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The heavy fermion contributions to the massive three loop form factors. (English) Zbl 1448.81470

Authors’ abstract: We compute the non-singlet \(n_h\) terms to the massive three loop vector-, axialvector-, scalar- and pseudoscalar form factors in a direct analytic calculation using the method of large moments. This method has the advantage, that the master integrals have to be dealt with only in their moment representation, allowing to also consider quantities which obey differential equations, which are not first order factorizable (elliptic and higher), already at this level. To obtain all the associated recursions, up to 8000 moments had to be calculated. A new technique has been applied to solve the associated differential equation systems. Here the decoupling is performed such, that only minimal depth \(\varepsilon\)-expansions had to be performed for non-first-order factorizing systems, minimizing the calculation of initial values. The pole terms in the dimensional parameter \(\varepsilon\) can be completely predicted using renormalization group methods, as confirmed by the present results. A series of contributions at \(O(\varepsilon^0)\) have first order factorizable representations. For a smaller number of color-zeta projections this is not the case. All first order factorizing terms can be represented by harmonic polylogarithms. We also obtain analytic results for the non-first-order factorizing terms by Taylor series in a variable \(x\), for which we have calculated at least 2000 expansion coefficients, in an approximation. Based on this representation the form factors can be given in the Euclidean region and in the region \(q^2 \approx 0\). Numerical results are presented.

MSC:

81V35 Nuclear physics
81R15 Operator algebra methods applied to problems in quantum theory
81T17 Renormalization group methods applied to problems in quantum field theory
41A58 Series expansions (e.g., Taylor, Lidstone series, but not Fourier series)
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References:

[1] Bernreuther, W.; Bonciani, R.; Gehrmann, T.; Heinesch, R.; Leineweber, T.; Mastrolia, P.; Remiddi, E., Nucl. Phys. B, 706, 245-324 (2005)
[2] Bernreuther, W.; Bonciani, R.; Gehrmann, T.; Heinesch, R.; Leineweber, T.; Mastrolia, P.; Remiddi, E., Nucl. Phys. B, 712, 229-286 (2005)
[3] Bernreuther, W.; Bonciani, R.; Gehrmann, T.; Heinesch, R.; Leineweber, T.; Remiddi, E., Nucl. Phys. B, 723, 91-116 (2005)
[4] Bernreuther, W.; Bonciani, R.; Gehrmann, T.; Heinesch, R.; Mastrolia, P.; Remiddi, E., Phys. Rev. D, 72, Article 096002 pp. (2005)
[5] Gluza, J.; Mitov, A.; Moch, S.; Riemann, T., JHEP, 07, Article 001 pp. (2009)
[6] Ablinger, J.; Behring, A.; Blümlein, J.; Falcioni, G.; De Freitas, A.; Marquard, P.; Rana, N.; Schneider, C., Phys. Rev. D, 97, 9, Article 094022 pp. (2018)
[7] Henn, J.; Smirnov, A. V.; Smirnov, V. A.; Steinhauser, M., JHEP, 1701, Article 074 pp. (2017)
[8] Henn, J. M.; Smirnov, A. V.; Smirnov, V. A., JHEP, 12, Article 144 pp. (2016)
[9] Ablinger, J.; Blümlein, J.; Marquard, P.; Rana, N.; Schneider, C., Phys. Lett. B, 782, 528-532 (2018)
[10] Lee, R. N.; Smirnov, A. V.; Smirnov, V. A.; Steinhauser, M., JHEP, 03, Article 136 pp. (2018)
[11] Lee, R. N.; Smirnov, A. V.; Smirnov, V. A.; Steinhauser, M., JHEP, 05, Article 187 pp. (2018)
[12] Ablinger, J.; Blümlein, J.; Marquard, P.; Rana, N.; Schneider, C., Nucl. Phys. B, 939, 253-291 (2019)
[13] Ahmed, T.; Henn, J. M.; Steinhauser, M., JHEP, 06, Article 125 pp. (2017)
[14] Blümlein, J.; Marquard, P.; Rana, N., Phys. Rev. D, 99, 1, Article 016013 pp. (2019)
[15] Blümlein, J.; Schneider, C., Phys. Lett. B, 771, 31-36 (2017)
[16] Lagrange, J., Nouvelles recherches sur la nature et la propagation du son, Miscellanea Taurinensis t. II (1760-61), 263 pp
[17] Gauß, C. F., Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo novo tractate, Commentationes societas scientiarum Gottingensis recentiores III, 5-7 (1813)
[18] Green, G., Essay on the Mathematical Theory of Electricity and Magnetism (1828), printed for the author, by T. Wheelhouse, Green Papers, pp. 1-115
[19] Ostrogradski, M., Mem. Ac. Sci. St. Peters., 6, 129-133 (1831)
[20] Chetyrkin, K. G.; Tkachov, F. V., Nucl. Phys. B, 192, 159-204 (1981)
[21] Laporta, S., Int. J. Mod. Phys. A, 15, 5087-5159 (2000)
[22] P. Marquard and D. Seidel, The algorithm, unpublished.
