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Physics Letters B 803 (2020) 135305

Contents lists available at ScienceDirect

Physics Letters B

www.elsevier.com/locate/physletb

HERA data and collinearly-improved BK dynamics

B. Ducloué

a,b

, E. Iancu

a,∗

, G. Soyez

, D.N. Triantafyllopoulos

Institut de physique théorique, Université Paris Saclay, CNRS, CEA, F-91191 Gif-sur-Yvette, France

Higgs Centre for Theoretical Physics, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK

European Centre for Theoretical Studies in Nuclear Physics and Related Areas (ECT*) and Fondazione Bruno Kessler,

Strada delle Tabarelle 286, I-38123 Villazzano, TN, Italy

a r t i c l e i n f o a b s t r a c t

Article history:

Received 3 January 2020

Accepted 13 February 2020

Available online 17 February 2020

Editor: A. Ringwald

Keywords:

QCD

Parton saturation

Deep inelastic scattering

Within the framework of the dipole factorisation, we use a recent collinearly-improved version of the

Balitsky-Kovchegov equation to ﬁt the HERA data for inclusive deep inelastic scattering at small Bjorken x.

The equation includes an all-order resummation of double and single transverse logarithms and running

coupling corrections. Compared to similar equations previously proposed in the literature, this work

makes a direct use of Bjorken x as the rapidity scale for the evolution variable. We obtain excellent ﬁts

for reasonable values for the four ﬁt parameters. We ﬁnd that the ﬁt quality improves when including

resummation effects and a physically-motivated initial condition. In particular, the resummation of the

DGLAP-like single transverse logarithms has a sizeable impact and allows one to extend the ﬁt up to

relatively large photon virtuality Q

(http://creativecommons.org/licenses/by/4.0/). Funded by SCOAP

1. Introduction

Derived via systematic approximations within perturbative QCD, the Colour Glass Condensate (CGC) effective theory [1–5]is a powerful

framework for computing high-energy processes in the presence of non-linear effects associated with high parton densities. There are

intense ongoing efforts towards extending this effective theory to next-to-leading order (NLO) accuracy, as required by realistic applications

to phenomenology [6–19]. These efforts refer both to the Balitsky-JIMWLK equations [20–26], which govern the high-energy evolution of

the scattering amplitudes, and to the impact factors, which represent cross-sections at relatively low energy. The ﬁrst NLO results, obtained

more than a decade ago [6–8], refer to the Balitsky-Kovchegov (BK) equation [20,27]. The latter is a non-linear equation emerging from

the B-JIMWLK hierarchy in the limit of a large number of colours (N

→∞). It describes the evolution of the elastic scattering amplitude

between a colour dipole and a dense hadronic target. Via appropriate factorisation schemes, like the “dipole factorisation” or the “hybrid

factorisation” [28], the BK equation also governs the high-energy evolution of the cross-sections for processes of phenomenological interest,

like deep inelastic scattering (DIS) at small Bjorken x, or forward particle production in proton-nucleus collisions.

A few years after the full NLO BK equation was ﬁrst presented [8], its numerical study in [29]showed that it is unstable, as anticipated

in [30,31]. Similar problems had been identiﬁed, and cured [32–37], for the NLO version of the BFKL equation [38–40]— the linearised

version of the BK equation, valid for weak scattering. The origin of this diﬃculty has been clearly identiﬁed, both numerically [29] and

conceptually [31,41]: it is associated with large and negative NLO corrections enhanced by a double “anti-collinear” logarithm, generated

by integrating out gluon emissions with small transverse momenta (see Sect. 2). Such double-logarithmic corrections spoil the convergence

of the ﬁxed-order perturbative expansion of the high-energy evolution, unless they are properly resummed to all orders. Refs. [31,41]

proposed

two different strategies for resumming this series of double-logarithmic corrections to all orders.

At a ﬁrst sight, these strategies seemed to be successful, leading to stable evolution equations [41,42] and allowing for good ﬁts

to the small-x HERA data [43,44]. However, a recent study [45]revealed some inconsistencies in the original analyses in [31,41,43,44].

In particular, there was a confusion concerning the meaning of the rapidity variable which plays the role of the evolution time. The

variable which a priori enters the perturbative calculations at NLO [8] and in the resummed equations proposed in [31,41]is the rapidity

Y ≡ ln(s/Q

) of the dipole projectile, with s the centre-of-mass energy squared and Q

a typical transverse momentum scale for the

Corresponding author.

E-mail addresses: bertrand.ducloue@ed.ac.uk (B. Ducloué),

edmond.iancu@ipht.fr (E. Iancu),

gregory.soyez@ipht.fr (G. Soyez),

trianta@ectstar.eu (D.N. Triantafyllopoulos).

https://doi.org/10.1016/j.physletb.2020.135305

SCOAP

2 B. Ducloué et al. / Physics Letters B 803 (2020) 135305

target. It is different from the rapidity η ≡ ln(1/x) of the hadronic target, which is the variable used in DIS (see Sect. 2 for details). When

translated to this physical variable, the results of the original resummations in [31,41]show a strong scheme dependence, preventing

any meaningful phenomenological applications [45]. The success of the corresponding ﬁts to the HERA data [43,44]was rather fortuitous

and can be attributed to several factors: (i) these ﬁts blindly assumed the physical rapidity variable η = ln(1/x) (although this was

inconsistent with the resummation scheme), (ii) the rapidity interval over which one can probe high-energy evolution is relatively

limited making it delicate to critically probe the effects of resummation. In other words, even though the ﬁts can hint at an evolution

being better than another (e.g. via a better χ

, or more physical ﬁt parameters), it remains diﬃcult to draw a ﬁrm conclusion.

