×

Maximal cuts in arbitrary dimension. (English) Zbl 1381.81146

Summary: We develop a systematic procedure for computing maximal unitarity cuts of multiloop Feynman integrals in arbitrary dimension. Our approach is based on the Baikov representation in which the structure of the cuts is particularly simple. We examine several planar and nonplanar integral topologies and demonstrate that the maximal cut inherits IBPs and dimension shift identities satisfied by the uncut integral. Furthermore, for the examples we calculated, we find that the maximal cut functions from different allowed regions, form the Wronskian matrix of the differential equations on the maximal cut.

MSC:

81U05 \(2\)-body potential quantum scattering theory
81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 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 [hep-ph/9403226] [INSPIRE]. · Zbl 1049.81644
[2] 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 [hep-ph/9409265] [INSPIRE]. · Zbl 1049.81644
[3] 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 [hep-th/0412103] [INSPIRE]. · Zbl 1178.81202 · doi:10.1016/j.nuclphysb.2005.07.014
[4] D. Forde, Direct extraction of one-loop integral coefficients, Phys. Rev.D 75 (2007) 125019 [arXiv:0704.1835] [INSPIRE].
[5] G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys.B 763 (2007) 147 [hep-ph/0609007] [INSPIRE]. · Zbl 1116.81067
[6] G. Ossola, C.G. Papadopoulos and R. Pittau, On the rational terms of the one-loop amplitudes, JHEP05 (2008) 004 [arXiv:0802.1876] [INSPIRE]. · doi:10.1088/1126-6708/2008/05/004
[7] Y. Zhang, Lecture notes on multi-loop integral reduction and applied algebraic geometry, arXiv:1612.02249 [INSPIRE]. · Zbl 1378.81039
[8] P. Mastrolia and G. Ossola, On the integrand-reduction method for two-loop scattering amplitudes, JHEP11 (2011) 014 [arXiv:1107.6041] [INSPIRE]. · Zbl 1306.81357 · doi:10.1007/JHEP11(2011)014
[9] S. Badger, H. Frellesvig and Y. Zhang, Hepta-cuts of two-loop scattering amplitudes, JHEP04 (2012) 055 [arXiv:1202.2019] [INSPIRE]. · Zbl 1348.81340 · doi:10.1007/JHEP04(2012)055
[10] P. Mastrolia, E. Mirabella, G. Ossola and T. Peraro, Integrand-reduction for two-loop scattering amplitudes through multivariate polynomial division, Phys. Rev.D 87 (2013) 085026 [arXiv:1209.4319] [INSPIRE]. · Zbl 1331.81218
[11] S. Badger, H. Frellesvig and Y. Zhang, A two-loop five-gluon helicity amplitude in QCD, JHEP12 (2013) 045 [arXiv:1310.1051] [INSPIRE]. · doi:10.1007/JHEP12(2013)045
[12] S. Badger, G. Mogull, A. Ochirov and D. O’Connell, A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory, JHEP10 (2015) 064 [arXiv:1507.08797] [INSPIRE]. · Zbl 1388.81274 · doi:10.1007/JHEP10(2015)064
[13] S. Badger, G. Mogull and T. Peraro, Local integrands for two-loop all-plus Yang-Mills amplitudes, JHEP08 (2016) 063 [arXiv:1606.02244] [INSPIRE]. · Zbl 1390.81278 · doi:10.1007/JHEP08(2016)063
[14] P. Mastrolia, T. Peraro and A. Primo, Adaptive integrand decomposition in parallel and orthogonal space, JHEP08 (2016) 164 [arXiv:1605.03157] [INSPIRE]. · Zbl 1390.81180 · doi:10.1007/JHEP08(2016)164
[15] S. Badger, C. Brønnum-Hansen, F. Buciuni and D. O’Connell, A unitarity compatible approach to one-loop amplitudes with massive fermions, JHEP06 (2017) 141 [arXiv:1703.05734] [INSPIRE].
[16] D.A. Kosower and K.J. Larsen, Maximal unitarity at two loops, Phys. Rev.D 85 (2012) 045017 [arXiv:1108.1180] [INSPIRE].
[17] H. Johansson, D.A. Kosower and K.J. Larsen, Two-loop maximal unitarity with external masses, Phys. Rev.D 87 (2013) 025030 [arXiv:1208.1754] [INSPIRE].
