# Defiducialization: Providing Experimental Measurements for Accurate Fixed-
Order Predictions

A. Glazov<EMAIL_ADDRESS>Deutsches Elektronen-Synchrotron DESY,
Hamburg, Germany

###### Abstract

An experimental procedure is proposed to perform measurements of differential
cross sections for vector boson production which can be compared to fixed-
order QCD predictions with improved accuracy. The procedure relies on applying
theoretical acceptance corrections computed as a function of the transverse
momentum of the $W/Z$ boson, $p_{T}$, to the experimental measurement, rather
than comparing data directly against fiducial fixed-order predictions. It is
demonstrated that, contrary to standard fiducial computations, these
acceptance factors vary little at low $p_{T}$, so they can be reliably
computed using fixed-order perturbation theory. An example analysis is
performed using the ATLAS measurement of the $Z$-boson production cross
section at center-of-mass energy of $8$ TeV. The resulting full phase space
measurement of the cross section differential in the boson rapidity is
compared to theoretical predictions computed with next-to-next-to leading-
order accuracy in QCD. Further extensions of the approach which include
different types of measurements and improved theoretical predictions are
discussed.

††preprint: DESY 20-001

##  I Introduction

Accurate knowledge of parton distribution functions of the proton (PDFs) is
essential for the physics program at the LHC. PDF uncertainties are the
leading source of systematics for precision measurements of the $W$-boson mass
and effective weak mixing angle, $\sin^{2}\theta_{W}$ Aaboud:2017svj ;
Sirunyan:2018swq . Reduction of the PDF uncertainties is important in
particular for the interpretation of high statistics measurements for the
future LHC data.

PDFs can be constrained using $W$\- and $Z$-boson production in the charged
and neutral current Drell-Yan processes. These processes can be measured with
sub-percent experimental accuracy and thus provide a valuable input for the
PDF determination. An example of an accurate measurement is the ATLAS result
on $\gamma^{*}/Z$\- and $W$-boson production at the center-of-mass energy of
$\sqrt{s}=7$ TeV Aaboud:2016btc . For the $Z$ boson, the measurement is
performed in bins of the invariant mass of the lepton pair, $m_{\ell\ell}$,
and of the lepton pair rapidity, $y_{\ell\ell}$. For the $W$ boson, the
results are reported as a function of the lepton pseudorapidity,
$\eta_{\ell}$. Excluding the global normalisation uncertainty, the measurement
reaches better than $0.5\%$ experimental uncertainty.

The cross sections differential in $y_{\ell\ell}$, $m_{\ell\ell}$ and
$\eta_{\ell}$ are known at next-to-next-to leading-order (NNLO) accuracy in
perturbative QCD Anastasiou:2003yy ; Anastasiou:2003ds ; Melnikov:2006di ;
Catani:2009sm ; Catani:2010en . For these observables, the corresponding
computations are inclusive in the boson transverse momentum $p_{T}$ and thus
insensitive to $\ln p_{T}/m_{\ell\ell}$ divergences, providing a robust input
for determination of collinear PDFs. However the ATLAS measurement is
performed in a fiducial volume with experimental cuts on the lepton transverse
momentum and lepton pseudorapidity which are required due to the detector
acceptance. These selection criteria introduce dependence on $p_{T}$ modelling
thereby spoiling accuracy of the fixed-order predictions. The study performed
in Ref. Aaboud:2016btc  shows that predictions become unstable with respect to
small variations of the cuts. In addition, calculations obtained using various
public NNLO codes differ as much as $1\%$ which is significant compared to the
experimental accuracy.

The impact of the fiducial cuts on fixed-order predictions is under
investigation since some time. It is known to be large when transverse
momentum of individual leptons (or jets) approaches half of the invariant mass
of the lepton pair (or jet pair). It has been proposed in Ref. Frixione:1997ks
to introduce an asymmetry for the cut values for the leading and sub-leading
objects. This procedure seems to stabilize the computation of the fixed-order
predictions with respect to small variation of experimental cuts, however a
closer analysis shows that it does not eliminate the uncertainty arising from
the logarithmic corrections which remains similar to the symmetric cuts cases
Salam? . Computation of higher order corrections and/or inclusion of the $\ln
p_{T}/m_{\ell\ell}$ resummation should reduce the uncertainty arising from the
fiducial cuts. However the full next-to-next-to-next-to leading-order (N3LO)
corrections for Drell-Yan processes are not available yet, while resummation
corrections may bring additional uncertainties from the recoil prescription
Catani:2015vma .

