{"id":"be2a9e7f-df33-45ff-9983-e9bc1cb82479","arxiv_id":"2411.12497","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":0,"one_line_summary":"The authors propose that the off-forward pomeron and odderon amplitudes evolve as the sum and difference, respectively, of the corresponding forward amplitudes evaluated at rescaled momenta.","lead":"This paper claims that the small-x evolution of off-forward dipole amplitudes in momentum space can be determined entirely from the evolution of forward amplitudes with shifted momenta. If valid, it would let physicists compute the high-energy evolution of gluon GTMDs from the better-studied TMDs, which is relevant for the Electron-Ion Collider.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The main result hinges on an unproven symmetry-restoration claim: numerical or analytic evidence is required before Eqs. (24)/(25) can be accepted.","rationale":"The paper's argument is algebraically clear and the final formulas are elegant, but the derivation rests entirely on the assumed translation symmetry of G. This is exactly the weakest point, and no independent support (e.g., numerical simulation of BK for off-forward amplitudes) is given. The reader's verdict of CONDITIONAL is appropriate: the result should be accepted only if the symmetry-restoration statement is substantiated, or clearly labeled as a conjecture pending numerical tests. I agree with the reader that this assumption is the load-bearing step between Eq. (19) and the claimed forward--off-forward correspondence.","tokens_in":10362,"tokens_out":1400,"duration_ms":13454,"concrete_test":"Choose a simple, analytically tractable initial condition, e.g. N(k,Δ)=N0(k) symmetric in Δ→−Δ, with N0(k) not translationally symmetric under the required momentum shift. Numerically evolve Eq. (12) or Eq. (19) in Y for several units, and directly test whether the resulting G(k,Δ) satisfies G(k,Δ)=G(k−a/2,Δ−a) for a=Δ and for general a. If the relation holds to good accuracy after sufficiently long evolution, the symmetry-restoration claim is supported; if it fails, Eqs. (24) and (25) are not established.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central result Eq. (24) follows only if G obeys the translation symmetry Eq. (21), G(k,Δ)=G(k−a/2,Δ−a), for all k,Δ. The integro-differential equation (19) is not explicitly shown to preserve this symmetry, and the paper's justification is a vague statement that 'the evolution depends very weakly on the initial conditions' and will 'eventually restore' the symmetry. BK evolution does not automatically restore this relation: the first term A1 does have the stated symmetry, but the nonlinear fourth term A4 is a convolution in Δ′ and can break the required relation. No concrete initial conditions, fixed-point analysis, or numerical test are given. Hence the derived formulas for N(k,Δ), O(k,Δ), and GTMDs are conditional on an unverified assumption, exactly as the reader noted.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper derives a forward–off-forward correspondence for the small-x evolution of dipole amplitudes in momentum space. The authors write the momentum-space BK equation for the off-forward pomeron amplitude N(k,Δ) as a sum G(k,Δ)+G(k,-Δ), then argue that G is invariant under a momentum-space translation symmetry. Setting a=Δ in that symmetry yields the central relation ∂_Y N(k,Δ) = 1/2 ∂_Y[N(k-Δ/2,0)+N(k+Δ/2,0)], and an analogous expression for the odderon. The same reasoning is extended to gluon GTMDs. The paper also gives estimates of the off-forward pomeron and odderon amplitudes in the linear and saturation regions.","tokens_in":10464,"tokens_out":10267,"duration_ms":96366,"significance":"If the central relation (24) were valid, it would be a powerful and practical simplification: off-forward small-x evolution would be fully determined by forward evolution with rescaled momenta, and gluon GTMD evolution could be obtained from TMD evolution. The algebraic manipulation from Eq. (18) to Eq. (24) is internally consistent conditional on the translation-symmetry assumption in Eq. (21), and the paper is well structured. However, the paper does not provide machine-checked proofs or numerical code, and the load-bearing symmetry claim is not established; the manuscript’s contribution is therefore best read as a proposal for a correspondence rather than a rigorous derivation.","major_comments":[{"comment":"Equation (21) is the load-bearing step of the derivation: setting a=Δ converts it into the main result Eq. (24), and the odderon result Eq. (48) in the appendix depends on the identical assumption. The justification offered in the text does not establish this relation. The invariance of the kernels in Eq. (19) under