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Differential Equations and Feynman Integrals

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Anti-Differentiation and the Calculation of Feynman Amplitudes

Part of the book series: Texts & Monographs in Symbolic Computation ((TEXTSMONOGR))

Abstract

The role of differential equations in the process of calculating Feynman integrals is reviewed. An example of a diagram is given for which the method of differential equations was introduced, the properties of the inverse-mass-expansion coefficients are shown, and modern methods based on differential equations are discussed.

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Notes

  1. 1.

    See also Ref. [3] and the reviews [4] and [5]. Note that multipoint massless FIs are as complex as massive 2-point FIs. For the relationship between 2-point massive FIs and 3-point massless FIs, cf. [6].

  2. 2.

    Investigations of hypergeometric functions related to the calculation of FI are recently presented [10] as a contribution to this volume.

  3. 3.

    Hereafter we consider only first order DEs. The consideration of the higher order DEs can be found in Section 7 of the review [33]. See also the recent papers [34].

  4. 4.

    In fact, the results for these two-point diagrams were found in the late eighties and early nineties, and were planned to be published in a long paper summarizing the results obtained in Refs. [13, 15]. However, this paper has not been published. These results, after verification, were published in Ref. [37].

  5. 5.

    The diagrams are complicated two-loop FIs that do not have cuts of three massive particles. Thus, their results should be expressible as combinations of polylogarithms. Note that we consider only three-point diagrams with independent upward momenta q 1 and q 2, which satisfy the conditions \(q_1^2q_2^2=0\) and (q 1 + q 2) q 2 ≠ 0, where q is a downward momentum.

  6. 6.

    In our previous papers [23, 25, 36, 37] the nested sums \(K_{a,b,...}(j)=\sum ^j_{m=1} \, \frac {(-1)^{m+1}}{m^a} S_{b,...}(m)= - S_{- a,b,...}(j)\) have been used together with their analytic continuations [25, 42].

  7. 7.

    The evaluation of the inverse mass expansion coefficients is demonstrated in Ref. [38].

  8. 8.

    A short review of many approaches has recently been presented as an introduction to this volume [77].

References

  1. K.G. Chetyrkin, F.V. Tkachov, Nucl. Phys. B 192, 159 (1981); F.V. Tkachov, Phys. Lett. B 100, 65 (1981); A.N. Vasiliev, Y.M. Pismak, Yu.R. Khonkonen, Theor. Math. Phys. 46, 104 (1981)

    Article  Google Scholar 

  2. K.G. Chetyrkin et al., Nucl. Phys. B 174, 345 (1980); A.V. Kotikov, Phys. Lett. B 375, 240 (1996)

    Article  Google Scholar 

  3. A.V. Kotikov, S. Teber, Phys. Rev. D 89(6), 065038 (2014)

    Article  Google Scholar 

  4. A. Grozin, Lectures on QED and QCD (2005), pp. 1–156 [hep-ph/0508242]; A.G. Grozin, Int. J. Mod. Phys. A 27, 1230018 (2012)

    Google Scholar 

  5. S. Teber, A.V. Kotikov, Theor. Math. Phys. 190(3), 446 (2017); A.V. Kotikov, S. Teber, Phys. Part. Nucl. 50(1), 1 (2019)

    Google Scholar 

  6. A.I. Davydychev, J.B. Tausk, Phys. Rev. D 53, 7381 (1996)

    Article  Google Scholar 

  7. D.J. Broadhurst, Z. Phys. C 47, 115 (1990)

    Google Scholar 

  8. B.A. Kniehl, A.F. Pikelner, O.L. Veretin, J. High Energy Phys. 1708, 024 (2017); A. Pikelner, Comput. Phys. Commun. 224, 282 (2018)

    Article  Google Scholar 

  9. L.H. Ryder, Quantum Field Theory (Cambridge University Press, Cambridge, 1996)

    Book  MATH  Google Scholar 

  10. M. Kalmykov, V. Bytev, B.A. Kniehl, S.O. Moch, B.F.L. Ward, S.A. Yost, Hypergeometric Functions and Feynman Diagrams. arXiv:2012.14492 [hep-th]

