Saturation Effects in Deep Inelastic Scattering at low $Q^2$ and its Implications on Diffraction

K. Golec-Biernat; M. Wusthoff

REVIEW

Saturation Effects in Deep Inelastic Scattering at low $Q^2$ and its Implications on Diffraction

K. Golec-Biernat, M. Wusthoff

High Energy Physics - Phenomenology · Submitted 27 Jul 1998 · 25 references

Pith's verdict: The original GBW paper: a genuinely useful three-parameter saturation fit, with a diffractive-ratio claim that is an asymptotic estimate rather than a result.

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arXiv IDhep-ph/9807513 (version 1)

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CitationK. Golec-Biernat, M. Wusthoff. "Saturation Effects in Deep Inelastic Scattering at low $Q^2$ and its Implications on Diffraction." arXiv:hep-ph/9807513 (1998). https://pith.science/paper/GZ73M34Y

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Editor's note

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This is the paper that gave everyone the saturating dipole cross section they still use as a baseline. The model is sigma-hat = sigma0{1 - exp[-r^2/(4R0^2(x))]} with R0^2(x) = (x/x0)^lambda/Q0^2, three parameters, fitted to all DIS data below x = 0.01. It works: chi2/ndf = 441/372 = 1.18 for light flavours, and the turnover in Fig. 3 sits where the critical line says it should. The lambda_eff plot (Fig. 9) interpolating between the soft value 0.08 and the hard 0.29 is the kind of thing people actually remember and reuse. The Mellin machinery in section 2.2 is clean and the simplified form in Eq. (29) is a nice touch. Credit where due: this is compact, transparent, and reproducible in principle from the formulae given, even without shipped tables or code. It is not a reparameterization of someone else's fit, and the diffractive ratio is not fitted to diffractive data, so it is a real model consequence rather than a circular one. I think the reader's circularity score of 4.0 is a bit harsh on that point; the mass tuning is disclosed in the text, not hidden. Where I do agree with the stress-test note is on abstract:s3. Eq. (35) is explicitly logarithmic in Q^2 and x, so 'roughly constant' is a claim about the weakness of that variation over the HERA range, and the paper never shows the variation is small across the fitted range. The normalization is worse: the quark piece gives 'a few percent', then a gluon-emission contribution of 'similar size' is asserted to reach 'of the order of 10%' without being computed. The body is honest about this, deferring precise diffraction to ref. [24], which is listed as in preparation, but the abstract states the result without the caveat. Secondary soft spots, both minor: no parameter uncertainties or error bands anywhere, and the charm fit at chi2/ndf = 1.50 is noticeably weaker than the light-flavour one. The photoproduction 'extrapolation' is really a tuned light-quark mass, which the authors say plainly in section 3; the abstract's phrasing oversells it slightly. Who should read it: anyone working on small-x DIS, dipole models, or diffraction at HERA energies, and anyone who needs the standard saturation baseline for comparison. Worth a serious referee, and worth citing.

Referee report

Major objections (4)

[Table 1 and §3.2] The charm fit degrades the description substantially (χ²/ndf rises from 1.18 to 1.50) and, more importantly, shifts the fitted parameters by a large amount: x0 moves from 3.04·10⁻⁴ to 0.41·10⁻⁴, nearly an order of magnitude, while λ changes from 0.288 to 0.277. Since the position of the critical line in Figs. 5 and 10 is a headline result and depends directly on x0 and λ, this shift needs to be discussed rather than summarized as 'still acceptable'. Please state where the additional χ² resides (which data sets, which Q² and x bins), and show the critical line for both fits on one plot so the reader can judge how much the saturation-scale prediction actually moves.
[§3 and abstract sentence [abstract:s2]] The abstract claims 'a successful extrapolation into the photoproduction regime', but §3 states that the light-quark mass of 140 MeV 'is chosen such that the photoproduction cross section is in rough agreement with the data without having them included in our fit'. The photoproduction agreement is therefore a one-parameter tuning to the very data it is said to describe, not a parameter-free extrapolation. The same caveat propagates to the effective slope: §3.1 concedes that λ_eff ≈ 0.08 at low Q² 'is dependent on the choice of the quark mass'. The authors should either qualify the abstract sentence or demonstrate the stability of the photoproduction prediction over a range of masses, showing how σ(γp) and λ_eff vary for, say, m_q = 100–200 MeV.
[§4, Eq. (35), abstract sentence [abstract:s3]] The abstract's claim of 'a roughly constant ratio of diffractive and inclusive cross section' is stronger than what §4 establishes. Eq. (35) gives σ_D/σ_tot ∝ 1/(6 ln Q² + 6λ ln(x/x0) + 6γ_E + 1), which varies logarithmically in both Q² and x; 'roughly constant' is a statement about the smallness of that variation over the HERA range, and no figure showing the ratio as a function of x and Q² across the fitted region is presented. In addition, the absolute normalization is an estimate: the quark piece gives 'a few percent', and the step to 'of the order of 10%' relies on a gluon-emission contribution that is asserted to have 'the same power behaviour in x as for the quarks' but is not computed. Please add a plot of σ_D/σ_tot versus Q² at several x (or versus x at several Q²) over the fitted range, and either compute or more carefully bound the gluon term. If that is deferred to ref. [24], the abstract sentence should be softened accordingly.
[Table 1 and §3] No uncertainties are quoted on σ0, λ or x0, and no error bands accompany any of the predictions (Figs. 3, 5, 6, 9, 12). For a three-parameter fit this is a significant omission, because several of the paper's qualitative conclusions — the location of the critical line, the value of the saturation scale at x = 10⁻⁴–10⁻⁵, and the predicted upward shift at a 1 TeV collider — are statements about central values whose significance depends on the fit errors. A covariance matrix or at least one-sigma ranges on the three parameters, propagated to the critical line in Fig. 5, would make the central claims assessable.

