij:2019kcg  at
${\sqrt{s}=13\mathrm{\,Te\kern-1.00006ptV}}$. An upgraded detector
lhcb:upgrade1  is foreseen to allow LHCb detector operation at a luminosity of
$2\times 10^{33}$ $\text{cm}^{-2}\text{s}^{-1}$ in the LHC Run 3 period
(2022-2024), which is an instantaneous luminosity five times higher than
before. By the end of the LHC Run 4 period (2026-2029), the LHCb detector is
expect to collect approximately 50$\mbox{\,fb}^{-1}$ $pp$ collision data
lhcb:upgrade2 . There is also a plan for a future LHCb Upgrade-II phase
(planned for 2031 data taking) lhcb:u2phys , to guarantee the LHCb detector
could run at an even higher luminosity ($2\times 10^{34}$
$\text{cm}^{-2}\text{s}^{-1}$) lhcb:highL . After the LHCb detector Upgrade-
II, by the end of the LHC operation, it is planned that the LHCb detector
should have collected a data sample corresponding to a minimum
300$\mbox{\,fb}^{-1}$ lhcb:upgrade2 . Therefore, in this article, the pseudo-
data samples used for the physics projections are set to either
5$\mbox{\,fb}^{-1}$ or 300$\mbox{\,fb}^{-1}$.

The article is organized as follows. In Sec. II, we discuss the error PDF
updating package ePump and pseudo-data samples used in the analysis. In Sec.
III, we study the impacts of the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ single
$W^{\pm}$ and $Z$ boson pseudo-data on the CT14HERA2 PDFs Dulat:2015mca ;
Hou:2016nqm . In Sec. IV, we discuss the choice of tolerance criteria in the
ePump update. Our conclusion is given in Sec. V.

##  II  Updating Error PDFs

Recently, the CTEQ-TEA global analysis group has released a tool named ePump
(Error PDF Updating Method Package) Schmidt:2018hvu ; Hou:2019gfw , which can
be used to explore the impact of new data on existing PDFs, without perform a
full global analysis. The ePump method has been demonstrated in Ref.
Willis:2018yln . In the ePump update, two inputs are needed: a measured result
from experiment and the corresponding theoretical predictions of the complete
set of error PDFs. Thus far, the LHCb Collaboration has only published the
single $Z$ boson production cross section Aaij:2016mgv  at
13$\mathrm{\,Te\kern-1.00006ptV}$, using a small fraction of its Run 2 data
sample (2015 data, with an integrated luminosity of 0.3$\mbox{\,fb}^{-1}$).
The comparisons between the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ data and
ResBos and FEWZ Li:2012wna  predictions are shown in Fig. 2, with good
agreement between data and ResBos prediction. FEWZ can provides fixed order
calculation at NNLO accuracy in QCD. In this paper, the FEWZ prediction is
calculated with the same parameter setting as the previous LHCb publication
Aaij:2016mgv , using the CT14NNLO PDF set. However, there is no publication
for the single $W^{\pm}$ boson production using the
13$\mathrm{\,Te\kern-1.00006ptV}$ LHCb data. Therefore, we shall use pseudo-
data in this analysis to emulate the impact of the upcoming LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ data on the PDFs.

The Monte Carlo events generated with the ResBos generator Balazs:1997xd ,
using the MMHT14 Harland-Lang:2014zoa  and CT14HERA2 Dulat:2015mca ;
Hou:2016nqm  PDFs, are taken as the pseudo-data and theoretical predictions,
respectively, in this work. The theoretical predictions in this work are
computed using the ResBos Balazs:1997xd  package at approximate NNLO plus
next-to-next-to-leading-logarithm (NNLL) in QCD interactions, in which the
canonical scales are used Landry:2002ix ; Su:2014wpa . For example, both the
renormalization and factorization scales are set to be the invariant mass of
the lepton pair in the Drell-Yan (DY) events.

To emulate the LHCb detector acceptance, the charged leptons ($\ell^{\pm}$,
electrons or muons) are required to have transverse momentum ($p_{T}$) greater
than $20{\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$ and pseudorapidity ($\eta$) in
the range of $2.0<\eta<4.5$. In the case of the $Z$ boson events, the
invariant mass of the dilepton pair is required to be in the range from
60${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$ to
120${\mathrm{\,Ge\kern-1.00006ptV\\!/}c^{2}}$.

