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From infinity to four dimensions: higher residue pairings and Feynman integrals. (English) Zbl 1435.81079
Summary: We study a surprising phenomenon in which Feynman integrals in \(D = 4 - 2\epsilon\) space-time dimensions as \(\epsilon \rightarrow 0\) can be fully characterized by their behavior in the opposite limit, \(\epsilon \rightarrow \infty\). More concretely, we consider vector bundles of Feynman integrals over kinematic spaces, whose connections have a polynomial dependence on \(\epsilon\) and are known to be governed by intersection numbers of twisted forms. They give rise to differential equations that can be obtained exactly as a truncating expansion in either \(\epsilon\) or \(1/ \epsilon\). We use the latter for explicit computations, which are performed by expanding intersection numbers in terms of Saito’s higher residue pairings (previously used in the context of topological Landau-Ginzburg models and mirror symmetry). These pairings localize on critical points of a certain Morse function, which correspond to regions in the loop-momentum space that were previously thought to govern only the large-\(D\) physics. The results of this work leverage recent understanding of an analogous situation for moduli spaces of curves, where the \(\alpha' \rightarrow 0\) and \(\alpha' \rightarrow \infty\) limits of intersection numbers coincide for scattering amplitudes of massless quantum field theories.

81Q30 Feynman integrals and graphs; applications of algebraic topology and algebraic geometry
81U05 \(2\)-body potential quantum scattering theory
Fuchsia; Macaulay2
Full Text: DOI arXiv
[1] F. Cachazo, S. He and E.Y. Yuan, Scattering equations and Kawai-Lewellen-Tye orthogonality, Phys. Rev.D 90 (2014) 065001 [arXiv:1306.6575] [INSPIRE].
[2] F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles in Arbitrary Dimensions, Phys. Rev. Lett.113 (2014) 171601 [arXiv:1307.2199] [INSPIRE]. · Zbl 1391.81198
[3] F. Cachazo, S. He and E.Y. Yuan, Scattering of Massless Particles: Scalars, Gluons and Gravitons, JHEP07 (2014) 033 [arXiv:1309.0885] [INSPIRE]. · Zbl 1391.81198
[4] S. Mizera, Scattering Amplitudes from Intersection Theory, Phys. Rev. Lett.120 (2018) 141602 [arXiv:1711.00469] [INSPIRE].
[5] S. Mizera, Aspects of Scattering Amplitudes and Moduli Space Localization, Ph.D. Thesis, Perimeter Inst. Theor. Phys. (2019) [arXiv:1906.02099] [INSPIRE].
[6] K. Saito, The higher residue pairings \({K}_F^{(k)}\) for a family of hypersurface singular points, in Singularities, Part 2, Arcata, California (1981), Amer. Math. Soc., Providence, RI, Proc. Symp. Pure Math.40 (1983) 441.
[7] K. Saito, On the Periods of Primitive Integrals, I, preprint Harvard (1980).
[8] K. Saito, Period mapping associated to a primitive form, Publ. Res. Inst. Math. Sci. Kyoto19 (1983) 1231. · Zbl 0539.58003
[9] K. Saito, Primitive forms for a universal unfolding of a function with an isolated critical point, J. Fac. Sci. Univ. Tokyo28 (1981) 775. · Zbl 0523.32015
[10] C. Vafa, Topological Landau-Ginzburg models, Mod. Phys. Lett.A 6 (1991) 337 [INSPIRE]. · Zbl 1020.81886
[11] S. Cecotti and C. Vafa, Topological antitopological fusion, Nucl. Phys.B 367 (1991) 359 [INSPIRE]. · Zbl 1136.81403
[12] B. Blok and A. Varchenko, Topological conformal field theories and the flat coordinates, Int. J. Mod. Phys.A 7 (1992) 1467 [INSPIRE]. · Zbl 0802.53034
[13] R. Dijkgraaf, Intersection theory, integrable hierarchies and topological field theory, hep-th/9201003 [INSPIRE]. · Zbl 1221.32005
[14] B. Dubrovin, Painlevé transcendents and two-dimensional topological field theory, math/9803107. · Zbl 1026.34095
[15] A. Chiodo, H. Iritani and Y. Ruan, Landau-Ginzburg/Calabi-Yau correspondence, global mirror symmetry and Orlov equivalence, Publ. Math. IHE^́S119 (2013) 127. · Zbl 1298.14042
[16] C. Li, S. Li and K. Saito, Primitive forms via polyvector fields, arXiv:1311.1659 [INSPIRE].
