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Reduction of $\epsilon$-expanded Feynman integrals

T0 review · 3 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read $\bar{R}$-operation removes ultraviolet divergences from Feynman integrals while preserving their infrared and collinear safety, and reduces the $\epsilon$-expanded master integrals to a minimal basis.

desk verdict A genuinely new method for reducing epsilon-expanded master integrals via a UV-subtraction scheme that preserves IR/CO safety; the main proof sketch needs tightening, but the work deserves a serious referee. read the letter →

arxiv 2504.16766 v1 pith:TERQJZ33 submitted 2025-04-23 hep-ph hep-th

Yan-Qing Ma , Cong-Hao Qin , Ao Tan , Kai Yan This is my paper · ORCID
classification hep-phhep-th MSC 81Q3081T1881T15
keywords Feynmanintegralsmasterepsilonexpansionultravioletsubtractioninfrareddivergencecollineardimensionalrecurrencerelationsintegrationbyparts
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

Feynman integrals are normally reduced to master integrals in general dimension $D=4-2\epsilon$, but those master integrals are not independent when only the first few terms of the $\epsilon$-expansion are needed. This paper establishes a systematic way to construct locally finite integrals, finite in every infrared, collinear, and ultraviolet region, by raising the spacetime dimension and then applying a modified ultraviolet subtraction, the $\bar{R}$-operation, whose counterterms do not reintroduce soft or collinear divergences. Expressed back in four dimensions through integration-by-parts identities and dimensional recurrence relations, these finite integrals impose constraints that cut the $\epsilon$-expansion coefficients of master integrals to a much smaller set of master coefficients. If correct, this substantially lowers the cost of high-order perturbative computations in quantum field theory, since only the reduced set of coefficients needs to be evaluated.

What carries the argument

The load-bearing object is the $\bar{R}$-operation: the standard $R$-operation subtraction formula $G^{UVCT} = \sum (-1)^k G_{r_1\cdots r_k}$, but with each counterterm $G_r$ built after the variable substitution $l_i\cdot l_j = \hat{l}_i\cdot \hat{l}_j + c_{ij}m^2$, $p_i\cdot l_j = p_i\cdot \hat{l}_j$, which puts a mass $m^2$ into the propagators of the large loop momenta while leaving the numerator alone. The counterterm is then the part of the large-momentum expansion (15) down to the superficial degree of divergence $wD$. Because the denominators of $G_r$ inherit their infrared and collinear scaling from the original integrand, and each term in the expansion inherits one homogeneous piece of the numerator, the counterterm cannot create new soft or collinear poles. Around this object, the method uses the strategy of regions with a small mass regulator to certify infrared and collinear finiteness, and Tarasov's dimensional recurrence relations to bring high-dimensional finite integrals back to $4-2\epsilon$.

What would settle it

Compute the full $\bar{R}$-subtracted integrand $G+G^{UVCT}$ for the two-loop double box at a generic kinematic point, introducing a small regulator $\eta$ into every propagator; if its expansion as $\eta\to 0$ contains any $1/\eta$ or $\ln\eta$ term, the subtraction has reintroduced an infrared or collinear divergence and the central claim of local finiteness is false.

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Extended reading notes

Core claim

Starting from any Feynman integral in a given family, one can first choose a sufficiently high spacetime dimension so that all infrared and collinear regions are power-counting finite. The paper's central proposal is the $\bar{R}$-operation, a variant of the $R$-operation in which each ultraviolet counterterm is constructed by expanding the integrand in the large loop momenta after shifting the squared perpendicular components of those momenta by a mass term, while leaving the numerator untouched. This mass substitution is what prevents the counterterms from developing new infrared or collinear singularities. The resulting locally finite integrals, once reduced to master integrals by IBP and converted to $D=4-2\epsilon$ by dimensional recurrence relations, yield relations of the form $\sum_i c_i M_i = O(\epsilon^0)$ or $O(\epsilon)$; expanding in $\epsilon$ gives linear constraints that reduce the divergent and low-order finite parts of the master integrals to a minimal set of master coefficients. The paper demonstrates the reduction on the one-loop pentagon, the two-loop double box and double pentagon, and a three-loop example, and packages the procedure in an automated tool.

