# Bjorken variable and scale dependence of quark transport coefficient in
semi-inclusive lepton-production of hadron off nuclei

Tian-Xing Bai<EMAIL_ADDRESS>College of Physics and Hebei Advanced Thin
Film Laboratory, Hebei Normal University, Shijiazhuang 050024, P.R.China
Chun-Gui Duan<EMAIL_ADDRESS>College of Physics and Hebei Advanced Thin
Film Laboratory, Hebei Normal University, Shijiazhuang 050024, P.R.China

###### Abstract

Nuclear modification of hadron production in deep inelastic lepton-nucleus
scattering can be applied to study the parton propagation mechanism in cold
nuclear matter. By means of the analytic parameterization of quenching weight
based on BDMPS formalism with the target nuclear geometry effect, the leading-
order computations for hadron multiplicity ratios are performed with
comparison to the HERMES charged pions production data on the quarks
hadronization occurring outside the nucleus. The relation is discovered
between quark transport coefficient and the measurable kinematic variables in
deep inelastic scattering. Four models are proposed on the quark transport
coefficient. The constant model, the power-law model and the double power-law
model can be ruled out because of the experimental fact that the transverse
momentum broadening increases as a function of the photon virtuality $Q^{2}$.
The quark transport coefficient is determined as a function of the Bjorken
variable $x$ and scale $Q^{2}$. The trend of quark transport coefficient in
respect of Bjorken variable $x$ and scale $Q^{2}$ is qualitatively in partial
agreement with HERMES experimental data on transverse momentum broadening. It
is hoped that our effort is conducive to understanding of jet quenching
phenomenon in relativistic heavy ion collisions.

Keywords: quark transport coefficient, deep inelastic scattering, hadron
production.

###### pacs:

12.38.-t 13.60.Le , 24.85.+p , 25.30.-c ,

##  I Introduction

Nuclear modification of hadron production in deep inelastic lepton-nucleus
scattering[1-9] can be used to study the parton propagation mechanism in cold
nuclear matter. For lepton-nucleus deep inelastic scattering, the interaction
point is well defined between the virtual photon and a bounded quark in a
nucleus, since the virtual photon energy is measured by the difference of the
incoming and outgoing lepton energies. In addition, the properties of target
nuclei in which the struck quark propagates, i.e. cold nuclear matter, is well
known. Therefore, the precise knowledge of parton propagation mechanism
obtained from the semi-inclusive lepton-production of hadron off nuclei under
”clean” conditions, is essential to explore the similar process appearing in
relativistic heavy ion collisions. The mechanisms by which the traversing
partons lose energy to, and interact with the quark-gluon plasma, are still
not fully understood[10-13].

According to the picture of multiple parton scattering by Baier, et
al.[14-15](BDMPS hereafter), the energy loss for an energetic parton
propagating in a nucleus is dominated by induced gluon bremsstrahlung. The
parton energy loss is determined by the so-called ”transport coefficient”

$\hat{q}_{R}=\rho\int dq^{2}q^{2}\frac{d\sigma_{R}}{dq^{2}},$ (1)

where $\rho$ is the nuclear matter density, $q^{2}$ is the two-dimensional
momentum transfer squared in a propagating parton-nucleon scattering, and $R$
is the color representation of the propagating parton in SU(3). By means of
the transport coefficient, the parton transverse momentum broadening due to
medium induced radiation reads[16]

$\left\langle q^{2}_{\perp}\right\rangle=\hat{q}_{R}L,$ (2)

with the length $L$ crossed by the propagating parton in the target. The
transport coefficient in the target nucleus can be calculated in terms of the
gluon distribution function in a nucleon[14],

$\hat{q}_{R}=\frac{4\pi^{2}\alpha_{\rm s}(Q^{2}_{\rm
G})C_{R}}{N^{2}_{c}-1}\rho x_{\rm G}G(x_{\rm G},Q^{2}_{\rm G}),$ (3)

where $C_{R}$ is the corresponding colour factor for quark or gluon, the
number of colors $N_{c}=3$, the virtuality $Q^{2}_{\rm G}=\hat{q}_{R}L$. The
strong coupling constant $\alpha_{\rm s}$ and the gluon distribution function
$G(x_{\rm G},\hat{q}_{R}L)$ depend on the scale $\hat{q}_{R}L$. The
theoretical calculation indicates that Bjorken variable $x_{\rm G}\ll 1$ in
the gluon distribution.

