Regge Limit of Gauge Theory Amplitudes beyond Leading Power Approximation

The paper is worth your time: it is the first all-order double-logarithmic resummation of a power-suppressed (O(t/s)) amplitude in the Regge limit, and it finds a surprising common function with the Sudakov limit. The one-loop calculation is transparent, and the two-loop cross-check against the Bhabha expansion gives real support. The main asymptotic result, Eq. (16), is notable: the subleading term can overtake the leading amplitude once |s/t| ~ exp(2π/α), before unitarity constraints kick in. The QCD color-singlet comparison to the BFKL pomeron is more qualitative, but it is honestly labelled as such.

The soft spot is exactly where the reader put it. The all-order step after Eq. (12) asserts without proof that for any number of extra Glauber bosons the double-logarithmic contribution factorizes: all vertices attach to the virtual fermion line carrying l, light-cone components order, and the n-fold contribution is (4zη(1−ξ))ⁿ/n!. Only the first insertion is actually checked via the two-loop Bhabha agreement. The O(z²) check verifies the n=1 case, not the factorization of multiple insertions. If n=3 (O(z³)) were computed and matched the formula, that would be much stronger. As it stands, Eq. (16) depends on a plausible but unproved pattern. This is a soft spot, not a fatal one: the pattern is natural, consistent with the known Sudakov factorization, and the two-loop check is nontrivial. But for a Letter that stakes its main claim on the all-order exponent, one would like the induction or a dedicated n=3 calculation.

The unitarity discussion is suggestive rather than rigorous, and the claim about the breakdown scale is tied to the asymptotic formula, so it inherits the same caveat. The citation pattern is fine; self-citations are used as cross-checks.

Verdict: this deserves a serious referee. I would engage with it, and I would cite it if I worked on Regge or Sudakov resummations. Ask the author to supply the missing check.

The central result M_LL = tau z g(2z) M^(0) and its asymptotic Eq. (16) follow only if, for every n, each additional leading-power Glauber boson attaches to the virtual fermion line carrying the loop momentum l, with the light-cone components ordered as q^2_perp < l'^2_{i,perp} < l^2_perp and l_- < l'_{1-} < ... < l'_{n-} < sqrt(s), so that each insertion gives the factor 4 z eta (1-xi) and the n-fold contribution is (4 z eta (1-xi))^n / n!. The text after Eq. (12) says this is 'straightforward to check' but does not check it. The agreement with the two-loop Bhabha small-angle expansion verifies only the first insertion, i.e. the O(z^2) term; it does not test the factorization of two or more insertions. In particular, one must show that diagrams with two Glauber bosons attached to different fermion lines, or with the contact vertex not located on the l-loop line, cancel or factor as claimed. If the ordered-integral pattern is altered, the exponent e^{4 z eta (1-xi)} and the asymptotic behavior in Eq. (16) do not follow.

The reader's weakest_assumption pinpoints the unproved all-order factorization after Eq. (12), and I agree that this is the most load-bearing gap. The two-loop agreement is genuine but only tests the first Glauber insertion; arbitrary n is exactly what Eq. (16) needs. The paper's own phrase 'straightforward to check' marks the absence of an induction or a diagrammatic proof, so the conditional verdict is appropriate. I also note a secondary, easily corrected issue: a stationary-phase evaluation of the integral representation Eq. (14) gives M_LL ~ (pi/sqrt(2 alpha)) ln^{-1}|s/t| |s/t|^{-1 + alpha/(2 pi) ln|s/t|} M^(0), whereas Eq. (16) displays ln|s/t| in the numerator. This changes the prefactor of the displayed main formula but not the exponential criterion |ln tau| ~ 1/alpha, so it does not alter the qualitative conclusion. The QCD color-singlet extension is schematic in the same way, since the effective color factor is stated without a derivation, but the abelian all-order claim is the load-bearing piece. Overall, the reader's CONDITIONAL verdict should stand.

The one-loop derivation and the two-loop comparison with the Bhabha small-angle expansion are clean, and the observation that subleading-power amplitudes can acquire double-logarithmic enhancement with an energy-dependent exponent is interesting and potentially important for the Regge-limit literature. The paper introduces no fitted parameters, and the abelian calculation is internally consistent as far as it is checked. However, the all-order factorization step and the QCD extrapolation are asserted rather than demonstrated, so the main asymptotic claims are not yet fully supported.

