On solutions of the Balitsky-Kovchegov equation with impact parameter

Worth your time if you work on saturation with b-dependence. This is the first full numerical solution of the BK equation keeping all three variables (ln r, ln b, cos θ), and it delivers three things the local-approximation literature does not have: a finite saturation window 1/Qs(b,Y) < r < R_H(b,Y), a b-dependent saturation scale whose large-b tail is power-like, Qs ~ 1/b^γ with γ ≈ 1.6–2, and a black-disc radius growing exponentially in rapidity. The headline claim at abstract:s5 — that the b-dependent BK cross section violates Froissart despite the nonlinear term — follows from that last number and lines up with the Kovner–Wiedemann analytic argument in [19]. I think it is correct in substance. What is done well: the method is described concretely enough to re-implement (grid, iteration scheme, αs = 0.2, Glauber–Mueller input), and the conformal-invariance consistency check λ_s = 2λ_BD actually holds within the quoted ranges. Fig. 7 is the right diagnostic — it splits short- from long-range contributions and shows the long-range piece owns the power tail. Fig. 6 makes the strongest single point: an exponential input exp(−b²/2) already comes out as 1/b^3.6 after one small step. So the power is kernel-generated; only the coefficient is initial-condition-dependent. I would push back on the worry that the Gaussian profile undermines the claim — the paper answers that itself, and the deferral to [31] matters for the coefficient, not the power. Where it is soft: the exponents are ranges, not numbers with errors. No grid-refinement study, no ΔY sensitivity, and the authors themselves say λ_BD has not stabilized for small r (pre-asymptotic). The tail amplitudes in Fig. 6 reach 1e-9, which is where a 100-point linear grid in ln b is thinnest. So exponential growth is consistent with the data but not established asymptotically. That is a calibration gap, not a flaw in the qualitative conclusion. Send it out; a referee should press on error bars and grid convergence, not on the physics.

The central claim (abstract:s5) requires the total cross section to grow faster than Y². Note first that with the reported tail exponent γ ≈ 2–3 > 2, the integral ∫ b db N converges at large b, so the total cross section is dominated by the black-disc region and σ ≈ 2πR²_BD (Eq. 26). Everything therefore hinges on Eq. 27, R_BD = R₀ exp(λ_BD α_s Y). That exponent is extracted from Y ≤ 11 on a (100 r × 100 b × 20 cosθ) grid with linear interpolation, with no stated error bars, no grid-refinement study, and no ΔY sensitivity analysis. The paper itself states in Sec. 5.3 that λ_BD 'increases rapidly for dipole sizes smaller than the given range above, mostly due to the pre-asymptotic effects' — a direct admission that the fitted exponent has not stabilized. The internal consistency relation λ_s = 2λ_BD (Eq. 30) does hold within the quoted ranges (2 × 0.9–1.0 versus 1.7–2.0), which is genuine support, but it constrains only the ratio, not whether either exponent is asymptotic. Separately, the tail amplitudes used to fix γ reach 10⁻⁹ (Fig. 6), precisely where a 100-point linear-interpolation grid in ln b is coarsest relative to the dynamic range. The numerical evidence is therefore consistent with, but does not by itself establish, asymptotic exponential growth. On the reader's specific framing: the Gaussian profile's support at all b is largely answered by the paper's own Figs. 6 and 7 — the input falls as exp(−b²/2), yet at Y = 0.1 the output already falls as 1/b^3.6, and Fig. 7 separates short- from long-range contributions. The power form is kernel-generated; what remains initial-condition-dependent is the coefficient, not the power.

The paper is a careful, re-implementable numerical study, and its qualitative picture — a kernel-generated power tail in b, a finite saturation window in r, and a black-disc radius growing exponentially in rapidity — is coherent and consistent with the analytic work of [19, 20]. The reader's verdict of CONDITIONAL is defensible and I would not move it. I agree with the reader that the quantitative exponents are uncalibrated, but I disagree with the specific mechanism proposed as the weakest assumption. The reader argues that the Gaussian profile S(b) = 10 exp(−b²/2) (Eq. 15) has support at every b, so the numerics cannot separate kernel-driven from initial-condition-driven tails. The paper largely closes that gap itself: Fig. 6 shows the input falling exponentially while the Y = 0.1 output already falls as 1/b^3.6, and Fig. 7 explicitly decomposes the right-hand side into short-range (|z − b| < r₀) and long-range (|z − b| > r₀) contributions, showing the latter dominates at large b. The analytic estimate in Eq. 22, r²πR₀²(Y)/b⁴, likewise makes the power form a property of the kernel, with only the coefficient set by the target profile. So the deferred companion paper [31] is a reasonable request for a systematic scan, but its absence does not by itself undercut the power-law claim. Where I would sharpen the reader's concern is elsewhere: the Froissart conclusion depends on the asymptotic behaviour of λ_BD, and the extraction window (Y ≤ 11) is one the authors describe as still pre-asymptotic. Combined with the absence of any grid- or step-size convergence study for a computation whose large-b amplitudes span nine orders of magnitude, this is the place where the numerical support is thinnest. That is a calibration gap, not an error, and the analytic argument of [19] independently supports the conclusion, so the verdict should remain CONDITIONAL rather than move toward rejection. Credit is due for the internal cross-check λ_s = 2λ_BD from conformal invariance (Eq. 30), which the quoted numbers do satisfy, and for the explicit lower-bound logic σ ≥ σ_BD that makes the Froissart argument valid even though only the black-disc piece is computed.