Figure 2:  Comparison of DY differential cross section as a function of the
$Z$ boson rapidity (top-left) and $p_{\mathrm{T}}$ (top-right), between theory
(ResBos, and FEWZ) and the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ data
Aaij:2016mgv . The blue points represent the LHCb results in muon channel, and
the red points the electron channel, while the black (green) line represents
the ResBos (FEWZ) prediction, the grey band represents the PDF uncertainty at
the 68% confidence level (CL) estimated using the CT14HERA2 PDF set. The ratio
of data to theoretical prediction are shown as a function of $Z$ boson
rapidity (bottom-left) and $p_{\mathrm{T}}$ (bottom-right).

In the ePump study, for the $Z\rightarrow\ell^{+}\ell^{-}$ pseudo-data input,
the statistical uncertainties are scaled to the 5$\mbox{\,fb}^{-1}$ and
300$\mbox{\,fb}^{-1}$ data samples, separately, by extrapolating the total
uncertainty of the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ publication
Aaij:2016mgv  to the pseudo-data sample. In the extrapolation, an assumption
is made that the ratio of statistical uncertainty to systematical uncertainty
will remain the same in all data samples. Similarly, for the
$W^{\pm}\rightarrow\ell^{\pm}\nu$ pseudo-data sample, uncertainties are
estimated using the LHCb 8$\mathrm{\,Te\kern-1.00006ptV}$ publication
Aaij:2015zlq ; Aaij:2016qqz .

##  III Impact of the LHCb 13 TeV $W^{\pm}$ and $Z$ pseudo-data on CT14HERA2
PDFs

In this section, we study the impact of the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ single $W^{\pm}$ and $Z$ boson pseudo-data
on the CT14HERA2 PDFs, to demonstrate the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ data sensitivity, and to further investigate
valuable observables for future measurements.

Since the pseudo-data sample generated with the MMHT14 PDFs is used to update
the CT14HERA2 PDF sets, and there are differences between the central PDF set
of MMHT14 and CT14HERA2, the central value of ePump updated-PDFs varies from
the CT14HERA2 central set. In this article, we are interested in variations of
PDF uncertainty, and thus we do not discuss variations of PDF central values
hereafter.

###  III.1 Update from LHCb 13 TeV $W^{\pm}$ pseudo-data

In the $W^{\pm}$ boson leptonic decay, there is a neutrino and a charged
lepton in the final state, where the neutrino will escape from detector, only
the charged lepton can be detected in a hadron collider experiment. This
feature makes a $W^{\pm}$ boson analysis complicated, since the irreducible
background contribution is difficult to model. On the other hand, the single
$W^{\pm}$ production rate is one order of magnitude larger than that of the
$Z$ boson at the LHCb. If we could model the background properly for the
$W^{\pm}$ events, such a sample with large statistics could allow us to
perform many precision measurements. In this study, we used the charged lepton
pseudorapidity distribution ($\eta$) as an observable in the ePump update,
with the same binning scheme as the previous LHCb publications Aaij:2015zlq ;
Aaij:2016qqz .

After the ePump update, the updated quark PDF distribution is compared with
the default one of CT14HERA2. The $d$ quark PDF distribution with its
uncertainty is shown in Fig. 3, which indicates that the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ $W^{\pm}$ boson data have a large impact on
the $d$ quark PDF, especially in the small-$x$ range from $10^{-5}$ to
$10^{-3}$. With a 300$\mbox{\,fb}^{-1}$ LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$
$W^{\pm}$ data sample, the $d$ quark PDF uncertainty can be reduced by a
factor of 30% around $x=10^{-3}$. The LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$
$W^{\pm}$ boson data have a smaller impact on the $u$ quark PDF as compared to
the $d$ quark PDF, but with a 300$\mbox{\,fb}^{-1}$ data sample, the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ $W^{\pm}$ boson data do have an impact on
the $u$ quark PDF in the small-$x$ region.

