Green's Functions in the Color Field of a Large Nucleus

Larry McLerran; Raju Venugopalan

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Green's Functions in the Color Field of a Large Nucleus

Larry McLerran, Raju Venugopalan

High Energy Physics - Phenomenology · Submitted 22 Feb 1994 · 7 references

Pith's verdict: The paper that makes the McLerran–Venugopalan model usable — explicit background-field propagators and a closed-form sea quark distribution — with one load-bearing step in Section 5 asserted rather than shown.

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arXiv IDhep-ph/9402335 (version 1)

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CitationLarry McLerran, Raju Venugopalan. "Green's Functions in the Color Field of a Large Nucleus." arXiv:hep-ph/9402335 (1994). https://pith.science/paper/XTTUM7P3

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Third paper in the McLerran–Venugopalan series, and the one that makes the model usable: explicit background-field Green's functions for scalars, fermions and gluons, plus a closed-form sea quark distribution. If you work on small-x nuclear partons, this is the reference for the propagators themselves, not just the source picture of the earlier papers (abstract:s3). What is new: Sections 3 and 4 do the matching across x⁻ = 0 carefully — the field is pure gauge on either side, only the step function matters — and the results (Eqs. 40, 47, 52, 54) are simple enough to check by hand. The fermion case reduces to the scalar one by a 2k⁺ factor, which is right. Section 5 then gets Eq. (61), and with γ(p_t) from the companion paper, Eq. (64): dN_sea/dy ∝ μ² α_s² ln(1/α_s), independent of quark mass for m << μ (abstract:s4). I checked the inner k_t integral of Eq. (62) — it is πp_t² — and the coefficient of Eq. (64) comes out exactly, as does the α_s/π suppression relative to glue. Soft spots, in order of how much they matter. The step from Eq. (57) to Eq. (58) is one sentence: 'It can also be shown that the pieces singular in the limit k⁺′→k⁺ either vanish or cancel out.' That is load-bearing. The whole point of Eq. (55) is that the k⁺′→k⁺ limit must be taken after the k⁻′ integral because of the 1/(k⁺′−k⁺+iε) pole, and the two surviving integrals run over disjoint k⁻′ ranges yet are assigned a common cutoff Z. If the singular pieces do not cancel identically, Eq. (61) is regulator-dependent and the distribution is an artifact of the subtraction. I would want that shown. Related: Eqs. (58)–(59) carry k⁺² while Eq. (61) carries 1/k⁺, a k⁺² mismatch that is either a transcription slip or an error in one of the two. Second, the normalization of Eq. (64) traces entirely to the Gaussian weight Eq. (2) and to γ(p_t) ∝ μ²/p_t⁴ in Eq. (63), which is quoted from Ref. [2] — listed there as 'to appear.' A reader cannot check Eq. (63) from this manuscript. The Gaussian ansatz is stated, not derived; a non-Gaussian charge distribution with the same μ² would leave the propagators intact and change the coefficient. The authors are otherwise candid about the limits: k_t << μ for the classical treatment, α_s ln(k_t/μ) corrections uncomputed, m ≥ μ dominated by k_t >> μ, and the appendix flags its own boundary-condition and light-cone IR problems. Who should read it: anyone doing small-x nuclear QCD or building on the MV model. It deserves referees — the propagator sections are solid and the sea quark result should be checked once and then cited for a long time.

Referee report

Significance

If the result holds, this is a foundational computation: it gives, for the first time within this framework, an explicit and parameter-free-in-form expression for the intrinsic sea quark content of the small-x nuclear wavefunction, and it makes the sharp, falsifiable statement that this content saturates and is mass-independent for m_quark << µ. The paper is also candid about its kinematic window (α_s²µ² << k_t² << µ²) and about where the method fails (m_quark ≳ µ, k_t ≳ µ). I give explicit credit for the parts of the chain I could check independently: the inner integral of Eq. (62), ∫d²k_t ln[((k_t+p_t)²+M²)/(k_t²+M²)], does equal πp_t², and combining that with Eq. (63) reproduces the coefficient 2(N_c²−1)/π² of Eq. (64) and the α_s/π ratio to Eq. (65) exactly. The continuity check at x⁻=0 in Eq. (36) also closes, since ∫d²q_t/(2π)² e^{iq_t·x_t}U(x_t)V†(q_t) = U(x_t)U†(x_t) = 1.