[17] K. Saito, From primitive form to mirror symmetry, arXiv:1408.4208 [INSPIRE].
[18] C. Li, S. Li, K. Saito and Y. Shen, Mirror symmetry for exceptional unimodular singularities, arXiv:1405.4530 [INSPIRE]. · Zbl 1387.14021
[19] W. Lerche, On Matrix Factorizations, Residue Pairings and Homological Mirror Symmetry, arXiv:1803.10333 [INSPIRE].
[20] S. Li and H. Wen, On the L^2-Hodge theory of Landau-Ginzburg models, arXiv:1903.02713 [INSPIRE].
[21] B. Dubrovin, Geometry of 2-D topological field theories, Lect. Notes Math.1620 (1996) 120 [hep-th/9407018] [INSPIRE]. · Zbl 0841.58065
[22] A. Losev, Descendants constructed from matter field in topological Landau-Ginzburg theories coupled to topological gravity, Theor. Math. Phys.95 (1993) 595 [hep-th/9211090] [INSPIRE]. · Zbl 0847.53057
[23] A. Losev, ‘Hodge strings’ and elements of K. Saito’s theory of the primitive form, in Topological field theory, primitive forms and related topics. Proceedings, 38th Taniguchi Symposium, Kyoto, Japan, 9-13 December 1996 and RIMS Symposium, Kyoto, Japan, 16-19 December 1996, pp. 305-335 (1998) [hep-th/9801179] [INSPIRE]. · Zbl 1059.14016
[24] A. Belavin, D. Gepner and Y. Kononov, Flat coordinates for Saito Frobenius manifolds and String theory, Theor. Math. Phys.189 (2016) 1775 [arXiv:1510.06970] [INSPIRE]. · Zbl 1358.81147
[25] S. Li, D. Xie and S.-T. Yau, Seiberg-Witten Differential via Primitive Forms, Commun. Math. Phys.367 (2019) 193 [arXiv:1802.06751] [INSPIRE]. · Zbl 1412.81166
[26] P. Deligne and G. Mostow, Monodromy of hypergeometric functions and non-lattice integral monodromy, Publ. Math. IHE^́S63 (1986) 5. · Zbl 0615.22008
[27] K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I, Nagoya Math. J.139 (1995) 67. · Zbl 0856.32015
[28] D.J. Gross and P.F. Mende, The High-Energy Behavior of String Scattering Amplitudes, Phys. Lett.B 197 (1987) 129 [INSPIRE].
[29] D.J. Gross and P.F. Mende, String Theory Beyond the Planck Scale, Nucl. Phys.B 303 (1988) 407 [INSPIRE].
[30] P. Mastrolia and S. Mizera, Feynman Integrals and Intersection Theory, JHEP02 (2019) 139 [arXiv:1810.03818] [INSPIRE]. · Zbl 1411.81093
[31] H. Frellesvig, F. Gasparotto, M.K. Mandal, P. Mastrolia, L. Mattiazzi and S. Mizera, Vector Space of Feynman Integrals and Multivariate Intersection Numbers, Phys. Rev. Lett.123 (2019) 201602 [arXiv:1907.02000] [INSPIRE]. · Zbl 1416.81198
[32] J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys.A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE]. · Zbl 1312.81078
[33] K. Hori et al., Mirror Symmetry, Clay Mathematics Monographs. American Mathematical Society (2003).
[34] H. Esnault, V. Schechtman and E. Viehweg, Cohomology of local systems on the complement of hyperplanes, Invent. Math.109 (1992) 557. · Zbl 0788.32005
[35] J. Milnor, Morse Theory, Annals of Mathematics Studies, vol. 51, Princeton University Press, (2016).
