The Wilson renormalization group for low x physics: towards the high density regime

Alex Kovner; Andrei Leonidov; Heribert Weigert; Jamal Jalilian - Marian

REVIEW

The Wilson renormalization group for low x physics: towards the high density regime

Jamal Jalilian - Marian, Alex Kovner, Andrei Leonidov, Heribert Weigert

High Energy Physics - Phenomenology · Submitted 16 Jun 1997 · 12 references

Pith's verdict: A short, honest step in the Wilson RG program: Eq. (32) is a real structural advance, but its kernels live in a paper still in preparation.

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arXiv IDhep-ph/9706377 (version 1)

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CitationAlex Kovner, Andrei Leonidov, Heribert Weigert, Jamal Jalilian - Marian. "The Wilson renormalization group for low x physics: towards the high density regime." arXiv:hep-ph/9706377 (1997). https://pith.science/paper/INHEOTYS

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Editor's note

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The thing to know is that Eqs. (31) and (32) are new. The earlier papers in this series set up the Wilson RG machinery and checked the BFKL limit, but neither did the functional integral over the color charge density at general F. Here they do it by steepest descent and get a closed nonlinear equation for the weight functional: a diffusion term built from the fluctuation correlator χ and a drift term from the mean shift σ. That is the object the saturation program needs, and it is written in a form you can actually read physically — real ladder emission in χ, virtual one-loop in σ. The algebra from (23) through (30) is laid out step by step and I could follow it; the reduction to BFKL in the weak-field limit is a genuine external benchmark against a published result, not a self-check. The paper also does not oversell itself, which I appreciate in a note this short. Two soft spots, in order of how much they matter. First, σ and χ are deferred to reference [10], listed as in preparation. Until those kernels exist, Eq. (32) is a template rather than an equation you can evaluate, and the authors say so plainly. That is the honest limitation and it is the main reason to read this as a progress report. Second, the stress-test note is right that the more load-bearing unstated assumption is Eq. (10): truncating the action at quadratic order in δA is what makes δρ Gaussian and kills the higher connected correlators in Eq. (19), and the paper asserts that rather than argues it. The cubic vertex is O(α_s^{1/2}) in the same counting, so the leading-log argument that it only enters at two loops should have been stated in a sentence. I would call this a real but mild gap, not a flaw: the same truncation underlies the BFKL derivation in [7], so the benchmark is partial evidence the approximation is fine at leading log. The fourth-order truncation the reader flagged is not a problem — the power counting does work out. This is for anyone working on small-x evolution or saturation; it is the paper the later JIMWLK literature descends from. Send it to a referee.

Referee report

Significance

If the derivation holds, this is a genuinely useful structural result: it reduces the full nonlinear Wilsonian RG for low-x physics to two functionals, σ and χ, with a transparent real/virtual interpretation, and it organizes the saturation problem as a closed flow of the weight functional rather than as an ad hoc modification of linear evolution. The paper is analytic throughout, contains no free parameters and no fitting, and it supplies an external benchmark of the right kind: in the weak-field limit, where σ is linear and χ quadratic in ρ, Eq. (34) is stated to reduce to the published BFKL equation of Ref. [8], as shown in the published companion Ref. [7]. The observation that a Gaussian F is not an eigenfunction of Eq. (31), and that quartic terms are generated even with weak-field σ and χ, is a substantive physical point that constrains how initial conditions for the flow may legitimately be chosen. I also checked the algebraic passage from Eq. (31) to Eq. (32) by direct substitution of Z' = -Z F' and Z'' = Z(F'² - F''), and the two equations are equivalent term by term; that consistency is a real strength of the presentation.