Load-bearing premise

The method rests on the assumption that putting a mass-like shift into the large loop momenta of each ultraviolet counterterm cannot create new infrared or collinear singular regions, so every subtraction term stays finite wherever the original integrand was safe.

Editorial extensions

If this is right

  • For a given integral family, the divergent parts of all master integrals can be expressed through a small set of master coefficients, many of which are simpler subsector or vacuum integrals, so only those coefficients need to be evaluated.
  • The one-loop pentagon relation $E_0 = \sum_i a_i D_{0i}+O(\epsilon)$ is recovered from the method, confirming that it generalizes a known simplification to higher loops.
  • In the two-loop double box, the 32 expansion coefficients of the eight master integrals through order $\epsilon^{-1}$ are cut by half; in the two-loop double pentagon, the divergent parts of 108 master integrals reduce to 127 master coefficients drawn from subsectors.
  • The method works at three loops in the tested family, where the divergent parts of all nine master integrals reduce to ten master coefficients, and the automated package makes the reduction turnkey once a family is defined.

Reading between the lines

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

  • If the $\bar{R}$-operation preserves infrared and collinear safety for all families, the number of master coefficients at each order in $\epsilon$ becomes a well-defined invariant of an integral family; computing this number across a set of standard topologies would provide a clean stress test of the method.
  • The appearance of vacuum-like ultraviolet-counterterm families among the master coefficients hints that the divergent structure of $\epsilon$-expanded integrals may be controlled by a small universal set of vacuum integrals, though the paper's examples do not prove this.
  • The same mass-shift idea could be transplanted to local subtraction schemes for physical cross sections, where one also needs to remove ultraviolet poles without disturbing soft and collinear safety; the authors do not explore that application.
  • A natural extension would be to apply the reduction to a family with known analytic results, such as a non-planar two-loop box, and compare the predicted master coefficients with direct integration; this would test the completeness of the generated constraints.
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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 / 5 minor

Summary. The manuscript proposes a systematic method to reduce the epsilon-expanded master integrals of a Feynman integral family to a smaller set of expansion coefficients. The strategy is to construct locally finite integrals by raising the spacetime dimension (which suppresses infrared and collinear divergences) and then subtracting ultraviolet divergences with a new operation called the R-bar operation, designed to preserve IR and CO safety. These finite integrals are expressed in terms of 4-dimensional master integrals using integration-by-parts identities and dimension recurrence relations, yielding constraints of the form sum_i c_i M_i = O(epsilon^0). Expanding in epsilon gives linear relations among master-integral expansion coefficients. The method is illustrated on a one-loop pentagon, a two-loop double box, a two-loop double pentagon, and a three-loop example, with the number of independent expansion coefficients substantially reduced. An open-source Mathematica package, FFI, is provided. The central technical claim is the R-bar construction in Section III.C, which asserts that the UV counterterms are IR and CO finite whenever the original integrand is.

Significance. If the central construction is valid, the paper offers a practical and general route to generating finite/evanescent relations among multiloop Feynman integrals, which could meaningfully reduce the cost of high-order perturbative computations. The explicit reductions in the double-box and three-loop examples, together with the numerical cross-check of the three-loop relations, demonstrate that the method works in nontrivial settings. The open-source package FFI is a useful deliverable. However, the proof of the key R-bar property is only sketched, and a technical issue with the dimension-recurrence formula for the five-point example is not addressed; these points prevent the paper from being fully convincing in its current form.

