{"id":"1983637d-36bf-4ee0-bbaa-83c971cdde51","arxiv_id":"2504.20977","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":0,"one_line_summary":"The integrated double-soft eikonal function for a massive and a massless emitter at arbitrary angle is derived analytically and cross-checked with a semi-numerical subtraction method.","lead":"This paper computes, in analytic form, the integrated double-emission eikonal function for one massive and one massless emitter at any relative angle, a required building block for next-to-next-to-leading-order QCD predictions. Two independent calculations are presented, together with fast downloadable code for numerical evaluation.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The pairwise integrated eikonal may inherit the acknowledged S^m_ij convention ambiguity from Ref [63] vs Ref [41]; only the color-summed object is guaranteed to agree, so the per-pair claim needs a check or qualification.","rationale":"The integration methodology itself is sound: two independent computational routes, numeric checks, small-beta expansion, and released C code give strong internal support, and I see no internal inconsistency in the IBP/DE or subtraction derivations. The load-bearing soft spot is at the input stage: the per-pair integrated eikonal that the abstract advertises is built from a particular S^m_ij whose acknowledged non-uniqueness under color conservation can change the pairwise function without changing the physical color-summed integral. The reader's weakest_assumption points to exactly this input and to the color-conservation reduction, and I agree that this is the least secure part of the central claim. Since the paper already discloses the different expression in footnote 1, this is not a hidden flaw; it is a qualification that should be closed before Eqs (4.36)-(4.37) are used as a universal pairwise ingredient. A targeted comparison with Ref [41]'s expression, or an explicit demonstration of pairwise convention independence, would settle it. Hence I would hold ACCEPT conditional on that check rather than reject the calculation, whose technical core appears reliable.","tokens_in":38669,"tokens_out":15170,"duration_ms":167461,"concrete_test":"Recompute SS[tilde S_ij] and SS[tilde I_ij] using the alternative S^m_ij from Ref [41] instead of Eq (2.9), keeping all other definitions identical and using an independent route, for example the rest-frame subtraction of Section 5 or direct integration in d=4-2ε with analytical pole extraction. If the pairwise results coincide, the ambiguity is benign. If they differ, form the color-summed combination sum_{i<j} SS[tilde S_ij] <M|T_i·T_j|M> for a minimal three-emitter Born state with one massive and one massless emitter plus a spectator, and check that it is identical for the two conventions. The paper should then either prove pairwise independence or explicitly state that Eqs (4.36)-(4.37) are convention-bound and only the color-summed combination is scheme-independent.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central results, Eqs (4.36)-(4.37), are per-pair functions SS[tilde S_ij] and SS[tilde I_ij]. The physical double-soft contribution is the color sum over all emitter pairs, and the pairwise tilde functions used in Eqs (2.12)-(2.14) depend on the chosen expression for the massive-emitter eikonal S^m_ij. Eq (2.9) takes S^m_ij from Ref [63]; footnote 1 states that Ref [41] gives a different expression that agrees only after summation over i,j thanks to color conservation. Because the reduction to pair sums rewrites the color sum as sum_{i<j} (2S_ij - S_ii - S_jj) T_i·T_j, and because the color operators T_i·T_j are linearly dependent whenever there are three or more emitters, a convention change in S^m_ij can change the pairwise tilde S_ij while leaving the total color-summed integral invariant. Sections 4 and 5 use the same Eq (2.9) as input, so their mutual agreement does not test this convention dependence. Thus the headline object, 'the integrated double-emission eikonal function of a massive and a massless emitter,' is not uniquely defined unless the convention dependence is shown to cancel in the summed physical quantity or the result is explicitly tied to the Ref [63] convention throughout the subtraction scheme.