the simultaneous shift (20) is insufficient: applying this shift to the first term of Eq. (19) and changing the integration variable k''=k'-a/2 yields an integrand N(k''+a/2,Δ-a), so equality with G(k,Δ) requires the translation symmetry of N itself, i.e. N(k+a/2,Δ-a)=N(k,Δ), which the authors explicitly deny for N(k,Δ) in the paragraph after Eq. (21). The nonlinear convolution term in Eq. (19) is not shown to restore the relation. No analytic proof or numerical demonstration is provided. Consequently, Eqs. (24) and (25) are not established.","section":"Exploiting the translation symmetry, Eq. (21)"},{"comment":"The statement that BK evolution 'depends very weakly on the initial conditions' and 'will eventually restore the translation symmetry in G' is not a known property of the BK equation in the requested form. BK evolution does produce geometric scaling and universality of the saturation scale at high rapidity, but that does not imply functional invariance of the kind in Eq. (21) for all k and Δ. The paper gives no fixed-point analysis or numerical test showing that solutions initialized without the symmetry converge to solutions satisfying Eq. (21). Without such evidence, the symmetry-restoration claim is a postulate, not a derivation.","section":"Exploiting the translation symmetry, paragraph on integro-differential equations"},{"comment":"The estimates in Eqs. (26)–(28) and the GTMD relation in Eq. (30) are applications of the central relations Eqs. (24) and (25). Because those relations are not established, these phenomenological estimates and the GTMD–TMD correspondence are not yet supported by the arguments in this paper. This is not a separate error but a direct consequence of the gap identified in the first major comment.","section":"Estimation of pomeron and odderon amplitudes and GTMD extension"}],"minor_comments":[{"comment":"The name 'Alterelli' in 'Dokshitzer-Gribov-Lipatov-Alterelli-Parisi' should be 'Altarelli'.","section":"Introduction, first sentence"},{"comment":"The word 'amlitudes' should be 'amplitudes'.","section":"Section title 'Evolution of Odderon amplitude'"},{"comment":"The object J(k,Δ) is introduced in Eq. (43) as a static decomposition of ∂_Y O, but Eq. (44) uses the notation ∂/∂Y J(k,Δ), which is not defined and appears to conflict with the earlier definition. The equation should either define J(k,Δ) directly or clarify the meaning of the derivative.","section":"Appendix, Eq. (44)"},{"comment":"The argument 'k + k′ − Δ / 2' in the second nonlinear term is ambiguous; parentheses should be added to clarify whether it means k + k' - Δ/2, (k + k' - Δ)/2, or k + (k' - Δ)/2.","section":"Eq. (19)"},{"comment":"The phrase 'completely determined' in the abstract and the claim in the introduction that the result holds 'within a reasonable assumption' are in tension with the later attempt to derive the symmetry as a consequence of evolution. The text should state the status of the translation-symmetry assumption explicitly and consistently.","section":"Abstract and introduction"}],"recommendation":"major_revision","confidential_remarks":"The manuscript presents a potentially useful correspondence, but its central step is an unproven and, as argued, formally under-justified symmetry statement. The authors should either prove the symmetry restoration, provide a numerical demonstration starting from non-symmetric initial conditions, or present the translation symmetry as an explicit ansatz with a clear discussion of its validity. The paper's current oscillation between 'assumption' and 'derived property' would benefit from editorial clarification. The topic is within the journal's scope and the algebraic structure is attractive; the key question is whether the load-bearing symmetry can be supported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper's central relation — Eqs. (24) and (25) — is clean and new, and the odderon mirror is a nice touch. The algebra from the definition of G to the forward expression is internally consistent, and the extension to gluon GTMDs is a reasonable corollary. If the correspondence held, it would save real work for EIC phenomenology.\n\nThe soft spot is exactly the one the reader flagged. The \"translation symmetry\" of the kernels in Eq. (19) does not imply Eq. (21). Those kernels are symmetric only when the integration variable and the external momenta are translated together, but then the amplitudes N inside sit at shifted arguments. So G is translationally invariant only when N itself satisfies N(k,Δ)=N(k−a/2,Δ−a), which for a=Δ is essentially the claimed result. The paper's statement that evolution \"will eventually restore\" this symmetry is not backed by any argument, analytic or numerical. The nonlinear term A4 is a convolution in Δ′, and nothing shows it preserves the relation; a simple test with N(k,Δ)=[F(k−Δ/2)+F(k+Δ/2)]/2 suggests the convolution does not factor into the forward pieces. So Eqs. (24) and (25) are a conjecture, not a proof.