Minor objections (8)

[§2, Eqs. (11) and (12)] The alternative Ansätze g(r̂²)=ln(1+r̂²) and g(r̂²)=r̂² ln(1+1/r̂²) are dismissed with 'It turns out, however, that they are disfavoured in our analysis', but no fit results are shown. Since the choice of g is one of the two defining ingredients of the model, a short table giving χ²/ndf for all three functional forms would substantiate the selection at negligible cost.
[Eq. (21) and §3.1] The modified Bjorken variable x̃ = x(Q²+4m_f²)/Q² is introduced in Eq. (21) and used inside the dipole cross section, while the wave function in Eqs. (2)–(3) still uses x. The text should state explicitly which variable enters where, and confirm that the same x̃ is the one held fixed in the λ_eff definition of Eq. (31) and in the dashed lines of Fig. 7.
[Eq. (29)] The simplified parametrization is advertised as a 'quick fit to the data', but the primed parameters σ'_0, λ' and x'_0 are never given numerical values. Please quote them, together with the quality of the fit they achieve, so the parametrization is actually usable.
[Table 1 caption] The column heading 'χ²/(no.exp.po.)' is not defined; presumably it means number of experimental points. Please spell it out, and state which data sets and Q² cuts enter the 372 points.
[Fig. 9 caption] The remark that 'the solid line appears twofold due to two different methods of calculation' is unclear from the figure; please distinguish the two curves visibly (e.g. solid versus dashed) and label them in the legend.
[Fig. 3] The critical line is drawn across the cross-section curves, but the vertical axis is σ(γ*p) rather than Q². Please explain in the caption how the line Q²=1/R0²(x) is mapped onto this plane at fixed W.
[Reference [8]] The citation 'J.R. Forshaw and D.A. Ross, QCD and the Pomeron, Cambridge University Press, 1986' appears to have the wrong year; the Cambridge monograph by these authors is 1997. Please check.
[§3.2 and §5] There are several typographical errors: 'eﬀected' should be 'affected' in §3.2, 'woud' should be 'would' in §5, and '1 T eV' should be '1 TeV' throughout.

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The paper

The paper claims

A three-parameter saturation model reproduces all small-$x$ deep inelastic data and predicts a nearly constant diffractive-to-inclusive ratio.

The reading

The authors want to show that two kinds of saturation — the obvious one where the photon's wavelength grows to the size of the proton at low $Q^2$, and the less obvious one where the parton cloud becomes dense at small $x$ — can be captured by a single dipole cross section with an $x$-dependent radius, and that this one object then describes the whole body of deep inelastic data below $x=0.01$. The payoff is economy: three numbers, an overall normalisation $1.2$–$1.5$ times the proton cross section, a power $1.28$–$1.29$ in $x$, and a reference $x_0$, do the work that elsewhere needs separate vector-meson, hard-Pomeron and Regge contributions stitched together. If the claim holds, the boundary between the scaling and saturation regimes is not a vague hand-wave but a definite curve in the $(x,Q^2)$ plane, $Q^2 = 1/R_0^2(x)$, along which the observed turnover in the cross section should sit — and that same curve should be visible in the $Q^2$-slope of $F_2$. The most striking corollary is that the ratio of diffractive to inclusive cross sections comes out roughly constant, of order a few percent from the quark part alone and around ten percent once gluon emission is added, rather than rising or falling steeply with $x$ and $Q^2$.

What carries the argument

The load-bearing object is the $x$-dependent saturation radius $R_0(x)$, which turns the dipole cross section into a two-regime function of $r/R_0$ and thereby defines a critical line $Q^2=1/R_0^2(x)$ in the $(x,Q^2)$ plane. The authors compute the photon-convolution integrals by Mellin transform in the variable conjugate to $ln r^2$, which reduces the numerical work to a single integral over $nu$ and, more importantly, exposes the pole structure in the complex $nu$-plane: a double pole at $nu=-i/2$ in the hard regime generates the logarithmic $ln(Q^2/Q_0^2)$ behaviour, a double pole at $nu=+i/2$ generates the soft behaviour, and stitching the two gives the simplified formula $sigma_{gamma^*p}=sigma_0'left{ln[(x_0'/x)^{lambda'}Q_0^2/Q^2+1]+(x_0'/x)^{lambda'}Q_0^2/Q^2ln[(x/x_0')^{lambda'}Q^2/Q_0^2+1]right}$ that reproduces the model's qualitative shape. For diffraction, the function $G_D(nu)=(2-2^{1/2+inu})G(nu)$ carries the whole argument: its zero at $nu=-i/2$ cancels the leading double pole down to a single pole, which is why the transverse diffractive cross section keeps the same power behaviour in $x$ as the inclusive one, while the longitudinal piece loses its pole entirely and rises twice as fast.

