Geodesic
New results for a two-loop massless propagator-type Feynman diagram
Teoretičeskaâ i matematičeskaâ fizika, Tome 194 (2018) no. 2, pp. 331-342 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a two-loop massless propagator-type Feynman diagram with an arbitrary (noninteger) index on the central line. We analytically prove the equality of two well-known results in the literature expressing this diagram in terms of hypergeometric functions ${}_3F_2$ of the respective arguments $-1$ and $1$. We also derive new representations for this diagram, which can be important in practical calculations.
Keywords: Feynman diagram, hypergeometric function.
Mots-clés : multiloop calculation 
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     author = {A. V. Kotikov and S. Teber},
     title = {New results for a~two-loop massless propagator-type {Feynman} diagram},
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}
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A. V. Kotikov; S. Teber. New results for a two-loop massless propagator-type Feynman diagram. Teoretičeskaâ i matematičeskaâ fizika, Tome 194 (2018) no. 2, pp. 331-342. http://geodesic.mathdoc.fr/item/TMF_2018_194_2_a7/

[1] A. G. Grozin, “Massless two-loop self-energy diagram: historical review”, Internat. J. Modern Phys. A, 27:19 (2012), 1230018, 22 pp., arXiv: 1206.2572 | DOI | MR | Zbl

[2] A. N. Vasilev, Yu. M. Pismak, Yu. R. Khonkonen, “$1/n$-Razlozhenie: raschet indeksov $\eta$ i $\nu$ v poryadke $1/n^2$ dlya proizvolnoi razmernosti”, TMF, 47:3 (1981), 291–306 | DOI

[3] K. G. Chetyrkin, F. V. Tkachov, “Integration by parts: the algorithm to calculate $\beta$-functions in 4 loops”, Nucl. Phys. B, 192:1 (1981), 159–204 | DOI

[4] S. J. Hathrell, “Trace anomalies and $\lambda \phi^4$ theory in curved space”, Ann. Phys., 139:1 (1982), 136–197 | DOI | MR

[5] K. G. Chetyrkin, A. L. Kataev, F. V. Tkachov, “New approach to evaluation of multiloop Feynman integrals: the Gegenbauer polynomial $x$-space technique”, Nucl. Phys. B, 174:2–3 (1980), 345–377 | DOI | MR

[6] D. I. Kazakov, “Vychislenie feinmanovskikh integralov metodom ‘unikalnostei’ ”, TMF, 58:3 (1984), 343–353 ; D. I. Kazakov, “The method of uniqueness, a new powerful technique for multiloop calculations”, Phys. Lett. B, 133:6 (1983), 406–410 | DOI | MR | DOI

[7] D. I. Kazakov, “Mnogopetlevye vychisleniya: metod unikalnostei i funktsionalnye uravneniya”, TMF, 62:1 (1985), 127–135 ; D. I. Kazakov, Analytical methods for multiloop calculations: two lectures on the method of uniqueness, Preprint JINR E2-84-410, JINR, Dubna, 1984 | DOI | MR

[8] A. V. Kotikov, “The Gegenbauer polynomial technique: the evaluation of a class of Feynman diagrams”, Phys. Lett. B, 375:1–4 (1996), 240–248, arXiv: hep-ph/9512270 | DOI | MR | Zbl

[9] D. J. Broadhurst, “Exploiting the $1{,}440$-fold symmetry of the master two-loop diagram”, Z. Phys. C, 32:2 (1986), 249–253 ; D. T. Barfoot, D. J. Broadhurst, “$Z_2\times S_6$ symmetry of the two-loop diagram”, Z. Phys. C, 41:1 (1988), 81–85 | DOI | DOI

[10] J. A. Gracey, “On the evaluation of massless Feynman diagrams by the method of uniqueness”, Phys. Lett. B, 277:4 (1992), 469–473 | DOI | MR

[11] N. A. Kivel, A. S. Stepanenko, A. N. Vasil'ev, “On the calculation of $2+\varepsilon$ RG functions in the Gross–Neveu model from large-$N$ expansions of critical exponents”, Nucl. Phys. B, 424:3 (1994), 619–627, arXiv: ; А. Н. Васильев, С. Э. Деркачев, Н. А. Кивель, А. С. Степаненко, “$1/n$-Разложение в модели Гросса–Нэве: расчет индекса $\eta$ в порядке $1/n^3$ методом конформного бутстрапа”, ТМФ, 94:2 (1993), 179–192, arXiv: hep-th/9308073hep-th/9302034 | DOI | DOI

