Edinburgh 2022/14

Nikhef-2022-013

TIF-UNIMI-2022-13

Parton Distributions and New Physics Searches:
the Drell-Yan Forward-Backward Asymmetry as a Case Study

Richard D. Ball,1 Alessandro Candido,2 Stefano Forte,2 Felix Hekhorn,2

Emanuele R. Nocera,3 Juan Rojo4,5, and Christopher Schwan6

1The Higgs Centre for Theoretical Physics, University of Edinburgh,
JCMB, KB, Mayfield Rd, Edinburgh EH9 3JZ, Scotland
2Tif Lab, Dipartimento di Fisica, Università di Milano and
INFN, Sezione di Milano, Via Celoria 16, I-20133 Milano, Italy
3 Dipartimento di Fisica, Università degli Studi di Torino and
INFN, Sezione di Torino, Via Pietro Giuria 1, I-10125 Torino, Italy
4Department of Physics and Astronomy, Vrije Universiteit, NL-1081 HV Amsterdam
5Nikhef Theory Group, Science Park 105, 1098 XG Amsterdam, The Netherlands
6Universität Würzburg, Institut für Theoretische Physik und Astrophysik, 97074
Würzburg, Germany

Abstract

We discuss the sensitivity of theoretical predictions of observables used in
searches for new physics to parton distributions (PDFs) at large momentum
fraction $x$. Specifically, we consider the neutral-current Drell-Yan
production of gauge bosons with invariant masses in the TeV range, for which
the forward-backward asymmetry of charged leptons from the decay of the gauge
boson in its rest frame is a traditional probe of new physics. We show that
the qualitative behaviour of the asymmetry depends strongly on the assumptions
made in determining the underlying PDFs. We discuss and compare the large-$x$
behaviour of various different PDF sets, and find that they differ
significantly. Consequently, the shape of the asymmetry observed at lower
dilepton invariant masses, where all PDF sets are in reasonable agreement
because of the presence of experimental constraints, is not necessarily
reproduced at large masses where the PDFs are mostly unconstrained by data. It
follows that the shape of the asymmetry at high masses may depend on
assumptions made in the PDF parametrization, and thus deviations from the
traditionally expected behaviour cannot be taken as a reliable indication of
new physics. We demonstrate that forward-backward asymmetry measurements could
help in constraining PDFs at large $x$ and discuss the accuracy that would be
required to disentangle the effects of new physics from uncertainties in the
PDFs in this region.

###### Contents

  1. 1 Introduction
  2. 2 Anatomy of Drell-Yan production
    1. 2.1 Drell-Yan kinematics and cross-sections at LO
    2. 2.2 Single-differential distributions and the forward-backward asymmetry
  3. 3 The forward-backward asymmetry and the large-$x$ PDFs
    1. 3.1 Qualitative features of $A_{\rm fb}$
    2. 3.2 Parton distributions
    3. 3.3 Parton luminosities
  4. 4 The Drell-Yan forward-backward asymmetry at the LHC
  5. 5 Summary and outlook
  6. A $A_{\rm fb}$ in NNPDF3.1

##  1 Introduction

An important direction for ongoing and future studies of new physics beyond
the Standard Model (BSM) at the Large Hadron Collider (LHC) is the search for
novel heavy resonances. The LHC is uniquely suited to direct searches for
these resonances, thanks to its unparalleled center of mass energy,
$\sqrt{s}=13.6$ TeV in the recently started Run III, and the high statistics
to be accumulated in the coming years, especially in the high-luminosity (HL)
phase. For instance, considering representative benchmark BSM scenarios, the
HL-LHC is sensitive [1] to searches for sequential Standard Model (SM)
$W^{\prime}$ gauge bosons up to $m_{W^{\prime}}=7.8$ TeV, $E_{6}$ model
$Z^{\prime}$ gauge bosons up to $m_{Z^{\prime}}=5.7$ TeV, and Kaluza-Klein
resonances decaying into a $t\bar{t}$ pair up to $m_{KK}=6.6$ TeV.

The production of such high-mass states proceeds via partonic scattering that
involves large values of the momentum fractions $x_{1}$ and $x_{2}$ of the
colliding partons, because the center of mass energy of the partonic collision
is $\hat{s}=x_{1}x_{2}s$. For instance, the on-shell production of a state
with invariant mass $m_{X}=8$ TeV requires $x_{1}x_{2}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.3$,
hence for central production at leading order $x_{1}=x_{2}\approx 0.6$. This
is problematic because parton distribution functions (PDFs) [2, 3] are poorly
known for $x\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.4$, as there is limited data
included in current PDF determinations to constrain this kinematic region.
Indeed, in the past, claims of possible BSM signals [4] were subsequently
traced to poor modeling of the PDFs in the large-$x$ region [5]. The impact of
lack of knowledge of the PDFs on BSM searches is thus a delicate issue [6].

Here we wish to further investigate this by specifically considering neutral-
current (NC) Drell-Yan (DY) dilepton production and associated observables,
frequently used for BSM searches at the LHC. NC Drell-Yan production is one of
the cleanest processes in the search for both narrow and broad heavy
resonances decaying into dileptons, $pp\to X\to\ell^{+}\ell^{-}$, since the
two charged leptons can be detected with excellent energy and angular
resolution. This also enables the search for smooth, non-resonant distortions
with respect to the SM backgrounds, such as those arising in the context of
contact interactions or, more generally, induced by Effective Field Theory
(EFT) higher-dimensional operators that lead to direct couplings between
quarks and leptons [7, 8, 9, 10]. Indeed, both ATLAS and CMS have extensively
explored this channel in their BSM search program [11, 12, 13, 14, 15, 16]. To
this purpose, it is mandatory to have a detailed understanding of the dominant
SM background, namely dilepton production from quark-antiquark annihilation
mediated by a virtual electroweak (EW) boson,
$q\bar{q}\to\gamma^{*}/Z\to\ell^{+}\ell^{-}$, with subleading processes
involving the quark-gluon and photon-photon initial states.

Drell-Yan production is one of the SM processes which is known to highest
perturbative accuracy: indeed, both N3LO QCD results [17] and the full mixed
QCD-EW corrections at NNLO [18, 19, 20, 21, 22] have become available
recently. Therefore, the main uncertainty on theoretical predictions for this
process is mostly due to the PDFs, which, as mentioned, are poorly known at
large $x$. Experimentally, uncertainties are minimized when considering
observables in which several systematics cancel in part or entirely. An
example relevant for the DY process is the forward-backward asymmetry $A_{\rm
fb}$ of the angular distribution of the dilepton pair in the center-of-mass
frame of the partonic collision, i.e. the asymmetry in the so-called Collins-
Soper angle $\theta^{*}$, recently measured from the Run II dataset by ATLAS
[23] and CMS [24]. The sensitivity of this observable to both PDFs and BSM
signals has been emphasized recently [25, 26, 27, 28, 29], as well as its
relevance to extractions of the weak mixing angle $\sin^{2}\theta_{W}$ at the
LHC [30]. These studies are mostly restricted to the vicinity of the $Z$-boson
peak, $m_{\ell\bar{\ell}}\sim m_{Z}$ with $m_{\ell\bar{\ell}}$ being the
dilepton mass, though in a recent study by CMS [24] the forward-backward
asymmetry has been used to obtain a lower mass limit (of 4.4 TeV) on a
hypothetical $Z^{\prime}$ heavy gauge boson.

In this work, we assess to which extent different assumptions on the large-$x$
behavior of PDFs, as well as different estimates of the PDF uncertainty in
this region, may affect BSM searches, by specifically studying neutral-current
Drell-Yan production, and the forward-backward asymmetry in particular. To
this purpose, we explain the dependence of the general qualitative features of
the asymmetry on the behavior of PDFs, based on an understanding of the
analytic dependence of the asymmetry on the partonic luminosities. We then
present detailed computations of the forward-backward asymmetry at the LHC,
with realistic experimental cuts, using a variety of PDF sets.

We find that first, the large-$x$ PDF shape and uncertainty can differ
considerably between different PDF sets, with NNPDF4.0 [31] generally
displaying a more flexible shape and a wider uncertainty. And second, that all
PDF sets except NNPDF4.0 lead to a qualitative behavior of the asymmetry which
in the large-mass multi-TeV region reproduces the shape found around the
$Z$-peak region, even though there is no fundamental reason why this should be
the case We will then trace the observed behavior of the asymmetry to that of
the underlying PDFs.

The structure of the paper is the following. First in Sect. 2 we review the
leading-order (LO) expressions for the Drell-Yan differential distributions
and forward-backward asymmetry, in order to explain how the leading
qualitative behavior of the asymmetry — specifically the reason for an
asymmetry, and its sign — is related to the underlying parton luminosities. We
will also show that this LO picture is not qualitatively modified by higher-
order perturbative corrections. Then in Sect. 3 we investigate the way the
shape of the asymmetry (and specifically its sign) is determined by the
large-$x$ behavior of the PDFs. After discussing this in a toy model, we
examine current PDF sets: ABMP16 [32], CT18 [33], MSHT20 [34], and NNPDF4.0.
Specifically, we compare the behavior of the PDFs and the asymmetry as the
final-state dilepton invariant mass is varied. Finally, in Sect. 4 we present
predictions for high-mass DY production, specifically the forward-backward
asymmetry, at the LHC with realistic experimental cuts, and accounting for NLO
QCD and electroweak corrections. For completeness, we present in App. A a
comparison to results obtained using the previous, widely used NNPDF3.1 PDF
set.

##  2 Anatomy of Drell-Yan production

The aim of this section is to scrutinize the PDF dependence of the neutral-
current Drell-Yan differential cross-section and of the associated forward-
backward asymmetry by reviewing the LO kinematics, determining LO analytic
expressions, and finally comparing these analytical calculations to the
results of LO and NLO numerical simulations obtained using MadGraph5_aMC@NLO
[35] interfaced to PineAPPL [36, 37]. Specifically, we will relate the
behavior of the differential distribution and asymmetry to the relevant parton
luminosities.

###  2.1 Drell-Yan kinematics and cross-sections at LO

We consider dilepton production via the exchange of an electroweak neutral
gauge boson $Z/\gamma^{*}$ in proton-proton collisions:

$\mathrm{p}(k_{1})+\mathrm{p}(k_{2})\to
Z/\gamma^{*}(q)\to\ell(p_{\ell})+\bar{\ell}(p_{\bar{\ell}})+X\text{.}$ (2.1)

The hadronic differential cross-section ${\rm
d}\sigma^{\mathrm{p}\mathrm{p}\to\ell\bar{\ell}}$ is factorized in terms of
PDFs $f_{i}$ and the partonic cross sections ${\rm d}\hat{\sigma}_{ij}$ for
incoming partons of species $i,\,j$ as

${\rm
d}\sigma^{\mathrm{p}\mathrm{p}\to\ell\bar{\ell}}=\sum_{ij}\int\limits_{0}^{1}\\!{\rm
d}x_{1}{\rm d}x_{2}f_{i}(x_{1},\mu_{F}^{2})f_{j}(x_{2},\mu_{F}^{2}){\rm
d}\hat{\sigma}_{ij}(\hat{k}_{1}=x_{1}k_{1},\hat{k}_{2}=x_{2}k_{2}).$ (2.2)

In the sequel we will set the factorization scale $\mu_{F}$ to the invariant
mass of the gauge boson, i.e. the dilepton invariant mass, so
$\mu^{2}_{F}=m_{\ell\bar{\ell}}^{2}=(p_{\ell}+p_{\bar{\ell}})^{2}$. The
kinematics and Feynman diagram of the LO partonic process in the quark-
antiquark channel are shown in Fig. 2.1. We do not consider photon-initiated
processes, as they do not affect the qualitative features of our discussion.