Figure 3: PDF uncertainties associated with the $d$ quark, as a function of
$x$, in the CT14HERA2 PDFs and ePump updated new PDFs. The denominator is the
central value of each PDF set. The blue (red) line represents the central
value of the PDF ratio before (after) the ePump update, the blue band
represents the CT14HERA2 PDF uncertainty, the red shaded band represents the
updated PDF uncertainty. The charged lepton pseudorapidity distributions of
$W^{\pm}$ events are used as inputs for ePump update. (top-left) The $d$ quark
result using 5$\mbox{\,fb}^{-1}$ $W^{+}$ events; (top-right) the $d$ quark
result using 5$\mbox{\,fb}^{-1}$ $W^{-}$ events; (bottom-left) the $d$ quark
result using 300$\mbox{\,fb}^{-1}$ $W^{+}$ events, and (bottom-right) the $d$
quark result using 300$\mbox{\,fb}^{-1}$ $W^{-}$ events.

The impacts of the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ $W^{+}/W^{-}$ data,
the ratio of $W^{+}$ and $W^{-}$ event rates, on the PDF ratios $d/u$ and
${\bar{d}}/{\bar{u}}$ are shown in Fig. 4. As we see, even with a
5$\mbox{\,fb}^{-1}$ data sample, the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$
$W^{+}/W^{-}$ data can already reduce the $d/u$ PDF uncertainty. Most of the
improvements are concentrated in the small-$x$ region, from $10^{-5}$ to
$10^{-3}$. The 300$\mbox{\,fb}^{-1}$ LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$
data sample could further reduce the uncertainties of both the PDF ratios
$d/u$ and ${\bar{d}}/{\bar{u}}$ (by about $\sim 20\%$) in the small-$x$
region, as well as giving some noticeable improvements in the large-$x$
region. In the current PDF global fitting, the DIS data provide the largest
constraint on the PDF ratio $d/u$, cf. Ref. Hou:2019gfw . In the future, the
LHCb data could provide additional information on the PDF ratios $d/u$ and
${\bar{d}}/{\bar{u}}$.

Figure 4:  PDF uncertainties associated with $d/u$ (left) and
${\bar{d}}/{\bar{u}}$ (right), as a function of $x$, in the CT14HERA2 PDFs and
the ePump updated PDFs. The charged lepton pseudorapidity distributions of
$W^{+}/W^{-}$ events are used as the input for the ePump update; The $d/u$
result (top-left) and the ${\bar{d}}/{\bar{u}}$ result (top-right) using a
5$\mbox{\,fb}^{-1}$ $W^{+}/W^{-}$ data sample; the $d/u$ result (bottom-left)
and the ${\bar{d}}/{\bar{u}}$ result (bottom-right) using a
300$\mbox{\,fb}^{-1}$ $W^{+}/W^{-}$ data sample.

###  III.2 Update from LHCb 13 TeV $Z$ pseudo-data

The single $Z$ boson leptonic decay has two charged leptons in the final
state. These two charged leptons have large transverse momentum, and are
isolated in the detector. Based on these features, the
$Z\rightarrow\ell^{+}\ell^{-}$ events are easy to reconstruct and identify in
a hadron collider, with small background contamination. Therefore, the
$Z\rightarrow\ell^{+}\ell^{-}$ channel is one of best channels to perform
precision EW measurements.

We used the $Z$ boson rapidity distribution as an observable for the ePump
update, explore other observables that could be used in future PDF fitting,
and propose a novel way to present $Z$ boson production measurement that
provides more valuable information for PDF fitting. In this study, a binning
scheme similar to the previous LHCb publication Aaij:2016mgv  is used.

The updated PDF results are shown in Fig. 5 for the $d$ quark. As shown in the
figure, with 5$\mbox{\,fb}^{-1}$ of data sample, the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ single $Z$ boson data is not as powerful as
$W^{\pm}$ data, mainly due to its smaller event rate. With
300$\mbox{\,fb}^{-1}$, however, the $Z$ boson data has an impact on $d$ quark
PDFs in the small-$x$ region, from $10^{-5}$ to $10^{-2}$.

The impacts from the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$
300$\mbox{\,fb}^{-1}$ single $Z$ boson data on $d/u$ and ${\bar{d}}/{\bar{u}}$
are shown in Fig. 6, where the LHCb $Z$ boson data could reduce the $d/u$ PDF
uncertainty in the small-$x$ region.

Figure 5: PDF uncertainties associated with the $d$ quark, as a function of
$x$, in the CT14HERA2 PDFs and ePump updated new PDFs. The denominator is the
central value of each PDF set. The rapidity distribution of $Z$ boson events
is used as input of the ePump update. (left) The $d$ quark result using
5$\mbox{\,fb}^{-1}$ data, and (right) the $d$ quark result using
300$\mbox{\,fb}^{-1}$ data.

