{"id":"1e69d376-ce74-46d4-be9c-5e03724a9ecc","arxiv_id":"2502.07576","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"The full set of one-loop matching coefficients for leading-twist GTMDs onto GPDs is computed, including new T-even/T-odd mixing in the ERBL region.","lead":"This paper computes the complete one-loop matching coefficients that connect generalized transverse-momentum-dependent distributions (GTMDs) to generalized parton distributions (GPDs) for all leading-twist polarizations. The results give phenomenologists a standard tool to reconstruct GTMDs, which encode the proton's 3D structure, from more established GPDs.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Imaginary part in ERBL matching and evolution may be regulator-dependent; the central T-even/T-odd mixing claim needs an independent-regulator check.","rationale":"The reader's weakest assumption already names regulator artifacts as a possible failure mode, so we partially agree. This stress-test sharpens that concern to the specific residual-function imaginary part in App. A.2, which is not covered by the regulator-independence argument in App. A.1. The paper itself flags the novelty and importance of the imaginary parts, so a check with a genuinely independent regulator is the natural gate for acceptance. No other load-bearing error was found: the forward-limit consistency check in Sec. 3, the explicit cancellation proof for the splitting-function imaginary parts in App. A.3, and the signed rotation structure in Eq. (71) are all internally coherent. The verdict is adjusted to CONDITIONAL rather than REJECT because the concern is about verification, not about a demonstrated contradiction.","tokens_in":28999,"tokens_out":32684,"duration_ms":287472,"concrete_test":"Recompute the one-loop gluon matching function r^{U/U}_{g/g}(y,κ) in the ERBL region with an independent regulator, e.g., off-the-light-cone Wilson lines parametrised by η (as in Refs. [17,28]), and verify that the δ(y−1/κ) coefficient equals −iπs Σ^{U/U,+,[1]}_{g/g}(κ)/κ and that the evolution-phase sign in Eq. (77) is −iπsθ(κ−1). If the sign or the coefficient differs, the imaginary part is regulator/scheme-dependent and the T-even/T-odd mixing picture is not universal.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central novelty is the perturbative T-even/T-odd mixing driven by imaginary parts in the one-loop matching functions (Eq. 21) and in the evolution kernel (Eq. 61). The derivation of the matching-function imaginary part in App. A.2 uses the δ-regulator and a light-cone-gauge boundary condition (Eqs. 87-94) to produce a δ(y−1/κ) term with coefficient −iπs. App. A.1 demonstrates regulator independence for the UV-renormalisation phase by citing off-the-light-cone Wilson lines, but the residual-function imaginary part itself is not shown to be regulator-independent; the identity in Eq. (94) is specific to the δ-regulator. If the coefficient of δ(y−1/κ) changes under another rapidity regulator (e.g., off-the-light-cone parameter η), the one-loop matching coefficients are not universal and the claimed T-odd mixing and Eq. (71) lose their predictive content for phenomenology. A sign error in Eq. (77) would also flip the rotation matrix in Eq. (67) and hence the sign of the mixing angle in the final reconstruction formula.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper computes the complete set of one-loop matching coefficients that connect all leading-twist generalised transverse-momentum-dependent distributions (GTMDs) of the proton to generalised parton distributions (GPDs). The coefficients are extracted from partonic bilocal correlators with staple-like Wilson lines, using the δ-regulator for rapidity divergences and a b-space OPE at small transverse separation. The paper presents the residual functions in Tab. 1, the singlet/non-singlet combinations in Tabs. 2-5, and the forward limit in Tab. 6. It further derives imaginary parts in the matching functions and in the evolution kernel, leading to a rotation between T-even and T-odd GTMDs in the ERBL region, and illustrates the effect with numerical results based on the GK model and the PARTONS/APFEL++/NangaParbat codes.","tokens_in":29197,"tokens_out":12180,"duration_ms":124414,"significance":"If the results are correct, this is the first complete one-loop matching kernel set for all leading-twist GTMDs, and the paper fills a genuine gap between TMD and GPD phenomenology. The central