$q$$\bar{q}$$\ell$$\bar{\ell}$$\hat{k}_{1}$$\hat{k}_{2}$$\gamma/Z$$q$$p_{\ell}$$p_{\bar{\ell}}$
Figure 2.1: Neutral-current Drell-Yan production at LO in the quark-antiquark
channel.

At LO, the momentum fractions of the two incoming partons are fully fixed by
knowledge of the invariant mass and rapidity of the gauge boson, i.e. of the
dilepton pair $y_{\ell\bar{\ell}}=(y_{\ell}+y_{\bar{\ell}})/2$:

$x_{1}=\frac{m_{\ell\bar{\ell}}}{\sqrt{s}}\exp(y_{\ell\bar{\ell}})\,,\quad
x_{2}=\frac{m_{\ell\bar{\ell}}}{\sqrt{s}}\exp(-y_{\ell\bar{\ell}})\,,$ (2.3)

where the center of mass energy of the hadronic collision is
$s=(k_{1}+k_{2})^{2}$ and at LO $m_{\ell\bar{\ell}}^{2}=\hat{s}=x_{1}x_{2}s$.
The absolute dilepton rapidity thus lies in the range
$|y_{\ell\bar{\ell}}|\leq\ln(\sqrt{s}/m_{\ell\bar{\ell}})$. Beyond LO there
might be extra radiation in the final state, so the LO kinematics provides a
lower bound on the momentum fractions of the incoming partons, and all values
of the momentum fractions such that $x_{1,2}\geq m_{\ell\bar{\ell}}/\sqrt{s}$
are allowed.

It is useful to define the so-called Collins-Soper angle $\theta^{*}$ [38],
which in the hadronic center-of-mass (CoM) frame is defined as

$\begin{split}\cos\theta^{*}&=\operatorname{sign}(y_{\ell\bar{\ell}})\cos\theta\,,\\\
\cos\theta&\equiv\frac{p_{\ell}^{+}p_{\bar{\ell}}^{-}-p_{\ell}^{-}p_{\bar{\ell}}^{+}}{m_{\ell\bar{\ell}}\sqrt{m_{\ell\bar{\ell}}^{2}+p_{\mathrm{T},\ell\bar{\ell}}^{2}}}\text{,}\quad
p^{\pm}=p^{0}\pm p^{3}\text{.}\end{split}$ (2.4)

It is easy to show that the Collins-Soper angle $\theta^{*}$ coincides with
the scattering angle of the lepton in the partonic CoM frame, $\bar{\theta}$.
The latter is defined in terms of the lepton momentum as

$\cos\bar{\theta}\equiv\frac{p^{z}_{\ell}}{m_{\ell\bar{\ell}}}\,,$ (2.5)

where the $z$ axis is along the direction of the incoming quark-antiquark
pair. In the partonic CoM frame, of course, $p^{z}_{\ell}=-p^{z}_{\bar{\ell}}$
and $y_{\ell\bar{\ell}}=0$, so

$p^{\pm}_{\ell}=p^{\mp}_{\bar{\ell}}=m_{\ell\bar{\ell}}\left(1\pm\cos{\bar{\theta}}\right)\,,$
(2.6)

and substituting in Eq. 2.4 it immediately follows that, taking the convention
$\operatorname{sign}(y_{\ell\bar{\ell}})=\operatorname{sign}(0)=+1$,
$\cos\theta^{*}=\cos\theta=\cos{\bar{\theta}}$. The expression of $\cos\theta$
in Eq. 2.4 is manifestly invariant upon boosts along the $z$ axis, so the
identification of $\theta$ with the CoM scattering angle $\bar{\theta}$
remains true in any reference frame.

Note that the definition Eq. 2.5 requires a choice for the positive direction
of the $z$ axis, which is usually taken along the direction of the incoming
fermion (quark). This direction is not experimentally accessible in proton-
proton collisions, so the Collins-Soper angle is defined by always taking the
positive $z$ axis in the direction of the boosted dilepton pair, i.e., at LO,
along the direction of the incoming quark with largest momentum fraction, i.e.
by supplementing in the definition a factor
$\operatorname{sign}(y_{\ell\bar{\ell}})$. Hence
$\cos\theta^{*}=\cos{\bar{\theta}}$ ($\cos\theta^{*}=-\cos{\bar{\theta}}$) if
the momentum fraction of the incoming quark (antiquark) is the largest.

The hard scattering matrix elements that enter the partonic cross-section in
Eq. 2.2 are the sum of a pure photon-exchange contribution, a photon-$Z$
interference term, and a pure $Z$-exchange contribution. Of course, in the
region $m_{\ell\bar{\ell}}\gtrsim m_{Z}$ these contributions are all of the
same order. Standard arguments [39] then imply that, because in the Standard
Model the photon coupling to leptons is vector while the $Z$ coupling is
chiral, the pure photon and pure $Z$ contributions to the cross-section are
necessarily even in $\cos\theta^{*}$ while the interference term is odd.

Specifically, at LO the fully differential hadronic cross-section can be
obtained from the well-known result [39] for $e^{+}e^{-}\to\mu^{+}\mu^{-}$ by
replacing the incoming lepton charges with those of the quarks, and accounting
for the PDFs, with the result

$\displaystyle\frac{{\rm d}^{3}\sigma}{{\rm d}m_{\ell\bar{\ell}}\,{\rm
d}y_{\ell\bar{\ell}}\,{\rm d}\cos\theta^{*}}$
$\displaystyle=\frac{\pi\alpha^{2}}{3m_{\ell\bar{\ell}}s}\left((1+\cos^{2}({\theta^{*}}))\sum_{q}S_{q}\left[f_{q}(x_{1},m_{\ell\bar{\ell}}^{2})f_{\bar{q}}(x_{2},m_{\ell\bar{\ell}}^{2})+f_{q}(x_{2},m_{\ell\bar{\ell}}^{2})f_{\bar{q}}(x_{1},m_{\ell\bar{\ell}}^{2})\right]\right.$
$\displaystyle\hskip
15.0pt+\left.\cos\theta^{*}\sum_{q}A_{q}\operatorname{sign}(y_{\ell\bar{\ell}})\left[f_{q}(x_{1},m_{\ell\bar{\ell}}^{2})f_{\bar{q}}(x_{2},m_{\ell\bar{\ell}}^{2})-f_{q}(x_{2},m_{\ell\bar{\ell}}^{2})f_{\bar{q}}(x_{1},m_{\ell\bar{\ell}}^{2})\right]\right)\,,$
(2.7)

where $\alpha$ is the QED coupling and the even (symmetric) and odd
(antisymmetric) couplings are given by

$\displaystyle S_{q}$ $\displaystyle=e_{l}^{2}e_{q}^{2}+P_{\gamma Z}\cdot
e_{l}v_{l}e_{q}v_{q}+P_{ZZ}\cdot(v_{l}^{2}+a_{l}^{2})(v_{q}^{2}+a_{q}^{2})$
$\displaystyle A_{q}$ $\displaystyle=P_{\gamma Z}\cdot
2e_{l}a_{l}e_{q}a_{q}+P_{ZZ}\cdot 8v_{l}a_{l}v_{q}a_{q}\,,$ (2.8)

in terms of the electric charges $e_{l}$, $e_{q}$ and the vector and axial
couplings $v_{l}$, $v_{q}$ and $a_{l}$, $a_{q}$ of the leptons and quarks, and
the propagator factors

$\displaystyle P_{\gamma Z}(m_{\ell\bar{\ell}})$
$\displaystyle=\frac{2m_{\ell\bar{\ell}}^{2}(m_{\ell\bar{\ell}}^{2}-m_{Z}^{2})}{\sin^{2}(\theta_{W})\cos^{2}(\theta_{W})\left[(m_{\ell\bar{\ell}}^{2}-m_{Z}^{2})^{2}+\Gamma_{Z}^{2}m_{Z}^{2}\right]}$
(2.9) $\displaystyle P_{ZZ}(m_{\ell\bar{\ell}})$
$\displaystyle=\frac{m_{\ell\bar{\ell}}^{4}}{\sin^{4}(\theta_{W})\cos^{4}(\theta_{W})\left[(m_{\ell\bar{\ell}}^{2}-m_{Z}^{2})^{2}+\Gamma_{Z}^{2}m_{Z}^{2}\right]},$
(2.10)

with $m_{Z}$ and $\Gamma_{Z}$ respectively the $Z$ mass and width and
$\theta_{W}$ the weak mixing angle. In Fig. 2.2 we display the symmetric
$S_{q}$ (left) and antisymmetric $A_{q}$ (right) couplings, Section 2.1, for
up-like and down-like quarks, as a function of the dilepton invariant mass
$m_{\ell\bar{\ell}}$. Both couplings are around a factor 2 larger for up-like
quarks than for down-like quarks, and become $m_{\ell\bar{\ell}}$-independent
for $m_{\ell\bar{\ell}}\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}1$ TeV, where they take the asymptotic
values $\bar{S}_{q}$, $\bar{A}_{q}$ obtained by substituting in Section 2.1
the large-mass expressions of the propagator factors

$\bar{P}_{\gamma
Z}=\frac{2}{\sin^{2}(\theta_{W})\cos^{2}(\theta_{W})},\qquad\bar{P}_{ZZ}=\frac{1}{\sin^{4}(\theta_{W})\cos^{4}(\theta_{W})},$
(2.11)

to which $P_{\gamma Z}$ and $P_{ZZ}$ respectively reduce up to
$O(m^{2}_{Z}/m_{\ell\bar{\ell}}^{2})$ corrections.

Figure 2.2: The symmetric $S_{q}$ (left) and antisymmetric $A_{q}$ (right)
couplings, Section 2.1, for up-like and down-like quarks, as a function of the
dilepton invariant mass $m_{\ell\bar{\ell}}$.

The interference term proportional to $A_{q}$ is odd in the Collins-Soper
angle $\cos\theta^{*}$, leading to a forward-backward scattering asymmetry. In
a proton-proton collision the initial state is completely symmetric, so the
quark and antiquark contributions to the cross-section Eq. 2.7 are necessarily
symmetric upon the interchange of the incoming quark and antiquark, with the
corresponding momentum fractions fixed at LO by Eq. 2.3. However, as
mentioned, there is a sign change in the relation between $\cos\theta^{*}$ and
$\cos\theta$ according to whether the incoming parton with largest momentum
fraction is a quark or an antiquark, i.e., when interchanging $x_{1}$ with
$x_{2}$ in the argument of the quark and antiquark PDFs, thereby leading to
the result of Eq. 2.7. This leads to a forward-backward asymmetry whenever the
quark and antiquark PDFs have different $x$ dependence.