Figure 6: PDF uncertainties associated with $d/u$ (left) and
${\bar{d}}/{\bar{u}}$ (right), as a function of $x$, in the CT14HERA2 PDFs and
ePump updated new PDFs. The denominator is the central value of each PDF set.
The rapidity distribution of $Z$ boson events is used as input of the ePump
update. (left) The $d/u$ quark result using 300$\mbox{\,fb}^{-1}$ data, and
(right) the ${\bar{d}}/{\bar{u}}$ quark result using 300$\mbox{\,fb}^{-1}$
data.

We have also explored the sensitivity of the $Z$ boson $p_{T}$, lepton
$\cos\theta^{*}$ (defined in the Collins-Soper frame Collins:1977iv ), and $Z$
boson rapidity distributions measured at the LHCb to further constrain the
PDFs. We consider their impacts one at a time in the ePump update. As shown in
Fig. 7, we see that each observable has a slightly different impact on the $u$
($d$) quark PDFs across the whole $x$ region, as expected.

Below, we propose a better way to extract useful information from the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ $Z$ data, by performing a multi-dimensional
analysis. With more $pp$ collision data to be collected by the LHCb detector
in the future, it is feasible to perform $Z$ boson production cross section
measurement with a multi-dimension binning, like double- or triple-
differential cross section measurements. Comparisons of updated PDF
uncertainties with different numbers of (input) experimental observables are
shown in Fig. 8. As shown in the figure, we compare the impacts of the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ $Z$ boson pseudo-data on PDFs by performing
a single differential ($Z$ boson $p_{\mathrm{T}}$, labeled as ‘1D’ in the
figure), a double-differential ($Z$ boson $p_{\mathrm{T}}$ and $Z$ boson
rapidity, labeled as ‘2D’ in the figure), and a triple-differential ($Z$ boson
$p_{\mathrm{T}}$, $Z$ boson rapidity, and lepton $\cos\theta^{*}$, labeled as
‘3D’ in the figure) cross section measurements. We find that with limited
statistics of $Z$ boson events (5$\mbox{\,fb}^{-1}$ data), the mulit-
dimensional measurements cannot significantly improve the PDF determination,
as compared to one-dimensional measurements on $Z$ boson $p_{\mathrm{T}}$,
lepton $\cos\theta^{*}$, and $Z$ boson rapidity, respectively. With a
300$\mbox{\,fb}^{-1}$ data sample, however, the mulit-dimensional measurement
has better constraints on the PDFs, across the whole $x$ range. The triple-
differential cross section gives the best constraints on $u$ and $d$ quark
PDFs, across the whole $x$ range. The improvements gained in going from ‘2D’
to ‘3D’ measurement is not as strong as that from ‘1D’ to ‘2D’ measurement.
From the experimental point of view, a triple-differential cross section
measurement could have limited statistics in extreme phase space, such as at
the boundaries of observables. Furthermore, it is complicated to calculate
correlated systematic uncertainties in a ‘3D’ measurement, compared to a ‘2D’
measurement. Therefore, with large future data samples, a double-dimensional
$Z$ boson cross section measurement (a double-differential $Z$ boson
$p_{\mathrm{T}}$ and $Z$ boson rapidity) is feasible and recommended, and
could provides more valuable information in PDF fitting than a single- or a
triple-dimensional measurement.

Figure 7:  PDF uncertainties associated with the $u$ quark (left) and $d$
quark (right), as a function of $x$, in the CT14HERA2 PDFs and ePump updated
new PDFs. The denominator is the central value of each PDF set. The single $Z$
boson events (300$\mbox{\,fb}^{-1}$) are used in the ePump update.

Figure 8: PDF uncertainties associated with the $u$ quark (left) and $d$ quark
(right), as a function of $x$, in the CT14HERA2 PDFs and ePump updated new
PDFs. The denominator is the central value of each PDF set. The single $Z$
boson multi-dimensional (1D, 2D, and 3D) differential cross section is used in
the ePump update. (top-left) The $u$ quark result using 5$\mbox{\,fb}^{-1}$
data; (top-right) the $d$ quark result using 5$\mbox{\,fb}^{-1}$ data;
(bottom-left) the $u$ quark result using 300$\mbox{\,fb}^{-1}$ data; and
(bottom-right) the $d$ quark result using 300$\mbox{\,fb}^{-1}$ data.

