A critical appraisal and evaluation of modern PDFs
Eur. Phys. J. C
A critical appraisal and evaluation of modern PDFs
A. Accardi 1 2
S. Alekhin 0 6
J. Blümlein 5
M. V. Garzelli 0
K. Lipka 4
W. Melnitchouk 1
S. Moch 0
J. F. Owens 3
R. Placˇakyte˙ 4
E. Reya 8
N. Sato 1
A. Vogt 7
O. Zenaiev 4
0 II. Institut für Theoretische Physik, Universität Hamburg , Luruper Chaussee 149, 22761 Hamburg , Germany
1 Jefferson Lab , Newport News, VA 23606 , USA
2 Hampton University , Hampton, VA 23668 , USA
3 Florida State University , Tallahassee, FL 32306 , USA
4 Deutsches Elektronensynchrotron DESY , Notkestraße 85, 22607 Hamburg , Germany
5 Deutsches Elektronensynchrotron DESY , Platanenallee 6, 15738 Zeuthen , Germany
6 Institute for High Energy Physics , 142281 Protvino, Moscow region , Russia
7 Department of Mathematical Sciences, University of Liverpool , Liverpool L69 3BX , UK
8 Institut für Physik, Technische Universität Dortmund , 44221 Dortmund , Germany
We review the present status of the determination of parton distribution functions (PDFs) in the light of the precision requirements for the LHC in Run 2 and other future hadron colliders. We provide brief reviews of all currently available PDF sets and use them to compute cross sections for a number of benchmark processes, including Higgs boson production in gluongluon fusion at the LHC. We show that the differences in the predictions obtained with the various PDFs are due to particular theory assumptions made in the fits of those PDFs. We discuss PDF uncertainties in the kinematic region covered by the LHC and on averaging procedures for PDFs, such as advocated by the PDF4LHC15 sets, and provide recommendations for the usage of PDF sets for theory predictions at the LHC. 1 Introduction . . . . . . . . . . . . . . . . . . . . . . 2 Data sets and results for PDF fits . . . . . . . . . . . 2.1 Data sets used in PDF fits . . . . . . . . . . . . . 2.2 Results for PDFs . . . . . . . . . . . . . . . . . 3 Theory for PDF fits . . . . . . . . . . . . . . . . . . . 3.1 Theory for analyses of DIS data . . . . . . . . . 3.1.1 Massless PDFs . . . . . . . . . . . . . . . 3.1.2 Heavyquark structure functions . . . . . . 3.2 Heavyflavor PDFs . . . . . . . . . . . . . . . . 3.3 Heavyquarks schemes . . . . . . . . . . . . . . 3.3.1 Variableflavor number schemes . . . . . . 3.3.2 Validation with DIS charmquark production 3.3.3 Charmquark mass . . . . . . . . . . . . . 3.4 Lightflavor PDFs . . . . . . . . . . . . . . . . . 3.4.1 Up and downquark distributions . . . . . 3.4.2 Strangequark distribution . . . . . . . . . 3.5 Nuclear corrections . . . . . . . . . . . . . . . . 3.6 Software and tools . . . . . . . . . . . . . . . . 4 Strong coupling constant . . . . . . . . . . . . . . . . 5 Cross section predictions for the LHC . . . . . . . . . 5.1 Higgs boson production . . . . . . . . . . . . . . 5.2 Hadroproduction of heavy quarks . . . . . . . . 5.2.1 Topquark hadroproduction: inclusive cross section . . . . . . . . . . . . . . . . . . . 5.2.2 Topquark hadroproduction: differential distributions . . . . . . . . . . . . . . . . . 5.2.3 Bottomquark hadroproduction . . . . . . 5.2.4 Charmquark hadroproduction . . . . . . . 5.3 W /Z production . . . . . . . . . . . . . . . . . 6 Recommendations for PDF usage . . . . . . . . . . . 6.1 The 2015 PDF4LHC recommendations: A critical appraisal . . . . . . . . . . . . . . . . . . . 6.2 New recommendations for the PDF usage at the LHC . . . . . . . . . . . . . . . . . . . . . . . . 7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . .

Contents
1 Introduction
In Run 2 of the Large Hadron Collider (LHC), the very details
of the Standard Model (SM), including cross sections of
different processes and Higgs bosons properties, are being
measured with very high precision. At the same time, the new data
at the highest centerofmass collision energies ever achieved
√
( s = 13 TeV) are used to search for physics
phenomena beyond the SM (BSM). The experimental data used to
perform those measurements are generally expected to have
percentlevel accuracy, depending on details such as the final
states and the acceptance and efficiency of the detectors in
particular kinematics ranges.
To further test the SM and to identify signals for new
physics, measurements need to be compared to precise
theoretical predictions, which need to incorporate higher order
radiative corrections in quantum chromodynamics (QCD)
and, possibly, the electroweak sector of the SM. In order
to reach the benchmark precision set by the accuracy of
the experimental data, nexttonexttoleading order (NNLO)
corrections in QCD are often required. At nexttoleading
order (NLO) in QCD, the residual theoretical uncertainty
from truncating the perturbative expansion commonly
estimated by variations of the renormalization and factorization
scales μr and μ f are often too large compared to the
experimental accuracy. Nonetheless, for observables with complex
final states, and indeed for many BSM signals, one must
still contend with NLO calculations, which will continue to
require corresponding NLO fits.
Parton distribution functions (PDFs) in the proton serve as
an essential input for any cross section prediction at hadron
colliders and have been measured with increasing precision
over the last three decades. Likewise, the strong coupling
constant αs (MZ ) at the Z boson mass scale MZ and the
masses mh of the heavy quarks h = c, b, t are well
constrained by existing data and their determination is accurate
at least to NNLO. However, despite steady improvements in
the accuracy of PDF determinations over the years, the
uncertainties associated with PDFs, the strong coupling αs (MZ ),
and quark masses still dominate many calculations of cross
sections for SM processes at the LHC. A particularly
prominent example is the cross section for the production of a SM
Higgs boson in the gluon–gluon fusion channel.
The currently available PDF sets are CJ15 [
1
], accurate
to NLO in QCD, as well as ABM12 [
2
], CT14 [
3
],
HERAPDF2.0 [
4
], JR14 [
5
], MMHT14 [
6
], and NNPDF3.0 [
7
] to
NNLO in QCD. These provide a detailed description of the
parton content of the proton, which depends on the chosen
sets of experimental data as well as on the theory
assumptions and the underlying physics models used in the analyses.
Both theoretical and experimental inputs have direct impact
on the obtained nonperturbative parameters, namely, the
fitted PDFs, the value of αs (MZ ) and the quark masses.
Moreover, they can lead to large systematic shifts compared to
the uncertainties of the experimental data used in the fit. For
precision predictions in Run 2 of the LHC it is therefore very
important to quantify those effects in detailed validations of
the individual PDF sets in order to reduce the uncertainties in
those nonperturbative input parameters. Moreover, this will
allow one to pinpoint problems with the determination of
certain PDFs. Any approach to determine the parton
luminosities at the LHC which implies mixing or averaging of
various PDFs or of their respective uncertainties, such as that
advocated in the recent PDF4LHC recommendations [
8
], is
therefore potentially dangerous in the context of precision
measurements, in particular, or when studying processes at
kinematic edges such as at large values of Bjorken x or small
scales Q2. The precision measurements of the LHC
experiments themselves help to constrain the different sets of PDFs
and may even indicate deviations from SM, cf. [
9
] for an
example. It is thus of central importance that comparisons
for all available PDF sets are performed in a quantitative
manner and with the best available accuracy.
In this paper we briefly discuss the available world data
used to constrain PDFs in Sect. 2 and stress the need to
include only compatible data sets in any analysis. The data
analysis relies on comparison with precise theoretical
predictions, with many of these implemented in software tools. In
this respect, we underline in Sect. 3 the importance of
opensource code to provide benchmarks and to facilitate theory
improvements through indication and reduction of possible
errors. In addition, Sect. 3 is devoted to a discussion of a
number of crucial theory aspects in PDF fits. These include the
treatment of heavy quarks and their masses, QCD corrections
for W ± and Z boson production applied in the fit of
lightflavor PDFs, and the importance of nuclear corrections in
scattering data off nuclei. The strong coupling constant is
correlated with the PDFs and is therefore an important parameter
to be determined simultaneously with the PDFs. The state of
the art is reviewed in Sect. 4. The need to address PDF
uncertainties for cross section predictions is illustrated in Sect. 5,
with the Higgs boson cross section in the gluon–gluon fusion
channel being the most prominent case. Other examples
include the production of heavy quarks at the LHC in
different kinematic regimes. Our observations illustrate important
shortcomings of the recent PDF4LHC recommendations [
8
]
which are addressed in Sect. 6, where alternative
recommendations for the usage of sets of PDFs for theory predictions
at the LHC are provided. Finally, we conclude in Sect. 7.
2 Data sets and results for PDF fits
We begin with an overview of the currently available data
which can be used to determine PDFs and present the fit
results of the various groups.
2.1 Data sets used in PDF fits The data used in the various PDF fits overlap to a large extent, as indicated in Table 1. However, there are also substantial differences which are related to the accuracy required in the
Refs. [
1–7
] and also the specific statistical analysis applied is described
in these papers. Note that different analyses use partly different data
sets for some processes
a CJ15 use χ 2 = 1 (for the 68 % c.l.) and the CJ15 PDF sets are provided with 90 % c.l. uncertainties ( χ 2 = 2.71)
b The CJ14 PDFs sets are provided with 90 % c.l. uncertainties. In addition, a twotier tolerance test has been applied in case of some data sets
cA Monte Carlo method is used to estimate the errors of the PDFs. This method has an interpretation with respect to a level of tolerance only in
the range in which the corresponding uncertainties are Gaussian, which applies to wide kinematic regions studied. In these regions the error bands
correspond to the 1 σ error obtained using the χ 2 method [
10
]
analysis, the feasibility of efficiently implementing the
corresponding theoretical computations, or the subjective
evaluation of the data quality, to name a few.
The core of all PDF fits comprises the deepinelastic
scattering (DIS) data obtained at the HERA electron–proton
(ep) collider and in fixedtarget experiments. While the
former has used only a proton target, the latter have collected
large amounts of data for the deuteron and heavier targets
as well. The analysis of nucleartarget data requires an
accurate account of nuclear effects. This is challenging already
in the case of the looselybound deuteron (cf. Sect. 3), and
even more so for heavier targets. Therefore, in general,
data sets for DIS on targets heavier than deuteron are not
used. Nonetheless, different combinations of data sets for
the neutrinoinduced DIS off iron and lead targets obtained
by the CCFR/NuTeV, CDHSW, and CHORUS experiments
are included in the CT14, MMHT14, and NNPDF3.0
analyses, but are not used by other groups to avoid any influence
of nuclear correction uncertainties. One can also point out
the abnormal dependence of the DIS structure functions on
the beam energy in the NuTeV experiment [
11
] and the poor
agreement of the CDHSW data with the QCD predictions on
the Q2 slope of structure functions [
12–14
] as an additional
motivation to exclude these data sets.
The kinematic cuts applied to the commonly used DIS
data also differ in various analyses in order to minimize the
influence of higher twist contributions. Another important
feature of the DIS data analyses in PDF fits concerns the
use of data for the DIS structure function F2 instead of the
data for the measured cross sections. These aspects will be
discussed in Sect. 3.
The inclusive DIS data are often supplemented by the
semiinclusive data on the neutralcurrent and
chargedcurrent DIS charmquark production. The neutralcurrent
sample collected by the HERA experiments provides a
valuable tool to study the heavyquark production mechanism.
This is vital for pinning down PDFs, in particular the gluon
PDF at small x , relevant for important phenomenological
applications at the LHC (cf. Sect. 5). The chargedcurrent
charm production data help to constrain the strange sea PDF,
which is strongly mixed with contributions from nonstrange
PDFs in other observables (cf. Sect. 3).
The Drell–Yan (DY) data are also a necessary ingredient
of any PDF analysis since DIS data alone do not allow for
a comprehensive disentangling of the quark and antiquark
distributions. Historically, for a long time only fixedtarget
DY data were available for PDF fits. In particular, this did not
allow for a modelindependent separation of the valence and
sea quarks at smallx . The high precision DY data obtained in
protonproton ( pp) and proton–antiproton ( p p¯) collisions
from the LHC and the Tevatron open new possibilities to
study the PDFs at small and large x . The LHC experiments
are quickly accumulating statistics and are currently
providing data samples at √s = 7 and 8 TeV for W  and Z
boson production with typical luminosities of over 20 fb−1
per experiment. The rapid progress in experimental
measurements causes a greatly nonuniform coverage of the recent
DY data in various PDF fits (cf. Tables 2, 3) and leads to
corresponding differences in the accuracy of the extracted PDFs.
Another issue here is the theoretical accuracy achieved for
the description of the DY data. This varies substantially and
will be discussed in Sect. 3.
Often, jet production in pp and p p¯ collisions is used as an
additional process to constrain the largex gluon PDF. Here,
the QCD corrections are known to NLO and the calculation
of the NNLO ones is in progress [
15
]. The incomplete
knowledge of the latter is problematic in view of a consistent PDF
analysis at NNLO when including those jet data. This will
be discussed in Sect. 4 in connection with the determination
of the value of the strong coupling constant αs .
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In addition to these major categories of data commonly
used to constrain PDFs, some complementary processes are
also employed in some cases, as indicated in Table 1. These
comprise the hadroproduction of topquark pairs from pp
and p p¯ collisions and the associated production of W bosons
with charm quarks in pp collisions. Sometimes, also jet
production in ep collisions and prompt photon (γ +jet)
production from pp and p p¯ collisions is considered. Except for
t t¯ production, the necessary QCD corrections are known to
NLO only, so that the same arguments as in the case of jet
hadroproduction data apply, if those data are included in a
fit at NNLO accuracy. For t t¯ production, only the inclusive
cross section is considered at the moment in the available
PDFs and there is a significant correlation with PDFs,
especially of the gluon PDF with the topquark mass.