In 2010, the HERMES collaboration[17] reported firstly their detailed
determination of the transverse momentum broadening of hadrons in semi-
inclusive deep inelastic scattering for charged pions and positively-charged
kaons on He, Ne, Kr, and Xe target. The average squared transverse momentum is
clearly observed to increase with atomic mass number $A$. The broadening
increases as a function of the photon virtuality $Q^{2}$ and Bjorken scaling
variable $x$, and remains constant as a function of both the virtual photon
energy $\nu$ and the fractional hadron energy $z$, except that it vanishes as
$z$ approaches unity. The experimental data shows that $\hat{q}L<1$ GeV2.
Hereafter we shall label $\hat{q}$ the quark transport coefficient.

As for the quark transport coefficient $\hat{q}$, although Bjorken variable
$x_{\rm G}$ is estimated in theory[14], $x_{\rm G}$ in the definition (3) is
an unobservable kinematic variable in experiment. The unmeasurable $x_{\rm G}$
makes $x_{\rm G}G(x_{\rm G},\hat{q}_{R}L)$ unknown. The unknown $x_{\rm
G}G(x_{\rm G},\hat{q}_{R}L)$ prevents us from making certain the
characteristic of quark transport coefficient $\hat{q}$. This often limits the
predictive power of theory. Therefore, in order to realize the details about
quark transport coefficient, one would like to know whether the unmeasurable
$x_{\rm G}$ can be replaced by the measurable kinematic variable in deep
inelastic scattering. The purpose of this paper is to establish the relation
between quark transport coefficient $\hat{q}$ and the measurable kinematic
variable in deep inelastic scattering.

In fact, as early as in 2014, the scale dependent of jet transport parameter
has been derived theoretically[18]. Following the similar idea, by means of
the experimental data on transverse momentum broadening from semi-inclusive
electron-nucleus deep inelastic scattering, Drell-Yan dilepton and heavy
quarkonium production in proton-nucleus collisions, as well as the nuclear
structure functions in deep inelastic scattering, the kinematic and scale
dependence of jet transport coefficient in cold nuclear matter was determined
in the framework of the generalized QCD factorization formalism[19]. By
fitting the world data on nuclear transverse momentum broadening from Drell-
Yan and quarkonium production for pion and proton incident on nucleus, the
transport coefficient parameterization as a function of Bjorken variable was
extracted[20]. In order to research the underlying structure of the quark-
gluon plasma (QGP), the jet transport coefficient was suggested to be
expressed in terms of a parton distribution function with energy and scale
dependence[21,22]. Recently, by using the nuclear parton distributions
determined only with lepton-nuclear deep inelastic scattering experimental
data and the analytic parameterization of quenching weight based on BDMPS
formalism[23,24], a phenomenological analysis of the nuclear Drell-Yan
differential cross section ratio as a function of Feynman variable was
performed from Fermilab E906[25] and E866[26] experimental data. The relation
was discovered between quark transport coefficient and the measurable
kinematic variables in Drell-Yan reaction. The Bjorken variable and scale
dependence of quark transport coefficient was explored[27].

In our preceding articles[28-30], we have researched the nuclear modifications
of hadron production in semi-inclusive deep inelastic scattering in a parton
energy loss model with the form of the mean energy loss. Our theoretical
results show that the nuclear effects on parton distribution functions can be
neglected. It is found that the theoretical results considering the nuclear
modification of fragmentation functions due to quark energy loss are in good
agreement with the selected experimental data. In the recent work[31], we
employed the two-dimensional data from the HERMES Collaboration on the
multiplicity ratios for positively charged pions production on neon nucleus
with respect to deuterium target[9]. The leading-order hadron multiplicity
ratios were calculated by means of Salgado-Wiedemann(SW) quenching weights[32]
and the analytic parameterizations of quenching weights[33] based on BDMPS
formalism. We studied the target nuclear geometry effect on hadron production,
and compared our calculated results with the HERMES positively charged pions
production data with the quarks hadronization occurring outside the nucleus.
The extracted transport parameters from the global fit are shown to be
$\hat{q}=0.74\pm 0.03$ GeV2/fm for the SW quenching weight without the finite
energy corrections, and $\hat{q}=0.20\pm 0.02$ GeV2/fm for the analytic
parameterization of BDMPS quenching weight without the quark energy $E$
dependence, respectively. We found that the nuclear geometry effect has a
significant impact on the quark transport coefficient in cold nuclear matter.

