# The Proton Gluon Distribution from
the Color Dipole Picture

G.R. Boroun  Department of Physics, Razi University,
Kermanshah 67149, Iran  M. Kuroda  Center for Liberal Arts, Meijigakuin
University
Yokohama, Japan  Dieter Schildknecht  Fakultät für Physik, Universität
Bielefeld
D-33501 Bielefeld, Germany
and
Max-Planck-Institut für Physik (Werner-Heisenberg-Institut)
Föhringer Ring 6, D-80805 München, Germany

###### Abstract

Employing the representation of the experimental data on deep inelastic
electron-proton scattering (DIS) in the color-dipole picture (CDP), we
determine the gluon distribution of the proton at small Bjorken $x$. At
sufficiently large momentum transfer, $Q^{2}$, the extracted gluon
distribution fulfills the standard evolution equation for the proton structure
function. For low values of $Q^{2}$, e.g. for $Q^{2}=1.9{\rm GeV}^{2}$, the
evolution equation for the proton structure function is violated. The standard
procedure of adopting a low-$Q^{2}$ starting scale for the extraction of the
gluon density is questionable and requires further investigations.

###### pacs:

##  I I. Introduction

The extraction of the gluon-distribution function Harland ; Dulat ; Ball ;
Buckley  of the proton from deep-inelastic electron-proton scattering (DIS)
Abramowicz  rests on the determination of the fit parameters in an ad-hoc
parametrization of the gluon x-distribution at a low-$Q^{2}$ input scale.111In
standard notation, the virtuality of the photon is denoted by $Q^{2}$, and $x$
is the Bjorken scaling variable, $x\cong Q^{2}/W^{2}$, where $W$ denotes the
virtual-photon-proton center of mass energy. It is well-known, and was
recently emphasized in ref.Pelicer  that the results on the gluon distribution
function from different collaborations Harland ; Dulat ; Ball ; Buckley  show
a significant spread 222See Fig.2 in ref.[6] in the low-x, low-$Q^{2}$ domain.
The quality of the fits is considered [6] not to be satisfactory. This led to
the introduction of phenomenological power corrections HM ; Cooper  to the
structure functions, and to modifications Pelicer  of the DGLAP evolution
equations Lipatov  for the gluon distribution by a non-linear term Gribov ;
Mueller .

In the color-dipole picture (CDP) [12], [13],[14], [15] of DIS, the
photoabsorption cross section at low $x$ is represented in terms of the quark-
antiquark proton ($q\bar{q}p$) cross section, $\sigma_{(q\bar{q})p}(W^{2})$.
The corresponding forward scattering amplitude is given by an ansatz for the
color-gauge-invariant interaction of $q\bar{q}$ states with the proton via
two-gluon exchange. The $q\bar{q}$ states form a massive continuum as a
function of the $q\bar{q}$ masses (as observed in $e^{+}e^{-}\to q\bar{q}$
annihilation), including a smooth interpolation of the low-lying vector meson
peaks. Accordingly, the massive $q\bar{q}$ continuum starts at a mass squared
of $m^{2}_{0}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}m^{2}_{\rho}$, where $m^{2}_{\rho^{0}}$ denotes the
square of the $\rho^{0}$ meson mass. The general structure (at leading order
of $\alpha_{s}(Q^{2})$) of the two-gluon exchange interaction of
$\gamma^{*}g\to q\bar{q}$ (where $\gamma^{*}$ denotes the photon of virtuality
$Q^{2}$) from the pQCD improved parton model is assumed to remain valid when
proceeding from large values of $Q^{2}$ (the genuine region of pQCD) to small
values of $Q^{2}$, for $Q^{2}$ towards $Q^{2}=0$. In the CDP, accordingly, the
representation of DIS is continued to include the $Q^{2}$ towards $Q^{2}=0$
limit, and the photoproduction cross section $\sigma_{\gamma p}(W^{2})$ may be
used as a normalization of the $\gamma^{*}p$ interaction cross section,
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})$.

The basic quantity of the CDP ansatz, the cross section
$\sigma_{(q\bar{q})p}(W^{2})$, is dependent on $W^{2}$ via
$\Lambda^{2}_{sat}(W^{2})\equiv C_{1}\left(\frac{W^{2}}{1{\rm
GeV}^{2}}\right)^{C_{2}}$, where $C_{1}$ and $C_{2}$ are adjustable
parameters.

The dominance of the $\gamma^{*}g\to q\bar{q}$ interaction is common to both,
the description of DIS at low $x$ in the CDP and in the pQCD-based parton
model, where instead of $\sigma_{(q\bar{q})p}(W^{2})$ the gluon distribution
function $G(x,Q^{2})$ enters as the basic quantity. The basic mechanism of
$\gamma^{*}g\to q\bar{q}$ being the same, the underlying gluon distribution
may be deduced from the CDP representation.

In the present paper, at leading order in the strong coupling
$\alpha_{s}(Q^{2})$, we accordingly determine the gluon distribution,
$G(x,Q^{2})$, from the CDP representation of the DIS experimental data.

Adopting an implicit validity of the $\gamma^{*}g\to q\bar{q}$ interaction, a
gluon distribution may also be extracted from a fit to the experimental data
not manifestly being based on the $\gamma^{*}g\to q\bar{q}$ interaction of the
CDP. Assuming implicit validity of the $\gamma^{*}g\to q\bar{q}$ interaction,
we in addition accordingly deduce $G(x,Q^{2})$ from the Froissart-bounded
representation of the DIS data. We find results consistent with the ones based
on the CDP. Our results on the gluon distribution do not exclusively depend on
the CDP representation of the DIS experimental data.

In the present paper, accordingly, the leading order gluon distribution of the
perturbative-QCD-improved (pQCD) parton model is extracted from the
representation of the proton structure functions $F_{2}(x,Q^{2})$ and
$F_{L}(x,Q^{2})$ in the CDP. We find that at low $x$, for $Q^{2}$ sufficiently
large ($Q^{2}{\gtrsim}10~{}{\rm GeV}^{2}$ to $20~{}{\rm GeV}^{2}$), the
experimental results on DIS parametrised by the CDP fulfill evolution in
distinction from low values of $Q^{2}$ (e.g. for $Q^{2}=1.9~{}{\rm GeV}^{2}$),
where due to hadronlike behaviour of DIS, the standard evoluation equation is
strongly violated. This result suggests that the unsatisfactory fits mentioned
in Ref.[6]333See Fig.2 in ref.[6] at low $Q^{2}$ are due to an improper use of
a low-$Q^{2}$ input scale, such as $Q^{2}=1.9~{}{\rm GeV}^{2}$, in the widely
employed analysis of the DIS experimental results.

In Section II, we briefly review the determination of the gluon distribution
at leading order of $\alpha_{s}(Q^{2})$ from the pQCD improved parton model.
In Section III, we review the relevant parts of the CDP. In Section IV, as an
alternative to the CDP, we consider and summarise the Froissart-bounded
representation of $F_{2}(x,Q^{2})$. In Section V, the results obtained for the
gluon distribution are presented for both the CDP and the Froissart-bounded
parametrization of the DIS data. At low $Q^{2}~{}~{}(Q^{2}=1.9{\rm GeV}^{2})$,
we observe drastic differences between our results on the gluon distribution
and the results published by several different collaborations. Note that for
the extraction of the gluon distribution in Sections II to V, no use is being
made of the pQCD evolution equations Lipatov .

In Sections VI and VII, we examine the consistency between the low-$Q^{2}$
dependence at low-$x$ of our results for the gluon distribution and the
validity of the evolution equation for the structure function
$F_{2}(x,Q^{2})$. At large $Q^{2}$, specifically for approximately $20~{}{\rm
GeV}^{2}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}Q^{2}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}100{\rm GeV}^{2}$,
the CDP gluon distribution is consistent with the validity of evolution,
leading to the important constraint of $C_{2}=0.29$ for the exponent in the
dependence of $F_{2}(x,Q^{2})$ on $(W^{2})^{C_{2}}=(Q^{2}/x)^{C_{2}}$. The
predicted value of $C_{2}\cong 0.29$ agrees with experiment. At low-$Q^{2}$,
the logarithmic derivative of the structure function $F_{2}(x,Q^{2})$ deviates
from the prediction of the evolution equation by a factor of magnitude up to
3. Final Conclusions are presented in Section VIII.

##  II II. The determination of the gluon distribution function in pQCD.

In the pQCD improved parton model, the longitudinal structure function
$F_{L}(x,Q^{2})$ to lowest order of $\alpha_{s}(Q^{2})$ is given by Martin

$F_{L}(x,Q^{2})=\frac{\alpha_{s}(Q^{2})}{4\pi}\left(\frac{16}{3}I_{F}+8\sum
Q^{2}_{q}I_{g}\right),$ (2.1)

where

$I_{F}\equiv
I_{F}(x,Q^{2})=\int^{1}_{x}\frac{dy}{y}\left(\frac{x}{y}\right)^{2}F_{2}(y,Q^{2}),$
(2.2)

and

$I_{g}\equiv
I_{g}(x,Q^{2})=\int^{1}_{x}\frac{dy}{y}\left(\frac{x}{y}\right)^{2}\left(1-\frac{x}{y}\right)G(x,Q^{2}),$
(2.3)

the gluon density being denoted by $G(x,Q^{2})=xg(x,Q^{2})$, and $\sum
Q^{2}_{q}=10/9$ for four flavors of quarks. The term proportional to $I_{F}$
in (2.1) is due to $\gamma^{*}$-quark (antiquark) interaction with gluon
emission,

$\gamma^{*}+q(\bar{q})\to q(\bar{q})+g,$ (2.4)

while the term proportional to $I_{g}$ in (2.1) originates from quark-
antiquark production,

$\gamma^{*}+g\to q\bar{q}.$ (2.5)

The successful representation of the experimental data on DIS in the CDP is
based on incorporating Cvetic  the $q\bar{q}$-color-dipole interaction (2.5)
with the gluon field of the proton

$\displaystyle(\gamma^{*}\to q\bar{q})+g\to$ $\displaystyle q\bar{q}$
$\displaystyle\mathrm{or}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\gamma^{*}+(g\to q\bar{q})\to$ $\displaystyle q\bar{q},$ (2.6)

into the mass-dispersion relation of Generalized Vector Dominance (GVD)
Sakurai . In the CDP the photon makes a transition to $q\bar{q}$ states that
interact with gluons, or, equivalently, the photon (exclusively) intracts with
the $q\bar{q}$ sea originating from $g\to q\bar{q}$ transitions, see (2.6).

Accordingly, motivated by the CDP representation of the experimental data, we
assume that the gluon term associated with reaction (2.5) is the dominant one
in the pQCD expression (2.1),

$\displaystyle
F_{L}(x,Q^{2})=\frac{2\alpha_{s}(Q^{2})}{\pi}\sum_{q}Q^{2}_{q}I_{g}(x,Q^{2}).$
(2.7)

In the CDP of low-$x$ DIS, the large-$Q^{2}$ pQCD equality (2.1) is replaced
by (2.7), thus excluding (2.4).

For a wide range of different gluon distributions, independently of their
specific form, the evaluation of $I_{g}(x,Q^{2})$ from (2.3) in (2.7) yields
the well-known approximate result [19, 14 ] for the longitudinal structure
function (2.7)

$F_{L}\left(\xi_{L}x,Q^{2}\right)=\frac{\alpha_{s}(Q^{2})}{3\pi}\sum_{q}Q^{2}_{q}G(x,Q^{2}).$
(2.8)

A given gluon distribution determines the longitudinal proton structure
function at a rescaled value of $x\to\xi_{L}x$. The rescaling factor $\xi_{L}$
in (2.8) has the preferred value of $\xi_{L}\cong 0.40$ Martin .

