Celestial closed strings at one-loop

Anthonny F. Canazas Garay; Francisco Rojas; Gaston Giribet; Yoel Parra-Cisterna

REVIEW 3 major objections 5 minor 30 references

Celestial closed strings at one-loop

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read At one loop, the field-theory limit of the closed superstring four-graviton amplitude commutes with the Mellin transform and reproduces the celestial N=8 supergravity amplitude for every cross-ratio.

desk verdict A plausible but not airtight one-loop closed-string celestial amplitude computation; the final commutativity claim is likely right, but the moduli-space asymptotics need real analytic support before publication. read the letter →

arxiv 2504.17989 v1 pith:6WN4OCCB submitted 2025-04-25 hep-th

Anthonny F. Canazas Garay , Gaston Giribet , Yoel Parra-Cisterna , Francisco Rojas This is my paper · ORCID

classification hep-th

keywords celestialholographyclosedsuperstringone-loopamplitudesfour-gravitonscatteringMellintransformconformalbasisN=8supergravityfieldtheorylimit

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

Celestial holography rewrites scattering amplitudes as correlation functions on the celestial sphere using a Mellin transform over the energies of the external particles. This paper asks whether the usual low-energy limit of string theory, $\alpha' \to 0$, survives that change of variables for four-graviton scattering at one loop in closed type II superstring theory. The authors show that it does: when the limit is taken by first isolating the region of the torus moduli space where $\operatorname{Im}\tau \to \infty$, the Mellin transform and the field-theory limit commute. The resulting celestial string amplitude is exactly the celestial one-loop $N=8$ supergravity amplitude, as distributions enforcing $\beta = D/2$, for all values of the conformal cross-ratio $r=-s/t$. A by-product is the direct computation of one-loop graviton amplitudes in the conformal basis.

What carries the argument

The argument is carried by the reduced integral $J(r,\beta)$ of equation (4.26), which contains all dependence on the torus moduli. Two logarithmic combinations of worldsheet $\theta$ functions, $X$ and $Y$ defined in (4.21), package the Mandelstam dependence; in the $\operatorname{Im}\tau\to\infty$ region they become linear in the rescaled vertex positions $\rho_i$ and in $T=\operatorname{Im}\tau$, reducing the integral to the parameter integral over the simplex. The change of variables $\eta=\alpha'\omega(rX-Y)$ turns the Mellin transform into the Gamma function $\Gamma(4-\beta)$ times $(rX-Y)^{\beta-4}$; then $\eta=(\alpha'T)^{-1}$ absorbs every remaining $\alpha'$ into the upper limit of a Schwinger-parameter integral. The distributional identity $I(x)=4\pi\delta(x)$ for real $x$, extended to complex $x$, is the mechanism that finally imposes $\beta=D/2$.

What would settle it

The equality of (4.37) and (4.42) would be falsified by any non-vanishing contribution to $J(r,\beta)$ from the part of the fundamental domain away from $\operatorname{Im}\tau\to\infty$ in the limit $\alpha'\to0$: for instance, a numerical evaluation of the full torus integral at fixed $r>1$ and fixed $\beta$ showing that the region near $\tau=e^{i\pi/3}$ or near the zero locus $rX-Y=0$ contributes at order $\alpha'^{\beta-5}$ or slower. A direct check would be to Mellin-transform the full string amplitude without the $\operatorname{Im}\tau\to\infty$ approximation and test numerically whether the difference from (4.42) vanishes as $\alpha'\to0$ for $r$ close to 1.

Watch

Extended reading notes

Core claim

The central claim of the paper is the equality of two expressions: equation (4.37), the $\alpha'\to0$ limit of the Mellin-transformed one-loop type II closed superstring amplitude, and equation (4.42), the Mellin transform of the one-loop $N=8$ supergravity amplitude. Both are written as the same rational function of $r$ and $1-r$ times the universal conformal prefactor, multiplied by an integral over the simplex $0\le\rho_1<\rho_2<\rho_3\le1$ with integrand $[(1-r)\rho_1\rho_3 + r\rho_1\rho_2 - \rho_2\rho_3 - (\rho_1-\rho_2)]^{\beta-4}$, up to permutations. The matching is enforced by the distributional kernel $I(\lambda-iD)=\int_0^\infty d\eta\,\eta^{\frac{i}{2}\lambda + \frac{D}{2} - 1}$, which behaves as a delta function setting $\beta=D/2$. Because this holds for every $r>1$, the one-loop graviton case does not need the forward-scattering, large-$r$ limit that was required at tree level.