[18] H. Johansson, D.A. Kosower and K.J. Larsen, Maximal unitarity for the four-mass double box, Phys. Rev. D 89 (2014) 125010 [arXiv:1308.4632] [INSPIRE].
[19] M. S0gaard, Global residues and two-loop hepta-cuts, JHEP09 (2013) 116 [arXiv:1306.1496] [INSPIRE].
[20] M. Søgaard and Y. Zhang, Multivariate residues and maximal unitarity, JHEP12 (2013) 008 [arXiv:1310.6006] [INSPIRE]. · Zbl 1342.81319 · doi:10.1007/JHEP12(2013)008
[21] M. Sogaard and Y. Zhang, Unitarity cuts of integrals with doubled propagators, JHEP07 (2014) 112 [arXiv:1403.2463] [INSPIRE]. · doi:10.1007/JHEP07(2014)112
[22] M. Sogaard and Y. Zhang, Massive nonplanar two-loop maximal unitarity, JHEP12 (2014) 006 [arXiv:1406.5044] [INSPIRE]. · doi:10.1007/JHEP12(2014)006
[23] M. Søgaard and Y. Zhang, Elliptic functions and maximal unitarity, Phys. Rev.D 91 (2015) 081701 [arXiv:1412.5577] [INSPIRE].
[24] H. Johansson, D.A. Kosower, K.J. Larsen and M. Søgaard, Cross-order integral relations from maximal cuts, Phys. Rev.D 92 (2015) 025015 [arXiv:1503.06711] [INSPIRE].
[25] S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier and B. Page, Subleading poles in the numerical unitarity method at two loops, Phys. Rev.D 95 (2017) 096011 [arXiv:1703.05255] [INSPIRE].
[26] S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier, B. Page and M. Zeng, Two-loop four-gluon amplitudes with the numerical unitarity method, arXiv:1703.05273 [INSPIRE]. · Zbl 1320.81059
[27] 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].
[28] A.V. Kotikov, Differential equation method: the calculation of N point Feynman diagrams, Phys. Lett.B 267 (1991) 123 [Erratum ibid.B 295 (1992) 409] [INSPIRE]. · Zbl 1020.81734
[29] A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett.B 254 (1991) 158 [INSPIRE]. · doi:10.1016/0370-2693(91)90413-K
[30] Z. Bern, L.J. Dixon and D.A. Kosower, Dimensionally regulated one loop integrals, Phys. Lett.B 302 (1993) 299 [Erratum ibid.B 318 (1993) 649] [hep-ph/9212308] [INSPIRE]. · Zbl 1007.81512
[31] E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim.A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].
[32] T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys.B 580 (2000) 485 [hep-ph/9912329] [INSPIRE]. · Zbl 1071.81089
[33] J. Ablinger et al., Calculating three loop ladder and V-topologies for massive operator matrix elements by computer algebra, Comput. Phys. Commun.202 (2016) 33 [arXiv:1509.08324] [INSPIRE]. · Zbl 1348.81034 · doi:10.1016/j.cpc.2016.01.002
[34] 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 [INSPIRE]. · doi:10.1016/0550-3213(81)90199-1
[35] J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett.110 (2013) 251601 [arXiv:1304.1806] [INSPIRE]. · doi:10.1103/PhysRevLett.110.251601
[36] J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys.A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE]. · Zbl 1312.81078
[37] M. Argeri et al., Magnus and Dyson series for master integrals, JHEP03 (2014) 082 [arXiv:1401.2979] [INSPIRE]. · Zbl 1333.81379 · doi:10.1007/JHEP03(2014)082
[38] R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP04 (2015) 108 [arXiv:1411.0911] [INSPIRE]. · Zbl 1388.81109 · doi:10.1007/JHEP04(2015)108
[39] C. Meyer, Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP04 (2017) 006 [arXiv:1611.01087] [INSPIRE]. · Zbl 1378.81064 · doi:10.1007/JHEP04(2017)006
[40] M. Prausa, epsilon: a tool to find a canonical basis of master integrals, Comput. Phys. Commun.219 (2017) 361 [arXiv:1701.00725] [INSPIRE]. · Zbl 1411.81019
[41] O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, arXiv:1701.04269 [INSPIRE]. · Zbl 1411.81015
[42] L. Adams, E. Chaubey and S. Weinzierl, Simplifying differential equations for multiscale feynman integrals beyond multiple polylogarithms, Phys. Rev. Lett.118 (2017) 141602 [arXiv:1702.04279] [INSPIRE]. · doi:10.1103/PhysRevLett.118.141602
[43] C.G. Papadopoulos, Simplified differential equations approach for master integrals, JHEP07 (2014) 088 [arXiv:1401.6057] [INSPIRE]. · doi:10.1007/JHEP07(2014)088
[44] C.G. Papadopoulos, D. Tommasini and C. Wever, The Pentabox master integrals with the simplified differential equations approach, JHEP04 (2016) 078 [arXiv:1511.09404] [INSPIRE].