Experimentally, it is sometimes possible to isolate regions in the phase space
which are not affected by fiducial cuts. For example, in Ref. Aaboud:2017ffb
the fiducial acceptance for the triple differential measurement of
$\frac{\mathrm{d}\sigma}{\mathrm{d}y_{\ell\ell}\mathrm{d}\cos\theta\mathrm{d}m_{\ell\ell}}$,
where $\cos\theta$ is the lepton polar angular variable, is above $99\%$ for
$|\cos\theta|<0.4$, $y<1$ and $m_{\ell\ell}>66$ GeV Walker:2019bjd . This
region can be safely compared to fixed-order predictions. Other experimental
methods include explicit correction to the full phase space during the data
analysis, as in e.g. Ref. Aaltonen:2011nr .

In this paper, another approach is proposed. It is based on an observation
that an acceptance correction from the fiducial to full phase space has
moderate dependence on $p_{T}$, including low $p_{T}$ region, as discussed in
section II, so that it can be determined using fixed-order calculations. This
is demonstrated in section III using the ATLAS measurement of $Z$-boson
production differential in $y_{\ell\ell}$ and $p_{T}$ performed using data
collected at $\sqrt{s}=8$ TeV Aad:2015auj . The acceptance correction is
computed at NLO for the $Z$ plus jet process ($O(\alpha_{S}^{2})$, where
$\alpha_{S}$ is the strong coupling constant), using the MCFM v6.8 program
Campbell:1999ah , interfaced to APPLGRID Carli:2010rw . In section IV, the
correction is used to determine the full phase space measurement differential
in $y_{\ell\ell}$ and $p_{T}$. The result is integrated in $p_{T}$ and the
inclusive $\mathrm{d}\sigma/\mathrm{d}y_{\ell\ell}$ differential cross-section
measurement is compared to the NNLO ($O(\alpha_{S}^{2})$) computation for
inclusive $Z$-boson production obtained using the MCFM v9 program
Boughezal:2016wmq ; Campbell:2019dru . To illustrate the importance of the
acceptance effects, another comparison is performed in which the fiducial data
are integrated in $p_{T}$ directly and compared to the fiducial predictions
computed to the same order. While these two comparisons are performed at the
same accuracy of the fixed order predictions in $O(\alpha_{S}^{2})$, a visible
difference is observed with the approach advocated in this paper giving a
better overall description of the data. The paper concludes with a discussion
of the results, possible applications of the method to other measurements, and
possible further studies of theoretical predictions.

##  II Formalism

Recent experimental measurements are usually reported in a fiducial phase
space, as determined by the acceptance of the experimental apparatus. This
phase space is defined by selection criteria applied to the final state
objects, such as lepton transverse momenta and pseudorapidities. The presence
of experimental cuts must be addressed by the theory predictions. In the
currently adopted procedure, the theoretical predictions are extended to be
fully differential, to match the data, so that they can be also computed in
the fiducial phase space. This procedure is also adapted for predictions
computed using fixed-order calculations. Schematically, the theory to data
comparison for $p_{T}$ integrated observables can be written in this case as

$\sigma_{\mathrm{fidu,theory}}=\int\frac{\mathrm{d}\sigma_{\mathrm{full,theory}}}{\mathrm{d}p_{T}}A(p_{T})\mathrm{d}p_{T}\leavevmode\nobreak\
\mbox{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
vs\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
}\sigma_{\mathrm{fidu,data}}=\int\frac{\mathrm{d}\sigma_{\mathrm{fidu,data}}}{\mathrm{d}p_{T}}\mathrm{d}p_{T}\,.$
(1)

Here a vector boson transverse meomentum dependent fiducial acceptance is
introduced as
$A(p_{T})=\sigma_{\mathrm{fidu,theory}}(p_{T})/\sigma_{\mathrm{full,theory}}(p_{T})$.

Fixed-order predictions for the cross sections become unreliable at low
$p_{T}$, varying rapidly as $p_{T}\to 0$. On the other hand, $A(p_{T})$ is
expected to have only a moderate dependence on $p_{T}$, driven by the changes
of the boost and polarisation. In general, $A(p_{T})$ is not expected to be
constant and deviations in $p_{T}$ distribution at low $p_{T}$ may lead to a
bias in $\sigma_{\mathrm{fidu,theory}}$.

For experimental measurements which are performed differentially in $p_{T}$,
the comparison between data and theory can be re-arranged such that the data
are corrected to the full phase space first, which is the central proposal of
this paper:

$\sigma_{\mathrm{full,theory}}=\int\frac{\mathrm{d}\sigma_{\mathrm{full,theory}}}{\mathrm{d}p_{T}}\mathrm{d}p_{T}\leavevmode\nobreak\
\mbox{\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
vs\leavevmode\nobreak\ \leavevmode\nobreak\ \leavevmode\nobreak\
}\sigma_{\mathrm{full,data}}=\int\frac{\mathrm{d}\sigma_{\mathrm{fidu,data}}}{\mathrm{d}p_{T}}\frac{1}{A(p_{T})}\mathrm{d}p_{T}\,.$
(2)

The main advantage of this approach is that the fully integrated in $p_{T}$
$\sigma_{\mathrm{full,theory}}$ can be computed safely using fixed-order
predictions. As it will be illustrated in the following, for $Z$-boson
production, $A(p_{T})$ can be also estimated using fixed-order $Z$ plus jet
calculations, including the low $p_{T}$ region. The success of this approach
relies on smallness of $A(p_{T})$ variation at low $p_{T}$, which is caused
mainly by kinematic effects, while polarisation effects are suppressed at low
$p_{T}$ as $p_{T}^{2}/m_{Z}^{2}$, where $m_{Z}$ is the $Z$-boson mass.

