Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 087, 26 pages      arXiv:2104.00593      https://doi.org/10.3842/SIGMA.2021.087
Contribution to the Special Issue on Algebraic Structures in Perturbative Quantum Field Theory in honor of Dirk Kreimer for his 60th birthday

Semiclassical Trans-Series from the Perturbative Hopf-Algebraic Dyson-Schwinger Equations: $\phi^3$ QFT in 6 Dimensions

Michael Borinsky a, Gerald V. Dunne b and Max Meynig b
a) Nikhef Theory Group, Amsterdam 1098 XG, The Netherlands
b) Department of Physics, University of Connecticut, Storrs CT 06269-3046, USA

Received April 07, 2021, in final form September 16, 2021; Published online September 23, 2021

Abstract
We analyze the asymptotically free massless scalar $\phi^3$ quantum field theory in 6 dimensions, using resurgent asymptotic analysis to find the trans-series solutions which yield the non-perturbative completion of the divergent perturbative solutions to the Kreimer-Connes Hopf-algebraic Dyson-Schwinger equations for the anomalous dimension. This scalar conformal field theory is asymptotically free and has a real Lipatov instanton. In the Hopf-algebraic approach we find a trans-series having an intricate Borel singularity structure, with three distinct but resonant non-perturbative terms, each repeated in an infinite series. These expansions are in terms of the renormalized coupling. The resonant structure leads to powers of logarithmic terms at higher levels of the trans-series, analogous to logarithmic terms arising from interactions between instantons and anti-instantons, but arising from a purely perturbative formalism rather than from a semi-classical analysis.

Key words: renormalons; resurgence; non-perturbative corrections; quantum field theory; renormalization; Hopf algebra; trans-series.

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References

  1. Álvarez G., Howls C.J., Silverstone H.J., Anharmonic oscillator discontinuity formulae up to second-exponentially-small order, J. Phys. A: Math. Gen. 35 (2002), 4003-4016.
  2. Anber M.M., Sulejmanpasic T., The renormalon diagram in gauge theories on $\mathbb R^3\times \mathbb S^1$, J. High Energy Phys. 2015 (2015), no. 1, 139, 34 pages, arXiv:1410.0121.
  3. Aniceto I., Başar G., Schiappa R., A primer on resurgent transseries and their asymptotics, Phys. Rep. 809 (2019), 1-135, arXiv:1802.10441.
  4. Aniceto I., Schiappa R., Vonk M., The resurgence of instantons in string theory, Commun. Number Theory Phys. 6 (2012), 339-496, arXiv:1106.5922.
  5. Bellon M.P., An efficient method for the solution of Schwinger-Dyson equations for propagators, Lett. Math. Phys. 94 (2010), 77-86, arXiv:1005.0196.
  6. Bellon M.P., Clavier P.J., Alien calculus and a Schwinger-Dyson equation: two-point function with a nonperturbative mass scale, Lett. Math. Phys. 108 (2018), 391-412, arXiv:1612.07813.
  7. Bellon M.P., Russo E.I., Resurgent analysis of Ward-Schwinger-Dyson equations, SIGMA 17 (2021), 075, 18 pages, arXiv:2011.13822.
  8. Bellon M.P., Russo E.I., Ward-Schwinger-Dyson equations in $\phi^3_6$ quantum field theory, Lett. Math. Phys. 111 (2021), 42, 31 pages, arXiv:2007.15675.
  9. Bellon M.P., Schaposnik F.A., Renormalization group functions for the Wess-Zumino model: up to 200 loops through Hopf algebras, Nuclear Phys. B 800 (2008), 517-526, arXiv:0801.0727.
  10. Bender C.M., Orszag S.A., Advanced mathematical methods for scientists and engineers. I. Asymptotic methods and perturbation theory, Springer-Verlag, New York, 1999.
  11. Benedetti D., Delporte N., Remarks on a melonic field theory with cubic interactions, J. High Energy Phys. 2021 (2021), no. 4, 197, 30 pages, arXiv:2012.12238.