In this paper, the experimental data with the quarks hadronization occurring
outside the target nucleus, are selected on the multiplicity ratios from
HERMES for charged pions production between neon and deuteron targets[9]. The
BDMPS quenching weight[14,33] will be employed with considering the quark
energy loss and nuclear geometry effect. The Bjorken variable and scale
dependence of quark transport coefficient is investigated in semi-inclusive
lepton-production of hadron off nuclei. It is hoped to gain new knowledge
about the characteristic of the quark transport coefficient $\hat{q}$ in a
cold nuclear medium.

The paper is organized as follows. In Section II we give details of the
formalism for the hadron multiplicity in semi-inclusive lepton-production of
hadron off the nucleus and nuclear modification of the fragmentation functions
owing to quark energy loss and nuclear geometry effect. The results and
discussion obtained are presented in Section III. Finally, we summarize our
findings in Section IV.

##  II The hadron multiplicity in semi-inclusive deep inelastic scattering on
the nucleus

According to the factorization theorem[34] and using leading order QCD
approximation, we can write the hadron multiplicity in the semi-inclusive deep
inelastic scattering off a nucleus,

$\frac{1}{N^{\rm DIS}_{A}}\frac{dN^{h}_{A}}{dzd\nu}=\frac{1}{\sigma^{\rm
lA}}\int dx\sum_{f}e^{2}_{f}q^{A}_{f}(x,Q^{2})\frac{d\sigma^{\rm
lq}}{dxd\nu}D^{A}_{f|h}(z,Q^{2}),$ (4) $\sigma^{\rm lA}=\int
dxd\nu\sum_{f}e^{2}_{f}q^{A}_{f}(x,Q^{2})\frac{d\sigma^{\rm lq}}{dxd\nu}.$ (5)

Here, $N_{A}^{h}$ ($N_{A}^{\rm DIS}$) is the yield of semi-
inclusive(inclusive) deep-inelastic scattering leptons on nucleus $A$,
$\sigma^{\rm lA}$ is the total cross section for the lepton-nucleus deep
inelastic scattering, $e_{f}$ is the electric charge of the quark with flavor
$f$, $q^{A}_{f}(x,Q^{2})$ is the nuclear quark distribution function,
$D^{A}_{f|h}(z,Q^{2})$ is the nuclear modified fragmentation function of a
quark of flavour $f$ into a hadron $h$. In above two equations, the
differential cross section for lepton-quark scattering at leading order,

$\frac{d\sigma^{\rm lq}}{dxd\nu}=M_{\rm N}x\frac{4\pi\alpha_{\rm
em}^{2}}{Q^{4}}[1+(1-y)^{2}],$ (6)

where $M_{\rm N}$ and $\alpha_{\rm em}$ are respectively the nucleon mass and
the electromagnetic coupling constant, $y$ is the fraction of energy of the
incoming lepton which is transferred to the virtual photon.

In semi-inclusive deep inelastic scattering, the energetic parton propagating
in a target nucleus suffers multiple scattering and radiate soft gluons. At
the time of hadronization, the quark energy is reduced from $E=\nu$ to
$E^{\prime}=\nu-\Delta E$ in which $\Delta E$ the quark energy loss in the
nuclear medium. The fragmentation function in the nuclear medium can be
expressed as[35],

$D^{A}_{f|h}(z,Q^{2})=\int_{0}^{(1-z)\nu}d(\Delta E)P(\Delta
E,\omega_{c},L)\frac{1}{1-\Delta E/\nu}D_{f|h}(z^{\prime},Q^{2}),$ (7)

with $z^{\prime}=z/(1-\Delta E/\nu)$. $D_{f|h}(z^{\prime},Q^{2})$ is the
standard(vacuum) fragmentation function of a quark of flavour $f$ into a
hadron $h$. $P(\Delta E,\omega_{c},L)$, the so-called quenching weight, is the
probability that the radiated gluons carry altogether a given energy $\Delta
E$ for a quark with energy $E$. The characteristic gluon frequency
$\omega_{c}=(1/2)\hat{q}L^{2}$ with the path length $L$ traversed by the
struck quark.