    Google Scholar 

  11. T. Regge, Algebraic Topology Methods in the Theory of Feynman Rela- tivistic Amplitudes, Battelle Rencontres: 1967 Lectures in Mathematics and Physics, ed. by C.M. DeWitt, J.A. Wheeler (W.A. Benjamin, New York, 1968), pp. 433–458

    Google Scholar 

  12. V.A. Golubeva, Some problems in the analytical theory of Feynman integrals. Russ. Math. Surv. 31, 139 (1976)

    Article  MATH  Google Scholar 

  13. A.V. Kotikov, Phys. Lett. B 254, 158 (1991)

    Article  MathSciNet  Google Scholar 

  14. E. Remiddi, Nuovo Cim. A 110, 1435 (1997)

    Article  Google Scholar 

  15. A.V. Kotikov, Phys. Lett. B 259, 314 (1991)

    Article  MathSciNet  Google Scholar 

  16. T. Gehrmann, E. Remiddi, Nucl. Phys. B 580, 485 (2000)

    Article  Google Scholar 

  17. A.V. Kotikov, Phys. Lett. B 267, 123 (1991); Mod. Phys. Lett. A 6, 3133 (1991)

    Article  MathSciNet  Google Scholar 

  18. Z. Bern, L.J. Dixon, D.C. Dunbar, D.A. Kosower, Nucl. Phys. B 425, 217 (1994)

    Article  Google Scholar 

  19. A.V. Kotikov, Ukr. Fiz. Zh. (Russ. Ed. ) 37, 303 (1992)

    Google Scholar 

  20. N.N. Bogoliubov, O.S. Parasiuk, Acta Math. 97, 227 (1957); K. Hepp, Commun. Math. Phys. 2, 301 (1966); W. Zimmermann, Commun. Math. Phys. 15, 208 (1969) [Lect. Notes Phys. 558, 217 (2000)]

    MathSciNet  Google Scholar 

  21. N.N. Bogolyubov, D.V. Shirkov, Intersci. Monogr. Phys. Astron. 3, 1 (1959)

    Google Scholar 

  22. D.I. Kazakov, Phys. Lett. 133B, 406 (1983); Theor. Math. Phys. 58, 223 (1984); D.I. Kazakov, Theor. Math. Phys. 62, 84 (1985); N.I. Usyukina, Theor. Math. Phys. 54, 78 (1983); V.V. Belokurov, N.I. Usyukina, J. Phys. A 16, 2811 (1983); Theor. Math. Phys. 79, 385 (1989)

    Article  Google Scholar 

  23. D.I. Kazakov, A.V. Kotikov, Theor. Math. Phys. 73, 1264 (1988); A.V. Kotikov, Theor. Math. Phys. 78, 134 (1989)

    Article  Google Scholar 

  24. J.A. Gracey, Conformal methods for massless Feynman integrals and large N f methods, in Computer Algebra in Quantum Field Theory, ed. by C. Schneider, J. Blümlein (Springer, Wien, 2013), pp. 97–118. arXiv:1301.7583 [hep-th]

    Google Scholar 

  25. D.I. Kazakov, A.V. Kotikov, Nucl. Phys. B 307, 721 (1988) [Nucl. Phys. B 345, 299 (1990)]

    Google Scholar 

  26. A.V. Kotikov, Mod. Phys. Lett. A 6, 677 (1991)

    Article  MathSciNet  Google Scholar 

  27. A.V. Kotikov, Int. J. Mod. Phys. A 7, 1977 (1992)

    Article  MathSciNet  Google Scholar 

  28. D.I. Kazakov, A.V. Kotikov, Phys. Lett. B 291, 171 (1992); Yad. Fiz. 46, 1767 (1987); D.I. Kazakov, A.V. Kotikov, G. Parente, O.A. Sampayo, J. Sanchez Guillen, Phys. Rev. Lett. 65, 1535 (1990)

