#### The photon PDF from high-mass Drell–Yan data at the LHC

Eur. Phys. J. C
The photon PDF from high-mass Drell-Yan data at the LHC
F. Giuli 3
the xFitter Developers' team: V. Bertone 0 1
D. Britzger 7
S. Carrazza 6
A. Cooper-Sarkar 3
A. Glazov 7
K. Lohwasser 5
A. Luszczak 4
F. Olness 9
R. Placˇakyte˙ 8
V. Radescu 3 6
J. Rojo 0 1
R. Sadykov 2
P. Shvydkin 2
O. Zenaiev 7
M. Lisovyi 10
0 Nikhef Theory Group Science Park 105 , 1098 XG Amsterdam , The Netherlands
1 Department of Physics and Astronomy, VU University , 1081 HV Amsterdam , The Netherlands
2 Joint Institute for Nuclear Research (JINR) , Joliot-Curie 6, 141980 Dubna, Moscow Region , Russia
3 University of Oxford , 1 Keble Road, Oxford, OX1 3NP , UK
4 T. Kosciuszko Cracow University of Technology , 30-084 Cracow , Poland
5 DESY Zeuthen , Platanenallee 6, 15738 Zeuthen , Germany
6 CERN , 1211 Geneva 23 , Switzerland
7 DESY Hamburg , Notkestrasse 85, 22609 Hamburg , Germany
8 Institut für Theoretische Physik, Universität Hamburg , Luruper Chaussee 149, 22761 Hamburg , Germany
9 SMU Physics , Box 0175, Dallas, TX 75275-0175 , USA
10 Physikalisches Institut, Ruprecht-Karls-Universität Heidelberg , Heidelberg , Germany
Achieving the highest precision for theoretical predictions at the LHC requires the calculation of hardscattering cross sections that include perturbative QCD corrections up to (N)NNLO and electroweak (EW) corrections up to NLO. Parton distribution functions (PDFs) need to be provided with matching accuracy, which in the case of QED effects involves introducing the photon parton distribution of the proton, x γ (x , Q2). In this work a determination of the photon PDF from fits to recent ATLAS measurements of high-mass Drell-Yan dilepton production at √s = 8 TeV is presented. This analysis is based on the xFitter framework, and has required improvements both in the APFEL program, to account for NLO QED effects, and in the aMCfast interface to account for the photon-initiated contributions in the EW calculations within MadGraph5_aMC@NLO. The results are compared with other recent QED fits and determinations of the photon PDF, consistent results are found.
1 Introduction
Precision phenomenology at the LHC requires theoretical
calculations which include not only QCD corrections, where
NNLO is rapidly becoming the standard, but also
electroweak (EW) corrections, which are particularly significant
for observables directly sensitive to the TeV region, where
EW Sudakov logarithms are enhanced. An important
ingredient of these electroweak corrections is the photon parton
distribution function (PDF) of the proton, x γ (x , Q2), which
must be introduced to absorb the collinear divergences
arising in initial-state QED emissions.
The first PDF fit to include both QED corrections and
a photon PDF was MRST2004QED [1], where the photon
PDF was taken from a model and tested on HERA data for
direct photon production. Almost 10 years later, the
NNPDF2.3QED analysis [2,3] provided a first model-independent
determination of the photon PDF based on Drell–Yan (DY)
data from the LHC. The resulting photon PDF was, however,
affected by large uncertainties due to the limited sensitivity
of the data used as input to that fit. The determination of
x γ (x , Q2) from NNPDF2.3QED was later combined with
the state-of-the-art quark and gluon PDFs from NNPDF3.0,
together with an improved QED evolution, to construct the
NNPDF3.0QED set [4,5]. The CT group has also released a
QED fit using a similar strategy as the MRST2004QED one,
named the CT14QED set [6].
A recent breakthrough concerning the determination of
the photon content of the proton has been the realisation that
x γ (x , Q2) can be calculated in terms of inclusive lepton–
proton deep-inelastic scattering (DIS) structure functions.
The photon PDF resulting from this strategy is called
LUXqed [7] and its residual uncertainties are now at the few
percent level, not too different from those of the quark and gluon
PDFs. A related approach by the HKR [8] group, denoted by
HKR16 in the following, also leads to a similar photon PDF
as compared to the LUXqed calculation, although in this case
no estimate for the associated uncertainties is provided.
The aim of this work is to perform a direct determination
of the photon PDF from recent high-mass Drell–Yan
measurements from ATLAS at √s = 8 TeV [9], and to compare it
with some of the existing determinations of x γ (x , Q2)
mentioned above. Note that earlier measurements of high-mass
DY from ATLAS and CMS were presented in Refs. [10–12].
The ATLAS 8 TeV DY data are provided in terms of both
single-differential cross-section distributions in the dilepton
invariant mass, mll , and of double-differential cross-section
distributions in mll and |yll |, the absolute value of
rapidity of the lepton pair, and in mll and ηll , the difference in
pseudo-rapidity between the two leptons. Using the Bayesian
reweighting method [13, 14] applied to NNPDF2.3QED, it
was shown in the same publication [9] that these
measurements provided significant information on x γ (x , Q2).
The goal of this study is therefore to investigate
further these constraints from the ATLAS high-mass DY
measurements on the photon PDF, this time by means of a
direct PDF fit performed within the open-source xFitter
framework [15]. State-of-the-art theoretical calculations are
employed, in particular the inclusion of NNLO QCD and
NLO QED corrections to the PDF evolution and the
computation of the DIS structure functions as implemented in
the APFEL program [16]. The implementation of NLO QED
effects in APFEL is presented here for the first time. The
inclusion of NLO QED evolution effects is cross-checked
using the independent QEDEVOL code [17] based on the
QCDNUM evolution program [18].
The resulting determination of x γ (x , Q2) represents an
important validation test of recent developments in theory
and data concerning our understanding of the nature and
implications of the photon PDF.
The outline of this paper is as follows. Section 2 reviews
the ATLAS 8 TeV high-mass DY data together with the
theoretical formalism of the DIS and Drell–Yan cross
sections used in the analysis. Section 3 presents the settings of
the PDF fit within the xFitter framework. The fit results
are then discussed in Sect. 4, where they are compared to
determinations by other groups. Finally, Sect. 5 summarises
and discusses the results and future lines of investigation.
“Appendix A” contains a detailed description of the
implementation and validation of NLO QED corrections to the
DGLAP PDF evolution equations and DIS structure
functions, which are available now in APFEL.