Major objections (4)

[§5, Eqs. (55)–(58)] The step from Eq. (57) to Eq. (58) is asserted rather than shown, and it is load-bearing for the well-definedness of Eq. (61). Eq. (55) defines the distribution through a k₊′ integral that must be taken to k₊′→k₊ only after the k₋′ integration, precisely because δD carries a 1/(k₊′−k₊+iε) singularity (Eq. 47). In Eq. (57) the limit is moved outside the k₋′ integrals, and the text then states in one sentence that “the pieces singular in the limit k₊′→k₊ either vanish or cancel out.” No demonstration is given. This matters because the two integrals being combined run over disjoint ranges (k₋′<0 and k₋′>0) yet are assigned a common cutoff Z; if the singular pieces do not cancel identically, the extracted distribution is regulator-dependent and Eq. (61) is an artifact of the subtraction rather than a property of the background field. Please show the cancellation explicitly, or state the subtraction prescription that makes it hold.
[§5, Eqs. (58)–(61)] The printed equations are not mutually consistent in their power of k₊. Eq. (58) gives ½(ln[(k_t+p_t)²+M²] − ln Z)k₊², and Eq. (59) carries the prefactor 4N_c/((2k₊)(2π)⁴), so the combination scales as k₊; Eq. (61) instead carries 4N_c/((2π)⁴k₊). Since dN/d³k must scale as 1/k₊ at fixed dN/dy, Eq. (61) is presumably the correct one and Eqs. (58)–(59) contain a transcription error. As printed, a reader cannot verify the normalization of the central result Eq. (64) from the preceding equations. Please correct the k₊ powers and check the overall factor.
[§5, Eq. (63) and Refs. [1], [2]] The key input γ(p_t) ∝ α_s²µ²/p_t⁴ in Eq. (63) is quoted from Ref. [2], which is listed as “to appear in Phys. Rev. D” and is therefore not available to the reader; the same is true of the framework result of Ref. [1] on which §1 rests. A short derivation of Eq. (63), or at least a statement of the approximations under which it holds, would make the paper self-contained. Relatedly, Eq. (63) is only the leading term in the range α_s²µ² << p_t² << µ² and by itself does not saturate the sum rule Eq. (45); the coefficient of Eq. (64) therefore inherits the uncertainty of the matching at p_t ~ α_sµ. The authors acknowledge this in §6, but the leading-log qualification should be stated at Eq. (64) itself, where the reader will first meet the result.
[§1, Eq. (2); §5, Eq. (64)] The entire normalization of Eq. (64) traces to the Gaussian model for the valence color charge, Eq. (2), which assigns a single variance µ² and no higher cumulants. Nothing in the paper derives this form from QCD; it is a modeling assumption about the small-x nuclear wavefunction. A non-Gaussian charge distribution with the same µ² would leave the propagator algebra of §§3–4 intact but change the coefficient of Eq. (64). The paper should say so explicitly at Eq. (64), and comment on whether higher cumulants are expected to be organized as a 1/ρ or 1/N_c² correction at large A.

Minor objections (8)

[§1, Eq. (1)] The numerical coefficient in µ² = 1.1 A^{1/3} fm⁻² is not derived or referenced; a one-line explanation of its origin would help.
[§3, Eq. (28)] The line reads “Aa− = = 0”, with a doubled equals sign; presumably one of these should be Aa+ = 0.
[§4, Eq. (53)] The factorization of the two-point function of the adjoint operators into the color structure times a single scalar function Γ(x_t−y_t) is an assumption (it holds in the Gaussian approximation) and should be stated as such.
[Appendix A, Eq. (70)] The bracket structure in the second line of Eq. (70) is unbalanced and the expression cannot be parsed as printed; also the text says “To obtain an expression for A₊, we integrate both sides of Eq. (67)” but Eq. (67) is the equation for A₋, and the boundary condition A₋(x⁻=0)=0 is imposed on a different field than the one being solved for. Please reconcile the labels.
[Appendix A] The statement that the solution is defined only up to a term Bx⁻, and the infrared singularities in p₊, are acknowledged as deferred; a forward pointer to the intended regularization scheme would be useful.
[§5, Eq. (64)] No numerical estimate is given. A sample value of µ² and of (1/πR²)dN_sea/dy for a heavy nucleus would let the reader judge whether the effect is phenomenologically relevant.
[§1 and §5] The renormalization scale at which α_s in Eqs. (63)–(65) is evaluated is not specified; presumably α_s(µ), but this should be stated.
[Throughout] Typographical: “explictly” (abstract and §1), “Wiezsäcker–Williams” for Weizsäcker–Williams (§1 and §5), “bremstrahlung” for bremsstrahlung (§1), “atleast” (§1), “degees” (§2), and “explictly” again in the abstract.