Taken together, the set of these data has a number of data
points (NDP) of the order of few thousand, and provides
sufficient information to describe the PDFs with an ansatz
of about O(30) free parameters. The parameters can include
the strong coupling constant αs (MZ ) and the heavyquark
masses mc, mb and mt , which are correlated with the PDFs,
as will be discussed in Sects. 3 and 4. This provides sufficient
flexibility for all PDF groups and it is routinely checked that
no additional terms are required to improve the quality of the
fit. The exception is the NNPDF group, which typically uses
O(250) free parameters in the neural network.
Apart from those considerations there is the general
problem of the quality of the experimental data, that is to say
whether or not the PDFs are extracted from a consistent data
set. The various groups have different approaches, which
roughly fall into two classes according to the different
confidence level (c.l.) criteria for the value of χ 2 in the
goodnessoffit test. One approach is to fit to a very wide (or even
the widest possible) set of data, while the other one rejects
inconsistent data sets. In the former case, a tolerance
criterion for χ 2 is introduced (e.g. χ 2 = 100), while the
latter approach maintains that χ 2 = 1. For the various
PDF groups this information is listed in Table 1.
For further reference, we quote here the definition of χ 2
used in data comparisons (Tables 4, 10, 11, 12, 14, 15, 16).
It follows the definition described in Refs. [
16–18
] and is
expressed as follows:
χ 2 =
i
+
i
μi − mi 1 − j γ ji b j
δi,uncmi2 + δi,stat μi mi
2 2
ln δi,uncmi2 + δi,stat μi mi
2 2
δi2,uncμi2 + δi2,statμi2 ,
2
+
j
b2j
where μi represents the measurement at the point i , mi is the
corresponding theoretical prediction and δi,stat, δi,unc are the
relative statistical and uncorrelated systematic uncertainties,
respectively. γ ji denotes the sensitivity of the measurement to
(1)
the correlated systematic source j and b j their shifts, with a
penalty term j b2j added. In addition, a logarithmic term is
introduced arising from the likelihood transition to χ 2 when
scaling of the errors is applied [
16
].
It is important to note that the χ 2 values obtained with
Eq. (1) will not necessarily correspond to numbers quoted
by PDF groups due to different χ 2 definitions, data treatment
and other parameters, see also Table 1.
2.2 Results for PDFs
Before we start a detailed discussion of the theoretical
aspects of the PDF determinations we would like to
illustrate the present status of PDF sets at NNLO in QCD and
discuss briefly some differences, which are clearly visible.
The currently available sets at NNLO in QCD are shown
in Figs. 1, 2, 3, 4, 5 and 6. The lightquark (u, d) valence
PDFs together with the gluon and the quark sea distributions
(x = 2x (u¯ + c¯ + d¯ + s¯) for four active flavors) with the
respective uncertainty bands are displayed in Figs. 1, 3 and 5
at the scales Q2 = 4 GeV2, 100 GeV2 and MZ2 in the range
10−4 ≤ x ≤ 1 for the sets ABM12 [
2
], HERAPDF2.0 [
4
] and
JR14 [
5
]. Likewise, Figs. 2, 4 and 6 show the sets CT14 [
3
],
MMHT14 [
6
] and NNPDF3.0 [
7
].
The main features of the present NNLO PDFs in Figs. 1,
2, 3, 4, 5 and 6 in the main kinematic region of x and Q2
relevant for hard scattering events at Tevatron and the LHC
can be characterized as follows. The agreement in the
distributions x uv, and to a slightly lesser extent , is very good
for ABM12, JR14 and HERAPDF2.0, as shown in Fig. 1.
For the valence PDF x dv there is also an overall reasonable
agreement, but the distribution deviates by more than 1σ
at x 0.1 in the case of HERAPDF2.0. One should note
that x dv is more difficult to measure in e± p DIS at HERA
than x uv and additional constraints from deuteron data are
important to fix the details of this PDF, as discussed in Sect. 3
below.
The results on the gluon momentum distribution xg are
clearly different at low values of x . Here, JR14 obtains the
largest values, followed by ABM12 and HERAPDF2.0, with
the latter displaying a valencelike shape below x = 10−3.
For CT14, MMHT14 and NNPDF3.0 there is very good
agreement for x uv, cf. Fig. 2. Some differences are
visible in case of x dv, where CT14 reports larger values than
NNPDF3.0 at x 5 · 10−3 and vice versa for smaller x .
The spread in for the sets in Fig. 2 is much greater than
those by ABM12, JR14 and HERAPDF2.0. This is true as
well for the gluon PDF xg with the CT14 uncertainty band
for the gluon PDF also covering the predictions for the
distributions by ABM12, and HERAPDF2.0. Note that the error
bands for CT14 in Figs. 2, 4 and 6 correspond to the c.l.
of 68 %.
The disagreement in x dv between HERAPDF2.0 and
ABM12 or JR14 persists through the evolution from Q2 =
4 GeV2 to Q2 = MZ2 , cf. Fig. 3 and 5. Likewise, the spread
in x dv between CT14, MMHT14 and NNPDF3.0 becomes
more pronounced, as shown in Fig. 4 and 6. On the other
hand, differences in the singlet PDFs and xg, while still
somewhat visible at Q2 = 100 GeV2, largely wash out at
scales Q2 = MZ2 which govern the physics of central
rapidity events at the LHC. Those remaining differences persist
at large scales (as in the case of the gluon PDFs at large
x > 0.1) and will have a significant impact. The crucial
test for all PDF sets comes through a detailed comparison of
cross section predictions to data. This will be discussed in
the remainder of the paper, in particular in Sects. 3 and 5.
3 Theory for PDF fits
In the following we describe the basic theoretical issues for
a consistent determination of the twisttwo PDFs from DIS
and other hard scattering data, on the basis of perturbative
QCD at NNLO using the MS scheme for renormalization
and factorization.
3.1 Theory for analyses of DIS data
The world DIS data are provided in terms of reduced cross
sections by the different experiments. QED and electroweak
radiative corrections [
45,46
] are applied, which requires
careful study of different kinematic variables [
46–49
]. In
this way also the contributions from the exchange of more
than one gauge boson to the partonic twist2 terms are taken
care of. In part, also the very small QED corrections to the
hadronic tensor are already accounted for. These have a flat
kinematic behavior and amount to O(1 %) or less [
50–53
].
The reduced cross sections are differential in either two of
the kinematic variables in the set {x , y, Q2}. The virtuality
Q2 = −q2 of the process is given by the 4momentum
transfer q to the hadronic system. The Bjorken variable is defined
as x = Q2/(sy), with y = 2 p · q/s, and s = ( p + l)2 the
squared centerofmass energy, where p and l denote the
4momenta of the nucleon and the lepton. At energies much
greater than the nucleon mass M , in the nucleon rest frame
y is the fractional energy of the lepton transferred to the
nucleon. The double differential cross sections used in the
QCD analyses are given by [
46,54,55
]
d2σ l± N
NC
d x d y =
2π α2s
Q4
2(1 − y) − 2x y
M 2
s
×F2N C (x , Q2) + Y−x F3N C (x , Q2)
+ y
2 1 −
2ml2
Q2
2x F1N C (x , Q2) ,
(2)
Fig. 1 The uvalence,
dvalence, gluon and sea quark
(x = 2x(u¯ + c¯ + d¯ + s¯))
PDFs with their 1 σ uncertainty
bands of ABM12 [
2
],
HERAPDF2.0 [
4
] and JR14 (set
JR14NNLO08VF) [
5
] at NNLO
at the scale Q2 = 4 GeV2;
absolute results (left) and ratio
with respect to ABM12 (right)
2 )
0.5
0
104
Q2 = 4.0 GeV2
ABM12 NNLO 4F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = 4.0 GeV2
ABM12 NNLO 4F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = 4.0 GeV2
ABM12 NNLO 4F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = 4.0 GeV2
ABM12 NNLO 4F
JR14NNLO08VF
HERAPDF2.0 NNLO
103
102
101
104
103
102
101
103
102
101
104
103
102
101
104
103
102
101
104
103
102
101
103
102
101
104
103
102
101
1
0.5
Q2 = 4.0 GeV2
ABM12 NNLO 4F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = 4.0 GeV2
ABM12 NNLO 4F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = 4.0 GeV2
ABM12 NNLO 4F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = 4.0 GeV2
ABM12 NNLO 4F
JR14NNLO08VF
HERAPDF2.0 NNLO
2 )
0.5
0
104
103
102
101
104
103
102
101
Q2 = 4.0 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = 4.0 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
103
102
101
Q2 = 4.0 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = 4.0 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = 4.0 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
104
103
102
101
Q2 = 4.0 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
1
0.5
Fig. 3 Same as Fig. 1 at the
scale Q2 = 100 GeV2 with the
sea x
= 2x(u¯ + c¯ + d¯ + s¯ + b¯)
2 )
,Q0.8
x
(
xuV0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
104
Q2 = 100 GeV2
ABM12 NNLO 5F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = 100 GeV2
ABM12 NNLO 5F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = 100 GeV2
ABM12 NNLO 5F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = 100 GeV2
ABM12 NNLO 5F
JR14NNLO08VF
HERAPDF2.0 NNLO
Fig. 4 Same as Fig. 2 at the
scale Q2 = 100 GeV2 with the
sea x
= 2x(u¯ + c¯ + d¯ + s¯ + b¯)
)
2 Q0.8
,
x
(
uV0.7
x
0.6
0.5
0.4
0.3
0.2
0.1
0
104
104
103
102
101
104
103
102
101
103
102
101
104
103
102
101
103
102
101
104
103
102
101
Q2 = 100 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = 100 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
103
102
101
104
103
102
101
Q2 = 100 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = 100 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = 100 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = 100 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = 100 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = 100 GeV2
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Fig. 5 Same as Fig. 3 at the
scale Q2 = MZ2
2 )
2 )
Q
,(x20
Σ
x
60
40
20
15
10
5
0
Q2 = M2
Z
ABM12 NNLO 5F
JR14NNLO08VF
HERAPDF2.0 NNLO
Q2 = M2
Z
ABM12 NNLO 5F
JR14NNLO08VF
HERAPDF2.0 NNLO
2 )
2 )
,Q20
x
(
Σ
x
15
10
5
0
103
102
101
104
103
102
101
104
103
102
101
104
103
102
101
Q2 = M2
Z
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
Q2 = M2
Z
CT14 NNLO (68%CL)
MMHT14nnlo68cl
NNPDF30 NNLO
d2σ ν(ν¯)N
NC
d x d y
G2F s
= 16π
d2σCC
d x d y =
G2F s
where α and G F denote the finestructure and Fermi
constants, Y± = 1 ± (1 − y)2 and we keep the dependence on
the masses of the nucleon (M ), the W and Z boson (MW ,
MZ ) and the lepton (ml ).
The structure functions FiN C and Wi are nonperturbative
quantities defining the hadronic tensor. They can be
measured by varying y at fixed Q2 and x and form the input to
the subsequent analysis. Note that in some previous
experiments, assumptions were made about the longitudinal
structure functions F N C and WL , where (in the massless limit)
L
FLN C (x , Q2) = F2N C (x , Q2) − 2x F1N C (x , Q2) ,
(5)
since at the time of the data analysis the corresponding QCD
corrections were still missing. Therefore, it is important to
use the differential cross sections in Eqs. (2)–(4) and to add
the correct longitudinal structure functions [
56,57
], cf. also
[
58,59
]. The structure functions are measured for DIS off
massive proton and deuteron targets and are, therefore,
subject to target mass corrections, which play an important role
in the region of lower values of Q2 and larger values of x .
They are available in Refs. [
54,60,61
].
The neutral and chargedcurrent structure functions
F N C , W N C and W CC consist of a sum of several terms, each
i i i
weighted by powers of the QED and electroweak couplings,
and F N C also include the γ − Z mixing, which has to be
i
accounted for, cf. [
46,54,55
]. Then, considering one specific
gauge boson exchange, one arrives at a representation for the
individual structure functions Fi , which are only subject to
QCD corrections. For example, for pure photon exchange,
they are given by
Fi (x , Q2) = Fiτ =2(x , Q2) +
k=2
∞ C τ =2k (x , Q2)
i
Q2(k−1)
,
(6)
where F τ =2 denotes the leadingtwist term and the
coeffii
cients Ciτ parametrize the higher twist contributions. The
latter terms are of relevance for many DIS data sets, see Sect. 2.
Present day QCD analyses are aimed at determining the
leadingtwist contributions to the structure functions. There
are two ways to account for the higher twist terms:
(i) One is fitting the higher twist terms in Fi . A rigorous
approach requires the knowledge of their scaling
violations (term by term) and of the various Wilson
coefficients to higher orders in αs , see e.g. Sect. 16 in Ref. [
54
].
Since at present this is practically out of reach, such fits
remain rather phenomenological. Moreover, the size of
the (nonsinglet) higher twist contributions to the
structure function F2 vary strongly with the correction applied
to the leadingtwist term up to
nexttonexttonexttoleading order (N3LO), as shown in Ref. [
58,62
]. Also,
the nonsinglet and singlet higher twist contributions are
different [
63,64
].
(ii) One has to find appropriate cuts to sufficiently reduce
the higher twist terms. For instance, in the flavor
nonsinglet analysis of Ref. [
58
] the cuts are taken to be Q2 ≥
4 GeV2, W 2 = M 2 + Q2(1− x )/x ≥ 12.5 GeV2. In the
combined singlet and nonsinglet analysis of Ref. [
64
],
Q2 ≥ 10 GeV2, W 2 ≥ 12.5 GeV2 have been used.
These bounds are found empirically by cutting on W 2
and/or Q2 starting from larger values. Applying these
cuts severely limits the amount of largex DIS data to
be fitted, and usually leads to an increase of the errors
of αs (MZ ) and other fitted fundamental parameters and
distributions.
Both methods (i) and (ii) allow to access the leadingtwist
contributions to the DIS structure functions, with some
qualifications, however. The cuts suggested in (ii) remove the
largex region potentially sensitive to the higher twist terms.
