### Nuclear parton distribution functions (nPDFs) and their uncertainties in the LHC Era

#### 1. Introduction

In the past decade, a tremendous experimental efforts have been carried out in order to study the partonic structure of proton.
In spite of the remarkable phenomenological success of Quantum Chromodynamics (QCD) in deep-inelastic-scattering (DIS) experiments, a detailed understanding of the partonic structure of bound nuclei is still lacking.
In the collinear factorized approach to perturbative QCD (pQCD), the structure and dynamics of nucleus are described by the nuclear parton distribution functions (nuclear PDFs) [1,2,3,4,5,6,7].
Analogously to the DIS data available for the free proton case, fixed-target
ℓ+A
DIS scattering has a huge potential to offer information on the nuclear PDFs.
In this work, we have reviewed the results of our recent KA15 NNLO nuclear PDFs and their uncertainties.
After first introducing to the nuclear DIS data sets, the analysis method are quickly recalled. Then the results of our NNLO nuclear PDFs discussed, and detailed comparison with nuclear DIS experimental data are presented.

#### 2. Nuclear DIS data sets

In our analysis, we have used a large variety of
ℓ±-A
DIS and
pA
Drell-Yan (DY) data sets. The data sets used in the KA15 analysis, listed in Ref. [1].
The neutral current charged-lepton (
ℓ±
+ nucleus) DIS data as well as DY cross-section ratios
σ DY A/σ DY A'
used in the KA15 analysis are shown in Figure. figxQdata. This plot nicely summarizes the universal
x
and
Q2
dependence of the nuclear DIS data.
However, the kinematic reach of currently available cross section measurements in
ℓ+A
DIS is much more restricted than in the case of
ℓ+p
DIS.
As a consequence, the nuclear PDFs are significantly less constrained than the proton PDFs.
As one can see from Fig. figxQdata, at small value of
x<0.01
, the obtainable constraints are limited by low statistics.

##### Figure 1

(Color online) Nominal coverage of the data sets used in KA15 global fits. The plot nicely summarizes the universal
Q2
and
x
dependence of the nuclear DIS data.

The LHC
p+pb
and
pb+pb
data open a new, high-
Q2
, kinematic regions for nuclear PDF studies.
Recently, the first global analysis of nuclear PDFs to include LHC proton-lead (
p+Pb
) Run-I data, EPPS16 [2], appeared.
They reported the impact of these data on the EPPS16 nuclear PDFs and shown that the CMS dijets data [8], are essential in constraining the nuclear effects in gluon distributions.
The Run-II
p+A
data as well as Relativistic Heavy-Ion Collider (RHIC) [9] will have significantly higher luminosities. Therefore, Run-II data are expected to provide much better constraints in the near future.

#### 3. Analysis Method

In KA15 framework [1] we parametrize the nuclear PDFs
fi(A,Z)(x,Q2)
which are to be multiplied with the free proton
PDFs
fip(x,Q2)
, taken here to be those of NNLO JR09 PDFs [10].
One can reconstruct the nuclear PDFs as follow at
Q02=2 GeV 2
:
Here we assume the following functional form for the nuclear modifications
In order to accommodate various nuclear target materials, we introduce
in which {
pi=b1,b2,c1,c2,d1,d2,...
} are free fit parameters.
To determine the best fit at NNLO, we need to minimize the
χ2
with respect to 16 free input nuclear PDFs parameters in above equations.
The global goodness-of-fit procedure follows the usual chi–squared method [1]. The optimization of the
χ2
values to determine the best parameters of nuclear PDFs is done by the CERN program library MINUIT [11]. The experimental errors are calculated from systematic
and statistical errors, added in quadrature. For the determination of nuclear PDFs uncertainties and corresponding observables, we use the standard Hessian method with a global tolerance
Δχ2=1
. This method are discussed in details in Refs. [1,12,13,14,15,16,17,18,19,20].

##### Figure 2

(Color online) Our theoretical predictions based on AT12 NLO and KA15 NNLO nuclear PDFs are compared with the data of the DY cross-section ratios
σ DY Fe/σ DY Be
. Data points are from the FNAL-E866 experiments at Fermilab [22].

#### 4. Results and discussion

In this section we present a comparison of the KA15 fit with the analyzed nuclear DIS data.
A detail comparison of the KA15 NNLO [1] and AT12 NLO [21]
x
dependence theoretical predictions of the structure function ratios
F2A(x,Q2)F2A'(x,Q2)
with the analyzed nuclear DIS data are presented in Figure figComparison. In this plot, ratios of structure functions for various nuclei as measured by the JLAB, E139, E140, NMC, and EMC collaborations, compared
with our fit. The error bars shown on the experimental data correspond to the systematic and statistical errors added in quadrature. As one can conclude from the figure, these data set into the KA15 fit without causing a significant tension.

##### Figure 3

(Color online) Comparison of the KA15 NNLO [1] and AT12 NLO [21] theory predictions with the analyzed nuclear DIS data.

In Figure figDYFEtoBe, our theoretical predictions based on AT12 NLO and KA15 NNLO nuclear PDFs are compared with the data of the DY cross-section ratios
σDYFe/σDYBe
measured by FNAL-E866 [22]. These plots have shown as a function of
x
for different Q
2
values of 4.5, 5.5, 6.5 and 7.5 GeV
2
.

As one can see from these plots, the FNAL-E866 data on DY cross-section are in a good agreement with both AT12 NLO and KA15 NNLO theory predictions.
As we already mentioned, the data from proton-lead (
p-pb
) and lead-lead (
pb-pb
) collisions at the RUN-I at CERN-LHC would be very desirable in order to determine the nuclear PDFs at low values of parton fractional momenta
x
[2,23].

#### 5. Summary and conclusion

As a short summary, we have presented our global analysis of NNLO nuclear PDFs, KA15, extracted from a larger variety
of nuclear DIS data sets. The obtained results are in good agreements with all data analyzed. However, the uncertainties are still significant for all components at small value of
x
and, clearly, more data is therefore required. In this respect, new data from the
p-pb
collisions at 13 TeV LHC will be available soon and will provide high-precision DIS constraints for all nuclear parton flavours.

#### Acknowledgments

Author is especially grateful Hamzeh Khanpour for reading this manuscript and for many useful discussions and comments.