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Resurgence and 1/N Expansion in Integrable Field Theories

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  • Published: 20 October 2021
  • Volume 2021, article number 166, (2021)
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Resurgence and 1/N Expansion in Integrable Field Theories
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  • Lorenzo Di Pietro1,2,
  • Marcos Mariño3,
  • Giacomo Sberveglieri2,4 &
  • …
  • Marco Serone  ORCID: orcid.org/0000-0002-3540-63642,4 

  • 458 Accesses

  • 24 Citations

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A preprint version of the article is available at arXiv.

Abstract

In theories with renormalons the perturbative series is factorially divergent even after restricting to a given order in 1/N, making the 1/N expansion a natural testing ground for the theory of resurgence. We study in detail the interplay between resurgent properties and the 1/N expansion in various integrable field theories with renormalons. We focus on the free energy in the presence of a chemical potential coupled to a conserved charge, which can be computed exactly with the thermodynamic Bethe ansatz (TBA). In some examples, like the first 1/N correction to the free energy in the non-linear sigma model, the terms in the 1/N expansion can be fully decoded in terms of a resurgent trans-series in the coupling constant. In the principal chiral field we find a new, explicit solution for the large N free energy which can be written as the median resummation of a trans-series with infinitely many, analytically computable IR renormalon corrections. However, in other examples, like the Gross-Neveu model, each term in the 1/N expansion includes non-perturbative corrections which can not be predicted by a resurgent analysis of the corresponding perturbative series. We also study the properties of the series in 1/N. In the Gross-Neveu model, where this is convergent, we analytically continue the series beyond its radius of convergence and show how the continuation matches with known dualities with sine-Gordon theories.

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References

  1. H.E. Stanley, Spherical model as the limit of infinite spin dimensionality, Phys. Rev. 176 (1968) 718 [INSPIRE].

    Article  ADS  Google Scholar 

  2. G. ’t Hooft, A Planar Diagram Theory for Strong Interactions, Nucl. Phys. B 72 (1974) 461 [INSPIRE].

  3. S.R. Coleman and E. Witten, Chiral Symmetry Breakdown in Large N Chromodynamics, Phys. Rev. Lett. 45 (1980) 100 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  4. E. Witten, Current Algebra Theorems for the U(1) Goldstone Boson, Nucl. Phys. B 156 (1979) 269 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  5. D.J. Gross and A. Neveu, Dynamical Symmetry Breaking in Asymptotically Free Field Theories, Phys. Rev. D 10 (1974) 3235 [INSPIRE].

    Article  ADS  Google Scholar 

  6. A. D’Adda, M. Lüscher and P. Di Vecchia, A 1/n Expandable Series of Nonlinear Sigma Models with Instantons, Nucl. Phys. B 146 (1978) 63 [INSPIRE].

    Article  ADS  Google Scholar 

  7. J. Koplik, A. Neveu and S. Nussinov, Some Aspects of the Planar Perturbation Series, Nucl. Phys. B 123 (1977) 109 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  8. E. Brézin, C. Itzykson, G. Parisi and J.B. Zuber, Planar Diagrams, Commun. Math. Phys. 59 (1978) 35 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. D.J. Broadhurst, Large N expansion of QED: Asymptotic photon propagator and contributions to the muon anomaly, for any number of loops, Z. Phys. C 58 (1993) 339 [INSPIRE].

    Article  ADS  Google Scholar 

  10. M. Beneke, Renormalons, Phys. Rept. 317 (1999) 1 [hep-ph/9807443] [INSPIRE].

  11. M. Mariño, Lectures on non-perturbative effects in large N gauge theories, matrix models and strings, Fortsch. Phys. 62 (2014) 455 [arXiv:1206.6272] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  12. I. Aniceto, G. Basar and R. Schiappa, A Primer on Resurgent Transseries and Their Asymptotics, Phys. Rept. 809 (2019) 1 [arXiv:1802.10441] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  13. A. Voros, The return of the quartic oscillator. The complex WKB method, Annales de l’I.H.P. Physique Théorique 39 (1983) 211.

  14. E. Delabaere, H. Dillinger, and F. Pham, Exact semiclassical expansions for one-dimensional quantum oscillators, J. Math. Phys. 38 (1997) 6126.

    Article  ADS  MathSciNet  MATH  Google Scholar 

  15. J. Zinn-Justin and U.D. Jentschura, Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions, Annals Phys. 313 (2004) 197 [quant-ph/0501136] [INSPIRE].