To overcome these diﬃculties, Ref. [45]proposed a reorganisation of perturbation theory in which the evolution time is the physical

rapidity η. This program lead to a new version of “collinearly-improved BK equation”, shown below in Eq. (6). This equation is non-local

in rapidity. In that sense it looks formally similar to the equation proposed in [31]but these two equations are fundamentally different:

(i) the evolution variable, η, occurring in Eq. (6)is the physical rapidity ln(1/x), and not the rapidity of the dipole projectile; (ii)

the rapidity shift in Eq. (6) accounts for an all-order resummation of double collinear logarithms, and not anti–collinear (see Sect. 2). The

collinear logarithms are generated by gluon emissions with relatively large momenta. Such emissions are atypical in the context of DIS.

Moreover, they are partially suppressed by non-linear effects, so their resummation is somewhat less critical. As a consequence, a com-

parison

between various resummation prescriptions shows only little scheme dependence [45], at the level of the expected perturbative

accuracy of the resummed equation.

The evolution equation we use in this work, Eq. (6), actually goes beyond the original proposal from Ref. [45]by additionally including

a class of DGLAP-like single transverse logarithms (both collinear and anti-collinear) which appears in the NLO BK equation. While in

principle one can also include [45]the full set of NLO BK corrections in our evolution equation, this is numerically cumbersome (see

however [42]) and it goes beyond the scope of this Letter. Instead, we directly confront the relatively simple equation (6)with the most

recent HERA data [46]for inclusive DIS at small Bjorken x ≤ 0.01. This dataset includes more data points as compared to the one used in

previous ﬁts [43,44].

We use the ﬁts to test the resummation of the aforementioned double logs and single transverse logarithms as well as various pre-

scriptions

for the running of the QCD coupling which enters the BK equation. We study two models for the initial condition to the BK

equation: the simple Golec-Biernat and Wüsthoff (GBW) model [47,48] and a running-coupling version of the McLerran-Venugopalan (MV)

model [49]. Both models include two free parameters which are ﬁtted to the experimental data. Two additional parameters (leaving aside

the quark masses that we keep ﬁxed) are associated with our ansatz for the running coupling and with the overall normalisation of the

DIS cross-section.

With this setup, we ﬁnd a good overall agreement with the HERA data at small x

Furthermore, it appears that the physically-motivated

MV model with running coupling is preferred over the GBW model, with the former giving both better χ

and more reasonable parameter

values than the latter. As with earlier studies, the ﬁts appear to favour an initial condition for the dipole scattering amplitude with a fast

and abrupt approach to the unitarity limit. This is the main reason why better ﬁts are obtained with the running coupling version of the

MV model as compared to its original ﬁxed-coupling version [49]. This ﬁnding is also in agreement with the fact that previous ﬁts [50,51]

using

the original MV model favoured a modiﬁed dependence of the initial amplitude on the dipole size, decreasing like r

2γ

with γ

> 1.

Another interesting observation of our ﬁts is that the inclusion of the DGLAP-like single logarithms signiﬁcantly improves the description

of the data at large Q

, allowing for good descriptions up to maximum Q

of 400 GeV

This Letter is organised as follows. Sect. 2 provides a qualitative and (hopefully) pedagogical summary of the arguments justifying the

replacement of the original collinear-improved BK equation formulated in terms of the rapidity of the dipole projectile [31,41], by a new

version formulated in terms of the rapidity of the hadron target, or Bjorken x. (We refer to Ref. [45]for more details.) Sect. 3 discusses the

main physical consequences of the various resummations on the solution to the collinearly-improved BK equation. Finally, Sect. 4 presents

the main original results of this paper: the setup and the results for the ﬁts together with their physical discussion.

2. Collinearly-improved BK evolution in the target rapidity

The main difference between our present approach and previous saturation ﬁts to DIS refers to our speciﬁc choice of evolution equation

used to describe the evolution of the dipole S-matrix with increasing energy. More precisely, we use a collinearly-improved version of

the BK equation — recently proposed in [45]— in which the rapidity variable playing the role of the evolution time is the proton rapidity

η = ln(1/x), with x the standard Bjorken variable. This contrasts previous studies (see e.g. [8,31,41,43]) where the evolution was formulated

in terms of the rapidity Y of the dipole projectile. Beyond leading order [20,27], the choice of η over Y has important consequences. To

make this clear, we ﬁrst summarise the main ﬁndings of [45] which are relevant for the ﬁt to DIS data described in the next section.

Basic kinematics, target and projectile rapidity We use a frame in which the virtual photon, γ

∗

, is an energetic right-mover with (light-cone)

4-momentum q

≡ (q

, q

−

, q

⊥

) = (q

, −

, 0

⊥

), whereas the proton target is a left-mover with P

= δ

μ−

−

In the high-energy or

small Bjorken x regime,

x ≡

2P · q

−

 1 , (1)

the coherence time x

 2q

of the virtual photon, i.e. the typical lifetime of its quark-antiquark (q

q) ﬂuctuation, is much larger

than the longitudinal extent 1/P

−

of the target. This justiﬁes the use of the dipole picture in which the γ

∗

ﬂuctuates into a q

q colour

dipole long before the collision, which then scatters inelastically off the proton. Via the optical theorem, the total dipole-hadron scattering

cross-section is related to the S-matrix for the elastic scattering. At high energy, one can work in the eikonal approximation where the

transverse coordinates x of the quark and y of the antiquark are not affected by the collision.

We neglect the proton mass M which is much smaller than all the other scales in the problem, M

 Q

 2p · q.

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