We have explicitly tested the validity of the replacement of the pQCD integral
representation (2.7) between the longitudinal structure function of the CDP
and the gluon distribution by the simple proportionality (2.8) for the
determination of the gluon distribution. Determining the gluon distribution
from (2.8) by inserting the explicit CDP expression for
$F_{L}\left(\xi_{L}x,Q^{2}\right)$ given by (5.1) below, and inserting the
result into (2.7), we indeed reproduce the CDP input for $F_{L}(x,Q^{2})$
within a deviation of up to at most 3.5 %. Compare the results in Table 1.

Table 1: The table shows the ratio of $F_{L}(W^{2},Q)$ from (2.7) upon insertion of the gluon distribution from the proportionality (2.8), to the CDP input for $F_{L}(x,Q^{2})$ from (5.1). The ratio is indeed close to unity. | $W^{2}=10^{5}$ | $W^{2}=10^{4}$ | $W^{2}=10^{3}$
---|---|---|---
| $[{\rm GeV}^{2}]$ | $[{\rm GeV}^{2}]$ | $[{\rm GeV}^{2}]$
$Q^{2}=100$ | 1.043 | 1.037 | 0.981
$[{\rm GeV}^{2}]$ |  |  |
$Q^{2}=50$ | 1.041 | 1.037 | 1.012
$[{\rm GeV}^{2}]$ |  |  |
$Q^{2}=10$ | 1.035 | 1.035 | 1.030
$[{\rm GeV}^{2}]$ |  |  |
$Q^{2}=2$ | 1.021 | 1.026 | 1.029
$[{\rm GeV}^{2}]$ |  |  |
$Q^{2}=1$ | 1.015 | 1.021 | 1.026
$[{\rm GeV}^{2}]$ |  |  |

The simple proportionality (2.8) indeed consistently provides a gluon
distribution that fulfills the pQCD relationship (2.7). The $Q^{2}$ dependence
of the gluon distribution in (2.7) is indeed determined by the $Q^{2}$
dependence of the proton structure function of the CDP in (5.1) upon insertion
into (2.8).

##  III III. The Color-Dipole Representation.

For a detailed representaion of the CDP, we refer to Refs.[14] and [15]. In a
brief summary, the photon interaction (2.5) with the gluon field in the proton
at low values of $x$ is interpreted as fluctuation of the photon into
$q\bar{q}$-color-dipole states, see (2.6), that interact with the gluon field
in the proton via color-gauge-invariant gluon couplings to quark and antiquark
states.

In Fig. 1444The photoabsorption cross section (LABEL:eq:3.1) and (3.2) in good
approximation is determined by $\eta(W^{2},Q^{2})$, i.e.
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})\cong\sigma_{\rho^{*}p}(\eta(W^{2},Q^{2}))$
except for a weak (logarithmic) $W$-dependence due to
$\sigma^{(\infty)}(W^{2})$ in (LABEL:eq:3.1) or $\sigma_{\gamma p}(W^{2})$ in
(3.2). The parameter $\xi$ (not to be confused with $\xi_{L}$) is fixed at
$\xi=\xi_{0}=130$, and for the range of $\eta(W^{2},Q^{2})$ of relevance for
the present paper, one may put $\xi\to\infty$., reproduced from refs. Ku-Schi
; Kuroda , we show the results for the photoabsorption cross section,
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})$, in the CDP. The results are obtained from
the explicit analytic expression

Figure 1: The theoretical results for the photoabsorption cross section
$\sigma_{\gamma^{*}p}(\eta(W^{2},Q^{2}),\xi)$ in the CDP as a function of the
low-x scaling variable
$\eta(W^{2},Q^{2})=(Q^{2}+m^{2}_{0})/\Lambda^{2}_{sat}(W^{2})$ for different
values of the parameter $\xi$ that determines the (squared) mass range
$M^{2}_{q\bar{q}}\leq m^{2}_{1}(W^{2})=\xi\Lambda^{2}_{sat}(W^{2})$ of the
$\gamma^{*}\to q\bar{q}$ fluctuations that are taken into account. The
experimental results for $\sigma_{\gamma^{*}p}(\eta(W^{2},Q^{2}),\xi)$ lie on
the full line corresponding to $\xi=\xi_{0}=130$, compare refs. Ku-Schi ;
Kuroda .

for $\sigma_{\gamma^{*}p}(W^{2},Q^{2})$,

$\displaystyle\sigma_{\gamma^{*}p}(W^{2},Q^{2})$ $\displaystyle=$
$\displaystyle\sigma_{\gamma^{*}_{L}p}(W^{2},Q^{2})+\sigma_{\gamma^{*}_{T}p}(W^{2},Q^{2})$
$\displaystyle=$ $\displaystyle\frac{\alpha
R_{e^{+}e^{-}}}{3\pi}\sigma^{(\infty)}(W^{2})$
$\displaystyle\times\left(I^{(1)}_{T}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)G_{T}(u)\right.$
$\displaystyle~{}~{}~{}~{}~{}\left.+I^{(1)}_{L}(\eta,\mu)G_{L}(u)\right),$

derived from an ansatz Ku-Schi ; Kuroda ; Schild  for the W-dependent dipole
cross section that essentially, via the structure of the coupling of the
quark-antiquark state to two gluons, represents the color-gauge-invariant
interaction of the $q\bar{q}$ dipole with the gluon-field in the nucleon. In
(LABEL:eq:3.1), $R_{e^{+}e^{-}}=3\sum_{q}Q_{q}^{2}$, where $q$ runs over the
active quark flavors, and $Q_{q}$ denotes the quark charge. The smooth
transition to $Q^{2}=0$ photoproduction in (LABEL:eq:3.1) allows one Kuroda
to replace (see (2.37) in ref. Kuroda ) $\sigma^{(\infty)}(W^{2})$, which
stems from the normalization of the $q\bar{q}$-dipole-proton cross
section555Note that the photon fluctuates [13,16] into on-mean-shell massive
$q\bar{q}$ states (see e.g. Ref.[18] for an explicit proof of this point)
implying a dipole-proton cross section that depends on $W^{2}$ (besides the
transverse dipole size) and not on $x$, a dependence on $x$ nevertheless being
adopted frequently without justification., by the photoproduction cross
section, and (LABEL:eq:3.1) becomes

$\displaystyle\sigma_{\gamma^{*}p}(W^{2},Q^{2})$
$\displaystyle=\frac{\sigma_{\gamma
p}(W^{2})}{\lim\limits_{\eta\to\mu(W^{2})}I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu(W^{2})}{\rho}\right)G_{T}(u)}$
(3.2)
$\displaystyle\times\left(I^{(1)}_{T}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)G_{T}(u)+I^{(1)}_{L}(\eta,\mu)G_{L}(u)\right).~{}~{}$

We note that $I_{L}^{(1)}(\eta,\mu)$ vanishes in the photoproduction $Q^{2}=0$
limit of
$\eta(W^{2},Q^{2}=0)=m_{0}^{2}/\Lambda^{2}_{sat}(W^{2})\equiv\mu(W^{2})$, and
$G_{T}(u\equiv{\xi\over\eta})\simeq 1$, and for later reference we also note

$\lim_{\eta\to\mu(W^{2})}I^{(1)}_{T}\left(\frac{\eta}{\rho},\frac{\mu(W^{2})}{\rho}\right)=\ln\frac{\rho}{\mu(W^{2})}.$
(3.3)

For the general explicit analytic expressions for the functions
$I^{(1)}_{T}\left(\frac{\eta}{\rho},\frac{\mu(W^{2})}{\rho}\right)$ and
$I^{(1)}_{L}(\eta,\mu)$ we refer to Appendix A. The functions
$G_{T}\left(u\equiv\frac{\xi}{\eta}\right)$ and
$G_{L}\left(u\equiv\frac{\xi}{\eta}\right)$ are given by

$G_{T}(u)=\frac{2u^{3}+3u^{2}+3u}{2(1+u)^{3}}\simeq\left\\{\begin{array}[]{l@{\quad,\quad}l}\frac{3}{2}\frac{\xi}{\eta}&(\eta\gg\xi),\\\
1-\frac{3}{2}\frac{\eta}{\xi}&(\eta\ll\xi),\end{array}\right.$ (3.4)

and

$G_{L}(u)=\frac{2u^{3}+6u^{2}}{2(1+u)^{3}}\simeq\left\\{\begin{array}[]{l@{\quad,\quad}l}3\left(\frac{\xi}{\eta}\right)^{2}&(\eta\gg\xi),\\\
1-3\left(\frac{\eta}{\xi}\right)^{2}&(\eta\ll\xi),\end{array}\right.$ (3.5)

where $u\equiv\xi/\eta$, and the constant parameter $\xi$ restricts the masses
of the contributing mass $q\bar{q}$ states via

$M^{2}_{qq}\leq m^{2}_{1}(W^{2})=\xi\Lambda^{2}_{sat}(W^{2}).$ (3.6)

The numerical results for the photoabsorption cross section in Fig. 1 are
obtained by numerical evaluation of (3.2) upon insertion of a $\left(\ln
W^{2}\right)^{2}$ fit to the experimental results for the photoproduction
cross section $\sigma_{\gamma p}(W^{2})$ from the Particle Data Group PDG .
The results in Fig. 1 were obtained for $W=275~{}{\rm GeV}$ from

$\displaystyle\sigma_{\gamma p}(W^{2})=$ $\displaystyle
0.003056\left(34.71+\frac{0.3894\pi}{M^{2}}\ln^{2}\frac{W^{2}}{M^{2}_{p}+M)^{2}}\right)$
(3.7) $\displaystyle+0.0128\left(\frac{(M_{p}+M)^{2}}{W^{2}}\right)^{0.462}.$

In (3.7), $M_{p}$ denotes the proton mass, $M=2.15~{}{\rm GeV}$ and
$\sigma_{\gamma\rho}(W^{2})$ is given in units of millibarn.

Before going into more detail, we note that the full curve in Fig. 1, which
for the parameter $\xi$ corresponds to the choice of $\xi=\xi_{0}=130$,
provides a representation of the full set of experimental data on
$\sigma_{\gamma^{*}p}(W^{2},Q^{2})$ at low $x\cong Q^{2}/W^{2}$, compare Fig.
8 in ref. Ku-Schi .

In (LABEL:eq:3.1) and (3.2), the low-x scaling variable $\eta(W^{2},Q^{2})$ is
given [13] by

$\eta\equiv\eta(W^{2},Q^{2})=\frac{Q^{2}+m^{2}_{0}}{\Lambda^{2}_{sat}(W^{2})},$
(3.8)

with

$\mu\equiv\mu(W^{2})=\eta(W^{2},Q^{2}=0)=\frac{m^{2}_{0}}{\Lambda^{2}_{sat}(W^{2})},$
(3.9)

the ”saturation scale”666The value of $\Lambda^{2}_{sat}(W^{2})$ via
$\eta(W^{2},Q^{2}){\cong}~{}5$ determines the transition from color
transparency of $\eta(W^{2},Q^{2}){\gtrsim}~{}5$ to hadronlike saturation of
$\eta{\lesssim}5$. , $\Lambda^{2}_{sat}(W^{2})$, being parametrized by

$\Lambda^{2}_{sat}(W^{2})=C_{1}\left(\frac{W^{2}}{1{\rm
GeV}^{2}}\right)^{\LARGE{C_{2}}},$ (3.10)

and numerically the results in Fig.1 are based on

$\displaystyle m^{2}_{0}$ $\displaystyle=$ $\displaystyle 0.15~{}{\rm
GeV}^{2},$ $\displaystyle C_{1}$ $\displaystyle=$ $\displaystyle 0.31~{}{\rm
GeV}^{2};~{}~{}~{}C_{2}=0.27.$ (3.11)

The parameter $\rho=const$ in (LABEL:eq:3.1) and (3.2) is related to the
longitudinal-to-transverse ratio $R(W^{2},Q^{2})$ of the photoabsorption cross
section, and approximately we have $R(W^{2},Q^{2})\simeq 1/2\rho$ for
$\eta(W^{2},Q^{2})\gg\mu(W^{2})$, while $R(W^{2},Q^{2})=0$ for $Q^{2}=0$. The
total cross section $\sigma_{\gamma^{*}p}(W^{2},Q^{2})$ is fairly insensitive
to the value of $\rho$ for realistic values of $\rho$ around $\rho\cong 1$,
and the evaluation presented in Fig. 1 is based on $\rho={4\over 3}$.