  16. J. Zinn-Justin and U.D. Jentschura, Multi-instantons and exact results II: Specific cases, higher-order effects, and numerical calculations, Annals Phys. 313 (2004) 269 [quant-ph/0501137] [INSPIRE].

  17. M. Serone, G. Spada and G. Villadoro, The Power of Perturbation Theory, JHEP 05 (2017) 056 [arXiv:1702.04148] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  18. M. Serone, G. Spada and G. Villadoro, Instantons from Perturbation Theory, Phys. Rev. D 96 (2017) 021701 [arXiv:1612.04376] [INSPIRE].

  19. M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. R. Couso-Santamaría, R. Schiappa and R. Vaz, Finite N from Resurgent Large N, Annals Phys. 356 (2015) 1 [arXiv:1501.01007] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. S. Gukov, M. Mariño and P. Putrov, Resurgence in complex Chern-Simons theory, arXiv:1605.07615 [INSPIRE].

  22. M. Mariño and T. Reis, Resurgence for superconductors, arXiv:1905.09569 [INSPIRE].

  23. M. Mariño and T. Reis, Resurgence and renormalons in the one-dimensional Hubbard model, arXiv:2006.05131 [INSPIRE].

  24. M. Borinsky and G.V. Dunne, Non-Perturbative Completion of Hopf-Algebraic Dyson-Schwinger Equations, Nucl. Phys. B 957 (2020) 115096 [arXiv:2005.04265] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  25. M.C. Abbott, Z. Bajnok, J. Balog and A. Hegedús, From perturbative to non-perturbative in the O (4) sigma model, Phys. Lett. B 818 (2021) 136369 [arXiv:2011.09897] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  26. M.C. Abbott, Z. Bajnok, J. Balog, A. Hegedús and S. Sadeghian, Resurgence in the O(4) sigma model, JHEP 05 (2021) 253 [arXiv:2011.12254] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  27. S. Garoufalidis, J. Gu and M. Mariño, Peacock patterns and resurgence in complex Chern-Simons theory, arXiv:2012.00062 [INSPIRE].

  28. F. David, Nonperturbative Effects and Infrared Renormalons Within the 1/N Expansion of the O(N) Nonlinear σ Model, Nucl. Phys. B 209 (1982) 433 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  29. F. David, On the Ambiguity of Composite Operators, IR Renormalons and the Status of the Operator Product Expansion, Nucl. Phys. B 234 (1984) 237 [INSPIRE].

    Article  ADS  Google Scholar 

  30. V.A. Novikov, M.A. Shifman, A.I. Vainshtein and V.I. Zakharov, Two-Dimensional Sigma Models: Modeling Nonperturbative Effects of Quantum Chromodynamics, Phys. Rept. 116 (1984) 103.

    Article  ADS  Google Scholar 

  31. A.M. Polyakov and P.B. Wiegmann, Theory of Nonabelian Goldstone Bosons, Phys. Lett. B 131 (1983) 121 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  32. P. Hasenfratz, M. Maggiore and F. Niedermayer, The Exact mass gap of the O(3) and O(4) nonlinear sigma models in d = 2, Phys. Lett. B 245 (1990) 522 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  33. P. Hasenfratz and F. Niedermayer, The Exact mass gap of the O(N) sigma model for arbitrary N is >= 3 in d = 2, Phys. Lett. B 245 (1990) 529 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  34. J. Balog, S. Naik, F. Niedermayer and P. Weisz, Exact mass gap of the chiral SU(N) × SU(N) model, Phys. Rev. Lett. 69 (1992) 873 [INSPIRE].

    Article  ADS  Google Scholar 

  35. P. Forgacs, F. Niedermayer and P. Weisz, The Exact mass gap of the Gross-Neveu model. 1. The Thermodynamic Bethe ansatz, Nucl. Phys. B 367 (1991) 123 [INSPIRE].

  36. P. Forgacs, F. Niedermayer and P. Weisz, The Exact mass gap of the Gross-Neveu model. 2. The 1/N expansion, Nucl. Phys. B 367 (1991) 144 [INSPIRE].