However, they do not affect the data at x 0.1, where
higher twist terms still play an important role [
64,65
]. To
some extent, the influence of higher twist can be dampened
by using the DIS data for the structure function F2 instead
of the cross section, since in this case the contribution to
the structure function FL need not be considered. It should
be kept in mind, though, that the experimental separation of
the structure functions F2 and FL in the full phase space of
common DIS experiments is very difficult without dedicated
longitudinal–transverse cross section separations. Therefore,
the data on F2 and F3 are typically extracted from the cross
section once a certain model for the structure function FL
is taken. This approach is justified only at large x , however,
where the contribution of FL is small and even large
uncertainties in the modeling of FL cannot affect the extracted
values of F2 and F3. The procedure is not applicable for HERA
kinematics, on the other hand, and introduces a bias into the
analysis of the data taken by the New Muon Collaboration
(NMC), in particular, a shift in the value of αs preferred by
the fit [
59,66
], cf. Sect. 4. Nonetheless, the MMHT14
analysis [
6
] is still based on the DIS structure function data, as are
the CJ15 and CT14 analyses [
1,3
]. The latter two use cross
section data for HERA, and for HERA and NMC,
respectively, and structure function data elsewhere. While CT14
performed this important change for the HERA and NMC
data, the authors of Ref. [67] report that the change has little
impact. Refs. [
59,64
], on the other hand, disagree with this
claim.
The deepinelastic structure functions are inclusive
quantities and contain massless parton and heavyquark
contributions,
F τ =2(x , Q2) = Fimassless(x , Q2) + Fimassive(x , Q2) .
i
Here the massless terms are given by
F massless(x , Q2) =
i
j
Ci, j x , μ2
Q2
⊗ f j (x , μ2) ,
(7)
(8)
where Ci, j denote the massless Wilson coefficients, f j the
massless PDFs and μ2 is the factorization scale. The Mellin
convolution is abbreviated by ⊗ and the sum over j is over
all contributing partons. The renormalization group equation
for F massless allows one to eliminate the dependence on μ2
i
orderbyorder in perturbation theory. This also applies to
F τ =2. Through the massive contributions F massive there is
i i
a dependence on the heavyquark masses mc and mb in the
present world DIS data. Note that Fimassive is not the structure
function of a tagged heavyflavor sample, which would be
infrared sensitive [
68
]. Rather, Fimassive is just given as the
difference of the complete structure function Fiτ =2 and the
massless one in Eq. (8).
3.1.1 Massless PDFs
For all QCD calculations we use perturbation theory. The
factorized representation in terms of Wilson coefficients and
PDFs is obtained using the lightcone expansion [
69–72
]. For
a proper definition of the Wilson coefficients and the PDFs
one has to use the LSZ formalism and refer to asymptotic
states at large times t → ±∞, given by massless partons. We
first describe the massless contributions in Eqs. (7) and (8),
and then discuss the contribution of heavy quarks. The
Wilson coefficients in Eq. (8) have a perturbative expansion in
the strong coupling constant. At one [73], two [
74–81
], and
threeloop order [
56,57,82–84
] they have been calculated for
the neutralcurrent structure functions Fi , with i = 1, 2, 3,
except for the γ − Z mixing contribution at three loops.
The structure functions in general depend on the following
three nonsinglet and singlet combinations of parton
densities:
q ±jk = f j ± f¯j − ( fk ± f¯k ), qv =
( fl − f¯l ),
n f
l=1
(9)
with the lightquark distributions fi of flavor i and n f the
number of massless flavors. These combinations evolve in
μ2 from an initial scale μ20 by the QCD evolution equations,
n f
l=1
qs =
( fl + f¯l ),
where the singlet distribution qs (x , μ2) mixes with the gluon
distribution g(x , μ2),
d
d ln(μ2) qi (x , μ2) = Pi (x ) ⊗ qi (x , μ2),
.
while the anomalous dimensions γi j corresponding to the
splitting functions Pi j are obtained by a Mellin transform,
0
1
γi j (N ) = −
d x x N Pi j (x ),
where we suppress for brevity the dependence of Pi j and
γi j on the strong coupling as (μ2) = αs (μ2)/(4π ). The Pi j
are known as well at one [
85–90
], two [
81,91–102
] and at
threeloop order [
103,104
] (see also [
105,106
] for checks of
Pps and Pgg at that order). The scale evolution of the strong
coupling constant in the MS scheme is given by
das (μ2)
d ln(μ2) = −
∞
k=0
βk ask+2(μ2),
where βk denote the expansion coefficients of the QCD
βfunction [
107–116
].
The evolution equations (10), (11) can be either solved in
x  or Mellin (or moment) N space. In Mellinspace, defined
by the transform Eq. (14), an analytic solution is
possible [
117–120
] by arranging the solution systematically in
powers of the coupling constants as (μ2) and as (μ20), and
even forming factorizationscheme invariant expressions. In
case of the x space solutions this is usually not done due to the
necessary iterative solution. In the smallx region the
iterative solution usually leads to a pileup of a few per cent [121].
This can be corrected for in x space solutions by applying
the method given in [
122
]. Likewise, the iterated solution can
be obtained in Mellin N space and is a standard option of the
evolution program QCDPegasus [
123
].
3.1.2 Heavyquark structure functions
Disregarding contributions from charm at the input scale
(“intrinsic charm”), cf. [
124–126
], the heavyflavor
corrections to the DIS functions are described by Wilson
coefficients. The leading order results are of O(as ). Higher order
corrections in the perturbative expansions are, therefore, of
O(as2) at NLO, and of O(as3) at NNLO, similar to the case
of the longitudinal structure function [
56,57
]. The
corrections in the neutral and chargedcurrent cases are available
in one [
127–134
] and twoloop order [
135–138
], where the
latter corrections were given in semianalytic form.
For the neutralcurrent exchange the heavyflavor
contributions to the structure functions Fi with i = 2, L are
[
139,140
]:
F massive(x , n f + 2, Q2, mc2, mb2)
i
n f
k=1
ek2 Lin,sq
x , n f + 1, μ2
They are determined by five massive Wilson coefficients,
Li{,nks, ps,s} and Hi{,kps,s}, where the electroweak current
couples either to a massless (Li,k ) or the massive (Hi,k ) quark
line. From threeloop order onwards there are contributions
containing both heavy flavors c and b in a nonseparable
form, denoted by F2massive,{c,b}, in Eq. (16). The PDFs and the
coupling constant in Eq. (16) are defined in the MS scheme,
while the heavy quark masses are taken either in the onshell
or MS schemes [
140,141
]. The relations of the heavy quark
masses between the pole mass (onshell scheme) and the MS
scheme are available to fourloop order [
142
]. Due to its
better perturbative stability, the MS scheme for the definition of
the heavyquark mass is preferred.
For Q2 mi2 the asymptotic corrections to FL are
available at threeloop order [
139,143
]. For F2, four out of the
five massive Wilson coefficients, Ln2,sq , L2p,sq , L2,g and H2p,sq
s
are known as well [
105,139,144–146
] at large scales Q2. For
the remaining coefficient, H2s,g, an estimate has been made in
Ref. [147] based on the anticipated smallx behavior [
148
], a
series of moments calculated in [
140
], and twoloop operator
matrix elements from Refs. [
149,150
]. This provides a good
approximation of the NNLO corrections.
3.2 Heavyflavor PDFs
An important issue in PDF fits concerns the number of
active quark flavors and the theoretical description of heavy
quarks such as charm and bottom. Due to the large range
of hard scales Q for the scattering processes considered,
different effective theories may be applied. At low scales,
when Q O(few) GeV, one typically works with n f = 3
massless quark flavors, setting n f = 3 in the hard
scattering cross section, the evolution kernels and the
anomalous dimensions. In this case, only the lightquark PDFs
for up, down and strange are taken into account. At higher
scales, e.g., for hadroproduction of jets at high transverse
momentum pt or top quarks, additional dynamical degrees
of freedom lead to theories with n f > 3. By means of the
renormalization group and matching these are related to
the case with n f = 3 massless quarks. Technically, one
has to apply decoupling relations [
151
] at some matching
scale μ, for instance in the transition of αs(n f ) → αs(n f +1).
This introduces some logarithmic dependence on the masses
of the heavy quarks mc, mb and mt for charm, bottom
and top. One should also note that the matching of the
effective theories for n f → n f + 1 does not need to be
smooth. In fact, it introduces discontinuities, such as for the
running coupling as a solution of the QCD βfunction at
higher order in the perturbative expansion, where αs(n f )(μ) =
αs(n f +1)(μ) in the MS scheme at the matching scale μ, see
e.g., [
152
].
In a similar manner, PDFs in theories with a fixed number
n f > 3 of quark flavors are related to those for n f = 3 with
the help of heavyquark operator matrix elements (OMEs)
Ai j at a chosen matching scale μ. Potential nonuniversal
nonlogarithmic heavyflavor effects are taken care of by the
Wilson coefficients. Starting with the PDFs in a socalled
fixedflavor number scheme (FFNS) with n f fixed, one has
fi(n f ) → fi(n f +1) for the lightquark distributions fi and
(qs, (n f ), g(n f )) → (qs, (n f +1), g(n f +1)) for the gluon and
the singlet quark distributions with operator mixing in the
singlet sector. In particular, one has [
140,153
]
fk (n f + 1, μ2) + fk¯ (n f + 1, μ2)
μ2
= Aqnsq,h n f , m2
⊗ fk (n f , μ2) + fk¯ (n f , μ2)
1 μ2
+ n f Aqpqs,h n f , m2
1 s μ2
+ n f Aqg,h n f , m2
μ2
n f , m2
+ Aqpqs,h
PDFs for charm and bottom (h = c, b) are then constructed
as
fh+h¯ (n f + 1, μ2) = Ahpqs
at the matching scale μ from the quark singlet and gluon
PDFs with h = h¯.
The matching conditions are typically imposed at the scale
μ = mh , and fh+h¯ = 0 is assumed for scales μ ≤ mh . The
necessary heavyquark OMEs Ai j depend logarithmically on
the heavyquark masses as αsl lnk (μ2/m2 ) with 0 ≤ k ≤ l in
h
the perturbative expansion. As discussed above, the OMEs
are known to NLO analytically [
149,154
] and at NNLO either
exactly or to a good approximation [
105,140,144,147,155
].
Thus, charm and bottom PDFs can be consistently extracted
in QCD with a fixed number n f = 3, 4 or 5.
It should be stressed, however, that the decoupling
relations for PDFs in Eqs. (17)–(20) assume the presence of one
heavy quark at a time upon moving from lower scales to
higher ones. Beginning at threeloop order, however, there are
graphs containing both charm and bottomquark lines, and
charm quarks cannot be treated as massless at the scale of the
bottomquark due to (mc/mb)2 ≈ 1/10. Such terms cannot
be attributed to either the charm or bottomquark PDFs, but
rather one has to decouple charm and bottom quarks together
at some large scale. The simultaneous decoupling of bottom
and charm quarks in the strong coupling constant αs is
discussed, for instance, in Ref. [
156
].
3.3 Heavyquarks schemes
3.3.1 Variableflavor number schemes
The hard scattering cross sections also depend on the
number of flavors n f and additional parton channels may open
up, which have to be included as well. In addition,
processes involving massive quarks depend logarithmically on
the ratio Q2/m2 , where Q is some hard scale associated
h
with the scattering. For the heavyflavor Wilson coefficients
in Eq. (16) these logarithms are of the type αsl lnk (Q2/m2 )
h
with 1 ≤ k ≤ l in perturbation theory. These originate
from collinear singularities screened by the heavyquark
qs (n f + 1, μ2) =
Aqnsq,h
+ Ahpqs
mass due to the constrained phase space for gluon
emission from massive quark lines, and as a prefactor of these
logarithms one has the standard splitting functions. In
addition to logarithmic terms, there are also power corrections
(m2h /Q2)l in the heavyflavor Wilson coefficients, usually
appearing in form of higher transcendental functions. In
the asymptotic regime of Q2 m2h the logarithms
dominate and the kinematic dependence is measured
experimentally, for instance in the tagged flavor case for
charmquark pairs in the structure function F2cc¯. Logarithms of a
similar kind are also experimentally observed in
differential distributions, e.g. due to the QED corrections
proportional to lnk (Q2/ml2) with ml being the charged lepton mass,
cf. [
45,46
].
The resummation of the logarithms αsl lnk (Q2/m2h ) to all
orders in perturbation theory is effectively carried out by the
transition n f → n f + 1 along with the introduction of new
heavyquark PDFs as described in Eqs. (17)–(20). Whether
such a transition is appropriate or not depends, of course, on
the detailed kinematics. If the hard scale is closer to threshold,
Q2 m2h , a description with n f light flavors is more suitable,
while for Q2 m2h one switches to a theory with n f + 1
massless flavors. In order to achieve a unified description for
hard scattering cross sections both at low scales Q2 m2h
and asymptotically for Q2 m2h , socalled variableflavor
number schemes (VFNS) have been constructed. Effectively,
these aim at an interpolation between the asymptotic limits
of the quarks being very light or very heavy relative to the
other hard scales of the process. At the LHC such
considerations apply to processes with bottom quarks in the initial
state such as single topquark production as well as
bottomquark initiated Higgs boson production (see Ref. [
157
] for
more recent studies and Ref. [
158
] for the socalled
Santander matching scheme for Higgs boson production in bb¯
annihilation).
Of particular interest for PDF fits is the reduced cross
section for the pairproduction of heavy quarks in DIS, which
is parametrized in terms of the DIS heavyquark structure
functions Fih for i = 2, L in Eq. (16) and with heavyflavor
Wilson coefficients which are known exactly at NLO [
135
],
and to a good approximation at NNLO [
147
] in QCD. For
the interpolation n f → n f + 1 of the heavyquark
structure functions Fih a number of socalled generalmass VFNS
(GMVFNS) have been discussed in the literature, such as
ACOT [
159–161
], BMSN [
153
], FONLL [
162
] or RT [
163
].
These keep mh = 0 and are to be distinguished from the
zeromass VFNS (ZMVFNS), which describes essentially
the massless case. Note that presently the GMVFNS are
applied only to one single heavy flavor at the time. That is
the sequential transition n f → n f + 1, so that the charm or
bottom quarks are not considered simultaneously and
charmquark mass effects in the bottomquark structure function Fib
are neglected as discussed above.
The various GMVFNS contain a number of additional
assumptions, and some come in more than one variety. The
GMVFNS differ, for instance, in the way the lowQ2 region
is modeled. This modeling is a necessary undertaking to
provide a reasonable behavior of the VFNS in the kinematical
regime of present DIS data. Additional assumptions in the
GMVFNS are related to the matching scale μ for the
transition n f → n f + 1 as the adopted choice μ = mh is not
unique, see [
164
] for an indepth discussion.