[36] K. Aomoto, Gauss-Manin connection of integral of difference products, J. Math. Soc. Jap.39 (1987) 191. · Zbl 0619.32010
[37] R. Silvotti, On a conjecture of Varchenko, Invent. Math.126 (1996) 235. · Zbl 0882.32022
[38] R. Hartshorne, Residues and Duality, Lecture Notes of a seminar on the work of A. Grothendieck, given at Harvard 1963/64, Springer (2014).
[39] E. Witten, A Note on the Antibracket Formalism, Mod. Phys. Lett.A 5 (1990) 487 [INSPIRE]. · Zbl 1020.81931
[40] K. Aomoto, On vanishing of cohomology attached to certain many valued meromorphic functions, J. Math. Soc. Jap.27 (1975) 248. · Zbl 0301.32010
[41] K. Matsumoto, Quadratic Identities for Hypergeometric Series of Type (k, l), Kyushu J. Math.48 (1994) 335. · Zbl 0839.33007
[42] K. Matsumoto, Intersection numbers for logarithmic k-forms, Osaka J. Math.35 (1998) 873. · Zbl 0937.32013
[43] K. Matsumoto, Intersection numbers for 1-forms associated with confluent hypergeometric functions, Funkcial. Ekvac.41 (1998) 291. · Zbl 1140.33303
[44] H. Majima, K. Matsumoto and N. Takayama, Quadratic relations for confluent hypergeometric functions, Tohoku Math. J.52 (2000) 489. · Zbl 1006.33004
[45] K. Ohara, Y. Sugiki and N. Takayama, Quadratic Relations for Generalized Hypergeometric Functions_pF_p−1 , Funkcial. Ekvac.46 (2003) 213. · Zbl 1162.33309
[46] Y. Goto, Twisted Cycles and Twisted Period Relations for Lauricella’s Hypergeometric Function F_C , Int. J. Math.24 (2013) 1350094 [arXiv:1308.5535]. · Zbl 1285.33011
[47] Y. Goto and K. Matsumoto, The monodromy representation and twisted period relations for Appell’s hypergeometric function F_4 , Nagoya Math. J.217 (2015) 61. · Zbl 1327.32001
[48] Y. Goto, Twisted period relations for Lauricella’s hypergeometric functions F_A , Osaka J. Math.52 (2015) 861. · Zbl 1336.33031
[49] Y. Goto, Intersection Numbers and Twisted Period Relations for the Generalized Hypergeometric Function_m+1F_m , Kyushu J. Math.69 (2015) 203. · Zbl 1318.33009
[50] W. Siegel, Amplitudes for left-handed strings, arXiv:1512.02569 [INSPIRE].
[51] K. Lee, S.-J. Rey and J.A. Rosabal, A string theory which isn’t about strings, JHEP11 (2017) 172 [arXiv:1708.05707] [INSPIRE]. · Zbl 1383.83183
[52] E. Casali and P. Tourkine, Windings of twisted strings, Phys. Rev.D 97 (2018) 061902 [arXiv:1710.01241] [INSPIRE]. · Zbl 1390.81493
[53] R. Lipinski Jusinskas, Chiral strings, the sectorized description and their integrated vertex operators, JHEP12 (2019) 143 [arXiv:1909.04069] [INSPIRE]. · Zbl 1431.83171
[54] F. Cachazo, S. Mizera and G. Zhang, Scattering Equations: Real Solutions and Particles on a Line, JHEP03 (2017) 151 [arXiv:1609.00008] [INSPIRE]. · Zbl 1377.81220
[55] Y. Namikawa, Higher Residues Associated with an Isolated Hypersurface Sigularity, in Algebraic Varieties and Analytic Varieties, Tokyo, Japan, pp. 181-193, Mathematical Society of Japan (1983) [DOI].