    Article  Google Scholar 

  29. F. Herzog, S. Moch, B. Ruijl, T. Ueda, J.A.M. Vermaseren, A. Vogt, Phys. Lett. B 790, 436 (2019)

    Article  Google Scholar 

  30. J. Blümlein, S. Klein, B. Tödtli, Phys. Rev. D 80, 094010 (2009)

    Article  Google Scholar 

  31. A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, K. Schönwald, C. Schneider, The Polarized Transition Matrix Element A gq(N) of the Variable Flavor Number Scheme at \(O(\alpha _s^3)\). arXiv:2101.05733 [hep-ph]

    Google Scholar 

  32. J. Ablinger, J. Blümlein, A. De Freitas, A. Goedicke, M. Saragnese, C. Schneider, K. Schönwald, Nucl. Phys. B 955, 115059 (2020)

    Article  Google Scholar 

  33. J. Blümlein, C. Schneider, Int. J. Mod. Phys. A 33(17), 1830015 (2018)

    Article  Google Scholar 

  34. A. Klemm, C. Nega, R. Safari, J. High Energy Phys. 2004, 088 (2020); K. Bönisch, F. Fischbach, A. Klemm, C. Nega, R. Safari, Analytic Structure of all Loop Banana Amplitudes. arXiv:2008.10574 [hep-th]

    Article  Google Scholar 

  35. J. Fleischer et al., Phys. Lett. B 462, 169 (1999)

    Article  Google Scholar 

  36. J. Fleischer et al., Phys. Lett. B 417, 163 (1998)

    Article  MathSciNet  Google Scholar 

  37. J. Fleischer et al., Nucl. Phys. B 547, 343 (1999); Acta Phys. Polon. B 29, 2611 (1998)

    Article  Google Scholar 

  38. A.V. Kotikov, Particles 3(2), 394 (2020)

    Article  Google Scholar 

  39. B.A. Kniehl et al., Nucl. Phys. B 738, 306 (2006); Nucl. Phys. B 948, 114780 (2019)

    Article  MathSciNet  Google Scholar 

  40. B.A. Kniehl, A.V. Kotikov, Phys. Lett. B 638, 531 (2006); Phys. Lett. B 712, 233 (2012)

    Article  Google Scholar 

  41. J.A.M. Vermaseren, Int. J. Mod. Phys. A 14, 2037 (1999); J. Blümlein, S. Kurth, Phys. Rev. D 60, 014018 (1999)

    Article  MathSciNet  Google Scholar 

  42. A.V. Kotikov, Phys. Atom. Nucl. 57, 133 (1994); A.V. Kotikov, V.N. Velizhanin, Analytic continuation of the Mellin moments of deep inelastic structure functions. hep-ph/0501274

    Google Scholar 

  43. A. Devoto, D.W. Duke, Riv. Nuovo Cim. 7N6, 1 (1984)

    Google Scholar 

  44. B.A. Kniehl et al., Phys. Rev. Lett. 97, 042001 (2006); Phys. Rev. D 79, 114032 (2009); Phys. Rev. Lett. 101, 193401 (2008); Phys. Rev. A 80, 052501 (2009)

    Article  Google Scholar 

  45. A.V. Kotikov, L.N. Lipatov, Nucl. Phys. B 661, 19 (2003); Proceedings of the XXXV Winter School. Repino, S’Peterburg, 2001 (hep-ph/0112346)

    Article  Google Scholar 

  46. L. Bianchi et al., Phys. Lett. B 725, 394 (2013)

    Article  MathSciNet  Google Scholar 

  47. A.V. Kotikov, L.N. Lipatov, Nucl. Phys. B582, 19 (2000)

    Article  Google Scholar 

  48. L.N. Lipatov, Sov. J. Nucl. Phys. 23, 338 (1976); V.S. Fadin et al., Phys. Lett. B 60, 50 (1975); E.A. Kuraev et al., Sov. Phys. JETP 44, 443 (1976); Sov. Phys. JETP 45, 199 (1977); I.I. Balitsky, L.N. Lipatov, Sov. J. Nucl. Phys. 28, 822 (1978); JETP Lett. 30, 355 (1979)