In the double- and triple-differential cross section measurements, the binning
schemes for $Z$ boson $p_{\mathrm{T}}$, rapidity ($y_{\ell\ell}$), and lepton
$\cos\theta^{*}$ are defined as:

  * •

$0<\mbox{$p_{\mathrm{T}}$}<250$: {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12,
13, 14, 15, 17.5, 20, 22, 25, 28, 33, 40, 50, 100,
250}${\mathrm{\,Ge\kern-1.00006ptV\\!/}c}$.

  * •

$2<y_{\ell\ell}<4.5$: {2.00, 2.14, 2.28, 2.42, 2.56, 2.69, 2.83, 2.97, 3.11,
3.25, 3.39, 3.53, 3.67, 3.81, 3.94, 4.08, 4.22, 4.36, 4.50}.

  * •

$-1<\cos\theta^{*}<1$: {-1, 0, 1}.

###  III.3 Update from LHCb 13 TeV $W^{+}\;+\;W^{-}\;+\;Z$ pseudo-data

In general, global fitting of PDFs should use all available experimental data
as inputs. Therefore, we checked the impact from the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ data on the PDF fitting, including both
single $W^{\pm}$ boson and $Z$ boson data samples. In reality, as $W^{\pm}$
and $Z$ boson results from one experiment are measured with the same data
sample, many systematic uncertainties are correlated. In any PDF fitting,
correlation matrices between different observables must be provided to avoid
potential data bias. In this study, without detector-level simulated events,
we cannot calculate the correlation matrices between the single $W^{\pm}$ and
$Z$ boson measurements. So we assume in this study that there is no
correlation between the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ $W^{\pm}$ and
$Z$ pseudo-data. In the study, the $W^{\pm}$ and $Z$ boson single differential
cross section results are used as the inputs of the ePump update, which are
the charged lepton $\eta$ distribution of $W^{\pm}$ boson events and the
rapidity distribution of $Z$ bosons.

The updated PDF results are shown in Fig. 9 for $u$, $d$, $c$ quark and gluon
PDFs, and the $d/u$ and $\bar{d}/\bar{u}$ ratio results are shown in Fig. 10.
Based on these figures, the following features are found:

  * •

The largest improvement is in the $d$ quark PDFs. The uncertainty of the $d$
quark PDFs can be improved significantly by the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ $W^{\pm}/Z$ data in the whole $x$ region. In
the small-$x$ region $10^{-5}<x<10^{-2}$ especially, the uncertainty would be
reduced by a factor of 60% at $x\sim 10^{-3}$.

  * •

The uncertainty of $u$, $s$, $c$ quark and gluon PDFs can be reduced across
the whole $x$ region, and significant improvements are expected in very small-
and larger-$x$ regions.

  * •

The uncertainties of $d/u$ and $\bar{d}/\bar{u}$ ratios can be significantly
reduced across the whole $x$ range, even with only 5$\mbox{\,fb}^{-1}$ data.
In the very larger-$x$ region, the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ data
could have a large impact on the $d/u$ ratio.

  * •

The LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ $W^{\pm}$ and $Z$ data also has a
large impact on the $\bar{u}$ and $\bar{d}$ quark PDFs, mainly in the
small-$x$ region.

Figure 9: PDF uncertainties associated with the $u$ quark (left) and $d$ quark
(right), as a function of $x$, in the CT14HERA2 PDFs and ePump updated new
PDFs. The denominator is the central value of each PDF set. The
300$\mbox{\,fb}^{-1}$ single $W^{\pm}$ and $Z$ boson differential cross
section are taken as ePump input. (top-left) The $u$ quark result; (top-right)
the $d$ quark result; (bottom-left) the gluon result; and (bottom-right) the
$c$ quark result.

For the $d/u$ and $\bar{d}/\bar{u}$ ratios, the future LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ data will provide the most important
constraints on them. In Fig. 10, the 300$\mbox{\,fb}^{-1}$ LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ pseudo-data provides valuable constraints on
the $d/u$ ratio in the very large-$x$ region ($>0.5$) and $\bar{d}/\bar{u}$ in
$x>0.2$. In these regions, the LHCb data would be the only clean data, as it
is free of nuclear corrections, as needed when describing the low energy
Drell-Yan data to constrain $d/u$.