Extended reading notes

Core claim

The paper proposes that the interaction of a quark–antiquark pair with the proton is governed by a single saturating dipole cross section, $hat{sigma}(x,r^2)=sigma_0left{1-exp[-r^2/(4R_0^2(x))]right}$, where the saturation radius scales as $R_0^2(x)=(x/x_0)^lambda/Q_0^2$ with $Q_0=1$ GeV. The exponential form gives colour transparency at short separations, $hat{sigma}propto r^2$, and a constant at large separations, which is what produces saturation of the total $gamma^*p$ cross section. Fitting only $sigma_0$, $lambda$ and $x_0$ to all available DIS data below $x=0.01$ yields $sigma_0=23.03$ mb, $lambda=0.288$, $x_0=3.04cdot10^{-4}$ for three light flavours with a common mass of 140 MeV ($chi^2=441$ for 372 data points), and $sigma_0=29.12$ mb, $lambda=0.277$, $x_0=0.41cdot10^{-4}$ when charm at 1.5 GeV is included without new parameters ($chi^2=558$ for 372 points). The same parameters extrapolate into photoproduction and give an effective slope $lambda_{rm eff}=partialln sigma_{gamma^*p}/partialln W^2$ that runs from about $0.08$, the soft-Pomeron value, at low $Q^2$, up to about $0.29$ in the deep region. Applying the identical wave-function convolution with $hat{sigma}$ replaced by $|hat{sigma}|^2$ gives a diffractive cross section whose leading singularity structure matches the inclusive one, so the ratio of diffractive to inclusive cross sections is roughly constant, with only mild logarithmic dependence on $x$ and $Q^2$.

Load-bearing premise

The central premise is that a single eikonal-like dipole cross section with a sharp-edged proton profile, and with the diffractive amplitude taken as the square of that same cross section, captures both the inclusive and the diffractive data without any additional gap-survival or resolved-Pomeron contribution.

Editorial extensions

If this is right

The turnover of $sigma_{gamma^*p}$ towards low $Q^2$ should track the critical line $Q^2=1/R_0^2(x)$, so the position of the maximum in the logarithmic $Q^2$-slope of $F_2$ is a measurable map of where saturation sets in.
The effective Pomeron intercept extracted from $W^2$ dependence should slide continuously from about 1.08 in photoproduction to about 1.29 in the deep inelastic region, with the low-$Q^2$ value following from the fit rather than being imposed.
The longitudinal structure function $F_L$ is predicted to be roughly 20 percent of $F_2$ near $Q^2=10$ GeV$^2$, a number that can be checked once $F_L$ is measured.
The diffractive-to-inclusive ratio should stay at the few-percent level from the quark contribution alone and near ten percent with gluon emission, with only logarithmic drift in $x$ and $Q^2$.
Raising the collision energy moves the saturation scale up only modestly: from about 1 to 2 GeV$^2$ between $x=10^{-4}$ and $10^{-5}$, and perhaps to 3 GeV$^2$ at a 1 TeV electron–proton collider.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The critical line is defined by the model rather than derived, and the authors themselves note that it sits slightly to the left of the maxima in the $Q^2$-slope plots; an alternative definition along the maxima would shift the quoted saturation scales by a modest but non-trivial amount, so the numerical values of $R_0$ should be read as convention-dependent.
Because the diffractive prediction uses the same convolution with $hat{sigma}^2$ and no gap-survival factor, the constant-ratio result is really a statement about the $x$-dependence of the two leading singularities; a measurement that found the ratio falling with $Q^2$ faster than a logarithm would not necessarily kill the saturation idea, only the specific identification of the diffractive amplit
The model has no explicit perturbative evolution, so its natural completion is to match the dipole cross section onto a DGLAP-resummed gluon density at large $Q^2$; the fact that a fixed power $lambda$ already fits down to $x=10^{-5}$ suggests the anomalous dimension is effectively frozen over the measured range, which is a testable statement once higher $Q^2$ precision data exist.
Since the three fitted parameters absorb the proton's transverse profile into a sharp-edged disc, a realistic profile function would change the normalisation and possibly the low-$Q^2$ turnover while leaving the small-$x$ power intact; comparing the two would separate the genuinely nonperturbative profile from the dynamical saturation.

Original abstract

We present a model based on the concept of saturation for small $Q^2$ and small $x$. With only three parameters we achieve a good description of all Deep Inelastic Scattering data below $x=0.01$. This includes a consistent treatment of charm and a successful extrapolation into the photoproduction regime. The same model leads to a roughly constant ratio of diffractive and inclusive cross section.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 3,517 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.18 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.74 · compares with prior results 0.20 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We present a model based on the concept of saturation for small $Q^2$ and small $x$. With only three parameters we achieve a good description of all Deep Inelastic Scattering data below $x=0.01$. This includes a consistent treatment of charm and a successful extrapolation into the photoproduction regime. The same model leads to a roughly constant ratio of diffractive and inclusive cross section.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We present a model based on the concept of saturation for small $Q^2$ and small $x$.	contribution 0.52	0.90	0.10	asserted 0.98
s1	With only three parameters we achieve a good description of all Deep Inelastic Scattering data below $x=0.01$.	result 0.96	0.93	0.22	asserted 1.00
s2	This includes a consistent treatment of charm and a successful extrapolation into the photoproduction regime.	result 0.93	0.89	0.11	asserted 1.00
s3	The same model leads to a roughly constant ratio of diffractive and inclusive cross section.	result 0.95	0.89	0.35	asserted 0.83

What kind of paper the classifier reports this is: theoretical result 0.99 · evidence from analysis of existing data 0.86.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** Diagrammatical representation of the basic proce

**Figure 2.** The profile of the dipole cross section for different

**Figure 3.** The γ ∗p-cross section for various energies. The solid lines show the fit results with a light quark mass of 140MeV. The dotted lines show the cross section with the same parameters but with zero quark mass. The line across the curves indicates the position of the critical line. 15

**Figure 4.** The logarithmic Q2 -slope of F2 for fixed energies W, plotted as a function of Q2 and x. The line across the curves shows the position of the critical line. 16