major comments (3)
  1. [III.C, Eqs. (13)-(17)] The proof that the counterterm G_r is IR and CO finite rests on two unproved assertions. First, the statement that 'IR and CO regions of G_r must inherit from those of G' is asserted without argument; the mass substitution (13) modifies propagators in a way that could in principle create new soft or collinear scaling regions in G_r that are not regions of G. Second, the claim that each homogeneous component φ_k independently satisfies φ_k = O(λ^a) 'because there cannot be cancellation between φ_k's' is not justified: the power-counting behavior applies to the full numerator N in region R, but the expansion (15) mixes φ_k with denominator factors, so individual terms g_k need not share the same scaling. The authors should either give a complete proof of these two points or verify the IR/CO finiteness of the counterterms numerically for every seed integral used in the main examples; Appendix A only checks the two-loop sunrise, which is too simple to test the double-box and double-pentagon cases where the subtraction is more intricate. This gap is load-bearing because all subsequent constraints assume the constructed integrals are actually finite.
  2. [III.D, Eq. (22), and IV.C] The dimension-recurrence relation (22) contains the Gram determinant V(p_1,...,p_E) in the denominator. For the double-pentagon example of Section IV.C, E=5 and the external momenta satisfy p_1+...+p_5=0 with four-dimensional kinematics, so V(p_1,...,p_5) vanishes identically. The manuscript does not explain how the dimension shift from D=6-2ε to D=4-2ε is implemented for this family. If Eq. (22) is used directly, the denominator is zero; if a different procedure is used, such as taking a limit or using an alternative recurrence, it should be spelled out. Since the double pentagon is one of the principal examples supporting the method, this technical point needs to be resolved or explicitly addressed.
  3. [IV.C and VI] The paper claims that the methods reduce the epsilon-expanded master integrals to a 'minimal basis', but it does not prove that the finite integrals generated by scanning a limited set of dimensions, ranks, and dot orders are sufficient to obtain the true minimal basis. The statement in Section IV.C that 'there are 108 MIs for the double pentagon' and that the divergent parts reduce to '127 MCs' describes the result of a particular finite scan, not a proof of completeness. Furthermore, the double-pentagon analysis is performed at a single numerical kinematic point; unless it is shown that the rank of the constraint system is independent of the point (or that the chosen point is generic), the reported MC counts could be special. The authors should either prove completeness of the generated constraints for the examples or soften the claims of minimality to 'minimal among the constraints generated by this procedure'.
minor comments (5)
  1. [IV.A, after Eq. (24)] The text contains the typo 'cen deduce' instead of 'can deduce' in the sentence following Eq. (24).
  2. [III.C, paragraph after Eq. (13)] The phrase 'cij's can be choses to satisfy' contains a grammatical error; it should read 'can be chosen to satisfy'.
  3. [V, second paragraph] The text reads 'FFI alse provides methods', which should be 'FFI also provides methods'.
  4. [Author affiliations] The affiliation line contains the typo 'Shanghai Jiao Tong Univeristy', which should be 'University'.
  5. [II, after Eq. (7)] The notation 'η−κ lniη' is printed without a superscript on the logarithm; this should be typeset as 'η^{-κ} ln^i η' or similar to be unambiguous.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the finite-integral constraints are derived by construction plus IBP/DRR identities, not by fitting or by definition.

full rationale

The paper's derivation chain is not circular. It constructs IR/CO-finite integrals by raising the spacetime dimension and then applies a UV-subtraction operation (the R-bar operation) to obtain locally finite integrals. These finite integrals are independent objects whose finiteness is asserted from power-counting and region analysis, not defined to equal the output relation. The later steps use IBP and DRR to express the finite higher-dimensional integrals as linear combinations of master integrals in 4-2epsilon dimensions; imposing that the left-hand side has no 1/epsilon poles gives genuine constraints on the epsilon-expanded master integrals. No fitted parameter is renamed as a prediction, and no equation is defined in terms of the quantity it is supposed to determine. The self-citations to Blade and related IBP packages (refs. [13,14,63]) are tool usage, not load-bearing premises of the reduction theorem. The most delicate point, the assertion in Sec. III.C that IR/CO regions of the counterterm G_r inherit from G and that each homogeneous numerator piece obeys the same power suppression, is a mathematical gap or correctness risk rather than a circular step: a failure there would invalidate the method, but it would not make the argument circular. The paper also checks an explicit case in Appendix A and reproduces the known one-loop relation, providing external consistency. Overall, the central reduction claim is self-contained with respect to the inputs of the construction.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

The free parameters are unphysical regulators and algorithmic degrees of freedom, not fitted physics constants. The axioms are standard tools of multiloop computation plus two ad hoc completeness/representativeness assumptions that underpin the 'minimal basis' and double-pentagon results.