For simplicity, we consider the case of the uniform hard-sphere nucleus of
mass number $A$ and radius $R_{A}=r_{0}{A^{1/3}}$ with $r_{0}=1.12$ fm[36].
Assuming that hadronization occurs after the struck quark has escaped from
target nucleus, the mean length $L$ covered by the hard quark is
$L=(3/4)R_{A}$. If the target nuclear geometry effect is added in the semi-
inclusive lepton-production of hadrons off nuclei, the colored quark produced
at $y$ the coordinate along the direction of the outgoing quark, will travel
the path length $L=\sqrt{R_{A}^{2}-b^{2}}-y$ with $\vec{b}$ its impact
parameter[31]. Then the averaged modified fragmentation function is given
by[31]

$D^{A}_{f|h}(z,Q^{2})=\int
d^{2}bdy\rho_{A}(\vec{b},y)\int_{0}^{(1-z)\nu}d(\Delta E)P(\Delta
E,\omega_{c},L)\frac{1}{1-\Delta E/\nu}D_{f|h}(z^{\prime},Q^{2}).$ (8)

In above equation,
$\rho_{A}(\sqrt{b^{2}+y^{2}})=(\rho_{0}/A)\Theta(R_{A}-\sqrt{b^{2}+y^{2}})$
with $\rho_{0}$ is the nuclear density.

Based on the above formalism, only the transport coefficient $\hat{q}$ remains
to be determined.

##  III Results and discussion

In order to explore the characteristic of the quark transport coefficient
$\hat{q}$ in a cold nuclear medium, the experimental data with quark
hadronization outside the nucleus are sorted out from the two-dimensional data
on the multiplicity ratio $R^{h}_{\rm M}$ for charged pions production on neon
nucleus in three $z$ slices as a function of $\nu$, and in five $\nu$ slices
as a function of $z$[9] under the condition $t>2R_{A}$ with the hadron
formation time $t=z^{0.35}(1-z)\nu/\kappa$ ($\kappa=1$ GeV/fm)[7,28,37,38].
The hadron formation time is defined as the time between the moment that the
quark is struck by the virtual photon and the moment that the prehadron is
formed. It is found that the number of points in selected experimental data
include 8 data points for $R^{{\pi}^{+}}_{\rm M}(\nu)$ and $R^{{\pi}^{-}}_{\rm
M}(\nu)$ in the $z$ region of $0.2<z<0.4$, and 30 data points for
$R^{{\pi}^{+}}_{\rm M}(z)$ and $R^{{\pi}^{-}}_{\rm M}(\nu)$ in three $\nu$
regions of $14<\nu<17$ GeV, $17<\nu<20$ GeV and $20<\nu<23.5$ GeV. In total,
our analysis includes 38 data points.

In calculating at leading order the hadron multiplicity ratios $R^{\pi}_{\rm
M}$,

$R^{\pi}_{\rm M}[\nu(z)]=\int\frac{1}{N^{\rm
DIS}_{A}}\frac{dN^{h}_{A}(\nu,z)}{dzd\nu}dz(\nu)\Bigg{/}\int\frac{1}{N^{\rm
DIS}_{\rm D}}\frac{dN^{h}_{\rm D}(\nu,z)}{dzd\nu}dz(\nu),$ (9)

we use the vacuum fragmentation functions given by Hirai, et al.[39], and
CTEQ6L parton density in the proton[40]. As for the quenching weight, Baier,
et al.(BDMPS)[23,24] have computed perturbatively the radiated gluon spectrum
by hard quarks produced in QCD media. By means of the BDMPS gluon spectrum,
the quenching weight $P(\Delta E,\omega_{c},L)$ can be determined[41]. The
analytic parameterization[33] on BDMPS quenching weight was given,