Major objections (3)

[Eq. (10) and Eq. (19)] The Gaussian form of the δρ weight in Eq. (21), and hence the entire closed form of Eqs. (31)-(32), rests on the truncation of the action at quadratic order in δA in Eq. (10) ('keeping terms of the first and second order in δA'). This truncation is what makes δA exactly Gaussian and therefore what makes all connected correlators of δρ beyond the two-point function vanish at leading order, as asserted in Eq. (19). The paper does not give the argument for why the cubic and quartic δA self-interactions, which are present in G² of Eq. (2), can be dropped at leading logarithmic accuracy. In the counting used elsewhere in the paper the cubic vertex is not obviously a higher-order effect, so the justification must be the leading-log one: that these vertices contribute only beyond O(α_s ln(1/x)). Please state that argument explicitly, or, if there is a region of phase space where the cubic vertex contributes at leading log, say how the Gaussian characterization of the δρ distribution survives it. As written, Eq. (19) is asserted from 'simple counting of powers of the coupling constant' following Eqs. (12)-(13), but the counting that matters here is in α_s ln(1/x), not in α_s alone.
[Eqs. (20)-(21) and the deferral to Ref. [10]] The equation that the paper advertises as the 'full nonlinear renormalization group equation' is expressed entirely in terms of σ[ρ] and χ[ρ], and those two functionals are not computed here; the paper states that the calculation 'is technically nontrivial [10] and is beyond the scope of this note', with Ref. [10] listed as 'in preparation'. This is an acknowledged limitation, and I do not ask that the full calculation be reproduced. But it does mean that the reader cannot verify the central equation in any concrete case from this manuscript alone. A short, self-contained remedy is available: the weak-field expressions for σ and χ are already given in the published companion Ref. [7], so the paper could display them explicitly and exhibit the reduction of Eq. (34) to BFKL as a worked check rather than as a citation. That would convert Eq. (32) from a formal identity into a demonstrated statement and would also make the paper usable independently of Ref. [10]. I would regard this as the single most valuable addition.
[Text preceding Eq. (23), and the abstract] The steepest-descent treatment assumes, in the authors' own words, that 'F[ρ] is a smooth functional of ρ and that the scale of its variation is independent of the strong coupling constant α_s'. The paper's stated target, however, is the high-density regime (title and abstract:s2), and it is precisely there that the curvature δ²F/δρ² may grow with density, potentially making the Gaussian width ηχ comparable to the scale of variation of F and invalidating the truncation. The paper should state the condition under which the saddle-point expansion holds, for instance a bound of the form η χ F'' << 1, and discuss whether that condition is expected to survive as ρ grows. If the expansion is in fact valid only in the dilute regime, the abstract's claim that the result 'should serve as the basis of the analysis of saturation effects' should be qualified accordingly.

Minor objections (6)

[Eqs. (26)-(27)] The statement that 'to reach this accuracy one needs to take into account up to the third order contributions in the steepest descent integration' is asserted rather than shown. The counting does appear to work out, since U'' and U^{(n)} for n ≥ 2 scale as 1/η while fluctuations scale as η^{1/2}, so the fifth-order term is O(η^{3/2}) and the cubic term vanishes at first order by symmetry, leaving only the quartic and the squared cubic at O(η). Please state this explicitly in one or two lines so the reader does not have to reconstruct it.
[Eq. (20) and Eqs. (30)-(32)] The symbol η is defined in Eq. (20) as α_s ln(1/x) and used through Eq. (23), but Eqs. (30)-(32) revert to writing α_s and ln(1/x) separately, with η absorbed into the derivative d/dln(1/x). The change is correct but the reader has to check it; a sentence noting that η is absorbed would help.
[Eq. (21)] The notation ∫D[ρ, ρ'] in Eq. (21) is loose, since ρ' is constrained by the delta function of Eq. (16) and is not an independent integration variable at that stage. Writing the constrained form explicitly, or stating that the ρ' integral is a bookkeeping device, would remove the ambiguity.
[Eq. (34) and the discussion following it] The paper correctly notes that Eq. (34) is not closed in general because σ and χ depend nonlinearly on ρ. It would strengthen the discussion to state explicitly what approximation is required to close it at second order in the gluon density, and to say whether the reduction to the GLR equation with the Mueller-Qiu coefficient is expected to follow in the double logarithmic approximation alone or requires further truncation.
[Eq. (36)] The normalization of the unintegrated gluon density φ(k_t, x) entering the initial condition μ is not specified, nor is the normalization of S(b_t). Since Eq. (36) is offered as the practical starting point for the flow, a normalization statement would make it usable.
[References and typography] There are several typographical errors: 'repesentation' and 'nolinear' in the third paragraph, and 'It argued a long time ago' (missing 'was') in the first paragraph. In the text Ref. [11] is cited as 'Levin and Laenen' while the reference list gives 'E. Laenen and E. Levin'; please make the two consistent.