Load-bearing premise

The calculation assumes that, in the $\alpha'\to0$ limit, the Mellin-transformed one-loop torus integral is dominated uniformly by the $\operatorname{Im}\tau\to\infty$ region of moduli space for every $r>1$, and that no other region, in particular the locus where $rX-Y=0$, contributes.

Editorial extensions

If this is right

The $\alpha'$ dependence of closed-superstring celestial amplitudes factorizes as an overall power: $(\alpha')^{\beta-1}$ at tree level and $(\alpha')^{\beta-5}$ at one loop, so the loop expansion is organized by the dimensionless ratio $\kappa_{10}^2/(\alpha')^4$.
The one-loop celestial string amplitude and the celestial $N=8$ supergravity amplitude coincide for every value of $r=-s/t$, not only in the forward-scattering region $r\to\infty$.
The features previously found for open-string gluons, namely factorization of $\alpha'$ and commutativity of the Mellin transform with the field-theory limit, also hold for closed-string gravitons, marking them as universal properties of string amplitudes in the conformal basis.
The worldline-formalism Schwinger parameters of the field-theory amplitude emerge from the modular parameter $T$ after the rescaling $\eta=(\alpha'T)^{-1}$, which explains why the field-theory integration domain is recovered.
The one-loop field-theory graviton amplitude in the conformal basis exhibits double poles in the $\beta$-plane, in agreement with the general expectation that logarithms of Mandelstam invariants produce such poles.

Reading between the lines

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

Editorial inference: the same mechanism, dominance by $\operatorname{Im}\tau\to\infty$ plus a rescaling of the modular parameter to a Schwinger parameter, is likely to generalize to higher loops whenever the Mandelstam dependence enters through exponentiated log-ratios with linear large-$T$ scaling; a two-loop graviton computation would test this.
Editorial inference: because the equality is distributional and enforces $\beta=D/2$, the conformal soft limits of the one-loop graviton amplitude should receive no string corrections in $\alpha'$; extracting the soft limits from (4.37) would be a concrete check.
Editorial inference: the different behaviour at tree level (large $r$ needed) and one loop (all $r$) raises a puzzle for open-closed duality in the celestial basis, since worldsheet duality would seem to require compatible $r$-dependence in the two descriptions.
Editorial inference: the commutativity result is sensitive to the order of limits; for celestial observables whose Mellin kernel is not a delta function, string corrections might survive the low-energy limit, so the result may be special to this basis and this observable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Here's my take on arXiv:2504.17989. It's a reasonable extension of the celestial-string program to one-loop closed strings, and the main claim—that the field-theory limit commutes with the Mellin transform for all r—is plausible. But the proof has real analytic gaps that need referee attention.

What's new: this is the first computation of the four-graviton one-loop closed superstring amplitude in the celestial basis. The α' factorization they find, with the dependence organized as an overall factor (α')^{β-5}, is consistent with earlier open-string work, and the paper makes a clean statement about the loop expansion. The matching between the string expression (4.37) and the N=8 supergravity celestial amplitude (4.42) is a non-trivial check, and having the supergravity result written down is itself useful.