  12. Berry M.V., Howls C.J., Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. London Ser. A 434 (1991), 657-675.
  13. Bersini J., Maiezza A., Vasquez J.C., Resurgence of the renormalization group equation, Ann. Physics 415 (2020), 168126, 15 pages, arXiv:1910.14507.
  14. Bogner C., Borowka S., Hahn T., Heinrich G., Jones S.P., Kerner M., von Manteuffel A., Michel M., Panzer E., Papara V., Loopedia, a database for loop integrals, Comput. Phys. Commun. 225 (2018), 1-9, arXiv:1709.01266.
  15. Borinsky M., Generating asymptotics for factorially divergent sequences, Electron. J. Combin. 25 (2018), 4.1, 32 pages, arXiv:1603.01236.
  16. Borinsky M., Graphs in perturbation theory: algebraic structure and asymptotics, Springer Theses, Springer, Cham, 2018.
  17. Borinsky M., Tropical Monte Carlo quadrature for Feynman integrals, arXiv:2008.12310.
  18. Borinsky M., Dunne G.V., Non-perturbative completion of Hopf-algebraic Dyson-Schwinger equations, Nuclear Phys. B 957 (2020), 115096, 17 pages, arXiv:2005.04265.
  19. Borinsky M., Gracey J.A., Kompaniets M.V., Schnetz O., Five-loop renormalization of $\phi^3$ theory with applications to the Lee-Yang edge singularity and percolation theory, Phys. Rev. D 103 (2021), 116024, 35 pages, arXiv:2103.16224.
  20. Borinsky M., Schnetz O., Graphical functions in even dimensions, arXiv:2105.05015.
  21. Brezin E., Le Guillou J.C., Zinn-Justin J., Perturbation theory at large order. I. The $\phi^{2N}$ interaction, Phys. Rev. D 15 (1977), 1544-1557.
  22. Broadhurst D.J., Kreimer D., Combinatoric explosion of renormalization tamed by Hopf algebra: 30-loop Padé-Borel resummation, Phys. Lett. B 475 (2000), 63-70, arXiv:hep-th/9912093.
  23. Broadhurst D.J., Kreimer D., Exact solutions of Dyson-Schwinger equations for iterated one loop integrals and propagator coupling duality, Nuclear Phys. B 600 (2001), 403-422, arXiv:hep-th/0012146.
  24. Caliceti E., Meyer-Hermann M., Ribeca P., Surzhykov A., Jentschura U.D., From useful algorithms for slowly convergent series to physical predictions based on divergent perturbative expansions, Phys. Rep. 446 (2007), 1-96, arXiv:0707.1596.
  25. Caprini I., Fischer J., Abbas G., Ananthanarayan B., Perturbative expansions in QCD improved by conformal mappings of the Borel plane, in Perturbation Theory: Advances in Research and Applications, Nova Science Publishers, Inc., Hauppauge, 2018, 211-254, arXiv:1711.04445.
  26. Cardy J.L., High-energy behavior in $\phi^3$ theory in six-dimensions, Nuclear Phys. B 93 (1975), 525-546.
  27. Clavier P.J., Borel-Écalle resummation of a two-point function, Ann. Henri Poincaré 22 (2021), 2103-2136, arXiv:1912.03237.
  28. Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem, J. High Energy Phys. 1999 (1999), no. 9, 024, 8 pages, arXiv:hep-th/9909126.
  29. Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. II. The $\beta$-function, diffeomorphisms and the renormalization group, Comm. Math. Phys. 216 (2001), 215-241, arXiv:hep-th/0003188.
  30. Cornwall J.M., Morris D.A., Toy models of nonperturbative asymptotic freedom in $\phi^3$ in six-dimensions, Phys. Rev. D 52 (1995), 6074-6086, arXiv:hep-ph/9506293.
  31. Costin O., Exponential asymptotics, transseries, and generalized Borel summation for analytic, nonlinear, rank-one systems of ordinary differential equations, Int. Math. Res. Not. 1995 (1995), 377-417, arXiv:math.CA/0608414.
  32. Costin O., On Borel summation and Stokes phenomena for rank-$1$ nonlinear systems of ordinary differential equations, Duke Math. J. 93 (1998), 289-344, arXiv:math.CA/0608408.