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents an analytic calculation of the integrated double-emission eikonal functions SS[\\tilde{S}_{ij}] and SS[\\tilde{I}_{ij}] for one massive and one massless emitter at arbitrary relative angle, which are needed to extend the nested soft-collinear subtraction scheme to processes with massive final-state particles. The main computation uses reverse unitarity extended to integrals with Heaviside functions, IBP reduction to 52 master integrals, and differential equations; the final result through O(ε^0) is expressed in terms of classical polylogarithms and the function Li_{2,2}. A complementary semi-numerical method is developed in the massive parton rest frame, where all divergences are extracted analytically and the finite remainder is given as a numerically integrable expression. The two methods agree, and additional checks include a small-β expansion up to β^4, benchmark points, and public C code for fast evaluation.","tokens_in":38939,"tokens_out":7864,"duration_ms":81339,"significance":"The result is a relevant technical ingredient for NNLO QCD calculations with massive final-state particles and appears to be correct. Strengths of the paper are the two independent computational routes (differential-equation/IBP and rest-frame subtraction) that agree, analytic boundary conditions at β=0, the simplification of the elliptic sectors, and the provided fast C code and ancillary files. The combination of these validation steps makes the central result credible and useful. The calculation is conditional on the accepted soft factorization input for the eikonal functions, which is standard and clearly referenced.","major_comments":[],"minor_comments":[{"comment":"The per-pair integrated functions in Eqs. (4.36) and (4.37) are defined through Eq. (2.9), i.e. the Ref. [63] expression for S^m_ij. Since footnote 1 notes that Ref. [41] gives a different expression for S^m_ij that agrees only after color summation, the pairwise integrated functions are convention-dependent, with only the color-summed combination entering the physical double-soft contribution being invariant. Please add an explicit sentence to this effect, so that readers do not apply the pairwise results in a different convention.","section":"Section 2 (footnote 1) and Section 6"},{"comment":"There is a typo in the integrand: 'Ξij({km,k n)' should read 'Ξij(km,kn)' with the braces removed.","section":"Eq. (2.20)"},{"comment":"The abstract calls the computation 'analytic', while Section 5 is semi-numerical and uses numerical integration for finite terms. Adjust the wording to indicate that an analytic result is provided and complemented by a semi-numerical cross-check.","section":"Abstract and Section 5"}],"recommendation":"minor_revision","confidential_remarks":"The stress-test concern about the convention dependence of the per-pair eikonal is real but already largely addressed by the explicit choice in Eq. (2.9) and footnote 1; a short clarifying statement would remove any remaining ambiguity. The paper is within the scope of JHEP and the technical work appears sound."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Heads up: this is a strong technical paper, and the stress-test concern about the S^m_ij convention is real but minor and mostly already acknowledged by the authors. The new result is the integrated double-emission eikonal for one massive and one massless emitter at arbitrary angle, which was missing for extending the nested soft-collinear subtraction scheme to NNLO with massive final-state particles. The calculation is hard: two elliptic sectors, a large system of master integrals, and the final expression is simplified to polylogs plus Li_{2,2}. The evidence is solid: two independent methods (IBP/differential equations and rest-frame subtraction) agree, the small-beta expansion matches to O(beta^4), benchmark points are given, and the ancillary files plus C code make the result usable. The elliptic complexity cancelling in the final answer is a nice structural observation, and the rest-frame method is a useful methodological addition.\n\nThe soft spot is genuinely a caveat, not a fatal flaw. The per-pair functions tilde S_ij and tilde I_ij are defined with the massive-emitter eikonal S^m_ij taken from Ref [63]. Footnote 1 notes Ref [41] has a different expression that agrees only after color summation. The stress-test note is correct that the pairwise integrated object is therefore convention-dependent, and the two computational methods share the same input, so they do not test that choice. However, the paper explicitly flags this, and the physical double-soft contribution is the color sum, which is invariant. For a subtraction-scheme ingredient this is par for the course; the result is what it is within the stated convention. It would help if the authors added one sentence in the conclusion saying that the per-pair function is convention-dependent and only the color-summed combination is scheme-independent. That is a minor revision.