[23] Schneider, C., (Schneider, C.; Blümlein, J., Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions. Computer Algebra in Quantum Field Theory: Integration, Summation and Special Functions, Texts and Monographs in Symbolic Computation (2013), Springer: Springer Wien), 325-360
[24] Blümlein, J.; Broadhurst, D. J.; Vermaseren, J. A.M., Comput. Phys. Commun., 181, 582-625 (2010)
[25] Kauers, M.; Jaroschek, M.; Johansson, F., (Gutierrez, J.; Schicho Josef, J.; Weimann, M., Computer Algebra and Polynomials. Computer Algebra and Polynomials, Lecture Notes in Computer Science, vol. 8942 (2015), Springer: Springer Berlin), 105-125
[26] Blümlein, J.; Kauers, M.; Klein, S.; Schneider, C., Comput. Phys. Commun., 180, 2143-2165 (2009)
[27] Schneider, C., Sém. Lothar. Combin., 56, 1-36 (2007)
[28] Blümlein, J.; Schneider, C., Int. J. Mod. Phys. A, 33, 17, Article 1830015 pp. (2018)
[29] Remiddi, E.; Vermaseren, J. A.M., Int. J. Mod. Phys. A, 15, 725-754 (2000)
[30] Nogueira, P., J. Comput. Phys., 105, 279-289 (1993)
[31] van Ritbergen, T.; Schellekens, A. N.; Vermaseren, J. A.M., Int. J. Mod. Phys. A, 14, 41-96 (1999)
[32] Harlander, R.; Seidensticker, T.; Steinhauser, M., Phys. Lett. B, 426, 125-132 (1998)
[33] Seidensticker, T., (Proc. of the 6th International Workshop on New Computing Techniques in Physics Research. Proc. of the 6th International Workshop on New Computing Techniques in Physics Research, AIHENP 99, Heraklion, Crete, Greece, April 12-16 (1999)), 12-16
[34] Vermaseren, J. A.M., New features of FORM
[35] Tentyukov, M.; Vermaseren, J. A.M., Comput. Phys. Commun., 181, 1419-1427 (2010)
[36] C. Schneider, in preparation.
[37] Melnikov, K.; van Ritbergen, T., Nucl. Phys. B, 591, 515-546 (2000)
[38] Marquard, P.; Piclum, J. H.; Seidel, D.; Steinhauser, M., Nucl. Phys. B, 758, 144-160 (2006)
[39] Kurz, A.; Liu, T.; Marquard, P.; Smirnov, A.; Smirnov, V.; Steinhauser, M., Phys. Rev. D, 93, 5, Article 053017 pp. (2016)
[40] Marquard, P.; Smirnov, A. V.; Smirnov, V. A.; Steinhauser, M., Phys. Rev. Lett., 114, 14, Article 142002 pp. (2015)
[41] Marquard, P.; Mihaila, L.; Piclum, J. H.; Steinhauser, M., Nucl. Phys. B, 773, 1-18 (2007)
[42] Bierenbaum, I.; Blumlein, J.; Klein, S., Nucl. Phys. B, 820, 417-482 (2009)
[43] Blümlein, J.; Ablinger, J.; Behring, A.; De Freitas, A.; von Manteuffel, A.; Schneider, C.; Schneider, C., PoS (QCDEV) (2017), 031
[44] Sage · Zbl 1312.68206
[45] Karr, M., J. ACM, 28, 305-350 (1981)
[46] Bronstein, M., J. Symb. Comput., 29, 6, 841-877 (2000)
[47] Schneider, C., Symbolic Summation in Difference Fields (2001), Johannes Kepler University, Ph.D. Thesis RISC, Linz technical report 01-17
[48] Schneider, C., An. Univ. Timisoara Ser. Mat.-Inform., 42, 163-179 (2004)
[49] Schneider, C., J. Differ. Equ. Appl., 11, 799-821 (2005)
[50] Schneider, C., Appl. Algebra Eng. Commun. Comput., 16, 1-32 (2005)
[51] Schneider, C., J. Algebra Appl., 6, 415-441 (2007)
[52] Schneider, C., (Carey, A.; Ellwood, D.; Paycha, S.; Rosenberg, S., Motives, Quantum Field Theory, and Pseudodifferential Operators. Motives, Quantum Field Theory, and Pseudodifferential Operators, Clay Mathematics Proceedings, vol. 12 (2010), Amer. Math. Soc.), 285-308
[53] Schneider, C., Ann. Comb., 14, 533-552 (2010)
[54] Schneider, C., (Gutierrez, J.; Schicho, J.; Weimann, M., Computer Algebra and Polynomials, Applications of Algebra and Number Theory. Computer Algebra and Polynomials, Applications of Algebra and Number Theory, Lecture Notes in Computer Science (LNCS), vol. 8942 (2015)), 157-191