In terms of the photoabsorption cross sections
$\sigma_{\gamma^{*}_{L,T}p}(W^{2},Q^{2})$, in (LABEL:eq:3.1) and (3.2), the
proton structure functions, relevant for the extraction of the gluon-density
distribution, see (2.8), are given by

$F_{L,T}(W^{2},Q^{2})=\frac{Q^{2}}{4\pi^{2}\alpha}\sigma_{\gamma^{*}_{L,T}p}(W^{2},Q^{2})$
(3.12)

and

$\displaystyle
F_{2}(W^{2},Q^{2})=\frac{Q^{2}}{4\pi^{2}\alpha}\sigma_{\gamma^{*}p}(W^{2},Q^{2})$
$\displaystyle=\frac{Q^{2}}{4\pi^{2}\alpha}\left(\sigma_{\gamma^{*}_{T}p}(W^{2},Q^{2})+\sigma_{\gamma^{*}_{L}p}(W^{2},Q^{2})\right).$
(3.13)

Upon introducing the longitudinal-to-transverse ratio $R(W^{2},Q^{2})$, the
longitudinal structure function becomes

$F_{L}(W^{2},Q^{2})=\frac{R}{1+R}F_{2}(W^{2},Q^{2}).$ (3.14)

Explicitly, according to (3.2), we have

$R(W^{2},Q^{2})=\frac{I_{L}^{(1)}(\eta,\mu)G_{L}(u)}{I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)G_{T}(u)}.$
(3.15)

For $Q^{2}=0$, as a consequence of electromagnetic gauge invariance,

$R(W^{2},Q^{2}=0)=0,$ (3.16)

while for $\eta(W^{2},Q^{2})$, restricted by the interval of
$1\ll\eta(W^{2},Q^{2})\ll\xi=\xi_{0}=130$ that will be of relevance
subsequently, we have

$R(W^{2},Q^{2})\approx\frac{1}{2\rho},$ (3.17)

with $\rho=const$ in the vicinity of $\rho\cong 1$, and, accordingly, (3.14)
becomes

$\displaystyle F_{L}(W^{2},Q^{2})$
$\displaystyle=\frac{1}{2\rho+1}F_{2}(W^{2},Q^{2}),$ (3.18)
$\displaystyle(1\ll\eta(W^{2},Q^{2})\ll\xi_{0}).$

While (3.18) holds strictly for sufficiently large $Q^{2}$, even for low
values of $Q^{2}{\gtrsim}1\textrm{GeV}^{2}$ it may be used as an approximation
of (3.14). The accuracy of (3.18) as approximation for low $Q^{2}$ is seen by
comparing the results in Fig.3 and Fig.4 in Section V below.

For large $Q^{2}$, specifically for $10{\rm GeV}^{2}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}Q^{2}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}100{\rm GeV}^{2}$
the range of $Q^{2}$ of particular relevance in connection with pQCD, the
experimental data on the proton structure function $F_{2}(W^{2},Q^{2})$ in
(3.13) depend on the single variable $W^{2}$, compare Fig. 2 quoted from ref.
Ku-Schi .

2a 2b

Figure 2: In Fig.2a we show the experimental data for $F_{2}(x\cong
Q^{2}/W^{2},Q^{2})$ as a function of $1/W^{2}$, and in Fig.2b, for comparison,
as a function of $x$. The theoretical prediction based on (3.19) with (3.20)
is also shown in Fig.2a.

A simple two-parameter eye-ball fit to the experimental data in Fig. 2 yields
Ku-Schi

$F_{2}(W^{2})=f_{2}\left(\frac{W^{2}}{1{\rm GeV}^{2}}\right)^{C_{2}},$ (3.19)

where 777The value of the exponent $C_{2}=0.29$ is consistent D21  with the
independent fit to the DIS experimental data based on the hard-Pomeron
conjecture leading to the exponent $C_{0}\cong 0.30\pm 0.1.$

$f_{2}=0.063,~{}~{}C_{2}=0.29.$ (3.20)

The fit result is understood as a consequence of the CDP in the
large-$\eta(W^{2},Q^{2})$ approximation. The structure function
$F_{2}(W^{2},Q^{2})$ for $Q^{2}\gg\Lambda^{2}_{sat}(W^{2})$ according to
(3.13) upon substitution of (LABEL:eq:3.1) takes the simple form

$\displaystyle F_{2}(W^{2},Q^{2})$ $\displaystyle=$
$\displaystyle\frac{R_{e^{+}e^{-}}\sigma^{(\infty)}(W^{2})}{24\pi^{3}}\frac{1+2\rho}{3}\Lambda^{2}_{sat}(W^{2})$
(3.21) $\displaystyle\times$
$\displaystyle\left(1+0\left(\frac{1}{\eta}\right)\right).$

Upon specifying $\Lambda^{2}_{sat}(W^{2})=C_{1}\left(\frac{W^{2}}{1{\rm
GeV}^{2}}\right)^{C_{2}}$ according to (3.10), the structure function (3.21)
can directly be compared with (3.19),

$\displaystyle F_{2}(W^{2},Q^{2})$
$\displaystyle=\frac{R_{e^{+}e^{-}}\sigma^{(\infty)}(W^{2})}{24\pi^{3}}\frac{1+2\rho}{3}C_{1}\left(\frac{W^{2}}{1{\rm
GeV}^{2}}\right)^{C_{2}}$
$\displaystyle\times\left(1+0\left(\frac{1}{\eta}\right)\right)$
$\displaystyle\equiv\hat{f}_{2}(W^{2})\left(\frac{W^{2}}{1{\rm
GeV}^{2}}\right)^{C_{2}}\left(1+0\left(\frac{1}{\eta}\right)\right).$

The product $R_{e^{+}e^{-}}\sigma^{(\infty)}(W^{2})$ is determined by the
photoproduction cross section (3.7) according to

$R_{e^{+}e^{-}}\sigma^{(\infty)}(W^{2})=\frac{3\pi}{\alpha}\frac{\sigma_{\gamma
p}(W^{2})}{\ln\frac{\rho\Lambda^{2}_{sat}(W)}{m^{2}_{0}}},$ (3.23)

compare to (LABEL:eq:3.1) to (3.3).

For definiteness, in Table 2, we explicitly present the input for the
evaluation of $F_{2}$ according to (III) that leads to

$\hat{f}_{2}(W^{2})=\left\\{\begin{array}[]{l@{\quad,\quad}l}0.067&{\rm
for}~{}W^{2}=10^{4}{\rm GeV^{2}},\\\ 0.068&{\rm for}~{}W^{2}=10^{5}{\rm
GeV^{2}}.\end{array}\right.$ (3.24)

Table 2: The evaluation of $\hat{f}_{2}(W^{2})$ defined by (III). The parameters $\rho$ and $m^{2}_{0}$ in (III) and (3.23) are given by $\rho=4/3$ and $m^{2}_{0}=0.15{\rm GeV}^{2}$. $W^{2}~{}[{\rm GeV}^{2}]$ | $10^{4}$ | $10^{5}$
---|---|---
$\sigma_{\gamma p}~{}[mb]$ | 0.146 | 0.175
$\Lambda^{2}_{sat}(W^{2})=C_{1}\left(\frac{W^{2}}{1{\rm GeV}^{2}}\right)^{C_{2}}$ |  |
$C_{1}=0.31~{}{\rm GeV}^{2}$, | 4.48 | 8.74
$C_{2}=0.29$ |  |
$R_{e^{+}e^{-}}\sigma^{(\infty)}(W^{2})~{}[mb]$ | 50.9 | 51.9
$\hat{f}_{2}(W^{2})$ | 0.067 | 0.068

These values of $\hat{f}_{2}(W^{2})$ are approximately 6 % to 7 % larger than
the values from the eye-ball fit (3.19) in (3.20). The structure function
$F_{2}(W^{2},Q^{2})$ is given by

$F_{2}\cong\left\\{\begin{array}[]{l@{\quad,\quad}l}0.96&{\rm
for}~{}W^{2}=10^{4}~{}{\rm GeV}^{2},\\\ 1.91&{\rm for}~{}W^{2}=10^{5}~{}{\rm
GeV}^{2}.\end{array}\right.$ (3.25)

For the determination of the gluon distribution according to (2.8), the
structure functions in (3.12) and (3.13) have to be evaluated at a rescaled or
shifted value of $x\to\xi_{L}x$, corresponding to a shift of
$W^{2}\to\xi^{-1}_{L}W^{2}$. In terms of the low-x scaling variable
$\eta(W^{2},Q^{2})$, the shift becomes

$\eta(W^{2},Q^{2})=\frac{Q^{2}+m^{2}_{0}}{\Lambda^{2}_{sat}(W^{2})}\to\frac{Q^{2}+m^{2}_{0}}{\Lambda^{2}_{sat}\left(\xi_{L}^{-1}W^{2}\right)}.$
(3.26)

For

$\Lambda^{2}_{sat}(W^{2})=C_{1}\left(\frac{W^{2}}{1~{}{\rm
GeV}^{2}}\right)^{C_{2}},$ (3.27)

to be employed subsequently, the shift becomes

$\eta(W^{2},Q^{2})\to\xi_{L}^{C_{2}}\eta(W^{2},Q^{2}).$ (3.28)

The photoabsorption cross section (3.2) essentially only depends on
$\eta(W^{2},Q)$,

$\sigma_{\gamma^{*}_{L,T}p}(W^{2},Q^{2})\cong\sigma_{\gamma^{*}_{L,T}p}\left(\eta(W^{2},Q^{2})\right),$
(3.29)

since the $Q^{2}=0$ photoproduction factor in (3.2) depends only weakly on
$W^{2}$. The rescaling shift (3.28) applied to the cross section (3.2),
accordingly, amounts to

$\sigma_{\gamma^{*}_{L,T}p}\left(\eta(W^{2},Q^{2})\right)\to\sigma_{\gamma^{*}_{L,T}p}\left(\xi_{L}^{C_{2}}\eta(W^{2},Q^{2})\right).$
(3.30)

Numerically, for $C_{2}=0.29$ and $\xi_{L}=0.4$, we have

$\xi_{L}^{C_{2}}\eta=0.4^{0.29}\eta\cong 0.77\eta(W^{2},Q^{2}).$ (3.31)

Explicitly, the gluon distribution (2.8) becomes

$\alpha_{s}(Q^{2})G(x,Q^{2})=\frac{3\pi}{\sum_{q}Q^{2}_{q}}\frac{Q^{2}}{4\pi^{2}\alpha}\sigma_{\gamma^{*}_{L}p}\left(\xi_{L}^{C_{2}}\eta(W^{2},Q^{2})\right).$
(3.32)

We note that (3.32) provides the connection between the gluon distribution
(the basic quantity of the pQCD- improved parton model) and the dipole cross
section (the basic quantity of the CDP), since $\sigma_{\gamma^{*}_{L}p}$ in
(3.32) may be replaced by the dipole cross section
$\sigma_{(q\bar{q})p}(r_{\bot},z(1-z),W^{2}=\frac{Q^{2}}{x})$, according to

$\displaystyle\sigma_{\gamma^{*}_{L}p}(W^{2}=\frac{Q^{2}}{x},Q^{2})$
$\displaystyle=$ $\displaystyle\int{dz}\int
d^{2}r_{\bot}|\Psi_{L}(r_{\bot},z(1-z),Q^{2})|^{2}$ (3.33)
$\displaystyle\times\sigma_{(q\bar{q})p}(r_{\bot},z(1-z),W^{2}=\frac{Q^{2}}{x}).$

The empirically verified low-x scaling of the photoabsorption cross section in
the variable $\eta(W^{2},Q^{2})$ translates from the photoabsorption cross
section to the gluon distribution. For sufficiently large $Q^{2}$, with
$\sigma_{\gamma^{*}_{L}p}\propto\Lambda^{2}_{sat}(W^{2})/Q^{2}$ [15],
according to (3.32) we obtain the asymptotic behavior of

$\alpha_{s}(Q^{2})G(x,Q^{2})\propto\Lambda^{2}_{sat}(W^{2}).$ (3.34)

The quantity $\Lambda^{2}_{sat}(W^{2})$, also known as saturation scale,
determines the gluon distribution function in the pQCD large-$Q^{2}$ limit.