In order to understand the relation of this forward-backward asymmetry in
terms of the behavior of the PDFs, it is convenient to rewrite the PDF
combinations that contribute to the differential cross-section Eq. 2.7 in
terms of symmetric and antisymmetric parton luminosities, defined as

$\displaystyle\mathcal{L}_{S,q}(m_{\ell\bar{\ell}},y_{\ell\bar{\ell}})$
$\displaystyle\equiv
f_{q}(x_{1},m_{\ell\bar{\ell}}^{2})f_{\bar{q}}(x_{2},m_{\ell\bar{\ell}}^{2})+f_{q}(x_{2},m_{\ell\bar{\ell}}^{2})f_{\bar{q}}(x_{1},m_{\ell\bar{\ell}}^{2})\,,$
$\displaystyle\mathcal{L}_{A,q}(m_{\ell\bar{\ell}},y_{\ell\bar{\ell}})$
$\displaystyle\equiv\operatorname{sign}(y_{\ell\bar{\ell}})\left[f_{q}(x_{1},m_{\ell\bar{\ell}}^{2})f_{\bar{q}}(x_{2},m_{\ell\bar{\ell}}^{2})-f_{q}(x_{2},m_{\ell\bar{\ell}}^{2})f_{\bar{q}}(x_{1},m_{\ell\bar{\ell}}^{2})\right]\,,$
(2.12)

where the momentum fractions $x_{1}$ and $x_{2}$ are given in terms of
$m_{\ell\bar{\ell}}$, $y_{\ell\bar{\ell}}$, and $\sqrt{s}$ in Eq. 2.3. Note
that both parton luminosities are invariant under the interchange
$x_{1}\leftrightarrow x_{2}$, upon which $y_{\ell\bar{\ell}}\to-
y_{\ell\bar{\ell}}$. In terms of these luminosities, the triple differential
cross-section Eq. 2.7 takes the compact form

$\frac{{\rm d}^{3}\sigma}{{\rm d}m_{\ell\bar{\ell}}\,{\rm
d}y_{\ell\bar{\ell}}\,{\rm
d}\cos\theta^{*}}=\frac{\pi\alpha^{2}}{3m_{\ell\bar{\ell}}s}\left((1+\cos^{2}({\theta^{*}}))\sum_{q}S_{q}\mathcal{L}_{S,q}(m_{\ell\bar{\ell}},y_{\ell\bar{\ell}})+\cos\theta^{*}\sum_{q}A_{q}\mathcal{L}_{A,q}(m_{\ell\bar{\ell}},y_{\ell\bar{\ell}})\right)\,,$
(2.13)

which explicitly displays its symmetry properties upon the transformation
$\cos\theta^{*}\to-\cos\theta^{*}$, equivalent to a charge conjugation
transformation $q\leftrightarrow\bar{q}$ and $\ell\leftrightarrow\bar{\ell}$.

The symmetric and antisymmetric parton luminosities Eq. 2.12 can also be
expressed in terms of the sum and difference of quark and antiquark PDFs,

$f_{q}^{\pm}\left(x,Q\right)=f_{q}\left(x,Q\right)\pm
f_{\bar{q}}\left(x,Q\right)\,,$ (2.14)

where $f_{q}^{-}$ is usually called the valence PDF combination, and
$f_{q}^{+}$ the total quark PDF. Note that at LO, and more generally in
factorization schemes in which PDFs are positive, such as $\overline{\rm MS}$
[40], $f^{+}_{q}$ is positive while $f_{q}^{-}$ in general is not, and
$f_{q}^{+}>|f_{q}^{-}|$. We can write the symmetric and antisymmetric parton
luminosities in Eq. 2.12 as

$\displaystyle\mathcal{L}_{S,q}(m_{\ell\bar{\ell}},y_{\ell\bar{\ell}})$
$\displaystyle=\frac{1}{2}\left(f_{q}^{+}(x_{1},m_{\ell\bar{\ell}}^{2})f_{q}^{+}(x_{2},m_{\ell\bar{\ell}}^{2})-f_{q}^{-}(x_{2},m_{\ell\bar{\ell}}^{2})f_{q}^{-}(x_{1},m_{\ell\bar{\ell}}^{2})\right)\,$
(2.15) $\displaystyle\mathcal{L}_{A,q}(m_{\ell\bar{\ell}},y_{\ell\bar{\ell}})$
$\displaystyle=\frac{\operatorname{sign}(y_{\ell\bar{\ell}})}{2}\left(f_{q}^{-}(x_{1},m_{\ell\bar{\ell}}^{2})f_{q}^{+}(x_{2},m_{\ell\bar{\ell}}^{2})-f_{q}^{-}(x_{2},m_{\ell\bar{\ell}}^{2})f_{q}^{+}(x_{1},m_{\ell\bar{\ell}}^{2})\,\right)\,.$
(2.16)

The symmetric luminosity $\mathcal{L}_{S,q}$ is of course positive, and it is
dominated by the
$f_{q}^{+}(x_{1},m_{\ell\bar{\ell}}^{2})f_{q}^{+}(x_{2},m_{\ell\bar{\ell}}^{2})$
term, which is always larger than the valence contribution
$f_{q}^{-}(x_{2},m_{\ell\bar{\ell}}^{2})f_{q}^{-}(x_{1},m_{\ell\bar{\ell}}^{2})$.
The sign of the antisymmetric combination, that in turn drives the sign of the
forward-backward asymmetry, is in general not determined uniquely. If $x_{1}$
is in the region of the valence peak, and $x_{2}$ in the small $x$ region,
then $f^{-}(x_{1},m_{\ell\bar{\ell}}^{2})\gg
f^{-}(x_{2},m_{\ell\bar{\ell}}^{2})$, and the antisymmetric luminosity is
positive provided only that the valence PDF is positive. As we will discuss in
Sect. 3, while this is indeed the case in the $Z$-peak region, it is actually
not necessarily the case in the high dilepton mass region relevant for BSM
searches.

###  2.2 Single-differential distributions and the forward-backward asymmetry

Starting from the triple differential cross section, Eq. 2.13, one can define
single differential distributions by integrating the other two kinematic
variables over the available phase space. In particular, the single-
differential distribution in the Collins-Soper angle $\theta^{*}$ is given by

$\frac{{\rm d}\sigma}{{\rm
d}\cos\theta^{*}}=\int\limits_{m_{\ell\bar{\ell}}^{{\rm
min}}}^{\sqrt{s}}\\!{\rm
d}m_{\ell\bar{\ell}}\\!\\!\int\limits_{\ln(m_{\ell\bar{\ell}}/\sqrt{s})}^{\ln(\sqrt{s}/m_{\ell\bar{\ell}})}\\!\\!{\rm
d}y_{\ell\bar{\ell}}\,\frac{{\rm d}^{3}\sigma}{{\rm d}m_{\ell\bar{\ell}}\,{\rm
d}y_{\ell\bar{\ell}}\,{\rm d}\cos\theta^{*}}\,,$ (2.17)

where $m_{\ell\bar{\ell}}^{{\rm min}}$ is a lower kinematic cut in the
dilepton invariant mass. Since Eq. 2.13 falls off steeply with
$m_{\ell\bar{\ell}}$, the region with $m_{\ell\bar{\ell}}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}m_{\ell\bar{\ell}}^{{\rm min}}$ will dominate the integral.
Given that the dependence of the fully differential cross-section Eq. 2.13 on
the Collins-Soper angle factorizes with respect to the PDF dependence, the
integration over rapidity and invariant mass does not affect the
$\cos\theta^{*}$ dependence, and the single-differential cross section Eq.
2.17 takes the simple form

$\frac{{\rm d}\sigma}{{\rm
d}\cos\theta^{*}}=(1+\cos^{2}\theta^{*})\sum_{q}g_{S,q}+\cos\theta^{*}\sum_{q}g_{A,q}\,,$
(2.18)

where the symmetric and antisymmetric coefficients $g_{S,q}$ and $g_{A,q}$
depend on the quark flavor and on the invariant mass cut
$m_{\ell\bar{\ell}}^{{\rm min}}$, but not on the Collins-Soper angle itself.
The contributions relevant for the forward-backward asymmetry, $g_{A,q}$, are
given at LO by

$g_{A,q}=\frac{\pi\alpha^{2}}{3s}\int\limits_{m_{\ell\bar{\ell}}^{{\rm
min}}}^{\sqrt{s}}\frac{{\rm
d}m_{\ell\bar{\ell}}}{m_{\ell\bar{\ell}}}A_{q}(m_{\ell\bar{\ell}})\\!\\!\int\limits_{\ln(m_{\ell\bar{\ell}}/\sqrt{s})}^{\ln(\sqrt{s}/m_{\ell\bar{\ell}})}\\!\\!{\rm
d}y_{\ell\bar{\ell}}\,\mathcal{L}_{A,q}(m_{\ell\bar{\ell}},y_{\ell\bar{\ell}})\,,$
(2.19)

which in the large-$m_{\ell\bar{\ell}}$ region, expressing the longitudinal
momentum integration in terms of $x_{1}$ (assuming $x_{1}~{}\geq x_{2}$),
becomes

$g_{A,q}=\frac{\pi\alpha^{2}\bar{A}_{q}}{3s}\int\limits_{m_{\ell\bar{\ell}}^{{\rm
min}}}^{\sqrt{s}}\frac{{\rm
d}m_{\ell\bar{\ell}}}{m_{\ell\bar{\ell}}}\\!\\!\int\limits_{m_{\ell\bar{\ell}}/\sqrt{s}}^{1}\\!\\!\frac{{\rm
d}x_{1}}{x_{1}}\,\mathcal{L}_{A,q}(m_{\ell\bar{\ell}},x_{1})+\mathcal{O}\left(\frac{m_{Z}^{2}}{m_{\ell\bar{\ell}}^{2}}\right)\,,$
(2.20)

where the $m_{\ell\bar{\ell}}$-independent effective couplings $\bar{A}_{q}$
are given substituting in Section 2.1 the expressions for the asymptotic
propagator factors Eq. 2.11.

Upon integration over the Collins-Soper angle, the antisymmetric contribution
vanishes: so for instance the rapidity distribution

$\frac{{\rm d}\sigma}{{\rm
d}y_{\ell\bar{\ell}}}=\int\limits_{m_{\ell\bar{\ell}}^{{\rm
min}}}^{\sqrt{s}}\\!{\rm d}m_{\ell\bar{\ell}}\int\limits_{-1}^{1}\\!\\!{\rm
d}\cos\theta^{*}\,\frac{{\rm d}^{3}\sigma}{{\rm d}m_{\ell\bar{\ell}}\,{\rm
d}y_{\ell\bar{\ell}}\,{\rm d}\cos\theta^{*}}\,,$ (2.21)

does not depend on terms proportional to $A_{q}$. Hence, for BSM searches in
which one is interested in the interference terms, as well as for PDF studies
in which one is interested in the valence-sea separation, the forward-backward
asymmetry is especially relevant. This observable is defined at the
differential level as

$A_{\rm fb}(\cos\theta^{*})\equiv\frac{\frac{{\rm d}\sigma}{{\rm
d}\cos\theta^{*}}(\cos\theta^{*})-\frac{{\rm d}\sigma}{{\rm
d}\cos\theta^{*}}(-\cos\theta^{*})}{\frac{{\rm d}\sigma}{{\rm
d}\cos\theta^{*}}(\cos\theta^{*})+\frac{{\rm d}\sigma}{{\rm
d}\cos\theta^{*}}(-\cos\theta^{*})}\,,\quad\cos\theta^{*}>0\,,$ (2.22)

which in terms of the coefficients introduced in Eq. 2.18 is given at LO by

$A_{\rm
fb}(\cos\theta^{*})=\frac{\cos\theta^{*}}{(1+\cos^{2}(\theta^{*}))}\frac{\sum_{q}g_{A,q}}{\sum_{q^{\prime}}g_{S,q^{\prime}}}\,,\quad\cos\theta^{*}>0\,.$
(2.23)

This shows that the dependence on $\cos\theta^{*}$ factorizes and the PDF
dependence only appears as an overall normalization factor depending on the
ratio of $\sum_{q}g_{A,q}$ and $\sum_{q}g_{S,q}$, which in turn depend on the
antisymmetric and symmetric partonic luminosities $\mathcal{L}_{A,q}$ and
$\mathcal{L}_{S,q}$ respectively. Note that the overall sign of $A_{\rm fb}$
remains in general undetermined.