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The paper

The paper claims

The paper claims that the full nonlinear Wilson renormalization-group equation for low-$x$ physics can be written in closed form using only the mean and fluctuation of the color charge density induced by integrating out gluons.

The reading

This paper asks whether the Wilsonian renormalization-group evolution of the low-$x$ color-charge weight functional can be written in closed form without first solving the full gluon dynamics. The authors argue yes: after integrating out intermediate-$x$ gluons, the change in the weight is controlled by two functionals of the charge density, the mean shift $sigma[rho]$ and the fluctuation correlator $chi[rho]$. The resulting equation, $d Z / d ln(1/x) = alpha_s { (1/2) delta^2 [Z chi^{uv}] / delta rho_u delta rho_v - delta [Z sigma^u] / delta rho_u }$, is the paper's central claim. If correct, it gives a single nonlinear evolution law whose weak-density limit is the BFKL equation and whose nonlinear terms are where saturation physics would enter.

What carries the argument

The machinery is the separation of the charge density after one renormalization-group step into a mean shift and a Gaussian fluctuation, $rho' = rho + eta sigma + delta rho$ with $langle delta rho delta rho rangle = eta chi$. Because the fluctuation weight is steep in $alpha_s$, the integral over $rho$ can be done by steepest descent around $rho_0$, and the Gaussian corrections can be re-exponentiated. The nontrivial step is keeping only the terms that survive at order $alpha_s$: the second derivative of the Gaussian width, the square of the first derivative of $F$, cross terms with $sigma$ and $chi$, and the cubic and quartic fluctuation corrections. The outcome is Eq. (32), a diffusion-drift equation in charge-density space: a second-derivative term with coefficient $chi$ plus a first-derivative term with velocity $sigma$.

Extended reading notes

Core claim

The paper's claim is that the full nonlinear Wilson renormalization-group equation for low-$x$ physics can be expressed through the average extra color charge density generated by integrating out gluons, $sigma[rho]$, and its connected two-point fluctuation, $chi[rho]$, with the expansion parameter $eta = alpha_s ln(1/x)$. Starting from the path integral over charge density and fluctuation field, the authors treat the Gaussian weight for the charge fluctuation by steepest descent, keep terms through first order in $alpha_s$, and obtain a closed evolution equation for $Z[rho] = exp(-F[rho])$ (Eq. 32), equivalently for $F$ (Eq. 31). The two terms are read as real and virtual contributions: $chi$ from ladder-like gluon fluctuations and $sigma$ from virtual one-loop diagrams. The paper does not compute $sigma$ or $chi$; it states that this calculation is presented elsewhere, so the result is a conditional closed form rather than a fully evaluated evolution kernel.

Load-bearing premise

The load-bearing premise is that $F[rho]$ varies on a scale independent of $alpha_s$, so that the Gaussian fluctuation weight is sharply peaked and the steepest-descent expansion truncated at fourth order in fluctuations captures every term through first order in $alpha_s$ for arbitrary $F$.

Editorial extensions

If this is right

In the weak-charge-density limit, with $sigma$ linear and $chi$ quadratic in $rho$, Eq. (34) reduces to the BFKL equation for the unintegrated gluon density.
At second order in gluon density and neglecting virtual contributions in the double logarithmic approximation, Eq. (34) should reduce to the GLR saturation equation with the coefficient computed in [12].
The evolution equation can be written directly for correlators $langle rho_{x_1} ... rho_{x_n} rangle$, giving a hierarchy in which the two-point function is sourced by $chi$ and $sigma$.
A Gaussian initial weight of the form $F = int rho mu^{-1} rho$ is not an eigenfunction; evolution generates non-quadratic terms even with weak-field $sigma$ and $chi$, so a Gaussian can only be an initial condition at moderate $x$.
The natural initial condition uses the unintegrated gluon density $varphi(k_t,x_0)$ multiplied by an impact-parameter profile $S(b_t).

