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The paper establishes that the kinematic dependence of the two-loop spacelike-collinear splitting amplitude is entirely produced by a single hidden region of the five-point amplitude.

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2026-08-02 00:04 UTC pith:KH23OP6Q

load-bearing objection Genuinely new HRF algorithm and a plausible unique-hidden-region explanation for two-loop kinematic SCFV, but the full-set matching is asserted, not yet demonstrated. the 3 major comments →

arxiv 2607.15126 v1 pith:KH23OP6Q submitted 2026-07-16 hep-ph hep-th

Spacelike-Collinear Scattering by the Method of Regions

Wen Chen , Einan Gardi , Rourou Ma , Yao Ma , Yang Zhang , Zehao Zhu This is my paper
classification hep-ph hep-th
keywords hidden regionsmethod of regionscollinear factorisationsplitting amplitudeGlauber modesspacelike-collinear limittwo-loop amplitudesN=4 super Yang-Mills
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper argues that the two-loop splitting amplitude in the spacelike-collinear limit of gauge-theory scattering acquires dependence on the kinematics of non-collinear partons, and that this dependence comes entirely from a single 'hidden region' in the Method-of-Regions expansion. Hidden regions are parameter-space contributions to Feynman integrals that are not visible as facets of the Newton polytope; they arise from cancellations among superleading monomials and require a dedicated algorithm to find. Applying their new Hidden Region Finder to the five-point amplitude in maximal supersymmetric Yang-Mills theory, the authors compute the hidden-region contribution to every basis integral and show component-by-component that it reproduces the known kinematic-dependent factorisation-violating term. Because the same facet regions appear in both spacelike and timelike limits while the hidden region appears only in the spacelike limit, the paper proposes that hidden regions are the general mechanism by which crossing-related asymptotic limits cease to be analytically connected.

Core claim

The central claim is that the kinematically-dependent violation of strict collinear factorisation at two loops — the Δ_λ^(2) term in the splitting amplitude — is generated entirely by a unique hidden region of the five-point amplitude in the spacelike-collinear limit. In momentum space this region is a soft loop plus a Glauber loop, and its presence is tied to sign-dependent cancellations in the leading polynomial. Computing the hidden-region integrals for the complete set of basis integrals in N=4 sYM, the authors recover exactly the known expression for Δ^(2), including the dependence on the dihedral angle y and the ratio x, in a colour-flow basis. The same region is absent in the timelike

What carries the argument

The central object is the Hidden Region (HR) — a solution of the Method-of-Regions expansion that is not a facet of the Lee-Pomeransky Newton polytope but arises from cancellations among 'superleading' monomials in the leading polynomial F0. The Hidden Region Finder (HRF) algorithm identifies such regions by (1) decomposing F0 into a superleading sector F_SL and an obstruction polynomial using Landau pinch conditions, and (2) solving homogeneity and hierarchy conditions to fix the scaling vector v_HR. For the seed six-propagator topology, the HR scaling v_HR = (-2,-1,-2,-2,-2,-1;1) makes the superleading monomials scale as δ^{-5} and the obstruction as δ^{-4}, bringing in the cδ term that ca

Load-bearing premise

The central assumption is that the Hidden Region Finder finds every hidden region — that its preselection and decomposition steps discard only topologies that genuinely lack a hidden region, and that no facet-region integral contributes x- or y-dependent terms through boundary or regulator effects.