The arguments in Section II and the present Section III may be summarized as
follows: In the CDP, at leading order of pQCD, DIS at low $x$ is
(successfully) described as interaction of the photon with the $q\bar{q}$ sea
according to (2.6). This implies that the pQCD equation (2.1) for the
longitudinal proton structure function can be approximated by the gluon
contribution, see (2.7), and, finally the gluon distribution can be determined
from (2.8). On the left-hand side, the proton structure function from the CDP
has to be substituted.

Independently from the CDP, the approximation of (2.1) by (2.8) as leading
term at large $Q^{2}$ was justified [19] by evaluating suitable models for the
gluon distribution.

Note that the pQCD equation (2.8) for its validity requires sufficiently large
values of $Q^{2}$. Since the CDP description by the proton structure functions
to be substituted into (2.8) includes the low-$Q^{2}$ limit, the use of (2.8)
contains a smooth extrapolation of the gluon density to low values of $Q^{2}$.

##  IV IV. The Froissart-bounded Representation of $F_{2}(x,Q)$.

A very accurate fit to the measured proton structure function of DIS, due to
Block et al. Block , is provided by a representation of the DIS data
satisfying the $(\log W^{2})^{2}$ bound derived from general-field-theory
principles by Froissart Froissart . Even though the determination of the gluon
distribution (2.8) contains the CDP representation for the proton structure
function, one may expect that different precise fits to the DIS data will at
least approximately lead to the same results for the gluon distribution. In
this spirit, we shall employ the Froissart-bounded representaion in addition
to the CDP representation for the proton structure function.

The results from DIS at low $x\leq 0.1$ and for a large range of $Q^{2}$ from
$0.15{\rm GeV}^{2}\leq Q^{2}\leq 3000{\rm GeV}^{2}$ are represented by the
structure function Block

$\displaystyle F_{2}(x,Q^{2})$ $\displaystyle=$ $\displaystyle
D(Q^{2})(1-x)^{n}\left[C(Q^{2})\right.$ (4.1) $\displaystyle+$ $\displaystyle
A(Q^{2})\ln\left(\frac{1}{x}\frac{Q^{2}}{Q^{2}+\mu^{2}}\right)$
$\displaystyle+$
$\displaystyle\left.B(Q^{2})\ln^{2}\left(\frac{1}{x}\frac{Q^{2}}{Q^{2}+\mu^{2}}\right)\right].$

The power $n$ and the scale $\mu^{2}$ are given by $n=11.49\pm 0.99$ and
$\mu^{2}=2.82\pm 0.290~{}{\rm GeV}^{2}$. The logarithmic dependence on $Q^{2}$
of the functions $A(Q^{2}),B(Q^{2}),C(Q^{2})$ and $D(Q^{2})$ is given in
Appendix B and Table B1 reproducing Table II from ref. Block .

In distinction from the CDP, the Froissart-bounded fit does not explicitly
provide a representation of the longitudinal-to-transverse ratio $R$. When
deducing the gluon distribution function, the value of $R$ from the CDP will
have to be adopted.

##  V V. The Results for the Gluon Distribution Function.

We turn to the explicit results for the gluon distribution which follow from
the evaluation of (2.8) upon substitution of the representation of the DIS
experimental results in the CDP and the Froissart-bounded representation, as
given in Sections III and IV, respectively.

Substitution into (2.8) of the CDP results (3.12) with (3.2) for
$F_{L}(W^{2},Q^{2})$ yields

$\displaystyle\alpha_{s}(Q^{2})G(x,Q^{2})=\frac{3\pi}{\sum_{q}Q^{2}_{q}}F_{L}\left(\xi_{L}x,Q^{2}\right)=\frac{9Q^{2}}{4\pi\alpha
R_{e^{+}e^{-}}}$ $\displaystyle\times\left(\frac{\sigma_{\gamma
p}(W^{2})}{\left(\ln\frac{\rho}{\mu}\right)G_{T}\left(\frac{\xi}{\eta}\right)}I_{L}^{(1)}(\eta,\mu)G_{L}\left(\frac{\xi}{\eta}\right)\right)\Bigg{|}_{W^{2}\to\xi_{L}^{-1}W^{2}}.$
(5.1)

The variables $\eta=\eta(W^{2},Q^{2})$ and $\mu(W^{2})$ are given by (see
(3.8), (3.9)),

$\displaystyle\eta(W^{2},Q^{2})$ $\displaystyle=$
$\displaystyle\frac{Q^{2}+m^{2}_{0}}{\Lambda^{2}_{sat}(W^{2})},$
$\displaystyle\mu(W^{2})$ $\displaystyle=$
$\displaystyle\frac{m^{2}_{0}}{\Lambda^{2}_{sat}(W^{2})},$ (5.2)

where

$\Lambda^{2}_{sat}(W^{2})=C_{1}(W^{2})^{C_{2}}=0.31\left(\frac{W^{2}}{1{\rm
GeV}^{2}}\right)^{0.29}~{}{\rm GeV}^{2},$ (5.3)

with $C_{2}=0.29$ from (3.20) and

$m^{2}_{0}=0.15~{}{\rm GeV}^{2}.$ (5.4)

For the assumed number of four flavors,

$R_{e^{+}e^{-}}\equiv 3\sum_{q}Q^{2}_{q}=\frac{10}{3}.$ (5.5)

The parameter $\rho$, denoting the size enhancement of transversely relative
to longitudinally polarized $q\bar{q}$ fluctuations, is given by [24]

$\rho=\frac{4}{3}.$ (5.6)

The parameter $\xi$ that stands for the upper limit,
$\xi\Lambda^{2}_{sat}(W^{2})$, of masses of $q\bar{q}$ fluctuations actively
contributing, is given by $\xi=130$. At the relevant (low) values of
$\eta(W^{2},Q^{2})$, the limit of $\xi\to\infty$ and
$G_{T}\left(\frac{\xi}{\eta}\right)=G_{L}\left(\frac{\xi}{\eta}\right)=1$ (see
(3.4) and (3.5)) may be taken without loss of generality, compare Fig. 1. The
$\log(W^{2})$ fit to the photoproduction experimental results is given by
(3.7), and, finally, the shift factor $\xi_{L}$ according to (2.8) is given by

$\xi_{L}=0.4.$ (5.7)

The essential effect of the shift $W^{2}\to\xi_{L}^{-1}W^{2}$, in (5.1)
amounts to

$\eta(W^{2},Q^{2})\to\xi_{L}^{C_{2}}\eta(W^{2},Q^{2}),$ (5.8)

compare (3.30), the $Q^{2}=0$ photoproduction term in (5.1), including its
normalization by the denominator in (5.1), being hardly affected due to its
weak dependence on $W^{2}$.

According to (2.8), the gluon distribution may equivalently be expressed in
terms of the structure function $F_{2}\left(\xi_{L}x,Q^{2}\right)$ instead of
$F_{L}\left(\xi_{L}x,Q^{2}\right)$, together with the longitudinal-to-
transverse ratio $R$. For sufficiently large $Q^{2}$, according to (2.8) and
(3.18), with $R=1/2\rho$,

$\alpha_{s}(Q^{2})G(x,Q^{2})=\frac{9\pi}{R_{e^{+}e^{-}}}\frac{1}{2\rho+1}F_{2}\left(\xi_{L}x,Q^{2}\right).$
(5.9)

The representation (5.9), even at small $Q^{2}$ may be used as an
approximation of the gluon distribution.

For $Q^{2}$ sufficiently large, specifically for $20~{}{\rm
GeV}^{2}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}Q^{2}\mathrel{\raise
1.29167pt\hbox{$<$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}100~{}{\rm
GeV}^{2}$, the large-$Q^{2}$ limit of (5.1) given by (3.19) becomes relevant.
It implies the simple large-$Q^{2}$ representation of

$\displaystyle\alpha_{s}(Q^{2})G(x,Q^{2})=\frac{9\pi}{R_{e^{+}e^{-}}}\frac{1}{(2\rho+1)}F_{2}(\xi_{L}x,Q^{2})$
(5.10) $\displaystyle=$
$\displaystyle\frac{9\pi}{R_{e^{+}e^{-}}}\frac{1}{(2\rho+1)}f_{2}\xi_{L}^{-0.29}\left(\frac{W^{2}}{1{\rm
GeV}^{2}}\right)^{C_{2}=0.29},$

where $f_{2}=0.063$, and $\rho=\frac{4}{3}$.

Figure 3: The gluon distribution
$\alpha_{s}(Q^{2})xg(x,Q^{2})\equiv\alpha_{s}(Q^{2})G(x,Q^{2})$ of the CDP,
compare (5.1), as a function of $W^{2}$ for various values of $Q^{2}$. The
solid line shows the asymptotic limit (5.10) that is reached at
$Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}30{\rm GeV}^{2}$.

In Fig. 3, we show the gluon distribution (2.8) deduced from
$F_{L}(W^{2},Q^{2})$ according to (5.1) as a function of $W^{2}$ for various
values of $Q^{2}$, where $1{\rm GeV}^{2}\leq Q^{2}\leq 100{\rm GeV}^{2}$. The
results in Fig. 3, for $Q^{2}$ sufficiently above $Q^{2}\cong
10{\rm~{}GeV}^{2}$, indeed converge towards the asymptotic representation
(5.10).

Figure 4: As Fig. 3, but based on (5.9), employing $R=const=1/2\rho=3/8$.
Compare text for details.

The results in Fig. 4, for low $Q^{2}$, $1{\rm GeV}^{2}\leq Q^{2}\leq 10{\rm
GeV}^{2}$, compared with the results in Fig. 3, show the enhanced gluon
distribution resulting from employing the large-$Q^{2}$ approximation (5.9).
For sufficiently large $Q^{2}$, the results in Fig. 4, based on
$F_{2}\left(\xi_{L}x,Q^{2}\right)$ according to (3.18) and (5.9), coincide
with the ones in Fig. 3 based on $F_{L}\left(\xi_{L}x,Q^{2}\right)$ according
to (5.1).

Figure 5: The gluon-distribution function $\alpha_{s}(Q^{2})xg(x,Q^{2})$ of
the CDP as a function of $x\cong Q^{2}/W^{2}$ at the (low) value of
$Q^{2}=1.9~{}{\rm GeV}^{2}$, the scale frequently used as input scale Pelicer
. The solid curve is due to $C_{1}=0.31$ and $C_{2}=0.29$ and the
uncertainties are due to $C_{1}=0.34{\pm}0.05$ and $C_{2}=0.27{\pm}0.01$
Schild  (G. Cvetic, D. Schildknecht, B. Surrow, M. Tentyukov, Eur. Phys. J. C
20, 77 (2001)). Figure 6: As Fig. 5, but for $xg(x,Q^{2})$ instead of
$\alpha_{s}(Q^{2})xg(x,Q^{2})$. The CDP results are compared with the results
from ref.[6] and the parametrization methods at the NLO approximation, CJ12 CJ
, NNPDF3.0 NNPDF  and MMHT14 MMHT  as accompanied with total errors.

For Figs. 5 and 6, the energy variable, $W$, of the CDP is replaced by
$x\simeq Q^{2}/W^{2}$. The increase of $x$ for $Q^{2}=1.9~{}{\rm GeV}^{2}$ in
the range of $10^{-4}\leq x\leq 10^{-1}$ corresponds to an increase of
$\eta(W^{2},Q^{2}=1.9~{}{\rm GeV}^{2})$ according to $0.35\leq\eta\leq 2.6$,
with $W^{2}$ decreasing according to $1.9\times 10^{4}{\rm GeV}^{2}\geq
W^{2}\geq 19{\rm GeV}^{2}$. Taking into account the proportionality of the
gluon distribution to the longitudinal photoabsorption cross section (3.32)
and consulting the results in Fig. 1, we expect a (hadronlike) increase of the
gluon distribution by a factor of approximately about 3.5 with decreasing $x$
in the interval $10^{-1}\geq x\geq 10^{-4}$. This factor of 3.5 for
$x=10^{-4}$ is seen in Figure 5.

