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Off-shell minimal form factors

T0 review · 3 major / 3 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper shows that two-loop off-shell minimal form factors in planar N=4 super-Yang-Mills exponentiate with the octagon anomalous dimension, not the cusp, and that their finite remainder shares the conformal symbol.

desk verdict Credible two-loop off-shell form factor calculation with a clean octagon-vs-cusp punchline; the main gap is the unproven no-1/p^2 observation in the superspace cross-check, which a referee should push on. read the letter →

arxiv 2411.16941 v1 pith:IDXKKDXT submitted 2024-11-25 hep-th hep-ph

A.V. Belitsky , L.V. Bork This is my paper · ORCID
classification hep-thhep-ph
keywords minimalformfactorsoff-shelloctagonanomalousdimensionplanarN=4super-Yang-MillsSudakovlogarithmsCoulombbranchN=1superspacesymboloffiniteremainder
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper computes two-loop off-shell minimal form factors — form factors of half-BPS operators whose field content matches the number of external scalars — in planar $N=4$ super-Yang-Mills, in the near-mass-shell limit where the small external virtuality $m = -p_i^2$ regulates infrared divergences. Its central claim is that the Sudakov double-logarithmic structure is governed by the octagon anomalous dimension $\Gamma_{\mathrm{oct}}(g)$, not the cusp anomalous dimension of on-shell physics, and that the finite remainder has a symbol identical to the conformal result found a decade ago while its beyond-the-symbol terms differ. The claim matters because it sharpens the emerging picture that the infrared 'universal' anomalous dimension is not universal: the coefficient of the double log changes when external legs go off the mass shell, with consequences for how off-shell partonic subprocesses enter factorization schemes. Two independent derivations — a Coulomb-branch uplift of massless integrands and an off-shell $N=1$ supergraph computation — produce the same set of scalar integrals, so the two routes cross-check each other.

What carries the argument

The load-bearing objects are the octagon anomalous dimension $\Gamma_{\mathrm{oct}}(g) = \frac{2}{\pi^2}\log\cosh(2\pi g)$, the exact function that replaces the cusp in the near-mass-shell regime, and two complementary ways of producing the off-shell integrands: the uplift, which reads the out-of-four-dimensional components of massless six-dimensional external momenta as four-dimensional virtualities, and the $N=1$ superspace formulation, whose algebra of covariant derivatives on unconstrained external superfields reduces the supergraphs to the same scalar integrals $G_1$–$G_7$. On the integral side the calculation runs on the near-mass-shell expansions of the ladder functions $\Phi_1$ and $\Phi_2$ for the triangle and double-triangle graphs, and on an integration-by-parts reduction of the three-leg TriBox and TriBox-red integrals into a basis of 62 master integrals solved in multiple polylogarithms. The symbol map then separates the kinematical content of the finite remainder — identical to the conformal remainder — from the beyond-the-symbol zeta-valued terms that distinguish the off-shell branch.

What would settle it

Evaluate the planar two-loop three-point off-shell integrals at equal ratios $u=v=w=1/3$ with $m/q^2 \sim 10^{-6}$, form $\log F_3$, and read off the coefficient of $g^4\log^2(m/q^2)$: the claim requires $-\frac{3}{4}\cdot 16\zeta_2 = -12\zeta_2$ (the octagon value), while substituting the cusp series would give $+6\zeta_4$. A second decisive check is the paper's own open question, Eq. (7.1): compute the three-loop remainder and test whether its symbol equals the conformal one, watching specifically for any $1/p_i^2$-enhanced integral in the off-shell supergraph reduction that would break the two-loop equivalence.