\n\nThe reliance on prior eikonal factorization is expected; the paper is about the integration, not about rederiving soft limits. The numerical performance near beta ~ 1 is noted as challenging, but the C code addresses it.\n\nWho this is for: anyone building NNLO subtraction schemes with massive partons, and people interested in phase-space integration with Heaviside functions and elliptic sectors. It deserves a serious referee; I would accept after a minor revision asking for the convention clarification.","headline":"Strong, careful NNLO ingredient: the integrated double-soft eikonal for a massive+massless pair is genuinely new, two independent methods agree, and the per-pair convention caveat is real but minor and already flagged by the authors.","tokens_in":39483,"tokens_out":2430,"would_cite":true,"duration_ms":26355,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper computes the integrated double-emission eikonal function for one massive and one massless emitter at an arbitrary angle, expressing the two soft-gluon and quark-antiquark channels analytically to order $\\varepsilon^0$ in terms…","keywords":["QCD corrections","NNLO calculations","hadronic colliders","double-soft limit","eikonal functions","massive emitters","reverse unitarity","polylogarithms"],"falsifier":"Numerically evaluate the defining ordered phase-space integrals in higher dimension $d=6$, where they converge, at a kinematic point from Table 2, then use the dimension-shift relation to continue to $d=4-2\\varepsilon$; a mismatch with the $\\varepsilon^0$ coefficient of Eqs. (4.36)-(4.37) would refute the analytic result.","tokens_in":2322,"feed_emoji":"📐","tokens_out":4238,"duration_ms":154577,"temperature":0.7,"pith_summary":"When two partons are emitted with very small momenta, the squared QCD amplitude factorizes into a universal eikonal function, and next-to-next-to-leading-order subtraction schemes need that function integrated over the unresolved phase space. The paper supplies this integrated quantity for the still-missing case of one massive and one massless emitter whose momenta form an arbitrary angle, covering both two-gluon and quark-antiquark emission. The result—explicit functions of the massive emitter's velocity and the cosine of the angle, expressed through ordinary logarithms and polylogarithms including $\\mathrm{Li}_{2,2}$—holds through the finite part in dimensional regularization. A second, semi-numerical computation in the massive parton's rest frame independently confirms the analytic answer.","feed_headline":"Massive-plus-massless soft integral is analytic at any angle","feed_subtitle":"Supplies a missing NNLO subtraction ingredient for massive final-state processes.","key_machinery":"The machinery is reverse unitarity extended to Heaviside functions: the energy-ordering theta-function becomes a delta-function when differentiated, enabling integration-by-parts reduction to 52 master integrals. Differential equations are solved and two elliptic sectors are transformed using complete elliptic integrals before symbol-level simplification removes elliptic functions entirely. A second approach works directly in the massive parton's rest frame, applying soft and collinear subtraction operators to the eikonal integrand so that divergent parts are computed analytically and the finite remainder is a four-dimensional numerical integral.","core_discovery":"The central claim is that the energy-ordered double-emission integrals $\\mathrm{SS}[\\widetilde{S}_{ij}]$ and $\\mathrm{SS}[\\widetilde{I}_{ij}]$ can be evaluated analytically for any relative angle. Equations (4.36) and (4.37) give their expansions through $O(\\varepsilon^0)$ as functions of $\\beta$ and $y=\\cos\\theta$ in terms of $\\mathrm{Li}_2$, $\\mathrm{Li}_3$, $\\mathrm{Li}_4$, and the two-variable function $\\mathrm{Li}_{2,2}$. All intermediate elliptic integrals cancel in the final combination, so the integrated eikonal has polylogarithmic complexity. The paper also develops an independent rest-frame subtraction computation that extracts all $1/\\varepsilon$ poles analytically and leaves a finite numerical remainder; the two methods agree.","pith_inferences":["The cancellation of elliptic sectors suggests that the integrated eikonal is simpler than the master integrals from which it is built; a direct derivation that avoids elliptic functions might exist for