2 Data and theory
In this work, the photon content of the proton x γ (x , Q2) is
extracted from a PDF analysis based on the combined
inclusive DIS cross-section data from HERA [19] supplemented
Fig. 1 Diagrams that contribute to lepton-pair production at hadron
colliders at the Born level
by the ATLAS measurements of high-mass Drell–Yan
differential cross sections at √s = 8 TeV [9]. The HERA inclusive
data are the backbone of modern PDF fits, providing
information on the quark and gluon content of the proton, while
the high-mass Drell–Yan data provide a direct sensitivity to
the photon PDF. As illustrated in Fig. 1, dilepton production
at hadron colliders can arise from either quark–antiquark
schannel scattering, or from photon–photon t - and u-channel
scattering mediated by a lepton.
The ATLAS high-mass Drell–Yan 8 TeV measurements
are presented in terms of both the single-differential (1D)
invariant-mass distribution, dσ/dmll , and the
double-differential (2D) distributions in mll and yll , namely d2σ/dmll d|yll |,
and in mll and ηll , d2σ/dmll ηll . For the invariant-mass
1D distribution, there are 12 bins between mll = 116 GeV
and 1.5 TeV; and for both double-differential distributions,
there are five different bins in invariant mass, from the lowest
bin with 116 GeV < mll <150 GeV to the highest bin with
500 GeV < mll < 1500 GeV.
The first three (last two) mll bins of the 2D distributions
are divided into 12 (6) bins with fixed width, extending up to
2.4 and 3.0 for the |yll | and | ηll | distributions, respectively.
The photons which undergo hard scattering in the γ γ →
ee process from Fig. 1 can be produced by either emission
from the proton as a whole (the “elastic” component) or
radiated by the constituent quarks (the “inelastic” component).
From the theory point of view, the photon PDF extracted from
the fit is by construction the sum of the elastic and inelastic
contributions.
For the calculation of NLO high-mass Drell–Yan cross
sections, the MadGraph5_aMC@NLO [20] program is used,
which includes the contribution from photon-initiated
diagrams, interfaced to APPLgrid [21] through aMCfast [22].
A tailored version of APPLgrid is used, accounting for the
contribution of the photon-initiated processes.1 The
calculation is performed in the n f = 5 scheme neglecting mass
effects of charm and bottom quarks in the matrix elements,
as appropriate for a high-scale process. These NLO
theoretical predictions match the analysis cuts of the data, with
mll ≥ 116 GeV, ηl ≤ 2.5, and plT ≥ 40 GeV (30) GeV
for the leading (sub-leading) lepton being the most
important ones. As discussed below, the NLO calculations are
then supplemented by NNLO/NLO K -factors obtained from
1 Modified version of APPLgrid available at: https://github.com/
scarrazza/applgridphoton.
FEWZ [23]. The NLO EW corrections to the DY processes are
also estimated using FEWZ. The photon-initiated process is
taken at LO since this corresponds to the APPLgrid
implementation and the NLO corrections are very small compared
to the data accuracy.
The DIS structure functions and PDF evolution are
computed with the APFEL program [16], which is currently
accurate up to NNLO in QCD and NLO in QED, including the
relevant mixed QCD + QED corrections. This means that,
on top of the pure QCD contributions, the DGLAP
evolution equations [24–26] are solved including the O (αs α) and
O α2 corrections to the splitting functions. Corrections of
O (α) are also included leading to a (weak) explicit
dependence of the predictions on the photon PDF. Details of the
implementation of these corrections and of their numerical
impact are given in “Appendix A”. Heavy-quark (charm and
bottom) mass effects to DIS structure functions are taken into
account using the FONLL-B (C) general-mass scheme [27]
for the NLO (NNLO) fits. The numerical values of the
heavyquark masses in the mass parameter scheme are taken to be
mc = 1.47 GeV and mb = 4.5 GeV as determined in [19],
consistent with the latest PDG averages [28]. The reference
values of the QCD and QED coupling constants are
chosen to be αs (m Z ) = 0.118 and α(mτ = 1.777 GeV) =
1/133.4, again consistent with the PDG recommended
values.
In the calculation of the Drell–Yan cross section, the
dynamical renormalisation μR and factorisation μF scales
are used, which are set equal to the scale of invariant mass
mll , both for the quark- and gluon-induced and for the
photon-induced contributions. The choice of other values
for these scales in the QED diagrams, such as a fixed scale
μR = μF = MZ , leads to variations of the photon-initiated
cross sections of at most a few percent. The choice of the
scale for the photon PDF is further discussed in [29,30]. For
the kinematics of the ATLAS DY data, the ratio between
the photon-initiated contributions and quark- and
gluoninduced dilepton production is largest for central rapidities
and large invariant masses. For the most central (forward)
rapidity bin, 0 < |yll | < 0.2 (2.0 < |yll | < 2.4), the ratio
between the QED and QCD contributions varies between
2.5% (2%) at low invariant masses and 12% (2.5%) for
the highest mll bin, based on MMHT14nnlo_68cl PDF set
for the QCD contribution and NNPDF30qed_nnlo_as_0118
PDF set for the LO QED contributions. Therefore, data
from the central region will exhibit the highest sensitivity to
x γ (x , Q2).
The MadGraph5_aMC@NLO NLO QCD and LO QED
calculations used in this work have been benchmarked
against the corresponding predictions obtained with the
FEWZ code [23], finding agreement within statistical
uncertainties of the predictions for both the 1D and the 2D
distributions.
Fig. 2 The NNLO/NLO K -factors, defined in Eq. (1), which account
for higher-order QCD and EW effects to the high-mass Drell–Yan cross
sections with the photon-induced contribution subtracted, as a function
of the dilepton rapidity |yll |. Each set of points corresponds to a different
bin in the dilepton invariant mass mll
In order to achieve NNLO QCD and NLO EW accuracy
in our theoretical calculations, the NLO QCD and LO QED
cross sections computed with MadGraph5_aMC@NLO have
been supplemented by bin-by-bin K -factors defined
by
K (mll , |yll |) ≡
NNLO QCD + NLO EW
NLO QCD + LO EW
using the MMHT2014 NNLO [31] PDF set both in the
numerator and in the denominator. The K -factors have
been computed using FEWZ with the same settings and
analysis cuts as the corresponding NLO calculations of
MadGraph5_aMC@NLO. This approximation is justified
since the NNLO K -factors as defined in Eq. (1) depend
very mildly on the input PDF set; see for example Ref. [32].