To motivate why this would be the case, it is instructive to decompose the
differential cross section in terms of harmonic polynomials, as discussed, for
example, in Ref. Gauld:2017tww . The fully differential $Z$-boson production
cross section can be expressed as

$\frac{\mathrm{d^{5}}\sigma}{\mathrm{d}p_{T}\mathrm{d}y_{\ell\ell}\mathrm{d}m_{\ell\ell}\mathrm{d}\cos\theta\mathrm{d}\phi}=\frac{3}{16\pi}\frac{\mathrm{d^{3}}\sigma^{U+L}}{\mathrm{d}p_{T}\mathrm{d}y_{\ell\ell}\mathrm{d}m_{\ell\ell}}\sum_{i=0}^{8}P_{i}(\cos\theta,\phi)\,.$
(3)

Here $\phi$ is the lepton azimuthal angular variable, $P_{i}(\cos\theta,\phi)$
are the nine harmonic polynomials, and $\sigma^{U+L}$ is the unpolarised cross
section. The harmonic polynomials depend on eight angular coefficients
$A_{i}(p_{T},y_{\ell\ell},m_{\ell\ell})$ which define fractions of helicity
cross sections with respect to the unpolarised case. Equation 3 relies on
factorization of the $Z$ boson production and decay processes. In essence, it
represents production of a spin one particle and its following decay. It is
violated by electroweak corrections which contain interaction of the initial
quarks and final state leptons. It is also modified for the
$\gamma\gamma\to\ell^{+}\ell^{-}$ scattering process. These corrections are
small at the $Z$-pole region and neglected in the following, however it is
possible to extend the proposed procedure and include them.

For the measurements insensitive to the azimuthal angle $\phi$ and forward-
backward asymmetry in $\cos\theta$, the most important coefficient is $A_{0}$.
This is the case for all fiducial measurements of $Z$-boson production at the
LHC, which use charge-symmetric cuts on lepton rapidities. The coefficient
$A_{0}$ vanishes for $p_{T}\to 0$ and saturates at $A_{0}=1$ for $p_{T}\gg
100$ GeV.

To understand better the $p_{T}$ dependence of the angular coefficients
$A_{i}$, and of the most important coefficient $A_{0}$ in particular, it is
useful to follow the geometrical approach of Refs. Peng:2015spa ;
Chang:2017kuv . For the coordinate system in which the $z$-axis is aligned
with the momentum direction of the $q\bar{q}$ pair producing the $Z$ boson,
$A_{0}=0$, as follows from helicity conservation arguments. The direction of
the $z$-axis in this coordinate system depends on radiation and in general
does not agree with the $z$-axis of the fixed physical rest frame, such as
given by the direction of the incoming hadrons (laboratory frame). Defining
$\theta_{1}$ as the angle between the two axes, it can be shown based on
simple geometrical arguments that $A_{0}$ can be calculated as
$A_{0}=\langle\sin^{2}\theta_{1}\rangle$ where the average is taken over all
event configurations. Given that $\theta_{1}\sim p_{T}/m_{Z}$ for small
$p_{T}$, the coefficient $A_{0}$ is expected to vanish as
$p^{2}_{T}/m^{2}_{Z}$ for $p_{T}\to 0$.

The coefficients $A_{i}(p_{T},y_{\ell\ell},m_{\ell\ell})$ can be calculated
using fixed-order predictions. It is possible that the $A_{2}$ coefficient is
sensitive to non-perturbative effects at low $p_{T}$ Chiappetta:1986yg ,
however this should have a small impact on the acceptance. It was demonstrated
in Ref. Gauld:2017tww  which compared predictions for the $A_{i}$ coefficients
to the data obtained by the ATLAS and CMS collaborations Khachatryan:2015paa ;
Aad:2016izn . The calculations were performed at LO ($O(\alpha_{S})$), NLO
($O(\alpha^{2}_{S})$) and NNLO ($O(\alpha^{3}_{S})$) order for the $Z$-boson
plus jet process, as provided by the NNLOJET program Ridder:2015dxa . For the
coefficient $A_{0}$, good perturbative convergence is observed with already LO
predictions being in excellent agreement with the data as shown in Ref.
Aad:2016izn  for $p_{T}$ down to $2.5$ GeV 111Note that Aad:2016izn  counts
order of the predictions based on inclusive $Z$ production, with NLO
corresponding to $O(\alpha_{S})$ correction.