    Google Scholar 

  49. A.V. Kotikov et al., Phys. Lett. B 557, 114 (2003)

    Article  MathSciNet  Google Scholar 

  50. A.V. Kotikov et al., Phys. Lett. B 595, 521 (2004)

    Article  MathSciNet  Google Scholar 

  51. A.V. Kotikov et al., J. Stat. Mech. 0710, P10003 (2007); Z. Bajnok, R.A. Janik, T. Lukowski, Nucl. Phys. B 816, 376 (2009)

    Article  Google Scholar 

  52. T. Lukowski et al., Nucl. Phys. B 831, 105 (2010)

    Article  MathSciNet  Google Scholar 

  53. C. Marboe et al., J. High Energy Phys. 1507, 084 (2015)

    Article  MathSciNet  Google Scholar 

  54. C. Marboe, V. Velizhanin, J. High Energy Phys. 1611, 013 (2016)

    Article  Google Scholar 

  55. M. Staudacher, J. High Energy Phys. 0505, 054 (2005); N. Beisert, M. Staudacher, Nucl. Phys. B 727, 1 (2005)

    Article  MathSciNet  Google Scholar 

  56. E. Remiddi, J.A.M. Vermaseren, Int. J. Mod. Phys. A 15, 725 (2000)

    Article  Google Scholar 

  57. A.I. Davydychev, M.Y. Kalmykov, Nucl. Phys. B 699, 3 (2004)

    Article  Google Scholar 

  58. A.V. Kotikov, The property of maximal transcendentality in the N = 4 supersymmetric Yang-Mills, in Subtleties in Quantum Field Theory, ed. by D. Diakonov (2010), pp. 150–174 [arXiv:1005.5029 [hep-th]]; Phys. Part. Nucl. 44, 374 (2013); A.V. Kotikov, A.I. Onishchenko, DGLAP and BFKL equations in \(\mathcal {N}=4\) SYM: from weak to strong coupling. arXiv:1908.05113 [hep-th]

    Google Scholar 

  59. A.V. Kotikov, Theor. Math. Phys. 176, 913 (2013); Theor. Math. Phys. 190(3), 391 (2017)

    Article  Google Scholar 

  60. T. Gehrmann et al., J. High Energy Phys. 1203, 101 (2012)

    Article  MathSciNet  Google Scholar 

  61. J.M. Henn, Phys. Rev. Lett. 110, 251601 (2013); J. Phys. A 48, 153001 (2015)

    Article  Google Scholar 

  62. L. Adams, S. Weinzierl, Phys. Lett. B 781, 270 (2018)

    Article  Google Scholar 

  63. L. Adams, C. Bogner, A. Schweitzer, S. Weinzierl, J. Math. Phys. 57(12), 122302 (2016)

    Article  MathSciNet  Google Scholar 

  64. C.G. Papadopoulos, J. High Energy Phys. 1407, 088 (2014)

    Article  Google Scholar 

  65. S. Weinzierl, Simple differential equations for Feynman integrals associated to elliptic curves. arXiv:1912.02578 [hep-ph]; arXiv:2012.08429 [hep-th]

    Google Scholar 

  66. M. Argeri, P. Mastrolia, Int. J. Mod. Phys. A 22, 4375 (2007)

    Article  Google Scholar 

  67. R.N. Lee, J. High Energy Phys. 1504, 108 (2015); J. High Energy Phys. 1810, 176 (2018); R.N. Lee, A.I. Onishchenko, J. High Energy Phys. 1912, 084 (2019)

    Article  Google Scholar 

  68. R.N. Lee, A.A. Pomeransky, Normalized Fuchsian form on Riemann sphere and differential equations for multiloop integrals. arXiv:1707.07856 [hep-th]

    Google Scholar 

  69. A.B. Goncharov, Multiple polylogarithms and mixed Tate motives. math/0103059 [math.AG]

    Google Scholar 

  70. C. Duhr, Mathematical aspects of scattering amplitudes. arXiv:1411.7538 [hep-ph]

    Google Scholar 

  71. A. Kotikov et al., Nucl. Phys. B 788, 47 (2008)

    Article  Google Scholar 

  72. U. Aglietti, R. Bonciani, Nucl. Phys. B 668, 3 (2003); U. Aglietti, R. Bonciani, G. Degrassi, A. Vicini, Phys. Lett. B 595, 432 (2004); Phys. Lett. B 600, 57 (2004); J. High Energy Phys. 0701, 021 (2007)