novelty is the derivation of an imaginary part in the matching functions and in the evolution kernel that induces perturbative mixing of T-even and T-odd GTMDs in the ERBL region; this is a concrete, falsifiable prediction for future exclusive and double-Drell-Yan phenomenology. The paper makes good use of internal consistency checks: the forward limit reproduces known TMD matching results (Tab. 6), the sign of the Wilson-line-dependent imaginary part is carefully tracked in App. A, and the numerical implementation is based on publicly available codes, which is a substantial reproducibility strength. There is no circularity in the derivation: the coefficients are computed from partonic matrix elements rather than fitted to data.","major_comments":[],"minor_comments":[{"comment":"In the expression for R^{U/U,-,[1]}_2(y,κ), the factor 'CF' appears twice, reading '2y(1−κ)CF'; this is presumably a typo and the intended expression is CF·2y(1−κ)/(1−κ²y²). Please correct it.","section":"Sec. 3, Eq. (25)"},{"comment":"The imaginary part of the residual function is derived with the δ-regulator and light-cone gauge, and the text explicitly demonstrates regulator independence only for the UV-renormalisation phase in A.1, not for the residual function. In my reading the concern is not fatal, because the sign of the iδ term is fixed by the Wilson-line boundary condition in Eq. (87), so any regulator preserving the staple direction gives the same δ-function coefficient; nevertheless, adding a sentence in A.2 stating this explicitly would preempt an otherwise natural objection.","section":"App. A.2, Eqs. (87)-(94)"},{"comment":"The identity in Eq. (77) relies on a specific branch of the logarithm; with the principal branch it is correct, but the branch convention should be stated explicitly for completeness.","section":"App. A.1, Eq. (77)"},{"comment":"The derivation of the entries in Tab. 1 is delegated to Refs. [30,56,57]. Since this table is the central result, including one explicit worked example for a new entry in an appendix or an ancillary file would improve the verifiability of the calculation.","section":"Sec. 3, Tab. 1"},{"comment":"The statement that evolution 'proves' to be the dominant source of T-odd effects is stronger than what a single model-dependent numerical study can establish; I suggest softening this to 'indicates' or 'demonstrates in this implementation'.","section":"Sec. 6 and Fig. 3"},{"comment":"The claim that the reconstructed GTMDs 'should be regarded as realistic' is somewhat overstrong given the ad hoc scaling factor c=10^{-1} for linearly polarised gluon GPDs and the absence of direct data validation; please qualify this statement.","section":"Sec. 7"}],"recommendation":"minor_revision","confidential_remarks":"I see no grounds for rejection. The paper is within the standard scope of a hep-ph journal, the novelty relative to Ref. [30] is clearly stated, and the overlap with the authors' previous work is properly cited. The main theoretical concern about regulator dependence of the imaginary part is, in my view, answered by the boundary-condition argument in App. A.2, though an explicit remark would make the paper more robust. The remaining issues are local and presentational."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nThis paper gives the complete one-loop matching of all leading-twist GTMDs onto GPDs, extending Ref. [30] which only had the U/U channel. That is the main result: the full set of residual functions for quarks and gluons in the U, L, T polarizations, with the forward limit reproducing the known TMD matching coefficients. The T-even/T-odd mixing in the ERBL region, from imaginary parts in both the matching functions and the evolution, is a genuinely new and interesting finding.\n\nThe paper is technically careful. The tables are laid out clearly, the internal checks are meaningful, and the derivation of the imaginary part in App. A is transparent. The proof that the GPD splitting functions stay real is a nice addition. The numerical section is explicitly exploratory, using an ad-hoc c = 0.1 for linearly polarized gluon GPDs, so that should not count against it.