Reading between the lines

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

Editorial inference: if the closed form survives explicit evaluation, it turns the saturation problem into a transport problem in the space of two-dimensional color-charge configurations, where $chi$ is a diffusion coefficient and $sigma$ a drift; numerical solution could then track the onset of saturation without resumming each gluon ladder separately.
Editorial inference: the equation suggests a practical diagnostic for where BFKL stops being adequate, namely the size of the term quadratic in $delta F / delta rho$ relative to the linear term, governed by the same $chi$ that drives real emission.
Editorial inference: because $sigma$ and $chi$ are themselves functionals of $rho$, the hierarchy for correlators is not a truncation in the number of charges but in the nonlinearity of the kernel; a mean-field closure might be testable against exact small-$x$ data in the moderate-density window.
Editorial inference: a lattice or light-cone simulation of the one-step charge-density reweighting could check Eq. (32) directly at small but nonzero $alpha_s$, independent of the weak-field expansion.

Original abstract

We continue the study of the effective action for low $x$ physics based on a Wilson renormalization group approach. We express the full nonlinear renormalization group equation in terms of the average value and the average fluctuation of extra color charge density generated by integrating out gluons with intermediate values of $x$. This form clearly exhibits the nature of the phenomena driving the evolution and should serve as the basis of the analysis of saturation effects at high gluon density at small $x$.

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, 2,970 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.03 · compares with prior results 0.08 · mentions code or data availability 0.01.

The abstract, sentence by sentence

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

We continue the study of the effective action for low $x$ physics based on a Wilson renormalization group approach. We express the full nonlinear renormalization group equation in terms of the average value and the average fluctuation of extra color charge density generated by integrating out gluons with intermediate values of $x$. This form clearly exhibits the nature of the phenomena driving the evolution and should serve as the basis of the analysis of saturation effects at high gluon density at small $x$.

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 continue the study of the effective action for low $x$ physics based on a Wilson renormalization group approach.	contribution 0.50	0.79	0.05	asserted 0.77
s1	We express the full nonlinear renormalization group equation in terms of the average value and the average fluctuation of extra color charge density generated by integrating out gluons with intermediate values of $x$.	method 0.50	0.91	0.11	asserted 1.00
s2	This form clearly exhibits the nature of the phenomena driving the evolution and should serve as the basis of the analysis of saturation effects at high gluon density at small $x$.	interpretation 0.93	0.43	0.45	asserted 0.70

What kind of paper the classifier reports this is: theoretical result 0.94 · evidence from proof or derivation 0.98.

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

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S sections
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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

References

Works cited (12)

[10]

J. Jalilian - Marian, A. Kovner and H. Weigert, in preparation

View on Pith
[7]

J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert “ The BFK L Equation from the Wilson Renormalization Group”, hep-ph/9701284

View on Pith Pith review arXiv
[6]

J. Jalilian-Marian, A. Kovner, L. McLerran and H. Weigert, Phys.Rev. D55, 5414 (1997) 13

View on Pith 1997
[8]

E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Sov. Phys. JETP 45 (1977), 199; Ya.Ya. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978), 22

View on Pith 1977
[1]

E.M. Levin, ”The Pomeron: Yesterday, Today and Tomorrow”, CBPF-NF- 010/95, hep-ph/9503399

View on Pith Pith review arXiv
[2]

H1 Collaboration, I. Abt et al. , Nucl. Phys. B407 (1993), 515; T. Ahmed et al. , Nucl. Phys. B439 (1995), 471; ZEUS Collaboration, M. Derrick et al. , Phys. Lett. B316 (1993), 412; Z. Phys. C65 (1995), 379

View on Pith 1993
[3]

A.D. Martin, W.J. Stirling and R.G. Roberts Phys.Rev. D50, 6734 (1994); CTEQ collaboration, H.L. Lai et. al. Phys.Rev. D51, 4763 (1995)

View on Pith 1994
[4]

L.V. Gribov, E.M. Levin and M.G. Ryskin, Phys. Rep 100 (1981)

