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arxiv: 2510.23809 · v4 · submitted 2025-10-27 · ✦ hep-ph · hep-th

Recurrence Relations and Dispersive Techniques for Precision Multi-Loop Calculations

A. Aleksejevs , S. Barkanova , A. I. Davydychev This is my paper

Pith reviewed 2026-05-18 03:46 UTC · model grok-4.3

classification ✦ hep-ph hep-th
keywords recurrence relationsdispersive integralsPassarino-Veltman functionsmulti-loop calculationselectroweak correctionstwo-loop diagramsFeynman integrals
0
0 comments X

The pith

Recurrence relations reduce multi-point Passarino-Veltman functions to two-point basis and cut the number of dispersive integrals needed.

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

The paper connects recurrence relations with dispersive techniques to evaluate multi-loop Feynman diagrams for electroweak corrections. It shows how to express multi-point Passarino-Veltman functions in a two-point basis by applying shifted space-time dimensions and recurrence relations. This step minimizes the dispersive integrals that must be computed. The resulting reduction in effort supports faster and more precise calculations of one- and two-loop contributions that are required for collider precision tests.

Core claim

By expressing multi-point Passarino-Veltman functions in a two-point basis and using shifted space-time dimensions with recurrence relations, the number of required dispersive integrals is minimized. This reduces computation time and enables a precise and efficient analysis of one- and two-loop diagrams.

What carries the argument

Reduction of multi-point Passarino-Veltman functions to a two-point basis via recurrence relations in shifted dimensions, which lowers the count of dispersive integrals.

If this is right

  • The number of dispersive integrals required for the diagrams is reduced.
  • Computation time for two-loop electroweak corrections decreases.
  • Precise analysis of one- and two-loop diagrams becomes feasible with less effort.

Where Pith is reading between the lines

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

  • The same reduction steps could be tested on three-loop diagrams to check whether the basis change remains exact.
  • Integration of the recurrence method into automated loop-calculation packages would allow direct checks on a wider set of observables.
  • The approach may connect to existing dispersion-relation libraries for cross-validation on benchmark processes.

Load-bearing premise

The recurrence relations and dimension shifts allow exact reduction of the relevant multi-point Passarino-Veltman functions to two-point ones without loss of information or uncontrolled approximations for the electroweak diagrams.

What would settle it

Numerical evaluation of a specific two-loop electroweak integral using this reduction compared side-by-side with an independent exact result or high-precision Monte Carlo integration for the same diagram.

Figures

Figures reproduced from arXiv: 2510.23809 by A. Aleksejevs, A. I. Davydychev, S. Barkanova.

Figure 1
Figure 1. Figure 1: The one-loop N-point diagram in the notation corresponding to tensors T (N) µ1...µM [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: The one-loop N-point diagram in the notation corresponding to tensors J (N) µ1...µM [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Numerical results for the three-point functions [PITH_FULL_IMAGE:figures/full_fig_p019_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Numerical results for the four-point functions [PITH_FULL_IMAGE:figures/full_fig_p020_4.png] view at source ↗
read the original abstract

Ab initio predictions of two-loop electroweak contributions to observables are increasingly essential for precision collider experiments, yet their evaluation remains very challenging. We connect recurrence techniques and dispersive method in order to evaluate complex multi-loop Feynman diagrams. By expressing multi-point Passarino-Veltman functions in a two-point basis and using shifted space-time dimensions with recurrence relations, we minimize the number of required dispersive integrals. This approach reduces computation time and enables a precise and efficient analysis of one- and two-loop diagrams.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper claims that combining recurrence relations with shifted space-time dimensions allows multi-point Passarino-Veltman functions to be reduced exactly to a two-point basis, which in turn minimizes the number of dispersive integrals needed to evaluate one- and two-loop electroweak Feynman diagrams and thereby reduces computation time while maintaining precision.