Figure 2.3: The single-inclusive differential distribution in the Collins-
Soper angle $\cos\theta^{*}$, Eq. 2.17, and the corresponding forward-backward
asymmetry computed at LO, where the analytic calculation Eq. 2.22 is compared
with the numerical simulation based on MadGraph5_aMC@NLO interfaced to
PineAPPL. The bottom panels display the difference between the analytic and
numerical calculations relative to the Monte Carlo integration uncertainty.
The blue band indicates the $3\sigma$ uncertainty interval. One of the
replicas of the NNPDF4.0 NNLO PDF set is used as input to the calculation.

In order to illustrate concretely these results, in Fig. 2.3 we display the
single-inclusive differential distribution in $\cos\theta^{*}$, Eq. 2.17, and
the corresponding forward-backward asymmetry, Eq. 2.22 evaluated at LO for
$m_{\ell\bar{\ell}}^{\rm min}=$5\text{\,}\mathrm{TeV}$$. The single-
differential rapidity distribution Eq. (2.21) is also shown for reference in
Fig. 2.4. We display both a numerical evaluation based on MadGraph5_aMC@NLO
interfaced to PineAPPL, as well as analytic results found using the form Eq.
2.13 of the triple differential luminosity, with all the values of the
parameters entering Sections 2.1 to 2.10 set to the values used in the
MadGraph5_aMC@NLO runcard, and performing numerically the integrals in Eqs.
2.17 and 2.21. For validation purposes, no kinematic cuts are applied to the
rapidities and transverse momenta of final-state leptons. The PDF input is
taken to be given, for illustrative purposes, by one of the replicas of the
NNPDF4.0 NNLO set. The relative difference between the analytic and numerical
calculation is shown in the bottom panels of Fig. 2.3 and demonstrates perfect
agreement.

While the discussion so far has been presented at LO, its qualitative features
are unaffected by higher-order corrections. To illustrate this, in Fig. 2.5 we
compare the LO result from Fig. 2.3 to the corresponding NLO QCD result. The
bottom panels display the NLO $K$-factor for the $\cos\theta^{*}$ distribution
and the forward-backward asymmetry. Whereas the NLO $K$-factor in the
$\cos\theta^{*}$ distribution is quite large (around 40%) it exhibits only a
mild dependence on the Collins-Soper angle. For $A_{\rm fb}$, the $K$-factor
is at the 10% level and essentially independent of the value of
$\cos\theta^{*}$.

Figure 2.4: Same as Fig. 2.3 but now for the absolute dilepton rapidity
distribution $|y_{\ell\bar{\ell}}|$

Figure 2.5: Same as Fig. 2.3 now comparing the LO result to the NLO QCD result
obtained using MadGraph5_aMC@NLO. The $K$-factor is shown in the lower panel.

##  3 The forward-backward asymmetry and the large-$x$ PDFs

After our general discussion of the Drell-Yan process, we now investigate
proton structure at large-$x$, focusing on its impact on the forward-backward
asymmetry $A_{\rm fb}\left(\cos\theta^{*}\right)$ at large invariant masses.
First, we discuss the dependence of the qualitative features of the asymmetry,
and specifically its sign, on the behavior of the underlying PDFs: we
illustrate this in a toy model, and compare results to a simple and commonly
used approximation. Subsequently, we study the large-$x$ behavior of the PDFs
from several recent PDF sets: we compare PDFs, luminosities and the LO
asymmetry $A_{\rm fb}$ as a function of the dilepton invariant mass
$m_{\ell\bar{\ell}}$.

###  3.1 Qualitative features of $A_{\rm fb}$

In order to understand the main qualitative features of the $\cos\theta^{*}$
distribution and of the asymmetry $A_{\rm fb}$ and their dependence on the
properties of the underlying PDFs, it is instructive to evaluate predictions
based on the same computational setup adopted in Sect. 2, namely LO matrix
elements without kinematic cuts, using toy PDFs as input. We consider toy
quark and antiquark PDF with the form

$xf_{q}(x)=A_{q}x^{-a_{q}}(1-x)^{b_{q}}\,,\quad
xf_{\bar{q}}(x)=A_{\bar{q}}x^{-a_{\bar{q}}}(1-x)^{b_{\bar{q}}}\,,$ (3.1)

where $A_{q}$ and $A_{\bar{q}}$ are normalization constants, irrelevant for
this discussion. For simplicity we neglect the scale dependence of the PDFs.
We then compute the single-differential distribution Eq. (2.17) and the
asymmetry Eq. (2.22) with different assumptions on the large $x$-behavior of
these toy PDFs, i.e. different values of the large-$x$ exponents $b_{q}$,
$b_{\bar{q}}$.

Since the overall normalization does not affect the shape of the distribution,
we set $A_{q}=A_{\bar{q}}=1$. Furthermore, since we are not interested in the
small-$x$ behavior, we set $a_{q}=a_{\bar{q}}=1$. Hence, we consider simple
scenarios in which

$\displaystyle xf_{q}^{+}(x;b_{q},b_{\bar{q}})$
$\displaystyle=xf_{q}(x)+xf_{\bar{q}}(x)=x^{-1}\left[(1-x)^{b_{q}}+(1-x)^{b_{\bar{q}}}\right]\,,$
(3.2) $\displaystyle xf_{q}^{-}(x;b_{q},b_{\bar{q}})$
$\displaystyle=xf_{q}(x)-xf_{\bar{q}}(x)=x^{-1}\left[(1-x)^{b_{q}}-(1-x)^{b_{\bar{q}}}\right]\,,$
(3.3)

with different choices of the parameters $b_{q}$ and $b_{\bar{q}}$.
Specifically, we consider a scenario with $b_{q}<b_{\bar{q}}$, in particular
$(b_{q},b_{\bar{q}})=(3,5)$; a scenario with $(b_{q},b_{\bar{q}})=(3,3)$; and
a third scenario in which the quark PDFs at large-$x$ fall off more rapidly
than the antiquarks, $(b_{q},b_{\bar{q}})=(5,3)$.

Figure 3.1: The single-inclusive $\cos\theta^{*}$ distribution Eq. (2.17)
(left) and the corresponding forward-backward asymmetry (right panel) Eq.
(2.22) evaluated using the toy PDFs of Eq. (3.1). No kinematic cuts are
applied except for $m_{\ell\bar{\ell}}^{\rm min}=5$ TeV.

In Fig. 3.1 we display both the $\cos\theta^{*}$ single-inclusive distribution
Eq. (2.17) and the asymmetry Eq. (2.22). It is apparent that if the antiquark
PDFs fall off at large-$x$ faster than the quarks, i.e. when
$b_{q}<b_{\bar{q}}$ the forward-backward asymmetry is positive, while if the
converse is true it is negative. Of course if the quark and antiquark PDFs
behave in the same way there is no asymmetry. Indeed, the condition for a
negative asymmetry is (assuming $x_{1}>x_{2}$)

$\operatorname{sign}\left[\mathcal{L}_{A,q}\right]=\operatorname{sign}\left[\frac{f_{q}^{+}(x_{2})}{f_{q}^{+}(x_{1})}-\frac{f_{q}^{-}(x_{2})}{f_{q}^{-}(x_{1})}\right]=\operatorname{sign}\left[\frac{f_{q}(x_{2})}{f_{q}(x_{1})}-\frac{f_{\bar{q}}(x_{2})}{f_{\bar{q}}(x_{1})}\right]\,,\quad
x_{1}>x_{2}\,.$ (3.4)

Namely, what determines the sign of the antisymmetric luminosity, and thus of
the forward-backward asymmetry, is the relative rate of decrease of the quark
and antiquark, or valence and total quark PDFs.

In the simple model that we discussed, this rate of decrease is controlled by
the values of the exponents $b_{q}$ and $b_{\bar{q}}$. The simple model has
unphysical features, in that a negative asymmetry corresponds to a negative
valence distribution, which conflicts with sum rules. It is easy to construct
a more contrived model, in which the valence drops faster than the total quark
PDF, yet it remains positive. Also one could argue that Brodsky-Farrar
counting rules [41, 42] imply that $b_{q}<b_{\bar{q}}$ as $x\to 1$, so a
faster dropping antiquark is favored. However, counting rules are supposed to
only hold asymptotically, so whether they apply in any given region of $x$ is
a priori unclear. Again, it is easy to construct more contrived models in
which the value of the exponent or effective exponent is $x$-dependent.
However, our purpose is to highlight which features determine the sign of the
asymmetry, and not to construct an explicit PDF model. In fact, in Sect. 3.2
we will explicitly exhibit PDFs that do lead to a negative asymmetry, while
being consistent with sum rules and not leading to contradiction with
asymptotic counting.

It is interesting to note that different conclusions on the asymmetry could be
reached by using an approximation to the asymmetry which is quite accurate in
the $Z$ peak region. This approximation however turns out to fail at high
invariant mass. Indeed, the expression Eq. (2.16) of the antisymmetric
luminosity in terms of the valence and total PDF combinations $f_{q}^{+}$ and
$f_{q}^{-}$ PDF combinations suggests an approximation based on the
expectation that the valence is dominant at large $x$ and the sea is dominant
at small $x$. Assuming $x_{1}>x_{2}$, one then expects that

$\mathcal{L}_{A,u}(y_{\ell\bar{\ell}},m_{\ell\bar{\ell}})\approx\frac{1}{2}f_{u}^{-}(x_{1},m_{\ell\bar{\ell}}^{2})f_{u}^{+}(x_{2},m_{\ell\bar{\ell}}^{2})\,\,,\quad
x_{1}>x_{2}.$ (3.5)

This is clearly true in the $Z$-peak region, which motivates the suggestion to
use the measurement of $A_{\rm fb}$ as a means to constrain the valence quark
combinations [27].

Figure 3.2: The antisymmetric partonic luminosity $\mathcal{L}_{A,q}$, Eq.
(2.16), for the up and down quarks compared to the approximation Eq. (3.5) in
the case of NNPDF4.0 at $m_{\ell\bar{\ell}}=m_{Z}$ (top) and
$m_{\ell\bar{\ell}}=5$ TeV (bottom panels).

However, while Eq. (3.5) provides a satisfactory approximation in the $Z$-peak
region, it fails at larger $m_{\ell\bar{\ell}}$ values. Indeed, for on-shell
$Z$ production, with $\sqrt{s}=14$ TeV, for a dilepton rapidity with
$y_{\ell\bar{\ell}}\sim 2.5$, the limit of the acceptance region of ATLAS and
CMS, the colliding partons have $x_{1}=0.09$ and $x_{2}=6\times 10^{-4}$. So
indeed the contribution in which the valence PDF is evaluated at the smallest
$x$ value is highly suppressed. But for $m_{\ell\bar{\ell}}=5$ TeV, the
smallest value of $x_{2}$, attained when $x_{1}=1$, is $x_{2}=0.35$: so both
momentum fractions are large and in fact to the right of the valence peak. In
such case, there is no obvious hierarchy between the different terms that
contribute to to antisymmetric luminosity $\mathcal{L}_{A,q}$.