Multiplication of the results in Fig. 5 by$\alpha_{s}(Q^{2}=1.9~{}{\rm
GeV}^{2})^{-1}=0.480^{-1}=2.083$ yields the results for $xg(x,Q^{2})\equiv
G(x,Q^{2})$ that are shown in Fig. 6 888The value of
$\alpha_{s}(Q^{2}=1.9~{}{\rm GeV}^{2})=0.480$ is obtained for
$\Lambda_{QCD}=437{\rm MeV}$ corresponding to $\alpha_{s}(M^{2}_{Z})=0.118.$.

The fairly small error bars of our results for the gluon distribution in Fig.
5 are a reflection of the small errors of the original fit to the experimental
data of DIS.

The comparison of our CDP results for the gluon distribution in Fig. 6 and
Fig. 10 below with published distributions reveals a significantly different
shape of the $x$ distribution at fixed $Q^{2}=1.9{\rm GeV}^{2}$ and a large
deviation in absolute normalization outside the total errors of these
distributions. Compare also the comment in the Conclusions on the
interpretation of the observed strong deviations.

In Fig. 6, we compare with the results obtained in ref. Pelicer  upon
introducing absorptive corrections to the measured proton structure functions,
and a modification of $\alpha_{s}(Q^{2})$ by $\alpha_{s}(Q^{2}+\mu^{2}_{0})$.
The gluon distributions from ref. Pelicer  in Fig. 6 are given by Pelicer

$\displaystyle
xg(x)=5.63x^{0.103}(1-x)^{10.261}~{}\mathrm{for}~{}\mu_{0}=0~{}\mathrm{GeV},$
$\displaystyle
xg(x)=4.21x^{0.021}(1-x)^{9.427}~{}\mathrm{for}~{}\mu_{0}=1~{}\mathrm{GeV}.$
(5.11)

As seen in Fig. 6, the results from ref. Pelicer  are drastically different
from ours 999Employing (2.8), we find that $F_{L}(x,Q^{2})$ resulting by
inserting the gluon distribution (5.11) differs from the CDP $F_{L}(x,Q^{2})$
by the same factor as observed for the gluon distribution, see Fig.6.. The
results obtained by exploiting the smooth low-$Q^{2}$ transition for
$F_{L}(x,Q^{2})$ of the CDP by substitution into the pQCD-improved parton
model do not support the modifications of evolution suggested and applied to
DIS in Ref.[6]. For further discussions on the consequences of our approach of
incorporating the CDP representation for $F_{L}(x,Q^{2})$ into pQCD, see
sections VI and VII.

For completeness, in Fig.7, we show the CDP gluon distribution for $Q^{2}$
between $Q^{2}=2~{}\textrm{GeV}^{2}$ and $Q^{2}=100~{}\textrm{GeV}^{2}$.

Figure 7: The CDP gluon distribution $xg(x,Q^{2})$ in a wide range of $Q^{2}$
values, $2~{}\textrm{GeV}^{2}{\leq}Q^{2}{\leq}100~{}\textrm{GeV}^{2}$

The analysis of the gluon distribution function that led to the results in
Figs. 3 to 7 is based on the representation of the proton structure functions
in the CDP. We expect that other precise and theoretically well-founded
representations (not manifestly including $q\bar{q}$ two-gluon couplings ) of
the measured structure functions will lead to (at least approximately)
identical results for the gluon distribution. Explicitly this equivalence is
tested by the example of employing the fit to the proton structure function in
the Froissart-bounded representation101010We evaluate the gluon distribution
from the pQCD approximation (2.8) as well as (6.5) below, in distinction from
the approach in ref. Block  that is based on an exact determination of
$G(x,Q^{2})$ Durand  upon converting (VI) below into an inhomogeneous second
order differential equation for $G(x,Q^{2})$. from ref. Block , described in
Section IV.

Since the theoretical representation in this approach does not isolate the
longitudinal structure function, the gluon distribution must be deduced from
$F_{2}\left(\xi_{L}x,Q^{2}\right)$ with $\rho=\frac{4}{3}$ in (5.9).

Figure 8: The gluon distribution $\alpha_{s}(Q^{2})xg(x,Q^{2})$ deduced from
the Froissart-bounded representation of the proton structure function
$F_{2}(x,Q^{2})$ given in ref. Block

The results in Fig. 8 for large $Q^{2}$, asymptotically, agree with the CDP
result. For $Q^{2}\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}10~{}{\rm GeV}^{2}$, there are acceptable deviations,
seen upon comparing the results in Fig. 8 with the ones in Figs. 3 and 4.

Figure 9: The gluon-distribution function $\alpha_{s}(Q^{2})xg(x,Q^{2})$ as a
function of $x\cong Q^{2}/W^{2}$ at $Q^{2}=1.9~{}{\rm GeV}^{2}$ from the CDP,
and from Froissart-bounded representations in refs. Block , Durand  and Block-
Durand .  Figure 10: Same as Fig. 9 except for showing $xg(x,Q^{2})$ instead
of $\alpha_{s}(Q^{2})xg(x,Q^{2})$. The CDP results and the results from the
Froissart-bounded representation are compared with the results from ref.
Pelicer  and with the results CJ12 ref. CJ  and NNPDF3.0 ref. NNPDF  at the LO
approximation as accompanied with total errors.

In Figs. 9 and 10, respectively, we present the comparison of the gluon
distribution from the Froissart-bounded representation of $F_{2}(x,Q^{2})$
with the results based on the CDP, and with the results from ref. Pelicer  at
$Q^{2}=1.9~{}{\rm GeV}^{2}$ based on pQCD modified by power law corrections
and absorptive effects. In addition to the results extracted from the fit to
the proton structure function described in Section IV., in Figs. 9 and 10, we
have employed Block  two additional somewhat different fits from refs. Durand
; Block-Durand  without entering into a detailed description of these fits.
The CDP result in this figure (i.e., Fig.10) are compared with the CJ12 ref.
CJ  and NNPDF3.0 ref. NNPDF  at the LO approximation. These parametrization
methods are accompanied with total errors.

By comparison of the results on the gluon distribution deduced from the
representation of the body of DIS experimental data of the proton structure
functions in the CDP, and the Froissart-bounded approach, at $Q^{2}=1.9~{}{\rm
GeV}^{2}$, we find a consistent behavior as expected.

The shape and the absolute magnitude of the CDP gluon distribution at
$Q^{2}=1.9~{}{\rm GeV}^{2}$ is significantly different from all published
results from several collaborations.

##  VI VI. The Consistency of pQCD and CDP evolution at large $Q^{2}$

In this Section, we examine the consistency of the $Q^{2}$ evolution thus
obtained for the gluon distribution from the CDP, with the $Q^{2}$ dependence
predicted from pQCD. To order $\alpha_{s}(Q^{2})$, only process (2.5)
contributes, and accordingly we can restrict ourselves to investigating the
consistency of the $Q^{2}$ dependence from the CDP with the $Q^{2}$ dependence
from the first DGLAP evolution equation Lipatov .

The change of $F_{2}(x,Q^{2})$ with $Q^{2}$, its evolution at large $Q^{2}$,
is given by Lipatov

$\displaystyle\frac{\partial}{\partial\ln
Q^{2}}F_{2}(x,Q^{2})=\frac{\alpha_{s}(Q^{2})}{2\pi}\int^{1}_{x}dzP_{qq}(z)F_{2}\left(\frac{x}{z},Q^{2}\right)$
$\displaystyle+\frac{R_{e^{+}e^{-}}}{3\pi}\alpha_{s}(Q^{2})\int^{1-x}_{0}dzP_{qg}(z)G\left(\frac{x}{1-z},Q^{2}\right),~{}~{}~{}~{}~{}$
(6.1)

where

$P_{qq}(z)=\frac{4}{3}\left(\frac{1+x^{2}}{(1-x)_{+}}+\frac{3}{2}\delta(1-x)\right),$
(6.2)

and

$P_{qg}(z)=\frac{1}{2}\left(z^{2}+(1-z)^{2}\right).$ (6.3)

According to our basic assumption (2.7), saying that the photon exclusively
interacts with $q\bar{q}$ pairs originating from a gluon, $g\to q\bar{q}$ see
(2.6), we exclude the first term on the right-hand side in (6.1) to obtain
Prytz

$\displaystyle\frac{\partial F_{2}(x,Q^{2})}{\partial\ln
Q^{2}}=\frac{R_{e^{+}e^{-}}}{3\pi}\int^{1-x}_{0}dzP_{qg}(z)G\left(\frac{x}{1-z},Q^{2}\right).$
(6.4)

Exploiting the symmetry of $P_{qg}(z)$ in (6.4) around $z=1/2$, the first
derivative of $G(x/(1-z),Q^{2})$ in a Taylor expansion of $G(x/(1-z),Q^{2})$
around $z=\frac{1}{2}$ yields a vanishing contribution to the integral in
(6.4), and approximately Prytz

$\frac{\partial F_{2}(x,Q^{2})}{\partial\ln
Q^{2}}=\frac{R_{e^{+}e^{-}}}{9\pi}\alpha_{s}(Q^{2})G\left(\frac{x}{\xi_{2}},Q^{2}\right).$
(6.5)

with $x/\xi_{2}=2x$, or $\xi_{2}=1/2$. Replacing the gluon distribution in
(6.5) by its proportionality (2.8) to the structure function $F_{L}$, (6.5)
becomes

$\frac{\partial F_{2}(x,Q^{2})}{\partial\ln
Q^{2}}=F_{L}\left(\frac{\xi_{L}}{\xi_{2}}x,Q^{2}\right)=\frac{1}{2\rho+1}F_{2}\left(\frac{\xi_{L}}{\xi_{2}}x,Q^{2}\right),$
(6.6)

where the second equality is due to (3.18).

So far no specific ansatz for the proton structure function was used.
Introducing the large-$Q^{2}$ power-law of the CDP (3.21) with (3.10),

$F_{2}(x,Q^{2})\sim(W^{2})^{C_{2}}=\left(\frac{Q^{2}}{x}\right)^{C_{2}},$
(6.7)

from (6.6) we obtain the important constraint

$C_{2}\left(\frac{Q^{2}}{x}\right)^{C_{2}}=\frac{1}{2\rho+1}\left(\frac{Q^{2}}{x}\right)^{C_{2}}\left(\frac{\xi_{2}}{\xi_{L}}\right)^{C_{2}},$
(6.8)

or

$C_{2}(2\rho+1)\left(\frac{\xi_{L}}{\xi_{2}}\right)^{C_{2}}=1.$ (6.9)

Expansion of the exponential to first order in $C_{2}$ yields the convenient
expression for $C_{2}$,

$C_{2}\cong\frac{1}{2\rho+1}\frac{1}{\left(1-\frac{1}{2\rho+1}\ln\frac{\xi_{2}}{\xi_{L}}\right)}.$
(6.10)

For $\xi_{2}/\xi_{L}=0.5/0.4=1.25$, with $\rho=\frac{4}{3}$, one finds
$C_{2}\cong 0.29$ for the exponent $C_{2}$ in (6.9).

The prediction of $C_{2}\cong 0.29$ is based on the assumption that the
evolution in $Q^{2}$ is entirely due to $q\bar{q}$ pairs originating from
gluons $g\to q\bar{q}$, see (6.4) and (6.5). The agreement of the prediction
of $C_{2}{\cong}~{}0.29$ from (6.9) with the experimentally determined value,
given in (3.20), provides empirical evidence for, or confirms the
approximation of reducing the complete evolution equation (6.1) to the reduced
form (6.5) employed to arrive at (6.9). The incoming photon dominantly
interacts with $q\bar{q}$ pairs originating from the $g\to q\bar{q}$
transition, see (2.6).