Briefly, the problem can be illustrated with the
heavyquark velocity, the leading order formula [
131
] being
(21)
v =
1 −
1
x ≤ 1 + 4m2h /Q2
.
4m2h
s
=
The transition n f → n f + 1 when the corresponding flavor
is considered as nearly massless requires lightlike
velocities v 1. That implies the absence of all power
corrections (m2h /Q2)l in the heavyflavor Wilson coefficients at the
matching scale μ2. In practice, the matching is often applied
at the scale μ2 = m2h and for kinematics Q2 m2h , where
this condition is not fulfilled, which implies restrictions on
the range in x in Eq. (21).
Finally, the logarithmic accuracy of the resummation
for large scales Q2 m2h , or the order of perturbation
theory in current implementations of GMVFNS, is often
not consistent. For example, NNLO evolution [
103,104
]
of the massless PDFs is sometimes combined with the
heavyquark OMEs at NLO [
149,154
], omitting NNLO
results [
105,140,144,147
].
Altogether, these facts introduce a significant model
dependence in any GMVFNS implementation. A sensitive
parameter to test this model dependence is the extraction of
the charm or bottomquark mass used in different versions
of GMVFNS and subsequent comparison with the Particle
Data Group (PDG) results [
55
]. In addition, the quality of
the various GMVFNS can be quantified with the
goodnessoffit for the description of HERA data on DIS charmquark
production obtained from the combination of individual H1
and ZEUS results [
165
].
3.3.2 Validation with DIS charmquark production
The H1 and ZEUS combined data for the DIS charm
production cross section are unique for tests of GMVFNS
and span the region of 2.5 ≤ Q2 ≤ 2000 GeV2 and
3 × 10−5 ≤ x ≤ 0.05. Values for the charmquark mass and
χ 2/NDP for the individual PDF sets ABM12, CJ15, CT14,
HERAPDF2.0, JR14, MMHT14, NNPDF3.0 as well as the
averaged set PDF4LHC15 are given in Table 4, along with the
information on the scheme choice for the heavyquark
structure functions and on the theoretical accuracy for the massive
quark DIS Wilson coefficients. For reference, Table 4 also
list the χ 2/NDP values for the HERA inclusive cross section
data [
4
]. Comparisons to data for the DIS charm production
cross section are shown in Figs. 7, 8, 9 and 10. Note that
Table 4 adopt the standard definition of perturbative orders
for the heavyquark structure functions. This is not shared by
CT14, MMHT14 and NNPDF3.0 in their GMVFNS. There
the Born contribution to the heavyquark Wilson coefficients
for ep → cc¯, which is proportional to O(αs ), is referred to
as being “NLO”. Analogously, the oneloop corrections of
order O(αs2) are denoted by “NNLO”.
Table 4 and Fig. 7 show that the ABM12 [
2
] and
JR14 [
5
] PDFs at NNLO, using charmquark masses in
the MS scheme, provide a good description of the data.
Both ABM12 and JR14 use the approximate massive
threeloop Wilson coefficients as obtained in [
147
] by
interpolating between existing O(αs3) softgluon threshold
resummation results and the O(αs3) asymptotic (Q2 mc2)
coefficients [
140,144
]. This is referred to as O(αs3)approx in
Table 4. The HERAPDF2.0 fit [4] also obtains a good
description of the data, cf. Fig. 8. This is the only set which has fitted
also to the HERA inclusive cross section data of Ref. [
4
]. On
the other hand, the SACOT [
160
] GMVFNS at NLO used
by CJ15 [
1
] does not describe the data too well, although we
should note that the HERA charm data were not included in
the CJ15 fit itself.
The remarkable fact in Table 4 and Fig. 9 is, however, that
the GMVFNS SACOT(χ ) [
161
] of CT14 [
3
] and RT
optimal [
163
] of MMHT14 [
6
] have difficulties in describing
the DIS charm production data. Note that MMHT14 models
the heavyquark Wilson coefficient functions at O(αs3) for
low Q2 as described in [
163
] using known leading
threshold logarithms [
168
] and ln(1/x ) terms [
148
], which have
been shown not to be leading. This is indicated as O(αs2)
in Table 4. Note that CT14 has applied a universal cut
of Q2 ≥ 4 GeV2 on all DIS data, excluding the bin at
Q2 = 2.5 GeV2 in the HERA data [
165
] from the fit (cf. the
upper left plot in Fig. 9). We have checked that including
the low Q2 bin leads to a dramatic deterioration of the fit
quality.
In addition, the schemes SACOT(χ ) and RT optimal as
well as FONLLC [
162
] of NNPDF3.0 [
7
] do not improve
the fit quality when comparing NLO and NNLO fits. We
note in this context that those fits do not include the
exact asymptotic [
105,140,144,146
] and approximate [147]
O(αs3) results for the heavyquark Wilson coefficients in their
theory predictions. The averaged set PDF4LHC15 [
8
], shown
in Fig. 10, mixes PDFs derived with different mass schemes
(ACOT, FONNL and RT) and does not describe the data very
well for virtualities up to Q2 20 GeV2.
ABM12 [
2
]a
Table 4 Values of the charmquark mass and renormalization scheme
used in the PDF fits together with a summary of schemes chosen for the
description of the charmquark structure function F2c and the
theoretical accuracy for the massive quark DIS Wilson coefficients. The values
of χ 2/NDP for the DIS charm production cross section data [
165
] and
HERA inclusive cross section data [
4
] are given in two columns with
the account of PDF uncertainties (with unc., where CT14 PDF errors
PDF sets
mc (GeV)
mc renorm. Theory method
scheme (F2c scheme)
scaled from 90 % c.l. to 68 % c.l., i.e., reduced by a factor 1.645) and for
the central prediction of each PDF set (nominal). In xFitter [
166,167
],
the values of electroweak parameters like the Fermi constant and W
boson mass are taken from Ref. [55]. The values for CT14 and for
PDF4LHC with the SACOT(χ ) scheme have been determined with a
cut on Q2 ≥ 5 GeV2 on the HERA data [
165
]
Theory accuracy
for heavy quark
DIS Wilson coeff.
χ 2/NDP for HERA χ 2/NDP for inclusive HERA data
charm data [
165
] with [
4
] (Q2min = 3.5 GeV2) with
xFitter [
166,167
] xFitter [
166,167
]
With unc. Nominal With unc. Nominal
a The value of mc in ABM12 is determined from a fit to HERA data [
165
]. ABM12 uses the approximate heavyquark Wilson coefficient functions
of Ref. [
147
]
b The data comparison always applies the SACOT(χ ) scheme at NLO as implemented in xFitter [
166,167
]. The implementation of this scheme
differs from the one used by CT14. Removing the Q2cut on the HERA data [165] one obtains χ 2/NDP = 158/52 (582/52) with PDF uncertainities
and 258/52 (648/52) for the central fit at NLO (NNLO)
c The χ 2/NDP values are determined for the dynamical set JR14NNLO08FF. JR14 uses the approximate heavyquark Wilson coefficient functions
of Ref. [
147
]
d MMHT14 uses the O(αs2) heavyquark Wilson coefficient functions together with some terms at O(αs3) for Q2 ∼ mc2 described in Ref. [
163
].
These terms at O(αs3) have been shown not to be leading
e The data comparision uses the xFitter [
166,167
] implementation of the schemes FONLLB, RT optimal and SACOT(χ ) with the set
PDF4LHC_100 at NLO
65/52
117/52
51/47
64/47
67/52
62/52
62/52
72/52
71/52
58/52
67/52
58/52
71/52
51/47
66/52
117/52
70/47
130/47
67/52
62/52
62/52
78/52
83/52
60/52
69/52
64/52
75/52
76/47
Dedicated studies of the charmquark mass dependence have
been performed by several groups. In the MS scheme, the
value of mc(mc) = 1.24 +− 00..0048 GeV has been obtained in
[169] together with χ 2/NDP=61/52 for the description of
the HERA data [
165
] as a variant of the ABM11 fit [
64
].
Other groups, which keep a fixed value of mc in the
analyses, cf. Table 4, have studied the effects of varying mc in
predefined ranges. This has been done, for example, in the
older NNPDF2.1 [
170
] and MSTW analyses [
171
] as well as
for the MMHT PDFs [
172
]. The latter yields a pole mass of
mcpole = 1.25 GeV as the best fit with χ 2/NDP = 75/52, while
the nominal fit uses mcpole = 1.4 GeV at the price of a
deterioration in the value of χ 2/NDP = 82/52. HERAPDF2.0 [
4
]
has performed a scan of the values of χ 2/NDP leading to
mcpole = 1.43 GeV at NNLO quoted in Table 4 as the best fit.
NNPDF3.0 computes heavyquark structure functions with
expressions for the pole mass definition, but adopts numerical
values for the charmquark pole mass, mcpole = 1.275 GeV,
which corresponds to the current PDG value for the MS mass.
This value is different from the one used in NNPDF2.3,
namely mcpole = √2 GeV. Within the framework of the CT10
PDFs [
173
] the charmquark mass in the MS scheme has
been determined in Ref. [
174
] using the SACOT(χ ) scheme
at order O(αs2), although with a significant spread in the
central values reported (mc(mc) = 1.12 − 1.24 GeV) depending
on assumption in the fit.
In this context, it is worth to point out that the running mass
mc(μ) in the MS scheme is free from renormalon
ambiguities and can therefore be determined with high precision. The
PDG [
55
] quotes mc(mc) = 1.275 ± 0.025 GeV based on
the averaging different mass determination in various
kinematics. DIS charmquark production analyzed in the FFNS
(n f = 3) leads to mc(mc) = 1.24 ± 0.03 +− 00..0033 GeV
at NNLO [
169
], while measurements of the MS mass in
e+e− annihilation give, for instance, mc(mc) = 1.279 ±
0.013 GeV [
175
] and mc(mc) = 1.288 ± 0.020 GeV [
176
].
The determination from quarkonium 1S energy levels yields
mc(mc) = 1.246 ± 0.023 GeV [
177
]. All these values are
consistent with each other within the uncertainties.
In contrast, the accuracy of the pole mass mcpole is
limited to be of the order of the QCD scale QCD and,
moreover, the conversion from the MS mass mc(mc) at low
scales to the pole mass mcpole does not converge. Using
αs (MZ ) = 0.1184, for example, the conversion yields for
Data Q2 = 2.5
δ uncorrelated
δ total
0.0001
a
ta 1.2
/D 1
ryeoh 00..86
T 0.00001 0.0001
H1 ZEUS Charm
H1 ZEUS Charm
0.1
0.1
x
0.1
0.1
x
ccσred00.3.45
cc red0.14 H1 ZEUS Charm
σ
0.1
0.1
x
0.1
0.1
x
Fig. 8 Same as Fig. 7 with QCD predictions at NLO and NNLO in the RT optimal [
163
] VFNS using the HERAPDF2.0 [
4
] PDF sets at NLO and
NNLO
the central value of the PDG mcpole = 1.47 GeV at one loop,
mcpole = 1.67 GeV at two loops, mcpole = 1.93 GeV at
three loops, and mcpole = 2.39 GeV at fourloops [
142
]. The
PDG quotes mcpole = 1.67 ± 0.07 GeV for conversion at two
loops.
The low values for the pole mass of the charm quark
assumed or obtained in some PDF fits as shown in Table 4
are thus not compatible with other determinations and with
the world average. The rigorous determination of the
charmquark mass discussed, for instance, in [
169
] provides a
more controlled way of determining mc from the world
DIS data, taking also into account its correlation with
αs (MZ ).
3.4 Lightflavor PDFs
3.4.1 Up and downquark distributions
The total quark contribution to nucleon matrix elements is
known fairly well due to constraints from the available DIS
data obtained in the fixedtarget and collider experiments in
the x range 10−4 x 0.8. However, a thorough
disentangling of the quark flavor structure is still a challenging
task in any PDF analysis. At moderate and large x values
this has been routinely achieved by using a combination of
the DIS data obtained on proton and deuteron targets.
However, uncertainties in the modeling of nuclear corrections in
the deuteron introduce a controllable source of theoretical
uncertainty on the dquark PDF obtained in this way,
especially at large x , as discussed below.
An alternative way to resolve the u and dquark
contributions is to use data on W  and Z boson production obtained
in pp and p p¯ collisions at the LHC and Tevatron,
respectively. Those experiments probe the W and Z rapidity
distributions up to rapidities of y = 3 − 4, depending on details of
the experiments, with an integrated luminosity of O (1) fb−1
achieved in each run. Such data samples are quite
competitive in accuracy with the ones obtained in fixedtarget DIS
experiments, and provide simultaneously constraints on the
quark and antiquark PDFs at large and small x .
Furthermore, the dquark PDF extracted from a combination of the
existing data on DIS off protons and W /Z boson production
in pp( p p¯) collisions is not sensitive to nuclear corrections.
Moreover, if DIS data with small hadronic invariant masses
W 2 are not used in the analyses in order to reduce the
sensitivity to higher twist contributions, the statistical potential of
deuteron targets [
1
] and using the ratio of heavy nuclei
to deuteron structure functions [
205
]. As mentioned in the
previous section, in general care should also be exercised
when using neutrinonucleus scattering data to obtain, for
example, constraints on strangequark PDFs, due to the
currently poor understanding of the interaction dynamics of
the final state heavy quark propagating through the target
nucleus [
192
].
3.6 Software and tools
Data used in the PDF fits cover a wide range of kinematics and
stem from a large number of different scattering processes. In
order to achieve an accurate theoretical description of both
the PDF evolution and the hard scattering cross sections,
welltested software and tools are necessary. Benchmark
numbers for the PDF evolution have long been established,
see e.g., the Les Houches report [
209
], and opensource
evolution codes such as QCDNUM [
210,211
] and Hoppet [212]
are available in Bjorken x space and QCDPegasus [
123
]
in Mellin N space. This is an important development as it
allows to expose the software used in the PDF fits to
systematic validation, the need of which can be illustrated with
recent theory improvements published by various groups. For
example, MSTW [
197
] has tested its NNLO evolution code
against QCDPegasus [
123
] and corrected the
implementation of one of the heavyquark OMEs.