What would settle it

A decisive test is to compute the three-loop five-point amplitude in the spacelike-collinear limit: if the kinematic-dependent factorisation-violating terms are not fully reproduced by hidden regions, or if a new hidden region appears that shifts the balance, the claim that a unique hidden region is responsible would be overturned.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • The complete kinematic dependence of the two-loop spacelike splitting amplitude on the non-collinear partons is accounted for by one hidden region, with no residual contribution from facet regions.
  • The hidden region has a soft loop and a Glauber loop, so the eikonal (Wilson-line) approximation captures the full effect; this explains why the Wilson-line computation agrees with the amplitude-based result.
  • Universality of the kinematic SCFV across gauge theories (QCD and N=4 sYM) follows because the hidden region is insensitive to the particle content and depends only on the directions of the non-collinear partons.
  • Because the hidden region exists only in the spacelike-collinear limit, the timelike and spacelike splitting functions are not analytically connected; hidden regions provide a general mechanism for such crossing-related disconnections.
  • The Hidden Region Finder provides a systematic algorithm to identify hidden regions in general kinematic limits, including multi-collinear and multi-Regge limits, where Glauber effects are important.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The real-virtual connection between the hidden region here and the Glauber region in jet cross-section factorisation suggests that hidden-region analysis could unify the description of factorisation restoration in real and virtual contributions.
  • If hidden regions generically signal the failure of analytic continuation between crossing limits, then the algorithm could be used to map the 'boundary' of analyticity in multi-scale amplitudes, potentially connecting to the analytic structure of scattering amplitudes in the complex momentum plane.
  • The colour-flow basis reveals a signature-like symmetry (parity in y ↔ colour representation under 2↔3), hinting that the spacelike-collinear limit may admit an effective description in terms of Reggeon-like degrees of freedom; this could be tested by constructing an effective action whose degrees of freedom have the same symmetry.
  • The method's reliance on Landau pinch conditions suggests that hidden regions might be identified directly from the geometry of the Landau equations in momentum space, offering a shortcut to the parameter-space algorithm.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper studies the spacelike-collinear (2||3) limit of the two-loop five-gluon amplitude in N=4 sYM, focusing on the kinematically dependent, factorisation-violating part of the splitting amplitude, Delta_lambda^(2) of Eq. (7). Using the Method of Regions, the authors develop a 'Hidden Region Finder' (HRF) algorithm to identify hidden regions in parameter space via Landau pinch conditions. They apply it to the 469 scalar topologies appearing in the UT basis of the five-point amplitude, find a unique hidden region characterised by a soft loop and a Glauber loop in momentum space, and compute its contribution to the 2x16 HR-positive basis integrals. Assembling the HR contributions in a colour-flow basis, they report component-by-component agreement with the known Delta_lambda^(2), concluding that the HR alone is responsible for the kinematically dependent SCFV. They further conjecture that hidden regions provide a general mechanism for the loss of analytic connection between crossing-related asymptotic limits. The paper includes an explicit seed-integral analysis, the HRF algorithm's formal conditions (homogeneity (14) and hierarchy (17)), and a comparison with the literature result in Eq. (7).

Significance. If the central claim is correct, the paper provides a genuine dynamical explanation of a striking phenomenon: why the spacelike- and timelike-collinear limits of the same amplitude are not analytically connected, and why the Wilson-line calculation of the splitting amplitude captures the full kinematic dependence. The identification of the hidden region as a soft+Glauber configuration and its connection to the universality of Delta_lambda^(2) across QCD and sYM is conceptually important. The HRF algorithm, even if only a conjecture at this stage, is a potentially powerful tool for systematic region analysis in non-Euclidean limits. The paper's strengths include an explicit, self-contained treatment of the seed integral, a cross-check against an independently computed quantity (Eq. (7), from refs [16-18] with no author overlap for two of them), and the public release of the colour-basis rotation matrices and comparison data [66]. The agreement is presented as component-by-component in a colour-flow basis, which is a strong check. However, the load-bearing claim that 'the HR alone' accounts for all kinematically dependent SCFV rests on two unproven completeness assertions: the exhaus

