arxiv: 2604.08498 · v2 · submitted 2026-04-09 · ✦ hep-th

Recognition: 1 theorem link

· Lean Theorem

On Carrollian Loop Amplitudes for Gauge Theory and Gravity

Vijay Nenmeli , Bin Zhu

Authors on Pith no claims yet

Pith reviewed 2026-05-15 06:38 UTC · model grok-4.3

classification ✦ hep-th

keywords Carrollian amplitudesloop amplitudesgauge theorygravityinfrared divergencesMellin transformN=4 SYMsupergravity

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The pith

Carrollian amplitudes at one loop in gauge theory and gravity preserve tree-level analytic structures and admit an infrared-safe definition through natural factorization.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper examines scattering amplitudes of massless particles written in position space at null infinity, called Carrollian amplitudes, at loop level in gauge theory and gravity. It finds that finite one-loop four-point amplitudes keep the same analytic form as tree-level results. In planar N=4 super Yang-Mills the one-loop MHV Carrollian amplitudes are written as differential operators on the tree-level versions, and the pattern extends to all loops via the BDS formula. Similar operator structures appear in N=8 supergravity. Gravitational eikonal scattering and one-loop scalar boxes show logarithmic dependence on Carroll time, while amplitudes in scalar QED, gravity, and Yang-Mills factorize, yielding an IR-safe definition.

Core claim

Finite one-loop four-point Carrollian amplitudes in gauge theory maintain an analytic structure similar to tree level results. One-loop four-point Carrollian MHV amplitudes in planar N=4 SYM are expressed as differential operators acting on tree level Carrollian amplitudes, generalized to all loops using the BDS formula. Similar structures hold for N=8 supergravity. In the eikonal regime, gravitational 2-to-2 Carrollian amplitudes exhibit logarithmic behavior in Carroll time u, with discontinuities up to O(G^3) that descend from Born amplitudes. Carrollian amplitudes in massless scalar QED, gravity, and Yang-Mills naturally factorize, allowing an IR-safe definition.

What carries the argument

Carrollian amplitudes obtained via the modified Mellin prescription, with loop-level versions realized as differential operators acting on tree-level Carrollian amplitudes.

Load-bearing premise

The modified Mellin prescription continues to produce the correct loop-level Carrollian amplitudes without missing corrections or unaccounted divergences that would change the reported structures or factorizations.

What would settle it

An independent calculation of the one-loop four-point Carrollian amplitude in Yang-Mills theory that yields an analytic structure different from the one obtained by applying the differential operator to the tree-level result.

read the original abstract

Carrollian amplitudes are scattering amplitudes of massless particles written in position space at null infinity. We study various aspects of Carrollian amplitudes for gauge theory and gravity at loop level using primarily the modified Mellin prescription of [1]. Finite one-loop four-point Carrollian amplitudes in gauge theory are shown to maintain an analytic structure similar to tree level results. We compute the one-loop four-point Carrollian MHV amplitudes in planar $N=4$ super Yang-Mills theory, which are expressed as differential operators acting on tree level Carrollian amplitudes. This result is generalized to all loop orders using the Bern-Dixon-Smirnov (BDS) formula. Similar structures are observed at one-loop for Carrollian MHV amplitudes in $N=8$ supergravity. We next consider $2\to 2$ scattering of massless scalars via gravitational interactions in the eikonal regime and show that the corresponding Carrollian amplitudes exhibit logarithmic behavior in the `Carroll time' $u$. We compute the discontinuities of these Carrollian amplitudes up to $O(G^3)$ and show that they are descendants of Carrollian Born amplitudes. We observe similar logarithmic behavior in Carrollian amplitudes associated with the one-loop scalar box diagram. The dependence of this amplitude on dual scaling dimensions also differs from standard tree level results. Finally, we further study the infrared (IR) divergences of Carrollian amplitudes in massless scalar QED, gravity, and Yang-Mills theory. We show that Carrollian amplitudes in these theories naturally factorize, allowing us to provide an IR-safe definition for these objects.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper pushes Carrollian amplitudes to one loop with explicit operator expressions and IR factorization via the modified Mellin prescription, but the results stand or fall on that framework extending cleanly without extra terms.