This paper claims that, in the high-energy small-angle Regge limit of fermion-antifermion scattering in gauge theories, the amplitude term suppressed by a power of $\tau=|t/s|$ does not stay subdominant. For an abelian gauge group, the author sums the double-logarithmic corrections $(\alpha\ln^2\tau)^n$ to all orders and obtains the asymptotic form $M_{\mathrm{LL}}\sim (\pi/\sqrt{2\alpha})\,\ln|s/t|\,|s/t|^{-1+\alpha/(2\pi)\ln|s/t|}\,M^{(0)}$. When $|\ln\tau|\sim 1/\alpha$, equivalently $|s/t|\approx e^{2\pi/\alpha}$, this formally subleading term becomes comparable to the leading-power amplitude, so the small-angle expansion breaks down. For the nonabelian color-singlet channel, the same effect can become comparable to the BFKL pomeron contribution. The result matters because it identifies a concrete energy scale above which the leading-power Regge analysis alone is not reliable.

What carries the argument. The argument is carried by an effective-theory expansion of the Born amplitude in the momentum transfer $q$, in which the power-suppressed $O(\tau)$ term appears as a local contact interaction while Glauber gauge bosons are exchanged in the static approximation. The decisive object is the factorization of $n$ additional leading-power Glauber bosons: their light-cone components reduce to delta functions, the remaining integrations are strictly ordered, and each insertion contributes $4z\eta(1-\xi)$, yielding the exponential $e^{4z\eta(1-\xi)}$ inside the one-loop integral. That exponential, after integration, produces the hypergeometric function $g(z)$, whose asymptotic form carries the main result.

Core claim. The central claim is that the leading power-suppressed piece of the amplitude, of order $\tau=|t/s|$, receives double-logarithmic corrections that resum into the universal function $g(z)={}_2F_2(1,1;3/2,2;z/2)$ with $z=(\alpha/2\pi)\ln^2\tau$. For large $z$, this function has the asymptotic behavior $g(z)\sim (2\pi e z/z^3)^{1/2}$, which turns the resummed amplitude into Eq. (16). The corresponding exponent $\gamma=(\alpha/2\pi)\ln|s/t|$ depends on the scattering angle, unlike the single-logarithmic leading-power exponents, and at $|s/t|\approx e^{2\pi/\alpha}$ the power suppression is overcome. The paper further shows that in the color-singlet channel of nonabelian theory the contribution can rival the BFKL pomeron, and it reports an unexpected universality: the same function $g(z)$ appears in the double-logarithmic asymptotics of mass-suppressed amplitudes in the Sudakov limit.

The premise it rests on. The entire all-order result relies on the unproved assertion that for any number $n$ of additional Glauber bosons the double-logarithmic correction factorizes as in the two-loop case, with the Glauber phase exponentiating and all double-log insertions attaching to the single virtual fermion line; if that factorization fails, the exponential $e^{4z\eta(1-\xi)}$ and the asymptotic formula Eq. (16) do not follow.

At $|s/t|\approx e^{2\pi/\alpha}$ the formally subleading term becomes comparable to the leading-power amplitude, so the small-angle expansion breaks down.

In QCD color-singlet quark-antiquark scattering, the power-suppressed contribution can become comparable to the BFKL pomeron contribution for sufficiently small scattering angles.

The exponent $\gamma=(\alpha/2\pi)\ln|s/t|$ grows with the logarithm of $s/t$, so the subleading term controls the high-energy growth in this regime rather than remaining a fixed correction.

The same universal function $g(z)$ appears in the Sudakov limit of mass-suppressed amplitudes, connecting double-logarithmic asymptotics of Regge and Sudakov limits.

The breakdown scale supplies a perturbative bound on the energy range where leading-power Regge analysis can be trusted.

The all-order factorization asserted after Eq. (12) could be tested at three loops: if the $z^3$ coefficient of $g(z)$ matches a full-theory computation, the mechanism is likely generic for subleading-power amplitudes.

The same effective-theory expansion may apply to higher-order power-suppressed terms, predicting a sequence of breakdown scales at which successive terms of the small-angle expansion overtake the leading amplitude.

The appearance of the same $g(z)$ in Sudakov and Regge limits suggests a common renormalization-group or exponentiation kernel for power-suppressed amplitudes, which would make the relation derived rather than accidental.

A direct observable probe could be ultra-forward high-energy electron scattering, where the angular distribution should deviate from the leading-power prediction near the scale $|s/t|\approx e^{2\pi/\alpha}$; the location of the deviation is a quantitative prediction of the resummation.

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