**Figure 5.** The position of the critical line in the (

**Figure 6.** F2 and FL structure functions plotted as a function of Q2 for fixed energies W. 18

**Figure 7.** The total γ ∗p-cross section as a function of W2 for different Q2 -values. The dashed lines represent the lines of constant ˜x as defined in eq.(21). 19

**Figure 8.** The same cross section as in Fig. 7 with the emphasis

**Figure 9.** The effective slope in W2 of the total cross section, λef f , plotted as a function of Q2 for fixed ˜x. The solid line appears twofold due to two different methods of calculation. One is based on the analytical expression (31), the second is calculated from fig.7 by taking a pair of points along the lines of constant ˜x = 10−2 and 10−4 . 21

**Figure 10.** The γ ∗p-cross section including charm. The charm contribution itself is plotted as dotted line. 22

**Figure 11.** The logarithmic Q2 -slope of F2 with charm. The dashed line shows the charm contribution. 23

**Figure 12.** The effective slope λef f based on the charm fit. The intercept is slightly higher at low Q2 as compared to fig.9. 24

Figures reproduced from arXiv:hep-ph/9807513 by the authors.

References

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A nonparametric PINN-constrained dipole amplitude simultaneously describes HERA total/charm DIS and J/psi photoproduction while maintaining Fourier positivity, unlike analytic MV-type fits.
Saturation effects in exclusive vector meson production in DIS
hep-ph 2025-11 conditional novelty 6.0 of 10

A dense-limit CGC hotspot calculation shows saturation mildly suppresses exclusive vector meson cross sections in ep, with suppression growing at higher color-charge density.
High-energy dynamics of QCD: Theoretical and phenomenological results
hep-ph 2025-06 conditional novelty 6.0 of 10

Real NLO corrections to the BFKL Higgs impact factor are derived with a physical top-quark mass, and updated tetraquark fragmentation functions are applied to predict rates at 14 and 100 TeV.
Initial condition for the Balitsky-Kovchegov equation at next-to-leading order
hep-ph 2025-05 conditional novelty 6.0 of 10

A Bayesian fit to HERA total and charm cross-section data constrains the NLO Balitsky-Kovchegov initial condition and provides posterior samples for uncertainty propagation.
Transverse and longitudinal spin alignment from color fields in heavy ion collisions
nucl-th 2024-11 conditional novelty 6.0 of 10

Spin alignment of phi mesons along the beam direction is predicted to exceed 1/3 for glasma fields and to show a sign-changing rapidity pattern for isotropic QGP color fields, offering a discriminating observable.
Nucleon Energy Correlators for the Color Glass Condensate
hep-ph 2023-01 conditional novelty 6.0 of 10

The angular distribution of energy deposited at small angles in electron-nucleus deep inelastic scattering is predicted to turn over at an angle set by the saturation momentum, a feature absent from collinear factorization.
Diffractive Dijet Production and Wigner Distributions from the Color Glass Condensate
hep-ph 2019-02 conditional novelty 6.0 of 10

Gluon Wigner and Husimi distributions computed from JIMWLK evolution predict a percent-level elliptic modulation in coherent diffractive dijet production at a future Electron-Ion Collider, decreasing with collision en...
The hadronic interaction model Sibyll-2.3c and inclusive lepton fluxes
hep-ph 2018-06 conditional novelty 6.0 of 10

An updated cosmic-ray interaction model, with new treatments of leading-particle and charm production, reproduces fixed-target and LHC data and yields atmospheric muon and neutrino fluxes that agree with other post-LH...
Paving the Way Towards Precision Physics in Saturation Studies Through Exclusive Diffractive Light Neutral Vector Meson Production
hep-ph 2016-12 conditional novelty 6.0 of 10

The first next-to-leading-order impact factor for the photon-to-vector-meson (rho, phi, omega) transition in the QCD shockwave approach is derived for arbitrary kinematics, with all divergences shown to cancel against...
CGC factorization for forward particle production in proton-nucleus collisions at next-to-leading order
hep-ph 2016-08 conditional novelty 6.0 of 10

A reorganization of the NLO Color Glass Condensate calculation removes the rapidity subtraction, which the authors argue is the source of unphysical negative cross-sections in prior work.
On the one loop $\gamma^{(*)}\to q\bar{q}$ impact factor and the exclusive diffractive cross sections for the production of two or three jets
hep-ph 2016-06 conditional novelty 6.0 of 10

The authors compute the one-loop photon-to-quark-antiquark impact factor in arbitrary kinematics and derive an infrared-finite cross section for exclusive diffractive two-jet production.
Direct numerical solution of the coordinate space Balitsky-Kovchegov equation at next to leading order
hep-ph 2015-02 conditional novelty 6.0 of 10

The first numerical solution of the next-to-leading-order BK equation shows the evolution is unstable and can drive the dipole amplitude negative for initial conditions relevant to phenomenology.
Impact factor for high-energy two and three jets diffractive production
hep-ph 2014-05 accept novelty 6.0 of 10

A tree-level derivation gives the gamma* to q qbar g impact factor with an arbitrary number of t-channel gluons, without soft or collinear approximations, enabling diffractive dijet and trijet predictions.
Exclusive $J/\psi$ and $\psi(2S)$ production in $pp$ collisions at $\sqrt{s}=7$ TeV
hep-ex 2013-01 conditional novelty 6.0 of 10