This is illustrated in Fig. 3.2, where we compare the antisymmetric luminosity
$\mathcal{L}_{A,q}$ for the up and down quarks to the approximation Eq. (3.5),
evaluated with NNPDF4.0 NNLO, in the $Z$-peak region
$m_{\ell\bar{\ell}}=m_{Z}$ and at $m_{\ell\bar{\ell}}=5$ TeV. While indeed for
$m_{\ell\bar{\ell}}=m_{Z}$ Eq. (3.5) reproduces the exact luminosity, this is
not the case for $m_{\ell\bar{\ell}}\gg m_{Z}$: both the magnitude and the
shape of the luminosity are very different. This qualitative behavior is
common to all PDF sets: the approximation fails equally badly regardless of
the PDF set.

We conclude that there is no simple relation between the sign of the asymmetry
and that of the valence PDF, and that the behavior of the asymmetry must be
determined by studying the large-$x$ behavior of the quark and antiquark PDFs.

###  3.2 Parton distributions

We assess now the large-$x$ behavior of the quark and antiquark PDFs in
different recent PDF determinations: specifically, we compare ABMP16, CT18,
NNPDF4.0, and MSHT20. For completeness, in App. A we also present results
obtained with the widely used NNPDF3.1 [43] set.

Figure 3.3: Comparison of the $xf^{+}_{q}$ (top) and $xf_{q}^{-}$ (bottom)
quark PDF combinations for the up, down, strange, and charm quarks, evaluated
at $m_{\ell\bar{\ell}}=5$ TeV for NNPDF4.0 NNLO. The right panels display the
relative 68% CL uncertainties. The two vertical lines indicate $x_{\rm
min}=m_{\ell\bar{\ell}}^{2}/s$, the smallest allowed value of $x$ for dilepton
DY production for a collider CoM energy $\sqrt{s}=14$ TeV, and the value of
$x$ corresponding to a symmetric partonic collision $x_{1}=x_{2}$, namely
$x_{\rm sym}=m_{\ell\bar{\ell}}/\sqrt{s}$.

First, we provide a qualitative assessment of the relative size of the PDFs
corresponding to individual quark flavors, both for the total and valence
PDFs. In Fig. 3.3 we compare the total $xf^{+}_{q}$ and valence $xf_{q}^{-}$
quark PDF combinations for the up, down, strange, and charm quarks, evaluated
at $m_{\ell\bar{\ell}}=5$ TeV with the NNPDF4.0 NNLO PDF set. The right panels
display the corresponding relative 68% CL uncertainties. Note that, because of
the way uncertainties are delivered by the various groups, in this and all
subsequent plots uncertainties for NNPDF are given are confidence levels (not
necessarily Gaussian) determined from the Monte Carlo replica sample, and thus
subject to point-to-point fluctuations, while for all other groups these are
one-$\sigma$ Gaussian intervals determined from a Hessian PDF representation.

The leftmost vertical line indicates $x_{\rm min}=m_{\ell\bar{\ell}}^{2}/s$,
the smallest allowed value of $x$ for dilepton DY production with invariant
mass $m_{\ell\bar{\ell}}=5$ TeV for a collider CoM energy $\sqrt{s}=14$ TeV.
The rightmost vertical line corresponds to the value of $x$ in a symmetric
partonic collision where $x_{1}=x_{2}$, namely $x_{\rm sym}\equiv
m_{\ell\bar{\ell}}/\sqrt{s}$.

From Fig. 3.3 one can observe that for $x\lesssim 0.3$ there is a clear
hierarchy $f_{u}^{+}>f_{d}^{+}>f_{s}^{+}>f_{c}^{+}$, while for larger $x$
values the strange and charm PDFs become of comparable magnitude. The up and
down quarks, both for $xf^{+}_{q}$ and $xf^{-}_{q}$, are significantly larger
than the second-generation quark PDFs until $x\simeq 0.7$, and hence dominate
the large-$m_{\ell\bar{\ell}}$ differential distributions in Drell-Yan
production. PDF uncertainties grow rapidly with $x$, reflecting the lack of
direct experimental constraints. The same qualitative behavior of the lighter
versus heavier flavor PDFs is observed for other PDF sets. Given the hierarchy
$f_{u}^{\pm},f_{d}^{\pm}\gg f_{s}^{\pm},f_{c}^{\pm}$, in the following we will
discuss only the behavior of the first-generation quark and antiquark PDFs
which are those relevant for the interpretation of neutral-current Drell-Yan
production in the kinematic region used for BSM searches.

Figure 3.4: The up and down quark and antiquark PDFs evaluated at
$m_{\ell\bar{\ell}}=5$ TeV for NNPDF4.0, CT18, MSHT20, and ABMP16 in the $x$
region relevant for high-mass Drell-Yan production. The upper panels display
the absolute PDFs, the middle ones their ratio to the central NNPDF4.0 value,
and the bottom panels the relative 68% CL uncertainties. The vertical lines in
the top row indicate the values of $x_{\rm min}=m_{\ell\bar{\ell}}^{2}/s$ and
in the central row those of $x_{\rm sym}=m_{\ell\bar{\ell}}/\sqrt{s}$ for
three different values $m_{\ell\bar{\ell}}=3,\,5,\,7$ TeV. Note that in the
second row the range on the $y$ axis is not the same for quarks and
antiquarks, and in the third row also for up and down quarks. Note also that
the PDFs, their ratios and their uncertainties are essentially unchanged in
the displayed large-$x$ region in the range $1~{}{\rm
TeV}<m_{\ell\bar{\ell}}<7$ TeV.

We next compare the large-$x$ behavior of the four PDF sets ABMP16, CT18,
MSHT20, and NNPDF4.0 in Fig. 3.4 for $m_{\ell\bar{\ell}}=5$ TeV. We display
from top to bottom the absolute PDFs, their ratio to the central NNPDF4.0
value, and their relative 68% CL uncertainties. As in the case of Fig. 3.3, we
indicate with two vertical lines the values of $x_{\rm min}$ and $x_{\rm
sym}$, both for $m_{\ell\bar{\ell}}=5$ TeV, and for a smaller and a larger
value of $m_{\ell\bar{\ell}}$, namely for $m_{\ell\bar{\ell}}=3$ TeV and
$m_{\ell\bar{\ell}}=7$ TeV. For clarity, the values of $x_{\rm min}$ are only
shown in the top row of plots, and the values of $x_{\rm sym}$ in the central
row. Note that the scale dependence of the PDFs in this range of $x$ and
invariant mass is very slight. Indeed, the PDFs shown in Fig. 3.3 are
essentially unchanged at $m_{\ell\bar{\ell}}=3$ TeV or $m_{\ell\bar{\ell}}=7$
TeV; only the corresponding ranges of $x_{1}$, $x_{2}$ vary significantly.

Good agreement between all PDF sets is found up to around $x\simeq 0.4$. For
$m_{\ell\bar{\ell}}=5$ TeV this corresponds to the value of $x_{\rm sym}$,
i.e. central rapidity. For larger values of $x\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.4$, the up quark
PDF $xf_{u}$ from the NNPDF4.0 set is somewhat suppressed in comparison to the
other three sets, which in turn agree among each other. A rather stronger
suppression of NNPDF4.0 in comparison to CT18 is observed for the down quark,
with MSHT20 and ABMP16 in a somewhat intermediate situation. The opposite
behavior is found in the same region $x\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.4$ for antiquark
PDFs $xf_{\bar{u}}$ and $xf_{\bar{d}}$: namely, the NNPDF4.0 PDF is
significantly larger than that of the other sets. It follows that for a lower
invariant mass value $m_{\ell\bar{\ell}}=3$ TeV, all PDF sets are in agreement
in the $x$ range in which they are probed, while for a higher value
$m_{\ell\bar{\ell}}=7$ TeV the disagreement between NNPDF4.0 and the other PDF
sets is present for most of the $x\geq x_{\rm min}$ range.

It is interesting to observe that in the region with $0.4\lesssim x\lesssim
0.6$ the PDFs are constrained by some fixed-target DIS structure functions and
by forward $W$ and $Z$ production data from LHCb. Hence, at the edge of the
data region NNPDF4.0 starts disagreeing with the other global PDF sets
considered here, with the disagreement getting more marked as $x$ grows
outside the region covered by the data. Qualitatively, NNPDF4.0 is
characterized by the fact that the quark PDFs drop faster as a function of
$x$, and the antiquark PDFs drop less fast as $x$ grows towards $x=1$. As we
will show next, this feature will lead to significant differences in the
antisymmetric PDF luminosities $\mathcal{L}_{A,q}$ as the value of the
dilepton invariant mass $m_{\ell\bar{\ell}}$ is increased.

The relative PDFs uncertainties, shown in the lower panels in Fig. 3.4 in all
cases grow with $x$ (see also Fig. 3.3). The largest PDF uncertainties
correspond to either CT18 or NNPDF4.0, depending on the $x$ range and the PDF
flavor. Specifically, the NNPDF4.0 uncertainties are largest for $f_{d}$ in
the region $x\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.6$ and for $f_{\bar{u}}$ and
$f_{\bar{d}}$ when $0.3\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}x\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.5$. The smallest
PDF uncertainties are displayed by ABMP16 and MSHT20.

The different behavior of the rate of decrease with $x$ of PDFs in the large
$x$ region, specifically comparing NNPDF4.0 to other PDF sets, can be seen
most clearly from a comparison off effective asymptotic exponents [44]

$\beta_{a,q}(x,Q)\equiv\frac{\partial\ln|xf_{q}(x,Q)|}{\partial\ln(1-x)}\,,$
(3.6)

which of course for PDFs of the form of Eq. (3.1) just coincide with the
exponent $b$ up to $O(1-x)$ corrections. In Fig. 3.5 we compare the values of
$\beta_{a,q}(x,m_{\ell\bar{\ell}})$ for ABMP16, CT18, MSHT20, and NNPDF4.0
evaluated at $m_{\ell\bar{\ell}}=5$ TeV for the up and down quark and
antiquark PDFs in the $x$ range of Fig. 3.3.

It is clear that while all PDF sets have a similar effective asymptotic
exponent for $x\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip
1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.35$, a different behavior of
NNPDF4.0 in comparison to other determinations sets in for $x\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.4$.
Specifically, for quarks the NNPDF4.0 exponents are always larger, and for
antiquarks smaller than those found with other PDF sets. Interestingly,
whereas for the up quark the effective exponent $\beta_{a,u}$ is approximately
constant for all PDF sets when $x\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.4$, with the
NNPDF4.0 value being just slightly higher and slowly increasing, for the down
quark and all antiquarks this approximately constant behavior is seen for
other PDF sets but not for NNPDF4.0. Specifically, for the NNPDF4.0 down quark
the exponent slowly but markedly increases for $x\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.3$, together with
its uncertainty. In the case of NNPDF4.0 for both antiquarks the exponent
rapidly drops in the region $0.3\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}x\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}0.4$.
This is consistent with the observation at the PDF level (Fig. 3.4) that for
NNPDF4.0 at large-$x$, as compared to the other groups, the up and especially
the down quark fall off more rapidly, while the antiquark PDFs drop more
slowly. Note in particular that for the down PDF the antiquark effective
exponent is significantly smaller than the quark effective exponent for all
$x\mathrel{\hbox to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise
1.0pt\hbox{$>$}}0.4$.

Figure 3.5: The large-$x$ asymptotic exponents
$\beta_{a,q}(x,m_{\ell\bar{\ell}})$, defined in Eq. (3.6), for ABMP16, CT18,
MSHT20, and NNPDF4.0 evaluated at $m_{\ell\bar{\ell}}=5$ TeV for the up and
down quark and antiquark PDFs.