For the hard scattering cross sections of the various
processes, fast fitting methods like fastNLO [
213,214
] and
APPLGrid [181] have been developed. In addition, some
groups have also published opensource code for the
theory predictions of all physical cross sections employed
in their analyses. The ABM11 and ABM12 fits [
2,64
]
use OPENQCDRAD [215] code, which is publicly
available. The HERAPDF2.0 fit [
4
] relies on the QCD fit
platform xFitter (formerly known as HERAFitter) [
166,
167
], which is an opensource package that provides a
framework for the determination of PDFs and enables the
choice of theoretical options for obtaining PDFdependent
cross section predictions. In particular, xFitter allows
for a choice of different available schemes for treatment
of heavy quarks in DIS. In Mellin N space, an efficient
method exists [
216,217
] which improves on that by [
218
]
and which has been widely used in analyses, e.g. [
217
].
However, no code has been made publicly available so
far.
Given the increasing precision of PDF analyses, which
is driven by the accuracy of the experimental data, there is
ongoing demand to provide theoretical predictions that are as
precise as possible. This has stimulated recent checks of the
analysis software used by various groups and has resulted in a
number of documented improvements. The list includes, for
example, the corrections to the different parts of the DIS cross
section calculations in the NNPDF2.1, MSTW and CT10
PDF analyses as mentioned in the discussion of the PDFs for
strange sea above.
This illustrates that there is a continued need for
benchmarking the hard scattering cross sections of relevance for
PDF determinations in order to consolidate the accuracy
of theory predictions for those observables. In this respect,
opensource software may facilitate future theory
improvements and may help to establish standards for precision
theory predictions.
4 Strong coupling constant
The value of the strong coupling constant αs (MZ ) has a direct
impact on the size of a number of cross sections at the LHC,
such as Higgs boson production, see Sect. 5, and is
therefore an important parameter. Due to QCD factorization, αs
exhibits a significant correlation with the gluon PDF and
also with the charmquark mass, as documented in the
published correlation matrices, see for instance [
64
]. Therefore,
the strong coupling constant has come to require particular
attention in the context of global PDF analyses.
Current precision determinations of αs (MZ ) require
NNLO accuracy in QCD because of the small
uncertainties in the experimental data analyzed and the significantly
reduced dependence from the variation of the
renormalization scale indicating the uncertainty due to the truncation
of the perturbative series. Extractions of αs at NLO
typically yield αs (MZ ) 0.118, however, the NLO scale
uncertainty is large, giving sizable variations αs (MZ ) = 0.005
for μr ∈ [Q/2, 2Q] in DIS analyses. Determinations of
αs to NNLO accuracy benefit from a significantly reduced
renormalization scale dependence, but generally result in
smaller central values for αs (MZ ), with shifts downwards
from NLO to NNLO of a few percent in DIS analyses.
Beyond NNLO, the perturbative expansion converges, as
illustrated in DIS in a valence analysis [
58
] at N3LO which
yields αs (MZ ) = 0.1141 +− 00..00002202, in agreement with the
NNLO values listed in Table 7.
Of course, measurements of αs (MZ ) are not limited to
global fits of PDFs, but stem from a large number of
different processes and methods at different scales, see, e.g.,
[
219–221
] for discussions and comparisons. Here we restrict
ourselves to issues of αs arising in PDF fits. In Table 6 we
give an overview of the αs values currently used in the PDF
analyses. There, two aspects are important. Firstly, some PDF
analyses leave αs as a free parameter in their fits, which
obviously allows one to control its correlation with other PDF
parameters and avoids potential biases. Secondly, among the
NNLO values of αs (MZ ) used there exists a large spread of αs
values, ranging from αs (MZ ) = 0.1132 to 0.1183. Some of
those fitted values of αs (MZ ) are significantly smaller than,
Page 26 of 46
Table 6 Values of αs (MZ ) obtained or used in the nominal PDF sets
of the various groups
Table 7 Determinations of αs (MZ ) values at NNLO from QCD
analyses of the deepinelastic world data and, partly, including other hard
scattering data. For recent compilations, see [
219–221
]
PDF sets
a In detail HERAPDF2.0Jets obtains at NLO αs (MZ ) = 0.1183
± 0.0009(exp) ± 0.0005(model/parameterisation) ± 0.0012
(hadronisation) +−00..00003370(scale), which have been added in
quadrature in the table entry. The HERAPDF2.0 central variant uses a fixed
value αs (MZ ) = 0.118
b MMHT14 obtains αs (MZ ) = 0.1172 ± 0.0013 at NNLO as a best fit
for example, an average provided by the PDG [
55
] in 2014,
which gives αs (MZ ) = 0.1185 ± 0.0006 at NNLO, and is
often quoted as a motivation for fixing αs (MZ ) = 0.118 as
in some entries in Table 6. In the recent 2015 update, the
PDG [
222
] reports the value αs (MZ ) = 0.1181 ± 0.0013
with the uncertainty increased by a factor of two.
While the potential agreement or disagreement with the
PDG average is beyond the scope of this study, it is instructive
to focus on αs (MZ ) measurements from PDF analyses as
listed in Table 7 which have been performed since the NNLO
QCD corrections in DIS first became available. This series
of measurements has led to αs (MZ ) values which are not
only mostly lower than the PDG average, but also exhibit a
large spread in the range αs (MZ ) = 0.1120 − 0.1175. This
spread is significant given the small size of the experimental
uncertainties in the data. As it turns out, the differences in
the values of αs (MZ ) can be traced back to different data sets
used or to different theory assumptions applied, as indicated
in Table 7.
For instance, the inclusion of data for the hadroproduction
of jets, e.g., from the LHC, does have an impact on the
value of αs (MZ ) and can therefore provide valuable
constraints. However, it is important to note that the perturbative
QCD corrections to the hard scattering cross section are only
known completely to NLO, while the exact NNLO result
for the gg channel [
15
] and approximations based on soft
gluon enhancement [
237–240
] indicate corrections as large
as 15–20 %. Those corrections and their magnitude depend,
of course, on the details of the kinematics, the choice of
the scale and on the jet parameters (e.g., jet radius R). For
MSTW
Thorne
(MSTW)
ABM11
ABM12
Thorne
(MSTW)
CT10
JR
CT14
MMHT
ABM11J
2010
NNPDF2.1 2011
Year
high pT they are dominated by threshold logarithms ln( pT )
accompanied by logarithms ln( R) for small jet radii [
240
].
The αs (MZ ) values in PDF analyses currently determined
with the help of jet data (cf. Table 7) are, strictly speaking,
valid to NLO accuracy only and therefore subject to
signifiTable 8 The jet data sets and the theory approximations used in the
NNLO PDF fits. The threshold corrections of Ref. [
237
] neglect the
dependence on the jet radius R. Ref. [
238
] has determined the regime
of validity (“safety cuts”) of the threshold approximation of Ref. [
240
]
by comparing to the exact NNLO result for the gg channel [
15
]
PDF set
cantly larger theory uncertainties due to the variation of the
renormalization scale. The various groups employ different
approaches in their NNLO analyses to cope with this
inconsistency, such as using dynamical scales or applying some
variant of threshold corrections, as detailed in Table 8. As a
result of these efforts, the gluon PDF and αs obtained, for
example, in the MMHT14 and NNPDF3.0 analyses are in a
good agreement.
Different modeling of important theory aspects, such as
whether or not to include target mass corrections, higher twist
contributions and nuclear corrections in the description of
DIS data, or whether or not to use a VFNS in the description
of DIS heavyquark data, can account for the range of αs (MZ )
values in Table 7. With largely similar model assumptions,
NNPDF2.1 [
234,235
], MSTW [231] and MMHT [
236
]
obtained the range αs (MZ ) = 0.1171 − 0.1174. All these
choices can lead to systematic shifts of the value of αs (MZ ).
Let us briefly mention some of the issues in detail.
Higher twist contributions do have a big impact, because
these terms are fitted within a combined analysis.
Alternatively, the part of the DIS data significantly affected
by these terms has to be removed by suitable kinematical
cuts on the scale Q2 and centerofmass energies W 2. In
a variant of the ABM11 analysis [
64
], higher twist terms
have been omitted and the cuts W 2 > 12.5 GeV2 and
Q2 > 2.5 GeV2 as used by MSTW [
231
] have been
applied. This resulted in a sizable shift upwards to αs (MZ2 ) =
0.1191 ± 0.0016 in line with earlier studies in [
241
]. Yet
more conservative cuts of W 2 > 12.5 GeV2 and Q2 >
10 GeV2 in the ABM11 variant with higher twist terms
set to zero led to αs (MZ2 ) = 0.1134 ± 0.0008, well in
agreement with the nominal value in the ABM11
analysis, cf. Table 7. Thus, in PDF analyses without account of
higher twist contributions to DIS data such tight cuts are
essential. In this regard we disagree with Refs. [
67,232,243
]
which claim higher twist effects to be negligible in the
framework of MSTW [
197
] and NNPDF2.3 [
250
]. We also
note that NNPDF3.0 [
7
] uses a cut of Q2 > 3.5 GeV2
which is too low to remove the higher twist
contributions.
Higher order constraints from fixedtarget DIS data can
also lead to shifts in αs (MZ ) [
59
]. For instance, NMC has
measured the DIS differential cross sections and extracted
the DIS structure functions F2NMC [
242
]. At the time of
the NMC analysis, however, the relevant DIS corrections to
O(αs3) [
57
] were not available (see discussion after Eq. (5)
above). This information is, however, important and has to
be taken into account now. In case of fitting F2NMC and
not describing FL (x , Q2) at NNLO, much larger values of
αs (MZ2 ) are obtained [
67
]. It is therefore strongly
recommended to fit the published differential scattering cross
sections using FL (x , Q2) at O(αs3). Presently, the MMHT [
236
]
analysis uses FL (x , Q2) only at NLO. One should note,
however, that the values of FL (x , Q2) at NNLO are significantly
different in the smallx region (see [
67
]).
Finally, great care needs to be exercised when DIS data off
nuclei are included in global fits, see Sect. 3. Details of
modeling of nuclear corrections can in fact also cause systematic
shifts in the value of αs (MZ ). Therefore, Table 7 indicates
if scattering data on heavy nuclei have been included in the
determination. For example, MMHT [
236
] has reported a
comparatively high value of αs (MZ ) as a consequence of
fitting the NuTeV νFe DIS data [
185
]. In general,
determinations of αs (MZ ) should be based upon, or at least
crosschecked with, fits using proton and deuteron DIS data only.
5 Cross section predictions for the LHC
5.1 Higgs boson production
The dominant production mechanism for the SM Higgs
boson at the LHC is the gluon–gluon fusion process. The
large size of the QCD radiative corrections to the inclusive
cross section at NLO, see, e.g. Ref. [
244
], together with the
sizable scale uncertainty have motivated systematic theory
improvements. In the effective theory based on the limit of
a large topquark mass (mt → ∞, integrating out the
topquark loop, but using the full mt dependence in the Born
cross section), this has led to the computation of the
corresponding corrections at NNLO [
245–247
] and even to N3LO
in QCD [
106,248
]. This shows an apparent, if slow,
convergence of the perturbative expansion, along with greatly
reduced sensitivity to the choice for the renormalization and
factorization scales μr and μ f . At N3LO the total scale
variation amounts to 3 % and estimates of the fourloop corrections
support these findings [249].
This leaves, as the largest remaining source of
uncertainties in the predictions of the physical cross section, the input
for the strong coupling constant αs and the PDFs. Despite
the impressive progress in theory and experiment, the
situation resembles that after the completion of the NLO QCD
Table 9 The Higgs cross section at NNLO in QCD (computed in
the effective theory) at √s = 13 TeV for m H = 125.0 GeV
at the nominal scale μr = μ f = m H with the PDF (and, if
available, also αs ) uncertainties. The columns correspond to
different choices for the central value of αs (MZ ) using the nominal
PDF set. The numbers in parenthesis are obtained using the PDF
sets CT14nnlo_as_0115, HERAPDF20_NNLO_ALPHAS_115,
MMHT2014nnlo_asmzlargerange and NNPDF30_nnlo_as
_0115
a The CJ15 PDFs have been determined at NLO accuracy in QCD. The PDF uncertainties quoted by CJ15 denote the 90 % c.l. and should be
reduced by a factor of 1.645 for comparison with the 68 % c.l. uncertainties quoted by other groups
b The PDF uncertainties quoted by CT14 denote the 90 % c.l. and should be reduced by a factor of 1.645 for comparison with the 68 % c.l.
uncertainties quoted by other groups
c The model uncertainities of the HERAPDF20_NNLO_VAR set are not included in the uncertainty estimates
corrections, when it was pointed out in Ref. [
244
] that one
of the main residual uncertainties in the predictions was due
to the gluon PDF.
In Table 9 we summarize the PDF dependence of the
inclusive cross section σ (H )NNLO in the effective theory (i.e., in
the limit of mt m H ) at √s = 13 TeV for a Higgs boson
mass m H = 125.0 GeV, μr = μ f = m H , and mtpole =
172.5 GeV with uncertainties σ (H )NNLO + σ (PDF + αs),
and compare the results for various PDF sets. The PDF
uncertainties are typically given at the 1σ c.l. We list the
results for σ (H )NNLO using either the values for the strong
coupling constant αs (MZ ) at NNLO, corresponding to the
respective PDF set, or fixed values of αs (MZ ) = 0.115 and
αs (MZ ) = 0.118. This is done to illustrate the fact that in
some PDFs the value of αs (MZ ) is not obtained from a fit to
data (including faithful uncertainties) but fixed beforehand,
e.g., to the world average [
55
]. Often the same fixed value
of αs (MZ ) is chosen at NLO and at NNLO independent of
the order of perturbation theory, see also Sect. 4. Table 9
shows a large spread for predictions from different PDFs
with a range σ (H )NNLO = 38.0 − 42.6 pb using the nominal
value of αs (MZ ). Specifically, the PDF and αs differences
between different sets are up to 11 % and are significantly
larger than the residual scale uncertainty due to N3LO QCD
corrections. In addition, the cross sections shift in the range
σ (H )NNLO = 39.0 − 44.7 pb if a fixed value of αs (MZ ) in
the range αs (MZ ) = 0.115 − 0.118 is used. This amounts to
a relative difference of more than 13 % and contradicts the
most recent estimates of the combined PDF and αs
uncertainties in the inclusive cross section [
106
], which quote 3.2 %.