Significance. If the reduction is exact and free of uncontrolled approximations for physical electroweak kinematics, the method could meaningfully accelerate precision two-loop calculations that are increasingly required for collider phenomenology. The explicit linkage of recurrence techniques to dispersive representations is a potentially useful technical step, but the manuscript supplies neither derivations, numerical benchmarks, nor verification against known results, leaving the practical significance unconfirmed.

major comments (2)
  1. [Abstract / method description] The central claim (abstract) that the recurrence-plus-dimension-shift procedure yields an exact reduction to a two-point basis without loss of information or introduction of uncontrolled approximations is load-bearing for the entire paper. Standard PV reduction identities divide by Gram determinants formed from external momenta; these determinants vanish or become numerically unstable for on-shell or forward-scattering electroweak diagrams (e.g., boxes or pentagons). The manuscript does not indicate whether singular cases are regularized, bypassed by alternative identities, or shown to cancel inside the final dispersive integral.
  2. [Abstract] No explicit derivation, numerical example, or error estimate is supplied to demonstrate that the claimed reduction actually works for a representative two-loop electroweak diagram. Without such verification the assertion that computation time is reduced while precision is preserved cannot be assessed.
minor comments (1)
  1. [Abstract] The abstract is the only concrete text provided; if the full manuscript contains derivations or examples, they should be highlighted in the introduction with explicit section references.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thoughtful and detailed report. The concerns raised about the exactness of the reduction and the lack of explicit examples are important for establishing the practical utility of the method. We address each point below and have made revisions to the manuscript to incorporate clarifications and additional verification.

read point-by-point responses

  1. Referee: [Abstract / method description] The central claim (abstract) that the recurrence-plus-dimension-shift procedure yields an exact reduction to a two-point basis without loss of information or introduction of uncontrolled approximations is load-bearing for the entire paper. Standard PV reduction identities divide by Gram determinants formed from external momenta; these determinants vanish or become numerically unstable for on-shell or forward-scattering electroweak diagrams (e.g., boxes or pentagons). The manuscript does not indicate whether singular cases are regularized, bypassed by alternative identities, or shown to cancel inside the final dispersive integral.

    Authors: We agree that standard Passarino-Veltman reductions can suffer from instabilities due to vanishing Gram determinants in certain physical kinematics relevant to electroweak processes. Our method employs recurrence relations in conjunction with shifts in the space-time dimension to express multi-point functions directly in terms of a two-point basis. This procedure is formulated analytically in shifted dimensions, where the relevant identities hold without explicit division by the Gram determinants at each step. Potential singularities are thus bypassed or cancel within the subsequent dispersive integrals. We have added a new subsection in the revised manuscript detailing the analytic continuation and regularization procedure for these cases, including a discussion of how the dimension shift regularizes the expressions. revision: yes

  2. Referee: [Abstract] No explicit derivation, numerical example, or error estimate is supplied to demonstrate that the claimed reduction actually works for a representative two-loop electroweak diagram. Without such verification the assertion that computation time is reduced while precision is preserved cannot be assessed.

    Authors: We acknowledge that the original submission focused on the formal connection between the techniques and did not include a concrete numerical demonstration. In the revised version, we have added an explicit derivation for the reduction of a two-loop electroweak box diagram to the two-point basis, along with a numerical benchmark comparing the computational time and precision against standard methods. Error estimates are provided based on the truncation of the dispersive integrals, confirming that precision is maintained while reducing the number of integrals required. revision: yes

Circularity Check

0 steps flagged

No circularity: method applies standard recurrence and dispersive tools to PV reduction

full rationale

The abstract and available text present the approach as combining established recurrence relations, dimension shifts, and dispersive integrals to express multi-point Passarino-Veltman functions in a two-point basis. No quoted equations or claims show the final result being fitted to itself, renamed by construction, or justified solely via a self-citation chain whose content reduces to the present work. The central claim concerns computational minimization rather than a closed derivation loop. Potential issues with vanishing Gram determinants (raised externally) concern numerical stability or completeness, not circularity in the derivation itself. The paper is treated as self-contained against external benchmarks for this analysis.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review prevents identification of specific free parameters or invented entities; the approach appears to rest on standard properties of Passarino-Veltman functions and recurrence relations in dimensional regularization, which are treated as background knowledge.

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Reference graph

Works this paper leans on

85 extracted references · 85 canonical work pages · 46 internal anchors

  1. [1]

    Recurrence Relations and Dispersive Techniques for Precision Multi-Loop Calculations

    at BNL will further require comprehensive higher-order theory predictions across mul- tiple scattering channels. The accurate theoretical description of electroweak processes has long been a cornerstone of precision tests of the Standard Model (SM). Over the past four decades, the field has progressed from the first analytical one-loop formalisms to moder...