Watch

Extended reading notes

Core claim

At two loops, the off-shell minimal form factor $F_n$ of the operator $\mathrm{tr}\,\phi_{12}^n$ with $n$ external scalars of equal virtuality $m$, in planar $N=4$ super-Yang-Mills, exponentiates in the near-mass-shell limit: $\log F_n = -\frac{3}{4}\Gamma_{\mathrm{oct}}(g)\log^2 m + \frac{1}{2}\Gamma_{\mathrm{oct}}(g)\log m\log(uvw) + f_n(u,v,w;g) + O(m)$, where $\Gamma_{\mathrm{oct}}(g) = \frac{2}{\pi^2}\log\cosh(2\pi g)$ is the exact octagon anomalous dimension rather than the cusp. The finite remainder $f_n$ carries the same symbol as the conformal (on-shell) case; for $n=3$ the explicit beyond-the-symbol difference is $f_3^{(2)} = R_3^{(2)} + \frac{327}{8}\zeta_4 + \zeta_3(\log u + \log v + \log w) + \frac{1}{4}\zeta_2(\log^2 u + \log^2 v + \log^2 w + 2\log u\log v + 2\log v\log w + 2\log w\log u)$. For $n>3$ the symbol identity with the conformal remainder is given by Eq. (6.15) in terms of the ratio variables $u_i, v_i, w_i$. Establishing this required expressing the form factor as a sum of independent scalar integrals, and the paper performs that reduction twice — with the uplift of six-dimensional integrands and with an off-shell $N=1$ superspace algebra of covariant derivatives — and the two representations coincide.

Load-bearing premise

The near-mass-shell superspace reduction hinges on the observed-but-unproved absence (Section 5) of integrals behaving as $1/p_i^2$ when $p_i^2 \to 0$, an absence that must persist for $n>3$ and at higher loops or else the leading $m$-dependence and the finite remainder of Eq. (6.6) would change.

Editorial extensions

If this is right

  • The near-mass-shell Sudakov exponentiation of minimal form factors is controlled by $\Gamma_{\mathrm{oct}}$, with no analog of the collinear anomalous dimension $G(g)$, and the conjectural all-order structure — a sum of two-leg Sudakov factors plus a remainder — now has two-loop confirmation for the minimal operators.
  • The off-shell remainder and its conformal counterpart share the same symbol at two loops for all $n$, while their difference is made explicit for $n = 3$ in terms of $\zeta_4$, $\zeta_3 \log(uvw)$, and $\zeta_2 \log^2$-type terms.
  • The authors conclude that off-shell Sudakov behavior in QCD needs dedicated studies, since off-shell partonic subprocesses are intrinsic building blocks of high-energy and $k_T$-factorization schemes and should not be assumed to be governed by the cusp.
  • The agreement between the Coulomb-branch uplift and the $N=1$ superspace reduction at two loops reinforces both techniques, and the same computation corrects earlier inaccuracies in the conformal superspace treatment.

Reading between the lines

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

  • If the two-loop pattern persists, the near-mass-shell limit becomes a direct probe of $\Gamma_{\mathrm{oct}}$: reading off the coefficient of $g^{2L}\log^2 m$ from any off-shell $L$-loop form factor should reproduce the exact series $(2/\pi^2)\log\cosh(2\pi g)$, order by order, independent of the operator.
  • The paper's own empirical caveat — absence of $1/p_i^2$ singular integrals observed but not proven at two loops, with the authors stating they do not expect the equivalence to survive higher loops — makes the conjectured three-loop symbol identity $S[f_3^{(3)}] = S[R_3^{(3)}]$ (Eq. (7.1)) a sharp diagnostic: its failure would show where the near-mass-shell dictionary first breaks.
  • The equal-weight, fully $u,v,w$-symmetric structure of the beyond-the-symbol terms ($\zeta_3 \log(uvw)$ plus $\zeta_2$ quadratic logs) is the shape one would expect from ultrasoft-mode physics; testing whether these terms can be absorbed into a subtraction scheme for the ultrasoft region would connect this result to the pinching-Sudakov analysis the paper cites.
  • A direct two-loop calculation of the off-shell quark form factor in QCD near the mass shell — extracting the coefficient of the Sudakov double log — would transfer the $N=4$ pattern to phenomenologically relevant territory, where off-shell partons enter $k_T$-factorization.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper computes two-loop off-shell minimal form factors F_n of half-BPS operators tr φ^n in planar N=4 SYM, with common off-shellness m=-p_i^2 for external legs while internal propagators remain massless. The integrand representation is obtained by two methods: (i) uplift of the known massless integrands to the Coulomb branch, and (ii) an N=1 superspace supergraph calculation with external superfields kept off-shell. The near-mass-shell (m→0) limit of the n=3 form factor is evaluated using the 62-master-integral basis of Belitsky-Smirnov and IBP reduction with FIRE, with high-precision numerical checks against FIESTA5/GiNaC. The authors find Sudakov exponentiation with the octagon anomalous dimension Γ_oct replacing the cusp, and a finite remainder whose symbol matches the conformal result of Ref. [38]. The beyond-symbol part is different and is given by Eq. (6.14). General n>3 is treated by cyclic permutations and a factorized C1×C1 contribution, yielding the symbol in Eq. (6.15).