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The paper

The paper claims

Inside the color field of a large nucleus, the sea quark distribution saturates at a value set by the nuclear color charge density alone, a factor $/alpha_s//pi$ below the glue.

The reading

The paper asks what the propagators for scalars, fermions and gluons look like inside the color field of a very large nucleus seen in the infinite momentum frame, where the valence quarks act as a thin sheet of color charge travelling at the speed of light. The authors solve the fluctuation equations in that background exactly, then average over the random color orientations of the sheet; the average collapses the problem onto a two-dimensional Euclidean, ultraviolet-finite field theory whose only input is one correlation function $/gamma(p_t)$. Feeding that propagator into the standard propagator-to-distribution relation gives the sea quark spectrum, and the answer is striking: for $m_{/rm quark} //ll /mu$ the sea quark distribution is the same as for massless quarks, so it saturates rather than falling with quark mass. The motivation is concrete: if the intrinsic strange and charm content of a nucleus is enhanced by the higher parton density, this is the calculation that would say by how much, and the answer is that it is fixed by $/mu^2 /sim A^{1/3}$ and suppressed relative to glue by $/alpha_s//pi$. The authors are explicit that the result holds only in the window $/alpha_s^2/mu^2 //ll k_t^2 //ll /mu^2$, and that for quarks with $m /gtrsim /mu$ the dominant contribution comes from $k_t //gg /mu$, outside the reach of the method.

What carries the argument

The load-bearing mechanism is the structure of the background field: it vanishes for $x^-<0$ and is a pure gauge transform $/alpha_i = -/frac{1}{ig} U /nabla_i U^/dagger$ for $x^->0$, so only the step function at $x^-=0$ prevents it from being entirely a gauge artifact. That lets the authors write down the scalar wavefunctions in each half-space and match them across the discontinuity, giving a closed-form Green's function (Eq. 40). Averaging over the random color orientations of the valence sheet replaces $/langle U(x_t)U^/dagger(y_t)/rangle$ by $/delta_{/alpha/beta}/Gamma(x_t-y_t)$, whose Fourier transform $/gamma(p_t)$ obeys the sum rule $/int d^2p_t/(2/pi)^2 /,/gamma(p_t) = 1$. The resulting correction to the propagator, Eq. (47), is the object everything else is built from; fermions follow by the relative factor $2p^+$ (Eq. 49) and gluons by the same algebra plus a Fierz identity in the adjoint representation. The sum rule is what kills the divergent constant in the sea quark integral and leaves a finite, cutoff-independent answer.

Extended reading notes

Core claim

The central claim is Eq. (64): the sea quark distribution per unit rapidity per unit transverse area is $$/frac{1}{/pi $R^{2}$}/frac{dN_{/rm sea}}{dy} = /frac{2($N_c^{2}$-1)}{/$pi^{2}$}/, /$mu^{2}$ /$alpha_s^{2}$ /ln(1//alpha_s),$$ and it does not depend on the sea quark mass as long as $m_{/rm quark} //ll /mu$. Compared with the gluon result of the same regime, $$/frac{1}{/pi $R^{2}$}/frac{dN_{/rm glue}}{dy} = /frac{2/alpha_s /ln(1//alpha_s)/$mu^{2}$ ($N_c^{2}$-1)}{/pi},$$ the intrinsic sea quark content of the nuclear wavefunction is suppressed by $/alpha_s//pi$. Both distributions carry the same $1/x$ scaling, so their ratio is flat in rapidity at this order. The authors present this as a leading-order statement in $/alpha_s$ but exact in $/alpha_s^2/mu^2$, valid where the classical-source treatment of the valence quarks is justified.

Load-bearing premise

The entire normalization traces back to modelling the valence color charge as a Gaussian random variable with a single variance $/mu^2$ and no higher cumulants; nothing in the paper derives that Gaussian form from QCD, and a different charge distribution with the same $/mu^2$ would leave the propagator algebra intact but change the coefficient of Eq. (64).

Editorial extensions

If this is right

The intrinsic sea quark density of a large nucleus is fixed by $/mu^2 /propto A^{1/3}$, so it grows with nuclear size while remaining a fixed fraction $/alpha_s//pi$ of the glue.
Strange and charm quarks contribute as much as massless quarks whenever their mass is below $/mu$, so heavy-quark enhancement in nuclei is a saturation statement rather than a mass-suppression story.
Sea quarks and gluons share the same $1/x$ behaviour in this regime, making their ratio rapidity-independent at leading order.
The propagators computed here are the ingredients needed for the first radiative corrections to the quark and gluon distributions, which is where a Lipatov-type enhancement would first appear.