$\omega_{c}P(\bar{\Delta E}=\Delta
E/\omega_{c})=\frac{1}{\sqrt{2\,\pi}\,\sigma\,\bar{\Delta
E}}\,\exp\left[-\frac{\left(\log{\bar{\Delta
E}}-\mu\right)^{2}}{2\,\sigma^{2}}\right],$ (10)

with $\mu=-1.5$ and $\sigma=0.73$ for the outgoing quark energy $E$
independence. Moreover, the path length $L=\sqrt{R_{A}^{2}-b^{2}}-y$ in view
of the target nuclear geometry effect.

In the next part of this effort, we focus on the quark transport coefficient
$\hat{q}$, which is obtained by minimizing $\chi^{2}$. The parabolic errors of
the fit parameters are given by MINUIT package[42].

If we do not consider any kinematic variable dependence of the quark transport
coefficient, $\hat{q}$ is treated as a constant $\hat{q}_{0}$, which is below
labelled the constant model. The global fit of all data gives
$\hat{q}_{0}=0.207\pm 0.017$ GeV2/fm, the relative uncertainty
$\delta\hat{q}_{0}/\hat{q}_{0}\simeq 8\%$ and $\chi^{2}/{\rm ndf}=0.774$. The
black solid curves in Fig.1 are our numeric results which are compared with
the selected experimental data. Our calculational result shows a good fit
quality.

Figure 1: The calculated multiplicity ratios $R^{\pi}_{\rm M}$ as a function
of $\nu$ in the region of $0.2<z<0.4$ and as a function of $z$ in $14<\nu<17$
GeV, $17<\nu<20$ GeV and $20<\nu<23.5$ GeV for charged pions production on
neon target from the parameterization of BDMPS quenching weigh for the quark
energy $E$ independence. The different lines correspond respectively to the
results from $\hat{q}$ in four different models. The HERMES data[9] are shown
with the total uncertainty (statistical plus systematic, added quadratically).

Although the quark transport coefficient $\hat{q}$ as a constant gives a high
quality fit, the detail of the quark transport coefficient $\hat{q}$ is still
unclear. From the formula (3) for the quark transport coefficient, $\hat{q}$
depends on the gluon distribution function $G(x_{\rm G},Q^{2}_{\rm G})$, which
is a function of the Bjorken variable $x_{\rm G}$ and the virtuality
$Q^{2}_{\rm G}$. Following Ref.[14], a fast parton of momentum $p$ traversing
cold nuclear matter interacts with a target nucleon of momentum $P$ by the
exchanged gluon of momentum $q$. In the target nucleus rest frame, using
light-cone variables, the Bjorken variable $x_{\rm G}$ in the gluon
distribution is given by

$x_{\rm G}\equiv\frac{q^{+}}{P^{+}}=\frac{q^{+}}{M_{\rm N}}.$ (11)

In the semi-inclusive deep inelastic lepton-nucleus scattering, the Bjorken
scaling variable $x$ is defined as

$x\equiv\frac{Q^{2}}{2M_{\rm N}\nu}.$ (12)

It is important to remember that $x_{\rm G}$ is an unobservable quantity, $x$
is a kinematic variable measured in the deep inelastic scattering experiment.
Thus, according to the above two definitions,

$x_{\rm G}=\frac{2\nu q^{+}}{Q^{2}}x.$ (13)

In this way, the unmeasurable $x_{\rm G}$ can be characterized by the
measurable Bjorken variable $x$.

As mentioned in Section I, $Q^{2}_{\rm G}<1$ GeV2 in the gluon distribution
from the quark transport coefficient. When the photon wavelength $1/Q_{\rm G}$
reaches the size of the nucleon with radius $R_{\rm N}$, the so-called
saturation occurs. At saturation scale $Q_{\rm S}$, the parton system becomes
to look dense to the probe. In this regime, the number of gluons is given
by[43]

$x_{\rm G}G(x_{\rm G},Q^{2}_{\rm S})\sim\frac{\pi}{\alpha_{\rm s}(Q^{2}_{\rm
S})}Q^{2}_{\rm S}R^{2}_{\rm N}.$ (14)

The saturation scale[44,45]

$Q^{2}_{\rm S}(x_{\rm G})\sim x^{-\lambda}_{\rm G}.$ (15)

Therefore, we can thus extract Bjorken variable $x$ dependence of the quark
transport coefficient by writing

$\hat{q}=\hat{q_{0}}x^{\alpha},$ (16)

which is referred to as the power-law model hereafter. Two parameters of the
fits with the power-law model are: $\hat{q_{0}}$ and $\alpha$.