We conclude: the CDP with the power-law ansatz of
$F_{2}(x,Q^{2})\sim(Q^{2}/x)^{C_{2}}$ for large $Q^{2}$ fulfills evolution
according to (6.4) with $Q^{2}$ at large $Q^{2}$, and it implies the
empirically successful prediction of $C_{2}\cong 0.29$.

##  VII VII. Evolution at low $Q^{2}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}1.9{\rm GeV}^{2}$.

In considering the validity of the pQCD evolution equation (6.5) for
$F_{2}(x,Q^{2})$, we so far restricted ourselves to large values of $Q^{2}$,
sufficiently large with respect to $\Lambda^{2}_{sat}(W^{2})$, effectively,
$Q^{2}\mathrel{\raise 1.29167pt\hbox{$>$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}10{\rm GeV}^{2}$ to $20{\rm GeV}^{2}$. Employing the
large-$Q^{2}$-representation for
$F_{2}(x,Q^{2})\sim\left(\frac{Q^{2}}{x}\right)^{C_{2}}$, the validity of the
evolution equation (6.6) implied the restriction (6.9) that predicts the
exponent $C_{2}$ in $\Lambda^{2}_{sat}(W^{2})\sim(W^{2})^{C_{2}}$ in terms of
the transverse-$q\bar{q}$-size parameter $\rho$, where $\rho$ has the
preferred value [24] of $\rho=4/3$.

In the present Section, we lift the restriction on the magnitude of $Q^{2}$ by
allowing for values of $Q^{2}$ as low as $Q^{2}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}1.9{\rm GeV}^{2}$,
the value frequently adopted [6] when applying pQCD to electron-proton DIS.

The CDP proton structure functions entering (6.6) according to (LABEL:eq:3.1),
(3.2) and (3.12), without restriction to large $Q^{2}$ are given by

$\displaystyle F_{2}(x,Q^{2})\equiv
F_{2}\left(\eta\left(W^{2}=\frac{Q^{2}}{x},Q^{2}\right),\mu\left(W^{2}=\frac{Q^{2}}{x}\right)\right)$
$\displaystyle=\frac{Q^{2}}{4\pi^{2}\alpha}\frac{\sigma_{\gamma
p}(W^{2})}{\ln\frac{\rho}{\mu(W^{2})}}\left(I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)G_{T}(u)+I^{(1)}_{L}(\eta,\mu)G_{L}(u)\right),$
(7.1)

and

$\displaystyle F_{L}(x,Q^{2})$ $\displaystyle\equiv
F_{L}\left(\eta\left(W^{2}=\frac{Q^{2}}{x},Q^{2}\right),\mu\left(W^{2}=\frac{Q^{2}}{x}\right)\right)$
(7.2) $\displaystyle=\frac{Q^{2}}{4\pi^{2}\alpha}\frac{\sigma_{\gamma
p}(W^{2})}{\ln\frac{\rho}{\mu(W^{2})}}I_{L}^{(1)}G_{L}(u).$

The fit to the experimental data of $\sigma_{\gamma p}(W^{2})$ is given by
(3.7). For the explicit expressions for $I_{T}^{(1)}$ and $I_{L}^{(1)}$ in
(7.1) and (7.2), we refer to Appendix A. In the present context we can
restrict ourselves to the range of $\eta\ll\xi=130$, and accordingly
$G_{L,T}(u)=1$ in (7.1) and (7.2), see (3.4) and (3.5).

Adopting (3.10),

$\Lambda^{2}_{sat}(W^{2})=C_{1}\left(\frac{W^{2}}{1{\rm
GeV}^{2}}\right)^{C_{2}},~{}~{}~{}W^{2}=\frac{Q^{2}}{x},$ (7.3)

the low-$x$ scaling variable $\eta(W^{2},Q^{2})$ from (3.8), expressed in
terms of the parton variables $x$ and $Q^{2}$, becomes

$\eta\equiv\eta(W^{2}=\frac{Q^{2}}{x},Q^{2})=\frac{x^{C_{2}}}{C_{1}}\frac{(Q^{2}+m^{2}_{0})}{(Q^{2})^{C_{2}}},$
(7.4)

and $\mu(W^{2})$ from (3.9) is given by

$\mu\equiv\mu(W^{2}=\frac{Q^{2}}{x})=\frac{x^{C_{2}}}{C_{1}}\frac{m^{2}_{0}}{(Q^{2})^{C_{2}}}.$
(7.5)

The numerical value of the exponent $C_{2}$, for a given value of $\rho$ is
fixed by the large-$Q^{2}$ constraint (6.9).

Turning to the examination of the validity of the evolution equation for
$F_{2}(x,Q^{2})$ in the form of the first equality in (6.6) relating the
logarithmic derivative of $F_{2}(x,Q^{2})$ to the longitudinal structure
function $F_{L}(x,Q^{2})$, we introduce the ratio

$\displaystyle{\rm Ratio}$ $\displaystyle\equiv\frac{\frac{\partial
F_{2}(x,Q^{2})}{\partial\ln
Q^{2}}}{F_{L}\left(\frac{\xi_{L}}{\xi_{2}}x,Q^{2}\right)}$
$\displaystyle=1+\Delta\left(\eta\left(W^{2}=\frac{Q^{2}}{x},Q^{2}\right),\mu\left(W^{2}=\frac{Q^{2}}{x}\right)\right),$

where, according to the validity of (6.6) under constraint (6.9), we have
$\Delta(\eta,\mu)\cong 0$ for sufficiently large $Q^{2}\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}10{\rm GeV}^{2}$.
Deviations from the validity of the evolution equation (6.5) are accordingly
parametrized by $\Delta(\eta,\mu)$ according to (VII).

Note that $\Delta(\eta,\mu){\neq}0$ in (7.6) may be due to either a
consequence of a violation of the evolution equation (6.5), under validity of
the relation between the gluon distribution and the longitudinal structure
function (2.8), or else to a violation of (2.8), not excluding both cases
simultanusely. In any case, $\Delta(\eta,\mu){\neq}0$ implies violation of the
underlying parton model. At low $\eta$, the interaction is dominantly due to
the interaction of low-lying $q\bar{q}$ vector states with the gluon field in
the proton, or, equivalently (see(2.6)) to the interaction of the photon with
low-lying $q\bar{q}$ vector states and not with freely moving quarks and
gluons. Nevertheless, one may formally define a gluon distribution function
according to (2.9) and (3.32).

From (7.1), the derivative of $F_{2}(x,Q^{2})$ is obtained as

$\displaystyle\frac{\partial F_{2}(x,Q^{2})}{\partial\ln Q^{2}}$
$\displaystyle=$
$\displaystyle\frac{Q^{2}}{4\pi^{2}\alpha}\frac{\sigma_{\gamma
p}(W^{2})}{\ln\frac{\rho}{\mu(W^{2})}}\left[\left(I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)+I_{L}^{(1)}(\eta,\mu)\right)\right.$
(7.7) $\displaystyle\times$
$\displaystyle\left(1+\frac{\ln\frac{\rho}{\mu(W^{2})}}{\sigma_{\gamma
p}(W^{2})}\frac{\partial}{\partial\ln W^{2}}\frac{\sigma_{\gamma
p}(W^{2})}{\ln\frac{\rho}{\mu(W^{2})}}\right)$ $\displaystyle+$
$\displaystyle\left.\frac{\partial}{\partial\ln
Q^{2}}\left(I^{(1)}_{T}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)+I_{L}^{(1)}(\eta,\mu)\right)\right],$

and the denominator of (VII) is given by

$F_{L}\left(\frac{\xi_{L}}{\xi_{2}}x,Q^{2}\right)=\frac{Q^{2}}{4\pi^{2}\alpha}\frac{\sigma_{\gamma
p}\left(\frac{\xi_{2}}{\xi_{L}}W^{2}\right)}{\ln\frac{\rho}{\mu\left(\frac{\xi_{2}}{\xi_{L}}W^{2}\right)}}I_{L}^{(1)}\left(\frac{\xi_{L}}{\xi_{2}}\eta,\frac{\xi_{L}}{\xi_{2}}\mu\right).$
(7.8)

In connection with the derivatives appearing in (7.7), we mention the equality
$\frac{\partial}{\partial\ln W^{2}}=\frac{\partial}{\partial\ln Q^{2}}$, and
moreover, we note the derivatives

$\displaystyle Q^{2}\frac{\partial\eta}{\partial Q^{2}}$
$\displaystyle=(1-C_{2})\eta-\mu,$ $\displaystyle
Q^{2}\frac{\partial\mu}{\partial Q^{2}}$ $\displaystyle=-C_{2}\mu.$ (7.9)

Since $\sigma_{\gamma p}(W^{2})$ only weakly depends on $W^{2}$, one may
approximate “Ratio” defined in (VII) by ignoring the derivative of
$\sigma_{\gamma p}(W^{2})$ in (7.7). Explicitly, Ratio from (VII)
becomes111111In (7.10), the shift $\xi_{2}/\xi_{L}W^{2}$ in $\sigma_{\gamma
p}\left(\frac{\xi_{2}}{\xi_{L}}W^{2}\right)/\ln\frac{\rho}{\mu\left(\frac{\xi_{2}}{\xi_{L}}W^{2}\right)}$
from (7.8) is ignored.

$\displaystyle
Ratio=\frac{1}{I^{(1)}_{L}\left(\frac{\xi_{L}}{\xi_{2}}\eta,\frac{\xi_{L}}{\xi_{2}}\mu\right)}$
(7.10)
$\displaystyle\times\left[I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)+I^{(1)}_{L}(\eta,\mu)\right.$
$\displaystyle\left.+\frac{\partial}{\partial\ln
Q^{2}}\left(I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)+I_{L}^{(1)}(\eta,\mu)\right)\right]$
$\displaystyle=\frac{1}{I^{(1)}_{L}\left(\frac{\xi_{L}}{\xi_{2}}\eta,\frac{\xi_{L}}{\xi_{2}}\mu\right)}\times\left[I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)+I^{(1)}_{L}(\eta,\mu)\right.$
$\displaystyle\left.+[(1-C_{2})\eta-\mu]\frac{\partial}{\partial\eta}\left(I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)+I_{L}^{(1)}(\eta,\mu)\right)\right].$

Moreover, since $\mu(W^{2})=m^{2}_{0}/\Lambda^{2}_{sat}(W^{2})$, with
$m^{2}_{0}=0.15{\rm GeV}^{2}$, for sufficiently large $W^{2},\mu(W^{2})$ is
small compared to unity, $\mu(W^{2})\ll 1$, and the approximation of
$I_{L}^{(1)}$ and $I_{T}^{(1)}$ in terms of $I_{0}(\eta)$ Kuroda  given by
(A3) to (A5) in Appendix A may be inserted when numerically evaluating Ratio
in (VII) and (7.10).

Making use of

$\frac{d}{d\eta}I_{0}(\eta)=\frac{-2}{1+4\eta}\left(I_{0}(\eta)-\frac{1}{2\eta}\right)$
(7.11)

together with (7.9), Ratio in (7.10) becomes a function that depends on
$\eta$, $\mu$ and $I_{0}(\eta)$. The different dependence of $I_{0}(\eta)$ on
$\eta$ at large $\eta\gg 1$, where

$I_{0}(\eta)\cong\frac{1}{2\eta}\left(1-\frac{1}{6\eta}\right)+0\left(\frac{1}{\eta^{3}}\right),$
(7.12)

and at small $\eta$ of $\mu\leq\eta\ll 1$, where

$I_{0}(\eta)\cong\ln\frac{1}{\eta}\left(1-2\eta\left(1-\frac{1}{\ln\frac{1}{\eta}}\right)\right)+...,$
(7.13)

determines the different behavior of the numerical results for Ratio to be
presented next.