The fact that a modification in behavior of the effective down quark and
especially antiquark PDFs is observed at the edge of the data region for
NNPDF4.0, but not for other PDF sets, suggests that this might be related to
the fact that NNPDF4.0 generally adopts a more flexible PDF parametrization in
comparison to other groups. Conversely, the fact that other groups display
similar behaviors suggests that this is related to their common choice of
parametrizing the large $x$ behavior of PDFs as $(1-x)^{\beta_{i}}$, with the
exponents $\beta_{i}$ fixed for each PDF flavor or combination of flavors.
Also, the uncertainties on the effective exponents
$\beta_{a,q}(x,m_{\ell\bar{\ell}})$ tend to be larger for NNPDF4.0 (and also
to a lesser extent for CT18) in comparison to those of other groups. Note
however that the full PDF uncertainty contains also a contribution from the
overall magnitude, which is not captured by the effective exponents displayed
here.

###  3.3 Parton luminosities

We finally turn to the behavior of parton luminosities, with particular regard
for the antisymmetric combination which is relevant for the forward-backward
asymmetry. As for PDFs, we first assess the qualitative features of the
luminosities corresponding to different quark flavors. Specifically, the
symmetric $\mathcal{L}_{S,q}$ and antisymmetric $\mathcal{L}_{A,q}$
luminosities Eq. (2.12) for individual flavors are displayed in Fig. 3.6,
evaluated with NNPDF4.0 NNLO for $m_{\ell\bar{\ell}}=5$ TeV and $\sqrt{s}=14$
TeV. The left panels display the absolute luminosities (in logarithmic and
linear scale respectively for the $y$ and $x$ axes) while the right panels
show the corresponding PDF uncertainties (relative and absolute for
$\mathcal{L}_{S,q}$ and $\mathcal{L}_{A,q}$, respectively). The bottom and top
$x$-axes in each plot show respectively the values of $x_{1}$ and $x_{2}$ at
which the luminosities are being evaluated, within the allowed range $x\geq
x_{\rm sym}=m_{\ell\bar{\ell}}/\sqrt{s}$, with the convention $x_{1}>x_{2}$.

Figure 3.6: The symmetric $\mathcal{L}_{S,q}$ (top) and antisymmetric
$\mathcal{L}_{A,q}$ (bottom) parton luminosities (left) and relative
uncertainties (right) evaluated with NNPDF4.0 NNLO at $m_{\ell\bar{\ell}}=5$
TeV and $\sqrt{s}=14$ TeV. The bottom and top $x$-axes in each plot show
respectively the values of $x_{1}$ and $x_{2}$ at which the luminosities are
being evaluated, within the allowed range $x\geq x_{\rm
sym}=m_{\ell\bar{\ell}}/\sqrt{s}$, with the convention $x_{1}>x_{2}$.

The symmetric parton luminosities exhibit of course the same hierarchy between
flavors as the corresponding PDF plots of Fig. 3.3. The luminosity
$\mathcal{L}_{S,q}$ drops rapidly for $x_{1}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.6$. PDF
uncertainties depend weakly on $x$ up to $x_{1}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}0.8$, after which
they blow up, and range between $\sim 20\%$ for the up quark luminosity to
$\sim 60\%$ for the charm quark one, with down and strange intermediate and of
similar magnitude.

As displayed in Fig. 3.7, the light quark symmetric luminosities of other
global PDF sets are qualitatively similar. We show $\mathcal{L}_{S,u}$,
$\mathcal{L}_{S,d}$, and their weighted sum that enters the enters the
symmetric coefficient $g_{S,q}$ in Eq. (2.18) for the NNPDF4.0, ABMP16, CT18,
and MSHT20 at $m_{\ell\bar{\ell}}=5$ TeV. The luminosities are multiplied by
the effective charges $S_{q}$ defined in Eq. (2.1), and the bottom panels
display the corresponding 68% CL PDF uncertainties. Good agreement between the
four sets, with a similar shape of $\mathcal{L}_{S,q}$, is observed. The PDF
luminosities for the dominant $\mathcal{L}_{S,u}$ contribution are the largest
for NNPDF4.0.

Figure 3.7: The symmetric parton luminosities
$\mathcal{L}_{S,q}(x_{1},m_{\ell\bar{\ell}})$ for the NNPDF4.0, ABMP16, CT18,
and MSHT20 NNLO PDF sets for dilepton invariant masses of
$m_{\ell\bar{\ell}}=5$ TeV. The luminosities are multiplied by the effective
charges $S_{q}$ defined in Eq. (2.1). From left to right, we display
$\mathcal{L}_{S,u}$, $\mathcal{L}_{S,d}$, and their weighted sum that enters
the coefficient $g_{S,q}$ in Eq. (2.18). The bottom panels display the
relative 68% CL PDF uncertainties.

Turning to the antisymmetric PDF luminosities $\mathcal{L}_{A,q}$, we note
that, for NNPDF4.0, while the up luminosity is positive, the central value of
the down luminosity is negative, though the luminosity is compatible with zero
at the one sigma level. Recalling from Fig. 3.3 that $xf_{d}^{-}$ itself is
positive for all values of $x$, this provides an explicit example in which the
condition Eq. (3.4) is satisfied without the valence combination being
negative. We conclude that for NNPDF4.0, the faster drop of the quark
distribution and slower drop of the antiquark distribution that was displayed
by the effective exponents of Fig. 3.5 leads to a negative antisymmetric
luminosity, in agreement with Eq. (3.4). The absolute PDF uncertainties are of
a similar size for $\mathcal{L}_{A,u}$ and $\mathcal{L}_{A,d}$, with a
different shape reflecting the underlying central values.

We compare in Fig. 3.8 the behavior of the antisymmetric luminosities for all
PDF sets for $m_{\ell\bar{\ell}}=3$ TeV (top) and $m_{\ell\bar{\ell}}=5$ TeV
(bottom). In order to facilitate the understanding of the way the PDF behavior
determines that of the asymmetry, we show both the contribution of individual
flavors and the total contribution to the antisymmetric coefficient $g_{A,q}$
of Eq. (2.19). Namely, in Fig. 3.8 the luminosities corresponding to
individual flavors are multiplied by the corresponding flavor-dependent
effective charges $A_{q}$ defined in Eq. (2.1): from left to right we display
$\mathcal{L}_{A,u}$, $\mathcal{L}_{A,d}$, and their weighted sum which
determines the sign and magnitude of the total forward-backward asymmetry. The
corresponding absolute PDF uncertainties for each of the four PDF sets are
displayed in Fig. 3.9.

Figure 3.8: The antisymmetric parton luminosities
$\mathcal{L}_{A,q}(x_{1},m_{\ell\bar{\ell}})$ for the NNPDF4.0, ABMP16, CT18,
and MSHT20 NNLO PDF sets for dilepton invariant masses of
$m_{\ell\bar{\ell}}=3$ TeV (top) and $m_{\ell\bar{\ell}}=5$ TeV (bottom). The
luminosities are multiplied by the effective charges $A_{q}$ defined in Eq.
(2.1). From left to right, we display $\mathcal{L}_{A,u}$,
$\mathcal{L}_{A,d}$, and their weighted sum that enters the coefficient
$g_{A,q}$ Eq. (2.19).

Figure 3.9: Same as Fig. 3.8 now for the absolute PDF uncertainties.

Fig. 3.8 shows that for ABMP16, CT18, and MSHT20 the antisymmetric parton
luminosities depend only mildly on $m_{\ell\bar{\ell}}$, whereas for NNPDF4.0
they exhibit a strong $m_{\ell\bar{\ell}}$ dependence. Indeed, for dilepton
invariant masses of $m_{\ell\bar{\ell}}=3$ TeV there is good agreement between
the three groups, but for $m_{\ell\bar{\ell}}=5$ TeV the NNPDF4.0 up quark
luminosity, while preserving a similar valence-like shape, is suppressed by a
factor 2 in comparison to other groups, and the down quark luminosity becomes
compatible with zero with a negative central value, as already noted. For all
PDF sets and $m_{\ell\bar{\ell}}$ values the weighted sum is dominated by the
up quark contribution. The strong scale dependence of $\mathcal{L}_{A,q}$ in
NNPDF4.0 reflects the underlying PDF behavior seen in Fig. 3.4 and highlighted
by the effective exponents Fig. 3.5. As the scale $m_{\ell\bar{\ell}}$
increases, a range of increasingly large $x$ values is probed, for which, in
the case of NNPDF4.0, the quark effective exponent slowly increases and the
antiquark exponent rapidly drops. This leads to a negative asymmetry,
following Eq. (3.4).

A comparison of the corresponding PDF uncertainties, displayed in Fig. 3.9,
clearly shows the transition from the data region to the extrapolation region.
For $m_{\ell\bar{\ell}}=3$ TeV the uncertainty $\delta\mathcal{L}_{A,u}$ is
generally small for all sets, with CT18 showing a somewhat larger uncertainty
for the up quark, and comparable uncertainties for the down quark for all PDF
sets. As the scale increases to $m_{\ell\bar{\ell}}=5$ TeV, where the
large-$x$ region is probed, the uncertainty increases, though more markedly
for NNPDF4.0. For all PDF sets but NNPDF4.0, the uncertainty is approximately
unchanged when the scale is further increased, while for NNPDF4.0 it grows
markedly.

Finally, in Fig. 3.10 we display for all PDF sets the ratio of antisymmetric
to symmetric couplings

$R_{\rm
fb}\equiv\frac{\sum_{q}g_{A,q}}{\sum_{q^{\prime}}g_{S,{q^{\prime}}}}\,,$ (3.7)

that, according to Eq. (2.23), determines at leading order the sign and
magnitude of the forward-backward asymmetry distribution $A_{\rm
fb}(\cos\theta^{*})$. The symmetric and antisymmetric coefficients are
obtained by integrating the corresponding symmetric $\mathcal{L}_{S,q}$ and
antisymmetric $\mathcal{L}_{A,q}$ partonic luminosities according to Eq.
(2.19), and the result is shown as a function of the lower integration cut
$m_{\ell\bar{\ell}}^{\rm min}$. In all cases the correlation between PDF
uncertainties in the numerator and the denominator are kept into account.

Figure 3.10: The coupling ratio $R_{\rm fb}$, Eq. (3.7), that enters the
forward-backward asymmetry $A_{\rm fb}(\cos\theta^{*})$ at LO, Eq. (2.23), for
different PDF sets, as a function of the lower cut in the dilepton invariant
mass $m_{\ell\bar{\ell}}^{\rm min}$.

Fig. 3.10 shows that, consistently with the behavior of the luminosity of Fig.
3.8, for $m_{\ell\bar{\ell}}^{{\rm min}}\mathrel{\hbox to0.0pt{\lower
4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$<$}}3$ TeV results agree
within uncertainties for all PDF sets. The situation is different for higher
dilepton invariant masses $m_{\ell\bar{\ell}}^{{\rm min}}\mathrel{\hbox
to0.0pt{\lower 4.0pt\hbox{\hskip 1.0pt$\sim$}\hss}\raise 1.0pt\hbox{$>$}}3$
TeV: the ratio $R_{\rm fb}$ starts to decrease for NNPDF4.0, while it remains
approximately constant for the other PDF sets. In particular, for NNPDF4.0 the
coupling ratio vanishes around $m_{\ell\bar{\ell}}^{{\rm min}}\sim 5$ TeV, and
it becomes negative for yet larger $m_{\ell\bar{\ell}}^{{\rm min}}$ values. It
follows that the forward-backward asymmetry in high-mass Drell-Yan production
should decrease and eventually vanish (and possibly even turn negative) in
NNPDF4.0 as the $m_{\ell\bar{\ell}}^{{\rm min}}$ cut is increased, while for
CT18, MSHT20, and ABMP16 it should remain positive with a similar magnitude
irrespective of the cut $m_{\ell\bar{\ell}}^{{\rm min}}$ adopted.