It is well-known that fixed-target Drell-Yan measurements provide an important
probe of the $x$ dependence of the nucleon (and nuclear) PDFs. This fact has
motivated a number of experiments, including the Fermilab E866/NuSea
experiment Towell:2001nh , which determined the normalized deuteron-to-proton
cross section ratio $\sigma_{pd}\big{/}2\sigma_{pp}$ out to relatively large
$x_{2}$, the momentum fraction of the target. As can be seen based upon a
leading-order quark-parton model analysis, this ratio is expected to have
especially pronounced sensitivity to the $x$ dependence of the PDF ratio
$\bar{d}/\bar{u}$. The E866 results stimulated an interest in performing a
similar measurement out to larger $x_{2}$ with higher precision — the main
objective of the subsequent SeaQuest/E906 experiment at Fermilab
Aidala:2017ofy , from which results are expected soon. The LHCb data could be
used to check the impact from the SeaQuest Tadepalli:2019ghd  result for
$\bar{d}/\bar{u}$ in the large-$x$ region. In Fig. 11, we compare the
theoretical prediction based on the update CT14HERA2 PDFs (with the
300$\mbox{\,fb}^{-1}$ LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ combined
$W^{\pm}$ and $Z$ pseudo-data as input) and the original CT14HERA2 PDFs. This
shows that the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ data could further
constrain the deuteron-to-proton ratio, $\sigma_{pd}\big{/}2\sigma_{pp}$, in
the large-$x$ region.

Figure 10: PDF uncertainties associated with $d/u$ (left) and
${\bar{d}}/{\bar{u}}$ (right), as a function of $x$, in the CT14HERA2 PDF sets
and ePump updated new PDF sets. The denominator is the central value of each
PDF set. The single $W^{\pm}$ and $Z$ boson differential cross section are
taken as ePump input. (top-left) The $d/u$ result using 5$\mbox{\,fb}^{-1}$
data; (top-right) the ${\bar{d}}/{\bar{u}}$ result using 5$\mbox{\,fb}^{-1}$
data; (bottom-left) the $d/u$ result using 300$\mbox{\,fb}^{-1}$ data; and
(bottom-right) the ${\bar{d}}/{\bar{u}}$ result using 300$\mbox{\,fb}^{-1}$
data. Figure 11: Theoretical predictions based on the updated CT14HERA2 (red
band) and original CT14HERA2 (blue band) for the fixed-target Drell-Yan cross
section, $\sigma_{pd}\big{/}2\sigma_{pp}$, in the region of larger
$x_{2}\gtrsim 0.1$ to be probed by the SeaQuest experiment Aidala:2017ofy  at
Fermilab. For comparison, the higher-$x_{2}$ portion of the older E866 data
Towell:2001nh  (black points) is also presented here.

###  III.4 Update from LHCb 13 TeV $(W^{+}+W^{-})/Z$ pseudo-data

At tree level, the $Z$ boson is produced via $q\bar{q}$ annihilation, where
$q$ could be $u$, $d$, $c$, $s$, and $b$, while the $W^{\pm}$ boson could
produced via $u\bar{s}$ and $\bar{u}s$. Therefore, the ratio of $W^{\pm}$
distribution to that of the $Z$ boson could be sensitive to the
$\frac{s+\bar{s}}{\bar{u}+\bar{d}}$ at first order Nadolsky:2008zw .

With a uniform binning (18 pseudorapidity/rapidity bins, from 2.0 to 4.5), a
$(W^{+}+W^{-})/Z$ ratio in each bin is calculated, where the muon
pseudorapidity of the $W^{\pm}$ boson and the rapidity of the $Z$ boson are
used. Correlations between the predicted LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ $(W^{+}+W^{-})/Z$ ratio and
$\frac{s+\bar{s}}{\bar{u}+\bar{d}}$ are shown in Fig. 12. With the calculated
$(W^{+}+W^{-})/Z$ ratio as ePump input, we checked the impact on
$\frac{s+\bar{s}}{\bar{u}+\bar{d}}$ from the LHCb single $W^{\pm}$ and $Z$
data, as shown in Fig. 12. With 5$\mbox{\,fb}^{-1}$ LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ pseudo-data as input, the $(W^{+}+W^{-})/Z$
data does not have a visible impact on the PDF ratio
$\frac{s+\bar{s}}{\bar{u}+\bar{d}}$. With 300$\mbox{\,fb}^{-1}$ LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ pseudo-data as input, the impact on the PDF
ratio $\frac{s+\bar{s}}{\bar{u}+\bar{d}}$ becomes significant in the $x$ range
of $10^{-2}$ to $10^{-1}$, which could be used to precisely determine the
strange quark PDFs in the future. As expected, larger correlations between
$(W^{+}+W^{-})/Z$ and $\frac{s+\bar{s}}{\bar{u}+\bar{d}}$ are seen in the same
$x$ range.