Figure 11: The derivative of the structure function
$F_{2}(\eta(W^{2}=Q^{2}/x),Q^{2})$ and the longitudinal structure function
$F_{L}(\eta^{*}=(\frac{\xi_{L}}{\xi_{2}})^{C_{2}}\eta,Q^{2})$ together with
the ratio of both of them. The results are based on evaluating (7.7) and
(7.8).

For the numerical evaluation of Ratio from (VII) with (7.7) and (7.8)
inserted, and of Ratio explicitly given in (7.10), we use the
parameters121212We add the comment that only the value of $C_{1}=0.31{\rm
GeV}^{2}$ is an entirely free fit parameter, whereas $\rho=\frac{4}{3}$ is a
consequence of the enhanced transverse size of $q\bar{q}$ states originating
from transversely relative to longitudinally polarized photons, and the
exponent $C_{2}=0.29$ is accordingly fixed by (6.9). The value of
$m^{2}_{0}=0.15{\rm GeV}^{2}<m^{2}_{\rho}$ is the starting point of the
effective $q\bar{q}$ continuum, including the $\rho^{0},\omega,\phi$ peaks. We
add the remark that $\sqrt{0.31}\rm{GeV}{\cong}0.57\rm{GeV}$ may be associated
with the level spacing between the vector mesons $\rho^{0}$ and
$\acute{\rho}^{0}$.

$\rho=\frac{4}{3},~{}C_{2}=0.29,~{}C_{1}=0.31~{}{\rm
GeV}^{2},~{}m^{2}_{0}=0.15{\rm GeV}^{2}.$ (7.14)

We also note $\xi_{L}=0.4$ (see (2.8)) and $\xi_{2}=0.5$ (see (6.5)).

In Fig. 11, we show the derivative of the structure function as well as the
longitudinal structure function according to (7.7) and (7.8) respectively, and
their ratio defined by (VII). For $W^{2}$, the value of $W^{2}=10^{5}~{}{\rm
GeV}^{2}$ was used 131313As a consequence of the dominant dependence on
$\eta(W^{2},Q^{2})$, the absolute value of $W^{2}$ in only relevant with
respect to $\sigma_{\gamma p}(W^{2})$ and $\mu(W^{2})$ in (7.7) and (7.8). .

Figure 12: As Fig. 11, however using the approximation $\sigma_{\gamma
p}(W^{2})\cong{\rm const}$, compare to (7.10).  Figure 13: The deviation from
validity of the evolution equation obtained according to Figs. 11 and 12,
respectively.

Since $\sigma_{\gamma p}(W^{2})$ depends weakly on $W^{2}$, we expect the
numerical results of Fig. 11 not being drastically affected, if in (7.7) the
term proportional to the derivative of $\sigma_{\gamma p}(W^{2})$ is put to
zero. The corresponding value of Ratio is explicitly given by (7.10).
Comparison of the results in Fig. 12 with the ones in Fig. 11 indeed confirms
that Ratio can reliably be evaluated according to the simple expression (7.10)
upon insertion of $I_{T}^{(1)}$ and $I_{L}^{(1)}$ from (A.3) to (A.5).

In Fig. 13, we show the deviation
$\Delta\left(\eta\left(W^{2}=\frac{Q^{2}}{x}\right),\mu(W^{2})\right)$ of
Ratio in (VII) from unity on a linear scale. Remember, a value of unity means
strict validity of the first DGLAP evolution equation (6.1) in the
approximation (6.4) and (6.5). A deviation smaller than
$\Delta\left(\eta\left(W^{2}=\frac{Q^{2}}{x}\right),~{}\mu(W^{2})\right)\cong
0.1$ to $0.2$ according to Fig. 13 requires $\eta(W^{2},Q^{2})\mathrel{\raise
1.29167pt\hbox{$>$\kern-7.5pt\lower 4.30554pt\hbox{$\sim$}}}5$. The weaker
increase, and finally the decrease of $F_{L}(\eta(x,Q^{2}),Q^{2})$ relative to
the derivative of $F_{2}(\eta(x,Q^{2}),Q^{2})$ with decreasing
$\eta(x,Q^{2})$, or decreasing $x$ at fixed $Q^{2}$, see (7.4), implies the
substantial violation of the evolution equation reaching
$\Delta(\eta(W^{2},Q^{2}),\mu(W^{2}))\cong 2.7$ at $\eta(W^{2},Q^{2})\cong
0.1$.

Figure 14: The extension of Fig. 11 to the $Q^{2}\to 0$ limit of
$\eta(W^{2},Q^{2})\to\mu(W^{2})$ for $W^{2}=10^{5}{\rm GeV}^{2}$

Fig. 14 explicitly shows that the increase of Ratio relative to unity with
decreasing $\eta$ is strongly correlated to the necessary vanishing of the
longitudinal structure function, $F_{L}(\eta(W^{2},Q^{2}),\mu(W^{2}))$, in the
denominator of Ratio in the limit of $\eta(W^{2},Q^{2})\to\mu(W^{2})$ for
$Q^{2}\to 0$.

Figure 15: The $(Q^{2},x)$ plane, showing the regions of $\eta(W^{2},Q^{2})>5$
(approximate validity of conventional evolution) with $\Delta(\eta,\mu)\cong
0$, and of $\eta(W^{2},Q^{2})<5$, where $\Delta(\eta,\mu)\not=0$ occurs.

The value of $\eta\cong 5$ quantifies the region in the $(Q^{2},x)$ plane, see
Fig. 15, below which the evolution equation becomes violated. It is amusing to
note the coincidence of this value with the value of $\eta\cong 5$, where the
behavior of $\sigma_{\gamma^{*}p}\sim 1/\eta$ of the CDP in Fig. 1 turns from
the region of color transparency to the region of hadronlike behavior. Above
$\eta\cong 5$, we have pQCD-evolution or CDP-color-transparency, while below
$\eta\cong 5$, we have a violation of conventional evolution that is
equivalent to hadronlike saturation.

Figure 16: The effect of the variation of $\rho$.

In terms of $x$ and $Q^{2}$, according to Fig.15, at fixed value $Q^{2}$ with
decreasing $x$, or at fixed value of $x$ with decreasing $Q^{2}$, the region
of hadronlike behavior is reached that in connected with a violation of
standard evolution.

In Fig. 16, we address the question of the dependence of the above conclusion
on the value of $\rho$ that specifies the asymptotic behavior of the
longitudinal-to-transverse ratio in photoabsorption via $R=1/2\rho$. As long
as precision experimental data on $R$, and accordingly on $\rho$, are lacking,
it is interesting to vary $\rho$ around its preferred value of $\rho=4/3$,
implying different values of the exponent $C_{2}$ according to (6.9). From
Fig. 16, we infer that a change of $\rho$ does qualitatively not significantly
change our general conclusion obtained for
$\Delta(\eta(W^{2}=Q^{2}/x),Q^{2})$.

In Fig.17, we show Ratio according to (7.10) as a function of $Q^{2}$ at fixed
$x$, upon replacing $\eta$ and $\mu$ in terms of $x$ and $Q^{2}$ according to
(7.4) and (7.5). The figure shows for decreasing fixed values of $x$ the
increasingly stronger deviation of Ratio from Ratio=1 with decreasing $Q^{2}$.

Figure 17: The Ratio (7.6) as a function of $Q^{2}$ in a wide range of x,
$x=10^{-5...-1}$.

In Figs. 11 to 14, we showed Ratio from (VII) as a function of
$\eta(W^{2},Q^{2})$. In connection with the extraction of the gluon
distribution from DIS experimental data, one frequently uses a fixed value of
$Q^{2}$, chosen as low as $Q^{2}=1.9~{}{\rm GeV}^{2}\cong 2~{}{\rm GeV}^{2}$,
as starting scale for the evolution of the gluon distribution with increasing
$Q^{2}$ at fixed $x$.

Table 3: Here $\Delta=\mathrm{Ratio}-1$ and $C_{2}=0.29$ . $Q^{2}[\mathrm{GeV}^{2}]$ | $x$ | $W^{2}[\mathrm{GeV}^{2}]$ | $\eta$ | $\Delta+1$ | $\frac{1}{\Delta+1}$
---|---|---|---|---|---
2 | $10^{-2}$ | 2.$10^{2}$ | 1.492 | 1.544 | 0.648
2 | $10^{-3}$ | 2.$10^{3}$ | 0.765 | 1.894 | 0.528
2 | $10^{-4}$ | 2.$10^{4}$ | 0.392 | 2.252 | 0.444
2 | $10^{-5}$ | 2.$10^{5}$ | 0.201 | 2.662 | 0.376
20 | $10^{-2}$ | 2.$10^{3}$ | 7.172 | 1.041 | 0.961
20 | $10^{-3}$ | 2.$10^{4}$ | 3.678 | 1.204 | 0.831
20 | $10^{-4}$ | 2.$10^{5}$ | 1.886 | 1.362 | 0.734
20 | $10^{-5}$ | 2.$10^{6}$ | 0.967 | 1.563 | 0.640
100 | $10^{-2}$ | 1.$10^{4}$ | 22.351 | 1.026 | 0.975
100 | $10^{-3}$ | 1.$10^{5}$ | 11.463 | 1.115 | 0.897
100 | $10^{-4}$ | 1.$10^{6}$ | 5.879 | 1.189 | 0.841
100 | $10^{-5}$ | 1.$10^{7}$ | 3.015 | 1.277 | 0.783

In Table III, we show the results for $(1+\Delta)^{-1}$ according to (VII)
with (7.7) and (7.8) for fixed $Q^{2}=2~{}{\rm GeV}^{2}$ and the typical range
of $10^{-2}\leq x\leq 10^{-5}$, corresponding to an interval of $\eta$
approximately given by $1.5\geq\eta\geq 0.2$. According to Table III, we
observe a dramatic correction factor to standard evolution of magnitude
$0.6\geq(1+\Delta)^{-1}\geq 0.4$. This correction to evolution is due to a
strong violation of the impulse approximation of the pQCD improved parton
model at low $Q^{2}$. The associated transition to hadronlike $(q\bar{q})p$
interactions is outside the range of validity of the standard evolution
equations. It comes without surprise that global DIS data fits based on
imposing evolution from a low-$Q^{2}$-starting input scale “do not describe
the deep inelastic scattering data in the low-$x$, low-$Q^{2}$ region very
well”, see ref. Pelicer . We add a comment on the interpretation of the huge
discrepancy between the CDP gluon distributions in Figs.6 and 10 and the
published results from several collaborations. The CDP gluon distribution is
deduced from color gauge-invariant CDP structure functions explicitly
incorporating the $Q^{2}{\rightarrow}0$ limit. The standard determinations
contain a relatively ad hoc continuation to low values of $Q^{2}$.

In Table III, in addition to the choice of $Q^{2}=2~{}{\rm GeV}^{2}$, we also
give the results for the correction factor $(1+\Delta)^{-1}$ at $Q^{2}=20{\rm
GeV}^{2}$. For $x=10^{-2}$, with $(1+\Delta)^{-1}=0.96\cong 1$, we have
consistency with perturbative evolution, as expected. For sufficiently large
$W^{2}$, or $10^{-3}>x>10^{-5}$, we observe the expected transition from
unmodified pQCD evolution to hadronlike saturation.

We turn to the discussion of the gluon distribution function. Rewriting (VII),
and substituting the gluon distribution from (2.8), we find

$\displaystyle\frac{\frac{\partial F_{2}(x,Q^{2})}{\partial\ln
Q^{2}}}{1+\Delta(\eta,\mu)}=F_{L}\left(\frac{\xi_{L}}{\xi_{2}}x,Q^{2}\right)$
$\displaystyle=\frac{\alpha_{s}(Q^{2})}{9\pi}R_{e^{+}e^{-}}G\left(\frac{x}{\xi_{2}},Q^{2}\right),$
(7.15)

where $R_{e^{+}e^{-}}=10/3$ for four flavors of quarks. The factor
$(1+\Delta)^{-1}$ in (7.15), for $\Delta\not=0$ represents a violation of, or
alternatively, a correction to the evolution equation for the gluon
distribution with decreasing $\eta$ for
$\eta(W^{2}=Q^{2}/x,Q^{2})\mathrel{\raise 1.29167pt\hbox{$<$\kern-7.5pt\lower
4.30554pt\hbox{$\sim$}}}5$.