Figure 3.11: The absolute (left) and relative (right panel) uncertainties in
the coupling ratio $R_{\rm fb}$ shown in Fig. 3.10.

Fig. 3.11 displays the absolute and relative uncertainties associated to the
coupling ratio $R_{\rm fb}$. We observe that NNPDF4.0 shows the most marked
increase of the uncertainties in $R_{\rm fb}$ as $m_{\ell\bar{\ell}}^{\rm
min}$ grows. For instance, for $m_{\ell\bar{\ell}}^{\rm min}\gtrsim 4$ TeV the
absolute PDF uncertainty in NNPDF4.0 is about twice as large as that found
using CT18 four times as large as MSHT20, and about one order of magnitude
larger than ABMP16. This trend is magnified for the relative uncertainties due
to the decrease in the central value of $R_{\rm fb}$ as
$m_{\ell\bar{\ell}}^{\rm min}$ increases.

##  4 The Drell-Yan forward-backward asymmetry at the LHC

After the qualitative discussion of the previous sections, here we present
results for the $\cos\theta^{*}$ distributions Eq. (2.17) and the forward-
backward asymmetry Eq. (2.22), with NLO QCD and electroweak corrections
included and with realistic selection and acceptance cuts for the LHC at
$\sqrt{s}=14$ TeV and different values of the invariant mass
$m_{\ell\bar{\ell}}$ relevant for SM studies and BSM searches.

Computations are performed using MadGraph5_aMC@NLO [35], interfaced to
PineAPPL [36, 37] to generate fast interpolation grids. In order to account
for realistic detector acceptances, we impose phase-space cuts on the
transverse momentum and the pseudo-rapidity of the two leading leptons,

$p_{T}^{\ell}>10~{}{\rm GeV}\,,\qquad|\eta_{\ell}|<2.4\,.$ (4.1)

We then consider various regions of dilepton invariant mass
$m_{\ell\bar{\ell}}$: either close to the $Z$-boson peak ($60$ GeV
$<m_{\ell\bar{\ell}}<120$ GeV), relevant for precision SM studies, or the
high-mass region relevant for BSM searches, with various choices of a lower
mass invariant cutoff ($m_{\ell\bar{\ell}}>3,4,5,6$ TeV). In all cases, in
order to facilitate the interpretation of hadron-level results and the
connection to the discussion of the PDF features from Sect. 3, we also provide
results for the two partonic channels that give the largest contribution to
the cross-section. As in Sect. 3, we compare results obtained using the
ABMP16, CT18, MSHT20, and NNPDF4.0 PDF sets. In all cases, we use the NNLO
sets corresponding to the value $\alpha_{s}(m_{Z})=0.118$ of the strong
coupling. Results obtained using the NNPDF3.1 PDF set are reported in App. A.

Figure 4.1: The differential distribution in absolute dilepton rapidity
$|y_{\ell\bar{\ell}}|$, given in Eq. (2.21), for dilepton invariant masses of
$m_{\ell\bar{\ell}}>5$ TeV for neutral current Drell-Yan production at the LHC
14 TeV, obtained using ABMP16, CT18, MSHT20, and NNPDF4.0 NNLO PDFs with
$\alpha_{s}(m_{Z})=0.118$. All uncertainties shown are 68% CL PDF
uncertainties, computed at NLO QCD with realistic cuts (see text). We show the
absolute distributions (top), relative uncertainties (normalized to the
central curve of each set, middle) and the pull with respect to the NNPDF4.0
result, Eq. (4.2) (bottom). For the central NNPDF4.0 prediction the
contributions of the $u\bar{u}+c\bar{c}$ and $d\bar{d}+s\bar{s}+b\bar{b}$
parton subchannels are also shown.

Before considering the angular distributions, in Fig. 4.1 we display the
differential distribution in absolute dilepton rapidity
$|y_{\ell\bar{\ell}}|$, defined in Eq. (2.21), for a dilepton invariant mass
of $m_{\ell\bar{\ell}}>5$ TeV. This is the kinematic region relevant for
searches of high-mass resonances in the dilepton channel at the LHC, e.g. [13,
45]. We display the absolute differential distributions with the 68% CL PDF
uncertainties (top), the relative PDF uncertainty (center) normalized for each
PDF set to the corresponding central prediction, and the pull between the
NNPDF4.0 result, taken as a reference, and other sets (bottom). This pull is
defined as

${\rm
Pull}_{i}=\frac{\sigma^{(0)}_{2,i}-\sigma^{(0)}_{1,i}}{\sqrt{\left(\delta\sigma_{2,i}\right)^{2}+\left(\delta\sigma_{1,i}\right)^{2}}}\,,\qquad
i=1,\ldots,n_{\rm bin}\,,$ (4.2)

where $\sigma^{(0)}_{1,i}$ and $\sigma^{(0)}_{2,i}$ are the central values of
the theory prediction in the $i$-th bin of the distribution and
$\delta\sigma_{1,i}$, $\delta\sigma_{2,i}$ are the corresponding PDF
uncertainties. For the central NNPDF4.0 prediction in the upper panel we also
display the contributions from the dominant parton subchannels, namely
$u\bar{u}+c\bar{c}$ and $d\bar{d}+s\bar{s}+b\bar{b}$.

As discussed in Sect. 2, the $|y_{\ell\bar{\ell}}|$ distribution depends on
the symmetric partonic luminosities $\mathcal{L}_{S,q}$, Eq. (2.15), which in
turn are driven by the total PDFs $xf^{+}_{q}$. The $|y_{\ell\bar{\ell}}|$
distribution is dominated by the $u\bar{u}$ contribution and its qualitative
behavior is found to be similar for the four PDF sets considered. PDF
uncertainties are the largest in NNPDF4.0, ranging between 25% and 50%, and
the pull between NNPDF4.0 and CT18 and MSHT20 is at most at the $1.5\sigma$
level, and slightly larger with ABMP16. The dependence of the
$|y_{\ell\bar{\ell}}|$ distribution on the dilepton mass $m_{\ell\bar{\ell}}$
is moderate, and the same qualitative features are obtained if
$m_{\ell\bar{\ell}}$ is lowered down to the $Z$-peak region, or raised to yet
higher values. Hence, for the absolute rapidity distribution there is a
reasonable agreement between all PDF sets for all scales considered.

Figure 4.2: Same as Fig. 4.1, now for the differential distribution in
$\cos\theta^{*}$ (left) and the corresponding forward-backward asymmetry
$A_{\rm fb}(\cos\theta^{*})$ (right), in the $Z$-peak region defined by
$60~{}{\rm GeV}<m_{\ell\bar{\ell}}<120$ GeV.

We now turn to the differential distribution in $\cos\theta^{*}$ and the
corresponding forward-backward asymmetry $A_{\rm fb}(\cos\theta^{*})$. We
first consider the $Z$-peak region, $60~{}{\rm GeV}<m_{\ell\bar{\ell}}<120$
GeV, in Fig. 4.2. The $\cos\theta^{*}$ distribution exhibits a small but non-
negligible asymmetry, and uncertainties are smallest for NNPDF4.0. The four
PDF sets predict a similar behavior and magnitude of the asymmetry
$A_{\mathrm{fb}}$. PDF uncertainties in the asymmetry are comparable for all
PDF sets when $\cos\theta^{*}\approx 0$, and actually largest for NNPDF4.0
when $\cos\theta^{*}\approx 1$. In all cases the predictions are compatible
within $2\sigma$, with ABMP16 showing larger differences of up to $2.8\sigma$
for the $\cos\theta^{*}$ distribution. Note that the sharp drop-off at the
edges $|\cos\theta^{*}|\approx 1$, appearing in all plots in this section, is
a consequence of the phase-space cuts which limit the phase-space volume.
Indeed, using LO kinematics

$|\cos\theta^{*}|=\tanh\left|\frac{\eta_{\ell}-\eta_{\bar{\ell}}}{2}\right|=\sqrt{1-\frac{4(p_{T}^{\ell})^{2}}{m_{\ell\bar{\ell}}^{2}}}\text{,}$
(4.3)

so $|\cos\theta^{*}|\approx 1$ requires a lepton pair with either a large
rapidity separation, or a very large invariant mass and small transverse
momenta.

As expected from the antisymmetric partonic luminosities studied in Sect. 3.3,
the situation is quite different when considering distributions with a higher
dilepton invariant mass range. The angular distribution and forward-backward
asymmetry in the high-mass region, for different values of the lower cut in
the dilepton invariant mass, namely $m_{\ell\bar{\ell}}^{\rm min}=3,4,5$ and 6
TeV, are respectively shown in Fig. 4.3 and Fig. 4.4.

Figure 4.3: Same as Fig. 4.2 (left) for different values of the lower cut in
the dilepton invariant mass: $m_{\ell\bar{\ell}}\geq 3,4,5,$ and 6 TeV
respectively.

Consistent with the underlying parton luminosities, the $\cos\theta^{*}$
distribution is dominated by $u\bar{u}$ scattering, while $d\bar{d}$ provides
a subdominant contribution. When the lower cut is $m_{\ell\bar{\ell}}^{(\rm
min)}=3$ TeV is used, the four PDF sets are in agreement at the $1\sigma$
level: they all display a positive forward-backward asymmetry, and exhibit PDF
uncertainties ranging between 10% and 15%. As the invariant mass cut is
raised, the qualitative behavior of the angular distribution and asymmetry
change substantially for NNPDF4.0, while they remain approximately the same
for all other PDF sets, consistent with the behavior of the PDFs and
luminosities discussed in Sect. 3.2-3.3. Specifically, raising the cut to
$m_{\ell\bar{\ell}}\geq 4$ TeV, for NNPDF4.0 the backwards cross-section
starts increasing, though the asymmetry remains positive.

For $m_{\ell\bar{\ell}}\geq 5$ TeV the central value of the NNPDF4.0
$\cos\theta^{*}$ distribution becomes symmetric, though the PDF uncertainty
band is rather asymmetric. Also, PDF uncertainties are now the largest for
NNPDF4.0, reaching up to 30%. Finally, for $m_{\ell\bar{\ell}}\geq 6$ TeV the
central value of forward-backward asymmetry for NNPDF4.0 becomes negative,
with the PDF uncertainties increasing further so the asymmetry remains
compatible with zero at about the 1.1 $\sigma$ level. For all other PDF sets
there is little change in the shape of the distribution as the dilepton
invariant mass cut is increased. Because of the large uncertainty on the
NNPDF4.0 result for the $\cos\theta^{*}$ distribution, even with the highest
value of the $m_{\ell\bar{\ell}}^{(\rm min)}$ cut, where NNPDF4.0 finds a
symmetric distributions while all other PDF sets find an asymmetry, the pull
is always below $2\sigma$.

Figure 4.4: Same as Fig. 4.2 (right) for different values of the lower cut in
the dilepton invariant mass: $m_{\ell\bar{\ell}}^{\rm min}=3,4,5,$ and 6 TeV.