Where I have concerns: the step from the full moduli integral to the Im τ→∞ asymptotic form is not justified. After the Mellin transform, the α' dependence has been factored out of the integrand, so the standard argument that the cusp dominates no longer applies. In fact, the T integral in (4.33) is dominated by T=1, not T→∞. The asymptotic expansion (4.29)-(4.30) is still accurate at T=1 since q=e^{-2π} is small, so this is not fatal, but the paper never proves that the replacement captures the full distributional limit. The measure equality (4.31) is asserted without derivation. The integrand (rX-Y)^{β-4} has a singular locus that is not analyzed; for imaginary β it is not classically integrable, and the paper relies entirely on the distributional identity from [27] without discussing how the ν-integral is defined. Finally, the field-theory side is computed in the worldline formalism derived from the string, so the cross-check is not fully independent.

These are fillable gaps, not a smoking gun. I don't see an obvious algebra error, and the final equality is consistent with the previous open-string result.

Who this is for: people working in celestial holography and string amplitudes. It extends a known series of results and gives a concrete example to test against.

Recommendation: send it to peer review. The referee should ask for a careful treatment of the moduli-space limit, including the finite-T contributions and the singular locus. With those fixed, this would be a solid contribution.

Referee Report

3 major / 5 minor

Summary. The manuscript studies the four-graviton one-loop amplitude in type II closed superstring theory in the celestial basis. After a Mellin transform in the external energies, the authors obtain an expression in which the α' dependence factorizes as an overall power (α')^{β−5}, Eq. (4.27). They then take the α'→0 limit of the Mellin-transformed amplitude by approximating the genus-one moduli integrand by its Im τ→∞ asymptotics, arriving at Eq. (4.37). This is compared with a direct calculation of the celestial one-loop N=8 supergravity amplitude in the worldline formalism, Eq. (4.42), and the two are claimed to coincide after the distributional identity I(λ−iD) is used to enforce β=D/2. The paper concludes that, at one loop, the field theory limit commutes with the Mellin transform for all r>1, extending earlier open-string gluon results to closed-string gravitons.

Significance. If the main claim were established, this would be a valuable contribution to celestial string amplitudes. The α' factorization is explicit and clean, the organization of the loop expansion in terms of κ_{10}²/(α')⁴ is natural, and the use of the distributional identity from [27] is a good idea. The paper also provides a useful worldline derivation of the celestial one-loop N=8 supergravity amplitude. However, the central commutativity result rests on an unverified cusp-dominance assumption; the concern raised below is load-bearing. With that gap filled, the paper would meet the standard for publication; as it stands the main claim is not fully supported.

major comments (3)

[Section 4.4, Eqs. (4.29)–(4.33) and (4.35)] The central step of the paper is the replacement of the moduli integral in (4.26) by its Im τ→∞ asymptotics, but this replacement is never justified by a uniform bound on the fundamental domain. The concern is not merely technical: after the change of variables (4.31) and η=(α'T)^{-1}, the remaining T integral in (4.33) is ∫_1^∞ dT T^{β−D/2−1}, whose absolute value for D=4 and β=−iλ/2 is ∫_1^∞ dT T^{−3}; this integral is dominated by the lower endpoint T=1, precisely the region where (4.29)–(4.30) are not valid. Therefore the assertion that the α'→0 limit of the Mellin-transformed amplitude is governed by the cusp is not supported. The authors need to either prove that finite-T contributions vanish or are absorbed into the same distributional identity, or revise the derivation of (4.37).
[Section 4.4, Eq. (4.31)] The displayed equality between the moduli measure and ∫_1^∞ dT/T² ∫_0^1[dρ] + perm is stated without derivation and is not an identity on the fundamental domain. The fundamental domain is |τ|≥1, so the lower limit on T should be √3/2 after the Re τ integration, not 1; more importantly, the full integrand in (4.26) depends on Re τ and Re ν through the theta functions in (3.8), so the reduction to T and ρ_i is valid only after the cusp approximation has been made. The constants coming from dτ dτ̄ and ∏ d²ν_l are also omitted. This step needs to be made precise, since it is the basis for the α' absorption in (4.35).
[Section 4.4, Eq. (4.25)] The analytic continuation of the ω-integral ignores the zero locus of rX−Y. Since χ in (3.8) is real and positive, rX−Y is real, and its zero locus is a real hypersurface; for β=−iλ/2 the naive power (rX−Y)^{β−4} has modulus |rX−Y|^{−4}, which is not locally integrable across that hypersurface. The paper does not analyze whether this locus contributes to the α'→0 limit or whether boundary terms from the analytic continuation in β survive in (4.33). This is load-bearing because the subsequent T-integral and the identification with (4.42) assume that only the cusp region matters.