The photon-induced contribution, as provided in Ref. [9],
has been explicitly subtracted from the FEWZ predictions.
Figure 2 shows the K -factors of Eq. (1) corresponding to
the double-differential (mll , |yll |) cross sections as a
function of the dilepton rapidity |yll |, where each set of points
corresponds to a different dilepton invariant mass mll bin.
The K -factors vary between 0.98 and 1.04, highlighting
the fact that higher-order corrections to the Drell–Yan
process are moderate, in particular at low values of mll and
in the central region. Even at forward rapidities, such K
factors modify the NLO QCD + LO EW results by at most
4%. Based on the definition in Eq. (1), in the rest of the
paper, unless explicitly specified, we will refer to NNLO
by actually meaning NNLO in QCD plus NLO EW
corrections, and to NLO by meaning NLO in QCD plus LO EW
corrections.
3 Settings
This section presents the settings of the PDF fits,
including the details of the parametrisation of the photon PDF
x γ (x , Q2), which have been carried out using the
opensource xFitter framework [15]. First of all, the scale Q20 at
which PDFs are parametrised is taken to be Q2
0 = 7.5 GeV2,
which coincides with the value Q2min that defines the
kinematic cut Q2 ≥ Q2min for the data points that are used as
input to the fits. The charm PDF is then generated
perturbatively from quarks and gluons by means of DGLAP
evolution, exploiting recent developments in APFEL which allow
the setting of heavy-quark thresholds μh differently from the
heavy-quark masses mh , such that μc = Q0 > mc. Hence
a high threshold can be used without having to parametrise
the charm PDF [33].
The expression for the χ 2 function used for the fits is that
of Ref. [34], which includes corrections for possible biases
from statistical fluctuations and treats the systematic
uncertainties multiplicatively. Alternative forms that do not include
these corrections, such as those defined in [19,35], have also
been studied, but no significant differences in the results have
been observed.
In this analysis, the parametrised PDFs are the valence
distributions x uv(x , Q20) and x dv(x , Q20), the gluon distribution
xg(x , Q20), and the u-type and d-type sea-quark distributions,
xU¯ (x , Q20), x D¯ (x , Q20), where xU¯ (x , Q20) = x u¯(x , Q20)
and x D¯ (x , Q20) = x d¯(x , Q20) + x s¯(x , Q20). In addition,
the photon distribution x γ (x , Q2) is also parametrised at
0
the starting scale. The following general functional form is
adopted:
x f (x , Q02) = Ax B (1 − x )C (1 + Dx + E x 2) ,
where some of the normalisation parameters, in
particular Auv , Adv and Ag, are constrained by the valence and
momentum sum rules (note that the photon PDF also enters
the momentum sum rule). The parameters BU¯ and BD¯ are
set equal to each other, so that the two quark sea
distributions share a common small-x behaviour. Since the
measurements used here are not sensitive to the strangeness content
of the proton, strangeness is fixed such that x s¯(x , Q2) =
0
rs x d¯(x , Q20), where rs = 1.0 is consistent with the ATLAS
analysis of inclusive W and Z production [36,37]. The
further constraint AU¯ = 0.5 AD¯ is imposed, such that
x u¯(x , Q2) → x d¯(x , Q2) as x → 0.
0 0
The explicit form of the PDF parametrisation Eq. (2) at
the scale Q20 is determined by the technique of saturation of
the χ 2, namely the number of parameters is increased one by
one until the χ 2 does not improve further, employing Wilks’
theorem [38]. Following this method, the optimal
parametrisation for the quark and gluon PDFs found for this analysis
is
x uv(x ) = Auv x Buv (1 − x )Cuv (1 + Euv x 2) ,
x dv(x ) = Adv x Bdv (1 − x )Cdv ,
while for the photon PDF it is used:
x γ (x ) = Aγ x Bγ (1 − x )Cγ (1 + Dγ x + Eγ x 2).
The parametrisation of the quark and gluon PDFs in Eq. (3)
differs from the one used in the HERAPDF2.0 analysis in
various ways. First of all, a higher value of the input
evolution scale Q20 is used, which is helpful to stabilise the fit of
the photon PDF. Second, an additional negative term in the
parametrisation of the gluon is not required here, because of
the increased value of Q20 which ensures the positiveness of
the gluon distribution. Third, the results of the
parametrisation scan are different because of the inclusion of the ATLAS
high-mass Drell–Yan cross-section data.
PDF uncertainties are estimated using the Monte Carlo
replica method [39–41], cross-checked with the Hessian
method [42] using χ 2 = 1. The former is expected to be
more robust than the latter, due to the potential non-Gaussian
nature of the photon PDF uncertainties [3]. In Sect. 4 it is
shown that these two methods to estimate the PDF
uncertainties on the photon PDF lead to similar results.
In addition, a number of cross-checks have been
performed to assess the impact of various model and
parametrisation uncertainties. For the model uncertainties, variations
of the charm mass between mc = 1.41 to 1.53 GeV, of the
bottom mass between mc = 4.25 to 4.75 GeV, of the strong
coupling constant αs (m Z ) between 0.116 to 0.120 are
considered, and additionally the strangeness fraction is decreased
down to rs = 0.75. For the parametrisation uncertainties, the
impact of increasing the input parametrisation scale up to
Q2
0 = 10 GeV2 is considered as well as the impact of
including additional parameters in Eq. (3). These extra parameters
make little difference to the χ 2 of the fit, but they can change
the shape of the PDFs in a non-negligible way. Such
additional parameters are Duv , Du , Ed¯, as well and the extra
¯
negative term in the gluon PDF used in HERAPDF2.0. The
impact of these model and parametrisation uncertainties on
the baseline results is quantified in Sect. 4.3.
4 Results
In this section the determination of the PDFs from a fit to
HERA inclusive structure functions and ATLAS high-mass
Drell–Yan cross sections, with an emphasis on the photon
PDF is presented. First the fit quality is assessed and the
Table 1 The χ 2/Ndat in the NNLO fits for the HERA inclusive structure
functions and for the various invariant mass mll bins of the ATLAS
highmass DY data. In the latter case, the contribution to the χ 2 arising from
the correlated and log-penalty terms are indicated, as well as the overall
χ 2/Ndof is provided, where Ndof is the number of degree of freedom
in the fit
fit results are compared with the experimental data. Then
the resulting photon PDF is shown and compared with other
recent determinations. The impact of the high-mass DY data
on the quark PDFs is also quantified. Thus, the robustness
of the fits of x γ (x , Q2) with respect to varying the model,
parametrisation, and procedural inputs is assessed. Finally,
perturbative stability is addressed by comparing NLO and
NNLO results.