[56] A. Matsuo, Summary of the Theory of Primitive Forms, Birkhäuser Boston, Boston, MA (1998) [DOI]. · Zbl 0933.32041
[57] R. Bott and L.W. Tu, The Čech-de Rham Complex, in Differential Forms in Algebraic Topology, chapter II, pp. 89-153, Springer New York, New York, NY (1982) [DOI].
[58] K. Aomoto and M. Kita, Theory of Hypergeometric Functions, Springer Monographs in Mathematics. Springer Japan (2011) [DOI]. · Zbl 1229.33001
[59] A. Varchenko, Multidimensional hypergeometric functions in conformal field theory, algebraic K-theory, algebraic geometry, in Proceedings of the International Congress of Mathematicians, vol. 1, pp. 281-300 (1990). · Zbl 0747.33002
[60] A. Varchenko, Multidimensional Hypergeometric Functions and Representation Theory of Lie Algebras and Quantum Groups, World Scientific (1995). · Zbl 0951.33001
[61] K. Mimachi and M. Yoshida, Intersection numbers of twisted cycles and the correlation functions of the conformal field theory. 2., Commun. Math. Phys.234 (2003) 339 [math/0208097] [INSPIRE]. · Zbl 1029.81062
[62] K. Mimachi and M. Yoshida, Intersection numbers of twisted cycles associated with the Selberg integral and an application to the conformal field theory, Commun. Math. Phys.250 (2004) 23 [INSPIRE]. · Zbl 1069.32015
[63] A. Varchenko, Bethe Ansatz for Arrangements of Hyperplanes and the Gaudin Model, math/0408001. · Zbl 1375.32050
[64] A. Schwarz and I. Shapiro, Twisted de Rham cohomology, homological definition of the integral and ‘Physics over a ring’, Nucl. Phys.B 809 (2009) 547 [arXiv:0809.0086] [INSPIRE]. · Zbl 1192.81311
[65] A. Varchenko, Quantum Integrable Model of an Arrangement of Hyperplanes, SIGMA7 (2011) 032. · Zbl 1217.82026
[66] S. Mizera, Combinatorics and Topology of Kawai-Lewellen-Tye Relations, JHEP08 (2017) 097 [arXiv:1706.08527] [INSPIRE].
[67] Z. Li and C. Zhang, Moduli Space of Paired Punctures, Cyclohedra and Particle Pairs on a Circle, JHEP05 (2019) 029 [arXiv:1812.10727] [INSPIRE].
[68] H. Frellesvig et al., Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers, JHEP05 (2019) 153 [arXiv:1901.11510] [INSPIRE].
[69] F. Brown and C. Dupont, Single-valued integration and superstring amplitudes in genus zero, arXiv:1910.01107 [INSPIRE].
[70] S. Abreu, R. Britto, C. Duhr, E. Gardi and J. Matthew, From positive geometries to a coaction on hypergeometric functions, JHEP02 (2020) 122 [arXiv:1910.08358] [INSPIRE].
[71] E. Casali, S. Mizera and P. Tourkine, Monodromy relations from twisted homology, JHEP12 (2019) 087 [arXiv:1910.08514] [INSPIRE]. · Zbl 1431.83163
[72] S. Caron-Huot and A. Pokraka, On the Poincaŕe dual of Feynman integrals, to appear.
[73] R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP11 (2013) 165 [arXiv:1308.6676] [INSPIRE]. · Zbl 1342.81139
[74] P. Mastrolia, Feynman Integrals and Intersection Theory, talk at The Mathematics of Linear Relations between Feynman Integrals, The MITP workshop, March 2019 [https://indico.mitp.uni-mainz.de/event/179/contributions/2834/attachments/2249/2377/mastrolia.pdf ].
[75] L. de la Cruz, Feynman integrals as A-hypergeometric functions, JHEP12 (2019) 123 [arXiv:1907.00507] [INSPIRE]. · Zbl 1431.81061
[76] R.D. Sameshima, On Different Parametrizations of Feynman Integrals, Ph.D. Thesis, The City University of New York (2019) [https://academicworks.cuny.edu/gc etds/3376].
[77] R.P. Klausen, Hypergeometric Series Representations of Feynman Integrals by GKZ Hypergeometric Systems, arXiv:1910.08651 [INSPIRE].