In general, the findings underpin the importance of
controlling the accuracy and the correlation of the strong coupling
constant with the PDF parameters in fits.
Of particular interest is the impact of additional
parameters in the PDF fits, such as the charmquark mass, on
the Higgs cross section. The differences in the treatment of
heavy quarks and the consequences for the quality of the
description of charmquark DIS data have already been
discussed in Sect. 3. ABM12 [
2
] fits the value of mc(mc) in the
MS scheme and the uncertainties in the charmquark mass
are included in the uncertainties quoted in Table 9. Other
groups keep a fixed value of the charmquark mass in the
onshell scheme, cf. Table 4, and vary the value of mcpole within
some range. Such studies have been performed in the past
by NNPDF2.1 [
170
] and MSTW [
171
] and more recently by
MMHT [
172
].
In Tables 10, 11 and 12 we display the results of these
fits together with the values of χ 2/NDP for the DIS
charmquark data [
165
], mostly computed with xFitter [
166,
167
], as well as the corresponding cross section for Higgs
boson production to NNLO accuracy. The MSTW analysis in
Table 10 shows a linear rise of the cross section for increasing
values mcpole = 1.05−1.75 GeV in the range σ (H ) = 40.6−
43.8 pb, which amounts to a variation of more than 7 %. Even
if αs (MZ ) = 0.1171 is kept fixed, the cross section varies
in the range σ (H ) = 41.6 − 42.6 pb, which is equivalent
to 2 %. The best fit in the MSTW analysis with χ 2/NDP =
63/52 leads to mcpole = 1.3 GeV and αs (MZ ) = 0.1166,
both of which are lower than the ones of the nominal fit with
mcpole = 1.4 GeV and αs (MZ ) = 0.1171. In Table 11 the
Table 10 The values of the charmquark mass (onshell scheme
mpole) and the strong coupling αs (MZ ) in the MSTW analysis [171]
using the set MSTW2008nnlo_mcrange together with the value
for χ 2/NDP for the HERA data [
165
] and the Higgs cross
section at NNLO in QCD (computed in the effective theory) at √s =
13 TeV for m H = 125.0 GeV at the nominal scale μr = μ f =
m H . The numbers in parentheses are obtained using the PDF set
MSTW2008nnlo_mcrange_fixasmz with the value of αs (MZ )
fixed to αs (MZ ) = 0.1171
Table 11 Same as Table 10 for the MMHT14 analysis [
172
] using the
set MMHT2014nnlo_mcrange_nf5 and setting αs (MZ ) to the best
fit value. The numbers of Ref. [
182
] keep full account of the
correlation between the PDFs and αs . The values of χ 2/NDP for the HERA
data [
165
] are those quoted in [
172
] for the best fit value of αs (MZ ). The
numbers in parentheses are obtained with the value of αs (MZ ) fixed to
αs (MZ ) = 0.118
αs (MZ ) (best fit)
χ 2/NDP (HERA data [
165
])
σ (H )NNLO (pb) best fit αs (MZ )
same study is performed for the MMHT PDFs [
172
], where
the reduced quark mass range mcpole = 1.15 − 1.55 GeV still
leads to cross section variations σ ( H ) = 40.5 − 42.1 pb (i.e.,
4 %) for the best fit αs (MZ ), or σ ( H ) = 42.1 − 42.6 pb (i.e.,
1 %) for a fixed αs (MZ ) = 0.118. The latter case leads to a
best fit of mcpole = 1.2 GeV with χ 2/NDP = 70/52, which
is significantly smaller than the nominal fit with mcpole =
1.4 GeV and χ 2/NDP = 82/52.
NNPDF has performed a study of the mc dependence
in [
170
], which shows the same trend as for MSTW and
MMHT, i.e., the smaller the chosen value of mcpole, the
better the goodnessoffit for the HERA data [
165
]. In addition,
Table 12 displays the changes in the charmquark mass
values from mcpole = √2 GeV to mcpole = 1.275 GeV in the
evolution of the NNPDF fits from v2.1 [
170
] and v2.3 [
250
]
to v3.0 [
7
], with the obvious correlation of smaller cross
sections for Higgs boson production with smaller chosen values
of mcpole.
As pointed out already in Sect. 3, onshell masses mcpole =
1.2 − 1.3 GeV, as preferred by the goodnessoffit
analyses in Tables 10, 11 and 12 for the charmquark data from
HERA [
165
], are not compatible with the world average of
the PDG [
55
]. Thus, in some PDF fits, the numerical value
of the charmquark mass effectively takes over the role of a
“tuning” parameter for the Higgs cross section. Note that the
three analyses are based on partly different data sets, theory
and methodology.
5.2.1 Topquark hadroproduction: inclusive cross section
The cross section for the hadroproduction of topquark pairs
has been measured with unprecedented accuracy at the LHC
in Run 1 with √s = 7 TeV and 8 TeV. The inclusive cross
section is known to NNLO in QCD [
251–254
], featuring
good convergence of the perturbation series and reduced
sensitivity to the renormalization and factorization scales μr and
μ f . These theory predictions adopt the onshell
renormalization scheme for the heavyquark mass. The conversion to the
MS scheme for the heavyquark mass has been discussed in
Refs. [
255–257
]. For observables such as the inclusive cross
section which are dominated by hard scales μr μ f mt ,
the theory predictions in terms of the MS mass for the top
quark show an even better scale stability and perturbative
convergence.
In a similar study as for Higgs boson production in Table 9
we illustrate in Table 13 the PDF dependence of the inclusive
cross section σ (t t¯)NNLO for various sets with uncertainties
σ (PDF +αs). The computation is performed in the
theoretical framework as implemented in the HATHOR code [
256
].
In Table 13 we choose √s = 13 TeV and fix the pole mass
mtpole = 172.0 GeV and the scales at μr = μ f = mtpole.
For this fixed value of mtpole, we show the impact of different
values for the strong coupling constant at NNLO. We choose
αs (MZ ) either corresponding to the respective PDF set or
fixed to the values 0.115 and 0.118. The results in Table 13
display a spread in a range σ (t t¯)NNLO = 715 − 834 pb
using the nominal value of αs (MZ ) for each PDF set, which
amounts to a relative range of more than 15 %. This decreases
to about 6 %, if the values of αs (MZ ) are fixed to 0.115 or
0.118.
The theoretical predictions at leading order depend
parametrically on the strong coupling constant and the topquark
mass to second power, as well as on the convolution of
the gluon PDFs, σ (t t¯)LO ∝ (αs2/mt2) (g ⊗ g). Therefore,
it is necessary to fully account for the correlations between
the topquark mass, the gluon PDF and the strong coupling
when comparing to experimental data. A number of
analyses have considered t t¯ hadroproduction data. ABM12 [
2
]
has included data for topquark pairproduction in a
variant of the fit to determine the MS mass mt (mt ), keeping the
full correlation with αs (MZ ) and the gluon PDF. On the
other hand, CMS has determined the topquark pole mass
as well as the strong coupling constant in a fit which kept
all other parameters mutually fixed [
258
], while Ref. [
259
]
has explored constraints on the gluon PDF from t t¯
hadroproduction data using fixed values for αs (MZ ) and the pole
mass mtpole.
In the global analyses by MMHT14 [
6
] and NNPDF3.0 [
7
]
those data were also used to fit αs (MZ ) and the gluon PDF.
These analyses employ a fixed value for the pole mass mtpole,
which is motivated by precisely measured topquark masses
from kinematic reconstructions, i.e., Monte Carlo masses,
but does not account for the above mentioned correlation
with αs (MZ ) and the gluon PDF. Moreover, the Monte Carlo
mass requires additional calibration [
260
].
For the inclusive topquark cross section we explore in
Tables 14 and 15 the implicit dependence of the cross
section on the charmquark mass mc used in the GMVFNS of
the PDF fits and list the corresponding values of χ 2/NDP
for the DIS charmquark data [
165
]. This is analogous to
the study for the Higgs cross section in Tables 11 and 12.
For MMHT [
172
] the best fit with mcpole = 1.25 GeV and
αs (MZ ) = 0.1167 leads to an inclusive cross section of
σ (t t¯)NNLO = 814 pb, which is 2 % lower than the value
obtained for the nominal MMHT fit, cf. Table 13. Likewise,
the changes in the NNPDF fits from v2.1 [
170
] and v2.3 [
250
]
to v3.0 [
7
] are documented in Table 15. The effects amount
to almost 2 % when comparing σ (t t¯)NNLO for the best fit of
NNPDF2.1 with mcpole = √2 GeV and αs (MZ ) = 0.119 to
the cross section computed with NNPDF3.0 with mcpole =
Table 13 The inclusive cross section for topquark pair production
at NNLO in QCD at √s = 13 TeV for a pole mass of mtpole =
172.0 GeV at the nominal scale μr = μ f = mtpole with the PDF
(and, if available, also αs ) uncertainties. The columns correspond to
different choices for the central value of αs (MZ ) using the
nominal PDF set. The numbers in parenthesis are obtained using PDF
sets CT14nnlo_as_0115, HERAPDF20_NNLO_ALPHAS_115,
MMHT2014nnlo_asmzlargerange and NNPDF30_nnlo_as
_0115
σ (t t¯)NNLO (pb) nominal αs (MZ )
σ (t t¯)NNLO (pb) αs (MZ ) = 0.115
σ (t t¯)NNLO (pb) αs (MZ ) = 0.118
1.275 GeV and αs (MZ ) = 0.118. In both Tables 14 and
15 there is a correlation showing decreasing cross sections
with decreasing values of mcpole, although less pronounced
than in the case of the Higgs production cross section. The
potential bias in the prediction of the inclusive topquark pair
production cross section due to a particular “tuning” of the
value of the charmquark mass for some PDFs is, however,
of the same order of magnitude or larger than the quoted PDF
uncertainties. Therefore, this needs to be accounted for as an
additional modeling uncertainty.
5.2.2 Topquark hadroproduction: differential distributions
The differential cross section of the topquark pair
production is also known to NNLO in QCD [
261
]. Publicly
availa The CJ15 PDFs have been determined at NLO accuracy in QCD. The PDF uncertainties quoted by CJ15 denote the 90 % c.l. and should be
reduced by a factor of 1.645 for comparison with the 68 % c.l. uncertainties quoted by other groups
b The PDF uncertainties quoted by CT14 denote the 90 % c.l. and should be reduced by a factor of 1.645 for comparison with the 68 % c.l.
uncertainties quoted by other groups
c The model uncertainities of the HERAPDF20_NNLO_VAR set are not included in the uncertainty estimates
Table 14 The values of the charmquark mass (onshell scheme mcpole)
and the strong coupling αs (MZ ) in the MMHT14 analysis [
172
] together
the inclusive cross section for topquark pair production at NNLO
in QCD computed with the set MMHT2014nnlo_mcrange_nf5 at
√s = 13 TeV for a pole mass of mtpole = 172.0 GeV at the nominal
scale μr = μ f = mtpole and setting αs (MZ ) to the best fit value. The
numbers of Ref. [
182
] keep full account of the correlation between the
PDFs and αs . The values of χ 2/NDP for the HERA data [
165
] are those
quoted in [
172
] for the best fit value of αs (MZ ). The numbers in
parentheses for the cross section and χ 2/NDP are obtained using the PDF set
with the value of αs (MZ ) fixed to αs (MZ ) = 0.118
Table 15 Same as Table 14 for various NNPDF analyses. The values
of the strong coupling αs (MZ ) have always been fixed in those fits.
The values of χ 2/NDP for the description of the HERA data have been
determined with the FONLLC [
162
] scheme
able codes such as Difftop [
262
] provide differential
distributions to approximate NNLO accuracy based on
softgluon threshold resummation results. We use Difftop to
calculate the distribution in the topquark rapidity yt for
protonproton collisions at √s = 13 TeV at NNLOapprox
accuracy using the ABM12, CT14, MMHT14, NNPDF3.0,
and the PDF4LHC15 PDF sets at NNLO with their
respective αs values. Here, we take the topquark pole mass to be
mtpole = 172.5 GeV, following the preferences in the LHC
analyses. The renormalization and factorization scales are set
to mtpole and the choice of a dynamical scale does not change
the following discussions.
By using differential cross sections, not only the
sensitivity of topquark pair production to the PDFs can be estimated,
but also possible effects on the experimental acceptance by
Difftop LHC √s=13 TeV, mt=172.5 GeV
CT14NNLO 68% CL NNPDF3.0NNLO
MMHT14NNLO ABM12NNLO
PDF4LHC_100
)
eV300
G
/
b
p
(
)
– tt+X200
→
p
p
(y100
d
/
σ
d
o
lnn 1.4
14 1.2
T
C 1
to 0.8
ito 0.6
a
r
3
2
1
0
1
2
3
yt
CT14
MMHT14 NNPDF30 ABM12 PDF4LHC
15_100
Fig. 15 (Left panel) Predictions for topquark pair production cross
sections at approximate NNLO as a function of the topquark
rapidity using different PDFs at NNLO with the respective PDF uncertainty
(depicted by bands of different style). (Right panel) The acceptance and
extrapolation estimators with the respective PDF uncertainties, obtained
by using different PDF sets
changing the PDF choice. In the experimental analysis, the
PDF dependent acceptance corrections arise mostly from the
PDF dependent normalization of the production cross section
and originate from the phase space regions uncovered by the
detector. Usually, the acceptances are determined by using
Monte Carlo simulations as a ratio of the number of
reconstructed events in the fiducial volume of the detector (visible
phase space) to the number of events generated in the full
phase space. In the case of topquark pair production, the
visible (full) phase space would correspond to the topquark
rapidity range of yt  < 2.5 (yt  < 3). Here, an acceptance
estimator and a related extrapolation factor are calculated by
using Difftop predictions for the respective cross section
ratios σvis/σtot and σunmeasured/σtot. Such estimators are not
expected to describe the true experimental efficiency, but are
helpful for drawing conclusions about PDF related effects.