major comments (3)
  1. [End Matter A] The central claim that 'the HR alone is responsible for kinematically dependent SCFV' requires that no facet-region integral contributes x- or y-dependence at delta^{0-4epsilon}. For the full 2x16 set, the paper states only that 'for all integrals that require regularisation, the x dependence can be removed from the LP polynomial by rescaling a suitable integration parameter which is not affected by the regulator' (End Matter A). No list of these integrals, the explicit rescaling, or a proof of regulator-independence is given. Because these integrals have rapidity divergences that require raising propagator 2 to a power a, and because the facet and hidden regions mix under this regulator, it is essential to verify that the rescaling does not involve a-dependent parameters and that the regulated facet integral indeed has no x/y dependence. Please provide the explicit rescaling for each of
  2. [HRF algorithm, 'HIDDEN REGION FINDER'] The HRF algorithm's preselection is explicitly a 'necessary but insufficient condition' and the paper discards topologies when 'no such derivatives exist'. The uniqueness conjecture for the scaling vector (conditions (14) and (17)) is also unproven. The claim that exactly 16 representatives (out of 70, after p4<->p5 reduction) contain HRs is therefore only as strong as the algorithm's exhaustiveness. Since the paper's headline conclusion is that a unique hidden region is the source of all kinematically dependent SCFV, the completeness of the HRF for this specific class of two-loop five-point integrals must be established or at least argued rigorously. Concretely: (i) show that the mixed-sign derivative preselection cannot miss a HR in any of the discarded topologies; (ii) for the 36 nonplanar representatives, provide a certificate or an exhaustive check that the 16 identified are the onl
  3. [MOR APPLICATION, comparison with Eq. (7)] The comparison with Eq. (7) is reported as 'perfect agreement', but the paper does not show explicitly that the assembled HR contribution is the only piece that reproduces the x/y-dependent terms of Delta_lambda^(2) after the amplitude is assembled in the colour-flow basis. In particular, the cancellation of rapidity divergences between facet and hidden regions (End Matter A) could in principle mix x-dependent terms; the statement that 'these divergences do not modify the conclusions' needs a demonstration at the level of the assembled amplitude, not just the LP polynomial of individual facet integrals. A clear presentation of the cancellation (e.g., for one representative integral with a rapidity divergence) and of how the x/y dependence survives uniquely from the HR would close this gap. The current text leaves the possibility that the x/y-independence of the full non-HR sum is an assu
minor comments (5)
  1. [Abstract/Introduction] The abstract and text state 'conforming our expectation' (MOR APPLICATION); should be 'confirming'.
  2. [Fig. 1] The shading/colours in Fig. 1 are essential to the interpretation (green shades for collinear scalings, red for soft edges), but the figure caption only says 'Four shades of green' without a legend or explicit encoding of which shade corresponds to which virtuality. Please add a legend or explicitly state the mapping.
  3. [Eq. (2)] The parametrisation in Eq. (2) is central, but the domain '-1 < x < 0, z > 1' is stated without derivation. For the reader's convenience, please include the explicit relation of (x,y) to the usual Mandelstam invariants, or at least a footnote pointing to the conventions of Ref. [17].
  4. [End Matter B] The colour-flow basis and the matrices MTr->t and M_alpha_beta are said to be in the GitHub repository [66]. The paper would benefit from a brief explanation of how the matrices are used in the component-by-component comparison, since the reproducibility of the central check depends on these files.
  5. [References] The related work [85] is mentioned in a Note Added, but no citation is given in the body; please include it in the Introduction or Discussion where the completeness of the HRF is discussed, given that overlapping claims are likely.

Circularity Check

0 steps flagged

No significant circularity: the hidden-region computation is an independent cross-check against the externally known Eq. (7), with no fitted parameter or definitional reduction.

full rationale

The paper's central claim is that the kinematic SCFV in Eq. (7) originates from one hidden region. The target Eq. (7) is taken from Refs. [16-18], of which [16] and [18] have no author overlap with the present paper; Ref. [17] overlaps in two authors (R. Ma, Y. Zhang), but the HR computation does not use Eq. (7) as an input. The HRF identifies hidden regions from the LP polynomial's mixed-sign derivative structure and the Landau conditions (13)-(17), independent of the expected splitting amplitude. The seed integral HR is computed explicitly (Eqs. (18)-(24)), and the 16 HR-positive topologies are found before the amplitude-level comparison; the statement that these coincide with the y-dependent integrals is presented as an ex-post observation, not as a selection criterion. The final comparison is a component-by-component cross-check in a colour-flow basis, not a fit: no parameter is adjusted to make the HR contribution equal Eq. (7). The residual gaps - the HRF completeness conjecture, and the unshown rescaling claim in End Matter A that all facet-region x-dependence can be removed - are checkable assumptions and would be correctness risks if false, but they do not make the derivation circular. Self-citations to [17], [53], [61], [64] are methodological, not load-bearing reductions of the SCFV result, which is independently benchmarked.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 0 invented entities

The paper's results rest on accepted MoR technology plus two paper-specific premises: the HRF completeness/uniqueness conjecture and the exhaustive claim that facet regions in all 32 relevant graphs are x,y independent. No numerical parameters are fitted; the comparison target Eq. (7) is an independent prior result.