free parameters (2)
  • auxiliary mass m in R-bar counterterms = m^2; appears in counterterms, set to 1 in I_UV definitions (Eqs. (26), (32))
    Introduced ad hoc to make counterterms Gr non-scaleless and IR/CO finite. The master-coefficient basis includes massive vacuum integrals, so the concrete reduction coefficients can depend on this unphysical regulator scale.
  • substitution coefficients c_ij = c_ii=1; c_12=c_21=-1/2 in Appendix A
    The free coefficients in the momentum substitution (13) are chosen by hand; the paper fixes them to demonstrate the method but does not give a canonical general prescription.
assumptions (6)
  • domain assumption Strategy of regions correctly identifies all IR/CO singular regions of the mass-regulated integral I(eta).
    Used in Sec. III.A to enumerate regions with asy2 and to judge when higher-dimensional integrals become IR/CO finite; a missed region would invalidate the finiteness claims.
  • standard math Power-counting theorem guarantees IR/CO finiteness in sufficiently high dimension and UV finiteness when superficial degrees of divergence are negative.
    Invoked in Sec. III.A-B to decide when a FI is locally finite after increasing dimension and applying R-bar subtraction.
  • standard math Dimensional recurrence relations (Tarasov) are valid and sufficient to convert master integrals from higher even dimensions to D=4-2 epsilon.
    Used in Sec. III.D and in the examples (e.g., Eqs. (28)-(29), Appendix B) to map 6-dimensional finite integrals back to 4-2 epsilon MIs.
  • domain assumption Integration-by-parts identities generate a complete finite basis of master integrals for each family.
    Assumed in Sec. II and used throughout to reduce arbitrary FIs to MIs before applying the new constraints.
  • ad hoc to paper The finite set of integrals scanned (by dimension, rank, and dot order) yields a complete set of constraints, so the resulting master-coefficient basis is minimal.
    The paper scans integrals with rank up to 3 and dots up to 2 in examples and concludes 'all' divergent parts reduce; no completeness proof is supplied, although the Abstract and Sec. VI call the result a 'minimal basis'.
  • ad hoc to paper For the double pentagon example, one numerical kinematic point is representative of the generic rank of the constraint system.
    Sec. IV.C states finite integral relations were generated 'at a numerical point'; the paper does not show a second point or a rational reconstruction proof.

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Pith. "Pith review of Reduction of $\epsilon$-expanded Feynman integrals." pith.science (2026). https://pith.science/paper/TERQJZ33

@misc{pith2026250416766,
  author       = {Pith},
  title        = {Pith review of: Reduction of $\epsilon$-expanded Feynman integrals},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TERQJZ33}},
  note         = {Machine review of arXiv:2504.16766}
}
abstract

Since Feynman integrals (FIs) at higher spacetime dimensions are free of infrared and collinear divergence--and their ultraviolet divergences can be systematically subtracted--this allows us to construct a wide range of locally finite Feynman integrals. Especially, we propose a method named $\bar{R}$-operation to subtract out ultraviolet divergences that at the same time preserves infrared and collinear safety of the original FI. By expressing these locally finite FIs in terms of master integrals and imposing constraints on their $\epsilon$-expanded forms, we reduce the $\epsilon$-expanded master integrals to a minimal basis. We provide an automated package to identify such constraints, offering a tool useful for high-order perturbative computations.

Figures

Figures reproduced from arXiv: 2504.16766 by the authors.

Figure 1
Figure 1. FIG. 1: One-loop pentagon [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Two-loop double box [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: Two-loop double pentagon [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (2 more)
Figure 4
Figure 4. Figure 4: FIG. 4: A simple 3-loop example [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Two-loop sunrise [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]

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Works this paper leans on

65 extracted references · 21 canonical work pages · cited by 1 Pith paper

  1. [1]

    Consequently, we find that the divergent parts of all MIs can be reduced to 14 MCs: I(1, 1, 0, 0, 1, 1, 1, 0, 0)−1, I(1, 1, 1, 0, 1, 0, 1, 0, 0)−1,0, I(1, 0, 1, 0, 1, 0, 1, 0, 0)0,1, I(1, 0, 1, 1, 0, 1, 0, 0, 0)1, I(0, 1, 0, 0, 1, 0, 1, 0, 0)0,1,2, I(1, 0, 0, 0, 0, 1, 1, 0, 0)1,2, IUV 1,−2,−1,IUV 4,−1, (31) where the integrals corresponding to the last 2 ...