  33. Costin O., Asymptotics and Borel summability, textitChapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics, Vol. 141, CRC Press, Boca Raton, FL, 2009.
  34. Costin O., Dunne G.V., Physical resurgent extrapolation, Phys. Lett. B 808 (2020), 135627, 7 pages, arXiv:2003.07451.
  35. Costin O., Dunne G.V., Uniformization and constructive analytic continuation of Taylor series, arXiv:2009.01962.
  36. Courtiel J., Yeats K., Next-tok leading log expansions by chord diagrams, Comm. Math. Phys. 377 (2020), 469-501, arXiv:1906.05139.
  37. Damburg R.J., Propin R.K., Graffi S., Grecchi V., Harrell E.M., Čívzek J., Paldus J., Silverstone H.J., $1/R$ expansion for $H_2^+$: analyticity, summability, asymptotics, and calculation of exponentially small terms, Phys. Rev. Lett. 52 (1984), 1112-1115.
  38. de Alcantara Bonfim O.F., Kirkham J.E., McKane A.J., Critical exponents to order $\epsilon^3$ for $\phi^3$ models of critical phenomena in six $\epsilon$-dimension, J. Phys. A: Math. Gen. 13 (1980), L247-L251, Erratum, J. Phys. A: Math. Gen. 13 (1980), 3785-3785.
  39. de Alcantara Bonfim O.F., Kirkham J.E., McKane A.J., Critical exponents for the percolation problem and the Yang-Lee edge singularity, J. Phys. A: Math. Gen. 13 (1981), 2391-2413.
  40. Delabaere E., Dillinger H., Pham F., Exact semiclassical expansions for one-dimensional quantum oscillators, J. Math. Phys. 38 (1997), 6126-6184.
  41. Delabaere E., Pham F., Resurgent methods in semi-classical asymptotics, Ann. Inst. H. Poincaré Phys. Théor. 71 (1999), 1-94.
  42. Delbourgo R., Elliott D., McAnally D.S., Dimensional renormalization in $\phi^{3}$ theory: ladders and rainbows, Phys. Rev. D 55 (1997), 5230-5233, arXiv:hep-th/9611150.
  43. Dondi N.A., Dunne G.V., Reichert M., Sannino F., Towards the QED beta function and renormalons at $1/N^2_f$ and $1/N^3_f$, Phys. Rev. D 102 (2020), 035005, 15 pages, arXiv:2003.08397.
  44. Dorigoni D., An introduction to resurgence, trans-series and alien calculus, Ann. Physics 409 (2019), 167914, 38 pages, arXiv:1411.3585.
  45. Dunne G.V., Ünsal M., Resurgence and trans-series in quantum field theory: the $\mathbb{CP}^{N-1}$ model, J. High Energy Phys. 2012 (2012), no. 11, 170, 85 pages, arXiv:1210.2423.
  46. Dunne G.V., Ünsal M., Generating nonperturbative physics from perturbation theory, Phys. Rev. D 89 (2014), 041701, 5 pages, arXiv:1306.4405.
  47. Dunne G.V., Ünsal M., New nonperturbative methods in quantum field theory: from large-$N$ orbifold equivalence to bions and resurgence, Ann. Rev. Nuclear Part. Sci. 66 (2016), 245-272, arXiv:1601.03414.
  48. Écalle J., Les fonctions résurgentes, Tome I, Les algèbres de fonctions résurgentes, Publications Mathématiques d'Orsay 81, Vol. 5, Université de Paris-Sud, Orsay, 1981.
  49. Fei L., Giombi S., Klebanov I.R., Critical $O(N)$ models in $6-\epsilon$ dimensions, Phys. Rev. D 90 (2014), 025018, 19 pages, arXiv:1404.1094.
  50. Fisher M.E., Yang-Lee edge singularity and $\phi^3$ field theory, Phys. Rev. Lett. 40 (1978), 1610-1613.
  51. Garoufalidis S., Its A., Kapaev A., Mariño M., Asymptotics of the instantons of Painlevé I, Int. Math. Res. Not. 2012 (2012), 561-606, arXiv:1002.3634.