LHCb observes exclusive J/psi and psi(2S) production in pp collisions at 7 TeV, measuring cross-sections of 307 and 7.8 pb and a photoproduction energy dependence consistent with HERA.
On the running coupling in the JIMWLK equation
hep-ph 2012-12 conditional novelty 6.0 of 10

Placing the running coupling in the noise correlator of the JIMWLK Langevin step yields a workable running-coupling JIMWLK equation whose evolution speed is slower than the previous square-root prescription.
Heavy-quarkonium suppression in p-A collisions from parton energy loss in cold QCD matter
hep-ph 2012-12 conditional novelty 6.0 of 10

A one-parameter model in which quarkonium energy loss in cold nuclear matter scales linearly with the quarkonium energy reproduces J/psi and Upsilon suppression data across energies and nuclear targets.
A Large Hadron Electron Collider at CERN: Report on the Physics and Design Concepts for Machine and Detector
physics.acc-ph 2012-06 conditional novelty 6.0 of 10

A design study arguing that adding a 60 GeV electron beam to the LHC would enable precision deep-inelastic scattering over a kinematic range roughly twenty times larger than HERA's, at about a hundred times HERA's int...
JIMWLK evolution in the Gaussian approximation
hep-ph 2011-12 accept novelty 6.0 of 10

A Gaussian mean-field approximation to the JIMWLK equations, valid at any number of colors, computes all n-point Wilson-line correlators from the dipole S-matrix alone.
Higher-point correlations from the JIMWLK evolution
hep-ph 2011-09 conditional novelty 6.0 of 10

Dropping the 'real' terms of the large-Nc JIMWLK evolution yields explicit expressions for the color quadrupole and sextupole as functionals of the dipole amplitude, claimed valid in both the dilute and saturated regimes.
Gluon spectrum in the glasma from JIMWLK evolution
hep-ph 2011-05 conditional novelty 6.0 of 10

Combining JIMWLK evolution with a classical Yang-Mills simulation shows that the gluon spectrum in the early glasma phase hardens with energy, with mean transverse momentum growing faster than the saturation scale.
Measurement of the Inclusive e{\pm}p Scattering Cross Section at High Inelasticity y and of the Structure Function FL
hep-ex 2010-12 conditional novelty 6.0 of 10

Using electron-proton collisions at 920, 575 and 460 GeV, H1 extracts the proton structure function FL at low Q2 and low x and finds R = FL/(F2 - FL) constant at about 0.26.
Wilson line correlator in the MV model: relating the glasma to deep inelastic scattering
hep-ph 2007-11 accept novelty 6.0 of 10

The Wilson line correlator computed with the standard glasma lattice prescription gives Qs approximately 0.6 g squared mu, revising the gluon liberation coefficient to c approximately 1.1.
Solving The High Energy Evolution Equation Including Running Coupling Corrections
hep-ph 2007-04 conditional novelty 6.0 of 10

Restoring the previously neglected subtraction term reconciles the Balitsky and Kovchegov-Weigert running-coupling kernels and yields a full numerical BK solution whose scaling function has anomalous dimension about 0.85.
Energy losses in the black disc regime and correlation effects in the STAR forward pion production in dAu collisions
nucl-th 2006-03 conditional novelty 6.0 of 10

Energy losses in the black disc QCD regime explain the STAR measurement of suppressed forward pion production in dAu collisions and imply that peripheral, not central, collisions dominate.
A Variable Flavour Number Scheme for Heavy Quark Production at Small x
hep-ph 2006-03 conditional novelty 6.0 of 10

White and Thorne define the DIS(χ) variable flavour number scheme, derive LL resummed heavy quark coefficient functions, and show that including them improves fits to low-x DIS data and stabilizes F_L.
Collinearly Improved Balitsky-Kovchegov Evolution of the Gluon Wigner Distribution
hep-ph 2026-06 unverdicted novelty 5.5 of 10

Collinearly improved IMST BK evolution moves the elliptic node and reshapes the rapidity and hard-scale dependence of coherent diffractive dijet ⟨cos2φ⟩ relative to LO BK, not just a normalization shift.
Diffractive Vector-Meson Production from the Light-Front Quark Model
hep-ph 2026-09 conditional novelty 5.0 of 10

In the color-dipole picture with light-front quark-model wave functions, the Melosh-Wigner spin structure (S-1) describes light-meson diffractive observables better, while the spin-improved structure (S-2) better desc...
The Pomeron loop as a perturbative correction to single Pomeron exchange revisited
hep-ph 2026-08 conditional novelty 5.0 of 10

A corrected re-derivation finds the QCD Pomeron loop is a moderate 24%-39% correction to single Pomeron exchange at small x and low scales, not the dominant correction claimed in 2013.
Femtoscopy-driven searches for saturated gluonic matter in inclusive photonuclear processes
hep-ph 2026-06 unverdicted novelty 5.0 of 10

Femtoscopy is presented as a method to isolate gluon saturation and nuclear shadowing effects via space-time structure in photonuclear collisions.
Maximum phase-space density of linearly polarized gluon TMDs in the saturation region
hep-ph 2026-05 unverdicted novelty 5.0 of 10

For the dipole gluon TMD, the Sudakov-limited maximum phase-space density of h_1^{⊥g} is 2 n_g^{max} ∼ 2 α_s^{-3/2} in the saturation region.
Dilepton angular distributions in the color-dipole $S$-matrix framework
hep-ph 2025-07 conditional novelty 5.0 of 10

First derivation of Z and W helicity density matrix elements in the color-dipole S-matrix framework, with forward-rapidity predictions at 14 TeV linking the Lam-Tung violation to the UGD tail.
Soft and semihard components of multiplicity distributions in the $k_T$ factorization approach
hep-ph 2025-06 conditional novelty 5.0 of 10