    work page

  2. [2]

    F. V. Tkachov, A theorem on analytical calculability of 4-loop renormalization group functions , Nucl. Phys. 100 (1981) 65–68 [ InSPIRE]

    work page 1981

  3. [3]

    A. V. Smirnov and A. V. Petukhov, The Number of Master Integrals is Finite , Lett. Math. Phys. 97 (2011) 37–44 [arXiv:1004.4199] [InSPIRE]. 10

    arXiv 2011

  4. [4]

    Laporta, High-precision calculation of multiloop Feynman integrals by difference equations, Int

    S. Laporta, High-precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087–5159 [ hep-ph/0102033] [InSPIRE]

    arXiv 2000

  5. [5]

    K. G. Chetyrkin and F. V. Tkachov, Integration by parts: The algorithm to calculate β-functions in 4 loops , Nucl. Phys. B 192 (1981) 159–204 [ InSPIRE]

    1981

  6. [6]

    R. M. Schabinger, A New Algorithm For The Generation Of Unitarity-Compatible Integration By Parts Relations, JHEP 01 (2012) 077 [arXiv:1111.4220] [InSPIRE]

    arXiv 2012

  7. [7]

    Gluza, K

    J. Gluza, K. Kajda, and D. A. Kosower, Towards a Basis for Planar Two-Loop Integrals , Phys. Rev. D 83 (2011) 045012 [ arXiv:1009.0472] [InSPIRE]

    arXiv 2011

  8. [8]

    R. N. Lee, LiteRed 1.4: a powerful tool for reduction of multiloop integrals, J. Phys. Conf. Ser. 523 (2014) 012059 [arXiv:1310.1145] [InSPIRE]

    arXiv 2014

Show all 65 references
  1. [9]

    von Manteuffel and C

    A. von Manteuffel and C. Studerus, Reduze 2 - Distributed Feynman Integral Reduction, [arXiv:1201.4330] [InSPIRE]

    arXiv

  2. [10]

    K. J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry , Phys. Rev. D 93 (2016) 041701 [ arXiv:1511.01071] [InSPIRE]

    2016 arXiv

  3. [11]

    von Manteuffel and R

    A. von Manteuffel and R. M. Schabinger, A novel approach to integration by parts reduction , Phys. Lett. B 744 (2015) 101–104 [ arXiv:1406.4513] [InSPIRE]

    2015 arXiv

  4. [12]

    Mastrolia and S

    P. Mastrolia and S. Mizera, Feynman Integrals and Intersection Theory, JHEP 02 (2019) 139 [arXiv:1810.03818] [InSPIRE]

    2019 arXiv

  5. [13]

    Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP 12 (2016) 030 [arXiv:1608.01902] [InSPIRE]

    T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP 12 (2016) 030 [arXiv:1608.01902] [InSPIRE]

    2016 arXiv

  6. [14]

    X. Guan, X. Liu, and Y.-Q. Ma, Complete reduction of integrals in two-loop five-light-parton scattering amplitudes, Chin. Phys. C 44 (2020) 093106 [arXiv:1912.09294] [InSPIRE]

    2020 arXiv

  7. [15]

    Liu and Y.-Q

    X. Liu and Y.-Q. Ma, Determining arbitrary Feynman integrals by vacuum integrals, Phys. Rev. D 99 (2019) 071501 [arXiv:1801.10523] [InSPIRE]

    2019 arXiv

  8. [16]

    Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs , JHEP 07 (2019) 031 [ arXiv:1905.08019] [InSPIRE]

    T. Peraro, FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs , JHEP 07 (2019) 031 [ arXiv:1905.08019] [InSPIRE]

    2019 arXiv

  9. [17]

    Klappert and F

    J. Klappert and F. Lange, Reconstructing rational functions with FireFly , Comput. Phys. Commun. 247 (2020) 106951 [ arXiv:1904.00009] [InSPIRE]

    2020 arXiv

  10. [18]

    Y. Wang, Z. Li, and N. Ul Basat, Direct reduction of multiloop multiscale scattering amplitudes , Phys. Rev. D 101 (2020) 076023 [ arXiv:1901.09390] [InSPIRE]