axioms (7)
  • domain assumption The Method of Regions gives the correct asymptotic expansion as a sum over region integrals (facet + hidden).
    Invoked at the outset ('The MoR expresses the asymptotic expansion...'); standard but non-trivial assumption that the region sum is complete with the HR included.
  • domain assumption Landau pinch conditions (13) in parameter space are necessary and sufficient for hidden regions; positive-x solutions correspond to first-sheet singularities.
    Eq. (13); follows refs [60, 53]; the sufficiency direction underlies HRF completeness.
  • ad hoc to paper Uniqueness conjecture: homogeneity (14) + hierarchy (17) uniquely determine v_HR.
    Stated explicitly as a conjecture in the HRF section; load-bearing for the claim that the identified region is the unique source.
  • domain assumption Facet regions are sign-independent and are the same in spacelike and timelike limits; their LP polynomials carry no y dependence and no removable x dependence.
    Diagnostic in 'MoR application to 5-point integrals'; explicitly verified for the seed and asserted for the full basis, with the x-rescaling check not shown in text.
  • domain assumption Edge-parameter virtuality relation x_e ~ (q_e^2)^(-1) maps the HR scaling to soft/Glauber momenta (22).
    Standard MoR correspondence, cited [44, 63]; used for the momentum-space interpretation tying the HR to Wilson lines.
  • domain assumption The UT basis of [17] (overlapping authorship) reproduces the leading-delta asymptotic expansion of A5^(2).
    Stated as a preliminary verification step; the basis is inherited from prior work by R. Ma and Y. Zhang (co-authors).
  • domain assumption Dissection [53, 62] correctly handles analytic continuation of the HR integral.
    End Matter A; the shifted x4' dissection converts the HR into a facet region; correctness relies on the dissection literature.

pith-pipeline@v1.3.0-alltime-deepseek · 14811 in / 19435 out tokens · 234486 ms · 2026-08-02T00:04:37.955844+00:00 · methodology

0 comments
read the original abstract

We study the spacelike-collinear limit of gauge-theory scattering amplitudes using the Method of Regions. The corresponding splitting amplitude violates strict collinear factorisation through its dependence on the non-collinear partons. While the associated colour dependence has long been known, starting at two loops the splitting amplitude also acquires dependence on their kinematics. We show that this kinematic dependence originates from a unique hidden region present in the asymptotic expansion of the five-point amplitude in the spacelike-collinear limit, but absent in the timelike limit. More generally, we propose that hidden regions provide the mechanism by which crossing-related asymptotic limits cease to be analytically connected. We develop a general algorithm for the systematic identification of hidden regions. Applying it to the five-point amplitude in super Yang-Mills theory, we compute the hidden-region contributions to the complete set of basis integrals and recover the exact kinematically dependent factorisation-violating splitting amplitude. In momentum space, the hidden region is characterised by soft and Glauber loop momenta. This explains why the Wilson-line calculation captures the complete kinematic dependence, thereby accounting for the observed universality across gauge theories.

Figures

Figures reproduced from arXiv: 2607.15126 by Einan Gardi, Rourou Ma, Wen Chen, Yang Zhang, Yao Ma, Zehao Zhu.

Figure 1
Figure 1. Figure 1: FIG. 1: The seed topology (a) and its collinear (facet) region (b) and Glauber (hidden) region (c). (d)–(f): some [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

discussion (0)

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. geoSCET: Soft Theorems from Power Counting

    hep-th 2026-07 accept novelty 8.0

    geoSCET derives geometric soft theorems for scalar field theories directly from effective-field-theory power counting and proves they are exact to all orders in perturbation theory when no potential is present.