\n\nMy soft spots are two. First, the regulator independence of the imaginary part in the residual function is asserted but not demonstrated for the matching functions. App. A.1 makes the case for the evolution phase, but App. A.2 does not carry the same argument to the matching. In practice the iε-type coefficient looks tied to the staple direction s, so I would expect it to survive any sensible regulator, but the paper should say so explicitly. A cross-check with an off-light-cone regulator would settle it. Second, Eq. (77) deserves a sign check. The sign of that imaginary part feeds directly into the rotation matrix in Eq. (67). I did not find a definite error, but the ordering of logs in the printed equation looked off to me. Worth verifying carefully before publication.\n\nWho is this for? The GTMD and TMD phenomenology community, especially anyone preparing for EIC-era studies of proton structure. It is a solid technical contribution with the right checks. I would definitely send it to peer review, and ask the authors to add a short remark on regulator independence and to double-check the sign.\n\nYours,","headline":"Complete one-loop GTMD-to-GPD matching with a new T-even/T-odd mixing claim; tables look right, but the regulator-independence of the imaginary part needs an explicit statement.","tokens_in":29766,"tokens_out":11601,"would_cite":true,"duration_ms":85182,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"All leading-twist GTMDs are matched onto GPDs at one loop, with T-even/T-odd mixing in the ERBL region.","keywords":["generalised transverse-momentum-dependent distributions","generalised parton distributions","one-loop matching","operator product expansion","T-odd effects","ERBL region","QCD factorization","proton tomography"],"falsifier":"Recompute the one-loop GTMD matching functions with a different rapidity regulator (for instance off-the-light-cone Wilson lines) and check that the imaginary term $-is\\pi/\\kappa \\, \\Sigma \\, \\delta(y - 1/\\kappa)$ in Eq. (21), and hence the T-even/T-odd rotation in Eq. (71), is reproduced with exactly the same phase; if the phase changes or disappears with the regulator, the claimed mixing is an artifact. Experimentally, a measurement of exclusive or double-Drell-Yan cross sections in ERBL kinematics that shows no dependence on the sign of the Wilson-line direction (e.g. no $\\sin(2\\theta_{k\\Delta})$ modulation in the $S_{1,1b}$-type distribution) would violate the predicted pattern.","tokens_in":28798,"feed_emoji":"⚛️","tokens_out":11867,"duration_ms":100418,"temperature":0.7,"pith_summary":"Generalised transverse-momentum-dependent distributions (GTMDs) are the most complete quark/gluon structure functions of the proton, depending on longitudinal momentum fraction, skewness, transverse momentum, and momentum transfer. This paper supplies the missing one-loop, $O(\\alpha_s)$, matching coefficients that express every leading-twist GTMD of the proton in terms of generalised parton distributions (GPDs) at small transverse separation, the analogue for GTMDs of the standard TMD-to-PDF matching used in data analyses. It establishes that polarisations mix: an unpolarised quark GTMD receives contributions from transversely polarised gluon GPDs, and other channels mix similarly. It also finds that in the ERBL region ($\\xi > x$) the gluon-induced matching functions and the evolution kernel pick up imaginary parts proportional to the Wilson-line direction, so time-reversal even (T-even) and time-reversal odd (T-odd) GTMD components rotate into each other. A sympathetic reader cares because this makes a complete, perturbatively controlled reconstruction of GTMDs feasible for the first time, which is needed before GTMDs can be extracted from or compared with data on exclusive processes.","feed_headline":"GTMD-to-GPD matching now complete at one loop","feed_subtitle":"All polarization channels covered, revealing T-even and T-odd mixing in the ERBL region that reshapes proton tomography.","key_machinery":"The central object is the one-loop matching function $C^{Y/\\Gamma}_{i/j}$ of Eq. (14), which factorises the small-$|b|$ GTMD correlator as a convolution of a universal coefficient with a GPD correlator. The argument is carried by the newly computed residual functions $r^{Y/\\Gamma,[1]}_{i/j}(y,\\kappa)$ in Table 1, together with the decomposition of Eq. (21) that isolates the principal-value singularity at $y = 1/\\kappa$ and the imaginary piece $-is\\pi/\\kappa \\, \\Sigma \\, \\delta(y - 1/\\kappa)$ coming from the staple Wilson-line direction. The evolution side is carried by the Sudakov factor $R_i$ of Eq. (63), whose complex phase $\\exp(i\\phi)$ with $\\phi = (\\pi/2)\\,\\theta(\\kappa-1)\\,K_i(b,\\mu)$ supplies the rotation that mixes the T-even