    2020 arXiv

  11. [19]

    Frellesvig, F

    H. Frellesvig, F. Gasparotto, S. Laporta, M. K. Mandal, P. Mastrolia, L. Mattiazzi, and S. Mizera, Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers , JHEP 05 (2019) 153 [arXiv:1901.11510] [InSPIRE]

    2019 arXiv

  12. [20]

    Klappert, F

    J. Klappert, F. Lange, P. Maierh¨ ofer, and J. Usovitsch, Integral reduction with Kira 2.0 and finite field methods , Comput. Phys. Commun. 266 (2021) 108024 [arXiv:2008.06494] [InSPIRE]

    2021 arXiv

  13. [21]

    A. V. Smirnov and F. S. Chukharev, FIRE6: Feynman Integral REduction with modular arithmetic , Comput. Phys. Commun. 247 (2020) 106877 [ arXiv:1901.07808] [InSPIRE]

    2020 arXiv

  14. [22]

    Heller and A

    M. Heller and A. von Manteuffel, MultivariateApart: Generalized partial fractions, Comput. Phys. Commun. 271 (2022) 108174 [ arXiv:2101.08283] [InSPIRE]

    2022 arXiv

  15. [23]

    Boehm, M

    J. Boehm, M. Wittmann, Z. Wu, Y. Xu, and Y. Zhang, IBP reduction coefficients made simple , JHEP 12 (2020) 054 [ arXiv:2008.13194] [InSPIRE]

    2020 arXiv

  16. [24]

    C. G. Bollini and J. J. Giambiagi, Dimensional Renormalization: The Number of Dimensions as a Regularizing Parameter, Nuovo Cim. B 12 (1972) 20–26 [InSPIRE]

    1972

  17. [25]

    Bendle, J

    D. Bendle, J. Boehm, M. Heymann, R. Ma, M. Rahn, L. Ristau, M. Wittmann, Z. Wu, H. Xu, and Y. Zhang, pfd-parallel, a Singular/GPI-Space package for massively parallel multivariate partial fractioning , Comput. Phys. Commun. 294 (2024) 108942 [arXiv:2104.06866] [InSPIRE]

    2024 arXiv

  18. [26]

    Z. Bern, L. J. Dixon, and D. A. Kosower, Dimensionally regulated pentagon integrals, Nucl. Phys. B 412 (1994) 751–816 [hep-ph/9306240] [InSPIRE]

    1994 arXiv

  19. [27]

    ’t Hooft and M

    G. ’t Hooft and M. J. G. Veltman, Regularization and Renormalization of Gauge Fields , Nucl. Phys. B 44 (1972) 189–213 [ InSPIRE]

    1972

  20. [28]

    Yan and X

    K. Yan and X. Zhang, Three-Point Energy Correlator in N=4 Supersymmetric Yang-Mills Theory , Phys. Rev. Lett. 129 (2022) 021602 [ arXiv:2203.04349] [InSPIRE]

    2022 arXiv

  21. [29]

    J. M. Henn, E. Sokatchev, K. Yan, and A. Zhiboedov, Energy-energy correlation in N=4 super Yang-Mills theory at next-to-next-to-leading order , Phys. Rev. D 100 (2019) 036010 [ arXiv:1903.05314] [InSPIRE]

    2019 arXiv

  22. [30]

    K. G. Wilson and J. B. Kogut, The Renormalization group and the epsilon expansion , Phys. Rept. 12 (1974) 75–199 [InSPIRE]

    1974

  23. [31]

    Chicherin, I

    D. Chicherin, I. Moult, E. Sokatchev, K. Yan, and Y. Zhu, Collinear limit of the four-point energy correlator in N=4 supersymmetric Yang-Mills theory , Phys. Rev. D 110 (2024) L091901 [ arXiv:2401.06463] [InSPIRE]

    2024 arXiv

  24. [32]

    Altarelli and G

    G. Altarelli and G. Parisi, Asymptotic Freedom in Parton Language, Nucl. Phys. B 126 (1977) 298–318 [InSPIRE]

    1977

  25. [33]

    V. N. Gribov and L. N. Lipatov, Deep inelastic e p scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438–450 [ InSPIRE]

    1972

  26. [34]