[78] T. Bitoun, C. Bogner, R.P. Klausen and E. Panzer, Feynman integral relations from parametric annihilators, Lett. Math. Phys.109 (2019) 497 [arXiv:1712.09215] [INSPIRE]. · Zbl 1412.81141
[79] A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett.B 254 (1991) 158 [INSPIRE]. · Zbl 1020.81734
[80] E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim.A 110 (1997) 1435 [hep-th/9711188] [INSPIRE]. · Zbl 0984.58004
[81] 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
[82] V.A. Smirnov, Feynman integral calculus, Springer (2006) [INSPIRE].
[83] M. Argeri and P. Mastrolia, Feynman Diagrams and Differential Equations, Int. J. Mod. Phys.A 22 (2007) 4375 [arXiv:0707.4037] [INSPIRE]. · Zbl 1141.81325
[84] J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett.110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].
[85] J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP07 (2013) 128 [arXiv:1306.2799] [INSPIRE]. · Zbl 1342.81352
[86] J.M. Henn and V.A. Smirnov, Analytic results for two-loop master integrals for Bhabha scattering I, JHEP11 (2013) 041 [arXiv:1307.4083] [INSPIRE].
[87] J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP03 (2014) 088 [arXiv:1312.2588] [INSPIRE].
[88] M. Barkatou, S.S. Maddah and H. Abbas, On the reduction of singularly-perturbed linear differential systems, arXiv:1401.5438. · Zbl 1325.68294
[89] S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, JHEP06 (2014) 114 [arXiv:1404.2922] [INSPIRE]. · Zbl 1333.81217
[90] M. Argeri et al., Magnus and Dyson Series for Master Integrals, JHEP03 (2014) 082 [arXiv:1401.2979] [INSPIRE]. · Zbl 1333.81379
[91] T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs, The two-loop master integrals for q \(\overline{q} → V V\) , JHEP06 (2014) 032 [arXiv:1404.4853] [INSPIRE].
[92] M. Höschele, J. Hoff and T. Ueda, Adequate bases of phase space master integrals for gg → h at NNLO and beyond, JHEP09 (2014) 116 [arXiv:1407.4049] [INSPIRE].
[93] R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP04 (2015) 108 [arXiv:1411.0911] [INSPIRE]. · Zbl 1388.81109
[94] O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, Comput. Phys. Commun.219 (2017) 329 [arXiv:1701.04269] [INSPIRE]. · Zbl 1411.81015
[95] R.N. Lee, A.V. Smirnov and V.A. Smirnov, Solving differential equations for Feynman integrals by expansions near singular points, JHEP03 (2018) 008 [arXiv:1709.07525] [INSPIRE]. · Zbl 1388.81927
[96] E. Herrmann and J. Parra-Martinez, Logarithmic forms and differential equations for Feynman integrals, JHEP02 (2020) 099 [arXiv:1909.04777] [INSPIRE].
[97] D.R. Grayson and M.E. Stillman, Macaulay2, a software system for research in algebraic geometry, available at http://www.math.uiuc.edu/Macaulay2/.
[98] A.V. Smirnov and F.S. Chuharev, FIRE6: Feynman Integral REduction with Modular Arithmetic, arXiv:1901.07808 [INSPIRE].
[99] 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
[100] E. Remiddi and L. Tancredi, Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph, Nucl. Phys.B 880 (2014) 343 [arXiv:1311.3342] [INSPIRE]. · Zbl 1284.81139
[101] 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
[102] 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
[103] M.Y. 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
[104] C. Bogner, S. Müller-Stach and S. Weinzierl, The unequal mass sunrise integral expressed through iterated integrals on \({\overline{\mathcal{M}}}_{1,3},\) arXiv:1907.01251 [INSPIRE].