\n\nThis is not a fatal flaw in the sense of wrong math — the manipulations are consistent. But the authors present the result as \"shown\" in the abstract, and the missing justification is load-bearing. The paper needs either a proof that the evolution preserves the required symmetry (or converges to it), or an honest re-labeling as a conjecture with numerical benchmarks.\n\nFor a referee: I'd send it to peer review because the question matters and the authors have a concrete, testable claim. But I'd ask for a major revision that addresses Eq. (21), and I would not accept it in the current form. If the symmetry check fails, the paper becomes a much weaker observation about a special class of initial conditions.","headline":"Clean, useful-looking correspondence between off-forward and forward small-x evolution, but it rests on an unproven symmetry-restoration claim that is close to the conclusion itself; currently a conjecture, not a proof.","tokens_in":11004,"tokens_out":8927,"would_cite":false,"duration_ms":78930,"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":"Off-forward dipole evolution reduces to forward amplitudes","keywords":["small-x evolution","dipole amplitude","off-forward","pomeron","odderon","momentum space","GTMD","TMD"],"falsifier":"Numerically evolve the leading-log BK equation in momentum space starting from an off-forward initial condition that breaks the translation symmetry of Eq. (20); if $\\partial_Y N(k,\\Delta)$ differs from $\\tfrac12\\partial_Y[N(k-\\Delta/2,0)+N(k+\\Delta/2,0)]$ at any $Y>0$, then the claimed symmetry restoration and the main correspondence fail.","tokens_in":10126,"feed_emoji":"⚛️","tokens_out":3092,"duration_ms":29919,"temperature":0.7,"pith_summary":"This paper tries to establish that the small-x evolution of off-forward dipole scattering amplitudes, for both the pomeron and the odderon, is completely determined by the evolution of the corresponding forward amplitudes evaluated at rescaled momenta. If true, it means that the complicated non-linear evolution at non-zero momentum transfer can be obtained for free from better-studied forward evolution. It also implies that high-energy gluon GTMD evolution can be constructed from TMD evolution without new dynamical input. A sympathetic reader would care because off-forward quantities (GPDs, GTMDs) are harder to compute and yet increasingly relevant for planned experiments.","feed_headline":"Off-forward pomeron evolution equals forward sum","feed_subtitle":"Pomeron and odderon with momentum transfer Δ evolve as sums and differences of forward amplitudes at k±Δ/2.","key_machinery":"The load-bearing object is the function $G(k,\\Delta)$ defined in Eq. (19), which groups the linear and non-linear terms of the momentum-space BK equation so that $\\partial_Y N(k,\\Delta) = G(k,\\Delta)+G(k,-\\Delta)$. The kernels in $G$ are invariant under the simultaneous translation $k\\to k-a/2$, $k'\\to k'-a/2$, $\\Delta\\to\\Delta-a$. The paper assumes this symmetry survives even when the initial condition lacks it, then sets $a=\\Delta$ to obtain $G(k-\\Delta/2,0)=\\tfrac12\\partial_Y N(k-\\Delta/2,0)$, which converts the sum into the forward-amplitude relation. The same construction is repeated for the odderon, where antisymmetry under $\\Delta\\to-\\Delta$ turns the sum into a difference.","core_discovery":"The central discovery is a forward--off-forward correspondence in momentum space. Writing $N(k,\\Delta)$ for the pomeron amplitude with momentum transfer $\\Delta$ and $k$ conjugate to the dipole separation, the paper argues that the leading-log Balitsky--Kovchegov evolution satisfies $\\partial_Y N(k,\\Delta) = \\tfrac12\\,\\partial_Y\\big[N(k-\\Delta/2,0)+N(k+\\Delta/2,0)\\big]$. For the C-odd odderon amplitude $O(k,\\Delta)$ the analogous relation is $\\partial_Y O(k,\\Delta) = \\tfrac12\\,\\partial_Y\\big[O(k-\\Delta/2,0)-O(k+\\Delta/2,0)\\big]$. The derivation relies on splitting the evolution into two pieces, $G(k,\\Delta)$ and $G(k,-\\Delta)$, that are exchanged under $\\Delta\\to-\\Delta$, and on imposing a translation symmetry of $G$ in the $(k,\\Delta)$ plane. A consequence is that the evolution of the unpolarised gluon GTMD $F_{1,1}$ is similarly fixed by the evolution of the corresponding TMD in the forward limit.","pith_inferences":["The translation symmetry of $G$ is asserted to be restored by evolution without a proof or numerical demonstration; a direct check of this restoration on a non-symmetric