View on Pith 1981
[5]

L. McLerran and R. Venugopalan, Phys. Rev. D49 (1994), 2233; D49 (1994), 3352

View on Pith 1994
[9]

A. Ayala, J. Jalilian-Marian, L. McLerran and R. Venugopalan, Phys. Rev. D52 (1995), 2935; Phys. Rev. D53 (1996), 458

View on Pith 1995
[11]

E. Laenen and E. Levin, Nucl. Phys. B451 (1995), 297

View on Pith 1995
[12]

A.H. Mueller and J.W. Qiu, Nucl. Phys. B268 (1986), 427. 14

View on Pith 1986

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A two-loop multi-Reggeon exchange contribution to 2->3 amplitudes is computed for the first time, enabling extraction of the two-loop Lipatov vertex.
Complete calculation of exclusive heavy vector meson production at next-to-leading order in the dipole picture
hep-ph 2022-04 conditional novelty 6.0 of 10

A complete next-to-leading-order calculation of exclusive heavy vector meson production in the Color Glass Condensate dipole picture, including the leading relativistic correction, reproduces HERA and LHC J/psi and Up...
Dijet photoproduction at low $x$ at next-to-leading order and its back-to-back limit
hep-ph 2022-04 conditional novelty 6.0 of 10

A full NLO CGC calculation of dijet photoproduction shows that the expected Sudakov double logarithm only emerges with the correct sign once the low-x evolution is kinematically improved.
High-energy Operator Product Expansion at sub-eikonal level
hep-ph 2021-01 conditional novelty 6.0 of 10

The high-energy OPE for deep inelastic scattering is extended to sub-eikonal order, yielding new impact factors and evolution equations for polarized structure functions at small x.
Forward dijets in proton-nucleus collisions at next-to-leading order: the real corrections
hep-ph 2020-09 conditional novelty 6.0 of 10

New simplified trijet cross-sections in the quark channel are integrated over one unmeasured parton to get the real NLO dijet corrections, reproducing B-JIMWLK and DGLAP evolution in the soft and collinear limits.
Color Glass Condensate at next-to-leading order meets HERA data
hep-ph 2020-07 conditional novelty 6.0 of 10

A first next-to-leading-order dipole-picture fit to HERA inclusive data shows that three improved Balitsky-Kovchegov evolution formulations describe the measurements equally well, and that light-quark data alone deman...
Helicity-dependent generalization of the JIMWLK evolution
hep-ph 2019-10 conditional novelty 6.0 of 10

The authors derive a helicity-dependent version of the JIMWLK evolution equation whose kernel adds spin-tracking sub-eikonal terms to the standard high-energy QCD kernel.
Non-linear evolution in QCD at high-energy beyond leading order
hep-ph 2019-02 conditional novelty 6.0 of 10

Evolving the NLO BK equation in target rapidity rather than projectile rapidity removes the large scheme dependence that plagued earlier collinear resummations and yields a stable, resummable evolution equation.
Subleading Power Rapidity Divergences and Power Corrections for $q_T$
hep-ph 2018-12 accept novelty 6.0 of 10

A new rapidity-only regulator makes subleading-power rapidity divergences tractable, and the resulting one-loop power correction to color-singlet q_T spectra is shown to involve derivatives of parton distribution functions.
Confronting impact parameter dependent JIMWLK evolution with HERA data
hep-ph 2018-06 conditional novelty 6.0 of 10

Full JIMWLK evolution of a finite-size proton, confronted with HERA structure function and diffractive J/psi data, indicates that gluon saturation is not accessible in electron-proton collisions at HERA energies.
Dipole factorization for DIS at NLO: Combining the $q\bar{q}$ and $q\bar{q}g$ contributions
hep-ph 2017-08 conditional novelty 6.0 of 10

The full next-to-leading-order corrections to the longitudinal and transverse photon-target cross sections are derived in dipole-factorized form, with the ultraviolet divergences of the quark-antiquark and quark-antiq...
Small-$x$ Asymptotics of the Gluon Helicity Distribution
nucl-th 2017-06 conditional novelty 6.0 of 10