In the power-law model, the global fit to the selected experimental data shows
that $\hat{q}_{0}=0.101\pm 0.003$ GeV2/fm with the relative uncertainty
$\delta\hat{q}_{0}/\hat{q}_{0}\simeq 3\%$, $\alpha=-0.288\pm 0.033$ with
$\delta\alpha/\alpha\simeq 11\%$, and $\chi^{2}/{\rm ndf}=0.825$. As can be
seen from Fig.1, our theoretical results(red dashed lines) are in good
agreement with the experimental data. From Fig.2, we can find that the quark
transport coefficient $\hat{q}$ decreases with the increasing Bjorken variable
$x$.

Figure 2: The extracted quark transport coefficient $\hat{q}$ as functions of
Bjorken variable $x$ from the power-law model and the double power-law model,
respectively.

Because of low $Q^{2}_{\rm G}$, perturbative QCD fails and non-perturbative
behavior become important. The gluon distribution in the quark transport
coefficient $\hat{q}$ is only an ”effective” distribution. After replacing the
unmeasurable $x_{\rm G}$ with the observable kinematic variable $x$ in the
deep inelastic scattering experiment, and considering the intermediate and
large $x$ effect, we can give Bjorken variable $x$ dependence of the quark
transport coefficient,

$\hat{q}=\hat{q_{0}}x^{\alpha}(1-x)^{\beta},$ (17)

which is hereafter referred to as the double power-law model. Three parameters
in the double power-law model are: $\hat{q_{0}}$, $\alpha$ and $\beta$.

In the double power-law model, the results of our fit to the experimental data
give that $\hat{q}_{0}=0.141\pm 0.001$ GeV2/fm with the relative uncertainty
$\delta\hat{q}_{0}/\hat{q}_{0}\simeq 1\%$, $\alpha=-0.298\pm 0.033$ with
$\delta\alpha/\alpha\simeq 11\%$, $\beta=3.348\pm 0.145$ with
$\delta\beta/\beta\simeq 4\%$, and $\chi^{2}/{\rm ndf}=0.875$. As shown in
Fig.1(blue dotted lines), we see a good fit quality by taking into account the
data precision. From Fig.2, we can find that the quark transport coefficient
$\hat{q}$ decreases rapidly with increasing $x$ in the region $x>0.1$.

The HERMES experiment on transverse momentum broadening of hadrons produced in
semi-inclusive deep inelastic scattering on nuclei[17] reveals that the
transverse momentum broadening is a function of Bjorken variable $x$ and
photon virtuality $Q^{2}$. In view of the relation (2) between transverse
momentum broadening and quark transport coefficient $\hat{q}$[14,46], we can
find that although the constant model, the power-law model and the double
power-law model indicate a high quality fit to the selected data, these three
models can not reflect the $Q^{2}$ dependence of quark transport coefficient
$\hat{q}$. Therefore, the constant model, the power-law model and the double
power-law model remain unable to give expression to complete information on
quark transport coefficient $\hat{q}$.

In contrast with $x_{\rm G}$ substituted by Bjorken variable $x$, we shall
conjecture that the photon virtuality $Q^{2}$ can replace $Q^{2}_{\rm G}$ in
the gluon distribution though the detail is yet unknown. Thus, the strong
coupling constant $\alpha_{\rm s}(Q^{2}_{\rm G})$ in quark transport
coefficient $\hat{q}$ formula (3) can be written as $\alpha_{\rm s}(Q^{2})$.
After further considering the evolution of gluon distribution with the photon
virtuality $Q^{2}$, the quark transport coefficient is assumed to be

$\hat{q}=\hat{q_{0}}\alpha_{\rm
s}(Q^{2})x^{\alpha}(1-x)^{\beta}\ln^{\gamma}(Q^{2}/Q^{2}_{0}),$ (18)

which is hereafter known as the evolution model. We make $Q^{2}_{0}=1$ GeV2 in
order to ensure the argument in the logarithm dimensionless. The
parametrization form is the same as that in Ref. [19]. Four parameters of the
fits with the evolution model are: $\hat{q_{0}}$, $\alpha$, $\beta$ and
$\gamma$.