4.1 Fit quality and comparison between data and fit results
In the following, the results that will be shown correspond to
those obtained from fitting the double-differential (mll , yll )
cross-section distributions. It has been verified that
comparable results are obtained if the (mll , ηll ) cross-section
distributions are fitted instead.
For the baseline NNLO fit, the value χ m2in/Ndof =
1284/1083 is obtained where Ndof is the number of degrees
of freedom in the fit which is equal to total number of data
points minus number of free parameters. The contribution
from the HERA inclusive data is χ 2/Ndat = 1236/1056 and
from the ATLAS high-mass DY data is χ 2/Ndat = 48/48,
where Ndat the number of the data points for the
corresponding data sample. These values for χ 2/Ndat, together with the
corresponding values for the various invariant mass mll bins
of the ATLAS data, are summarised in Table 1. The quality of
the agreement with the HERA cross sections is of
comparable quality to that found in the HERAPDF2.0 analysis. Note
that in the calculation of the total χ 2 for the ATLAS data, the
correlations between the different mll bins have been taken
into account.
Figures 3, 4 and 5 then show the comparison between the
results of the NNLO fit, denoted by xFitter_epHMDY,
Fig. 3 Comparison between the results of the fit and the ATLAS data
for the (mll , |yll |) double-differential Drell–Yan cross sections as
functions of |yll |, for the first two mll bins. The comparisons are shown both
in an absolute scale (upper plots) and as ratios to the central value of
the experimental data in each yll bin (lower plots). The error bars on
the data points correspond to the bin-to-bin uncorrelated uncertainties,
while the yellow bands indicate the size of the correlated
uncertainties. The solid lines indicate the theory calculations obtained using the
results of the fit xFitter_epHMDY
and the ATLAS data for the (mll , |yll |) double-differential
Drell–Yan cross sections as functions of |yll |, for the five
bins in mll separately.
The comparisons are shown both on an absolute scale and
as ratios to the central value of the experimental data.
The error bars on the data points correspond to the
bin-tobin uncorrelated uncertainties, while the bands indicate the
size of the correlated systematic uncertainties.
The solid lines indicate the theory calculations obtained
using the results of the fit.
Figures 3, 4 and 5 demonstrate good agreement between
ATLAS data and the NNLO theory predictions obtained from
the xFitter_epHMDY fit. This agreement is also
quantitatively expressed by the values of the χ 2 reported in Table 1,
where for the ATLAS data a χ 2/Ndat = 1 is found. This is
Fig. 4 Same as Fig. 3 for the third and fourth mll bins
Fig. 5 Same as Fig. 3 for the highest mll bin
particularly remarkable given the high precision of the data,
with total experimental uncertainties at the few percent level
in most of the kinematic range.
Q2 = 10000 GeV2
xFitter_epHMDY
LUXqed
HKR16
NNPDF30qed
Q2 = 10000 GeV2
xFitter_epHMDY
LUXqed
HKR16
NNPDF30qed
pp → l+l-; s = 8 TeV
Fig. 6 Upper plot Comparison between the photon xγ (x, Q2) at Q2 =
104 GeV2 from the present NNLO analysis (xFitter_epHMDY) with
the corresponding results from NNPDF3.0QED, LUXqed and HKR16.
Lower plot The same comparison, now with the results normalised
to the central value of xFitter_epHMDY. For the present fit, the
PDF uncertainties are shown at the 68% CL obtained from the MC
method, while model and parametrisation uncertainties are discussed
below. For HKR16 only the central value is shown, while for LUXqed
the associated PDF uncertainty band [7] is included
4.2 The photon PDF from LHC high-mass DY data
In Fig. 6, the photon PDF, x γ (x , Q2), is shown at Q2 = 104
GeV2, and it is compared to the corresponding LUXqed,
HKR16 and NNPDF3.0QED results. In the upper plot the
comparison is presented in an absolute scale, while in
the lower plot the ratio of different results normalised to
the central value of the fit is shown. For the present fit,
xFitter_epHMDY, the experimental PDF uncertainties at
the 68% confidence level (CL) are obtained from the Monte
Carlo method, while model and parametrisation uncertainties
are discussed below.
Likewise, the NNPDF3.0QED PDF set is shown the 68%
CL uncertainty band, while for LUXqed the associated PDF
uncertainty band is computed according to the prescription
of Ref. [7].
For HKR16, only the central value is available. The x
range in Fig. 6 has been restricted to the region 0.02 ≤ x ≤
0.9, since beyond that region there is only limited sensitivity
to x γ (x , Q2).
Figure 6 shows that for x ≥ 0.1 the four determinations
of the photon PDF are consistent within PDF uncertainties.
For smaller values of x , the photon PDF from LUXqed and
HKR16 is somewhat smaller than xFitter_epHMDY, but
still in agreement at the 2σ level. This agreement is further
improved if the PDF uncertainties in xFitter_epHMDY
arising from variations of the input parametrisation are added
to experimental uncertainties, as discussed in Sect. 4.3.
Moreover, the results of this work and NNPDF3.0QED agree at
the 68% CL for x ≥ 0.03, and the agreement extends to
smaller values of x once the parametrisation uncertainties
in xFitter_epHMDY are accounted for. The LUXqed and
the HKR16 calculations of x γ (x , Q2) are very close to each
other across the entire range of x .
Figure 6 shows that for 0.04 ≤ x ≤ 0.2 the present
analysis exhibits smaller PDF uncertainties as compared to those
from NNPDF3.0QED. Indeed, the experimental uncertainty
on the xFitter_epHMDY turns out to be at the ∼30% level
for x ≤ 0.1. At larger x it increases rapidly specially in the
positive direction. The reason for this behaviour at large x
can be understood by recalling that variations of x γ (x , Q2)
in the negative direction are constrained by positiveness. The
limited sensitivity of the ATLAS data does not allow a
determination of x γ (x , Q2) with uncertainties competitive with
those of LUXqed, which are at the few percent level.
It is also interesting to assess the impact of the high-mass
Drell–Yan 8 TeV measurements on the light quark and gluon
PDFs. For this purpose, the fits have been repeated freezing
the photon PDF to the xFitter_epHMDY shape. This is
necessary because HERA inclusive data alone, which are the
benchmark for this comparison, have no sensitivity to the
photon PDF. This way, a meaningful comparison between
the quark and gluon PDFs from a HERA-only baseline and
the HERA + HMDY fit can be performed.