    Article  Google Scholar 

  73. R.N. Lee, M.D. Schwartz, X. Zhang, The Compton Scattering Total Cross Section at Next-to-Leading Order (2021). [arXiv:2102.06718 [hep-ph]]

    Google Scholar 

  74. J. Ablinger, J. Blümlein, C. Schneider, J. Math. Phys. 52, 102301 (2011)

    Article  MathSciNet  Google Scholar 

  75. J. Ablinger, J. Blümlein, C. Schneider, Iterated Integrals Over Letters Induced by Quadratic Forms (2021) [arXiv:2103.08330 [hep-th]]

    Google Scholar 

  76. J. Henn, A.V. Smirnov, V.A. Smirnov, M. Steinhauser, R.N. Lee, J. High Energy Phys. 1703, 139 (2017); J. Ablinger, J. Blümlein, P. Marquard, N. Rana, C. Schneider, Phys. Lett. B 782, 528 (2018); J. Ablinger, J. Blümlein, P. Marquard, N. Rana, C. Schneider, Nucl. Phys. B 939, 253 (2019)

    Article  Google Scholar 

  77. J. Blümlein, Analytic Integration Methods in Quantum Field Theory: An Introduction (2021). arXiv:2103.10652 [hep-th]

    Google Scholar 

  78. D.D. Canko, C.G. Papadopoulos, N. Syrrakos, J. High Energy Phys. 2101, 199 (2021); D.D. Canko, N. Syrrakos, Resummation Methods for Master Integrals (2020). arXiv:2010.06947 [hep-ph]; N. Syrrakos, Pentagon Integrals to Arbitrary Order in the Dimensional regulator (2020). arXiv:2012.10635 [hep-ph]

    Article  Google Scholar 

  79. R.N. Lee, A.V. Smirnov, V.A. Smirnov, J. High Energy Phys. 1803, 008 (2018)

    Article  Google Scholar 

  80. R.N. Lee, Libra: A Package for Transformation of Differential Systems for Multiloop Integrals (2020). arXiv:2012.00279 [hep-ph]

    Google Scholar 

  81. B. Kol, Symmetries of Feynman Integrals and the Integration by Parts Method (2015). arXiv:1507.01359 [hep-th]

    Google Scholar 

  82. B. Kol, A. Schiller, R. Shir, Numerator Seagull and extended symmetries of Feynman integrals. J. High Energy Phys. 2101, 165 (2021)

    Article  MathSciNet  Google Scholar 

  83. P. Vanhove, Feynman Integrals, Toric Geometry and Mirror Symmetry (2018). arXiv:1807.11466 [hep-th]

    Google Scholar 

  84. J. Broedel, C. Duhr, F. Dulat, B. Penante, L. Tancredi, From Modular Forms to Differential Equations for Feynman Integrals (2018). arXiv:1807.00842 [hep-th]; J. Broedel, A. Kaderli, J. Phys. A 53(24), 245201 (2020)

    Google Scholar 

  85. M. Walden, S. Weinzierl, Numerical Evaluation of Iterated Integrals Related to Elliptic Feynman Integrals (2020). arXiv:2010.05271 [hep-ph]; L. Adams, E. Chaubey, S. Weinzierl, PoS LL 2018, 069 (2018). [arXiv:1807.03599 [hep-ph]]

    Google Scholar 

  86. J. Campert, F. Moriello, A. Kotikov, Sunrise Integral with Two Internal Masses and Pseudo-Threshold Kinematics in Terms of Elliptic Polylogarithms (2020). arXiv:2011.01904 [hep-ph]