Reading between the lines

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

Because the final answer depends on $/gamma(p_t)$ only through its behaviour at small transverse momentum, the prediction is insensitive to the short-distance details of the charge distribution but sensitive to its cumulant structure; that asymmetry is a useful handle when comparing models of the nuclear wavefunction.
The mass independence suggests a practical test that needs no absolute normalization: the ratio of heavy to light sea quark yields in the same nucleus should approach one as $A$ grows, since $/mu /sim A^{1/6}$ rises with nuclear size.
The whole reduction hinges on the source being infinitely thin in $x^-$; any correction that gives the sheet a finite thickness would break the pure-gauge matching argument, so the leading correction in $/alpha_s$ is the natural place to look for the breakdown.
An observed nuclear enhancement of intrinsic strangeness larger than the $/alpha_s//pi$ fraction of glue would point to the $k_t /gtrsim /mu$ region that this paper explicitly cannot reach, rather than to a failure of the classical-source picture itself.

Original abstract

We compute the Green's functions for scalars, fermions and vectors in the color field associated with the infinite momentum frame wavefunction of a large nucleus. Expectation values of this wavefunction can be computed by integrating over random orientations of the valence quark charge density. This relates the Green's functions to correlation functions of a two dimensional, ultraviolet finite, field theory. We show how one can compute the sea quark distribution functions, and explictly compute them in the kinematic range of transverse momenta, $\alpha_s^2 \mu^2 << k_t^2 << \mu^2$, where $\mu^2$ is the average color charge squared per unit area. When $m_{quark}^2 << \mu^2 \sim A^{1/3}$, the sea quark contribution to the infinite momentum frame wave function saturates at a value that is the same as that for massless sea quarks.

The compiled record

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,412 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.43 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.02 · states quantitative result 0.44 · compares with prior results 0.10 · 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 compute the Green's functions for scalars, fermions and vectors in the color field associated with the infinite momentum frame wavefunction of a large nucleus. Expectation values of this wavefunction can be computed by integrating over random orientations of the valence quark charge density. This relates the Green's functions to correlation functions of a two dimensional, ultraviolet finite, field theory. We show how one can compute the sea quark distribution functions, and explictly compute them in the kinematic range of transverse momenta, $\alpha_s^2 \mu^2 << k_t^2 << \mu^2$, where $\mu^2$ is the average color charge squared per unit area. When $m_{quark}^2 << \mu^2 \sim A^{1/3}$, the sea quark contribution to the infinite momentum frame wave function saturates at a value that is the same as that for massless sea quarks.

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 compute the Green's functions for scalars, fermions and vectors in the color field associated with the infinite momentum frame wavefunction of a large nucleus.	contribution 1.00	0.95	0.11	asserted 1.00
s1	Expectation values of this wavefunction can be computed by integrating over random orientations of the valence quark charge density.	method 0.97	0.39	0.05	asserted 0.98
s2	This relates the Green's functions to correlation functions of a two dimensional, ultraviolet finite, field theory.	interpretation 0.54	0.72	0.18	asserted 1.00
s3	We show how one can compute the sea quark distribution functions, and explictly compute them in the kinematic range of transverse momenta, $\alpha_s^2 \mu^2 << k_t^2 << \mu^2$, where $\mu^2$ is the average color charge squared per unit area.	contribution 0.64	0.94	0.24	asserted 1.00
s4	When $m_{quark}^2 << \mu^2 \sim A^{1/3}$, the sea quark contribution to the infinite momentum frame wave function saturates at a value that is the same as that for massless sea quarks.	result 1.00	0.87	0.22	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 0.92.