[12] D. J. Broadhurst, J. A. Gracey, D. Kreimer, “Beyond the triangle and uniqueness relations: non-zeta counterterms at large $N$ from positive knots”, Z. Phys. C, 75:3 (1997), 559–574, arXiv: hep-th/9607174 | DOI | MR

[13] D. J. Broadhurst, A. V. Kotikov, “Compact analytical form for non-zeta terms in critical exponents at order $1/N^3$”, Phys. Lett. B, 441:1–4 (1998), 345–353, arXiv: hep-th/9612013 | DOI

[14] D. J. Broadhurst, “Where do the tedious products of $\zeta$'s come from?”, Nucl. Phys. B Proc. Suppl., 116 (2003), 432–436, arXiv: hep-ph/0211194 | DOI | Zbl

[15] I. Bierenbaum, S. Weinzierl, “The massless two-loop two-point function”, Eur. Phys. J. C, 32:1 (2003), 67–78, arXiv: hep-ph/0308311 | DOI | Zbl

[16] S. Teber, “Electromagnetic current correlations in reduced quantum electrodynamics”, Phys. Rev. D, 86:2 (2012), 025005, 9 pp., arXiv: 1204.5664 | DOI

[17] A. V. Kotikov, S. Teber, “Note on an application of the method of uniqueness to reduced quantum electrodynamics”, Phys. Rev. D, 87:8 (2013), 087701, 5 pp., arXiv: 1302.3939 | DOI

[18] A. V. Kotikov, S. Teber, “Two-loop fermion self-energy in reduced quantum electrodynamics and application to the ultrarelativistic limit of graphene”, Phys. Rev. D, 89:6 (2014), 065038, 24 pp., arXiv: 1312.2430 | DOI

[19] A. N. Vasilev, Kvantovopolevaya renormgruppa v teorii kriticheskogo povedeniya i stokhasticheskoi dinamike, Izd-vo PIYaF, SPb., 1988 | MR | Zbl

[20] S. Teber, A. V. Kotikov, “Interaction corrections to the minimal conductivity of graphene via dimensional regularization”, Europhys. Lett., 107:5 (2014), 57001, 7 pp., arXiv: 1407.7501 | DOI

[21] S. Teber, A. V. Kotikov, “Metod unikalnostei i opticheskaya provodimost grafena: novoe primenenie moschnoi tekhniki mnogopetlevykh vychislenii”, TMF, 190:3 (2017), 519–532, arXiv: 1602.01962 | DOI | DOI | MR

[22] S. G. Gorishnii, A. P. Isaev, “Ob odnom podkhode k vychisleniyu mnogopetlevykh bezmassovykh feinmanovskikh integralov”, TMF, 62:3 (1985), 345–358 | DOI | MR

[23] A. V. Kotikov, L. N. Lipatov, “NLO corrections to the BFKL equation in QCD and in supersymmetric gauge theories”, Nucl. Phys. B, 582:1–3 (2000), 19–43, arXiv: hep-ph/0004008 | DOI

[24] A. V. Kotikov, L. N. Lipatov, “DGLAP and BFKL equations in the $N=4$ supersymmetric gauge theory”, Nucl. Phys. B, 661:1–2 (2003), 19–61 ; Erratum, 685, no. 1–3, 2004, 405–407, arXiv: hep-ph/0208220 | DOI | MR | Zbl | DOI

[25] W. N. Bailey, Generalized Hypergeometric Series, Cambridge Tracts in Mathematics and Mathematical Physics, 32, Cambridge Univ. Press, Cambridge, 1935 | MR

[26] A. V. Kotikov, V. I. Shilin, S. Teber, “Critical behavior of ($2+1$)-dimensional QED: $1/N_f$ corrections in the Landau gauge”, Phys. Rev. D, 94:5 (2016), 056009, 6 pp., arXiv: 1605.01911 | DOI

[27] A. V. Kotikov, S. Teber, “Critical behavior of ($2+1$)-dimensional QED: $1/N_f$ corrections in an arbitrary nonlocal gauge”, Phys. Rev. D, 94:11 (2016), 114011, 9 pp., arXiv: 1609.06912 | DOI

[28] A. V. Kotikov, S. Teber, “Critical behaviour of reduced QED$_{4,3}$ and dynamical fermion gap generation in graphene”, Phys. Rev. D, 94:11 (2016), 114010, 8 pp., arXiv: 1610.00934 | DOI