Using a transverse momentum cutoff in kT factorization, the authors fix the soft and semihard mean multiplicities in a double negative binomial fit and find that KNO scaling holds only for narrow rapidity windows and ...
Constraining hot and cold nuclear matter properties from heavy-ion collisions and deep-inelastic scattering
nucl-th 2025-04 unverdicted novelty 5.0 of 10

Global analysis of HERA DIS and ALICE Pb+Pb data using saturation initial-state model determines early-time non-equilibrium η/s of QGP.
Precise determination of pomeron intercept via scaling entropy analysis
hep-ph 2024-12 conditional novelty 5.0 of 10

Measuring the entropy of final-state hadron multiplicities in H1 data gives a Pomeron intercept of 0.322 ± 0.007, consistent with the value from inclusive DIS cross-section scaling.
TMDlib2 and TMDplotter: a platform for 3D hadron structure studies
hep-ph 2021-03 accept novelty 5.0 of 10

TMDlib2 packages about fifty published TMD and unintegrated PDF sets behind a single C++/Fortran interface, adds library-level access to their uncertainty replicas, and ships a web plotter for comparing them with coll...
New CTEQ global analysis of quantum chromodynamics with high-precision data from the LHC
hep-ph 2019-12 unverdicted novelty 5.0 of 10

New CT18 PDFs at NLO and NNLO from global fit to HERA plus LHC jet, Drell-Yan, top-pair and Z data, with Hessian errors, Lagrange-multiplier studies, and alternate sets for data tensions and scale choices.
In depth analysis of the combined HERA data in the dipole models with and without saturation
hep-ph 2018-04 conditional novelty 5.0 of 10

Fitting a saturation model and its linearized (non-saturating) counterpart to the same combined HERA data yields equally good descriptions, so saturation effects are not resolvable in existing electron-proton data but...
Energy dependence of dissociative $\mathrm{J/}\psi$ photoproduction as a signature of gluon saturation at the LHC
hep-ph 2016-08 conditional novelty 5.0 of 10

A proton model with an energy-growing number of gluon hot spots reproduces existing J/psi photoproduction data and predicts that the dissociative cross section peaks near 500 GeV before falling, which the authors read...
Prompt atmospheric neutrino fluxes: perturbative QCD models and nuclear effects
hep-ph 2016-07 conditional novelty 5.0 of 10

Prompt atmospheric muon neutrino fluxes computed in three QCD frameworks with nuclear shadowing come out 10 to 50 percent below free-nucleon predictions at high energy, and the dipole-model variants sit at or above th...
Single inclusive forward hadron production at next-to-leading order
hep-ph 2016-04 conditional novelty 5.0 of 10

The negative NLO forward-hadron cross sections come from subtracting too much of the rapidity divergence; making the subtraction cutoff depend on the produced particle's transverse momentum restores positivity over a ...
Collinearly-improved BK evolution meets the HERA data
hep-ph 2015-07 conditional novelty 5.0 of 10

A resummed small-x QCD evolution equation, with double and single collinear logarithms plus running coupling, describes the HERA reduced cross-section at roughly one chi-squared per data point with four free parameters.
Gluon saturation in dijet production in p-Pb collisions at Large Hadron Collider
hep-ph 2012-05 conditional novelty 5.0 of 10

Using a nonlinear QCD evolution equation fitted to HERA structure-function data, the authors predict that central-forward dijet pairs in p-Pb collisions are suppressed by about a factor of two near back-to-back azimut...
Single Inclusive Hadron Production at RHIC and the LHC from the Color Glass Condensate
hep-ph 2010-01 conditional novelty 5.0 of 10

A running-coupling BK gluon distribution, with two parameters fitted to RHIC forward data, describes p+p and d+Au hadron spectra and predicts roughly factor-two suppression of forward hadrons in p+Pb at the LHC.
Prompt neutrino fluxes from atmospheric charm
hep-ph 2008-06 accept novelty 5.0 of 10

Gluon saturation suppresses the prompt atmospheric neutrino flux above 10^6 GeV, keeping it well below the level invoked for detectable stau production in the Earth.
Nuclear enhancement of universal dynamics of high parton densities
hep-ph 2007-05 conditional novelty 5.0 of 10

The saturation scale in large nuclei grows faster than the naive A^{1/3} estimate because of DGLAP evolution and the impact-parameter profile, producing a strong, measurable A-dependence in diffractive deeply inelasti...

See all 90 Pith citations

How later papers used it

249 later papers cite it

Evidence: substantial. 247 of the citing papers share no author with it, and 90 of those carry a confident label from Pith's citation classifier.

Pith read 40 of the 404 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

20 later papers by other authors are listed below: 11 used its method or tool, 8 used its data or values and 1 compared its own results with it. 1 more paper by the same authors is listed: 1 used its data or values.

Uses its method or tool 11 of 50 papers with this label

Used the approach of this paper to set the variable x-tilde with an effective light-quark mass of 140 MeV when computing F2.

Energy dependence of dissociative $\mathrm{J/}\psi$ photoproduction as a signature of gluon saturation at the LHC (2016) · 1.00 · paraphrased: the licence does not allow quoting
We work at leading order in the number of colours and use the GW model for the target average of two Wilson lines [ref] .

Correlations between azimuthal asymmetries and multiplicity and mean transverse momentum in small collisions systems in the CGC (2021) · $v_2^2$ and correlations · 0.99
We chose the form of $N(x,r)$ given by the model of Golec-Biernat and Wusthoff [ref] : [equation] with the saturation scale, $Q_s(x)$, given by the parameters $λ$, $x_0$ and $Q^2_0$.