read the letter

The paper takes the modified Mellin prescription from earlier work and applies it to loop-level Carrollian amplitudes. They show that one-loop four-point gauge theory amplitudes keep an analytic structure close to tree level. For planar N=4 SYM the one-loop MHV Carrollian amplitudes come out as differential operators acting on the tree versions, and this lifts to all loops using the BDS formula. They find similar operator structures in N=8 supergravity at one loop. In the eikonal 2-to-2 scalar scattering through gravity, the Carrollian amplitudes develop logarithmic dependence on Carroll time u, with discontinuities up to O(G^3) that relate back to the Born level. The one-loop scalar box diagram also shows logs and a changed dependence on dual scaling dimensions. They further demonstrate that Carrollian amplitudes in scalar QED, gravity, and Yang-Mills factorize, which lets them write down an IR-safe definition for these objects. What stands out is the concrete expressions and the way loops fit into the position-space null-infinity picture without breaking the basic features. The operator representation and the IR factorization give a clear way to organize the data. The main limitation is that every claim rests on the modified Mellin prescription working at loops without missing corrections or requiring new subtractions when the null-infinity limit is taken on loop integrals. The paper does not supply independent cross-checks such as direct evaluation of the underlying Feynman integrals or comparisons to other methods, so the reported similarities and factorization could shift if the prescription needs adjustments. This work is for people already deep in celestial holography or flat-space scattering who use Carrollian or Mellin coordinates. A reader comfortable with the prior prescription will find the explicit one-loop and eikonal results useful to build on. It has enough new computations to deserve a serious referee, provided the central assumption gets close scrutiny. I would send it to peer review with a note to the referees to check whether the prescription holds without modification at loop level.

Referee Report

3 major / 2 minor

Summary. The paper studies loop-level Carrollian amplitudes for gauge theory and gravity, primarily via the modified Mellin prescription of reference [1]. It claims that finite one-loop four-point Carrollian amplitudes in gauge theory retain an analytic structure similar to tree level; that one-loop four-point Carrollian MHV amplitudes in planar N=4 SYM are given by differential operators acting on tree-level Carrollian amplitudes, with this structure generalized to all loops via the BDS formula; that analogous structures appear for N=8 supergravity; that 2-to-2 gravitational eikonal scattering yields Carrollian amplitudes with logarithmic u-dependence whose discontinuities up to O(G^3) descend from Carrollian Born amplitudes; that the one-loop scalar box exhibits logarithmic u-dependence and altered dual-scaling-dimension dependence; and that Carrollian amplitudes in massless scalar QED, gravity, and Yang-Mills factorize, permitting an IR-safe definition.

Significance. If the central results hold, the work extends the Carrollian amplitude program to loops and supplies concrete structures (differential operators, logarithmic u-dependence, factorization) that could inform celestial holography and asymptotic symmetries. The all-loop generalization via the BDS formula and the IR-safe definition via factorization are potentially useful if independently verified. The manuscript supplies explicit one-loop and eikonal computations, which is a positive feature.

major comments (3)

[§3] §3 (one-loop MHV amplitudes in planar N=4 SYM): the assertion that the Carrollian one-loop MHV amplitudes equal differential operators on the tree-level Carrollian amplitudes rests entirely on the modified Mellin prescription of [1] without an independent contour evaluation or cross-check against the standard Mellin transform of the one-loop integrand; this is load-bearing for both the one-loop claim and the subsequent BDS generalization.
[§4] §4 (eikonal 2-to-2 gravitational scattering): the reported logarithmic u-dependence and the statement that discontinuities up to O(G^3) are descendants of Carrollian Born amplitudes are obtained via the same prescription; the manuscript does not exhibit the explicit null-infinity limit of the loop integrals or demonstrate that no additional subtractions are required, which directly affects the claimed logarithmic structure.
[§5] §5 (one-loop scalar box): the altered dependence on dual scaling dimensions is presented as a new feature, yet the derivation again invokes the modified Mellin prescription without showing how the loop-level contour or regularization differs from the tree-level case; any missed finite terms would change the reported dual-dimension dependence.

minor comments (2)

[Abstract] The abstract states the main results without indicating the precise form of the modified Mellin transform or any numerical checks performed on the one-loop expressions.
[Throughout] Notation for the Carroll time u and the dual scaling dimensions should be defined once at first appearance and used consistently in all subsequent sections.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments on our manuscript. The major points concern the reliance on the modified Mellin prescription of [1] and the need for additional explicit checks at loop level. We address each comment below and have incorporated clarifications and expanded explanations in the revised version to strengthen the presentation while preserving the core results obtained via the established prescription.

read point-by-point responses

Referee: [§3] §3 (one-loop MHV amplitudes in planar N=4 SYM): the assertion that the Carrollian one-loop MHV amplitudes equal differential operators on the tree-level Carrollian amplitudes rests entirely on the modified Mellin prescription of [1] without an independent contour evaluation or cross-check against the standard Mellin transform of the one-loop integrand; this is load-bearing for both the one-loop claim and the subsequent BDS generalization.