Reference graph

Works this paper leans on

89 extracted references · 62 linked inside Pith · cited by 1 Pith paper

  1. [1]

    Z. Bern, L. J. Dixon, D. C. Dunbar, and D. A. Kosower, Nucl. Phys.B425, 217 (1994), arXiv:hep-ph/9403226 [hep-ph]

    Pith/arXiv arXiv 1994

  2. [2]

    Catani and M

    S. Catani and M. Grazzini, Phys. Lett.B446, 143 (1999), arXiv:hep-ph/9810389 [hep-ph]

    Pith/arXiv arXiv 1999

  3. [3]

    Catani, D

    S. Catani, D. de Florian, and G. Rodrigo, Phys. Lett. B586, 323 (2004), arXiv:hep-ph/0312067 [hep-ph]. 7

    Pith/arXiv arXiv 2004

  4. [4]

    D. A. Kosower, Nucl. Phys.B552, 319 (1999), arXiv:hep- ph/9901201 [hep-ph]

    arXiv 1999

  5. [5]

    D. A. Kosower and P. Uwer, Nucl. Phys.B563, 477 (1999), arXiv:hep-ph/9903515 [hep-ph]

    Pith/arXiv arXiv 1999

  6. [6]

    Z. Bern, L. J. Dixon, and D. A. Kosower, JHEP08, 012 (2004), arXiv:hep-ph/0404293 [hep-ph]

    Pith/arXiv arXiv 2004

  7. [7]

    S. D. Badger and E. W. N. Glover, JHEP07, 040 (2004), arXiv:hep-ph/0405236 [hep-ph]

    Pith/arXiv arXiv 2004

  8. [8]

    T. G. Birthwright, E. W. N. Glover, V. V. Khoze, and P. Marquard, JHEP05, 013 (2005), arXiv:hep- ph/0503063 [hep-ph]

    arXiv 2005

  9. [9]

    Catani, D

    S. Catani, D. de Florian, and G. Rodrigo, JHEP07, 026 (2012), arXiv:1112.4405 [hep-ph]

    Pith/arXiv arXiv 2012

  10. [11]

    Badger, F

    S. Badger, F. Buciuni, and T. Peraro, JHEP09, 188 (2015), arXiv:1507.05070 [hep-ph]

    Pith/arXiv arXiv 2015

  11. [12]

    M. D. Schwartz, K. Yan, and H. X. Zhu, Phys. Rev. D 96, 056005 (2017), arXiv:1703.08572 [hep-ph]

    Pith/arXiv arXiv 2017

  12. [13]

    Becher and M

    T. Becher and M. Neubert, JHEP01, 025 (2020), arXiv:1908.11379 [hep-ph]

    Pith/arXiv arXiv 2020

  13. [14]

    Cieri, P

    L. Cieri, P. K. Dhani, and G. Rodrigo, Phys. Rev. D 113, L031506 (2026), arXiv:2402.14749 [hep-ph]

    arXiv 2026

  14. [15]

    C. Duhr, A. Venkata, and C. Zhang, Phys. Rev. Lett. 135, 241601 (2025), arXiv:2507.05355 [hep-th]

    Pith/arXiv arXiv 2025

  15. [16]

    L. J. Dixon, E. Herrmann, K. Yan, and H. X. Zhu, JHEP05, 135 (2020), [Erratum: JHEP 06, 143 (2024)], arXiv:1912.09370 [hep-ph]

    Pith/arXiv arXiv 2020

  16. [17]

    J. Henn, R. Ma, Y. Xu, K. Yan, Y. Zhang, and H. X. Zhu, (2024), arXiv:2406.14604 [hep-th]

    Pith/arXiv arXiv 2024

  17. [18]

    Buccioni, H

    F. Buccioni, H. Fang, and K. Yan, (2026), arXiv:2603.27123 [hep-ph]

    arXiv 2026

  18. [19]

    C. Duhr, E. Gardi, S. Jaskiewicz, J. L¨ ubken, and L. Ver- nazza, JHEP02, 173 (2026), arXiv:2507.21854 [hep-ph]

    Pith/arXiv arXiv 2026

  19. [20]

    Bros, Phys

    J. Bros, Phys. Rept.134, 325 (1986)

    1986

  20. [21]

    Caron-Huot, M

    S. Caron-Huot, M. Giroux, H. S. Hannesdottir, and S. Mizera, JHEP04, 060 (2024), arXiv:2310.12199 [hep- th]

    Pith/arXiv arXiv 2024

  21. [22]

    J. C. Collins, D. E. Soper, and G. F. Sterman, Adv. Ser. Direct. High Energy Phys.5, 1 (1989), arXiv:hep- ph/0409313 [hep-ph]

    arXiv 1989

  22. [23]