and T-odd parts of a GTMD.","core_discovery":"The paper computes the complete set of one-loop matching functions $C^{Y/\\Gamma,[1]}_{i/j}$ that connect $b$-space GTMD correlators to collinear GPD correlators for all leading-twist quark and gluon polarisations. The new results are the residual functions $R^{Y/\\Gamma,[1]}_{i/j}$ listed in Table 1 (and their singlet/non-singlet combinations in Tables 2-5), which supplement the known logarithmic and splitting-function terms in Eq. (14). Two structural discoveries follow. First, GTMD and GPD polarisations mix under matching: for example, unpolarised quark GTMDs receive contributions from transversely/linearly polarised gluon GPDs. Second, for gluon GPD channels in the ERBL region ($\\kappa > 1$), the matching functions develop an imaginary part proportional to the Wilson-line direction $s$, of the form $-is\\pi/\\kappa \\, \\Sigma^{Y/\\Gamma,+,[1]}_{i/j} \\, \\delta(y - 1/\\kappa)$ in the singlet convolution; combined with an analogous imaginary phase in the Sudakov evolution of Eq. (63), this produces a rotation matrix in Eq. (71) that mixes T-even and T-odd components of each GTMD. Numerically the T-odd component is small but non-negligible in the ERBL region, and evolution, not matching, is the dominant source.","pith_inferences":["If the matching is process-universal as claimed, existing high-precision TMD fits (which already constrain the $b$-space non-perturbative input) could be adapted to seed GTMD models for the DGLAP region, though the ERBL region also needs the new imaginary matching terms; this is my inference, not a proposal in the paper.","The complex phase in the Sudakov factor implies that GTMDs reconstructed in $b$-space are genuinely complex-valued even when the input GPDs are real; an observable built from ERBL-kinematics GTMDs in forward or diffractive processes should therefore show a T-odd modulation with a characteristic $\\log(\\mu/\\mu_b)$ growth, which would be a testable quantitative prediction of this one-loop picture.","A natural extension the paper does not attempt is to compute the two-loop matching functions to check whether the imaginary part and the T-even/T-odd rotation persist beyond $O(\\alpha_s)$; the paper's claim that one-loop coefficients are the only relevant ones for now is a scope judgement, not a proof of convergence."],"forward_implications":["With these coefficients, all leading-twist proton GTMDs can be reconstructed from GPDs at small $b$ (large $k_T$) at one-loop accuracy, in the same way TMDs are matched onto PDFs in global fits.","Polarisation mixing means any extraction of one GTMD flavour must account for GPDs of different polarisations; e.g. unpolarised quark GTMDs receive contributions through the U/T channel from linearly polarised gluon GPDs.","T-even and T-odd components of a GTMD mix both under $O(\\alpha_s)$ matching onto gluon GPDs and under DGLAP-type evolution in the ERBL region, so T-odd contributions cannot be treated as purely non-perturbative there.","The matching functions are real in the quark-GPD channels and acquire an imaginary part only through gluon GPDs in the ERBL region; the imaginary part scales with the sign $s$ of the Wilson line, making the mixing process-dependent.","The kinematic point $x = \\xi$ is out of reach of the transverse-momentum factorisation used here: the $\\log|1-\\kappa^2|$ divergence at $x = \\xi$ is argued to persist to all orders because one parton carries zero longitudinal momentum, so double-Drell-Yan observables must avoid this region."],"supporting_citations":[{"why":"Previous one-loop GTMD-to-GPD matching, restricted to the unpolarised U/U channel; this paper completes, expands, and amends it.","marker":"[30]"},{"why":"Supplies the OPE formalism for transverse-momentum-dependent operators that the GTMD matching relation Eq. (11) is built on.","marker":"[11]"},{"why":"Gives the forward-limit TMD matching coefficients that the new results must reduce to, verified in Table 6.","marker":"[16]"},{"why":"Provides the proper definition and evolution of GTMD correlators with staple Wilson lines and soft factors used throughout.","marker":"[35]"},{"why":"The GTMD parametrisation (modified in Sec. 4) that organises the matching onto individual distributions.","marker":"[33]"},{"why":"The factorisation proof for exclusive double Drell-Yan that motivates the OPE and fixes why $x = \\xi$ lies outside the factorisation region.","marker":"[50]"},{"why":"One-loop