This comparison is shown in Fig. 7 for the up and down
antiquarks x u¯(x , Q2) and x d¯(x , Q2), for which the effect of
the high-mass Drell–Yan data is expected to be most
pronounced, since HERA inclusive cross sections provide
little information on quark flavour separation. In Fig. 7, the
x u¯(x , Q2) and x d¯(x , Q2) together with the associated MC
uncertainties have been computed at the initial
parametrisation scale of Q2 = 7.5 GeV2 and are shown as ratios to
the central value of the xFitter_epHMDY fit. The
modifications in the medium and large-x antiquark distributions
from the high-mass DY data are rather moderate. It has
been verified that the same conclusions can be derived from
fits obtained by switching off the QED effects for both the
Q2 = 7.5 GeV2
xFitter_epHMDY
HERAI+II
Q2 = 7.5 GeV2
xFitter_epHMDY
HERAI+II
u
x
/xu1.2
δ
d
x
/
d
x
δ1.5
Fig. 7 The impact of the ATLAS high-mass 8 TeV Drell–Yan
measurements on the xu¯ and xd¯ sea-quark PDFs at the input parametrisation
scale Q2 = 7.5 GeV2. The results are shown normalised to the central
value of xFitter_epHMDY.
HERA-only fits and the HERA+HMDY fits. Therefore, while
the ATLAS high-mass Drell–Yan measurements have a
significant constraint on the photon PDF, their impact on the
quark and gluon PDFs is moderate.
4.3 Robustness and perturbative stability checks
Following the presentation of the main result of this work,
the xFitter_epHMDY determination of the photon PDF
x γ (x , Q2), the robustness of this determination with respect
to a number of variations is assessed. Firstly, variations in
the values of the input physical parameters, such as αs or
the charm mass are explored. Secondly, variations of the
choices made for the PDF input parametrisation are
considered. Finally, variations associated to different
methodological choices in the fitting procedure are quantified. In
each case, one variation at a time is performed and compared
with the central value of x γ (x , Q2) and its experimental PDF
uncertainties computed using the Monte Carlo method.
2 )Q
,
x
(
γ
x0.08
Q2 = 10000 GeV2
xFitter_epHMDY
rs2= 0.75
Qmin = 5 GeV2
Fig. 8 Comparison between the baseline determination of xγ (x, Q2)
at Q2 = 104 GeV2 in the present analysis, xFitter_epHMDY, with
the central value of a number of fits for which one input parameter has
been varied. The following variations have been considered: rs = 0.75,
Q2min = 5 GeV2, αs = 0.116 and 0.118 (upper plot); and mc = 1.41
and 1.53 GeV, mb = 4.25 and 4.75 GeV, and Q2
0 = 10 GeV2 (lower
plot). The curves are indistinguishable because they overlap due to their
negligible impact on photon PDF fit. Only the impact of the variation
of the strange fraction assumption is visible by eye. See text for more
details of these variations
First the impact of uncertainties associated to either the
choice of input physical parameters or of specific settings
adopted in the fit is considered. Figure 8 shows the
comparison between the xFitter_epHMDY determination of
x γ (x , Q2) at Q2 = 104 GeV2, including the
experimental MC uncertainties, with the central value of those fits for
which a number of variations have been performed.
Specifically:
– The strong coupling constant is varied by δαs = ±0.002
around the central value.
– The ratio of strange to non-strange light quark PDFs is
decreased to rs = 0.75 instead of rs = 1.
– The value of the charm mass is varied between mc =
1.41 GeV and mc = 1.53 GeV, and that of the bottom
mass between mb = 4.25 GeV and mb = 4.75 GeV.
– The minimum value Q2min of the fitted data is decreased
down to 5 GeV2.
– The input parametrisation scale Q20 is raised to 10 GeV2
as compared to the baseline value of Q2
0 = 7.5 GeV2.
The results of Fig. 8 highlight that in all cases effect of the
variations considered here is contained within (and typically
much smaller than) the experimental PDF uncertainty bands
of the reference fit. The largest variation comes from the
strangeness ratio rs , where the resulting central value turns
out to be at the bottom end of the PDF uncertainty band for
x ≥ 0.1.
Another important check of the robustness of the present
determination of x γ (x , Q2) can be obtained by comparing
the baseline fit with further fits where a number of new free
parameters are allowed in the PDF parametrisation, in
addition to those listed in Eq. (3). Figure 9 shows the impact of
three representative variations (others have been explored,
leading to smaller differences): more flexibility to the gluon
distribution, allowing it to become negative at the initial scale
(labeled by “neg”), in addition to Duv , and then Du¯ + Dd¯. As
before, all variations are contained within the experimental
PDF uncertainty bands, though the impact of the
parametrisation variations is typically larger than that of the model
variations: in the case of the neg + Du¯ + Dd¯ variations, the
central value is at the lower edge of the PDF uncertainty band
in the entire range of x shown.
A cross-check of the robustness of the estimated
experimental uncertainty of the photon PDF in this analysis is
provided by the comparison of the Monte Carlo and Hessian
methods. Figure 9 shows this comparison indicating a
reasonable agreement between the two methods. In particular,
the central values of the photon obtained with the two fitting
techniques are quite similar to each other. As expected, the
MC uncertainties tend to be larger than the Hessian ones,
specially in the region x 0.2, indicating deviations with
respect to the Gaussian behaviour of the photon PDF.