Given the $p_{T},y_{\ell\ell},m_{\ell\ell}$ values and the $A_{i}$
coefficients, the kinematics of the final state leptons and thereby the
fraction of events passing fiducial cuts is fully determined. Any fixed-order
prediction for the $Z$ boson plus jet process must obey decomposition of
equation 3. Therefore, the fiducial acceptance for each
$p_{T},y_{\ell\ell},m_{\ell\ell}$ bin can be determined at fixed order,
provided it is narrow enough to neglect the $p_{T}$ dependence inside the bin.
The residual theoretical uncertainties arising in this procedure can be
estimated the usual way, by PDF and scale variations.

An estimate of the uncertainty based on the scale variation may not be
perfect, especially for the quantities computed as ratios of cross sections.
An attempt to improve it by a partial decorrelation is used in the current
analysis, however it may be still incomplete. It is also possible that $A_{i}$
coefficients may be affected significantly by the resummation effects, but
study of these is beyond the scope of the current paper. Further studies may
bring additional insights on this issue. It should be noted, however, that
fixed order analyses usually ignore uncertainties from fiducial acceptance
effects altogether.

##  III Validation

Figure 1: Fiducial acceptance calculated at LO ($O(\alpha_{S})$) and NLO
($O(\alpha^{2}_{S})$) using $Z$-boson plus jet process. The error bars show
total uncertainties of the calculation.

The correction procedure is tested using the ATLAS $\gamma^{*}/Z$ data at
$\sqrt{s}=8$ TeV Aad:2015auj . The cross sections in this analysis are
measured differentially in $p_{T}$ and $y_{\ell\ell}$ for the
$66<m_{\ell\ell}<116$ GeV invariant mass range. There are $20$ variable-size
bins in $p_{T}$, starting with narrow $2$ GeV bins for low $p_{T}$ and ending
with a wide $700$ GeV bin from $200$ to $900$ GeV. The $6$ bins in
$y_{\ell\ell}$ are equidistant spanning from $0$ to $2.4$ with a step of
$\Delta y_{\ell\ell}=0.4$. The measurement is performed in the fiducial space
defined by the lepton transverse momentum $p_{T}^{\ell}>20$ GeV and
$|\eta_{\ell}|<2.4$ requirements. All data tables are taken from the HEPDATA
record of the ATLAS publication. Following Refs. Aaboud:2016hhf ;
Aaboud:2016zpd , the measurement is re-scaled by the original and updated
luminosity ratio: $20.3/20.2$. The total normalization uncertainty of the
measurement is $1.9\%$.

###  III.1 Estimation of $A(p_{T})$ at LO and NLO

The fiducial acceptance is estimated using the MCFM v6.8 program interfaced to
APPLGRID, for fast evaluation of theoretical uncertainties. The $Z$ plus jet
process is computed at LO and NLO yielding acceptance estimates
$A_{\mathrm{LO}}$ and $A_{\mathrm{NLO}}$, respectively. It is also possible to
use NNLO calculations for the $Z$ plus jet process which became available
recently Ridder:2016rzm , however they are not used in the present analysis.
Note that in order of $\alpha_{S}$, the NLO calculations for the $Z$ plus jet
process match NNLO calculations for inclusive $Z$-boson production.

The renormalization and factorization scales are set to $m_{\ell\ell}$ and the
CT14NNLO PDF set Dulat:2015mca  is used for the calculations. The jet
transverse momentum is required to be above $1$ GeV. The APPLGRID is binned in
the $Z$-boson transverse momentum and rapidity, following the binning used in
the ATLAS measurement. Note that while at NLO jet and $Z$-boson transverse
momenta do not agree, the $1$ GeV jet transvserse momentum cut has no impact
for the bins with the $Z$-boson $p_{T}\geq 2$ GeV.

The resulting fiducial acceptance is shown in Figure 1. The statistical
uncertainties for the LO calculations are negligible and for NLO they are
below $0.1\%$ for the majority of the bins. To achieve this accuracy,
computational resources of about $20000$ CPU hours are required.

The fiducial acceptance is very similar for the three central rapidity bins
with $y<1.2$ and it is always above 70%. For the forward rapidity, the
acceptance starts to decrease and drops below $20\%$ for the
$2.0<y_{\ell\ell}<2.4$ bin at low $p_{T}$. The $p_{T}$ dependence of the
acceptance is also different for forward compared to central rapidities. The
first bin in $p_{T}$ spans between $0$ and $2$ GeV and NLO calculations of the
acceptance become unstable, thus only the LO result is presented. For all
other bins, LO and NLO calculations agree very well.