At small x the gluon helicity distribution rises as (1/x)^α with α = 13/(4√3)√(α_s N_c/2π) ≈ 1.88√(α_s N_c/2π), about 20% below the quark helicity exponent.
Small-$x$ Asymptotics of the Quark Helicity Distribution: Analytic Results
hep-ph 2017-03 conditional novelty 6.0 of 10

The small-x quark helicity distribution grows as (1/x)^alpha_h with alpha_h = (4/sqrt(3)) sqrt(alpha_s N_c / 2 pi), derived analytically and matching the prior numerical value 2.31 sqrt(alpha_s N_c / 2 pi).
Helicity Evolution at Small $x$: Flavor Singlet and Non-Singlet Observables
hep-ph 2016-10 conditional novelty 6.0 of 10

The large-Nc flavor non-singlet helicity evolution equation yields an intercept sqrt(alpha_s N_c/pi) that matches Bartels-Ermolaev-Ryskin, while the flavor-singlet intercept comes out about 35 percent below theirs.
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.
Evidence of strong proton shape fluctuations from incoherent diffraction
hep-ph 2016-03 conditional novelty 6.0 of 10

Incoherent J/psi photoproduction at HERA can only be reproduced if the proton's small-x gluon density is strongly lumpy, with geometric fluctuations dominating over saturation-scale and colour-charge fluctuations.
Next-to-leading order Balitsky-Kovchegov equation with resummation
hep-ph 2016-01 conditional novelty 6.0 of 10

Resumming large transverse logarithms stabilizes the next-to-leading-order Balitsky-Kovchegov equation, and the remaining fixed-order corrections are small at high saturation scales but numerically important near phen...
Helicity Evolution at Small-x
hep-ph 2015-11 conditional novelty 6.0 of 10

The paper constructs small-x evolution equations that resum double logarithms of 1/x for quark and gluon helicity, close them in the large-Nc and large-Nc & Nf limits, and check the ladder limit against prior work.
Tracing the origin of azimuthal gluon correlations in the color glass condensate
hep-ph 2015-09 conditional novelty 6.0 of 10

The color field domain model used to explain azimuthal correlations in proton-nucleus collisions is shown to be equivalent to non-Gaussian correlations of color electric fields in the color glass condensate.
Resummation of non-global logarithms and the BFKL equation
hep-ph 2015-01 conditional novelty 6.0 of 10

A color density matrix functional resums non-global logarithms to all orders, and its two-loop anomalous dimension reproduces the NLO BFKL and Balitsky-JIMWLK equations.
Non-global Logarithms at 3 Loops, 4 Loops, 5 Loops and Beyond
hep-ph 2014-03 accept novelty 6.0 of 10

The leading non-global logarithm series for the hemisphere mass distribution is computed analytically through five loops at large Nc, with the five-loop coefficient given in closed form.
The non-linear evolution of jet quenching
hep-ph 2014-03 accept novelty 6.0 of 10

Extending JIMWLK/BK evolution beyond the eikonal (shockwave) limit produces a functional equation for jet quenching whose dominant double-logarithmic corrections resum into a renormalized jet quenching parameter.
Improving the kinematics for low-x QCD evolution equations in coordinate space
hep-ph 2014-01 conditional novelty 6.0 of 10

A kinematically constrained version of the BK equation in mixed space is derived, with the constraint shown to be local in the dipole cascade and with explicit counter-terms for NLO observables and NLL evolution.
Analysing radial flow features in p-Pb and p-p collisions at several TeV by studying identified particle production in EPOS3
nucl-th 2013-12 conditional novelty 6.0 of 10

EPOS3, a viscous-hydrodynamics event generator, reproduces identified-particle spectra in p-Pb and p-p collisions and is used to argue that radial flow is present even in small systems.
When does the gluon reggeize?
hep-th 2013-09 conditional novelty 6.0 of 10

In the strict planar limit, high-energy amplitudes are governed by a finite, process-dependent set of Wilson lines rather than reggeized gluons, and BFKL dynamics breaks the dipole formula for infrared divergences at ...
Resummation of non-global logarithms at finite $N_c$
hep-ph 2013-04 conditional novelty 6.0 of 10

Simulating the finite-Nc Langevin dynamics of Wilson lines yields the first numerical resummation of non-global logarithms, and the result tracks the large-Nc mean-field answer closely while individual trajectories fl...
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...