  52. Giombi S., Huang R., Klebanov I.R., Pufu S.S., Tarnopolsky G., $O(N)$ model in $4Phys. Rev. D 101 (2020), 045013, 25 pages, arXiv:1910.02462.
  53. Gracey J.A., Four loop renormalization of $\phi^3$ theory in six dimensions, Phys. Rev. D 92 (2015), 025012, 31 pages, arXiv:1506.03357.
  54. Gracey J.A., Asymptotic freedom from the two-loop term of the $\beta$ function in a cubic theory, Phys. Rev. D 101 (2020), 125022, 13 pages, arXiv:2004.14208.
  55. Gracey J.A., Ryttov T.A., Shrock R., Renormalization-group behavior of $\phi^3$ theories in $d = 6$ dimensions, Phys. Rev. D 102 (2020), 045016, 9 pages, arXiv:2007.12234.
  56. Grinstein B., Stone D., Stergiou A., Zhong M., Challenge to the $a$ theorem in six dimensions, Phys. Rev. Lett. 113 (2014), 231602, 5 pages, arXiv:1406.3626.
  57. Houghton A., Reeve J.S., Wallace D.J., High order behavior in $\phi^3$ field theories and the percolation problem, Phys. Rev. B 15 (1977), 2956-2964.
  58. Ishikawa K., Morikawa O., Shibata K., Suzuki H., Takaura H., Renormalon structure in compactified spacetime, Prog. Theor. Exp. Phys. (2020), 013B01, 14 pages, arXiv:1909.09579.
  59. Jentschura U.D., Soff G., Improved conformal mapping of the Borel plane, J. Phys. A: Math. Gen. 34 (2001), 1451-1457, arXiv:hep-ph/0006089.
  60. Kompaniets M.V., Panzer E., Minimally subtracted six-loop renormalization of $O(n)$-symmetric $\phi^4$ theory and critical exponents, Phys. Rev. D 96 (2017), 036016, 26 pages, arXiv:1705.06483.
  61. Kreimer D., On the Hopf algebra structure of perturbative quantum field theories, Adv. Theor. Math. Phys. 2 (1998), 303-334, arXiv:q-alg/9707029.
  62. Kreimer D., Yeats K., An étude in non-linear Dyson-Schwinger equations, Nuclear Phys. B Proc. Suppl. 160 (2006), 116-121, arXiv:hep-th/0605096.
  63. Kreimer D., Yeats K., Recursion and growth estimates in renormalizable quantum field theory, Comm. Math. Phys. 279 (2008), 401-427, arXiv:hep-th/0612179.
  64. Krüger O., Log expansions from combinatorial Dyson-Schwinger equations, Lett. Math. Phys. 110 (2020), 2175-2202, arXiv:1906.06131.
  65. Lapedes A., Mottola E., Complex path integrals and finite temperature, Nuclear Phys. B 203 (1982), 58-92.
  66. Lipatov L.N., Divergence of the perturbation theory series and the quasiclassical theory, Sov. Phys. JETP 45 (1977), 216-223.
  67. Ma E., Asymptotic freedom and a 'quark' model in six-dimension, Progr. Theoret. Phys. 54 (1975), 1828-1832.
  68. Macfarlane A.J., Woo G., $\phi^3$ theory in six dimensions and the renormalization group, Nuclear Phys. B 77 (1974), 91-108, Erratum, Nuclear Phys. B 86 (1975), 548-548.
  69. Mahmoud A.A., Yeats K., Connected chord diagrams and the combinatorics of asymptotic expansions, arXiv:2010.06550.
  70. Maiezza A., Vasquez J.C., Renormalons in a general quantum field theory, Ann. Physics 394 (2018), 84-97, arXiv:1802.06022.
  71. Mariño M., Lectures on non-perturbative effects in large $N$ gauge theories, matrix models and strings, Fortschr. Phys. 62 (2014), 455-540, arXiv:1206.6272.