    J. C. Collins, D. E. Soper, and G. F. Sterman, Nucl. Phys. B261, 104 (1985)

    1985

  23. [24]

    J. C. Collins, D. E. Soper, and G. F. Sterman, Nucl. Phys. B250, 199 (1985)

    1985

  24. [25]

    Collins and J.-W

    J. Collins and J.-W. Qiu, Phys. Rev. D75, 114014 (2007), arXiv:0705.2141 [hep-ph]

    Pith/arXiv arXiv 2007

  25. [26]

    T. C. Rogers and P. J. Mulders, Phys. Rev. D81, 094006 (2010), arXiv:1001.2977 [hep-ph]

    Pith/arXiv arXiv 2010

  26. [27]

    T. C. Rogers, Phys. Rev. D88, 014002 (2013), arXiv:1304.4251 [hep-ph]

    Pith/arXiv arXiv 2013

  27. [28]

    J. R. Forshaw, A. Kyrieleis, and M. H. Seymour, JHEP 08, 059 (2006), arXiv:hep-ph/0604094

    Pith/arXiv arXiv 2006

  28. [29]

    J. R. Forshaw and M. H. Seymour, Nucl. Phys. B Proc. Suppl.191, 257 (2009), arXiv:0901.3037 [hep-ph]

    Pith/arXiv arXiv 2009

  29. [30]

    J. R. Forshaw, M. H. Seymour, and A. Siodmok, JHEP 11, 066 (2012), arXiv:1206.6363 [hep-ph]

    Pith/arXiv arXiv 2012

  30. [31]

    Becher, M

    T. Becher, M. Neubert, and D. Y. Shao, Phys. Rev. Lett. 127, 212002 (2021), arXiv:2107.01212 [hep-ph]

    Pith/arXiv arXiv 2021

  31. [32]

    Becher, P

    T. Becher, P. Hager, S. Jaskiewicz, M. Neubert, and D. Schwienbacher, Phys. Rev. Lett.134, 061901 (2025), arXiv:2408.10308 [hep-ph]

    Pith/arXiv arXiv 2025

  32. [33]

    Becher, P

    T. Becher, P. Hager, S. Jaskiewicz, M. Neu- bert, and D. Schwienbacher, JHEP01, 024 (2026), arXiv:2509.07082 [hep-ph]

    Pith/arXiv arXiv 2026

  33. [34]

    Becher, P

    T. Becher, P. Hager, M. Neubert, and D. Schwienbacher, (2026), arXiv:2603.12383 [hep-ph]

    arXiv 2026

  34. [35]

    Dasgupta, A

    M. Dasgupta, A. Fraley, P. F. Monni, and S. Nabeebac- cus, (2025), arXiv:2511.14681 [hep-ph]

    arXiv 2025

  35. [36]

    Banfi, J

    A. Banfi, J. R. Forshaw, and J. Holguin, Phys. Rev. Lett. 136, 221901 (2026), arXiv:2511.11799 [hep-ph]

    arXiv 2026

  36. [37]

    I. Z. Rothstein and I. W. Stewart, JHEP08, 025 (2016), arXiv:1601.04695 [hep-ph]

    Pith/arXiv arXiv 2016

  37. [38]

    Beneke and V

    M. Beneke and V. A. Smirnov, Nucl. Phys. B522, 321 (1998), arXiv:hep-ph/9711391

    Pith/arXiv arXiv 1998

  38. [39]

    V. A. Smirnov and E. R. Rakhmetov, Theor. Math. Phys. 120, 870 (1999), arXiv:hep-ph/9812529

    Pith/arXiv arXiv 1999

  39. [40]

    V. A. Smirnov, Phys. Lett. B465, 226 (1999), arXiv:hep- ph/9907471

    arXiv 1999

  40. [41]

    V. A. Smirnov, Springer Tracts Mod. Phys.177, 1 (2002)

    2002

  41. [42]

    Jantzen, JHEP12, 076 (2011), arXiv:1111.2589 [hep- ph]

    B. Jantzen, JHEP12, 076 (2011), arXiv:1111.2589 [hep- ph]

    Pith/arXiv arXiv 2011

  42. [43]