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
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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 (7)

[2]

L. McLerran and R. Venugopalan, University of Minnesota preprint, TPI– MINN–93–52/T, to appear in Phys. Rev. D

View on Pith
[1]

L. McLerran and R. Venugopalan, University of Minnesota preprint, TPI– MINN–93–44/T, to appear in Phys. Rev. D

View on Pith
[3]

G. Leibbrandt and S. L. Nyeo, Phys. Rev. D33 (1986) 3135

View on Pith 1986
[4]

S. Mandelstam, Nucl. Phys. 213 (1983) 149. 20

View on Pith 1983
[5]

A. Bassetto, M. Dalbesco, I. Lazzizzera and R. Soldati, Phys. Rev. D31 (1985) 2012; A. Bassetto, in 14th Workshop on Problems in High Energ y Physics and Field Theory, Protvino, U.S.S.R. (1991), Padua preprint DF PD–91–TH–19

View on Pith 1985
[6]

Hung-Hsiang Lin and D. Soper, Phys. Rev. D48 (1993) 1841

View on Pith 1993
[7]

E. A. Kuraev, L. N. Lipatov and V. S. Fadin, Sov. Phys. JETP 45, 199 (1977). 21

View on Pith 1977

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Color decorrelation in the evolving glasma melts roughly half of initially singlet charm and bottom quark pairs within about 0.4 to 0.5 fm/c after their formation in proton-nucleus collisions at the LHC.
Simple Scaling Laws for Energy Correlators in Nuclear Matter
hep-ph 2024-11 conditional novelty 6.0 of 10

The leading nuclear modification of the two-point energy correlator in A-A and p-A collisions is a twist-4 light-ray operator contribution, giving an EEC ratio of 1 + a theta^2.
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.
Gluon Saturation Effects in Exclusive Heavy Vector Meson Photoproduction
hep-ph 2024-11 conditional novelty 6.0 of 10

Exclusive J/psi photoproduction off Pb targets prefers BK evolution with gluon saturation over linear BFKL evolution, and Upsilon production is predicted to distinguish them only at higher energies.
Finite-size effects on small-$x$ evolution and saturation in proton and nuclear targets
hep-ph 2024-11 conditional novelty 6.0 of 10

Including the proton's finite transverse size in small-x evolution weakens saturation effects compared with impact-parameter-averaged treatments, while lead-target predictions are insensitive to scheme choices once fi...
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.
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.
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.
Deeply inelastic scattering structure functions on a hybrid quantum computer
hep-th 2019-08 conditional novelty 6.0 of 10

A worldline representation of the QCD quark determinant is used to write a quantum circuit that computes F2 in a small-x dipole model, with a road map for generalizing to real-time DIS.
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...
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...
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.
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.
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.
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...
NLO corrections for the dipole factorization of DIS structure functions at low x
hep-ph 2011-12 conditional novelty 6.0 of 10

The NLO dipole factorization formula for the DIS structure functions F2 and FL at low x is derived in mixed space, yielding explicit transverse and longitudinal impact factors ready for phenomenological use.
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.
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.
Multiplicity Distribution in the Color Glass Condensate Formalism
hep-ph 2026-10 conditional novelty 5.0 of 10

The double negative binomial distribution, constrained by kT factorization, describes LHC multiplicity distributions with four free parameters instead of six.
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.
Shrinking the Quark Gluon Plasma
nucl-th 2019-01 conditional novelty 5.0 of 10

In hydrodynamic simulations of PbPb, XeXe, ArAr and OO collisions, the map from initial geometry to final flow becomes more purely linear as the system shrinks, and the centrality window where flow fluctuations can be...
Small-$x$ Helicity Evolution: an Operator Treatment
hep-ph 2018-08 conditional novelty 5.0 of 10

The paper gives explicit operator expressions for polarized quark and gluon Wilson lines and uses them to rederive, with one asserted correction, the double-logarithmic small-x helicity evolution equations.
Long-Range Rapidity Correlations in Heavy-Light Ion Collisions
hep-ph 2012-12 conditional novelty 5.0 of 10

Two-gluon production in heavy-light ion collisions generates near-side and away-side long-range rapidity correlations of identical azimuthal shape, contributing only even harmonics to the di-gluon correlator.
Triumvirate of Running Couplings in Small-x Evolution
hep-ph 2006-09 conditional novelty 5.0 of 10

Resumming quark-loop corrections to the JIMWLK and BK small-x kernels replaces the fixed coupling by a product of three running couplings, alpha_s(r1) alpha_s(r2) / alpha_s(R), with an explicit formula for the scale R.
Some Features of the Glasma
hep-ph 2006-02 conditional novelty 5.0 of 10

High-energy heavy-ion collisions create, immediately after impact, longitudinal color electric and magnetic fields, giving a large Chern-Simons charge density that decays as the Glasma turns into quark-gluon plasma.
Jets as differential probes of hot QCD matter
hep-ph 2026-10 accept novelty 4.0 of 10

A comprehensive theoretical framework for extracting directional and multi-scale properties of the quark-gluon plasma from differential jet substructure observables.
Entropy from decoherence: a case study using glasma-based occupation numbers
hep-ph 2025-07 reject novelty 4.0 of 10