To complete these studies, an interesting exercise is to
quantify the perturbative stability of the xFitter_epHMDY
determination of the photon PDF x γ (x , Q2) with respect to
the inclusion of NNLO QCD corrections in the analysis. To
study this, Fig. 10 shows a comparison between the baseline
fit of x γ (x , Q2), based on NNLO QCD and NLO QED
theoretical calculations, with the central value resulting from
a corresponding fit based instead on NLO QCD and QED
theory. In other words, the QED part of the calculations is
identical in both cases. For the NNLO fit, only the
experimental PDF uncertainties, estimated using the Monte Carlo
method, are shown. From the comparison of Fig. 10, it is clear
that the fit of x γ (x , Q2) exhibits a reasonable perturbative
2 )
,Q0.07
x
(
γ
x0.06
Q2 = 10000 GeV2
xFitter_epHMDY
+neg
+neg+Duv
+neg+Duv+DUbar
Q2 = 10000 GeV2
MC 68CL xFitter_epHMDY
Hessian xFitter_epHMDY
Fig. 9 Upper plot The impact on the photon PDF xγ (x, Q2) from
xFitter_epHMDY in fits where a number of additional free
parameters are allowed in the PDF parametrisation Eq. (3). The parametrisation
variations that have been explored are: more flexibility to the gluon
distribution, allowing it to become negative (labeled by “neg”), adding on
top Duv , and then adding Du¯ + Dd¯. Lower plot Comparison between
the xFitter_epHMDY determinations obtained with the Monte Carlo
(baseline) and with the Hessian methods, where in both cases the PDF
error band shown corresponds to the 68% CL uncertainties
stability, since the central value of the NLO fit is always
contained in the one-sigma PDF uncertainty band of the baseline
xFitter_epHMDY fit. The agreement between the two fits
is particularly good for x 0.1, where the two central
values are very close to each other. This comparison is shown at
low scale, Q2 = 7.5 GeV2 and high scales Q2 = 104 GeV2,
indicating that perturbative stability is not scale dependent.
5 Summary
In this work, a new determination of the photon PDF from a
fit of HERA inclusive DIS structure functions supplemented
by ATLAS data on high-mass Drell–Yan cross sections has
been presented, based on the xFitter framework. As
sug2 )0.06
Q
,
x
(
γ
x0.05
2 )0.08
Q
,
x
γ(0.07
x
0.06
Fig. 10 Upper plot comparison between the reference
xFitter_epHMDY fit of xγ (x, Q2), based on NNLO QCD and
NLO QED theoretical calculations, with the central value of the
corresponding fit based on NLO QCD and QED theory, at Q2 = 7.5
GeV2. In the former case, only the experimental Monte Carlo PDF
uncertainties are shown. Lower plot Same comparison, now presented
at the higher scale of Q2 = 104 GeV2
gested by a previous reweighting analysis [9], this high-mass
DY data provides significant constraints on the photon PDF,
allowing a determination of x γ (x , Q2) with uncertainties at
the 30% level for 0.02 ≤ x ≤ 0.1. The results of the present
study, dubbed xFitter_epHMDY, are in agreement and
exhibit smaller PDF uncertainties that the only other existing
photon PDF fit from LHC data, the NNPDF3.0QED analysis,
based on previous LHC Drell–Yan measurements.
The results are in agreement within uncertainties with
two recent calculations of the photon PDF, LUXqed and
HKR16. For x ≥ 0.1, the agreement is at the 1σ level already
including only the experimental MC uncertainties, while for
0.02 ≥ x ≥ 0.1 it is important to account for parametrisation
uncertainties. The findings indicate that a direct
determination of the photon PDF from hadron collider data is still far
from being competitive with the LUXqed and HKR
calculations, which are based instead on precise measurements of
the inclusive DIS structure functions of the proton.
The results of this study, which are available upon request
in the LHAPDF6 format [43], have been made possible
by a number of technical developments that should be
of direct application for future PDF fits accounting for
QED corrections. First of all, the full NLO QED
corrections to the DGLAP evolution equations and the DIS
structure functions have been implemented in the APFEL
program. Moreover, our results illustrate the flexibility of the
xFitter framework to extend its capabilities beyond the
traditional quark and gluon PDF fits. Finally, the
extension of aMCfast and APPLgrid to allow for the
presence of photon-initiated channels in the calculations
provided by MadGraph5_aMC@NLO, significantly streamlines
the inclusion of future LHC measurements in PDF fits with
QED corrections by consistently including diagrams with
initial-state photons. All these technical improvements will
certainly be helpful for future studies of the photon content
of the proton.
Acknowledgements We thank L. Harland-Lang for providing us
the LHAPDF6 grid of the HKR16 photon determination. We thank
A. Sapronov for providing support to the xFitter platform. We
thank P. Starovoitov for discussions related to the APPLgrid files.
We thank M. Zinser for discussions related to the ATLAS data. We
thank M. Dyndal for discussions on the scale choices for the
photoninduced contributions. The work of V. B., F. G., J. R. has been supported
by the European Research Council Starting Grant “PDF4BSM”. The
work of S. C. is supported by the HICCUP ERC Consolidator grant
(614577). A. L. thanks for the support from the Mobility Plus Grant
No. 1320/MOB/IV/2015/0. The work of F. O. has been supported by
the US DoE Grant DE-SC0010129. The work of P. S and R. S. has
been supported by the BMBF-JINR cooperation. We are grateful to the
DESY IT department for their support of the xFitter developers.
Open Access This article is distributed under the terms of the Creative
Commons Attribution 4.0 International License (http://creativecomm
ons.org/licenses/by/4.0/), which permits unrestricted use, distribution,
and reproduction in any medium, provided you give appropriate credit
to the original author(s) and the source, provide a link to the Creative
Commons license, and indicate if changes were made.
Funded by SCOAP3.
Appendix A: Implementation of NLO QED corrections
in APFEL
In this appendix, the details of the implementation of the
combined NLO QCD + QED corrections in the APFEL
program are presented. As discussed in Ref. [16], the
implementation of the LO QED corrections to the DGLAP evolution
equations presents many simplifications, in particular the fact
that QED and QCD corrections do not mix and therefore the
DGLAP equations, as well as the evolution equations for the
running of αs and α, are decoupled. When considering NLO
corrections, this property does not hold anymore, and QED
and QCD contributions mix both in the DGLAP and in the
coupling evolution equations. On top of this, NLO QED
corrections induce the presence of diagrams with a real photon in
the initial state, and these have to be consistently included in
the computation of the DIS structure functions resulting in an
explicit dependence on the photon PDF of these observables.
In the following, the discussion starts by discussing the
generalisation of the equations for the running of the QCD
and QED couplings, finding that the mixed QCD + QED
terms have a negligible impact. Then the extension of the
DGLAP evolution equations to account for the complete
NLO QCD + QED effects is discussed. Finally, the
modifications introduced by the NLO QED corrections in both the
neutral-current and the charged-current DIS structure
functions are discussed.
Appendix A.1: Evolution of the couplings
As mentioned above, the NLO QCD+QED corrections
induce the presence of mixed terms in the evolution
equations of αs and α. In practice, the QCD β-function receives
corrections proportional to α and, vice versa, the QED
βfunction receives corrections proportional to αs , in such a
way that the coupling evolution equations read
μ2 ∂∂μαs2 = βQCD(αs , α),
μ2 ∂∂μα2 = βQED(αs , α).