Significance. If correct, the result provides a nontrivial two-loop confirmation that on the Coulomb branch the infrared double logarithms of off-shell minimal form factors are controlled by the octagon anomalous dimension rather than the cusp, and that the finite remainder shares its symbol with the conformal phase while differing in beyond-symbol terms. The paper's strengths are the cross-check between two independent derivation methods, the use of IBP reduction with FIRE, the reduction to a 62-integral canonical basis, and high-precision numerical verification with FIESTA5 and GiNaC, with results bundled in a Mathematica notebook. The main value lies in the explicit n=3 remainder (6.14) and the general-n symbol statement (6.15).

major comments (3)
  1. [Section 5 (near p. 12, paragraph beginning 'Of course, there is a possibility...')] The entire matching of the N=1 off-shell supergraph calculation to the uplifted Coulomb-branch representation, and hence Eqs. (5.11), (6.6), (6.14), and (6.15), relies on the assertion that no Feynman integral in the off-shell N=1 supergraph calculation behaves as 1/p_i^2 in the p_i^2→0 limit. The paper states 'we observed no such contributions in the two-loop order' and cites Ref. [62] for one loop, but provides neither a proof nor a list of checked topologies or denominator structures. If, for example, a supergraph such as C3 or C5/C6 produced such an integral, the leading m-behavior and the finite remainder would change. This is a load-bearing point and needs to be fixed, either by a general argument or by an explicit exhaustive check of all two-loop supergraphs.
  2. [Section 6.2, Eqs. (6.13) and (6.14)] The beyond-symbol coefficients in Eq. (6.14), including 327/8 ζ4, are fixed by high-precision numerical fitting of f_3^(2)-R_3^(2) against the ansatz (6.13). The completeness of this ansatz is not demonstrated: no argument is given that the coefficient functions are constant rational numbers rather than functions of the cross ratios, nor that no additional weight-4 beyond-symbol structures built from the same alphabet can appear. Since Eq. (6.14) is the central quantitative result distinguishing the off-shell from the conformal remainder, this should be justified analytically or the ansatz should be proven complete.
  3. [Sections 4, 5, and 6.3] The generalization to n>3 is asserted rather than shown. Eq. (6.15) is stated to be the symbol of f_n^(2) without an explicit calculation, and Section 6.3 gives no reduction of the factorized C1×C1 contribution or the coefficient matching for generic n. Moreover, the n>3 claim inherits both the unproven absence of 1/p_i^2 singular integrals from Section 5 and the potential failure of the naive uplift noted in Section 3. The authors should either provide the explicit reduction for generic n or state precisely which assumptions allow Eq. (6.15) to follow from the n=3 computation.
minor comments (3)
  1. [Section 5 and Figure captions] The manuscript contains unedited draft fragments: the passage with triple question marks near Eq. (4.5) ('???It will be interesting...') should be removed, and the caption 'Figure 2' appears twice while Eq. (5.9)/(5.11) are duplicated in the text around Figure 6.
  2. [Throughout] There are several typos that should be corrected in a final version: 'Comlomb' for 'Coulomb', 'void' for 'avoid' in Section 6, 'Madelstam' for 'Mandelstam', and 'T able 1' for 'Table 1'.
  3. [Notation] The shift operator P is used in Eqs. (3.5)-(3.6) and (4.1)-(4.2) before it is explicitly defined; please define it once near its first use.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the two-loop remainder and the octagon coefficient are benchmarked against independent external results; the self-citations are not load-bearing.