Figure 2: Percentage difference between acceptance corrections calculated at
NLO and LO. The error bars indicate total uncertainties of the NLO
calculations.

Figure 2 presents quantitative comparison of the acceptances computed at
different orders in terms of $\delta=A_{\mathrm{NLO}}/A_{\mathrm{LO}}-1$. The
NLO correction is below $0.1\%$ for $p_{T}<m_{Z}$ for all rapidity bins with
$y_{\ell\ell}<2.0$. For all bins the correction does not exceed $1\%$,
indicating good convergence of the perturbative series. Based on this good
agreement and moderate dependence of the correction on $p_{T}$, the correction
for the bin $0<p_{T}<2$ GeV is estimated using LO calculation, re-scaled by
the ratio of NLO to LO for the bin $2<p_{T}<4$ GeV.

Moderate variation of the estimated fiducial acceptance as a function of
$p_{T}$ and very good agreement of predictions computed at LO and NLO provide
good confidence that fixed-order predictions can be safely used. Additional
support comes from the studies of the angular coefficient $A_{0}$ at NNLO
Ridder:2015dxa , mentioned above, which indicate good perturbative
convergence. Note, however, that good perturbative convergence does not
necessarily lead to small theoretical uncertainties and furhter in-depth
theoretical studies are required to fully validate this proposal. With these
observations, the analysis proceeds to an estimate of the uncertainties for
the NLO-based predictions.

###  III.2 Estimation of uncertainties

Figure 3: PDF and scale variation uncertainties of the acceptance correction
calculated at NLO.

PDF and scale uncertainties of the acceptance computed at NLO are shown in
Figure 3. The PDF uncertainties are evaluated using the CT18ANNLO set
Hou:2019qau ; Hou:2019efy  scaled to $68\%$ c.l. The scale uncertainties are
estimated by varying factorization and renormalization scale by factor of two
and taking the envelope of resulting acceptance factors. Both PDF and scale
uncertainties evaluated this way are typically below $0.1\%$. The PDF
uncertainties increase to about $0.15\%$ for the highest rapidity bin. As an
additional check of the PDF dependence, the acceptance is calculated using
CT14NNLO, MMHT14NNLO Harland-Lang:2014zoa , NNPDF31 Ball:2017nwa , ABMP16
Alekhin:2017kpj  and ATLASepWZ16 Aaboud:2016btc  PDF sets. For all sets except
ATLASepWZ16, the calculated acceptance is found to be within the the PDF error
bands of the CT18ANNLO set. The ATLASepWZ16 based acceptance agrees with the
other sets for $y_{\ell\ell}<2$. For the highest rapidity bin, the acceptance
based on ATLASepwz16 deviates by as much as $-0.2\%$ for low $p_{T}$ and up to
$+0.35\%$ for $p_{T}$ around $m_{Z}$. No additional uncertainty is introduced
due to this deviation.

Figure 4: Fiducial acceptance calculated for $q\bar{q}$ and $qg$ contributions
for $0<y_{\ell\ell}<0.4$.

The smallness of the uncertainties is related to the fact that the main effect
is purely kinematic variation of the fiducial acceptance as a function of
$p_{T}$ and theoretical uncertainties arise only from modeling of the
polarisation. To gain some insight on polarisation uncertainties, one can
again follow the intuitive geometrical approach of Refs. Peng:2015spa ;
Chang:2017kuv . For the predictions at LO, the contributions from $q\bar{q}$
and $qg$ subprocesses lead to different average mis-alignment angle
$\theta_{1}$ yielding in different $A_{0}$. It is shown in Refs. Peng:2015spa
; Chang:2017kuv  that the polarisation coefficients observed in data do indeed
lie between the two predictions. APPLGRID based calculations allow to separate
contributions of $q\bar{q}$ and $qg$ subprocesses, fiducial acceptances for
which can be compared to the full result. The resulting comparison is shown in
Figure 4 for $|y|<0.4$. Scale variations yield only to small changes of the
acceptances of the individual contributions, and a common scale variation for
both of them yields to a moderate change in their relative size. This may
explain the smallness of the acceptance uncertainty due to the scale variation
seen in Figure 3. It is possible that higher order corrections may have
different impact on $q\bar{q}$ and $qg$ subprocesses, and to be conservative
the scale variations are considered as independent for them with the total
scale uncertainty evaluated as a sum in quadrature. The resulting uncertainty
is shown in Figure 5 is always larger compared to the standard approach shown
in Figure 3.

Figure 5: Bin size and QCD scale uncertainties calculated assuming independent
variations of $q\bar{q}$ and $qg$ contributions.