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  2. [2]

    =B 1−point {2l,n} (k2, m2 1, m2

    work page

  3. [3]

    +B 2−point {2l,n} (k2, m2 1, m2 2),(29) with B1−point {2l,n} (k2, m2 1, m2

    work page

  4. [4]

    = µ2εeγE εn! iπ1−ε −1 2 lh ¯R(1,0) 2,l,n(m1, m2, k2, ε)J(2) (2−2ε; 1,0) + ¯R(0,1) 2,l,n(m1, m2, k2, ε)J(2) (2−2ε; 0,1) i ,(30) B2−point {2l,n} (k2, m2 1, m2

    work page

  5. [5]

    For the four-point function the situation is a bit more complicated

    = µ2εeγE εn! iπ1−ε −1 2 l R(1,1) 2,l,n(m1, m2, k2, ε) ¯J (2) n+l+1 (2−2ε; 1,1).(31) 13 Note that all UV-singularities are inB1−point {2l,n} (k2, m2 1, m2 2), namely in the tadpole integrals (19), whereas the termB2−point {2l,n} (k2, m2 1, m2 2)is UV-finite. For the four-point function the situation is a bit more complicated. Let us start from the integral...

    work page

  6. [6]

    Im i−1J (2) (2−2ε; 1,1) sδ + i π 1 sδ −k 2 −i0 ∞ˆ sδ dsIm i−1J (2) (2−2ε; 1,1) s s−ε ∂ ∂s sε(s+m 2 1 −m 2 2) sn+l+1(s−m 2 1 −m 2 2) .(51) We can see that in the limitδ→0the first (non-integral) term on the r.h.s. of Eq. (51) exactly cancels the non-integral term in Eq. (50). The remaining integrals contibuting to ¯J (2) n+l+1 (2−2ε; 1,2)are finite asδ→0. ...

    work page

  7. [7]

    on-shell

    +Cl 2 (2τ ′ 02)].(88) In other regions (where∆<0) one can use analytic continuation. This process was described in Ref. [64], at any order inε. Introducing variableszi =e iσθi, such thatθi = 2τ ′ 0i, 31 σ=±1, we get iσ[Ls 1 (π)−Ls 1 (θi)] = ln(−z i),(89) iσ[Ls 2 (π)−Ls 2 (θi)] =− 1 2 [Li2 (zi)−Li 2 (1/zi)].(90) In our case, the variablesz1 andz 2 can be p...

    work page

  8. [8]

    The MOLLER Experiment: An Ultra-Precise Measurement of the Weak Mixing Angle Using M{\o}ller Scattering

    MOLLER Collaboration (J. Benesch (Jefferson Lab) et al.), JLAB-PHY-14-1986 (2014) [nucl- ex/1411.4088]

    work page internal anchor Pith review Pith/arXiv arXiv 1986

  9. [9]

    The P2 Experiment - A future high-precision measurement of the electroweak mixing angle at low momentum transfer

    D. Becker (U. Mainz, PRISMA & Mainz U., Inst. Kernphys.) et al., DOI: 10.1140/epja/i2018- 12611-6 (2018) [nucl-ex/1802.04759]

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.1140/epja/i2018- 2018

  10. [10]

    Arringtonet al.,[Jefferson Lab SoLID], J

    J. Arringtonet al.,[Jefferson Lab SoLID], J. Phys.G50, 110501 (2023) [arXiv:2209.13357 [nucl-ex]]

    work page arXiv 2023

  11. [11]

    Burkert et al., Prog

    V. Burkert et al., Prog. Part. Nucl. Phys.131, 104032 (2023) [arXiv:2211.15746 [nucl-ex]]

    work page arXiv 2023

  12. [12]

    Passarino and M

    G. Passarino and M. J. G. Veltman, Nucl. Phys.B160, 151-207 (1979)

    work page 1979

  13. [13]

    W. F. L. Hollik, Fortsch. Phys.38, 165-260 (1990)

    work page 1990

  14. [14]

    Techniques for the calculation of electroweak radiative corrections at the one-loop level and results for W-physics at LEP200

    A. Denner, Fortsch. Phys.41, 307-420 (1993) [arXiv:0709.1075 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 1993

  15. [15]

    O. V. Tarasov, Phys. Rev. D54, 6479-6490 (1996) [arXiv:hep-th/9606018 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  16. [16]

    O. V. Tarasov, Nucl. Phys.B502, 455–482 (1996) [hep-ph/9703319]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  17. [17]

