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On the maximal cut of Feynman integrals and the solution of their differential equations. (English) Zbl 1356.81136
Summary: The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in $$\epsilon = (4 - d) / 2$$, where $$d$$ are the space-time dimensions. The differential equations are, in general, coupled and can be solved using Euler’s variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exists no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations.

##### MSC:
 81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry 41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series) 35F35 Systems of linear first-order PDEs 35G35 Systems of linear higher-order PDEs
##### Keywords:
Euler’s variation of constants
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##### References:
 [1] Kotikov, A., Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B, 254, 158-164, (1991) [2] Bern, Z.; Dixon, L. J.; Kosower, D. A., Dimensionally regulated pentagon integrals, Nucl. Phys. B, 412, 751-816, (1994) · Zbl 1007.81512 [3] Remiddi, E., Differential equations for Feynman graph amplitudes, Nuovo Cimento A, 110, 1435-1452, (1997) [4] Gehrmann, T.; Remiddi, E., Differential equations for two loop four point functions, Nucl. Phys. B, 580, 485-518, (2000) · Zbl 1071.81089 [5] Tkachov, F., A theorem on analytical calculability of four loop renormalization group functions, Phys. Lett. B, 100, 65-68, (1981) [6] Chetyrkin, K.; Tkachov, F., Integration by parts: the algorithm to calculate beta functions in 4 loops, Nucl. Phys. B, 192, 159-204, (1981) [7] Laporta, S., High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A, 15, 5087-5159, (2000) · Zbl 0973.81082 [8] Argeri, M.; Mastrolia, P., Feynman diagrams and differential equations, Int. J. Mod. Phys. A, 22, 4375-4436, (2007) · Zbl 1141.81325 [9] Henn, J. M., Lectures on differential equations for Feynman integrals, J. Phys. A, 48, 15, 153001, (2015) · Zbl 1312.81078 [10] Remiddi, E.; Tancredi, L., Schouten identities for Feynman graph amplitudes; the master integrals for the two-loop massive sunrise graph, Nucl. Phys. B, 880, 343-377, (2014) · Zbl 1284.81139 [11] Tancredi, L., Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations, Nucl. Phys. B, 901, 282-317, (2015) · Zbl 1332.81065 [12] Henn, J. M., Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett., 110, (2013) [13] Argeri, M.; Di Vita, S.; Mastrolia, P.; Mirabella, E.; Schlenk, J.; Schubert, U.; Tancredi, L., Magnus and Dyson series for master integrals, J. High Energy Phys., 1403, (2014) · Zbl 1333.81379 [14] Lee, R. N., Reducing differential equations for multiloop master integrals, J. High Energy Phys., 1504, (2015) · Zbl 1388.81109 [15] Ablinger, J.; Behring, A.; Blümlein, J.; De Freitas, A.; von Manteuffel, A.; Schneider, C., Calculating three loop ladder and V-topologies for massive operator matrix elements by computer algebra, Comput. Phys. Commun., 202, 33-112, (2016) · Zbl 1348.81034 [16] Lee, R. N.; Smirnov, V. A., Evaluating the last missing ingredient for the three-loop quark static potential by differential equations, J. High Energy Phys., 10, (2016) [17] Gehrmann, T.; von Manteuffel, A.; Tancredi, L.; Weihs, E., The two-loop master integrals for $$q \overline{q} \rightarrow V V$$, J. High Energy Phys., 1406, (2014) [18] Meyer, C., Transforming differential equations of multi-loop Feynman integrals into canonical form · Zbl 1378.81064 [19] Arkani-Hamed, N.; Bourjaily, J. L.; Cachazo, F.; Trnka, J., Local integrals for planar scattering amplitudes, J. High Energy Phys., 06, (2012) · Zbl 1348.81339 [20] Bonciani, R.; Del Duca, V.; Frellesvig, H.; Henn, J. M.; Moriello, F.; Smirnov, V. A., Two-loop planar master integrals for $$\text{Higgs} \rightarrow 3$$ partons with full heavy-quark mass dependence [21] Laporta, S.; Remiddi, E., Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B, 704, 349-386, (2005) · Zbl 1119.81356 [22] Anastasiou, C.; Melnikov, K., Higgs boson production at hadron colliders in NNLO QCD, Nucl. Phys. B, 646, 220-256, (2002) [23] Larsen, K. J.; Zhang, Y., Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D, 93, 4, (2016) [24] Lee, R. N.; Smirnov, V. A., The dimensional recurrence and analyticity method for multicomponent master integrals: using unitarity cuts to construct homogeneous solutions, J. High Energy Phys., 12, (2012) · Zbl 1397.81073 [25] Cutkosky, R. E., Singularities and discontinuities of Feynman amplitudes, J. Math. Phys., 1, 429-433, (1960) · Zbl 0122.22605 [26] Veltman, M. J.G., Unitarity and causality in a renormalizable field theory with unstable particles, Physica, 29, 186-207, (1963) · Zbl 0127.20001 [27] Remiddi, E., Dispersion relations for Feynman graphs, Helv. Phys. Acta, 54, 364, (1982) [28] Abreu, S.; Britto, R.; Duhr, C.; Gardi, E., From multiple unitarity cuts to the coproduct of Feynman integrals, J. High Energy Phys., 10, (2014) · Zbl 1333.81148 [29] Abreu, S.; Britto, R.; Grönqvist, H., Cuts and coproducts of massive triangle diagrams, J. High Energy Phys., 07, (2015) · Zbl 1388.83151 [30] Mastrolia, P.; Mirabella, E.; Ossola, G.; Peraro, T., Scattering amplitudes from multivariate polynomial division, Phys. Lett. B, 718, 173-177, (2012) [31] Baikov, P. A., Explicit solutions of the multiloop integral recurrence relations and its application, Nucl. Instrum. Methods A, 389, 347-349, (1997) [32] Adams, L.; Bogner, C.; Weinzierl, S., The two-loop sunrise graph with arbitrary masses, J. Math. Phys., 54, (2013) · Zbl 1282.81193 [33] Adams, L.; Bogner, C.; Weinzierl, S., The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys., 55, 10, 102301, (2014) · Zbl 1298.81204 [34] Adams, L.; Bogner, C.; Weinzierl, S., The two-loop sunrise integral around four space-time dimensions and generalisations of the clausen and glaisher functions towards the elliptic case · Zbl 1320.81059 [35] Adams, L.; Bogner, C.; Weinzierl, S., The iterated structure of the all-order result for the two-loop sunrise integral · Zbl 1333.81283 [36] Aglietti, U.; Bonciani, R.; Grassi, L.; Remiddi, E., The two loop crossed ladder vertex diagram with two massive exchanges, Nucl. Phys. B, 789, 45-83, (2008) · Zbl 1151.81364 [37] Caron-Huot, S.; Larsen, K. J., Uniqueness of two-loop master contours, J. High Energy Phys., 10, (2012) [38] Remiddi, E.; Tancredi, L., Differential equations and dispersion relations for Feynman amplitudes. the two-loop massive sunrise and the kite integral, Nucl. Phys. B, 907, 400-444, (2016) · Zbl 1336.81038 [39] Adams, L.; Bogner, C.; Schweitzer, A.; Weinzierl, S., The kite integral to all orders in terms of elliptic polylogarithms · Zbl 1353.81097 [40] A. von Manteuffel, L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, in preparation.
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