As it is mentioned above, the difference in $p_{T}$ distribution between the
data and predictions should not matter for fiducial acceptance estimation, as
long as bin sizes are small enough such that variation of the acceptance
within the bin can be neglected. The bins are however rather large in this
analysis compared to variations of the acceptance at low $p_{T}$, see Figure
1. There is also a large difference between data and fixed order predictions
at low $p_{T}$. The effect of acceptance variation within the bin is studied
at LO. Predictions are generated from $p_{T}>0.1$ GeV, $p_{T}>10$ GeV, and
$p_{T}>100$ GeV with $0.1$ GeV, $1$ GeV, and $20$ GeV bin size, respectively.
For these bin sizes the variation of the fiducial acceptance can be neglected
since it is much smaller compared to the data statistical uncertainty for the
corresponding enclosing analysis bins. The full phase-space $p_{T}$
distribution is then reweighted to follow one estimated in the data. The
latter is approximated by first correcting the measured result to the full
phase space using acceptance as in Figure 1 and then fitting by a
phenomenological function $f(p_{T})=p_{T}\sum_{i=1}^{N}a_{i}\exp(-p_{T}b_{i})$
with $N=3$ and $a_{i},b_{i}$ treated as free parameters. The fits are
performed in each $y$ bin independently. The function describes the data well
for the whole $p_{T}$ range. Its shape is very different from the fixed order
predictions for $p_{T}=0$ GeV with $f(0\,\mathrm{G}eV{})=0$. The reweighted
full-phase space distribution is corrected for the fiducial acceptance in
$0.1,1,20$ GeV $p_{T}$ bins. The resulting fiducial and full phase-space
distributions are integrated following the analysis binning and the binned
fiducial acceptance is determined from their ratio. For the lowest $p_{T}<2$
GeV bin, the fixed order distributions are integrated starting form $1$ GeV,
to emulate the analysis procedure, while the reweighted distributions are
integrated from $0$ GeV with fiducial acceptance for $p_{T}<0.1$ GeV estimated
using a linear extrapolation.

The difference between acceptances determined using reweighted and original LO
$p_{T}$ distribution is taken as systematic uncertainty which is shown in
Figure 5. The uncertainty reaches about $0.1\%$ for the two lowest $p_{T}$
bins for $y<0.4$, where $p_{T}$ variation of the acceptance is significant.
The uncertainty for the lowest $p_{T}$ bin is reduced by using $p_{T}>1$ GeV
for the fixed order predictions. For $p_{T}>100$ GeV, the uncertainty starts
to increase and reaches few percent level, however it remains smaller than the
statistical uncertainty of the data.

To summarise the study of uncertainties, the largest effect arises from the
decorrelated scale variation (up to $0.5\%$ for $p_{T}\sim m_{Z}/2$ ) followed
by PDFs (up to $0.15\%$ for $y_{\ell\ell}>2$). The bin-size effects remain
below $0.1\%$ for the bulk of the distribution. It should be noted, however,
as it was already discussed, that the current analysis ignores the effect of
electroweak corrections and potential resummation effects.

##  IV Application and Results

Figure 6:  Ratios of differential
$\mathrm{d}\sigma/\mathrm{d}p^{\ell\ell}_{T}$ full phase space distributions
for different $y_{\ell\ell}$ bins to the $0<y_{\ell\ell}<0.4$ bin. The error
bars show total uncertainties of individual
$\mathrm{d}\sigma/\mathrm{d}p^{\ell\ell}_{T}$ measurements. The horizontal
lines shown for $p_{T}>35$ GeV represent NLO predictions for $Z$ plus jet
process using CT14NNLO PDF set.

With the caveat discussed above that the estimate may be not complete, the
size of the theoretical uncertainties on the acceptance corrections suggests
that it is possible to proceed with applying them to the data. Figure 6 shows
the ratio of the $\mathrm{d}\sigma/\mathrm{d}p_{T}$ distributions corrected to
the full phase space for a given $y_{\ell\ell}$ bin to the most central
$0<y_{\ell\ell}<0.4$ bin. The ratio is close to one for small $p_{T}$
decreasing almost linearly for large $p_{T}$ and large $y_{\ell\ell}$. For
$p_{T}>35$ GeV, the data are compared to fixed-order predictions at NLO for
the $Z$-boson plus jet process computed using the same APPLGRID as for the
acceptance correction which is convoluted with the CT14NNLO PDF set. The data
and predictions are found to be in reasonable agreement.

Figure 7:  Decomposition of the uncertainties for the
$\mathrm{d}\sigma/\mathrm{d}y_{\ell\ell}$ measurement. The global
normalization uncertainty due to the luminosity measurement of $1.9\%$ is not
shown.

As the next step the data are integrated in $p_{T}$. The correlated
uncertainties are propagated linearly while the statistical uncertainties are
combined in quadrature. The uncertainty decomposition is shown in figure 7.
The uncertainties are dominated by the correlated experimental errors. The
statistical uncertainties are at around $0.1\%$ except the highest rapidity
bin. The acceptance correction uncertainties are comparable to the statistical
uncertainties.