  37. T.J. Hollowood, The Exact mass gaps of the principal chiral models, Phys. Lett. B 329 (1994) 450 [hep-th/9402084] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  38. J.M. Evans and T.J. Hollowood, The Exact mass gap of the supersymmetric o(N) sigma model, Phys. Lett. B 343 (1995) 189 [hep-th/9409141] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  39. J.M. Evans and T.J. Hollowood, The Exact mass gap of the supersymmetric CP**(n-1) sigma model, Phys. Lett. B 343 (1995) 198 [hep-th/9409142] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  40. J.M. Evans and T.J. Hollowood, Exact results for integrable asymptotically - free field theories, Nucl. Phys. B Proc. Suppl. 45 (1996) 130 [hep-th/9508141] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  41. D. Volin, From the mass gap in O(N) to the non-Borel-summability in O(3) and O(4) sigma-models, Phys. Rev. D 81 (2010) 105008 [arXiv:0904.2744] [INSPIRE].

    Article  ADS  Google Scholar 

  42. D. Volin, Quantum integrability and functional equations: Applications to the spectral problem of AdS/CFT and two-dimensional sigma models, J. Phys. A 44 (2011) 124003 [arXiv:1003.4725] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  43. M. Mariño and T. Reis, Renormalons in integrable field theories, JHEP 04 (2020) 160 [arXiv:1909.12134] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  44. M. Mariño and T. Reis, Exact perturbative results for the Lieb-Liniger and Gaudin-Yang models, arXiv:1905.09575 [INSPIRE].

  45. M. Mariño and T. Reis, Three roads to the energy gap, arXiv:2010.16174 [INSPIRE].

  46. V.A. Fateev, P.B. Wiegmann and V.A. Kazakov, Large N chiral field in two-dimensions, Phys. Rev. Lett. 73 (1994) 1750 [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  47. V.A. Fateev, V.A. Kazakov and P.B. Wiegmann, Principal chiral field at large N, Nucl. Phys. B 424 (1994) 505 [hep-th/9403099] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  48. K. Zarembo, Quantum Giant Magnons, JHEP 05 (2008) 047 [arXiv:0802.3681] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  49. V. Kazakov, E. Sobko and K. Zarembo, Double-Scaling Limit in the Principal Chiral Model: A New Noncritical String?, Phys. Rev. Lett. 124 (2020) 191602 [arXiv:1911.12860] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  50. M. Mariño, R. Miravitllas and T. Reis, Testing the Bethe ansatz with large N renormalons, arXiv:2102.03078 [INSPIRE].

  51. F.W.J. Olver et al. eds., NIST Digital Library of Mathematical Functions, http://dlmf.nist.gov/, release 1.1.1 of 2021-03-15.

  52. Z. Bajnok, J. Balog, B. Basso, G.P. Korchemsky and L. Palla, Scaling function in AdS/CFT from the O(6) sigma model, Nucl. Phys. B 811 (2009) 438 [arXiv:0809.4952] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  53. E. Brézin and J. Zinn-Justin, Spontaneous Breakdown of Continuous Symmetries Near Two-Dimensions, Phys. Rev. B 14 (1976) 3110 [INSPIRE].

    Article  ADS  Google Scholar 

  54. M. Mariño, Instantons and large N . An introduction to non-perturbative methods in quantum field theory. Cambridge University Press, Cambridge, U.K. (2015).

  55. P. Biscari, M. Campostrini and P. Rossi, Quantitative Picture of the Scaling Behavior of Lattice Nonlinear σ Models From the 1/N Expansion, Phys. Lett. B 242 (1990) 225 [INSPIRE].

    Article  ADS  Google Scholar 

  56. C. Bonet, D. Sauzin, T. Seara and M. València, Adiabatic invariant of the harmonic oscillator, complex matching and resurgence, SIAM J. Math. Anal. 29 (1998) 1335.

    Article  MathSciNet  MATH  Google Scholar 

  57. T.M. Seara and D. Sauzin, Resumació de Borel i teoria de la ressurgencia, Butl. Soc. Catalana Mat. 18 (2003) 131.

    MathSciNet  Google Scholar 

  58. I. Aniceto and R. Schiappa, Nonperturbative Ambiguities and the Reality of Resurgent Transseries, Commun. Math. Phys. 335 (2015) 183 [arXiv:1308.1115] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  59. M. Serone, G. Spada and G. Villadoro, λϕ4 Theory I: The Symmetric Phase Beyond NNNNNNNNLO, JHEP 08 (2018) 148 [arXiv:1805.05882] [INSPIRE].

  60. C. Hunter and B. Guerrieri, Deducing the properties of singularities of functions from their Taylor series coefficients, SIAM J. Appl. Math. 39 (1980) 248.

    Article  MathSciNet  MATH  Google Scholar 

  61. H. Stahl, The convergence of Padé approximants to functions with branch points, Journal of Approximation Theory 91 (1997) 139.