full rationale

The paper's central claims are checked against independent, parameter-free results rather than derived from them. The infrared coefficient is compared with the exact null-octagon anomalous dimension of Refs. [45,46], which comes from a different observable; the finite-remainder symbol is benchmarked against the conformal result of Ref. [38]; and the n=2 supergraph coefficients are fixed by matching to Refs. [63,64]. The exponentiation formula (2.13) originates in the authors' earlier work, but it is tested here by the explicit cancellation of the top two powers of log m in the two-loop combination (6.9), not assumed. The only load-bearing technical assumption is the statement in Section 5 that no supergraph integral behaves as 1/p_i^2 in the near-mass-shell limit ('we observed no such contributions in the two-loop order'); this is unproved and could affect the remainder if wrong, but it is a correctness risk, not a circular reduction of the result to its inputs. Consequently, no step in the derivation is equivalent by construction to its assumed output.

Assumptions & free parameters 1 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the validity of the Coulomb-branch uplift for low legs, the absence of singular 1/p^2 integrals in the off-shell superspace calculation, and the completeness of the beyond-symbol ansatz. These are stated assumptions rather than proven theorems. The beyond-symbol coefficients are fitted numerically, which is a free-parameter-like step.

free parameters (1)
  • Beyond-symbol coefficients in f3^(2) = 327/8, 1, 1/4 (rational coefficients for zeta4, zeta3 log, zeta2 log^2 terms)
    Fixed by high-precision numerical comparison of f3^(2) - R3^(2) against an ansatz of transcendental terms (section 6.2, Eqs. (6.13)-(6.14)). These coefficients are outputs of a fit, not derived analytically.
assumptions (5)
  • domain assumption Planar N=4 sYM on the Coulomb branch with massive external legs and massless internal propagators correctly represents off-shell minimal form factors in the near-mass-shell limit.
    This is the setup of section 3, uplift procedure; it relies on the equivalence of the higher-dimensional and Higgs/Coulomb-branch constructions stated in the introduction.
  • domain assumption The naive uplift of four-dimensional integrands to higher dimensions is valid for the number of legs and loop order considered (no (D-4)-dimensional mu-terms).
    Section 3 and 4 use uplift for n=3 and n>3; the paper notes uplift is 'destined to fail' for higher legs, but assumes its validity here.
  • ad hoc to paper In the N=1 superspace off-shell calculation, no integrals singular as 1/p_i^2 appear up to two loops, so the near-mass-shell limit is controlled by the same scalar integrals as the on-shell case.
    Section 5 states 'we observed no such contributions in the two-loop order', an empirical observation without proof; if such terms existed, the integral representation would differ.
  • ad hoc to paper The ansatz (6.13) of zeta2 log z_i log z_j, zeta2 Li2(z_i), zeta3 log z_i, zeta4 terms is complete for the beyond-symbol difference f3^(2)-R3^(2).
    Used to fix the coefficients in (6.14); completeness of the alphabet is assumed but not proven.
  • standard math Multiple polylogarithms and the symbol map capture the full analytic structure of the two-loop integrals in the near-mass-shell limit; no elliptic or other non-MPL functions appear.
    The authors rely on MPL results from [65,66] and the symbol technique [73,74]; this is standard in the field.

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Cite this review

Pith. "Pith review of Off-shell minimal form factors." pith.science (2026). https://pith.science/paper/IDXKKDXT

@misc{pith2026241116941,
  author       = {Pith},
  title        = {Pith review of: Off-shell minimal form factors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IDXKKDXT}},
  note         = {Machine review of arXiv:2411.16941}
}
read the original abstract

We study off-shell n-particle form factors of half-BPS operators built from n complex scalar fields at the two-loop order in the planar maximally supersymmetric Yang-Mills theory (sYM). These are known as minimal form factors. We construct their representation as a sum of independent scalar Feynman integrals relying on two complementary techniques. First, by going to the Coulomb branch of the theory by employing the spontaneous symmetry breaking which induces masses, but only for external particles while retaining masslessness for virtual states propagating in quantum loops. For a low number of external legs, this entails an uplift of massless integrands to their massive counterparts. Second, utilizing the N=1 superspace formulation of the N=4 sYM and performing algebra of covariant derivatives off-shell. Both techniques provide identical results. These form factors are then studied in the near-mass-shell limit with the off-shellness regularizing emerging infrared divergences. We observe their exponentiation and confirm the octagon anomalous dimension, not the cusp, as the coefficient of the Sudakov double logarithmic behavior. By subtracting these singularities and defining a finite remainder, we verified that its symbol is identical to the one found a decade ago in the conformal case. Beyond-the-symbol contributions are different in the two cases, however.