    Google Scholar 

  87. J. Broedel, C. Duhr, F. Dulat, L. Tancredi, J. High Energy Phys. 1805, 093 (2018)

    Article  Google Scholar 

  88. M.Y. Kalmykov, B.A. Kniehl, Nucl. Phys. B 809, 365 (2009)

    Article  Google Scholar 

  89. M.A. Bezuglov, A.I. Onishchenko, O.L. Veretin, Nucl. Phys. B 963, 115302 (2021)

    Article  Google Scholar 

  90. J.C. Collins, J.A.M. Vermaseren, Axodraw Version 2 (2016). [arXiv:1606.01177 [cs.OH]]

    Google Scholar 

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Acknowledgements

The author thanks Johannes Blümlein for invitation to present a contribution to the Proceedings of International Conference “Antidifferentiation and the Calculation of Feynman Amplitudes” (4–9 October 2020, Zeuthen, Germany) and Andrey Pikelner for help with Axodraw2 [90]. The author thanks Kay Schönwald also for a strong improvement in article style.

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Appendix: Massive Part of J 1(q 2, m 2) in Eq. (5)

Appendix: Massive Part of J 1(q 2, m 2) in Eq. (5)

In this appendix we consider the following diagrams

(43)

The IBP relations for the internal loop of the diagram produce two equations:

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle (d-1-2\alpha) \, I_2^{(\alpha)}(q^2,m^2) = \alpha \, J_2^{(\alpha+1)}(q^2,m^2) - m^2 \, \alpha \, I_2^{(\alpha+1)}(q^2,m^2) \, ,\quad {} \end{array} \end{aligned} $$
(44)
(45)

where

$$\displaystyle \begin{aligned} J_2^{(\alpha)}(q^2,m^2) = T_{0,\alpha}(m^2) \, L_{1,1}(q^2) \, - S^{(1,2)}(q^2,m^2) \, . {} \end{aligned} $$
(46)

We note that T 0,2(m 2 = 0) = 0 in dimensional regularization and

$$\displaystyle \begin{aligned} T_{0,2}(m^2) L_{1,1}(q^2) = \frac{1}{(4\pi)^{d}} \, \frac{R(0,2)A(1,1)}{m^{2(2-d/2)}q^{2(2-d/2)}} \, , {} \end{aligned} $$
(47)

where R(α 1, α 2) and A(α 1, α 2) are given in Eqs. (13) and (12), respectively.

The IBP relations for internal triangles of the diagram \(I_2^{(1)}(q^2,m^2)\) produce two additional equations:

(48)
(49)

Using Eqs. (45) and (49) in the combination: 2 ×(45) + (49) , we have

(50)

So, we have for the mass-dependent part of J 1(q 2, m 2), see Eq. (5),

(51)

i.e. the mass-dependent combination is expressed through the diagram \(I_2^{(1)}(q^2,m^2)\) and its derivative.

Using Eq. (44) one obtains

$$\displaystyle \begin{aligned} \left[d-2-\alpha - m^2 \frac{d}{dm^2}\right] \, I_2^{(\alpha)}(q^2,m^2) = \alpha \, J_2^{(\alpha+1)}(q^2,m^2) \, , {} \end{aligned} $$
(52)

i.e. diagram \(I_2^{(\alpha )}(q^2,m^2)\) obeys the differential equation with the inhomogeneous term \(J_2^{(\alpha +1)}(q^2,m^2)\) having a very simple form: it contains only one-loop diagrams. We see that the last term in \(J_2^{(\alpha )}(q^2,m^2)\), see Eq. (46), is expressed through the massive one loop integral \(M_{\alpha _1,\alpha _2}(q^2,m^2)\):

(53)

Indeed,

$$\displaystyle \begin{aligned} S^{(1,\alpha)}(q^2,m^2) = \, A(1,1) \, M_{2-d/2,\alpha}(q^2,m^2) \, . {} \end{aligned} $$
(54)

The one-loop diagram M 2−d∕2,α(q 2, m 2) can be evaluated by one of some effective methods, for example, by Feynman parameters.