    J. M. Henn, C. Milloy, and K. Yan, A new method for calculating the soft anomalous dimension matrix for massive particle scattering, JHEP 04 (2024) 117 [arXiv:2310.10145] [InSPIRE]

    2024 arXiv

  27. [35]

    Y. L. Dokshitzer, Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics. , Sov. Phys. JETP 46 (1977) 641–653 [ InSPIRE]

    1977

  28. [36]

    Caron-Huot and J

    S. Caron-Huot and J. M. Henn, Iterative structure of finite loop integrals, JHEP 06 (2014) 114 [arXiv:1404.2922] [InSPIRE]

    2014 arXiv

  29. [37]

    Br¨ user, C

    R. Br¨ user, C. Dlapa, J. M. Henn, and K. Yan, Full Angle Dependence of the Four-Loop Cusp Anomalous Dimension in QED , Phys. Rev. Lett. 126 (2021) 021601 [arXiv:2007.04851] [InSPIRE]

    2021 arXiv

  30. [38]

    Dubovyk, A

    I. Dubovyk, A. Freitas, J. Gluza, K. Grzanka, M. Hidding, and J. Usovitsch, Evaluation of multiloop 11 multiscale Feynman integrals for precision physics , Phys. Rev. D 106 (2022) L111301 [ arXiv:2201.02576] [InSPIRE]

    2022 arXiv

  31. [39]

    von Manteuffel, E

    A. von Manteuffel, E. Panzer, and R. M. Schabinger, A quasi-finite basis for multi-loop Feynman integrals , JHEP 02 (2015) 120 [ arXiv:1411.7392] [InSPIRE]

    2015 arXiv

  32. [40]

    Hidding, DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions , Comput

    M. Hidding, DiffExp, a Mathematica package for computing Feynman integrals in terms of one-dimensional series expansions , Comput. Phys. Commun. 269 (2021) 108125 [ arXiv:2006.05510] [InSPIRE]

    2021 arXiv

  33. [41]

    Hidding and J

    M. Hidding and J. Usovitsch, Feynman parameter integration through differential equations, Phys. Rev. D 108 (2023) 036024 [ arXiv:2206.14790] [InSPIRE]

    2023 arXiv

  34. [42]

    Armadillo, R

    T. Armadillo, R. Bonciani, S. Devoto, N. Rana, and A. Vicini, Evaluation of Feynman integrals with arbitrary complex masses via series expansions , Comput. Phys. Commun. 282 (2023) 108545 [arXiv:2205.03345] [InSPIRE]

    2023 arXiv

  35. [43]

    Maierh¨ ofer, J

    P. Maierh¨ ofer, J. Usovitsch, and P. Uwer,Kira—A Feynman integral reduction program, Comput. Phys. Commun. 230 (2018) 99–112 [ arXiv:1705.05610] [InSPIRE]

    2018 arXiv

  36. [44]

    O. V. Tarasov, Connection between Feynman integrals having different values of the space-time dimension , Phys. Rev. D 54 (1996) 6479–6490 [ hep-th/9606018] [InSPIRE]

    1996 arXiv

  37. [45]

    Borowka, G

    S. Borowka, G. Heinrich, S. Jahn, S. P. Jones, M. Kerner, and J. Schlenk, A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec , Comput. Phys. Commun. 240 (2019) 120–137 [arXiv:1811.11720] [InSPIRE]

    2019 arXiv

  38. [46]

    de la Cruz, D

    L. de la Cruz, D. A. Kosower, and P. P. Novichkov, Finite Integrals from Feynman Polytopes , [arXiv:2410.18014] [InSPIRE]

    arXiv

  39. [47]

    Gambuti, D

    G. Gambuti, D. A. Kosower, P. P. Novichkov, and L. Tancredi, Finite Feynman Integrals, [arXiv:2311.16907] [InSPIRE]

    arXiv

  40. [48]

    K. G. Chetyrkin and V. A. Smirnov, R* OPERATION CORRECTED, Phys. Lett. B 144 (1984) 419–424 [InSPIRE]

    1984

  41. [49]

    K. G. Chetyrkin and F. V. Tkachov, INFRARED R OPERATION AND ULTRAVIOLET COUNTERTERMS IN THE MS SCHEME , Phys. Lett. B 114 (1982) 340–344 [ InSPIRE]