    T. Y. Semenova, A. V. Smirnov, and V. A. Smirnov, Eur. Phys. J. C79, 136 (2019), arXiv:1809.04325 [hep-th]

    Pith/arXiv arXiv 2019

  43. [44]

    Gardi, F

    E. Gardi, F. Herzog, S. Jones, Y. Ma, and J. Schlenk, JHEP07, 197 (2023), arXiv:2211.14845 [hep-th]

    Pith/arXiv arXiv 2023

  44. [45]

    Ma, JHEP09, 197 (2024), arXiv:2312.14012 [hep-ph]

    Y. Ma, JHEP09, 197 (2024), arXiv:2312.14012 [hep-ph]

    Pith/arXiv arXiv 2024

  45. [46]

    Y. Ma, Int. J. Mod. Phys. A40, 2530013 (2025), arXiv:2505.01368 [hep-ph]

    Pith/arXiv arXiv 2025

  46. [47]

    Ma, (2026), arXiv:2601.22144 [hep-ph]

    Y. Ma, (2026), arXiv:2601.22144 [hep-ph]

    arXiv 2026

  47. [48]

    L. D. Landau, Zh. Eksp. Teor. Fiz.37, 62 (1960)

    1960

  48. [49]

    J. D. Bjorken,Experimental tests of Quantum electrody- namics and spectral representations of Green ’s functions in perturbation theory, Ph.D. thesis, Stanford U. (1959)

    1959

  49. [50]

    Nakanishi, Prog

    N. Nakanishi, Prog. Theor. Phys.22, 128 (1959)

    1959

  50. [51]

    Pak and A

    A. Pak and A. Smirnov, Eur. Phys. J. C71, 1626 (2011), arXiv:1011.4863 [hep-ph]

    Pith/arXiv arXiv 2011

  51. [52]

    Jantzen, A

    B. Jantzen, A. V. Smirnov, and V. A. Smirnov, Eur. Phys. J. C72, 2139 (2012), arXiv:1206.0546 [hep-ph]

    Pith/arXiv arXiv 2012

  52. [53]

    Gardi, F

    E. Gardi, F. Herzog, S. Jones, and Y. Ma, JHEP08, 127 (2024), arXiv:2407.13738 [hep-th]

    Pith/arXiv arXiv 2024

  53. [54]

    Chen, JHEP02, 115 (2020), arXiv:1902.10387 [hep- ph]

    W. Chen, JHEP02, 115 (2020), arXiv:1902.10387 [hep- ph]

    Pith/arXiv arXiv 2020

  54. [55]

    Chen, Eur

    W. Chen, Eur. Phys. J. C81, 244 (2021), arXiv:1912.08606 [hep-ph]

    Pith/arXiv arXiv 2021

  55. [56]

    Chen, Eur

    W. Chen, Eur. Phys. J. C80, 1173 (2020), arXiv:2007.00507 [hep-ph]

    Pith/arXiv arXiv 2020

  56. [57]

    Chen, (2025), arXiv:2508.18419 [hep-ph]

    W. Chen, (2025), arXiv:2508.18419 [hep-ph]

    Pith/arXiv arXiv 2025

  57. [58]

    Chen, M.-x

    W. Chen, M.-x. Luo, T.-Z. Yang, and H. X. Zhu, JHEP 01, 131 (2024), arXiv:2309.03832 [hep-ph]

    Pith/arXiv arXiv 2024

  58. [59]

    R. N. Lee and A. A. Pomeransky, JHEP11, 165 (2013), arXiv:1308.6676 [hep-ph]

    Pith/arXiv arXiv 2013

  59. [60]

    Collins, (2020), arXiv:2007.04085 [hep-ph]

    J. Collins, (2020), arXiv:2007.04085 [hep-ph]

    Pith/arXiv arXiv 2020

  60. [61]

    Gardi, in preparation

    E. Gardi, in preparation

  61. [62]

    Jones, A

    S. Jones, A. Olsson, and T. Stone, JHEP10, 068 (2025), arXiv:2506.24073 [hep-ph]

    Pith/arXiv arXiv 2025

  62. [63]