this quantity.","The rest-frame subtraction strategy likely applies to the remaining two-massive-emitter arbitrary-angle case, where conventional reduction is even harder.","Because the energy-ordering constraint is what forces the laboratory frame, replacing it with another slicing variable may change the polylogarithmic content; testing this would clarify the role of the Heaviside function in the final complexity.","Embedding $\\mathrm{SS}[\\widetilde{S}_{ij}]$ and $\\mathrm{SS}[\\widetilde{I}_{ij}]$ in a subtraction code for a specific massive process and verifying infrared pole cancellation would provide a direct phenomenological validation."],"forward_implications":["Removes the missing massless-massive ingredient for extending the nested soft-collinear subtraction scheme to processes with massive final-state particles.","The analytic form uses only classical polylogarithms and $\\mathrm{Li}_{2,2}$, allowing fast, high-precision evaluation in milliseconds via the supplied C implementation.","The rest-frame subtraction method provides an independent check and is expected to be reusable for other measurement constraints, including the massive-massive emitter case.","The provided benchmark points and small-$\\beta$ expansions give concrete validation targets for future implementations."],"supporting_citations":[{"why":"Defines the nested soft-collinear subtraction scheme and the energy-ordered phase-space integrals computed here.","marker":"[43]"},{"why":"Extends reverse unitarity to integrands with Heaviside functions, enabling the IBP reduction of Section 4.","marker":"[57]"},{"why":"Provides the massive-emitter eikonal function used as input in Eq. (2.9).","marker":"[63]"},{"why":"Establishes the double-soft factorization of tree-level QCD amplitudes from which the eikonal functions are taken.","marker":"[61]"},{"why":"Computes the analogous integrated double-soft integral for two massless emitters, the case this paper generalizes.","marker":"[54]"},{"why":"Computes the analogous integrated double-soft terms for massive back-to-back emitters.","marker":"[55]"},{"why":"Introduces reverse unitarity, the underlying method for real-emission integrals.","marker":"[56]"},{"why":"Describes how to bring elliptic differential-equation sectors into epsilon-form, used for the two irreducible master-integral sectors.","marker":"[78]"},{"why":"Supplies the rest-frame subtraction idea adapted in Section 5.","marker":"[60]"},{"why":"Establishes the canonical epsilon-form method for differential equations used to solve the master-integral system.","marker":"[67]"}],"fun_headline_variants":["Polylog solution for mixed-mass emitters at any angle","Massive-massless soft eikonal: analytic at any angle","Closed form for arbitrary-angle massive-massless emission","Two-emitter soft integral yields polylogs at any angle"],"cache_read_input_tokens":41600,"weakest_assumption_plain":"The load-bearing premise is that the massive-emitter eikonal function taken from Ref. [63] is the correct soft factor; the paper notes that another reference gives a different expression that agrees only after color summation, and both independent computational routes share this same input, so an error there would invalidate the integrated result.","fun_headline_variants_meta":{"raw":{"variants":["Polylog solution for mixed-mass emitters at any angle","Massive-massless soft eikonal: analytic at any angle","Closed form for arbitrary-angle massive-massless emission","Two-emitter soft integral yields polylogs at any angle"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000933,"raw_usage":{"total_tokens":3944,"prompt_tokens":848,"completion_tokens":3096,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":3027}},"tokens_in":464,"tokens_out":3096,"duration_ms":22600,"temperature":1.0,"reasoning_tokens":3027,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T05:15:18.922911+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically evaluate the defining ordered phase-space integrals in higher dimension $d=6$, where they converge, at a kinematic point from Table 2, then use the dimension-shift relation to continue to $d=4-2\\varepsilon$; a mismatch with the $\\varepsilon^0$ coefficient of Eqs. (4.36)-(4.37) would refute the analytic result.","supporting_citations":[{"cited_title":"Analytic double-soft integrated subtraction terms for two massive emitters in a back-to-back kinematics","cited_arxiv_id":"2004.01663","evidence_quote":"Computes the analogous integrated double-soft terms for massive back-to-back emitters."}],"review_version":1}