Figure 12:  (left) $\cos\phi$ between the PDF ratio
$\frac{s+\bar{s}}{\bar{u}+\bar{d}}$ and the $(W^{+}+W^{-})/Z$ ratio. Lines
with different color represent different bins, and the same index as Fig. 1 is
used. (right) The PDF uncertainties associated with
$\frac{s+\bar{s}}{\bar{u}+\bar{d}}$ distribution, using the
300$\mbox{\,fb}^{-1}$ data, as a function of $x$, in the CT14HERA2 PDFs and
ePump updated new PDFs. The denominator is the central value of each PDF set.
The single $(W^{+}+W^{-})/Z$ boson differential cross sections are taken as
the ePump input.

##  IV Tolerance criteria in ePump update

The tolerance criterion (i.e., the choice of total $\Delta\chi^{2}$ value in a
global analysis) is an important parameter in PDF fitting.

It was extensively discussed in Ref. Hou:2019efy  that in order to best
reproduce CT14HERA2 global fit, one should use dynamical tolerance in ePump.
If the tolerance is set to be $\Delta\chi^{2}=1$ at 68% CL, or equivalently,
$(1.645)^{2}$ at 90% CL, instead of using a dynamical tolerance, a very large
weight is effectively assigned to the new input data, when updating the
existing PDFs in the CT PDF global analysis framework.

To illustrate the differences between the ePump updated PDFs using a dynamical
tolerance and a fixed tolerance with $\Delta\chi^{2}=(1.645)^{2}$, we used the
single LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ $Z$ data as the ePump input. The
result is shown in Fig. 13. The impact from the new data is enhanced when the
PDFs are updated with $\Delta\chi^{2}=(1.645)^{2}$, which could introduce bias
in the new updated PDF set.

This conclusion also holds for using MMHT2014 Harland-Lang:2014zoa  and
PDF4LHC15 Butterworth:2015oua  PDFs in profiling analysis to study the impact
of a new (pseudo-) data on updating the existing PDFs.

Figure 13: PDF uncertainties associated with the $d$ quark distribution, as a
function of $x$, in the CT14HERA2 PDFs and ePump updated new PDFs. The
denominator is the central set of each PDF sets. The rapidity distribution of
$Z$ boson events (300$\mbox{\,fb}^{-1}$) is used as input to the ePump update.
(left) The result with fixed tolerance of $\Delta\chi^{2}=(1.645)^{2}$ at the
90% CL, and (right) the result with dynamical tolerance.

##  V Conclusion

With detector instrumented in the forward region, the $pp$ collision data
collected by the LHCb experiment provides essential and complementary
information for a global analysis of experimental data to determine the PDFs
of the proton, as compared to data collected by the ATLAS and CMS detectors.

In this article, we have studied the potential of the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ single $W^{\pm}$ and $Z$ boson pseudo-data
for constraining the PDFs in the proton. As an example, we demonstrated the
sensitivity of the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ data, collected with
an integrated luminosity of 5$\mbox{\,fb}^{-1}$ and 300$\mbox{\,fb}^{-1}$,
respectively, to reducing the PDF uncertainty bands of the CT14HERA2 PDFs. We
have also investigated the sensitivities of various experimental observables.