initial condition would be the minimal test of the paper's main claim.","If the correspondence holds beyond leading log, it could simplify resummation schemes for off-forward observables, but the present derivation is explicitly leading-log and large-$N_c$.","The paper suggests a practical recipe: measure or compute TMD evolution, shift the argument by $\\Delta/2$, and read off GTMD evolution; experimental data from future electron-ion collisions could indirectly test this by comparing diffractive di-jet observables with TMD-based predictions."],"forward_implications":["Off-forward pomeron and odderon amplitudes can be evolved in $Y=\\ln(1/x)$ using only forward-limit evolution codes or analytic results at rescaled momenta, without solving the full $\\Delta$-dependent equation.","The odderon amplitude does not evolve in the exact forward limit $\\Delta=0$, since the difference of the two forward odderon amplitudes vanishes, consistent with the expectation that odderons appear only in non-forward scattering.","The unpolarised dipole GTMD $F_{1,1}$ satisfies the same forward--off-forward correspondence, so its small-x evolution can be obtained from the corresponding TMD evolution, going beyond the small-$\\Delta$ approximation.","Known dilute-regime and saturated-regime forward amplitudes can be immediately translated into closed-form estimates for $N(k,\\Delta)$ and $O(k,\\Delta)$ at non-zero momentum transfer.","Future numerical studies of the full off-forward evolution can be benchmarked against the identities in Eq. (24) and Eq. (25) at leading-log accuracy."],"supporting_citations":[{"why":"Balitsky's derivation of the non-linear small-x evolution equation for the dipole S-matrix supplies the starting position-space evolution.","marker":"[10]"},{"why":"Kovchegov's independent derivation gives the same equation in the Mueller dipole model, establishing the BK equation used here.","marker":"[11]"},{"why":"Hatta and Zhou provide the Fourier-transformed momentum-space form of the BK equation, Eq. (12), which is the basis for defining $G(k,\\Delta)$.","marker":"[15]"},{"why":"Motyka's study of the odderon in momentum space supplies the odderon forward amplitude form and the exponentials used in estimates at non-zero $\\Delta$.","marker":"[14]"},{"why":"Hatta, Iancu, Itakura, and McLerran supply the odderon-odderon to pomeron non-linear term structure needed for the evolution of $O(k,\\Delta)$.","marker":"[16]"}],"fun_headline_variants":["Off-forward dipole evolution from shifted forward amplitudes","Momentum-space correspondence: off-forward reduces to forward","Pomeron and odderon off-forward evolution fixed by forward","Small-x evolution: off-forward determined by forward with k shift","GTMD evolution flows from TMD at shifted momenta"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes that the function $G(k,\\Delta)$, after evolution, is invariant under the specific translation $k\\to k-a/2$, $\\Delta\\to\\Delta-a$ even for initial conditions that do not have this symmetry, and this restoration is asserted rather than proven.","fun_headline_variants_meta":{"raw":{"variants":["Off-forward dipole evolution from shifted forward amplitudes","Momentum-space correspondence: off-forward reduces to forward","Pomeron and odderon off-forward evolution fixed by forward","Small-x evolution: off-forward determined by forward with k shift","GTMD evolution flows from TMD at shifted momenta"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000343,"raw_usage":{"total_tokens":1881,"prompt_tokens":933,"completion_tokens":948,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":549,"completion_tokens_details":{"reasoning_tokens":867}},"tokens_in":549,"tokens_out":948,"duration_ms":9800,"temperature":1.0,"reasoning_tokens":867,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T17:28:02.988202+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically evolve the leading-log BK equation in momentum space starting from an off-forward initial condition that breaks the translation symmetry of Eq. (20); if $\\partial_Y N(k,\\Delta)$ differs from $\\tfrac12\\partial_Y[N(k-\\Delta/2,0)+N(k+\\Delta/2,0)]$ at any $Y>0$, then the claimed symmetry restoration and the main correspondence fail.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Balitsky's derivation of the non-linear small-x evolution equation for the dipole S-matrix supplies the starting position-space evolution."},{"cited_title":"Conformal symmetry and the Balitsky-Kovchegov equation","cited_arxiv_id":"1102.4040","evidence_quote":"Motyka's study of the odderon in momentum space supplies the odderon forward amplitude form and the exponentials used in estimates at non-zero $\\Delta$."}],"review_version":1}