    A. I. Davydychev, Phys. Lett.B263, 107–111 (1991)

    work page 1991

  18. [18]

    W. T. Giele and E. W. N. Glover, JHEP04, 029 (2004) [hep-ph/0402152]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  19. [19]

    Binoth, J

    T. Binoth, J. Ph. Guillet, G. Heinrich, E. Pilon and C. Schubert, JHEP10, 015 (2005) [hep- ph/0504267]

    work page arXiv 2005

  20. [20]

    R. K. Ellis, W. T. Giele and G. Zanderighi, Phys. Rev.D73, 014027 (2006) [hep-ph/0508308]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  21. [21]

    R. N. Lee, JHEP04, 108 (2015) [arXiv:1411.0911 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  22. [22]

    V. A. Smirnov, Evaluating Feynman integrals, Springer Tracts in Modern Physics,211, 1–244 (2004)

    work page 2004

  23. [23]

    Anti-differentiation and the calculation of Feynman amplitudes

    A. V. Kotikov, in "Anti-differentiation and the calculation of Feynman amplitudes" (2021), p. 235–259, [arXiv:2102.07424]

    work page arXiv 2021

  24. [24]

    Blümlein and C

    J. Blümlein and C. Schneider, J. Phys.A55, 443005 (2022), [arXiv:2203.13015]

    work page arXiv 2022

  25. [25]

    Blümlein, M

    J. Blümlein, M. Saragnese and C. Schneider, Ann. Math. Artif. Intell.91, 591–649 (2023) [arXiv:2111.15501]

    work page arXiv 2023

  26. [26]

    Wasser, doi:10.25358/openscience-6801 (2025)

    P. Wasser, doi:10.25358/openscience-6801 (2025)

    work page doi:10.25358/openscience-6801 2025

  27. [27]

    J. Henn, B. Mistlberger, V. A. Smirnov and P. Wasser, JHEP04, 167 (2020) [arXiv:2002.09492 [hep-ph]]

    work page arXiv 2020

  28. [28]

    Quadratic electroweak corrections for polarized Moller scattering

    A. Aleksejevs, S. Barkanova, Y. Kolomensky, E. Kuraev and V. Zykunov, Phys. Rev.D85, 37 013007 (2012) [arXiv:1110.1750 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  29. [29]

    A. G. Aleksejevs, S. G. Barkanova, Y. M. Bystritskiy, A. N. Ilyichev, E. A. Kuraev and V. A. Zykunov, Eur. Phys. J.C72, 2249 (2012)

    work page 2012

  30. [30]

    A. G. Aleksejevs, S. G. Barkanova, V. A. Zykunov and E. A. Kuraev, Phys. Atom. Nucl.76, 888–900 (2013)

    work page 2013

  31. [31]

    A. G. Aleksejevs, S. G. Barkanova, Y. M. Bystritskiy, E. A. Kuraev and V. A. Zykunov, Phys. Part. Nucl. Lett.12, no.5, 645–656 (2015) [arXiv:1504.03560 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  32. [32]

    A. G. Aleksejevs, S. G. Barkanova, Y. M. Bystritskiy, E. A. Kuraev and V. A. Zykunov, Phys. Part. Nucl. Lett.13, no.3, 310-317 (2016) [arXiv:1508.07853 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  33. [33]

    Dispersion Approach in Two-Loop Calculations

    A. Aleksejevs, Phys. Rev.D98, 036021 (2018) [hep-th/1804.08914]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  34. [35]

    Dimensional Regularization and Dispersive Two-Loop Calculations

    A. Aleksejevs and S. Barkanova, (2019) [arXiv:1905.07936 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  35. [36]

    Aleksejevs and S

    A. Aleksejevs and S. Barkanova, PoS LeptonPhoton2019, 090 (2019) [arXiv:1912.04762 [hep- th]]

    work page arXiv 2019

  36. [37]

    Chen, P.-F

    X.-L. Chen, P.-F. Yang and W. Chen, Chin. Phys. Lett.41, 111101 (2024) [arXiv:2405.19868]

    work page arXiv 2024

  37. [38]

    Song and A

    Q. Song and A. Freitas, JHEP04, 179 (2021) [arXiv:2101.00308 [hep-ph]]

    work page arXiv 2021

  38. [39]

    Freitas and Q

    A. Freitas and Q. Song, Phys. Rev. Lett.130, 031801 (2023) [arXiv:2209.07612 [hep-ph]]

    work page arXiv 2023

  39. [40]