Table 1: The parameter values of $\hat{q}$ and $\chi^{2}/{\rm ndf}$ extracted in four different models from the selected data on the hadron multiplicity ratios for charged pions produced on neon target. $\hat{q}$(GeV2/fm) | $\hat{q_{0}}$ | $\alpha$ | $\beta$ | $\gamma$ | $\chi^{2}/{\rm ndf}$
---|---|---|---|---|---
$\hat{q_{0}}$ | $0.207\pm 0.017$ |  |  |  | $0.774$
$\hat{q_{0}}x^{\alpha}$ | $0.101\pm 0.003$ | $-0.288\pm 0.033$ |  |  | $0.825$
$\hat{q_{0}}x^{\alpha}(1-x)^{\beta}$ | $0.141\pm 0.001$ | $-0.298\pm 0.033$ | $3.348\pm 0.145$ |  | $0.875$
$\hat{q_{0}}\alpha_{\rm s}(Q^{2})x^{\alpha}(1-x)^{\beta}\ln^{\gamma}(Q^{2}/Q^{2}_{0})$ | $0.401\pm 0.001$ | $-0.297\pm 0.022$ | $3.186\pm 0.055$ | $0.590\pm 0.192$ | $0.967$

After performing the global fit of $\hat{q}$ to the selected experimental
data. The theoretical results are compared with the selected experimental data
in Fig.1. It is found that the expected multiplicity ratios shown by green
dash-dotted curves are in good agreement with the experimental data. The
obtained parameter values of $\hat{q}$ and $\chi^{2}/{\rm ndf}$ in our fits
are summarized in Table.1. As for the evolution model, $\hat{q}_{0}=0.401\pm
0.001$ GeV2/fm with the relative uncertainty
$\delta\hat{q}_{0}/\hat{q}_{0}\simeq 0.2\%$, $\alpha=-0.297\pm 0.022$ with
$\delta\alpha/\alpha\simeq 7\%$, $\beta=3.186\pm 0.055$ with
$\delta\beta/\beta\simeq 2\%$, $\gamma=0.590\pm 0.192$ with
$\delta\gamma/\gamma\simeq 33\%$, and $\chi^{2}/{\rm ndf}=0.967$. The rather
small value of $\chi^{2}/{\rm ndf}$ indicates a good fit quality.

In Table.1, the determined parameter values of $\hat{q}$ and $\chi^{2}/{\rm
ndf}$ are also given from the other three models. From this table and Fig.2,
we can find that in the evolution model, the factor $x^{\alpha}$ reflects the
small $x$ effect on the quark transport coefficient $\hat{q}$. The factor
$(1-x)^{\beta}$ indicates the correction to $\hat{q}$ in the middle and large
$x$ region. A narrow different of the extracted parameter $\alpha$ and $\beta$
between the evolution model and double power-law model indicates that Bjorken
scaling variable $x$ part of $\hat{q}$ varying with $x$ in the evolution model
follows the same trend as that from double power-law model(see the red dashed
line in Fig.2).

Figure 3: The obtained quark transport coefficient $\hat{q}$ as a function of
Bjorken variable $x$ and scale $Q^{2}$. Figure 4: The part
$\hat{q}_{0}\alpha_{\rm s}\ln^{\gamma}(Q^{2}/Q_{0}^{2})$ in $\hat{q}$ varying
with scale $Q^{2}$ in the evolution model.

In Fig.3, Bjorken variable $x$ and photon virtuality $Q^{2}$ dependence is
shown from the quark transport coefficient $\hat{q}$ in the evolution model.
In order to investigate $\hat{q}$ dependence on the single kinematic variable,
the scale $Q^{2}$ part of $\hat{q}$ varying with $Q^{2}$ is given in Fig.4. It
is found that the factor $\hat{q_{0}}\alpha_{\rm
s}(Q^{2})\ln^{\gamma}(Q^{2}/Q^{2}_{0})$ increases with $Q^{2}$ rapidly in the
region $Q^{2}<6.0$ GeV2, afterwards has a slow rising trend.