A large impact from the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ data on various
quark flavor PDFs has been seen across the $x$ region, and significant
contributions in the small-$x$ ($<10^{-3}$) region are expected. In
particular, the $d$ and $\bar{d}$ quark PDF uncertainties are reduced
dramatically, $\sim$60% improvement at momentum fraction ($x$) around
$10^{-3}$. Due to its large event rate, the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ ${W^{+}}/{W^{-}}$ data can already further
reduce the $d/u$ PDF uncertainty, even with only a 5$\mbox{\,fb}^{-1}$ data
sample, as shown in Fig. 4. Most of the improvements are concentrated in the
small-$x$ region, from $10^{-5}$ to $10^{-3}$. The 300$\mbox{\,fb}^{-1}$ LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ data sample could further reduce the
uncertainties of both the PDF ratios $d/u$ and ${\bar{d}}/{\bar{u}}$ (by $\sim
20\%$) in the small-$x$ region, as well as giving some noticeable improvements
in the large-$x$ region, which is currently dominated by DIS data in global
analysis.

Although it has a smaller event rate than the $W^{\pm}$ events, the LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ $Z$ data can also provide important
constraints on PDFs, particularly when considering a double-differential
distribution (as a function of $Z$ boson $p_{\mathrm{T}}$ and rapidity), as
the experimental observable for updating the existing PDFs. We have compared
the impact of the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ $Z$ boson pseudo-data
on PDFs by performing single differential ($Z$ boson $p_{\mathrm{T}}$),
double-differential ($Z$ boson $p_{\mathrm{T}}$ and $Z$ boson rapidity), and
triple-differential ($Z$ boson $p_{\mathrm{T}}$, $Z$ boson rapidity, and
lepton $\cos\theta^{*}$ ) cross section measurements. We found that with
limited statistics of $Z$ boson events (5$\mbox{\,fb}^{-1}$ data), the mulit-
dimensional measurements cannot significantly improve the PDF determination,
as compared to one-dimensional measurements of $Z$ boson $p_{\mathrm{T}}$,
lepton $\cos\theta^{*}$, and $Z$ boson rapidity, respectively. With a
300$\mbox{\,fb}^{-1}$ data sample, however, the mulit-dimensional measurement
has better constraints on the PDFs, across the whole $x$ range.

It is evident that the combined data sample of $W^{\pm}$ and $Z$ boson events
can provide further constraints on PDFs. Examining the ratio of the event
rates $(W^{+}+W^{-})/Z$ could directly probe the PDF ratio
$\frac{s+\bar{s}}{\bar{u}+\bar{d}}$, as previously noted in Ref.
Nadolsky:2008zw . With 5$\mbox{\,fb}^{-1}$ LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ pseudo-data as input, the $(W^{+}+W^{-})/Z$
data does not have a visible impact on the PDF ratio
$\frac{s+\bar{s}}{\bar{u}+\bar{d}}$. With 300$\mbox{\,fb}^{-1}$ LHCb
13$\mathrm{\,Te\kern-1.00006ptV}$ pseudo-data as input, the impact on the PDF
ratio $\frac{s+\bar{s}}{\bar{u}+\bar{d}}$ becomes significant in the $x$ range
of $10^{-2}$ to $10^{-1}$, which could be used to precisely determine the
strange quark PDFs in the future. The above features suggest that the LHCb
single $W^{\pm}$ and $Z$ data taken at the LHC
13$\mathrm{\,Te\kern-1.00006ptV}$ will provide very important and unique
information in a global analysis, complementary to the ATLAS and CMS results.
With an integrated luminosity of 300$\mbox{\,fb}^{-1}$ data to be collected in
the future, the impact from the LHCb 13$\mathrm{\,Te\kern-1.00006ptV}$ data on
the PDF fitting could be enhanced significantly by performing a multi-
differential cross section measurement.

Before concluding our work, we also pointed out the important role of
tolerance criteria in the PDF updating. As discussed in Ref. Hou:2019efy , one
should use dynamical tolerance in ePump. Setting the tolerance to be 1 at 68%
CL, or equivalently, $(1.645)^{2}$ at 90% CL, will greatly overestimate the
impact of a given new data set when updating the existing PDFs in the CT PDF
global analysis framework. This conclusion also holds for using MMHT2014
Harland-Lang:2014zoa  and PDF4LHC15 Butterworth:2015oua  PDFs in profiling
analysis to study the impact of a new (pseudo-) data on updating the existing
PDFs.

##  VI Acknowledgments

This work is supported by the National Natural Science Foundation of China
under the Grant No. 11875142, No. 11965020, and by the U.S. National Science
Foundation under Grant No. PHY-2013791. C.-P. Yuan is also grateful for the
support from the Wu-Ki Tung endowed chair in particle physics.

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