  72. Mariño M., Reis T., Renormalons in integrable field theories, J. High Energy Phys. 2020 (2020), no. 4, 160, 36 pages, arXiv:1909.12134.
  73. Mariño M., Reis T., A new renormalon in two dimensions, J. High Energy Phys. 2020 (2020), no. 7, 216, 34 pages, arXiv:1912.06228.
  74. Mariño M., Reis T., Resurgence and renormalons in the one-dimensional Hubbard model, arXiv:2006.05131.
  75. Marie N., Yeats K., A chord diagram expansion coming from some Dyson-Schwinger equations, Commun. Number Theory Phys. 7 (2013), 251-291, arXiv:1210.5457.
  76. McKane A.J., Vacuum instability in scalar field theories, Ph.D. Thesis, University of Southampton, 1978.
  77. McKane A.J., Vacuum instability in scalar field theories, Nuclear Phys. B 152 (1979), 166-188.
  78. McKane A.J., Perturbation expansions at large order: results for scalar field theories revisite, J. Phys. A: Math. Theor. 52 (2019), 055401, 23 pages, arXiv:1807.00656.
  79. Misumi T., Nitta M., Sakai N., Resurgence in sine-Gordon quantum mechanics: exact agreement between multi-instantons and uniform WKB, J. High Energy Phys. 2015 (2015), no. 9, 157, 41 pages, arXiv:1507.00408.
  80. Mitschi C., Sauzin D., Divergent series, summability and resurgence. I. Monodromy and resurgence, Lecture Notes in Math., Vol. 2153, Springer, Cham, 2016.
  81. Panzer E., Algorithms for the symbolic integration of hyperlogarithms with applications to Feynman integrals, Comput. Phys. Commun. 188 (2015), 148-166, arXiv:1403.3385.
  82. Sauzin D., Introduction to 1-summability and resurgence, arXiv:1405.0356.
  83. Schnetz O., Graphical functions and single-valued multiple polylogarithms, Commun. Number Theory Phys. 8 (2014), 589-675, arXiv:1302.6445.
  84. Schnetz O., The Galois coaction on the electron anomalous magnetic moment, Commun. Number Theory Phys. 12 (2018), 335-354, arXiv:1711.05118.
  85. Schnetz O., Numbers and functions in quantum field theory, Phys. Rev. D 97 (2018), 085018, 20 pages, arXiv:1606.08598.
  86. Sloane N.J.A., The on-line encyclopedia of integer sequences, http://oeis.org/.
  87. Stahl H., The convergence of Padé approximants to functions with branch points, J. Approx. Theory 91 (1997), 139-204.
  88. van Baalen G., Kreimer D., Uminsky D., Yeats K., The QED $\beta$-function from global solutions to Dyson-Schwinger equations, Ann. Physics 324 (2009), 205-219, arXiv:0805.0826.
  89. van Baalen G., Kreimer D., Uminsky D., Yeats K., The QCD $\beta$-function from global solutions to Dyson-Schwinger equations, Ann. Physics 325 (2010), 300-324, arXiv:0906.1754.
  90. Yeats K., Growth estimates for Dyson-Schwinger equations, Ph.D. Thesis, University of Waterloo, 2008, arXiv:0810.2249.
  91. Yeats K., A combinatorial perspective on quantum field theory, SpringerBriefs in Mathematical Physics, Vol. 15, Springer, Cham, 2017.
  92. Zinn-Justin J., Expansion around instantons in quantum mechanics, J. Math. Phys. 22 (1981), 511-520.
  93. Zinn-Justin J., Quantum field theory and critical phenomena, International Series of Monographs on Physics, Vol. 77, The Clarendon Press, Oxford University Press, New York, 1989.
  94. Zinn-Justin J., Jentschura U.D., Multi-instantons and exact results. I. Conjectures, WKB expansions, and instanton interactions, Ann. Physics 313 (2004), 197-267, arXiv:quant-ph/0501136.
  95. Zinn-Justin J., Jentschura U.D., Multi-instantons and exact results. II. Specific cases, higher-order effects, and numerical calculations, Ann. Physics 313 (2004), 269-325.

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