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Forward citations

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Reference graph

Works this paper leans on

74 extracted references · 17 canonical work pages · cited by 3 Pith papers

  1. [38]

    Brandhuber, B

    A. Brandhuber, B. Penante, G. Travaglini and C. Wen, The last of the simple remainders , JHEP 08 (2014) 100, [ 1406.1443]. – 21 –

    arXiv 2014

  2. [62]

    A. V. Santini, Off-shell amplitudes in the N=4 Super Yang-Mills theory . PhD thesis, Rome U., Tor Vergata, 2012

    work page 2012

  3. [1]

    Kinoshita, Mass singularities of Feynman amplitudes , J

    T. Kinoshita, Mass singularities of Feynman amplitudes , J. Math. Phys. 3 (1962) 650–677

    work page 1962

  4. [2]

    T. D. Lee and M. Nauenberg, Degenerate Systems and Mass Singularities , Phys. Rev. 133 (1964) B1549–B1562. – 19 –

    work page 1964

  5. [3]

    J. C. Collins, D. E. Soper and G. F. Sterman, Factorization of Hard Processes in QCD , Adv. Ser. Direct. High Energy Phys. 5 (1989) 1–91, [ hep-ph/0409313]

    arXiv 1989

  6. [4]

    Catani, M

    S. Catani, M. Ciafaloni and F. Hautmann, High-energy factorization and small x heavy flavor production, Nucl. Phys. B 366 (1991) 135–188

    work page 1991

  7. [5]

    J. C. Collins and R. K. Ellis, Heavy quark production in very high-energy hadron collisions , Nucl. Phys. B 360 (1991) 3–30

    work page 1991

  8. [6]

    L. F. Abbott, The Background Field Method Beyond One Loop , Nucl. Phys. B 185 (1981) 189–203

    1981

Show all 74 references
  1. [7]

    L. F. Abbott, Introduction to the Background Field Method , Acta Phys. Polon. B 13 (1982) 33

    1982

  2. [8]

    S. J. Gates, M. T. Grisaru, M. Rocek and W. Siegel, Superspace Or One Thousand and One Lessons in Supersymmetry , hep-th/0108200

    arXiv

  3. [9]

    Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits , Nucl. Phys. B 425 (1994) 217–260, [hep-ph/9403226]

    1994 arXiv

  4. [10]

    Z. Bern, L. J. Dixon, D. C. Dunbar and D. A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes , Nucl. Phys. B 435 (1995) 59–101, [ hep-ph/9409265]

    1995 arXiv

  5. [11]

    Britto, F

    R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N=4 super-Yang-Mills, Nucl. Phys. B 725 (2005) 275–305, [ hep-th/0412103]

    2005 arXiv

  6. [12]

    Cachazo, Sharpening The Leading Singularity , 0803.1988

    F. Cachazo, Sharpening The Leading Singularity , 0803.1988

    1988 arXiv

  7. [13]

    Gastmans and T

    R. Gastmans and T. T. Wu, The Ubiquitous photon: Helicity method for QED and QCD , vol. 80. 1990

    1990

  8. [14]

    Xu, D.-H

    Z. Xu, D.-H. Zhang and L. Chang, Helicity Amplitudes for Multiple Bremsstrahlung in Massless Nonabelian Gauge Theories , Nucl. Phys. B 291 (1987) 392–428

    1987

  9. [15]

    S. J. Parke and T. R. Taylor, An Amplitude for n Gluon Scattering, Phys. Rev. Lett. 56 (1986) 2459

    1986

  10. [16]

    L. J. Dixon, A brief introduction to modern amplitude methods , in Theoretical Advanced Study Institute in Elementary Particle Physics: Particle Physics: The Higgs Boson and Beyond, pp. 31–67, 2014. 1310.5353. DOI

    2014 arXiv

  11. [17]

    W. T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes , JHEP 04 (2008) 049, [ 0801.2237]

    2008 arXiv

  12. [18]