As a consequence, these evolution equations form a set of
coupled differential equations. Up to three loops (i.e. NLO),
the β-functions can be expanded as
where the mixed terms, β1(αs α) and β(ααs ), and the pure NLO
1
QED term, β(α2), can be found in Ref. [44]. Taking into
1
account a factor 4 due to the different definitions of the
expansion parameters, one finds
i=1
i=1
i=1
where Nc = 3 is the number of colours, eq is the electric
charge of the quark flavour q, and n f and nl are the number
of active quark and lepton flavours, respectively.
Equation (A.1) can be written in vectorial form:
Equation (A.5) is an ordinary differential equation that can be
numerically solved using, for example, Runge–Kutta
methods.
The first two terms in Eq. (A.4) are responsible for the
coupling of the evolution of αs and α, and thus they
introduce a complication that affects both the implementation and
the performance of the code. One can then ask what is the
effect of their presence and whether their removal makes a
substantial difference.
Figure 11 shows the comparison between the evolution at
NLO of both couplings αs and α including and excluding the
mixed terms in the respective β-functions. The evolution is
performed between the Z mass scale MZ and 10 TeV with 5
active quark flavours and 3 active lepton flavours and uses as
boundary conditions αs (MZ ) = 0.118 and α(MZ ) = 1/128.
The two curves in Fig. 11 are normalised to the respective
curves without mixed terms. It is clear that the mixed terms
lead to tiny relative differences that are at most of O(10−4) at
10 TeV for αs and O(10−3) at the same scale for α. Thus it is
safe to conclude that the mixed terms in the β-functions have
a negligible effect on the evolution of the couplings and thus
they are excluded to simplify the code and to improve the
performance without introducing any significant inaccuracy.
Appendix A.2: PDF evolution with NLO QED corrections
Next the implementation of the full NLO QCD+QED
corrections to the DGLAP evolution equations is considered.
The discussion is limited to consider of the photon PDF
only, while the possible presence of the lepton PDFs is not
addressed here. The first step towards an efficient
implementation of the solution of the DGLAP equations in the presence
of QED corrections is the adoption of a suitable PDF basis
that diagonalises the splitting function matrix, decoupling as
Fig. 11 Comparison between the running with the scale Q of the QCD
and QED couplings, αs and α, including or not the mixed terms in the
corresponding β-functions. The curves are normalised to the result of
the respective coupling running without the mixed terms included in
the β functions
many equations as possible. Such a basis was already
introduced in Appendix A of Ref. [45] and will be used also here.
Excluding the lepton PDFs, this basis contains 14
independent PDF combinations and reads
3 : = u + d , 9 : V = Vu + Vd ,
4 : = u − d , 10 : V = Vu − Vd ,
5 : T1u = u+ − c+, 11 :V1u = u− − c−,
7 : T1d = d+ − s+, 13 :V1d = d− − s−,
u = u+ + c+ + t +, Vu = u− + c− + t −,
d = d+ + s+ + b+, Vd = d− + s− + b−.
The second step is the construction of the splitting function
matrix that determines the evolution of each of the
combinations listed in Eq. (A.7). To this end, the splitting
function matrix P is split into a pure QCD term P, which only
depends on αs , and a mixed QCD+QED correction term P,
which instead contains contributions proportional to at least
one power of the QED coupling α. In practice, this means
that
P = P + P,
where the pure QCD term reads
P = αs P(1,0) + αs2P(2,0) + · · · ,
while the term containing the QED coupling is given by
P = αP(0,1) + αs αP(1,1) + α2P(0,2) + · · · .
Note that in the r.h.s. of Eqs. (A.10) and (A.11) the convention
of Refs. [46,47] is followed to indicate the power of αs and
α that each splitting function multiplies.
The structure of the pure QCD splitting function matrix
P as well as the first term in P, which represents the pure
LO QED correction, were already discussed in Ref. [45]. It is
now necessary to analyse the structure of the two additional
terms, namely P(1,1) and P(0,2). Starting with the O(αs α)
term, the resulting evolution equations read
⎜⎜ e2 Pγ(1g,1) e2 Pγ(1γ,1) η+Pγ(1q,1) η−Pγ(1q,1) ⎟⎟
= ⎜⎜ 2e2 Pq(1g,1) 2e2 Pq(1γ,1) η+P+(1,1) η−P+(1,1)⎟⎟
⎝ 2δe2Pq(1g,1) 2δe2Pq(1γ,1) η−P+(1,1) η+P+(1,1)⎠
= eu2P+(1,1) ⊗ T1u,2,
= ed2 P+(1,1) ⊗ T1d,2,
= eu2P−(1,1) ⊗ V1u,2,
= ed2 P−(1,1) ⊗ V1d,2,
with eu and ed the electric charges of the up- and down-type
quarks, and nu and nd the number of up- and down-type
active quark flavours (such that nu + nd = n f ).
Next the O(α2) corrections as considered. The
expressions of the splitting functions at this order have been
presented in Ref. [47]. There are two relevant new features that
distinguish these corrections from the O(α) and the O(αs α)
ones. The first one is that, contrary to the other cases in which
the electric charges appears to the second power at most, here
they appear up to the fourth power. As a consequence, new
couplings must be introduced:
The second feature is that the dependence on the electric
charges of some of the O(α2) splitting functions is not
factorisable as was the case for all the O(α) and O(αs α) ones
and therefore a distinction must be made between up- and
down-type splitting functions. Taking into account these
features, it is possible to show that the O(α2) contributions to
the DGLAP equations take the following form:
1 eu4Pu−u(0,2) + ed4 Pd−d(0,2) eu4Pu−u(0,2) − ed4 Pd−d(0,2)
= 2
eu4Pu−u(0,2) − ed4 Pd−d(0,2) eu4Pu−u(0,2) + ed4 Pd−d(0,2)
= eu4Pu+u(0,2) ⊗ T1u,2,
= ed4 Pd+d(0,2) ⊗ T1d,2,
= eu4Pu−u(0,2) ⊗ V1u,2,
= ed4 Pd−d(0,2) ⊗ V1d,2.
It should be noted that, as compared to the expressions for
P(0,2) presented in Ref. [47], the electric charges have been
factored out in such a way that the expressions of the splitting
functions are either independent from the electric charges
themselves or depend on them only through the ratio e2 /eq2.