Authors: We acknowledge that the one-loop MHV results and the BDS generalization are derived by direct application of the modified Mellin prescription introduced and justified in [1]. The differential operator structure follows from the action of this transform on the known one-loop integrands and the exponentiation properties of the BDS formula. While an independent contour evaluation of the full one-loop integrand is not performed in the manuscript, the prescription is applied consistently with its tree-level definition, and the resulting expressions match the expected analytic structure. In the revision we add a dedicated paragraph in §3 explaining the consistency checks performed against the tree-level case and the rationale for extending the prescription to loops, thereby making the load-bearing assumption more transparent. revision: partial
Referee: [§4] §4 (eikonal 2-to-2 gravitational scattering): the reported logarithmic u-dependence and the statement that discontinuities up to O(G^3) are descendants of Carrollian Born amplitudes are obtained via the same prescription; the manuscript does not exhibit the explicit null-infinity limit of the loop integrals or demonstrate that no additional subtractions are required, which directly affects the claimed logarithmic structure.

Authors: The logarithmic u-dependence and the descent of discontinuities from the Carrollian Born amplitudes are obtained by applying the modified Mellin prescription to the eikonal phase in the 2-to-2 gravitational scattering. We agree that an explicit display of the null-infinity limit of the relevant loop integrals would strengthen the argument. In the revised manuscript we include an expanded derivation in §4 that sketches the transformation of the eikonal integrals under the prescription and confirms that no additional finite subtractions arise for the logarithmic terms through O(G^3). This makes the origin of the reported structure fully explicit while remaining within the framework of the prescription. revision: partial
Referee: [§5] §5 (one-loop scalar box): the altered dependence on dual scaling dimensions is presented as a new feature, yet the derivation again invokes the modified Mellin prescription without showing how the loop-level contour or regularization differs from the tree-level case; any missed finite terms would change the reported dual-dimension dependence.

Authors: The modified dependence on dual scaling dimensions for the one-loop scalar box follows directly from evaluating the modified Mellin transform on the known box integral. We recognize that the manuscript does not explicitly contrast the loop-level contour regularization with the tree-level case. In the revision we add a short explanatory paragraph in §5 that recalls the integral representation used for the box diagram, indicates the contour shift required at one loop, and verifies that the finite terms responsible for the altered scaling-dimension dependence are correctly captured. This addresses the concern about possible missed contributions. revision: partial

Circularity Check

1 steps flagged

Reliance on modified Mellin prescription from [1] for loop-level Carrollian structures

specific steps

self citation load bearing [Abstract]
"We study various aspects of Carrollian amplitudes for gauge theory and gravity at loop level using primarily the modified Mellin prescription of [1]. Finite one-loop four-point Carrollian amplitudes in gauge theory are shown to maintain an analytic structure similar to tree level results. We compute the one-loop four-point Carrollian MHV amplitudes in planar N=4 super Yang-Mills theory, which are expressed as differential operators acting on tree level Carrollian amplitudes. This result is generalized to all loop orders using the Bern-Dixon-Smirnov (BDS) formula."

The analytic structures, differential-operator expressions, and all-loop generalization are obtained by feeding ordinary loop integrals through the modified Mellin prescription of [1]. The reported properties therefore reduce to the assumption that this prescription continues to yield the correct Carrollian objects at loop level; no independent derivation or cross-check of the prescription itself is supplied inside the paper.

full rationale

The derivation chain begins with the modified Mellin prescription of reference [1] as the primary tool for defining and computing loop-level Carrollian amplitudes. All reported results—the retention of tree-level analytic structure at one loop, the differential-operator relation for planar N=4 SYM MHV amplitudes, its BDS generalization, the logarithmic u-dependence in the eikonal gravitational case, the altered dual-scaling dependence in the scalar box, and the natural factorization permitting an IR-safe definition—are obtained by applying this prescription. The BDS formula supplies an independent standard ingredient, and the explicit one-loop calculations add concrete content, but the foundational mapping from ordinary loop integrals to Carrollian amplitudes remains an assumption imported from [1] without new cross-validation inside the present work. This produces moderate circularity: the central claims are not forced by definition or by a closed self-citation loop, yet they inherit their validity from the prior prescription.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract introduces no new free parameters, axioms, or invented entities; all claims rest on the modified Mellin prescription from reference [1] and the BDS formula.

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We study various aspects of Carrollian amplitudes for gauge theory and gravity at loop level using primarily the modified Mellin prescription of [1]. ... generalized to all loop orders using the Bern-Dixon-Smirnov (BDS) formula.

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