Mass dependence of vector meson photoproduction off protons and nuclei within the energy-dependent hot-spot model (2018) · Review of the formalism · 0.98
Used the well-known GBW parametrization as a model for the dipole scattering amplitude.

Low projectile density contributions in the dilute-dense CGC framework for two-particle correlations (2023) · 0.98 · paraphrased: the licence does not allow quoting
Chose this paper's model for its purpose of calculating distributions from HERA data.

Ioffe Times in DIS from a Dipole Model Fit (2009) · 0.97 · paraphrased: the licence does not allow quoting
Used this paper's saturation model to do its calculation, which assigns the neutrino a fixed fraction of the mother meson's momentum.

Prompt neutrino fluxes from atmospheric charm (2008) · 0.88 · paraphrased: the licence does not allow quoting
Employed an s-dependent dipole cross section parametrization similar to this paper's x-dependent GBW parametrization at small Q squared.

Neutrino-production of single pions: dipole description (2011) · 0.84 · paraphrased: the licence does not allow quoting
Used this paper's model together with geometrical scaling to parametrize the dipole gluon distribution.

Drell-Yan Lepton-Pair-Jet Correlation in pA collisions (2012) · 0.93 · paraphrased: the licence does not allow quoting
The dipole amplitude $N(r⃗, x)$ is computed with the Golec-Biernat and Wusthoff model [ref] with the value of the parameters as in Ref. [ref] .

Diffractive photoproduction of dark photons (2024) · Brief overview of the formalism · 0.93
Evaluated structure functions through an integral from this paper, with the dipole cross section as the imaginary part of the dipole-nucleon scattering amplitude.

Studying low-$x$ structure function models with astrophysical tau neutrinos: double bang, lollipop and sugardaddy topologies (2019) · 0.97 · paraphrased: the licence does not allow quoting
Gave the dipole scattering amplitude by this paper's model, with the saturation scale defined by Q0 squared, x0 and lambda.

Incoherent ${\rm J}/\psi$ production at large $|t|$ identifies the onset of saturation at the LHC (2023) · 0.90 · paraphrased: the licence does not allow quoting

Uses its data or values 9 of 33 papers with this label · 1 by the same authors

Used the numerical value of sigma0 of 29.12 mb, obtained from fitting HERA data in this paper.

Pseudo and quasi quark PDF in the BFKL approximation (2023) · 1.00 · paraphrased: the licence does not allow quoting
Took its value of lambda from this paper's fits to deep-inelastic e-p data from HERA, which determined Q0 squared times x0 to the lambda of about 0.097 GeV squared together with lambda.

Relativistic diffusion model for hadron production in p-Pb collisions at the LHC (2024) · 1.00 · paraphrased: the licence does not allow quoting
The values of the involved parameters are obtained from a fit to HERA data as $λ=0.277$, $x_0=0.41·10^-4$ and $Q_0=1 GeV$ [ref] .

The impact of glasma on heavy flavor azimuthal correlations and spectra (2024) · Numerical $R_AA$ (glasma simulations) · 1.00
As for the light quarks, a massive case with $m_l=0.14$ GeV, the value used originally in Ref. [ref] , and the massless cases were studied.

Sudakov effects and the dipole amplitude (2022) · Fits of the Sudakov-improved saturation models · 0.99
Obtained the parameters sigma0 of 29 mb, lambda of 0.28 and x0 of 4 times 10 to the minus 5 from a fit including charm quarks to inclusive deep-inelastic scattering data in this paper.

Impact parameter dependent colour glass condensate dipole model (2007) · 0.99 · paraphrased: the licence does not allow quoting
Fixed its parameters Lambda of 0.6 GeV and lambda of 0.3 by DIS data from this paper.

Forward hadron production in high energy pA collisions: from RHIC to LHC (2007) · 0.98 · paraphrased: the licence does not allow quoting
The free parameters in this model are chosen as $λ= 0.29$, $x_0 = 3 ×10^-4$ and $S_⊥=1/2 ×23 mb$ for proton targets following Ref. [ref] .

Transverse Energy-Energy Correlators in the Color-Glass Condensate at the Electron-Ion Collider (2023) · Quark distribution · 0.98
Gave the parameters lambda, sigma0 and x0 in a table, citing this paper.

Charmed Meson Production in Deep Inelastic Scattering (2007) · 0.90 · paraphrased: the licence does not allow quoting

By the same authors

Took the dipole cross section parameters sigma0 of 29 mb, x0 of 4 times 10 to the minus 5 and lambda of 0.28 from this paper's Fit 2 with charm.

Diffractive structure function F_L^D from fits with higher twist (2007) (same authors) · 1.00 · paraphrased: the licence does not allow quoting

Compares with it 1 of 12 papers with this label

Compared its numerical results for the integrated diffractive gamma-p to V-p photoproduction cross section obtained with this paper's original GBW model and with an updated GBW fit accounting for heavy quarks.

Exclusive photoproduction of excited quarkonia in ultraperipheral collisions (2020) · 0.98 · paraphrased: the licence does not allow quoting

How the literature uses it

225 arXiv papers cite this in 374 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 225 sentences without a confident label

Contests it 1 paper

We found that inserting popular parametrizations such as the GBW model [ref] into higher correlations of the dipole operator, <ref>, leads to inconsistencies for higher-order $N$-point functions, where e.g. the absence of cancellatiosn in the quadrupole correlation leads to a non-vanishing constant even for $||→∞$, resulting in divergent amplitudes.