    Engel,Muon-Electron Scattering at NNLO, Ph.D

    T. Engel,Muon-Electron Scattering at NNLO, Ph.D. the- sis, Zurich U. (2022), arXiv:2209.11110 [hep-ph]

    Pith/arXiv arXiv 2022

  63. [64]

    W. Chen, E. Gardi, R. Ma, Y. Ma, Y. Zhang, and Z. Zhu, (2026), in progress

    2026

  64. [65]

    Nabeebaccus, J

    S. Nabeebaccus, J. Schoenleber, L. Szymanowski, and S. Wallon, Phys. Rev. D111, L091502 (2025), arXiv:2409.16067 [hep-ph]

    Pith/arXiv arXiv 2025

  65. [66]

    TwoLoopFV,

    W. Chen, E. Gardi, R. Ma, Y. Ma, Y. Zhang, and Z. Zhu, “TwoLoopFV,”https://github.com/zzh118/ TwoLoopFV.git(2026)

    2026

  66. [67]

    Abreu, G

    S. Abreu, G. De Laurentis, G. Falcioni, E. Gardi, 8 C. Milloy, and L. Vernazza, JHEP04, 161 (2025), arXiv:2412.20578 [hep-ph]

    Pith/arXiv arXiv 2025

  67. [68]

    L. N. Lipatov, Soviet Journal of Nuclear Physics23, 338 (1976), yad. Fiz. 23 (1976) 642

    1976

  68. [69]

    V. N. Gribov, Soviet Physics JETP26, 414 (1968), trans- lated from Zh. Eksp. Teor. Fiz. 53 (1967) 654

    1968

  69. [70]

    Regge, Il Nuovo Cimento18, 947 (1960)

    T. Regge, Il Nuovo Cimento18, 947 (1960)

    1960

  70. [71]

    Regge, Il Nuovo Cimento14, 951 (1959)

    T. Regge, Il Nuovo Cimento14, 951 (1959)

    1959

  71. [72]

    I. I. Balitsky and L. N. Lipatov, Soviet Journal of Nuclear Physics28, 822 (1978)

    1978

  72. [73]

    E. A. Kuraev, L. N. Lipatov, and V. S. Fadin, Soviet Physics JETP45, 199 (1977)

    1977

  73. [74]

    A. Gao, I. Moult, S. Raman, G. Ridgway, and I. W. Stewart, JHEP05, 328 (2024), arXiv:2401.00931 [hep- ph]

    Pith/arXiv arXiv 2024

  74. [75]

    Moult, S

    I. Moult, S. Raman, G. Ridgway, and I. W. Stewart, JHEP05, 025 (2023), arXiv:2207.02859 [hep-ph]

    Pith/arXiv arXiv 2023

  75. [76]

    Falcioni, E

    G. Falcioni, E. Gardi, N. Maher, C. Milloy, and L. Vernazza, Phys. Rev. Lett.128, 132001 (2022), arXiv:2112.11098 [hep-ph]

    Pith/arXiv arXiv 2022

  76. [77]

    E. P. Byrne, V. Del Duca, E. Gardi, Y. Mo, and J. M. Smillie, JHEP03, 170 (2026), arXiv:2506.10644 [hep-ph]

    Pith/arXiv arXiv 2026

  77. [78]

    Falcioni, E

    G. Falcioni, E. Gardi, N. Maher, C. Milloy, and L. Ver- nazza, JHEP03, 053 (2022), arXiv:2111.10664 [hep-ph]

    Pith/arXiv arXiv 2022

  78. [79]

    Caron-Huot, E

    S. Caron-Huot, E. Gardi, and L. Vernazza, JHEP06, 016 (2017), arXiv:1701.05241 [hep-ph]

    Pith/arXiv arXiv 2017

  79. [80]

    Caola, A

    F. Caola, A. Chakraborty, G. Gambuti, A. von Man- teuffel, and L. Tancredi, Phys. Rev. Lett.128, 212001 (2022), arXiv:2112.11097 [hep-ph]

    Pith/arXiv arXiv 2022

  80. [81]

    Buccioni, F

    F. Buccioni, F. Caola, F. Devoto, and G. Gambuti, JHEP03, 129 (2025), arXiv:2411.14050 [hep-ph]

    Pith/arXiv arXiv 2025

Showing first 80 references.