    R. K. Ellis, W. T. Giele, Z. Kunszt and K. Melnikov, Masses, fermions and generalized D-dimensional unitarity , Nucl. Phys. B 822 (2009) 270–282, [ 0806.3467]

    2009 arXiv

  13. [19]

    Z. Bern, J. J. Carrasco, T. Dennen, Y.-t. Huang and H. Ita, Generalized Unitarity and Six-Dimensional Helicity , Phys. Rev. D 83 (2011) 085022, [ 1010.0494]

    2011 arXiv

  14. [20]

    Boughezal, K

    R. Boughezal, K. Melnikov and F. Petriello, The four-dimensional helicity scheme and dimensional reconstruction, Phys. Rev. D 84 (2011) 034044, [ 1106.5520]. – 20 –

    2011 arXiv

  15. [21]

    Davies, One-Loop QCD and Higgs to Partons Processes Using Six-Dimensional Helicity and Generalized Unitarity , Phys

    S. Davies, One-Loop QCD and Higgs to Partons Processes Using Six-Dimensional Helicity and Generalized Unitarity , Phys. Rev. D 84 (2011) 094016, [ 1108.0398]

    2011 arXiv

  16. [22]

    Accettulli Huber, A

    M. Accettulli Huber, A. Brandhuber, S. De Angelis and G. Travaglini, Complete Form Factors in Yang-Mills from Unitarity and Spinor Helicity in Six Dimensions , Phys. Rev. D 101 (2020) 026004, [ 1910.04772]

    2020 arXiv

  17. [23]

    Cheung and D

    C. Cheung and D. O’Connell, Amplitudes and Spinor-Helicity in Six Dimensions , JHEP 07 (2009) 075, [ 0902.0981]

    2009 arXiv

  18. [24]

    Caron-Huot and F

    S. Caron-Huot and F. Coronado, Ten dimensional symmetry of N = 4 SYM correlators , JHEP 03 (2022) 151, [ 2106.03892]

    2022 arXiv

  19. [25]

    K. G. Selivanov, An Infinite set of tree amplitudes in Higgs-Yang-Mills , Phys. Lett. B 460 (1999) 116–118, [ hep-th/9906001]

    1999 arXiv

  20. [26]

    R. H. Boels, No triangles on the moduli space of maximally supersymmetric gauge theory , JHEP 05 (2010) 046, [ 1003.2989]

    2010 arXiv

  21. [27]

    Brink, J

    L. Brink, J. H. Schwarz and J. Scherk, Supersymmetric Yang-Mills Theories, Nucl. Phys. B 121 (1977) 77–92

    1977

  22. [28]

    Gliozzi, J

    F. Gliozzi, J. Scherk and D. I. Olive, Supersymmetry, Supergravity Theories and the Dual Spinor Model, Nucl. Phys. B 122 (1977) 253–290

    1977

  23. [29]

    L. V. Bork, N. B. Muzhichkov and E. S. Sozinov, Infrared properties of five-point massive amplitudes in N = 4 SYM on the Coulomb branch , JHEP 08 (2022) 173, [ 2201.08762]

    2022 arXiv

  24. [30]

    A. V. Belitsky, L. V. Bork, A. F. Pikelner and V. A. Smirnov, Exact off-shell Sudakov form factor in N=4 SYM , 2209.09263

    arXiv

  25. [31]

    A. V. Belitsky, L. V. Bork and V. A. Smirnov, Off-shell form factor in N =4 sYM at three loops, JHEP 11 (2023) 111, [ 2306.16859]

    2023 arXiv

  26. [32]

    A. V. Belitsky, L. V. Bork, J. M. Grumski-Flores and V. A. Smirnov, Three-leg form factor on Coulomb branch , 2402.18475

    arXiv

  27. [33]

    Huang, Non-Chiral S-Matrix of N=4 Super Yang-Mills , 1104.2021

    Y.-t. Huang, Non-Chiral S-Matrix of N=4 Super Yang-Mills , 1104.2021

    2021 arXiv

  28. [34]

    Plefka, T

    J. Plefka, T. Schuster and V. Verschinin, From Six to Four and More: Massless and Massive Maximal Super Yang-Mills Amplitudes in 6d and 4d and their Hidden Symmetries , JHEP 01 (2015) 098, [ 1405.7248]