If correct, the result is significant: it shows that saturation in the b-dependent BK equation is local in impact parameter but not global, so that unitarity of the total cross section requires physics outside the equation. The paper is a careful and honest numerical study that goes beyond the local approximation of earlier work, and it is consistent with the analytic arguments of Ref. [19]. Strengths include the full three-variable discretization, the explicit iterative scheme in Eqs. (12)-(13), the consistency check lambda_s = 2 lambda_BD from conformal invariance (Eq. 30), and the clean demonstration in Fig. 7 that the long-range kernel contribution dominates at large b. The authors acknowledge the main limitations (no running coupling, no confinement, alternative initial conditions deferred to Ref. [31]).

The Balitsky-Kovchegov (BK) equation is the standard mean-field description of how a small colour dipole scatters off a dense nuclear target as energy rises: linear BFKL growth is tamed by a nonlinear term that saturates the amplitude at one. Almost all earlier work solved it while pretending the target is an infinite uniform slab, so the answer depended only on the dipole size $r$. This paper keeps the impact parameter $b$ and solves the equation numerically on a three-dimensional grid in $(r, b, cos theta)$ with a Glauber-Mueller initial profile. The authors find that the exponential fall-off in $b$ they start with is not preserved: the evolution replaces it with a power tail $N sim 1/b^{gamma}$, which makes the black disc radius grow exponentially with rapidity and pushes the integrated cross section past the Froissart bound. If that is right, the nonlinear term enforces unitarity only locally in $b$, and global unitarity has to come from physics the equation does not contain.

What carries the argument. The load-bearing object is the BK equation itself, the large-$N_c$ closed equation for the two-point dipole amplitude $N(x,y;Y)$, whose kernel describes a parent dipole splitting into two with a measure invariant under Möbius transformations of the transverse plane. The paper exploits two consequences of that symmetry structure. Translation invariance is deliberately broken by the initial condition, which lets the $b$-dependence propagate; and conformal invariance of the kernel implies that the amplitude depends on the combination $r r_0/b^2$, which yields the relation $lambda_s = 2 lambda_{BD}$ between the growth rates of the saturation scale and of the black disc radius, a relation the numerics confirm. The saturation scale is extracted by solving $langle N(r=1/Q_s, b, theta, Y) rangle_theta = 1/2$, and the black disc radius by the same condition with $b$ as the unknown.

Core claim. The paper's central claim is that the b-dependent BK equation violates the Froissart bound even though it contains the nonlinear term responsible for saturation. Two numerical facts carry it. First, the amplitude as a function of dipole size is not monotone: because the initial profile is localised in $b$, a very large dipole with both ends outside the target has $N to 0$, so saturation occupies only a finite window $1/Q_s(b,Y) < r < R_H(b,Y)$ whose width grows with rapidity. Second, the steep Gaussian profile in $b$ is washed out immediately, and already after one small rapidity step the amplitude behaves as $N sim 1/b^{3.6}$, softening to $1/b^{gamma}$ with $gamma$ between 2 and 3 at higher rapidity, with the growth at large $b$ matching the hard Pomeron intercept. The authors trace this to long-range contributions in the kernel, where one end of the parent dipole sits in the strong-field region and the other far outside it. Fitting the black disc radius to $R_{BD} = R_0 exp(lambda_{BD} alpha_s Y)$ gives $lambda_{BD} approx 0.6$ to $1.0$, far faster than the linear-in-$Y$ growth an exponential profile would give, hence the bound is violated.

The premise it rests on. The demonstration starts from a Gaussian profile in impact parameter that is nonzero everywhere, so long-range contributions are present from the first evolution step, and the paper's own check of whether the kernel rather than the initial condition generates the power tail is deferred to a companion paper listed as work in progress.

The saturation region in dipole size is bounded on both sides, $1/Q_s(b,Y) < r < R_H(b,Y)$, and that window widens with rapidity, so geometric scaling in $r Q_s(b,Y)$ holds only for $r$ well below the outer scale $R_H$.

The impact-parameter dependence of the saturation scale factorises at high rapidity, $Q_s(b,Y) = exp(lambda_s alpha_s Y) g(b)$ with $lambda_s approx 1.7$ to $2.0$ nearly independent of $b$, and the large-$b$ tail becomes a power $Q_s sim 1/b^{gamma}$ with $gamma approx 1.6$ to $2.0$ rather than the input Gaussian.

The growth rates satisfy $lambda_s = 2 lambda_{BD}$, so the exponential expansion of the black disc is not an independent parameter but a consequence of conformal invariance of the kernel.

Because the power tail comes from long-range Coulomb-like contributions, restoring the Froissart bound requires breaking the conformal symmetry of the kernel, for instance by a naive cutoff that drops the long-range piece of the integration region.

The result reframes saturation as a local statement in impact parameter: the nonlinear term caps $N$ at each $b$, but nothing in the equation prevents the black disc from spreading sideways, so global unitarity is an emergent constraint on the kernel rather than a property of the equation.

This suggests a sharp numerical test that the paper defers: rerun with a compact-support profile in $b$ and check whether the $1/b^{gamma}$ tail still appears at the same rate. If the tail survives, it is kernel-driven; if it tracks the input support, the Froissart-violation claim is an artefact of the Gaussian profile.

The same mechanism should show up in the onium-onium Monte Carlo amplitudes, where multiple scattering is included but the target is not a uniform slab, so comparing the extracted $lambda_{BD}$ across those two approaches would test whether the effect is universal or specific to the mean-field truncation.

If confinement-modified kernels restore a faster-than-power fall-off in $b$, the practical consequence is a ceiling on how much of the observed small-$x$ rise can be attributed to BFKL-type evolution at large impact parameter, which is directly relevant to interpreting forward-physics and diffractive data.

This review was written by a machine. No person has reviewed this paper on Pith yet. A scientist listed on Pith can review it for a fee, and their review is published here whatever it concludes.

Request a human review