[45] H. Frellesvig and C.G. Papadopoulos, Cuts of Feynman integrals in Baikov representation, JHEP04 (2017) 083 [arXiv:1701.07356] [INSPIRE]. · Zbl 1378.81039 · doi:10.1007/JHEP04(2017)083
[46] P.A. Baikov, Explicit solutions of n loop vacuum integral recurrence relations, hep-ph/9604254 [INSPIRE].
[47] P.A. Baikov, Explicit solutions of the three loop vacuum integral recurrence relations, Phys. Lett.B 385 (1996) 404 [hep-ph/9603267] [INSPIRE].
[48] P.A. Baikov, A Practical criterion of irreducibility of multi-loop Feynman integrals, Phys. Lett.B 634 (2006) 325 [hep-ph/0507053] [INSPIRE]. · Zbl 1247.81314
[49] S. Abreu, R. Britto, C. Duhr and E. Gardi, Cuts from residues: the one-loop case, JHEP06 (2017) 114 [arXiv:1702.03163] [INSPIRE]. · Zbl 1380.81421 · doi:10.1007/JHEP06(2017)114
[50] S. Abreu, R. Britto, C. Duhr and E. Gardi, The algebraic structure of cut Feynman integrals and the diagrammatic coaction, arXiv:1703.05064 [INSPIRE]. · Zbl 1383.81321
[51] A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys.B 916 (2017) 94 [arXiv:1610.08397] [INSPIRE]. · Zbl 1356.81136 · doi:10.1016/j.nuclphysb.2016.12.021
[52] M. Zeng, Differential equations on unitarity cut surfaces, JHEP06 (2017) 121 [arXiv:1702.02355] [INSPIRE]. · Zbl 1380.81135 · doi:10.1007/JHEP06(2017)121
[53] R.N. Lee, Modern techniques of multiloop calculations, in the proceedings of the 49th Rencontres de Moriond on QCD and High Energy Interactions, March 22-29, La thuile, Italy (2014), arXiv:1405.5616 [INSPIRE].
[54] H. Ita, Two-loop integrand decomposition into master integrals and surface terms, Phys. Rev.D 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE].
[55] 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].
[56] R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP11 (2013) 165 [arXiv:1308.6676] [INSPIRE]. · Zbl 1342.81139 · doi:10.1007/JHEP11(2013)165
[57] A. Georgoudis, K.J. Larsen and Y. Zhang, Azurite: an algebraic geometry based package for finding bases of loop integrals, arXiv:1612.04252 [INSPIRE]. · Zbl 1498.81007
[58] A.V. Smirnov and V.A. Smirnov, Applying Grobner bases to solve reduction problems for Feynman integrals, JHEP01 (2006) 001 [hep-lat/0509187] [INSPIRE].
[59] A.V. Smirnov, An algorithm to construct Grobner bases for solving integration by parts relations, JHEP04 (2006) 026 [hep-ph/0602078] [INSPIRE].
[60] A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP10 (2008) 107 [arXiv:0807.3243] [INSPIRE]. · Zbl 1245.81033 · doi:10.1088/1126-6708/2008/10/107
[61] A.V. Smirnov, FIRE5: a C++ implementation of Feynman Integral REduction, Comput. Phys. Commun.189 (2015) 182 [arXiv:1408.2372] [INSPIRE]. · Zbl 1344.81030 · doi:10.1016/j.cpc.2014.11.024
[62] R.N. Lee, Presenting LiteRed: a tool for the Loop InTEgrals REDuction, arXiv:1212.2685 [INSPIRE].