We would like to note that \(I_2^{(1)}(q^2,m^2)\) satisfies Eq. (52) with α = 1 that is not of the type of (35). But the integral \(I_2^{(2)}(q^2,m^2)\) satisfies Eq. (52) with α = 2 and is of the type of (35). So, it is convenient to rewrite (51) with \(I_2^{(2)}(q^2,m^2)\) in its r.h.s.:

(55)

Now we should compare the IBP-based equations for \(J_2^{(2)}(q^2,m^2)\) and \(J_2^{(3)}(q^2,m^2)\) obtained in the right-hand sides of (55) and (52), respectively, with Eq. (35). Since \(J_2^{(3)}(q^2,m^2)=-(d/dm^2) \, J_2^{(2)}(q^2,m^2)\), consider only \(J_2^{(2)}(q^2,m^2)\).

So, we should prepare the IBP-based equations for the massive one-loop diagrams M ε,2(q 2, m 2) and M ε,3(q 2, m 2). Applying the IBP procedure with massive distinguished line to M ε,2(q 2, m 2), we have

$$\displaystyle \begin{aligned} \begin{array}{rcl} (-3\varepsilon) M_{\varepsilon,2}(q^2,m^2) & =&\displaystyle \varepsilon \bigl[M_{1+\varepsilon,1}(q^2,m^2) - (q^2+m^2) \, M_{1+\varepsilon,2}(q^2,m^2)\bigr]\\ & &\displaystyle -4m^2 \, M_{\varepsilon,3}(q^2,m^2) \, . {} \end{array} \end{aligned} $$
(56)

The corresponding applications of the IBP procedure with massless distinguished line to M 1+ε,1(q 2, m 2) and M 1+ε,2(q 2, m 2) leads to the following results:

$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle (1-4\varepsilon) M_{1+\varepsilon,1}(q^2,m^2) = M_{\varepsilon,2}(q^2,m^2) - (q^2+m^2) \, M_{1+\varepsilon,2}(q^2,m^2) \, ,\quad {} \end{array} \end{aligned} $$
(57)
$$\displaystyle \begin{aligned} \begin{array}{rcl} & &\displaystyle -4\varepsilon M_{1+\varepsilon,2}(q^2,m^2) = 2 \, M_{\varepsilon,3}(q^2,m^2) - 2 \, (q^2+m^2) \, M_{1+\varepsilon,3}(q^2,m^2). {} \end{array} \end{aligned} $$
(58)

The last equation has the following form

$$\displaystyle \begin{aligned} \left[-4\varepsilon - (q^2+m^2) \,\frac{d}{dm^2} \, \right] \, M_{1+\varepsilon,2}(q^2,m^2) = - \,\frac{d}{dm^2} \, M_{\varepsilon,2}(q^2,m^2). {} \end{aligned} $$
(59)

Putting (57) to (56), we have after little algebra

$$\displaystyle \begin{aligned} \begin{array}{rcl} -4\varepsilon(1-3\varepsilon) M_{\varepsilon,2}(q^2,m^2) & =&\displaystyle -2\varepsilon(1-4\varepsilon)\, (q^2+m^2) \, M_{1+\varepsilon,2}(q^2,m^2)\big] \\ & &\displaystyle -4(1-4\varepsilon)\,m^2 \, M_{\varepsilon,3}(q^2,m^2) \, , {} \end{array} \end{aligned} $$
(60)

which transforms to

$$\displaystyle \begin{aligned} \begin{array}{rcl} -\left[4\varepsilon(1-3\varepsilon) + 2(1-4\varepsilon) \frac{d}{dm^2} \right] M_{\varepsilon,2}(q^2,m^2) = -2\varepsilon(1-4\varepsilon)(q^2+m^2) M_{1+\varepsilon,2}(q^2,m^2)\\ {} \end{array} \end{aligned} $$
(61)

So, Eqs. (59) and (61) can be considered as a system of equations having a form similar to Eq. (35).

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Kotikov, A.V. (2021). Differential Equations and Feynman Integrals. In: Blümlein, J., Schneider, C. (eds) Anti-Differentiation and the Calculation of Feynman Amplitudes. Texts & Monographs in Symbolic Computation. Springer, Cham. https://doi.org/10.1007/978-3-030-80219-6_10

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