minor comments (5)

[Section 2, after Eq. (2.10)] There is a typo in the condition 's + t + u+ = 0'; it should read s+t+u=0.
[References] References [24] and [29] appear in the bibliography but are not cited in the body of the text; please cite them in the relevant discussion or remove them.
[Abstract and Section 5] The claim that the field theory limit commutes with the Mellin transform 'for all values of the conformally invariant cross-ratio' should be qualified: the derivation explicitly restricts to r>1 via the Heaviside function after Eq. (4.22) and to the distributional support β=D/2.
[Section 4.4, Eq. (4.31)] The lower limit of the T-integration is written as 1, but the fundamental domain (3.4) gives T≥√3/2 after integrating over Re τ; if this replacement is an approximation, it should be stated as such.
[Eqs. (4.37) and (4.42)] The exponent notation such as r^{14−β}/3 is ambiguous without parentheses; the authors may wish to write r^{(14−β)/3}.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the commutativity claim is a substantive distributional identity, not a repackaging of inputs.

full rationale

The paper's central claim is that at one loop the α′→0 limit commutes with the Mellin transform. Both sides of the comparison are ultimately evaluated through the same Im τ→∞ asymptotic that defines the field-theory limit, but the computations are performed in different orders, and the agreement is not imposed by definition. The string side (4.26)–(4.37) first performs the Mellin transform over ω, factors out (α′)β−5, and then takes the α′→0 limit via the change η=(α′T)−1, producing the distributional kernel I(λ−iD). The field-theory side (4.38)–(4.42) starts from the known worldline expression (3.12), benchmarked to N=8 supergravity [15,16], and obtains the same kernel from the independent energy integral (4.40). The equality (4.37)=(4.42) then follows because I(λ−iD) enforces β=D/2, making the ρ-integrand exponents coincide; the paper states this explicitly rather than hiding it. The ρ-polynomials on both sides agree only after carrying out the respective asymptotics, so this is a genuine consistency check rather than a tautology. The paper's self-citations [7,8,19] are to prior method and context, but the closed-string computation is performed here, and the distributional property of I is attributed to external ref. [27]. The main weakness is the unproved assumption that the Mellin-transformed moduli integral is dominated uniformly by Im τ→∞, together with the unanalyzed singular locus of (rX−Y)β−4; that is a convergence and analyticity risk, not a circular reduction of the result to its inputs.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new free parameters or entities. It relies on string-theory background and the celestial holography framework from prior literature. The load-bearing assumptions are the correctness of the known type II one-loop amplitude, the dominance of the Imτ to ∞ region in the field theory limit, the distributional identity of [27], and the validity of exchanging the Mellin transform with the worldsheet and α' limits.

assumptions (5)

domain assumption The type II superstring four-graviton one-loop amplitude is given by the integral over the genus-1 fundamental domain in eqs. (3.3)-(3.7).
Taken from Green-Schwarz-Brink [15] and Schwarz [16]; the paper does not re-derive it.
domain assumption The α' to 0 field theory limit of the one-loop string amplitude is controlled by the Im τ to ∞ region of moduli space, with F2(a,τ) ≈ a(Imτ)^(1/2).
Cited to [15,16]; used in Section 4.4 to evaluate the celestial integral. No analysis of other moduli regions or singular loci is given.
standard math The Mellin transform formula (4.25), ∫_0^∞ dω ω^(-β+3) e^(-α'ω(rX-Y)) = (α')^(β-4)(rX-Y)^(β-4)Γ(4-β), and its exchange with the worldsheet integrals.
Standard gamma integral; the exchange of integrals is assumed valid by analytic continuation.
domain assumption The distributional identity I(x) = ∫_0^∞ dη η^(ix/2-1) = 4πδ(x), extended to complex argument x = λ-iD as in Borji-Pano [27].
Provides the support condition β = D/2 that makes (4.37) and (4.42) match. Not derived in the paper.
domain assumption The celestial holography dictionary: Mellin transform definition (4.7), delta function (4.8), conformal weights (4.11), and the conformal correlator form (4.10).
Standard framework from Pasterski-Shao-Strominger [1-3]; the paper uses it without re-derivation.