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How later papers used it

305 later papers cite it

Evidence: some. 301 of the citing papers share no author with it, and 25 of those carry a confident label from Pith's citation classifier.

Pith read 5 of the 417 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

1 later paper by other authors is listed below: 1 used its method or tool.

Uses its method or tool 1 of 17 papers with this label

To contrast with experimental data, it was selected deep inelastic processes using the CGC/saturation equations from Ref. [ref] in a simple version of the non-linear Balitsky-Kovchegov (BK) equation [ref] .

CGC and saturation approach: Impact-parameter dependent model of perturbative QCD and combined HERA data (2023) · Introduction · 0.99

How the literature uses it

292 arXiv papers cite this in 404 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 · 199 sentences without a confident label

Contests it 1 paper

Such a picture not only contradicts the QCD expectations [ref] , but also contradicts available experimental data.

Inelastic processes in DIS and N=4 SYM · Introduction · contest 0.71

Builds on it 2 papers

In this paper we construct a helicity analogue of JIMWLK evolution [ref] , or, more precisely, a generalization of the JIMWLK evolution that includes helicity dependence.

Helicity-dependent generalization of the JIMWLK evolution · Introduction · 0.98
In this paper we continue our effort [ref] to comprehend such interactions, by constructing a model that incorporates the advantages of two theoretical approaches to high energy QCD: the CGC/saturation approach [ref] and the BFKL Pomeron calculus [ref] .

CGC/saturation approach for high energy soft interactions: $\mathbf{v_2}$ in proton-proton collisions · Introduction · 0.80

Uses its method or tool 15 papers

In this work, we incorporate the energy evolution based on the perturbative JIMWLK equations [ref] into the IP-Glasma initial state description.

Collision energy dependence in heavy ion collisions from nonlinear QCD evolution · Introduction · 0.95
To contrast with experimental data, it was selected deep inelastic processes using the CGC/saturation equations from Ref. [ref] in a simple version of the non-linear Balitsky-Kovchegov (BK) equation [ref] .

CGC and saturation approach: Impact-parameter dependent model of perturbative QCD and combined HERA data · Introduction · 0.99
We will first work in the quasi-classical GGM/MV approximation [ref] , then we will include LO small-$x$ BK/JIMWLK evolution [ref] .

Novel Cross Section Ratios as Possible Signals of Saturation in UPCs · Estimates of the double ratio in UPCs · 0.97

Uses its data or values 1 paper

(The relevant $x$–variable in this diffractive context is the longitudinal momentum fraction $x_ℙ$ lost by the nuclear target and taken by the Pomeron.) The smallest values of $x_ℙ$ explored by the dijet measurements in [ref] , in the ballpark of $x_ℙ=0.01$, are only marginally favourable for saturation — for instance, they do not allow one to test the high-energy evolution in QCD, as described by the BK/JIMWLK equations [ref] .

Probing gluon saturation via diffractive jets in ultra-peripheral nucleus-nucleus collisions · Introduction · 0.97

Compares with it 3 papers

We will determine the second, third, and fourth azimuthal Fourier coefficients of the two particle correlation function within these two approximation schemes and compare the results to numerical lattice simulations of the full correlator in the McLerran–Venugopalan (MV) model [ref] and after renormalization group (JIMWLK) evolution [ref] of the MV model initial conditions to higher rapidities [ref] .

Tracing the origin of azimuthal gluon correlations in the color glass condensate · Introduction · 0.95
In this respect the treatment of [ref] is quite similar to the unpolarized small-$x$ evolution [ref] .

Helicity Evolution at Small $x$: the Single-Logarithmic Contribution · Evolution Equations · 0.93
We note that the unpolarized evolution resummation parameter $ln(1/x)$ at $x=0.01$ is approximately equal to the polarized evolution parameter $ln^2 (1/x)$ at $x=0.1$, suggesting comparable accuracy for our helicity evolution with $x_0 = 0.1$ and the unpolarized small-$x$ evolution [ref] with the commonly used value of $x_0=0.01$ [ref] .