[55] Schneider, C., J. Symb. Comput., 43, 611-644 (2008)
[56] Schneider, C., J. Symb. Comput., 72, 82-127 (2016)
[57] Schneider, C., J. Symb. Comput., 80, 616-664 (2017)
[58] Vermaseren, J., Int. J. Mod. Phys. A, 14, 2037-2076 (1999)
[59] Blümlein, J.; Kurth, S., Phys. Rev. D, 60, Article 014018 pp. (1999)
[60] Ablinger, J.; Blümlein, J.; Schneider, C., J. Math. Phys., 54, Article 082301 pp. (2013)
[61] Ablinger, J.; Blümlein, J.; Schneider, C., J. Math. Phys., 52, Article 102301 pp. (2011)
[62] Ablinger, J.; Blümlein, J.; Raab, C. G.; Schneider, C., J. Math. Phys., 55, Article 112301 pp. (2014)
[63] Ablinger, J., PoS, LL2014, Article 019 pp. (2014)
[64] Ablinger, J., A Computer Algebra Toolbox for Harmonic Sums Related to Particle Physics (2009), JKU Linz, Diploma Thesis
[65] Ablinger, J., Computer Algebra Algorithms for Special Functions in Particle Physics (2012), Linz U, Ph.D. Thesis
[66] Ablinger, J., PoS, RADCOR2017, Article 001 pp. (2017)
[67] Kummer, E. E., J. Reine Angew. Math. (Crelle), 21, 74-90 (1840), 193-225, 328-371
[68] Poincaré, H., Acta Math., 4, 201-312 (1884)
[69] Moch, S.; Uwer, P.; Weinzierl, S., J. Math. Phys., 43, 3363-3386 (2002)
[70] Catani, S., Phys. Lett. B, 427, 161-171 (1998)
[71] Sterman, G. F.; Tejeda-Yeomans, M. E., Phys. Lett. B, 552, 48-56 (2003)
[72] Becher, T.; Neubert, M., Phys. Rev. Lett.. Phys. Rev. Lett., Phys. Rev. Lett., 111, 19, Article 199905 pp. (2013), Erratum:
[73] Gardi, E.; Magnea, L., JHEP, 0903, Article 079 pp. (2009)
[74] Ravindran, V.; Smith, J.; van Neerven, W. L., Nucl. Phys. B, 704, 332-348 (2005)
[75] Moch, S.; Vermaseren, J. A.M.; Vogt, A., Phys. Lett. B, 625, 245-252 (2005)
[76] Mitov, A.; Moch, S., JHEP, 0705, Article 001 pp. (2007)
[77] Penin, A. A., Nucl. Phys. B, 734, 185-202 (2006)
[78] Becher, T.; Melnikov, K., JHEP, 0706, Article 084 pp. (2007)
[79] Ravindran, V., Nucl. Phys. B, 746, 58-76 (2006)
[80] Ravindran, V., Nucl. Phys. B, 752, 173-196 (2006)
[81] Ahmed, T.; Mahakhud, M.; Rana, N.; Ravindran, V., Phys. Rev. Lett., 113, 11, Article 112002 pp. (2014)
[82] Ahmed, T.; Rana, N.; Ravindran, V., JHEP, 1410, Article 139 pp. (2014)
[83] Becher, T.; Neubert, M., Phys. Rev. D. Phys. Rev. D, Phys. Rev. D, 80, Article 109901 pp. (2009), Erratum:
[84] Weinberg, S., Phys. Lett. B, 91, 51-55 (1980)
[85] Ovrut, B. A.; Schnitzer, H. J., Phys. Lett. B, 100, 403-406 (1981)
[86] Wetzel, W., Nucl. Phys. B, 196, 259-272 (1982)
[87] Bernreuther, W.; Wetzel, W., Nucl. Phys. B. Nucl. Phys. B, Nucl. Phys. B, 513, 758-236 (1998), Erratum:
[88] Bernreuther, W., Ann. Phys., 151, 127-162 (1983)
[89] Chetyrkin, K. G.; Kniehl, B. A.; Steinhauser, M., Nucl. Phys. B, 510, 61-87 (1998)
[90] Korchemsky, G. P.; Radyushkin, A. V., Nucl. Phys. B, 283, 342-364 (1987)
[91] Korchemsky, G. P.; Radyushkin, A. V., Phys. Lett. B, 279, 359-366 (1992)
[92] Grozin, A.; Henn, J. M.; Korchemsky, G. P.; Marquard, P., Phys. Rev. Lett., 114, 6, Article 062006 pp. (2015)
[93] Grozin, A.; Henn, J. M.; Korchemsky, G. P.; Marquard, P., JHEP, 1601, Article 140 pp. (2016)
[94] Grozin, A. G.; Marquard, P.; Piclum, J. H.; Steinhauser, M., Nucl. Phys. B, 789, 277-293 (2008)
[95] Grozin, A. G.; Höschele, M.; Hoff, J.; Steinhauser, M., JHEP, 1109, Article 066 pp. (2011)
[96] S.A. Larin, F.V. Tkachov, J.A.M. Vermaseren, The FORM version of MINCER NIKHEF-H-91-18.