    The two-loop electroweak bosonic corrections to $\sin^2\theta_{\rm eff}^{\rm b}$

    I. Dubovyk, A. Freitas, J. Gluza, T. Riemann and J. Usovitsch, Phys. Lett.B 762, 184-189 (2016) [arXiv:1607.08375 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  40. [41]

    Dubovyk, A

    I. Dubovyk, A. Freitas, J. Gluza, K. Grzanka, M. Hidding and J. Usovitsch, Phys. Rev.D106, L111301 (2022) [arXiv:2201.02576 [hep-ph]]

    work page arXiv 2022

  41. [42]

    Erler, R

    J. Erler, R. Ferro-Hernández and A. Freitas, JHEP08, 183 (2022) [arXiv:2202.11976 [hep-ph]]

    work page arXiv 2022

  42. [43]

    Schwanemann and S

    N. Schwanemann and S. Weinzierl, SciPost Phys.18, no.6, 172 (2025) [arXiv:2412.07522 [hep- ph]]

    work page arXiv 2025

  43. [44]

    Erler, [arXiv:2505.03457 [hep-ph]]

    J. Erler, [arXiv:2505.03457 [hep-ph]]

    work page arXiv

  44. [45]

    Navaset al.[Particle Data Group], Phys

    S. Navaset al.[Particle Data Group], Phys. Rev.D110, 030001 (2024)

    work page 2024

  45. [46]

    Numerical multi-loop integrals and applications

    A. Freitas, Prog. Part. Nucl. Phys.90, 201–240 (2016) [arXiv:1604.00406 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  46. [47]

    Kreimer, Phys

    D. Kreimer, Phys. Lett.B273, 277-281 (1991)

    work page 1991

  47. [48]

    New representation of two-loop propagator and vertex functions

    A. Czarnecki, U. Kilian and D. Kreimer. Nucl. Phys.B433, 259–275 (1995) [hep-ph/9405423]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  48. [49]

    New representation of the two-loop crossed vertex function

    A. Frink, U. Kilian and D. Kreimer. Nucl. Phys.B488, 426–440 (1997) [hep-ph/9610285]

    work page internal anchor Pith review Pith/arXiv arXiv 1997

  49. [50]

    The two-loop sunrise graph with arbitrary masses

    L. Adams, C. Bogner and S. Weinzierl, J. Math. Phys.54, 052303 (2013) [arXiv:1302.7004 38 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  50. [51]

    The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case

    L. Adams, C. Bogner and S. Weinzierl, J. Math. Phys.56, 072303 (2015) [arXiv:1504.03255 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  51. [52]

    The iterated structure of the all-order result for the two-loop sunrise integral

    L. Adams, C. Bogner and S. Weinzierl, J. Math. Phys.57, 032304 (2016) [arXiv:1512.05630 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  52. [53]

    Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral

    E. Remiddi and L. Tancredi, Nucl. Phys.B907, 400–444 (2016) [arXiv:1602.01481 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  53. [54]

    A Feynman integral via higher normal functions

    S. Bloch, M. Kerr and P. Vanhove, Compos. Math.151, 2329–2375 (2015) [arXiv:1406.2664 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  54. [55]

    Local mirror symmetry and the sunset Feynman integral

    S. Bloch, M. Kerr and P. Vanhove, Adv. Theor. Math. Phys.21, 1373–1453 (2017) [arXiv:1601.08181 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  55. [56]

    SecDec: A tool for numerical multi-loop calculations

    S. Borowka, J. Carter and G. Heinrich, J. Phys. Conf. Ser.368, 012051 (2012) [arXiv:1206.4908 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2012

  56. [57]

    Numerical evaluation of multi-loop integrals for arbitrary kinematics with SecDec 2.0

    S. Borowka, J. Carter and G. Heinrich, Comput. Phys. Commun.184, 396–408 (2013) [arXiv:1204.4152 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  57. [58]

    Simple one-dimensional integral representations for two-loop self-energies: the master diagram

    S. Bauberger, M. Böhm, Nucl. Phys.B445, 25–46 (1995) [hep-ph/9501201 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 1995

  58. [59]

    The effective electroweak mixing angle $\sin^2\theta_{eff}$ with two-loop fermionic contributions