First analysis of world polarized DIS data with small-$x$ helicity evolution · Constraints from Polarized DIS Data · 0.90

Cites it as support 1 paper

It has been demonstrated in Refs. [ref] , that the CGC equations [ref] as well as all other approaches, based on perturbative QCD, lead to the amplitude that increases as a power of energy, resulting in the violation of the Froissart theorem [ref] [It should be noted that NLO corrections to the kernel of CGC evolution equations does not change the power-like behaviour of the scattering amplitude but improve the situation moving the violation of the Froissart theorem to larger values of $b$ [ref] .].

Non-linear evolution in the re-summed next-to-leading order of perturbative QCD:\confronting the experimental data · Introduction · 0.97

Cites it as background 160 papers

However, high density QCD [ref] leads to a completely different picture of inclusive hadron production.

Gluon saturation and inclusive hadron production at LHC · Introduction · 0.84
In recent years, a novel approach to small-$x$ evolution for helicity distributions has been developed in [ref] , based on the $s$-channel evolution/shock wave formalism previously constructed in [ref] for unpolarized high-energy scattering.

Orbital Angular Momentum at Small $x$ Revisited · Introduction · 0.85
Its energy evolution is governed by the nonlinear Balitsky–Kovchegov (BK) equation [ref] or, more generally, by the Jalilian-Marian–Iancu–McLerran–Weigert–Leonidov–Kovner (JIMWLK) hierarchy [ref] .

Unbiased Data-Driven Determination of the Nuclear Dipole Amplitude in the Color Glass Condensate · Introduction · 0.97

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [10]

J. Jalilian - Marian, A. Kovner and H. Weigert, in preparation

View on Pith

[2] [7]

J. Jalilian-Marian, A. Kovner, A. Leonidov and H. Weigert “ The BFK L Equation from the Wilson Renormalization Group”, hep-ph/9701284

View on Pith Pith review arXiv

[3] [6]

J. Jalilian-Marian, A. Kovner, L. McLerran and H. Weigert, Phys.Rev. D55, 5414 (1997) 13

View on Pith 1997

[4] [8]

E.A. Kuraev, L.N. Lipatov and V.S. Fadin, Sov. Phys. JETP 45 (1977), 199; Ya.Ya. Balitsky and L.N. Lipatov, Sov. J. Nucl. Phys. 28 (1978), 22

View on Pith 1977

[5] [1]

E.M. Levin, ”The Pomeron: Yesterday, Today and Tomorrow”, CBPF-NF- 010/95, hep-ph/9503399

View on Pith Pith review arXiv

[6] [2]

H1 Collaboration, I. Abt et al. , Nucl. Phys. B407 (1993), 515; T. Ahmed et al. , Nucl. Phys. B439 (1995), 471; ZEUS Collaboration, M. Derrick et al. , Phys. Lett. B316 (1993), 412; Z. Phys. C65 (1995), 379

View on Pith 1993

[7] [3]

A.D. Martin, W.J. Stirling and R.G. Roberts Phys.Rev. D50, 6734 (1994); CTEQ collaboration, H.L. Lai et. al. Phys.Rev. D51, 4763 (1995)

View on Pith 1994

[8] [4]

L.V. Gribov, E.M. Levin and M.G. Ryskin, Phys. Rep 100 (1981)

View on Pith 1981

[9] [5]

L. McLerran and R. Venugopalan, Phys. Rev. D49 (1994), 2233; D49 (1994), 3352

View on Pith 1994

[10] [9]

A. Ayala, J. Jalilian-Marian, L. McLerran and R. Venugopalan, Phys. Rev. D52 (1995), 2935; Phys. Rev. D53 (1996), 458

View on Pith 1995

[11] [11]

E. Laenen and E. Levin, Nucl. Phys. B451 (1995), 297

View on Pith 1995

[12] [12]

A.H. Mueller and J.W. Qiu, Nucl. Phys. B268 (1986), 427. 14

View on Pith 1986