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Cite this review

Pith. "Pith review of Celestial closed strings at one-loop." pith.science (2026). https://pith.science/paper/6WN4OCCB

@misc{pith2026250417989,
  author       = {Pith},
  title        = {Pith review of: Celestial closed strings at one-loop},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6WN4OCCB}},
  note         = {Machine review of arXiv:2504.17989}
}

abstract

In this paper we continue our investigation of superstring scattering amplitudes in the conformal basis. We focus on the case of four graviton scattering processes at 1-loop in \emph{closed} superstring theory. We write the expression for such a process in the celestial variables and confirm previous expectations. In particular, we find the adequate overall factorization of the $\alpha'$ dependence which organizes the loop expansion of closed string celestial amplitudes. We also show that, at 1-loop, the field theory limit, when properly defined, commutes with the Mellin transform of the amplitudes for all values of the conformally invariant cross-ratio, something that had already been observed for gluon processes at 1-loop in open string theory and is to be compared with the tree-level computations. This indicates that many of the features satisfied for open string gluon amplitudes at tree and 1-loop levels are also universal properties of graviton celestial amplitudes in closed string theory. As a by-product, we also compute field theory graviton amplitudes at 1-loop in the conformal basis.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.

[1] [27]

Borji and Y

M. Borji and Y. Pano, Distributional celestial amplitudes, JHEP 07, 120 (2024); [arXiv:2401.08877 [hep-th]]

arXiv 2024

[2] [1]

Pasterski, S

S. Pasterski, S. H. Shao and A. Strominger, Flat Space Amplitudes and Conformal Symmetry of the Celestial Sphere , Phys. Rev. D 96, 065026 (2017); [arXiv:1701.00049 [hep-th]]

arXiv 2017

[3] [2]

Pasterski and S

S. Pasterski and S. H. Shao, Conformal basis for ﬂat space amplitudes, Phys. Rev. D 96, 065022 (2017); [arXiv:1705.01027 [hep-th]]

arXiv 2017

[4] [3]

Pasterski, S

S. Pasterski, S. H. Shao and A. Strominger, Gluon Amplitudes as 2d Conformal Cor- relators, Phys. Rev. D 96, 085006 (2017); [arXiv:1706.03917 [hep-th]]

arXiv 2017

[5] [4]

Stieberger and T

S. Stieberger and T. R. Taylor, Strings on Celestial Sphere , Nucl. Phys. B 935, 388 (2018); [arXiv:1806.05688 [hep-th]]

arXiv 2018

[6] [5]

Superstring/Supergravity Mellin Correspondence in Grassmannian Formulation

S. Stieberger and T. R. Taylor, “Superstring/Supergravity Me llin Correspon- dence in Grassmannian Formulation,” Phys. Lett. B 725, 180-183 (2013) doi:10.1016/j.physletb.2013.07.012 [arXiv:1306.1844 [hep-th]]

work page Pith review arXiv 2013

[7] [6]

Superstring Amplitudes as a Mellin Transform of Supergravity

S. Stieberger and T. R. Taylor, Nucl. Phys. B 873, 65-91 (2013) doi:10.1016/j.nuclphysb.2013.03.016 [arXiv:1303.1532 [hep-th]]

work page Pith review arXiv 2013

[8] [7]

Castiblanco, G

L. Castiblanco, G. Giribet, G. Marin and F. Rojas, Celestial strings: ﬁeld theory, conformally soft limits, and mapping the worldsheet onto th e celestial sphere, Phys. Rev. D 110, 126001 (2024); [arXiv:2405.01643 [hep-th]]

arXiv 2024