    2015 arXiv

  29. [35]

    Z. Bern, L. J. Dixon, D. A. Kosower, R. Roiban, M. Spradlin, C. Vergu et al., The Two-Loop Six-Gluon MHV Amplitude in Maximally Supersymmetric Yang-Mills Theory , Phys. Rev. D 78 (2008) 045007, [ 0803.1465]

    2008 arXiv

  30. [36]

    L. F. Alday, J. M. Henn, J. Plefka and T. Schuster, Scattering into the fifth dimension of N=4 super Yang-Mills , JHEP 01 (2010) 077, [ 0908.0684]

    2010 arXiv

  31. [37]

    J. L. Bourjaily, P. Heslop and V.-V. Tran, Amplitudes and Correlators to Ten Loops Using Simple, Graphical Bootstraps, JHEP 11 (2016) 125, [ 1609.00007]

    2016 arXiv

  32. [39]

    B. Eden, P. Heslop, G. P. Korchemsky and E. Sokatchev, The super-correlator/super-amplitude duality: Part I , Nucl. Phys. B 869 (2013) 329–377, [1103.3714]

    2013 arXiv

  33. [40]

    A. H. Mueller, On the Asymptotic Behavior of the Sudakov Form-factor , Phys. Rev. D 20 (1979) 2037

    1979

  34. [41]

    Magnea and G

    L. Magnea and G. F. Sterman, Analytic continuation of the Sudakov form-factor in QCD , Phys. Rev. D 42 (1990) 4222–4227

    1990

  35. [42]

    G. F. Sterman and M. E. Tejeda-Yeomans, Multiloop amplitudes and resummation , Phys. Lett. B 552 (2003) 48–56, [ hep-ph/0210130]

    2003 arXiv

  36. [43]

    Z. Bern, L. J. Dixon and V. A. Smirnov, Iteration of planar amplitudes in maximally supersymmetric Yang-Mills theory at three loops and beyond , Phys. Rev. D 72 (2005) 085001, [hep-th/0505205]

    2005 arXiv

  37. [44]

    Coronado, Bootstrapping the Simplest Correlator in Planar N = 4 Supersymmetric Yang-Mills Theory to All Loops , Phys

    F. Coronado, Bootstrapping the Simplest Correlator in Planar N = 4 Supersymmetric Yang-Mills Theory to All Loops , Phys. Rev. Lett. 124 (2020) 171601, [ 1811.03282]

    2020 arXiv

  38. [45]

    A. V. Belitsky and G. Korchemsky, Exact null octagon , JHEP 05 (2020) 070, [ 1907.13131]

    2020 arXiv

  39. [46]

    A. V. Belitsky, Null octagon from Deift-Zhou steepest descent , Nucl. Phys. B 980 (2022) 115844, [2012.10446]

    2022 arXiv

  40. [47]

    V. V. Sudakov, Vertex parts at very high-energies in quantum electrodynamics , Sov. Phys. JETP 3 (1956) 65–71

    1956

  41. [48]

    Jackiw, Dynamics at high momentum and the vertex function of spinor electrodynamics , Annals Phys

    R. Jackiw, Dynamics at high momentum and the vertex function of spinor electrodynamics , Annals Phys. 48 (1968) 292–321

    1968

  42. [49]

    P. M. Fishbane and J. D. Sullivan, Asymptotic behavior of the vertex function in quantum electrodynamics, Phys. Rev. D 4 (1971) 458–475

    1971

  43. [50]

    A. H. Mueller, Perturbative QCD at High-Energies , Phys. Rept. 73 (1981) 237

    1981

  44. [51]

    Korchemsky, Sudakov Form-factor in QCD , Phys

    G. Korchemsky, Sudakov Form-factor in QCD , Phys. Lett. B 220 (1989) 629–634

    1989

  45. [52]

    A. V. Belitsky, L. V. Bork and V. A. Smirnov, Pinching Sudakov , 2409.05945

    arXiv

  46. [53]

    Dennen, Y.-t

    T. Dennen, Y.-t. Huang and W. Siegel, Supertwistor space for 6D maximal super Yang-Mills , JHEP 04 (2010) 127, [ 0910.2688]