    Article  MathSciNet  MATH  Google Scholar 

  62. L. Di Pietro and M. Serone, Looking through the QCD Conformal Window with Perturbation Theory, JHEP 07 (2020) 049 [arXiv:2003.01742] [INSPIRE].

    Article  MathSciNet  Google Scholar 

  63. G.A. Baker Jr. and P. Peter Graves-Morris, Padé Approximants, Encyclopedia of Mathematics and its Applications, Cambridge University Press, Cambridge, U.K. (1996)

  64. A.B. Zamolodchikov, Mass scale in the sine-Gordon model and its reductions, Int. J. Mod. Phys. A 10 (1995) 1125 [INSPIRE].

    Article  ADS  Google Scholar 

  65. D.J. Amit, Y.Y. Goldschmidt and G. Grinstein, Renormalization Group Analysis of the Phase Transition in the 2D Coulomb Gas, sine-Gordon Theory and xy Model, J. Phys. A 13 (1980) 585 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  66. S.R. Coleman, The Quantum sine-Gordon Equation as the Massive Thirring Model, Phys. Rev. D 11 (1975) 2088 [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

  67. C. Kozçaz, T. Sulejmanpasic, Y. Tanizaki and M. Ünsal, Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems, Commun. Math. Phys. 364 (2018) 835 [arXiv:1609.06198] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  68. G.V. Dunne and M. Ünsal, Resurgence and Dynamics of O(N) and Grassmannian Sigma Models, JHEP 09 (2015) 199 [arXiv:1505.07803] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  69. A. Cherman, D. Dorigoni, G.V. Dunne and M. Ünsal, Resurgence in Quantum Field Theory: Nonperturbative Effects in the Principal Chiral Model, Phys. Rev. Lett. 112 (2014) 021601 [arXiv:1308.0127] [INSPIRE].

  70. G. Sberveglieri, M. Serone and G. Spada, Renormalization scheme dependence, RG flow, and Borel summability in ϕ4 Theories in d < 4, Phys. Rev. D 100 (2019) 045008 [arXiv:1905.02122] [INSPIRE].

  71. G. Sberveglieri, M. Serone and G. Spada, Self-Dualities and Renormalization Dependence of the Phase Diagram in 3d O(N) Vector Models, JHEP 02 (2021) 098 [arXiv:2010.09737] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  72. M. Mariño and T. Reis, A new renormalon in two dimensions, JHEP 07 (2020) 216 [arXiv:1912.06228] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  73. M. Berry and C. Howls, Hyperasymptotics for integrals with saddles, Proceedings of the Royal Society A: Mathematical and Physical Sciences A434 (1991) 657.

    ADS  MathSciNet  MATH  Google Scholar 

  74. S. Hikami and E. Brézin, Large Order Behavior of the 1/N Expansion in Zero-dimensions and One-dimensions, J. Phys. A 12 (1979) 759 [INSPIRE].

    Article  ADS  Google Scholar 

  75. E.H. Lieb and W. Liniger, Exact analysis of an interacting Bose gas. 1. The General solution and the ground state, Phys. Rev. 130 (1963) 1605 [INSPIRE].

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Authors and Affiliations

  1. Dipartimento di Fisica, Università di Trieste, Strada Costiera 11, I-34151, Trieste, Italy

    Lorenzo Di Pietro

  2. INFN, Sezione di Trieste, Via Valerio 2, I-34127, Trieste, Italy

    Lorenzo Di Pietro, Giacomo Sberveglieri & Marco Serone

  3. Département de Physique Théorique et Section de Mathématiques, Université de Genève, CH-1211, Genève, Switzerland

    Marcos Mariño

  4. SISSA, Via Bonomea 265, I-34136, Trieste, Italy

    Giacomo Sberveglieri & Marco Serone

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  1. Lorenzo Di Pietro
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Di Pietro, L., Mariño, M., Sberveglieri, G. et al. Resurgence and 1/N Expansion in Integrable Field Theories. J. High Energ. Phys. 2021, 166 (2021). https://doi.org/10.1007/JHEP10(2021)166

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  • Received: 23 August 2021

  • Accepted: 01 October 2021

  • Published: 20 October 2021

  • DOI: https://doi.org/10.1007/JHEP10(2021)166

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Keywords

  • 1/N Expansion
  • Integrable Field Theories
  • Field Theories in Lower Dimensions
  • Renormalization Regularization and Renormalons
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