GPD splitting functions that enter the universal part of the matching functions in Eq. (14).","marker":"[56]"},{"why":"Computational framework and detailed expressions for the GPD splitting-function pieces used in the extraction of the residual functions.","marker":"[57]"}],"fun_headline_variants":["One-loop GTMD matching now covers all polarizations","GTMD-GPD one-loop matching reveals T-odd mixing","Complete one-loop GTMD matching: new T-even/T-odd mixing","At one loop: GTMD polarizations mix, T-odd emerges","One-loop GTMD matching: ERBL region mixes T-even and T-odd"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole construction rests on a single set of universal one-loop coefficients — extracted from simple quark and gluon states — controlling how every polarisation of a proton's GTMD is built from GPDs at small transverse separation.","fun_headline_variants_meta":{"raw":{"variants":["One-loop GTMD matching now covers all polarizations","GTMD-GPD one-loop matching reveals T-odd mixing","Complete one-loop GTMD matching: new T-even/T-odd mixing","At one loop: GTMD polarizations mix, T-odd emerges","One-loop GTMD matching: ERBL region mixes T-even and T-odd"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00088,"raw_usage":{"total_tokens":3834,"prompt_tokens":1004,"completion_tokens":2830,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":620,"completion_tokens_details":{"reasoning_tokens":2748}},"tokens_in":620,"tokens_out":2830,"duration_ms":18509,"temperature":1.0,"reasoning_tokens":2748,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T12:14:50.001367+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the one-loop GTMD matching functions with a different rapidity regulator (for instance off-the-light-cone Wilson lines) and check that the imaginary term $-is\\pi/\\kappa \\, \\Sigma \\, \\delta(y - 1/\\kappa)$ in Eq. (21), and hence the T-even/T-odd rotation in Eq. (71), is reproduced with exactly the same phase; if the phase changes or disappears with the regulator, the claimed mixing is an artifact. Experimentally, a measurement of exclusive or double-Drell-Yan cross sections in ERBL kinematics that shows no dependence on the sign of the Wilson-line direction (e.g. no $\\sin(2\\theta_{k\\Delta})$ modulation in the $S_{1,1b}$-type distribution) would violate the predicted pattern.","supporting_citations":[{"cited_title":"Matching generalised transverse-momentum-dependent distributions onto gen- eralised parton distributions at one loop,","cited_arxiv_id":null,"evidence_quote":"Previous one-loop GTMD-to-GPD matching, restricted to the unpolarised U/U channel; this paper completes, expands, and amends it."},{"cited_title":"Calculation of transverse momentum dependent distributions beyond the leading power,","cited_arxiv_id":null,"evidence_quote":"Supplies the OPE formalism for transverse-momentum-dependent operators that the GTMD matching relation Eq. (11) is built on."},{"cited_title":"Twist-2 matching of transverse momentum dependent distributions,","cited_arxiv_id":null,"evidence_quote":"Gives the forward-limit TMD matching coefficients that the new results must reduce to, verified in Table 6."},{"cited_title":"Proper definition and evolution of generalized transverse momentum dependent distribu- tions,","cited_arxiv_id":null,"evidence_quote":"Provides the proper definition and evolution of GTMD correlators with staple Wilson lines and soft factors used throughout."},{"cited_title":"Structure analysis of the generalized correlator of quark and gluon for a spin-1/2 target,","cited_arxiv_id":null,"evidence_quote":"The GTMD parametrisation (modified in Sec. 4) that organises the matching onto individual distributions."},{"cited_title":"GTMDs and the factorization of exclusive double Drell-Yan,","cited_arxiv_id":null,"evidence_quote":"The factorisation proof for exclusive double Drell-Yan that motivates the OPE and fixes why $x = \\xi$ lies outside the factorisation region."},{"cited_title":"One-loop evo- lution of twist-2 generalized parton distributions,","cited_arxiv_id":null,"evidence_quote":"One-loop GPD splitting functions that enter the universal part of the matching functions in Eq. (14)."},{"cited_title":"Revisiting evolu- tion equations for generalised parton distributions,","cited_arxiv_id":null,"evidence_quote":"Computational framework and detailed expressions for the GPD splitting-function pieces used in the extraction of the residual functions."}],"review_version":1}