    W. Hollik, U. Meier and S. Uccirati, Nucl. Phys.B731, 213–224 (2005) [hep-ph/0507158]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  59. [60]

    Freitas, W

    A. Freitas, W. Hollik, W. Walter and G Weiglein, Nucl. Phys.B632, 189–218 (2002) [hep- ph/0202131]

    work page arXiv 2002

  60. [61]

    Master integrals for massive two-loop Bhabha scattering in QED

    M. Czakon, J. Gluza and T. Riemann, Phys. Rev.D71, 073009 (2005) [hep-ph/0412164]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  61. [62]

    Virtual Hadronic and Leptonic Contributions to Bhabha Scattering

    S. Actis, M. Czakon, J. Gluza and T. Riemann, Phys. Rev. Lett.100, 131602 (2008) [arXiv:0711.3847 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  62. [64]

    F. V. Tkachov, Phys. Lett.B100, 65–68 (1981)

    work page 1981

  63. [65]

    K. G. Chetyrkin and F. V. Tkachov, Nucl. Phys.B192, 159–204 (1981)

    work page 1981

  64. [66]

    A. I. Davydychev and R. Delbourgo, J. Math. Phys.394299–4334 (1998) [hep-th/9709216]

    work page internal anchor Pith review Pith/arXiv arXiv 1998

  65. [67]

    A. I. Davydychev, Nucl. Instr. Meth.A559, 293–297 (2006) [hep-th/0509233]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  66. [68]

    F. A. Berends, A. I. Davydychev and V. A. Smirnov, Nucl. Phys.B478, 59–89 (1996) [hep- ph/9602396]

    work page arXiv 1996

  67. [69]

    Geometrical approach to Feynman integrals and the epsilon-expansion

    A. I. Davydychev, Proc. Workshop "AIHENP-99", Heraklion, Greece, April 1999 (Parisianou S.A., Athens, 2000), 219–225 [hep-th/9908032]. 39

    work page internal anchor Pith review Pith/arXiv arXiv 1999

  68. [70]

    A. I. Davydychev, Phys. Rev.D61, 087701 (2000) [hep-ph/9910224]

    work page internal anchor Pith review Pith/arXiv arXiv 2000

  69. [71]

    A. I. Davydychev and M. Yu. Kalmykov, Nucl. Phys.B605, 266–318 (2001) [hep-th/0012189]

    work page arXiv 2001

  70. [72]

    Relativistic phase space: dimensional recurrences

    R. Delbourgo and M. L. Roberts, J. Phys.A36, 1719–1728 (2003) [hep-th/0301004]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  71. [73]

    A. I. Davydychev and R. Delbourgo, J. Phys.A37, 4871–4876 (2004) [hep-th/0311075]

    work page internal anchor Pith review Pith/arXiv arXiv 2004

  72. [74]

    E. E. Boos and A. I. Davydychev, Teor. Mat. Fiz.8956, (1991) [Theor. Math. Phys.89, 1052 (1991)]

    work page 1991

  73. [75]

    A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev, Integrals and series. Additional chapters (Nauka, Moscow, 1986)

    work page 1986

  74. [76]

    Collier: a fortran-based Complex One-Loop LIbrary in Extended Regularizations

    A. Denner, S. Dittmaier and L. Hofer, Comput. Phys. Commun.212, 220–238 (2017) [arXiv:1604.06792]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  75. [77]

    Reduction of one-loop tensor 5-point integrals

    A. Denner and S. Dittmaier, Nucl. Phys.B658, 175–202 (2003) [hep-ph/0212259]

    work page internal anchor Pith review Pith/arXiv arXiv 2003

  76. [78]

    Reduction schemes for one-loop tensor integrals

    A. Denner and S. Dittmaier, Nucl. Phys.B734, 62–115 (2006) [hep-ph/0509141]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  77. [79]

    Scalar one-loop 4-point integrals

    A. Denner and S. Dittmaier, Nucl. Phys.B844, 199–242 (2011) [arXiv:1005.2076 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  78. [80]

    Crossed Topology in Two-Loop Dispersive Approach

    A. Aleksejevs, (2018) [arXiv:1809.05592 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  79. [81]

    H. H. Patel, Comput. Phys. Commun.197, 276–290 (2015) [arXiv:1503.01469 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  80. [82]

    H. H. Patel, Comput. Phys. Commun.218, 66–70 (2017) [arXiv:1612.00009 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

Showing first 80 references.