The predictions of the topquark rapidity and the
acceptance estimates obtained by using Difftop with
different PDFs are shown in Fig. 15. The largest difference in
the global normalization of the predicted cross sections is
observed if the ABM12 PDFs are used instead of the CT14,
NNPDF3.0 or MMHT14 sets. The origin of this effect is
again the smaller nominal value of αs in ABM12 in
combination with a smaller gluon PDF in the x range relevant to
topquark pair production at √s = 13 TeV. The
corresponding acceptance estimators and their uncertainties, obtained
from the error propagation of the corresponding PDF
uncertainties at 68 % c.l., however, demonstrate significant
differences also in the expected acceptance corrections, obtained
by using ABM12 alternative to other PDFs.
The recent PDF4LHC recommendation [
8
] for calculation
of the acceptance corrections for precision observables, such
as the topquark pairproduction cross section in the LHC
Difftop LHC √s=13 TeV, mt=172.5 GeV
Extrapolation estimated as σunmeas/σtot
Acceptance estimated as σvis/σtot
Run 2 data taking period, is to use the set PDF4LHC15_100,
which is obtained by averaging the CT14, MMHT14 and
NNPDF3.0 PDFs. While the central prediction obtained by
using PDF4LHC15 is indeed very close to those obtained
with the CT14, MMHT14 or NNPDF3.0 PDFs, the error on
the corresponding acceptance estimator somewhat
underestimates the acceptance spread of the individual PDFs with their
uncertainties. Furthermore, it does not cover the difference in
the acceptances to the one using the ABM12 PDF. Therefore,
for the conservative estimate of the acceptance correction
and its uncertainty, as demanded in the measurement of SM
precision observables, the use of the PDF4LHC15_100 set
would lead to a significant underestimation of the uncertainty
on the resulting cross section measurement.
A further conclusion from Fig. 15 is that in the case
of topquark pair production, once calculational speed is
needed, it seems to be sufficient to consider a reduced
choice of PDF sets. For instance, instead of using the
averaged set PDF4LHC15_100 one can take just one of
the three PDFs, CT14, MMHT14 or NNPDF3.0.
Alternative PDF choices can then always be studied to some
approximation with a reweighting method. In spite of the
valiant effort in Ref. [
8
] to provide a uniform solution,
the PDF choice for measurements of precision observables
must be decided on a casebycase basis for each particular
process.
5.2.3 Bottomquark hadroproduction
Bottomquark production in protonproton collisions at the
LHC is also dominated by the gluon–gluon fusion process.
Therefore, the LHCb measurements of Bmeson production
in the forward region [
263
] with rapidities 2.0 < y < 4.5
at √s = 7 TeV probe the gluon distributions
simultaneously at small x up to x ∼ 2 × 10−5 and at large x 1.
The smallx region is not accessible with HERA DIS data,
for example. The potential improvements of PDFs near the
edges of the currently covered kinematical region, namely, at
small x and low scales, was first illustrated in [
264,265
] using
differential LHCb data on hadroproduction of cc¯ and bb¯
pairs.
In the present comparison in Table 16, the normalized
cross sections, (dσ/dy)/(dσ/dy0), for bottomquark
production are calculated from the absolute measurements
published by LHCb, with dσ/dy0 being the cross section in the
central bin, 3 < y0 < 3.5, of the measured rapidity range
in each pT bin [
264
]. In the absence of NNLO QCD
corrections, the theoretical predictions are obtained at NLO in
QCD [
266–268
] using a fixed number of flavors, n f = 3, for
the hard scattering cross sections. Since data for the
hadroproduction of heavy quarks other than top have not been
considered for publicly available PDF fits thus far, issues
of any model dependence such as in [
158
] due to the use of
Table 16 The values of χ2/NDP for the normalised bottomquark cross
sections measured at LHCb [
263
] using the NLO PDFs of the individual
groups. The left column accounts for the quoted PDF uncertainties (with
the CJ15 and CT14 PDF uncertainties rescaled to 68 % c.l.), while the
right column uses the central prediction of each PDF set
χ2/NDP (with unc.)
χ2/NDP (nominal)
a The set ABM11 fit [
64
] is used here, because ABM12 [
2
] sets are only
available at NNLO
GMVFNS cannot be quantified. In the calculation of the
normalized cross sections, the theoretical uncertainty is strongly
reduced, since variations of the renormalization and
factorization scales as well as of the fragmentation parameters do
not significantly affect the shape of the y distributions for
heavyflavor production, while this shape remains sensitive
to PDFs.
The values for χ 2/NDP given in Table 16 are computed
with the QCD fit platform xFitter for the individual
PDF sets obtained at NLO, namely, ABM11 [
64
], CJ15 [
1
],
CT14 [
3
], HERAPDF2.0 [
4
], JR14 [
5
], MMHT14 [
6
],
NNPDF3.0 [
7
], as well as the averaged set PDF4LHC15 [
8
].
All PDFs provide a good description of the data, despite the
fact that none of the groups use any data sensitive to the
gluons at very low x , in the region directly probed by the
LHCb Bmeson measurement. Remarkably, one finds that
χ 2/NDP < 1 for the vast majority of the groups (left
column in Table 16), suggesting that the derived PDF
uncertainties at the edges of the so far measured regions might be
inflated.
5.2.4 Charmquark hadroproduction
Charmquark hadroproduction offers another possibility to
illustrate the consistency of the theory predictions for the
various PDF sets. The exclusive production of charmed mesons
in the forward region at LHCb probes the gluon distribution
down to smallx values of x ∼ 5 × 10−6 at √s = 7 TeV,
and data can be confronted with QCD predictions at NLO
accuracy, see, e.g., [
269,270
].
For the inclusive cross section of the reaction pp → cc¯ the
QCD predictions are known up to NNLO in the MS scheme
for the charmquark mass and display good convergence of
the perturbative expansion and stability under variation of
the renormalization and factorization scales [
269
]. In Figs. 16
10
101
102
101
102
103
σpp → cc [mb]
ABM12
102
103
102
102
Fig. 16 Theoretical predictions for the total pp → cc¯ cross section
as a function of the centerofmass energy √s at NLO (dashed lines)
and NNLO (solid lines) QCD accuracy in the MS mass scheme with
mc(mc) = 1.27 GeV and scale choice μR = μF = 2mc(mc) using
the central PDF sets (solid lines) of ABM12 [
2
], CJ15 [
1
], CT14 [
3
] and
JR14 [
5
] and the respective PDF uncertainties (dashed lines). The
predictions for ABM12 (CJ15) use the NNLO (NLO) PDFs independent
of the order of perturbation theory. See text for details and references
on the experimental data from fixed target experiments and colliders
(STAR, PHENIX, ALICE, ATLAS, LHCb)
and 17 we compare the theory predictions at NLO and NNLO
with mc(mc) = 1.27 GeV in the MS scheme, see Sect. 3,
for the scale choice μr = μ f = 2mc(mc) as a function of
the centerofmass energy √s to available experimental data.
These data span a large range in √s, which starts with fixed
target experiments at energies up to √s = 50 GeV
summarized in [
271
] and HERAB data [
272
] (purple points in
Figs. 16, 17). At higher energies RHIC data from PHENIX
and STAR [
273,274
] (black points in Figs. 16, 17) are
available and the LHC contributes measurements at energies
√s = 2.76 TeV from ALICE [275], at √s = 7 TeV from
ALICE [
275
], ATLAS [
276
] and LHCb [
277
], and at the
highest available energy √s = 13 TeV from LHCb [
278
]
(blue points in Figs. 16, 17). The total cross sections of LHCb
have been obtained from charmed hadron production
measurements in a limited phase space region [
277,278
] using
extrapolations based on NLO QCD predictions matched with
parton shower Monte Carlo generators.
The theory predictions for the PDF sets ABM12, CJ15,
CT14 and JR14 at NLO and NNLO are shown in Fig. 16,
together with the respective PDF uncertainties. For all these
PDF sets the perturbative expansion is stable, the theory
computations agree well with the data and predictions, e.g., for
a future collider with √s 100 TeV, yield positive cross
sections. The PDF uncertainties obtained for CT14, however,
do increase significantly above energies of √s 1 TeV.
The same information for the sets HERAPDF2.0,
MMHT14, NNPDF3.0 and PDF4LHC15 is displayed in
Fig. 17. These predictions all agree with data at low
energies but start to behave very differently for
HERAPDF2.0, MMHT14 or NNPDF3.0 at energies above √s
O(10) TeV and for PDF4LHC15 above √s O(100) TeV.
At the same time, the associated PDF uncertainties in this
region of phase space become very large, thereby
limiting the predictive power. Typically, the PDF uncertainties
of the NNLO sets are even larger than at NLO. In the
case of MMHT14 the consistency of the NNLO
predictions with LHC data from ALICE [
275
], ATLAS [
276
] and
LHCb [
277,278
] at energies of √s = 7 TeV and 13 TeV
deteriorates. For NNPDF3.0 the central prediction at NNLO
displays a change in slope for energies above √s 3 TeV
leading to a steeply rising cross section. The most striking
feature, however, are the negative cross sections for
HERAPDF2.0, MMHT14 and PDF4LHC15 at energies above
√s O(30 − 100) TeV, depending on the chosen set. This
is an effect of the negative gluon PDF for those sets at values
10 2
10
1
101
102
103
10 2
10
1
101
102
103
σpp → cc [mb]
HERAPDF20
10
102
σpp → cc [mb]
NNPDF30
NNLO
103
10 2
102
103
10 2
10
1
101
102
103
σpp → cc [mb]
MMHT
σpp → cc [mb]
PDF4LHC15
NLO
NNLO
103
NNLO
103
NLO
Fig. 17 Same as Fig. 16 using the central PDF sets of HERAPDF2.0 [
4
], MMHT14 [
6
], NNPDF3.0 [
7
] and PDF4LHC15 [
8
] together with the
respective PDF uncertainties
of x within the kinematic reach of current or future hadron
colliders up to √s 100 TeV. This results in an instability
of the perturbative expansion of the σ pp→cc¯ cross section at
high energies when the contribution from the quark–gluon
channel dominates. The reason for a negative gluon PDF in
the NNLO set of PDF4LHC15 (being some average of the
CT14, MMHT14 and NNPDF3.0 sets) is unclear. In
contrast, other PDFs shown in Fig. 16 demonstrate stability of
the perturbative expansion through NNLO up to very high
energies and good consistency of the predictions with the
experimental data.
Cross sections sensitive to largex parton distributions
typically fall rapidly with increasing x values, leading to
limitations in the quantity and precision of experimental data and
the kinematic range over which they can be obtained.
Consequently, the precision to which one can constrain
largex PDFs decreases with x , and systematic uncertainties due
to extrapolations into unmeasured regions of x (or those
excluded by cuts) increase. Similarly, the theoretical
uncertainties due to various approximations in the treatment of
nuclear corrections for deuterium data, or target mass and
higher twist effects, also become larger.
To illustrate this, consider the production of a heavy W
boson as a function of the W rapidity yW [
279
].
Assuming Standard Model couplings, the parton luminosity for a
produced negatively charged W − boson is given by
2π G F
LW − = 3√2 x1x2 cos2 θC u¯(x2)d(x1) + c¯(x2)s(x1)
+ sin2 θC u¯(x2)s(x1) + c¯(x2)d(x1)
+ (x1 ↔ x2) , (22)
where G F is the Fermi constant and θC the Cabibbo angle.
The uncertainty δLW − in the luminosity is shown in Fig. 18
for various PDF sets as a function of yW , for several fixed
values of the boson mass from the SM W up to MW = 7 TeV.
Note that as the rapidity or mass of the produced
boson increases, so does the momentum fraction x1,2 =
(MW /√s) e±yW of one or both partons, in which case the
luminosity behaves as L− ∼ u¯(x2)d(x1). Except for the
highest MW values, the PDF uncertainty typically remains
small up to large values of yW , corresponding to x1 ≈ 0.65,
beyond which it rises dramatically for all MW . This is
precisely the region where data constraining the dquark
PDF are scarce, and theoretical assumptions play an
important role [
1
]. This is particularly pronounced for fits that
exclude DIS data at low invariant masses, such as the three
fits included in the PDF4LHC combination [
8
]. For large
W masses, the u¯ PDF is evaluated at x2 ∼ 0.2 − 0.5,
where data are either nonexistent or have large errors, giving
rise to the increased uncertainties in some of the PDF sets
at yW ∼ 0.
Page 36 of 46
LCH 2.5
4F
PD WL2.0
/
W 1.5
L
1.0
PDF4LHC15 (68% c.l.)
CJ15
MMHT14
CT14
NNPDF3.0
500 GeV
PDF4LHC15
CJ15
MMHT14
CT14
80 GeV
3
yW
1
2
4
5
6
Fig. 18 Relative uncertainty δLW − /LW − in the W − luminosity as
a function of rapidity yW for the combined PDF4LHC15 set (dotted),
the CJ15 (solid), MMHT14 (dotdashed), and CT14 (dashed) PDFs for
various W masses from 80 GeV (SM) to 7.0 TeV. All PDF uncertainties
have been scaled to a common 68 % c.l. as provided by the various
groups
The relative uncertainties in the luminosities in Fig. 18
have been scaled to a common 68 % c.l., as in the tables in
the previous sections. One observes a very large range of
uncertainties for the various PDF sets, which stems from
different tolerance criteria used and different
methodologies employed for the treatment of data at high values of
x . The smallest uncertainty is obtained for the CJ15 PDF
set, which makes use of low invariant mass data to
constrain the highx region, and does not employ additional
tolerance factors inflating the uncertainties. The MMHT and
CT14 PDF sets have larger errors, due to stronger cuts
on lowmass DIS data and larger tolerances, and
consequently the averaged PDF4LHC15 set gives similarly large
uncertainties.
This example illustrates the problematic nature of
statistically combining PDF sets that have been determined using
very different theoretical treatments of the highx region,
leading to an overestimate of the uncertainties at these
kinematics. Using the PDF4LHC15 set as the sole basis for
background estimates, for example, one could potentially miss
signals of new physics in regions such as at high
rapidity yW . A more meaningful PDF uncertainty would be
obtained when combining PDF sets obtained under similar
conditions and inputs; if large differences are found, these
should be investigated further rather than simply averaged
over.