    2010 arXiv

  47. [54]

    Basso, L

    B. Basso, L. J. Dixon and A. G. Tumanov, The Three-Point Form Factor of Tr ϕ3 to Six Loops, 2410.22402

    arXiv

  48. [55]

    G. Lin, G. Yang and S. Zhang, Three-Loop Color-Kinematics Duality: A 24-Dimensional Solution Space Induced by New Generalized Gauge Transformations , Phys. Rev. Lett. 127 (2021) 171602, [ 2106.05280]

    2021 arXiv

  49. [56]

    G. Lin, G. Yang and S. Zhang, Full-color three-loop three-point form factors in N = 4 SYM, JHEP 03 (2022) 061, [ 2111.03021]

    2022 arXiv

  50. [57]

    J. M. Henn, J. Lim and W. J. Torres Bobadilla, Analytic evaluation of the three-loop three-point form factor of trϕ3 in N = 4 sYM, 2410.22465. – 22 –

    arXiv

  51. [58]

    N. I. Usyukina and A. I. Davydychev, An Approach to the evaluation of three and four point ladder diagrams, Phys. Lett. B 298 (1993) 363–370

    1993

  52. [59]

    N. I. Usyukina and A. I. Davydychev, Exact results for three and four point ladder diagrams with an arbitrary number of rungs , Phys. Lett. B 305 (1993) 136–143

    1993

  53. [60]

    L. V. Bork, D. I. Kazakov and G. S. Vartanov, On form factors in N=4 SYM , JHEP 02 (2011) 063, [ 1011.2440]

    2011 arXiv

  54. [61]

    M. T. Grisaru, M. Rocek and W. Siegel, Superloops 3, Beta 0: A Calculation in N = 4 Yang-Mills Theory, Nucl. Phys. B 183 (1981) 141–156

    1981

  55. [63]

    W. L. van Neerven, Infrared Behavior of On-shell Form-factors in a N = 4 Supersymmetric Yang-Mills Field Theory , Z. Phys. C 30 (1986) 595

    1986

  56. [64]

    Gehrmann, J

    T. Gehrmann, J. M. Henn and T. Huber, The three-loop form factor in N=4 super Yang-Mills, JHEP 03 (2012) 101, [ 1112.4524]

    2012 arXiv

  57. [65]

    A. V. Belitsky and V. A. Smirnov, Near mass-shell double boxes , JHEP 05 (2024) 155, [2312.00641]

    2024 arXiv

  58. [66]

    A. B. Goncharov, Multiple polylogarithms and mixed Tate motives , math/0103059

    arXiv

  59. [67]

    A. V. Smirnov and M. Zeng, FIRE 6.5: Feynman integral reduction with new simplification library, Comput. Phys. Commun. 302 (2024) 109261, [ 2311.02370]

    2024 arXiv

  60. [68]

    J. M. Henn, Multiloop integrals in dimensional regularization made simple , Phys. Rev. Lett. 110 (2013) 251601, [ 1304.1806]

    2013 arXiv

  61. [69]

    C. W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language , J. Symb. Comput. 33 (2002) 1–12, [cs/0004015]

    2002 arXiv

  62. [70]

    Duhr and F

    C. Duhr and F. Dulat, PolyLogTools — polylogs for the masses , JHEP 08 (2019) 135, [1904.07279]

    2019 arXiv

  63. [71]

    A. V. Smirnov, FIESTA 3: cluster-parallelizable multiloop numerical calculations in physical regions, Comput. Phys. Commun. 185 (2014) 2090–2100, [ 1312.3186]

    2014 arXiv

  64. [72]

    A. V. Smirnov, N. D. Shapurov and L. I. Vysotsky, FIESTA5: numerical high-performance Feynman integral evaluation, 2110.11660

    arXiv

  65. [73]

    A. B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical Polylogarithms for Amplitudes and Wilson Loops , Phys. Rev. Lett. 105 (2010) 151605, [ 1006.5703]

    2010 arXiv

  66. [74]

    A. B. Goncharov, A simple construction of Grassmannian polylogarithms , 0908.2238. – 23 –

    arXiv

Pith tools

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