The violation of the evolution with decreasing $Q^{2}$ may be seen directly in
terms of the gluon distribution by showing the gluon distribution as a
function of $x$ for e.g. $Q^{2}=100~{}{\rm GeV}^{2}$ and $Q^{2}=2~{}{\rm
GeV}^{2}$ i.e. rewriting (7.15) as

$\displaystyle G\left({x},Q^{2}\right)=\frac{\frac{\partial
F_{2}(\xi_{2}{x},Q^{2})}{\partial\ln
Q^{2}}}{1+\Delta(\eta,\mu)}\frac{9\pi}{\alpha_{s}(Q^{2})R_{e^{+}e^{-}}},$
(7.16)

where $\Delta(\eta,\mu)$ is inserted from (7.6). In Fig.18 we show the gluon
distribution according to (7.16). In addition, we show the gluon distribution
under the ad hoc assumption of $\Delta(\eta,\mu){\equiv}0$ corresponding to
validity of evolution in violation of the CDP result where
$\Delta(\eta,\mu)\neq 0$.

Figure 18: The gluon distribution $G(x,Q^{2})$ deduced from Eq.(7.16) for
$\Delta(\eta,\mu){\neq}0$ and $\Delta(\eta,\mu){\equiv}0$ .

Fig.18 confirms our previous conclusion that the experimental data described
by the CDP obey evolution at $Q^{2}$ large,
$Q^{2}{\cong}100~{}\textrm{GeV}^{2}$, while evolution is violated at low
$Q^{2}$, i.e., specifically for $Q^{2}{=}1.9~{}\textrm{GeV}^{2}$. Since the
experimental data, described by the $\gamma^{*}g{\rightarrow}q\bar{q}$
interaction of the CDP, violate the evolution equation at low $Q^{2}$, the
assumption of a universal validity of the evolution equation at low $Q^{2}$ is
excluded. Relying on the universal absolute validity of the evolution equation
at a low-$Q^{2}$ input scale in e.g. global fits to the experimental data is
highly questionable.

The Ratio may be explicitly interpreted as a modification of the evolution
equation (6.6). Upon substitution of (7.10) equation (6.6) becomes

$\displaystyle{\frac{\partial F_{2}(x,Q^{2})}{\partial\ln
Q^{2}}}=F_{L}\left(\frac{\xi_{L}}{\xi_{2}}x,Q^{2}\right){\times}Ratio$
$\displaystyle=F_{L}\left(\frac{\xi_{L}}{\xi_{2}}x,Q^{2}\right)\frac{1}{I^{(1)}_{L}\left(\frac{\xi_{L}}{\xi_{2}}\eta,\frac{\xi_{L}}{\xi_{2}}\mu\right)}$
$\displaystyle\times\left[I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)+I^{(1)}_{L}(\eta,\mu)\right.$
$\displaystyle\left.+\frac{\partial}{\partial\ln
Q^{2}}\left(I_{T}^{(1)}\left(\frac{\eta}{\rho},\frac{\mu}{\rho}\right)+I_{L}^{(1)}(\eta,\mu)\right)\right],$
(7.17)

where the simple equations for $I_{L,T}^{(1)}$ in Appendix (A3) to (A5) are to
be inserted. At small $\mu$, the CDP evolution equation (7.17) becomes

$\displaystyle\frac{\partial F_{2}(x,Q^{2})}{\partial\ln
Q^{2}}{\simeq}F_{L}(\frac{\xi_{L}}{\xi_{2}}x,Q^{2})$
$\displaystyle{\times}\bigg{\\{}\frac{I_{0}(\eta)-\frac{(1-C_{2})\eta}{1+4\eta}(2I_{0}(\eta)+\frac{1}{\eta})}{1-2\eta^{*}I_{0}(\eta^{*})}\bigg{\\}}$
(7.18)

where $I_{0}$ is defined in (A.5). The factor multiplying the longitudinal
structure function continuously converges towards unity at large $Q^{2}$. We
note that the CDP evolution equation (7.17), differs from an ad hoc
modification of conventional evolution, since it is an analytically determined
smooth extrapolation to low $Q^{2}$ consistent with DIS measurements.

##  VIII VIII. Conclusions.

Employing the pQCD relation between the longitudinal proton structure function
and the gluon distribution function allows one to define a gluon distribution
associated with the proton structure functions of the CDP at low $x$ and any
$Q^{2}{\geq}0$. The CDP gluon distribution depends on $W^{2}=Q^{2}/x$ and
$Q^{2}$. For $Q^{2}$ sufficient large
($10~{}\mathrm{GeV}^{2}{\lesssim}Q^{2}{\lesssim}100~{}\mathrm{GeV}^{2}$ at
presently available energies) the CDP gluon distribution, multiplied by
$\alpha_{s}(Q^{2})$, converges towards the asymptotic limit proportional to
the saturation scale
$\Lambda^{2}_{\mathrm{sat}}(W^{2}){\sim}(\frac{W^{2}}{1\mathrm{GeV}^{2}})^{C_{2}=0.29}$.

The deviation between the CDP gluon distribution and several published
distributions, at low $Q^{2}$ in particular, is a consequence of the
representation of the structure functions of DIS in the CDP. The CDP structure
functions based on two-gluon exchange explicitly include large as well as
small values of $Q^{2}$, including the $Q^{2}\to 0$ limit. The associated CDP
gluon distribution is valid for large as well as small $Q^{2}$. The usual
determination of the gluon distribution however is not based on a very
detailed justification of deducing the gluon distribution from the
experimental data at a low-$Q^{2}$ input scale. The CDP explicitly includes
the $Q^{2}\to 0$ structure functions, in distinction from the standard
approach based on a low $Q^{2}$ input scale that lacks a very detailed
justification for a low $Q^{2}$ input scale.

Concerning evolution, the dependence of the logarithmic derivative of the
proton structure function on $x$ and $Q^{2}$, we find a CDP evolution equation
that agrees with the standard model evolution equation in the limit of large
$Q^{2}$ at any fixed low $x$.

At low $Q^{2}$, where photoabsorption cross section becomes hadronlike, the
CDP evolution differs significantly from the standard one by an explicitly
analytically given factor. Relying on a universal validity of the parton model
and standard evolution at a low-$Q^{2}$ starting scale, such as
$Q^{2}{\cong}1.9~{}\mathrm{GeV}^{2}$, as frequently employed in global fits,
is questionable and requires further investigations.

## Appendix A

The explicit expressions for the functions
$I_{L,T}^{(1)}\left(\eta(W^{2},Q^{2}),\mu(W^{2})\right)$ in (3.2) are given
by:

$\displaystyle I_{L,T}(\eta(W^{2},Q^{2}),\mu(W^{2}))$
$\displaystyle=I^{(1)}_{L,T}(\eta(W^{2},Q^{2}),\mu(W^{2}))$ (A.1)
$\displaystyle\times\left(1+0\left(\mu(W^{2})\right)\right),$

where $I^{(1)}_{L}(\eta(W^{2},Q^{2}),\mu(W^{2}))~{}{\rm
and}~{}I^{(1)}_{T}(\eta(W^{2},Q^{2}),\mu(W^{2}))$ are given by

$\displaystyle I^{(1)}_{L}(\eta,\mu)=\frac{\eta-\mu}{\eta}$
$\displaystyle\times\left(1-\frac{\eta}{\sqrt{1+4(\eta-\mu)}}\right.$
$\displaystyle\left.\times\ln\frac{\eta(1+\sqrt{1+4(\eta-\mu)})}{4\mu-1-3\eta+\sqrt{(1+4(\eta-\mu))((1+\eta)^{2}-4\mu)}}\right),$
$\displaystyle
I^{(1)}_{T}(\eta,\mu)=\frac{1}{2}\ln\frac{\eta-1+\sqrt{(1+\eta)^{2}-4\mu}}{2\eta}$
$\displaystyle-\frac{\eta-\mu}{\eta}+\frac{1+2(\eta-\mu)}{2\sqrt{1+4(\eta-\mu)}}$
(A.2)
$\displaystyle~{}~{}\times\ln\frac{\eta(1+\sqrt{1+4(\eta-\mu)})}{4\mu-1-3\eta+\sqrt{(1+4(\eta-\mu))((1+\eta)^{2}-4\mu)}}.$

For $\mu(W^{2})\ll 1$ (and $\eta(W^{2},Q^{2})\geq\mu(W^{2})$) the functions
$I_{L}^{(1)}(\eta,\mu)$ and $I_{T}^{(1)}(\eta\mu)$ can be simplified to become

$I_{L}^{(1)}(\eta,\mu)=\frac{\eta-\mu}{\eta}\left(1-2\eta I_{0}(\eta)\right)$
(A.3)

and

$I_{T}^{(1)}(\eta,\mu)=I_{0}(\eta)-\frac{\eta-\mu}{\eta}\left(1-2\eta
I_{0}(\eta)\right),$ (A.4)

where

$I_{0}(\eta)=\frac{1}{\sqrt{1+4\eta}}\ln\frac{\sqrt{1+4\eta}+1}{\sqrt{1+4\eta}-1}.$
(A.5)

We also note that for the relevant range of $\eta\gg\mu$, or $Q^{2}\gg
m^{2}_{0}$, we may put $\mu=0$ in (A.3) and (A.4).

## Appendix B

In this Appendix we show the coefficients of the structure functions
$F_{2}(x,Q^{2})$ given in (4.1) as taken from ref. Block . The coefficients in
(4.1),

$\displaystyle A(Q^{2})$ $\displaystyle=$ $\displaystyle
a_{0}+a_{1}\ln\left(1+\frac{Q^{2}}{\mu^{2}}\right)+a_{2}\ln^{2}\left(1+\frac{Q^{2}}{\mu^{2}}\right),$
$\displaystyle B(Q^{2})$ $\displaystyle=$ $\displaystyle
b_{0}+b_{1}\ln\left(1+\frac{Q^{2}}{\mu^{2}}\right)+b_{2}\ln^{2}\left(1+\frac{Q^{2}}{\mu^{2}}\right),$
$\displaystyle C(Q^{2})$ $\displaystyle=$ $\displaystyle
c_{0}+c_{1}\ln\left(1+\frac{Q^{2}}{\mu^{2}}\right),$ $\displaystyle D(Q^{2})$
$\displaystyle=$ $\displaystyle\frac{Q^{2}(Q^{2}+\lambda
M^{2})}{(Q^{2}+M^{2})^{2}},$

depend on the fit parameters listed in Table II.

Table B.1: The effective parameters at low $x$ for $0.15{\rm GeV}^{2}<Q^{2}<3000{\rm GeV}^{2}$ provided by the following values. The fixed parameters are defined by the Block-Halzen fit to the real photon-proton cross sections as $M^{2}=0.753\pm 0.068~{}{\rm GeV}^{2}$, $\mu^{2}=2.82\pm 0.290~{}{\rm GeV}^{2}$ and $c_{0}=0.255\pm 0.016$. parameters | value
---|---
$a_{0}$ | $8.205\times 10^{-4}\pm 4.62\times 10^{-4}$
$a_{1}$ | $-5.148\times 10^{-2}\pm 8.19\times 10^{-3}$
$a_{2}$ | $-4.725\times 10^{-3}\pm 1.01\times 10^{-3}$
$b_{0}$ | $2.217\times 10^{-3}\pm 1.42\times 10^{-4}$
$b_{1}$ | $1.244\times 10^{-2}\pm 8.56\times 10^{-4}$
$b_{2}$ | $5.958\times 10^{-4}\pm 2.32\times 10^{-4}$
$c_{1}$ | $1.475\times 10^{-1}\pm 3.025\times 10^{-2}$
$n$ | $11.49\pm 0.99$
$\lambda$ | $2.430\pm 0.153$
$\chi^{2}$ (goodness of fit) | $0.95$

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