In Fig.5, the curves of quark transport coefficient $\hat{q}$ to Bjorken
variable $x$ are drawn in four fixed photon virtuality $Q^{2}$ (1.4, 2.5, 3.9,
and 6.5 GeV2). Four $(x,Q^{2})$ points, (0.06, 1.4), (0.10, 2.5), (0.15, 3.9)
and (0.25, 6.5), correspond to the HERMES experimental measurements(see Table
2 in Ref.[17]). As shown in Fig.5, the quark transport coefficient $\hat{q}$
increases in the region $Q^{2}<2.5$ GeV2 and $x<0.1$ as a function of $Q^{2}$
and $x$. Therefore, the trend of $\hat{q}$ in respect of the Bjorken variable
$x$ and scale $Q^{2}$ is qualitatively in partial agreement with HERMES
experimental data on transverse momentum broadening[17].

Figure 5: The quark transport coefficient $\hat{q}$ to Bjorken variable $x$ in
four fixed $Q^{2}$ with four $(x,Q^{2})$ points from the HERMES experimental
measurements.

In our present effort, the used experimental data on the hadron multiplicity
ratios $R^{\pi}_{\rm M}$ are from HERMES Collaboration[9] by using 27.6 GeV
positron and electron beams stored in HERA at DESY. The experimental data were
acquired by the following kinematic conditions: $x=0.02-1.0$ and $Q^{2}=1-45$
GeV2. Ru, et al.[19] employed the HERMES experimental data[17] on the
transverse momentum broadening for positively charged pion as a function of
atomic mass number $A$, and as a function of $Q^{2}$, $x$ and the virtual
photon energy $\nu$ from Xe target with excluding He, Ne and Kr targets.
Therefore, our global fit on quark transport coefficient $\hat{q}$ has
different kinematic region from those in Ref.[19]. The reason remains to be
further explored on the obtained parameter values existing difference for the
quark transport coefficient in the future.

##  IV Summary

The investigation into the detailed characteristic of the quark transport
coefficient in a cold nuclear medium, prompts us to address the question how
the quark transport coefficient depend on the measurable kinematic variables
in deep inelastic scattering. We sort out the experimental data with the quark
hadronization occurring outside the nucleus by the following criterion: the
hadron formation time is more than twice the target nuclear radius. We take
advantage of the analytic parameterization of quenching weight based on BDMPS
formalism with considering the target nuclear geometry effect. Four models are
provided on the quark transport coefficient $\hat{q}$. The leading order
calculation of the hadron multiplicity ratio has been done, and compared with
the selected HERMES experimental data. It is found that all four models
display high quality fit. However, the constant model, the power-law model and
the double power-law model can be ruled out because of the experimental fact
that the transverse momentum broadening increases a function of the photon
virtuality $Q^{2}$. The quark transport coefficient is determined as a
function of Bjorken variable $x$ and scale $Q^{2}$. The quark transport
coefficient as a function of Bjorken variable $x$ and scale $Q^{2}$
demonstrates qualitatively the partial behavior in HERMES experimental data on
transverse momentum broadening.

In order to gain precise knowledge about the quark transport coefficient in a
cold nuclear medium, we suggest that in the follow-up experiment on the semi-
inclusive hadron production in deep inelastic scattering on nuclear target,
the hadron multiplicity ratios should be measured in multi-dimensional
representation, which consists in a fine binning in one variable and coarser
bins in other variables. Meanwhile, the future measurements on the hadron
multiplicity ratio and transverse momentum broadening are needed with wider
kinematic coverage (e.g. at a future Electron Ion Collider[47]). Furthermore,
on the basis of the existing researches, a better understanding of the parton
propagation mechanism can be achieved in the future.

Acknowledgments We thank Professor Zhi-Hui Guo for interesting and useful
discussions. This work is supported in part by the National Natural Science
Foundation of China(11575052, 11975090).

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