Figure 8:  Full phase space measurement of
$\mathrm{d}\sigma/\mathrm{d}y_{\ell\ell}$ compared to predictions using
various PDFs computed at NNLO using MCFM v9. Global normalization uncertainty
of $1.9\%$ is not shown. Error bars show total data uncertainties. Bands
indicate PDF uncertainties of the predictions. For ATLASepWZ16, the
uncertainties are computed using the ATLASepWZ16-EIG set only.

The measurements are compared to NNLO predictions obtained using the MCFM v9
program. Two PDF sets are used for the comparison: CT14NNLO and ATLASepWZ16.
CT14NNLO is representative of other global PDF sets with reduced compared to
$\bar{u}$ and $\bar{d}$ strange-quark distribution. ATLASepWZ16, on the other
hand, is obtained by fitting the ATLAS fiducial $W,Z$ data at $\sqrt{s}=7$ TeV
from Ref. Aaboud:2016btc  and has enhanced strangeness. The MCFM program is
used with nominal settings, electroweak corrections disabled, and scales set
to $m_{\ell\ell}$. The central value of the total cross section for
$\sqrt{s}=8$ TeV and $66<m_{\ell\ell}<116$ GeV obtained for the CT14NNLO PDF
set is $1114.9(1)$ pb, which is $0.4\%$ larger than the value of $1110(1)$ pb
obtained in Ref. Aaboud:2016zpd  using the DYTURBO program Camarda:2019zyx .

The comparison between the full phase space measurement of
$\mathrm{d}\sigma/\mathrm{d}y_{\ell\ell}$ at $\sqrt{s}=8$ TeV and predictions
is shown in figure 8. The global normalization uncertainty of $1.9\%$ is
omitted from the figure and the predictions are shown with PDF uncertainties
only. The uncertainties are larger for the CT14NNLO compared to
ATLASepWZ16-EIG PDF set due to usage of increased tolerance factors compared
to the $\Delta\chi^{2}=1$ criterion adapted for the ATLASepWZ16-EIG set. The
predictions based on CT14NNLO (ATLASepWZ16) underestimate (overestimate) the
data. Given the significant global normalization uncertainty the difference is
however not significant.

Figure 9:  Ratio of the predicted to the measured
$\mathrm{d}\sigma/\mathrm{d}y_{\ell\ell}$ cross section. The ratios are
calculated in the full or fiducial phase space using the CT14NNLO and ATLAS-
epWZ PDF sets, as indicated in the legend. The error band (bars) centered at
one shows the data total experimental (statistical) uncertainty. The error
bars for the ratios of the data to predicted cross sections show estimated
statistical uncertainties of the predictions. Global normalization uncertainty
of $1.9\%$ is not shown.

More quantitatively, the comparison can be performed by taking ratios of the
predictions to the data. It is also interesting to compare the ratios of the
full cross sections to the ratios of the fiducial cross sections, to see the
impact of the $p_{T}$-dependent acceptance corrections. The fiducial cross
section in data is computed by simple integration over the differential cross
section. It is verified that the total fiducial cross section agrees with the
one reported in the ATLAS publication. The NNLO predictions are obtained using
the MCFM v9 program with fiducial cuts enabled. The resulting full and
fiducial cross-section ratios for the two PDF sets are shown in figure 9. For
the CT14NNLO based prediction, the ratio of the full phase space cross
sections is closer to the unity compared to the ratio of the fiducial cross
sections. The dependence of the ratio on $y_{\ell\ell}$ is also reduced
significantly. For the ATLASepWZ16 based prediction, the fiducial ratio is
closer to unity, which is not too surprising since the set is fitted to the
fiducial ATLAS measurement at $\sqrt{s}=7$ TeV. The difference between full
and fiducial ratios is similar for both predictions for the five
$y_{\ell\ell}<2.0$ bins. For the first three bins the fiducial ratio is lower
than the full one, by as much as $2\%$, for the two lowest $y$ bins. For bins
with $1.2<y_{\ell\ell}<2.0$, they roughly agree. For the highest rapidity bin,
the behavior is different depending on the PDF set. For the CT14NNLO based
prediction the two ratios agree while for the ATLASepWZ16 based prediction the
fiducial ratio is below the full one by almost $2\%$.

For the full phase space comparison based on the CT14NNLO and ATLASepWZ16
sets, the ratios of $\sigma_{\mathrm{theory}}/\sigma_{\mathrm{data}}$ show
opposite trend as a function of $y_{\ell\ell}$. Given that the strange-quark
distribution affects low rapidity region more, based on this observation, it
is possible to make a conjecture that the ATLAS data may prefer somewhat
larger strangeness content than in the CT14NNLO set and somewhat smaller than
in the ATLASepWZ16 set. The overshoot of the predictions based on the
ATLASepWZ16 set is likely to be caused by the bias in the fiducial NNLO
prediction which was used in the analysis of the ATLAS data at $\sqrt{s}=7$
TeV.