[97] Steinhauser, M., Comput. Phys. Commun., 134, 335-364 (2001)
[98] Boughezal, R.; Czakon, M.; Schutzmeier, T., Nucl. Phys. B, Proc. Suppl., 160, 160-164 (2006)
[99] Maier, A.; Maierhofer, P.; Marquard, P., Nucl. Phys. B, 797, 218-242 (2008)
[100] Maier, A.; Marquard, P., Nucl. Phys. B, 859, 1-12 (2012)
[101] Maier, A.; Marquard, P., Nucl. Phys. B, 899, 451-462 (2015)
[102] Maierhöfer, P.; Marquard, P., Phys. Lett. B, 721, 131-135 (2013)
[103] Lee, R. N.; Smirnov, A. V.; Smirnov, V. A., JHEP, 1803, Article 008 pp. (2018)
[104] Kudashkin, K.; Melnikov, K.; Wever, C., JHEP, 1802, Article 135 pp. (2018)
[105] Ablinger, J.; Behring, A.; Blümlein, J.; De Freitas, A.; von Manteuffel, A.; Schneider, C., Comput. Phys. Commun., 202, 33-112 (2016)
[106] Gerhold, S., Uncoupling Systems of Linear Ore Operator Equations (2002), J. Kepler University: J. Kepler University Linz, Master’s thesis, RISC
[107] Zürcher, B., Rationale Normalformen von pseudo-linearen Abbildungen (1994), ETH Zürich, Master’s thesis, Mathematik
[108] Blümlein, J.; Klein, S.; Schneider, C.; Stan, F., J. Symb. Comput., 47, 1267-1289 (2012)
[109] Ablinger, J.; Blümlein, J.; De Freitas, A.; van Hoeij, M.; Imamoglu, E.; Raab, C. G.; Radu, C. S.; Schneider, C., J. Math. Phys., 59, 6, Article 062305 pp. (2018)
[110] Broadhurst, D. J.; Gray, N.; Schilcher, K., Z. Phys. C, 52, 111-122 (1991)
[111] Marquard, P.; Smirnov, A. V.; Smirnov, V. A.; Steinhauser, M.; Wellmann, D., Phys. Rev. D, 94, 7, Article 074025 pp. (2016)
[112] Marquard, P.; Smirnov, A. V.; Smirnov, V. A.; Steinhauser, M., Phys. Rev. D, 97, 5, Article 054032 pp. (2018)
[113] Tarasov, O. V.; Vladimirov, A. A.; Zharkov, A. Y., Phys. Lett. B, 93, 429-432 (1980)
[114] Larin, S. A.; Vermaseren, J. A.M., Phys. Lett. B, 303, 334-336 (1993)
[115] van Ritbergen, T.; Vermaseren, J. A.M.; Larin, S. A., Phys. Lett. B, 400, 379-384 (1997)
[116] Czakon, M., Nucl. Phys. B, 710, 485-498 (2005)
[117] Baikov, P. A.; Chetyrkin, K. G.; Kühn, J. H., Phys. Rev. Lett., 118, 8, Article 082002 pp. (2017)
[118] Herzog, F.; Ruijl, B.; Ueda, T.; Vermaseren, J. A.M.; Vogt, A., JHEP, 02, Article 090 pp. (2017)
[119] Luthe, T.; Maier, A.; Marquard, P.; Schröder, Y., JHEP, 10, Article 166 pp. (2017)
[120] Davydychev, A. I.; Kalmykov, M. Y., Nucl. Phys. B, 699, 3-64 (2004)
[121] Broadhurst, D. J., Z. Phys. C, 54, 599-606 (1992)
[122] Broadhurst, D. J.; Fleischer, J.; Tarasov, O. V., Z. Phys. C, 60, 287-302 (1993)
[123] Blümlein, J.; Hasselhuhn, A.; Schneider, C., PoS, RADCOR2011, Article 032 pp. (2019)
[124] Schneider, C., J. Phys. Conf. Ser., 523, Article 012037 pp. (2014)
[125] J. Vermaseren, private communication, July 2018.
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