[105] A. Strominger, The Inverse Dimensional Expansion in Quantum Gravity, Phys. Rev.D 24 (1981) 3082 [INSPIRE]. · Zbl 1267.83043
[106] N.E.J. Bjerrum-Bohr, Quantum gravity at a large number of dimensions, Nucl. Phys.B 684 (2004) 209 [hep-th/0310263] [INSPIRE]. · Zbl 1107.83308
[107] H.W. Hamber and R.M. Williams, Quantum gravity in large dimensions, Phys. Rev.D 73 (2006) 044031 [hep-th/0512003] [INSPIRE].
[108] F.C.S. Brown, On the periods of some Feynman integrals, arXiv:0910.0114 [INSPIRE].
[109] F. Brown and O. Schnetz, A K 3 in 𝜙^4 , Duke Math. J.161 (2012) 1817 [arXiv:1006.4064] [INSPIRE]. · Zbl 1253.14024
[110] J.L. Bourjaily, Y.-H. He, A.J. Mcleod, M. Von Hippel and M. Wilhelm, Traintracks through Calabi-Yau Manifolds: Scattering Amplitudes beyond Elliptic Polylogarithms, Phys. Rev. Lett.121 (2018) 071603 [arXiv:1805.09326] [INSPIRE].
[111] P. Vanhove, Feynman integrals, toric geometry and mirror symmetry, in Proceedings, KMPB Conference: Elliptic Integrals, Elliptic Functions and Modular Forms in Quantum Field Theory, Zeuthen, Germany, 23-26 October 2017, pp. 415-458 (2019) [DOI] [arXiv:1807.11466] [INSPIRE].
[112] D. Festi and D. van Straten, Bhabha Scattering and a special pencil of K 3 surfaces, Commun. Num. Theor. Phys.13 (2019) 463 [arXiv:1809.04970] [INSPIRE]. · Zbl 07060218
[113] J.L. Bourjaily, A.J. McLeod, M. von Hippel and M. Wilhelm, Bounded Collection of Feynman Integral Calabi-Yau Geometries, Phys. Rev. Lett.122 (2019) 031601 [arXiv:1810.07689] [INSPIRE].
[114] M. Besier, D. Festi, M. Harrison and B. Naskrecki, Arithmetic and geometry of a K 3 surface emerging from virtual corrections to Drell-Yan scattering, arXiv:1908.01079 [INSPIRE].
[115] J.L. Bourjaily, A.J. McLeod, C. Vergu, M. Volk, M. Von Hippel and M. Wilhelm, Embedding Feynman Integral (Calabi-Yau) Geometries in Weighted Projective Space, JHEP01 (2020) 078 [arXiv:1910.01534] [INSPIRE].
[116] D.E. Roberts, Mathematical structure of dual amplitudes, Ph.D. Thesis, Durham University (1972).
[117] D.B. Fairlie and D.E. Roberts, Dual Models Without Tachyons — A New Approach, PRINT-72-2440 (1972) [INSPIRE].
[118] E. Witten, The Feynman i𝜖 in String Theory, JHEP04 (2015) 055 [arXiv:1307.5124] [INSPIRE].
[119] F. Pham, Vanishing homologies and the n variables saddlepoint method, in Singularities, Part 2, Proc. Symp. Pure Math.40 (1983) 310. · Zbl 0519.49026
[120] E. Arnold, S. Gusein-Zade and A. Varchenko, Singularities of Differentiable Maps, Volume 2: Monodromy and Asymptotics of Integrals, Modern Birkhäuser Classics, Birkhäuser Boston (2012).
[121] M.V. Berry, Infinitely Many Stokes Smoothings in the Gamma Function, Proc. Roy. Soc. Lond.434 (1991) 465. · Zbl 0729.33002
[122] W.G.C. Boyd, Gamma function asymptotics by an extension of the method of steepest descents, Proc. Roy. Soc. Lond.A 447 (1994) 609. · Zbl 0829.33001
[123] E. Plahte, Symmetry properties of dual tree-graph n-point amplitudes, Nuovo Cim.A 66 (1970) 713 [INSPIRE].
[124] J.-C. Lee and Y. Yang, Review on High energy String Scattering Amplitudes and Symmetries of String Theory, arXiv:1510.03297 [INSPIRE].
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.