In Fig. 4.5 the forward-backward asymmetry with $m_{\ell\bar{\ell}}^{(\rm
min)}=5$ TeV shown in Fig. 4.4 (bottom left) is shown again, now also
including a prediction obtained using the PDF4LHC21 combination of parton
distributions [46], specifically its compressed Monte Carlo representation
[47], based on the CT18, MSHT and NNPDF3.1 PDF sets. Because the PDF
uncertainties for NNPDF3.1 are generally, and in particular at large $x$,
rather larger than those on NNPDF4.0 (see also App. A) the uncertainty on the
$A_{\rm fb}$ distribution found using the PDF4LHC21 combination is extremely
large, and no signal for the asymmetry can be seen. PDF4LHC recommends [46]
usage of the combination for BSM searches, and that of individual PDF sets for
comparison between data and theory for SM measurements. The results presented
here suggest that the uncertainty estimate of NNPDF4.0 in the extrapolation
region, which is rather more conservative than that of the other PDF sets
shown here, might be desirable and lead to more robust predictions for the
forward-backward asymmetry in the high-mass region which is relevant for new
physics searches.

Figure 4.5: Same as Fig. 4.4 (bottom left), now also including the PDF4LHC21
prediction. In the left plot the PDF uncertainty is not shown, in the right
plot the scale on the $y$ axis is suitably expanded. Note that the PDF4LHC
percentage PDF uncertainty does not show as it falls outside the plot.

##  5 Summary and outlook

In this work we have scrutinised the PDF dependence of neutral current Drell-
Yan production at large dilepton invariant masses $m_{\ell\bar{\ell}}$,
focusing on the behavior of the forward-backward asymmetry $A_{\rm fb}$ in the
Collins-Soper angle $\cos\theta^{*}$, an observable frequently considered in
the context of searches for new physics beyond the SM. We have demonstrated
that while theoretical predictions for the sign and magnitude of $A_{\rm fb}$
are very similar for all PDF sets in the $Z$ peak region, they depend markedly
on the choice of PDF set for large values of $m_{\ell\bar{\ell}}$. We have
traced this behavior to that of the PDFs, which agree in the data region, but
differ in the large-$x$ region, where PDFs are mostly unconstrained by data.

We have specifically shown that the uncertainty on the asymmetry differs
substantially between PDF sets, with NNPDF4.0 displaying a more marked
increase as $m_{\ell\bar{\ell}}$ grows, leading to an absolute uncertainty
that e.g. for $m_{\ell\bar{\ell}}^{\rm min}\gtrsim 4$ TeV is about twice as
large as that found using CT18, four times as large as MSHT20, and about one
order of magnitude larger than ABMP16. Also, whereas other PDF sets predict a
shape of the asymmetry which is unchanged when $m_{\ell\bar{\ell}}$ increases
from the $Z$-peak region to the TeV range, namely a positive asymmetry
implying a larger cross-section for $\cos\theta^{*}\geq 0$, NNPDF4.0 finds
that as $m_{\ell\bar{\ell}}$ increases, the asymmetry is reduced, and the
$\cos\theta^{*}$ distribution becomes symmetric when $m_{\ell\bar{\ell}}^{\rm
min}\sim 5$ TeV.

We have traced this behavior to that of the underlying PDFs in the large-$x$
region, where PDFs are mostly unconstrained by data. Specifically we have seen
that in this region NNPDF4.0 has generally wider uncertainties. Also, while
for all PDF sets the quark and antiquark distributions vanish as a power of
$(1-x)$ as $x\to 1$, for all groups but NNPDF4.0 this power is constant for
light quarks to the right of the valence peak, while for NNPDF4.0 it changes
as $x$ increases, slowly for up quarks, more rapidly for down quarks and even
more rapidly for antiquarks. All this suggests that the different behavior of
NNPDF4.0 is due to its more flexible PDF parametrization.

Our general conclusion is that the behavior of the forward-backward asymmetry
observed at lower invariant masses is not necessarily reproduced at large
masses if flexible enough PDFs are used: the characteristic positive asymmetry
observed for low $m_{\ell\bar{\ell}}$ values can be washed out in the high-
mass region. Hence, deviations from the traditional expectation of a positive
forward-backward asymmetry in high-mass Drell-Yan cannot be taken as an
indication of BSM physics, at least based on our current understanding of
proton structure in the large-$x$ region.

Turning the argument around, future measurements of the $\cos\theta^{*}$
distribution and the associated forward-backward asymmetry $A_{\rm fb}$ when
included in PDF determinations could help in constraining PDFs at large $x$.
For instance, Fig. 4.3 indicates that for $m_{\ell\bar{\ell}}^{\rm min}=5$ TeV
and $\sqrt{s}=14$ TeV the asymmetry $A_{\rm fb}$ can be as large as 50% for
ABMP16 while it vanishes (within large uncertainties) in the case of NNPDF4.0.
By rebinning the $\cos\theta^{*}$ distribution, for an integrated luminosity
of $\mathcal{L}=6$ ab-1, corresponding to the combination at ATLAS and CMS at
the end of the HL-LHC data-taking period, $\mathcal{O}(10)$ events are
expected in the backward region, with an statistical uncertainty of
$\delta_{\rm stat}\sim 30\%$ which could be sufficient to discriminate between
these two limiting scenarios at the $2\sigma$ level.

Higher event counts are expected if the $m_{\ell\bar{\ell}}$ cut is loosened,
though one is then less sensitive to the large-$x$ region where differences
between PDF sets and their uncertainties are the largest. Ultimately, the
constraining power of high-mass Drell-Yan in general and of the forward-
backward asymmetry in particular can only be addressed by means of a dedicated
projections based on binned pseudo-data such as those carried out for the HL-
LHC and the Electron Ion Collider in e.g. [48, 49]. While we leave this
exercise for a future study, the investigations presented in this work
indicate that $A_{\rm fb}$ at high-invariant masses represents a promising and
mostly unexplored channel to pin down large-$x$ light quark and antiquark PDFs
at the HL-LHC.

While in this work we have focused on the forward-backward asymmetry in
neutral-current Drell-Yan production, similar considerations apply for other
processes relevant for BSM searches at high mass at the LHC. Indeed, the HL-
LHC will be sensitive to a broad range of hypothetical new massive particles,
from resonances in the $m_{jj}$ dijet invariant mass distribution up to 11
TeV, heavy vector triplet resonances decaying into a diboson $VV^{\prime}$
pair up to 5 TeV, and gluinos with masses up to $m_{\tilde{g}}=3$ TeV in the
minimal supersymmetric standard model (MSSM) with a massless lightest SUSY
particle [1].

For all these channels, a robust understanding of PDFs and their uncertainties
at large $x$, including the role of methodological and model assumptions, will
be necessary to fully exploit the HL-LHC discovery potential for BSM
signatures. Conversely, once BSM phenomena have been excluded in some high-
energy channel, the corresponding search can be unfolded into a measurement to
provide direct constraints on the PDFs in this key large-$x$ region, which in
turn will enhance the reach of other searches. It would be very interesting to
perform a detailed study, in the same vein as Ref. [48], of the impact of
future HL-LHC data on large-$x$ PDFs and the prospect of asymmetry
measurements in searches for new physics, but this will be left for future
work.

### Acknowledgments

We are grateful to Dimitri Bourilkov, Alexander Grohsjean, Meng Lu, and Jan
Schulte for raising with us the issue of the PDF dependence of $A_{\rm fb}$ at
high invariant masses and for the subsequent discussions. A. C., S. F., and F.
H. are supported by the European Research Council under the European Union’s
Horizon 2020 research and innovation Programme (grant agreement n.740006). R.
D. B. is supported by the U.K. Science and Technology Facility Council (STFC)
grant ST/P000630/1. E. R. N. is supported by the Italian Ministry of
University and Research (MUR) through the “Rita Levi-Montalcini” Program. J.
R. is partially supported by NWO (Dutch Research Council). C. S. is supported
by the German Research Foundation (DFG) under reference number DE 623/6-2.

##  Appendix A $A_{\rm fb}$ in NNPDF3.1

In this appendix we compare partonic luminosities and LHC differential
distributions obtained with NNPDF4.0 in Sects. 3 and 4 with those based on its
predecessor NNPDF3.1, as well as with a variant of NNPDF4.0 where positivity
is imposed at the level of observable cross-sections but not at the PDF level,
as was the case in NNPDF3.1, which we will denote NNPDF4.0(3.1pos).

Fig. A.1 compares the symmetric partonic luminosities $\mathcal{L}_{S,q}$
evaluated for $m_{\ell\bar{\ell}}=5$ TeV. The three sets are found to agree
within uncertainties, with NNPDF4.0 having the smallest uncertainties. This
increase in precision arises only marginally due to the more restrictive
positivity constraints, since predictions with the NNPDF4.0(3.1pos) variant
are close to the baseline NNPDF4.0, especially for the $u\bar{u}$
contribution, for both central values and uncertainties. The comparison in
Fig. A.1 indicates that phenomenological predictions for high-mass Drell-Yan
production based on NNPDF3.1 are expected to be consistent within errors with
those of NNPDF4.0 for the contributions symmetric in $\cos\theta^{*}$, such as
the $|y_{\ell\bar{\ell}}|$ distribution.

Figure A.1: Same as Fig. 3.7 (upper panels) comparing NNPDF4.0,
NNPDF4.0(3.1pos), and NNPDF3.1.

The antisymmetric luminosities $\mathcal{L}_{A,q}$, relevant for the forward-
backward asymmetry, are displayed in Fig. A.2 for $m_{\ell\bar{\ell}}=3$ and 5
TeV respectively. Their qualitative behavior is similar for all PDF sets, with
a marked decrease of PDF uncertainties first from NNPDF3.1 to NNPDF4.0(3.1pos)
then to NNPDF4.0. Specifically, the qualitative $m_{\ell\bar{\ell}}$
dependence of $\mathcal{L}_{A,q}$ remains unchanged. Namely, the positive
$A_{\rm fb}$ found for $m_{\ell\bar{\ell}}=3$ TeV decreases as the dilepton
invariant mass is increased. Hence also for the component of the Drell-Yan
cross-section which is odd in $\cos\theta^{*}$ we expect LHC predictions based
on NNPDF3.1 to be consistent with those obtained from NNPDF4.0.

Figure A.2: Same as Fig. 3.8 for the antisymmetric partonic luminosities
$\mathcal{L}_{A,q}$, comparing NNPDF4.0, NNPDF4.0(3.1pos), and NNPDF3.1.

These expectations are confirmed by Fig. A.3, which shows the dilepton
rapidity $|y_{\ell\bar{\ell}}|$ and the Collins-Soper angle $\cos\theta^{*}$
distributions for neutral-current DY production at the LHC 14 TeV for dilepton
invariant masses of $m_{\ell\bar{\ell}}\geq 5$ TeV, comparing the baseline
NNPDF4.0 predictions with those from NNPDF3.1 and NNPDF4.0(3.1pos). Indeed,
good agreement within the three PDF sets is observed with a significant
reduction of PDF uncertainties between NNPDF3.1 and NNPDF4.0, consistent with
the behaviour exhibited by the corresponding partonic luminosities.

Figure A.3: Same as Figs. 4.1 and 4.3 for the absolute dilepton rapidity
$|y_{\ell\bar{\ell}}|$ (left) and the $\cos\theta^{*}$ (right) distributions
for dilepton invariant masses of $m_{\ell\bar{\ell}}\geq 5$ TeV comparing
NNPDF4.0, NNPDF4.0(3.1pos), and NNPDF3.1.

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