To check this, predictions based on CT18NNLO, CT18ANNLO and CT18ZNNLO PDF sets
Hou:2019qau ; Hou:2019efy  are compared to the data in the full phase space.
The results are shown in figure 10. Compared to the CT18NNLO set, both
CT18ANNLO and CT18ZNNLO sets include the ATLAS $W,Z$ data at $\sqrt{s}=7$ TeV.
As a result, the CT18ANNLO and CT18ZNNLO sets have an enhanced strange-quark
distribution, with no suppression compared to other light quarks for Bjorken
$x<0.01$. The enhancement is however smaller compared to the ATLASepWZ16 set
since the CT18 sets use other data which favor smaller strangeness. The
strange-quark distribution in CT18NNLO is similar to CT14NNLO. The CT18ANNLO
set in addition uses a charm-quark mass that is increased compared to other
PDF sets. This leads to suppression of the charm-quark contribution,
compensated by an increase of the light-quark sea, yielding in turn an
increase of the Drell-Yan cross sections at the LHC, as it was demonstrated in
Ref. Abramowicz:1900rp . The predictions based on the CT18ANNLO and CT18ZNNLO
sets show better agreement with the ATLAS data compared to those based on the
CT18NNLO set: the ratio shows no dependence on $y_{\ell\ell}$ and the overall
normalisation of the ratio is closer to unity.

Figure 10:  Ratio of the predicted to the measured
$\mathrm{d}\sigma/\mathrm{d}y_{\ell\ell}$ cross section. The ratios are
calculated in the full phase space using the CT18, CT18A and CT18Z NNLO PDF
sets, as indicated in the legend. The error bands and error bars centered at
one show the data normalisation, total experimental (excluding normalisation)
and statistical uncertainty.

##  V Discussion

In the discussion above, it is argued that for comparison of fixed-order
predictions with data, it is better to use full (equation 2) instead of
fiducial (equation 1) cross sections. Both formula require accurate prediction
for the acceptance $A(p_{T})$, however equation 2 does not require accurate
modeling of the $p_{T}$ distribution by the prediction.

The accuracy of the fixed order predictions for $A(p_{T})$ may be improved
further by using existing higher order calculations. Furthermore, electroweak
corrections should be included as well. Comparison of $A(p_{T})$ computed
using pure fixed order predictions and ones which include resummation effects
can be used to quantify importance of the latter for the data analysis. If
needed, the PDF uncertainty can be reduced further, by using the same PDF in
the calculation of $\sigma_{\mathrm{full,theory}}$ and $A(p_{T})$. This could
be relevant for PDF fits in particular and can be arranged without difficulty
since the PDF-dependent acceptance $A(p_{T})$ can be computed using exact and
fast methods such as APPLGRID.

The method is applicable to the measurements differential in the boson
transverse momentum $p_{T}$. It is natural for $Z$-boson production, but can
be extended for the $W$ boson too. Given that $A(p_{T})$ has only mild
variation at low $p_{T}$ (see figure 1), it is probably possible to use
coarser $p_{T}$ binning, which is required by the experimental resolution.
Since the rapidity of the $W$ boson can not be reconstructed, this may lead to
increased PDF dependence of the acceptance correction factor, requiring its
update when performing comparisons for predictions based on different PDFs.
Studies of the defiducialization method for $W$-boson production are however
beyond the scope of the current analysis and they await for experimental
measurements, such as of the lepton pseudorapidity distribution, reported in
bins of the $W$-boson transverse momentum.

##  VI Summary

In summary, a method is proposed to perform comparisons of experimental data
on Drell-Yan production, performed in fiducial phase space, and fixed-order
QCD predictions with improved accuracy. The method requires the data to be
measured differentially in the boson transverse momentum $p_{T}$ in addition
to other variables of interest such as $y_{\ell\ell}$ and $m_{\ell\ell}$. It
relies on the ability to calculate the fiducial acceptance correction for a
given $p_{T}$ value at fixed order. The method should be extended in future by
inclusion of electroweak corrections and estimation of resummation effects.

The method is applied to the ATLAS $Z$-boson production data at a center-of-
mass energy of $8$ TeV. The data are integrated in $p_{T}$ and compared to the
NNLO predictions. To check impact of the new approach, the comparison is also
performed using fiducial cross sections. A significant, compared to
experimental uncertainties, improvement in the data description by the
predictions based on CT14NNLO set is observed when performing comparison in
the full phase space.

##  VII Acknowledgements

The author would like to thank Simone Amoroso, Ludovica Aperio Bella, Maarten
Boonenkamp, Stefano Camarda, and Jan Kretzschmar for discussions and comments
to the paper draft.

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