As an illustration, the effects of the O(αs α) and O(α2)
corrections to the DGLAP evolution equations on the γ γ
luminosity at √s = 13 TeV are quantified. This luminosity
is defined by
as a function of the final-state invariant mass MX . Figure 12
illustrates the behaviour of Lγ γ computed using the photon
PDF from the NNPDF30QED NLO set as an input at Q0 = 1
GeV and evolved to Q = MX including, on top of the pure
QCD NLO evolution, the following corrections:
– the O(α) corrections only,
– same as above, adding also the mixed O(αs α)
corrections, and
– the complete NLO QCD+QED corrections accounting
for the O(α + αs α + α2) effects.
The results are shown normalised to the predictions obtained
with LO QED corrections only. It is clear that the O(αs α) and
O(α2) corrections have a small but non-negligible impact on
the γ γ -luminosity. In particular, these corrections suppress
Lγ γ by around 10% at relatively small values of MX , while
the suppression gradually shrinks to 1–2% as MX increases.
Fig. 12 The photon–photon PDF luminosity Lγ γ at √s = 13 TeV
as a function of the final-state invariant mass MX . The results with the
photon evolved with only the O(α) corrections are compared with the
corresponding results taking into account the O(α + αs α) corrections
and the complete O(α + αs α + α2) effects, normalised in all three cases
to the O(α) result. The calculation has been performed using the central
value of the NNPDF3.0QED NLO fit
As expected, most of this effect comes from the O(αs α)
corrections, while the impact of the O(α2) ones is
substantially smaller. These corrections to the DGLAP evolution
have more recently been implemented also in the QEDEVOL
package [17] based on the QCDNUM evolution code [18].
APFEL and QEDEVOL have been found to be in excellent
agreement.
Appendix A.3: DIS structure functions
When considering NLO QCD + QED corrections to the DIS
structure functions, it becomes necessary to include into the
hard cross sections all the O(α) diagrams where one photon
is either in the initial state or emitted from an incoming quark
(or possibly an incoming lepton). Such diagrams, being of
purely QED origin, have associated coefficient functions that
can easily be derived from the QCD expressions by properly
adjusting the colour factors. This correspondence holds
irrespective of whether mass effects are included.
The main complication of the inclusion of these
corrections arises from their flavour structure. In fact, in the case of
quarks the isospin symmetry is broken due to the fact that the
coupling of the photon is proportional to the squared charge
of the parton to which it couples (a quark or a lepton). In
the following, the neutral-current (NC) case, where lepton
and proton exchange a neutral boson γ ∗/Z , and the
chargedcurrent (CC) case, where instead lepton and proton exchange
a charged W boson, are addressed separately.
First the O(α) contributions to a generic NC structure
function F are considered. Due to the fact that to this order
there is no mixing between QCD and QED, such corrections
can easily be derived from the O(αs ) coefficient functions
In order to construct the corresponding structure functions,
considering that the coupling between a photon and a quark
of flavour q is proportional to eq2 , the electroweak couplings
should be adjusted as follows:
Dq = Dq eq
where Bq and Dq are defined, e.g., in Ref. [49]. Following
this prescription, it is possible to write the O (α) contributions
to the NC structure functions:
= x
= x
Bq C2(α,L);q ⊗ (q + q ) + C2(α,L);γ ⊗ γ ,
Dq C3(α;q) ⊗ (q − q ) + C3(α;γ) ⊗ γ .
This structure holds for both massless and massive structure
functions. This aspect is relevant to the construction of the
FONLL general-mass structure functions.
For the CC case the procedure to obtain the expressions
of the O (α) coefficient functions is exactly the same as in
the NC case (see Eq. (A.21)). However, this case is more
complicated because the flavour structure of CC structure
functions is more complex. Taking into account the presence
of a factor eq2 every time that a quark of flavour q couples to
a photon, the O (α) corrections to the CC structure functions
F2 and FL for the production of a neutrino or an anti-neutrino
take the form
|VU D|2 C2(α,L);q ⊗ e2D D + eU2 U
|VU D|2 C2(α,L);q ⊗ e2D D + eU2 U
where VU D are the elements of the CKM matrix. The flavour
structure of F3 is instead slightly different:
|VU D|2 C3(α;q) ⊗ −e2D D + eU2 U
just by adjusting the colour factors by setting C F = TR = 1
and C A = 0. Referring, e.g., to the expressions reported in
Ref. [48], the coefficient functions become
Fig. 13 The effects of the NLO QED corrections on the neutral-current
(upper) and charged-current (lower plot) DIS structure functions F2, FL
and x F3, normalised to the pure QCD results. The calculation has been
performed in the FONLL-B general-mass scheme using the central
NNPDF3.0QED NLO set as input. Note that QED effects enter both
via DGLAP evolution and the O(α) DIS coefficient functions. The
behaviour of x F3 in the lower plot for x ∼ 0.007 is explained by
the fact that this structure function exhibits a node in that region
In order to simplify the implementation, it is advantageous to
assume that, in these particular corrections, the CKM matrix
is a 3 × 3 unitary matrix. Note, however, that the exact CKM
matrix is still used in the QCD part of the structure functions.
This approximation introduces an inaccuracy of the order
of the QED coupling α times the value of the off-diagonal
elements of the CKM matrix and therefore it is numerically
negligible.
As an illustration of the impact of the O (α) correction
on the DIS structure functions, Fig. 13 shows the effect of
introducing these contributions on top of the pure QCD
computation at NLO. The plots are produced by evolving the
NNPDF3.0QED NLO set from Q0 = 1 GeV to Q = 100
GeV including the full NLO QCD + QED corrections
discussed in the previous section and using the resulting evolved
PDFs to compute the NC (upper panel) and the CC (lower
panel) DIS structure functions in the FONLL-B scheme,
including the O(α) corrections to the coefficient functions
discussed above. The predictions are shown normalised to
the pure QCD computation where the QED corrections are
absent both in the evolution and in the computation of the
structure functions.
It is clear that the impact of the full NLO QCD + QED
corrections is pretty small especially in the low-x region where
it is well below 1%. In the large-x region, instead, the
presence of a photon-initiated contribution has a more significant
effect because of the suppression of the QCD distributions
(quarks and gluon) relative to the photon PDF and the impact
of the QED corrections reaches the 2% level. It should be
noticed that the behaviour around x = 10−2 of the CC x F3
(green curve in the lower panel) is driven by a change of sign
of the predictions so that the ratio diverges. Finally, we stress
that the general features observed in Fig. 13 do not depend
on the input PDF set. In particular, we have checked that the
same picture holds using the xFitter_epHMDY PDF set
presented in this work.
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