Saturation effects in exclusive vector meson production in DIS · Proton hotspot model · contest 0.84

Builds on it 2 papers

First of all, we made evaluations in the GBW model [ref] extended to the $b$-dependent partial amplitude $f_q̅q^N(r⃗,b⃗)$.

DVCS via color dipoles: Nonperturbative effects · GBW based partial dipole amplitude · 0.98
In particular, we first study the Golec-Biernat Wusthoff (GBW) model [ref] , extended to include a spatially dependent target and angular correlations.

Diffractive dijet production in impact parameter dependent saturation models · Introduction · 0.96

Uses its method or tool 45 papers

This calculation takes fragmentation into account by assigning the neutrino a fixed fraction of the momentum of the mother meson, and is done using the saturation model of Golec-Biernat and Wüstoff [ref] described above.

Prompt neutrino fluxes from atmospheric charm · Results and discussion · 0.88
We use the $s$-dependent dipole cross section proposed in [ref] , which saturates at large separations in analogy to the $x$-dependent one proposed in [ref] .

Real Compton Scattering via Color Dipoles · Photoabsorption · 0.84
At small $Q^2$ we employ the $s$-dependent parametrization of the dipole cross section [ref] , which is similar to the $x$-dependent GBW parametrization [ref] , but is more suitable for soft processes [equation] These parameters and the scale $s_0=1000$ GeV$^2$ are fitted to data on DIS, real photoproduction and $πp$ scattering.

Neutrino-production of single pions: dipole description · Numerical results · 0.84

Uses its data or values 31 papers

The dipole cross section parameters, $σ_0=29 mb, x_0=4·10^-5$ and $λ=0.28$, are taken from [ref] (Fit 2 with charm).

Diffractive structure function F_L^D from fits with higher twist · Twist–4 contribution · 1.00
The initial condition is given by a form motivated by the momentum space Golec-Biernat, Wusthoff (GBW) model [ref] , and the fit is performed to the reduced cross section data from HERA [ref] , and consistent description is achieved.

Revisiting gluon density from the BK equation with kinematical constraint and large x terms · Introduction · 0.86
The parameters $λ, σ_0$ and $x_0$ are given in Table ([ref]) [ref] .

Charmed Meson Production in Deep Inelastic Scattering · $D$-Meson production in the color-dipole model · 0.90

Compares with it 12 papers

The refitted parameters are close to the original values from [ref] with around $10 %$ decrease of the power $λ$ in the saturation scale ([ref]), see Table 1.

Saturation model of DIS : an update · Conclusions · 0.97
First, data are compared with the energy-dependent hot-spot model by Cepila et al. [ref] , where the energy evolution is given by two sources: the Golec-Biernat Wüsthoff model [ref] and the growth in the number of hot spots inside a nucleon to mimic the growth of the gluon density with energy [ref] .

Evidence for J/$\psi$ suppression in incoherent photonuclear production · Introduction · 0.95
In a separate dedicated analysis, we have compared the numerical results for the integrated diffractive $γp→V p$ photoproduction cross section obtained with the original GBW model [ref] discussed above and with the updated GBW fit accounting for heavy quarks [ref] .

Exclusive photoproduction of excited quarkonia in ultraperipheral collisions · Numerical results for $γp→V p$ cross sections · 0.98

Cites it as support 3 papers

Such models successfully described all the data on deep inelastic scattering, including the behavior of inclusive and diffractive cross sections, at a low value of photon virtuality [ref] .

Soft processes at high energy without soft Pomeron: a QCD motivated model · Introduction · 0.97
For example, in the small-$x$ region, we expect that $q̂$ depends on the gluon saturation scale, which exhibits a power-law behavior as $Q_s^2 ∝x^-1/3$ [ref] .

Probing the jet transport coefficient of cold nuclear matter in electron-ion collisions · Framework and previous results of the analysis · 0.98
This result supports the numerous saturation models that describes well the experimental data [ref] .

Summing Pomeron loops in the dipole approach · Conclusions · 0.97

Cites it as background 36 papers

Since the publication [ref] , numerous fits with the dipole picture of DIS were performed.

Saturation model of DIS : an update · Introduction · 0.93
In Ref. [ref] , following Golec-Biernat–Wüsthoff (GBW) model [ref] , function $Q_s(x)$ – called saturation scale – was taken in the following form

Quantitative Study of Different Forms of Geometrical Scaling in Deep Inelastic Scattering at HERA · Introduction · 0.80
Here [equation] is Bjorken $x$ variable, $M_p$ stands for the proton mass, and the saturation scale $Q_s$ takes the following form [ref] [equation] Here $Q_0$ and $x_0$ are free parameters which can be fitted to the data within some specific model of DIS, and exponent $λ$ is a dynamical quantity of the order of $λ∼0.3$.

Quantitative Study of Geometrical Scaling in Deep Inelastic Scattering at HERA · Introduction · 0.90

Discussion

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J.Jalilian-Marian, A.Kovner, A.Leonidov, H.Weigert , preprint Cavendish-HEP- 98/12, hep-ph/9807462. 14 W=245 (x128) W=210 (x64) W=170 (x32) W=140 (x16) W=115 (x8) W=95 (x4) W=75 (x2) W=60 (x1) ZEUS BPC H1 H1 low Q2 Q2 (GeV2) σγ*p (µb) Q2 (GeV2) σγ*p (µb) 1 10 10 2 10 3 10 4 10 -2 10 -1 1 10 10 2 Figure 3: The γ∗p-cross section for various energies. The so...

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