[63] 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].
[64] A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE]. · Zbl 1219.81133
[65] P. Maierhoefer, J. Usovitsch and P. Uwer, Kira — A Feynman Integral Reduction Program, arXiv:1705.05610 [INSPIRE].
[66] J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun.83 (1994) 45 [INSPIRE]. · Zbl 1114.68598 · doi:10.1016/0010-4655(94)90034-5
[67] D. Binosi, J. Collins, C. Kaufhold and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams. Version 2.0 release notes, Comput. Phys. Commun.180 (2009) 1709 [arXiv:0811.4113] [INSPIRE].
[68] R.N. Lee and V.A. Smirnov, The dimensional recurrence and analyticity method for multicomponent master integrals: using unitarity cuts to construct homogeneous solutions, JHEP12 (2012) 104 [arXiv:1209.0339] [INSPIRE]. · Zbl 1397.81073 · doi:10.1007/JHEP12(2012)104
[69] T. Huber and D. Maitre, HypExp: a Mathematica package for expanding hypergeometric functions around integer-valued parameters, Comput. Phys. Commun.175 (2006) 122 [hep-ph/0507094] [INSPIRE]. · Zbl 1196.68326
[70] T. Huber and D. Maitre, HypExp 2, expanding hyper geometric functions about half-integer parameters, Comput. Phys. Commun.178 (2008) 755 [arXiv:0708.2443] [INSPIRE]. · Zbl 1196.81024 · doi:10.1016/j.cpc.2007.12.008
[71] S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys.B 704 (2005) 349 [hep-ph/0406160] [INSPIRE]. · Zbl 1119.81356
[72] S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor.148 (2015) 328 [arXiv:1309.5865] [INSPIRE]. · Zbl 1319.81044 · doi:10.1016/j.jnt.2014.09.032
[73] L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys.54 (2013) 052303 [arXiv:1302.7004] [INSPIRE]. · Zbl 1282.81193 · doi:10.1063/1.4804996
[74] L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys.55 (2014) 102301 [arXiv:1405.5640] [INSPIRE]. · Zbl 1298.81204 · doi:10.1063/1.4896563
[75] L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case, J. Math. Phys.56 (2015) 072303 [arXiv:1504.03255] [INSPIRE]. · Zbl 1320.81059 · doi:10.1063/1.4926985
[76] S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, arXiv:1601.08181 [INSPIRE]. · Zbl 1390.14123
[77] G. Passarino, Elliptic polylogarithms and basic hypergeometric functions, Eur. Phys. J.C 77 (2017) 77 [arXiv:1610.06207] [INSPIRE]. · doi:10.1140/epjc/s10052-017-4623-1
[78] M.Yu. Kalmykov and B.A. Kniehl, Counting the number of master integrals for sunrise diagrams via the Mellin-Barnes representation, JHEP07 (2017) 031 [arXiv:1612.06637] [INSPIRE]. · Zbl 1380.81423 · doi:10.1007/JHEP07(2017)031
[79] A. von Manteuffel and L. Tancredi, A non-planar two-loop three-point function beyond multiple polylogarithms, JHEP06 (2017) 127 [arXiv:1701.05905] [INSPIRE]. · doi:10.1007/JHEP06(2017)127
[80] L. Adams, C. Bogner and S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral, J. Math. Phys.57 (2016) 032304 [arXiv:1512.05630] [INSPIRE]. · Zbl 1333.81283 · doi:10.1063/1.4944722
[81] L. Tancredi, Simplifying systems of differential equations. The case of the Sunrise graph, Acta Phys. Polon.B 46 (2015) 2125. · Zbl 1371.81131 · doi:10.5506/APhysPolB.46.2125
[82] E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys.B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE]. · Zbl 1336.81038
[83] E.T. Whittaker, G.N. Watson, A course of modern analysis, Cambridge University Press, Cambridge U.K. (1996). · Zbl 0951.30002 · doi:10.1017/CBO9780511608759
[84] Z.X. Wang, D.R. Guo, Special functions, World Scientific Publishing Co., U.S.A. (1989). · doi:10.1142/0653
[85] M.J. Schlosser, Multiple hyper geometric series: appell series and beyond, Springer, Germany (2013). · Zbl 1310.33013
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.