    1982

  42. [50]

    Beneke and V

    M. Beneke and V. A. Smirnov, Asymptotic expansion of Feynman integrals near threshold, Nucl. Phys. B 522 (1998) 321–344 [ hep-ph/9711391] [InSPIRE]

    1998 arXiv

  43. [51]

    Herzog and B

    F. Herzog and B. Ruijl, The R∗-operation for Feynman graphs with generic numerators , JHEP 05 (2017) 037 [arXiv:1703.03776] [InSPIRE]

    2017 arXiv

  44. [52]

    L. D. Landau, On analytic properties of vertex parts in quantum field theory , Nucl. Phys. 13 (1959) 181–192 [InSPIRE]

    1959

  45. [53]

    J. D. Bjorken, Experimental tests of Quantum electrodynamics and spectral representations of Green’s functions in perturbation theory . PhD thesis, Stanford U., 1959 [ InSPIRE]

    1959

  46. [54]

    Jantzen, A

    B. Jantzen, A. V. Smirnov, and V. A. Smirnov, Expansion by regions: revealing potential and Glauber regions automatically, Eur. Phys. J. C 72 (2012) 2139 [arXiv:1206.0546] [InSPIRE]

    2012 arXiv

  47. [55]

    Nakanishi, Ordinary and anomalous thresholds in perturbation theory, Progress of Theoretical Physics 22 (07, 1959) 128–144

    N. Nakanishi, Ordinary and anomalous thresholds in perturbation theory, Progress of Theoretical Physics 22 (07, 1959) 128–144

    1959

  48. [56]

    G. F. Sterman, Mass Divergences in Annihilation Processes. 1. Origin and Nature of Divergences in Cut Vacuum Polarization Diagrams, Phys. Rev. D 17 (1978) 2773 [InSPIRE]

    1978

  49. [57]

    Anastasiou and G

    C. Anastasiou and G. Sterman, Removing infrared divergences from two-loop integrals, JHEP 07 (2019) 056 [arXiv:1812.03753] [InSPIRE]

    2019 arXiv

  50. [58]

    S. B. Libby and G. F. Sterman, Mass Divergences in Two Particle Inelastic Scattering , Phys. Rev. D 18 (1978) 4737 [ InSPIRE]

    1978

  51. [59]

    S. B. Libby and G. F. Sterman, Jet and Lepton Pair Production in High-Energy Lepton-Hadron and Hadron-Hadron Scattering, Phys. Rev. D 18 (1978) 3252 [InSPIRE]

    1978

  52. [60]

    Weinberg, High-energy behavior in quantum field theory, Phys

    S. Weinberg, High-energy behavior in quantum field theory, Phys. Rev. 118 (1960) 838–849 [ InSPIRE]

    1960

  53. [61]

    F. J. Dyson, The S matrix in quantum electrodynamics , Phys. Rev. 75 (1949) 1736–1755 [ InSPIRE]

    1949

  54. [62]

    Singular 4-4-0 — A computer algebra system for polynomial computations

    W. Decker, G.-M. Greuel, G. Pfister, and H. Sch¨ onemann, “Singular 4-4-0 — A computer algebra system for polynomial computations.” http://www.singular.uni-kl.de, 2024

    2024

  55. [63]

    Level" -> n,

    for IBP reduction, and FiniteFlow [16] to solve lin- ear equations. The full package can be downloaded from https://github.com/Tanao-pku/FFI To use FFI, the first step is to use DefineFamily to define a family object: DefineFamily[ Family, Propagators, ISPs, PropagatorMomenta,...

  56. [64]

    R. N. Lee, Space-time dimensionality D as complex variable: Calculating loop integrals using dimensional recurrence relation and analytical properties with respect to D, Nucl. Phys. B 830 (2010) 474–492 [arXiv:0911.0252] [InSPIRE]

    2010 arXiv

  57. [65]

    X. Guan, X. Liu, Y.-Q. Ma, and W.-H. Wu, Blade: A package for block-triangular form improved Feynman integrals decomposition, [arXiv:2405.14621] [InSPIRE]

    arXiv

Pith tools

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