Decoherence of a glasma-like coherent state yields an entropy per particle below the thermal gluon gas value, except in proton-nucleus collisions at small g mu.
An Experimental Exploration of the QCD Phase Diagram: The Search for the Critical Point and the Onset of De-confinement
nucl-ex 2010-07 conditional novelty 4.0 of 10

STAR proposes a RHIC beam energy scan from 5 to 39 GeV, with event-count targets for critical-point and deconfinement measurements and test-run evidence that the detector performs at those energies.
Deriving the Jalilian-Marian-Iancu-McLerran-Weigert-Leonidov-Kovner equation with classical and quantum source terms
hep-ph 2006-03 conditional novelty 4.0 of 10

A functional-integral derivation obtains the JIMWLK equation from a density-of-states representation and shows that losing non-commutativity of color charges corresponds to keeping only the shift term in the Wilson-li...

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How later papers used it

188 later papers cite it

Evidence: thin. 164 of the citing papers share no author with it, and 40 of those carry a confident label from Pith's citation classifier.

Pith read 1 of the 296 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 38 papers with this label

Obtained its gauge fields as gauge rotations of the vacuum confined to the transverse plane by solving this paper's equations in light cone gauge.

Evolution of eccentricities induced by geometrical and quantum fluctuations in proton-nucleus collisions (2023) · 0.93 · paraphrased: the licence does not allow quoting

How the literature uses it

166 arXiv papers cite this in 272 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 · 148 sentences without a confident label

Builds on it 1 paper

We take this initial condition to be a generalization of the McLerran–Venugopalan (MV) model [ref] by introducing a dependence on the transverse coordinate $$ to the color correlator.

Finite-size effects on small-$x$ evolution and saturation in proton and nuclear targets · Impact-parameter-dependent initial condition · 0.98

Uses its method or tool 33 papers

We apply the LCOT [ref] , rewriting the matrix element as a small-$x$ quasi-classical average in the `target' proton state [ref] [equation] Here we have inserted a sum over the complete set of states $|X⟩$, and the outer angular brackets indicate the quasi-classical averaging over the proton wave function.

Spin-Spin Coupling at Small $x$: Worm-Gear and Pretzelosity TMDs · Worm-Gear $g_1T$ · 1.00
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
We introduce both a simple impact parameter dependent Golec-Biernat-Wüsthoff (GBW) model [ref] with angular correlations, and a more realistic CGC dipole computed from the spatially dependent McLerran-Venugopalan (MV) [ref] initial color charge configurations coupled to small $x$ JIMWLK[JIMWLK is an acronym for Jalilian-Marian, Iancu, McLerran, Weigert, Leonidov, Kovner.] evolution [ref] .

Gluon imaging using azimuthal correlations in diffractive scattering at the Electron-Ion Collider · Introduction · 0.93

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
First, we derive expressions for the Gaussian approximation of the TMD distributions, which turn out to agree with the explicit numerical evaluation for the McLerran–Venugopalan (MV) initial condition [ref] in a wide range of separations.

Small-x TMD distributions initial condition: Nc-dependence and Gaussian approximations · Introduction · 0.80
We note that these vertices are identical to the ones in the conventional CGC calculations [ref] , but now derived via a different approach [ref] .

Single Inclusive Jet Production in $pA$ Collisions at NLO in the small-$x$ regime · Power counting, Modes, and Feynman Rules · 0.91

Cites it as background 74 papers

In approaches that account for saturation phenomena, namely the Colour Glass Condensate [ref] (where the saturation momentum sets the scale for other physical quantities) and the String Percolation Model [ref] (where Gauss' theorem links $ρ$ to $⟨p_T^2 ⟩$) there is a simple relation between $dn/dy$ and $⟨p_T^2 ⟩$, [equation] where $S_A∼A^2/3$ is the size of the overlapping area in the collision.

Energy conservation and scaling violations in particle production · Introduction · 0.81
An important ingredient of the MV model is the 2-dimensional color charge density [ref] [equation]

Helicity-dependent extension of the McLerran-Venugopalan model · Charge density · 0.84
The existence of this scale allows for a weak coupling description of the gluon dynamics in this high density regime of Quantum Chromodynamics (QCD) in a semi-classical effective field theory (EFT) framework known as the Color Glass Condensate (CGC) [ref] .

Gluon imaging using azimuthal correlations in diffractive scattering at the Electron-Ion Collider · Introduction · 0.93

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.

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