This is also illustrated in Fig. 19, where the central
values for the W − luminosity for several PDF sets are
compared relative to the luminosity computed from the central
PDF4LHC15 distributions. The different theoretical
assumptions utilized in the fits produce systematic differences in the
largex PDFs, which give rise to ratios of central values that
are of the same order as the overall PDF4LHC15 68 % c.l.
MW = 3.5 TeV
0.2
Fig. 19 Ratio of central values of the W − luminosity LW − to the
PDF4LHC value (dotted, 68 % c.l. shaded band) as a function of rapidity
yW . The PDF sets CJ15 (red solid curve), MMHT14 (blue dotdashed
curve), CT14 (blue dashed curve), and NNPDF3.0 (green dashed curve)
are compared for a W mass MW = 3.5 TeV
uncertainty, and in the case of the NNPDF3.0 set are about
twice as large.
The fact that the uncertainty bands of the individual sets
overlap with that of the PDF4LHC15 set is not, however, an
indication that the latter is a good estimate of the PDF
uncertainties in this extrapolation region. Rather, the PDF4LHC15
band effectively represents a statistical envelope of the
systematic theoretical differences between the sets included in
the combination. A comparison with the luminosity
computed using the CJ15 PDF set, which is not included in the
PDF4LHC15 combination, is instructive in this respect. The
two main theoretical assumptions affecting the W −
luminosity are the nuclear corrections in deuterium (applied or
fitted in the CJ15 and MMHT14 analyses, as well as in
JR14 and ABM12), and the parametrization of the dquark
PDF.
For the latter, the traditional choice has been to assume
a behavior ∝ (1 − x )β as x → 1 for both the d and
uquark PDFs (as, e.g., in the MMHT14 and NNPDF3.0
analyses), in which case the d/u ratio either vanishes or
becomes infinite in the x → 1 limit depending on whether
the exponent β is larger for d or u. Alternatively,
including an additive term in the dquark PDF proportional to
u(x ) (as in CJ15) or constraining βu = βd (as in CT14)
allows the d/u ratio to reach a finite, nonzero limiting
value at x → 1. Furthermore, the CJ15 distributions were
also fitted to low invariant mass (3.5 GeV2 < W 2 <
12.5 GeV2) DIS data, which were excluded by kinematic
cuts in the MMHT14, CT14 and NNPDF3.0 analyses.
Consequently, the following features can be observed in
Fig. 19:
• The MMHT14 curve follows CJ15 closely until yW ≈
1(x ≈ 0.65), after which the dquark PDF turns upwards
relative to CJ15, in the region not constrained by the
largex and lowW 2 SLAC data utilized in CJ15.
• The CT14 curve is lower than CJ15 at yW 0.6
(x 0.45), and higher at larger yW , because of the
neglect of nuclear corrections. At yW > 1 the dquark
PDF is essentially unconstrained since neither the
lowW 2 SLAC data nor the reconstructed Tevatron W boson
production data are included in the fit.
• The NNPDF3.0 fit, which excludes lowW 2 DIS data and
does not utilize nuclear or hadronic corrections,
consistently deviates from all others. It is, however, compatible
with those within its own uncertainties, which at large x
are about four times larger than that of the other fits.
6 Recommendations for PDF usage
Recommendations for the usage of PDFs generally aim in
providing guidance for estimates of the magnitude and the
uncertainties of cross sections in a reliable but also
efficient way. First recommendations have been provided by
the PDF4LHC Working Group in the Interim
Recommendations [
280
]. There, the MSTW [
197
] PDF was used as a
central set for predictions at NNLO in QCD and the procedure
for calculation of the PDF uncertainties, based on an
envelope of several PDF sets, was proposed. This approach has
been criticized for being impractical. The 2015 PDF4LHC
recommendations [
8
] have evolved from related discussions
and aim in improving the efficiency of cross section
computations by averaging several PDFs along with their respective
uncertainties. Here, we briefly recall these suggestions and
put them into context of the findings of the previous sections.
We comment on several shortcomings of the
recommendations [
8
] and propose an alternative for the PDF usage at the
LHC.
6.1 The 2015 PDF4LHC recommendations: A critical
appraisal
The 2015 PDF4LHC recommendations [
8
] distinguish four
cases: (i) Comparisons between data and theory for Standard
Model measurements, (ii) Searches for Beyond the Standard
Model phenomena, (iii) Calculation of PDF uncertainties in
situations when computational speed is needed, or a more
limited number of error PDFs may be desirable and (iv)
Calculation of PDF uncertainties in precision observables.
For the case (i), the recommendation is to use the
individual PDF sets ABM12 [
2
], CJ12 [
191
], CT14 [
3
], JR14 [
5
],
HERAPDF2.0 [
4
], MMHT14 [
6
], and NNPDF3.0 [
7
]. It
is not clear, why the full account of the PDF
dependence should be limited to SM processes only. Deviations
observed in the theory predictions obtained with the
various PDFs can often be traced back to the differences in
the underlying theoretical assumptions and models in the
PDF fits. With more LHC data available, tests of the
compatibility of those data sets in the individual PDF fits will
become more stringent. Studies to quantify the
constraining power of processes like hadroproduction of t t¯ pairs,
jets or W ± and Z bosons become possible at high
precision.
For the case (ii), it is recommended to employ the
PDF4LHC15 sets [
8
], which represent the combination of
the CT14 [
3
], MMHT14 [
6
], and NNPDF3.0 [
7
]. The
combination is performed using the Monte Carlo approach at
different levels of precision, leading to the recommended
sets PDF4LHC15_30 and PDF4LHC15_100. The
restriction to CT14, MMHT14 and NNPDF3.0 implies a bias
both for the central value and for the PDF uncertainties of
BSM cross section predictions. For example, a bias is
introduced by fixing the central value of αs (MZ ) to an agreed
common value, currently chosen to be αs (MZ ) = 0.118
at both NLO and NNLO. This choice is in contradiction
with the precision determinations of αs (MZ ) at different
orders in perturbation theory, as summarized in Sect. 4.
Further, for searches at the highest energies, the PDFs are
probed close to the hadronic threshold near x 1, where
nuclear corrections and other hadronic effects, considered for
instance in the CJ15 [
1
] and JR14 [
5
] analyses, are
important.
For the case (iii), the PDF4LHC15_30 sets are
recommended to use. We would like to note, that here the
balance between the computational speed and the precision
of the result (in e.g. MC simulation) has to be determined
by the analysers. The problem rises from the large
deviations between data and theory predictions at low scales
and also at the edges of the kinematical ranges of data
currently used in PDF fits as illustrated in Sects. 3 and 5.
The average of various GMVFNS for heavy quark
production, such as ACOT [
159
], FONLL [
162
] and RT [
163
],
leaves a large degree of arbitrariness in the theory
predictions, cf. Fig. 10. Note that the PDF4LHC15_30 sets
were updated in December 2015 [
281
] to account for
an extension of their validity range below the original
Q > 8 GeV as only discussed in the later
publication [
282
].
For the case (iv), the set PDF4LHC15_100 is
recommended. Recalling that this case concerns measurements of
the precision observables, it is unclear why PDFs should
be treated differently than in the case (i). The differences
between individual PDF sets propagate the cross section
measurements directly through the acceptance corrections or
extrapolation factors, as illustrated in Figs. 15, 17 and 19. Use
of the PDF4LHC15_100 is worrysome, since these
differences are smeared out in the combination, which, in addition,
is limited to only three PDF sets. The SM parameters,
determined using the precision observables obtained in this way,
may be biased.
In summary, the recent PDF4LHC recommendations [
8
]
cannot be viewed as definitive in the case of precision
theory predictions, as the advocated averaging procedure
introduces bias, artificially inflates the uncertainties, and makes
it difficult to quantify potential discrepancies between the
individual PDF sets.
6.2 New recommendations for the PDF usage at the LHC
Based on the considerations above, we propose modifications
to the recommendations for PDF usage at the LHC in order
to retain the predictive capability of the individual PDF sets.
Two cases can be distinguished:
1. Precise theory predictions, addressing a class of
predictions, within or beyond the SM, which encompasses
any type of cross section prediction including radiative
corrections of any kind, whether at fixedorder or via
resummation to some logarithmic accuracy. This class
also includes the MC simulations used for the
calculation of the acceptance corrections for precision
observables, e.g. cross sections which might be used further for
determination of SM parameters.
• Recommendation: Use the individual recent PDF
sets, currently ABM12 [
2
], CJ15 [
1
], CT14 [
3
],
JR14 [
5
], HERAPDF2.0 [
4
], MMHT14 [
6
], and
NNPDF3.0 [
7
] (or as many as possible), together
with the respective uncertainties for the chosen PDF
set, the strong coupling αs (MZ ) and the heavy quark
masses mc, mb and mt . Once a PDF set is updated,
the most recent version should be used.
• Rationale: Precise theory predictions as needed for
any comparisons between theory and data for
processes in the SM or beyond (such as hadroproduction
of jets, W ± or Z boson production, either singly
or in pairs, heavyquark hadroproduction, or
generally the production of new massive particles at the
TeV scale) often depend on details of the PDF fits
and the underlying theory assumptions and schemes
used. Differences in the theory predictions based
on the individual sets can give an indication of
residual systematic uncertainties or shed light on
drawbacks and need for potential improvements
in the physics models used in the extraction of
those PDFs. This applies in particular to
measurements used for the determination of SM
parameters such as the strong coupling αs (MZ ), heavy
quark masses mc, mb and mt or the W boson mass,
because these parameters are directly correlated to the
PDFs used in their extraction from the experimental
observables.
2. Theory predictions for feasibility studies, the
complementary class containing all other cross section
predictions where high precision is not required, such as those
based on Born approximations and/or order of magnitude
estimates, or in cases where precision may be sacrificed
in favor of computational speed. Here, also studies of
novel accelerators and detectors are addressed.
• Recommendation: Use any of the recent PDF sets
(listed in LHAPDFv6 or later versions).
• Rationale: Often in phenomenological applications
for the modern and future facilities one is interested
in a quick order of magnitude estimate for the
particular cross sections. These are directly proportional
to the parton luminosity and to the value of αs (MZ ).
In these cases, one may be willing to sacrifice
precision in favor of computational speed. Here, the usage
of the sets PDF4LHC15_30 and PDF4LHC15_100
may provide an efficient estimate of PDF
uncertainties, although care must be taken in their
interpretation depending on the observable and covered
kinematic range. Restricting the recommendation to
PDFs listed in the LHAPDF(v6) [
283
] interface
excludes parton luminosities with lesser precision
in the interpolation of the underlying grids (e.g., in
LHAPDF(v5) [
284
]) or “partonometers” [
285
] with
outdated calibration.
In the Monte Carlo generators, for example,
MadGraph5_aMC@NLO [
286
], POWHEGBOX (v2) [
287,
288
] and SHERPA (v2) [
289,290
], or other recently
developed generators, like Geneva [
291
], different PDF sets can
be efficiently studied with reweighting methods. This allows
to generate weighted events for a given setup, and to reweight
aposteriori each event in a fast and efficient way, by
generating new weights associated with different choices of
renormalization and factorization scales and/or PDFs. Please note,
that at present, PDF reweighting is performed by assuming
the linear PDF weight dependence, which is not correct, since
PDFs are also present in the Sudakov formfactor. Efforts to
extend the reweighting to the entire Sudakov formfactor and
to the full parton shower are ongoing. The reweighting
technique turns out to be particularly useful to compute in a fast
(although at the moment approximate) way PDF
uncertainties affecting the predictions.
7 Conclusion
In this report we have reviewed recent developments in the
determination of PDFs in global QCD analyses. Thanks to
high precision experimental measurements and continuous
theoretical improvements, the parton content of the proton is
generally well constrained and PDFs, along with the strong
coupling constant αs (MZ ) and the heavyquark masses mc,
mb and mt , have been determined with good accuracy, at least
at NNLO in QCD. This forms the foundation for precise cross
section predictions at the LHC in Run 2.
We have briefly discussed the available data used in PDF
extractions and the kinematic range covered, and emphasized
the importance of selecting mutually consistent sets of data
in PDF fits in order to achieve acceptable χ 2 values for the
goodnessoffit estimate. The main thrust of the study has
been the computation of benchmark cross sections for a
variety of processes at hadron colliders, including Higgs boson
production in gluon–gluon fusion. We have illustrated how
different choices for the theoretical description of the hard
scattering process and choices of parameters have an impact
on the predicted cross sections, and lead to systematic shifts
that are often significantly larger than the associated PDF
and αs (MZ ) uncertainties. A particular example has been
the treatment of heavy quarks in DIS, where the quality of
the various scheme choices has been quantified in terms of
χ 2/NDP values when comparing predicted cross sections to
data. We have also pointed out the inconsistently low values
for the pole mass of the charm quark used in some fits, and
have stressed the correlation of the strong coupling constant
αs (MZ ) with the PDF parameters. Ideally, αs (MZ ) should
be determined simultaneously with the PDFs, and we have
summarized here the state of the art in the context of PDF
analyses.
Our findings expose a number of shortcomings in the
recent PDF4LHC recommendations [
8
]. We have shown that
these do not provide sufficient control over some theoretical
uncertainties, and may therefore be problematic for precision
predictions in Run 2 of the LHC. Instead, we suggest new
recommendations for the usage of PDFs based on a
theoretically consistent procedure necessary to meet the precision
requirements of the LHC era.
Acknowledgments We would like to thank S. Alioli, M. Botje, E.W.N.
Glover and K. Rabbertz for discussions, K. Rabbertz also for valuable
comments on the manuscript, and L. HarlandLang and R. Thorne for
providing us with the Higgs and tt¯ cross sections in Tables 11 and
14. This work has been supported by Bundesministerium für Bildung
und Forschung through contract (05H15GUCC1), by the DOE
contract No. DEAC0506OR23177, under which Jefferson Science
Associates, LLC operates Jefferson Lab, and by the European Commission
through PITNGA2012316704 (HIGGSTOOLS). The work of A.A.
and J.F.O. was supported in part by DOE contracts No. DESC0008791
and No. DEFG0297ER41922, respectively. Two of the authors (J.B.
and S.M.) would like to thank the Mainz Institute for Theoretical Physics
(MITP) for its hospitality and support.
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.
Page 40 of 46
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