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Nonperturbative Ambiguities and the Reality of Resurgent Transseries

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Abstract

In a wide range of quantum theoretical settings—from quantum mechanics to quantum field theory, from gauge theory to string theory—singularities in the complex Borel plane, usually associated to instantons or renormalons, render perturbation theory ill-defined as they give rise to nonperturbative ambiguities. These ambiguities are associated to choices of an integration contour in the resummation of perturbation theory, along (singular) Stokes directions in the complex Borel plane (rendering perturbative expansions non-Borel summable along any Stokes line). More recently, it has been shown that the proper framework to address these issues is that of resurgent analysis and transseries. In this context, the cancelation of all nonperturbative ambiguities is shown to be a consequence of choosing the transseries median resummation as the appropriate family of unambiguous real solutions along the coupling-constant real axis. While the median resummation is easily implemented for one-parameter transseries, once one considers more general multi-parameter transseries the procedure becomes highly dependent upon properly understanding Stokes transitions in the complex Borel plane. In particular, all Stokes coefficients must now be known in order to explicitly implement multi-parameter median resummations. In the cases where quantum-theoretical physical observables are described by resurgent functions and transseries, the methods described herein show how one may cancel nonperturbative ambiguities, and define these observables nonperturbatively starting out from perturbation theory. Along the way, structural results concerning resurgent transseries are also obtained.

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References

  1. Zinn-Justin J.: Perturbation series at large orders in quantum mechanics and field theories: application to the problem of resummation. Phys. Rep. 70, 109 (1981)

    Article  ADS  MathSciNet  Google Scholar 

  2. Beneke, M.: Renormalons. Phys. Rep. 317, 1 (1999). arXiv:hep-ph/9807443

  3. Bender C.M., Wu T.T.: Anharmonic oscillator. Phys. Rev. 184, 1231 (1969)

    Article  ADS  MathSciNet  Google Scholar 

  4. Bender C.M., Wu T.: Anharmonic oscillator 2: a study of perturbation theory in large order. Phys. Rev. D 7, 1620 (1973)

    Article  ADS  MathSciNet  Google Scholar 

  5. Bogomolny E.: Calculation of instanton–anti-instanton contributions in quantum mechanics. Phys. Lett. B 91, 431 (1980)

    Article  ADS  Google Scholar 

  6. Zinn-Justin J.: Multi-instanton contributions in quantum mechanics. Nucl. Phys. B 192, 125 (1981)

    Article  ADS  Google Scholar 

  7. Zinn-Justin J.: Multi-instanton contributions in quantum mechanics 2. Nucl. Phys. B 218, 333 (1983)

    Article  ADS  MathSciNet  Google Scholar 

  8. Dunne, G.V., Ünsal, M.: Resurgence and trans-series in quantum field theory: the \({\mathbb{c}\mathbb{p}^{N-1}}\) model. JHEP 1211, 170 (2012). arXiv:1210.2423

  9. Zinn-Justin J.: Instantons in quantum mechanics: numerical evidence for a conjecture. J. Math. Phys. 25, 549 (1984)

    Article  ADS  MathSciNet  Google Scholar 

  10. Zinn-Justin, J., Jentschura, U.D. : Multi-instantons and exact results I: conjectures, WKB expansions, and instanton interactions. Ann. Phys. 313, 197 (2004a). arXiv:quant-ph/0501136

  11. Zinn-Justin, J., Jentschura, U.D. : Multi-instantons and exact results II: specific cases, higher-order effects, and numerical calculations. Ann. Phys. 313, 269 (2004b). arXiv:quant-ph/0501137

  12. Ambroziński, Z., Wosiek, J.: Resummation of not summable series. arXiv:1210.3554

  13. Candelpergher B., Nosmas J., Pham F.: Premiers Pas en Calcul Étranger . Ann. Inst. Fourier 43, 201 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  14. Delabaere E., Pham F.: Resurgent methods in semi-classical asymptotics. Ann. Inst. Henri Poincaré 71, 1 (1999)

    MATH  MathSciNet  Google Scholar 

  15. Seara T., Sauzin D.: Resumació de Borel i Teoria de la Ressurgència. Butl. Soc. Catalana Mat. 18, 131 (2003)

    MathSciNet  Google Scholar 

  16. Edgar, G.A.: Transseries for beginners real anal. Exchange 35, 253 (2009). arXiv:0801.4877

  17. Mariño, M.: Nonperturbative effects and nonperturbative definitions in matrix models and topological strings. JHEP 0812, 114 (2008). arXiv:0805.3033

  18. Aniceto, I., Schiappa, R., Vonk, M.: The resurgence of instantons in string theory. Commun. Number Theory Phys. 6, 339 (2012). arXiv:1106.5922

  19. Mariño M.: Lectures on non-perturbative effects in large N Gauge theories, matrix models and strings. Fortschr. Phys. 62, 455–540 (2014)

    Article  MathSciNet  Google Scholar 

  20. Voros A.: The return of the quartic oscillator: the complex WKB method. Ann. Inst. Henri Poincaré 39, 211 (1983)

    MATH  MathSciNet  Google Scholar 

  21. Delabaere E., Dillinger H., Pham F.: Développements semi-classiques exacts des niveaux d’énergie d’un oscillateur à une dimension. Compt. Rend. Acad. Sci. 310, 141 (1990)

    MATH  MathSciNet  Google Scholar 

  22. Voros A.: Résurgence quantique. Ann. Inst. Fourier 43, 1509 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  23. Voros A.: Exact quantization condition for anharmonic oscillators (in one dimension). J. Phys. A Math. Gen. 27, 4653 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  24. Zinn-Justin J.: From multi-instantons to exact results. Ann. Inst. Fourier 53, 1259 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  25. Jentschura, U.D., Zinn-Justin, J.: Instantons in quantum mechanics and resurgent expansions. Phys. Lett. B596, 138 (2004). arXiv:hep-ph/0405279

  26. Dunne, G.V., Ünsal, M.: Generating energy eigenvalue trans-series from perturbation theory. arXiv:1306.4405v1

  27. ’t Hooft, G.: Can we make sense out of quantum chromodynamics? Subnucl. Ser. 15, 943 (1979)

  28. Argyres, P., Ünsal, M.: A semiclassical realization of infrared renormalons. Phys. Rev. Lett. 109, 121601 (2012). arXiv:1204.1661

  29. Argyres, P.C., Ünsal, M.: The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects. JHEP 1208, 063 (2012). arXiv:1206.1890

  30. Dunne, G.V., Ünsal, M.: Continuity and resurgence: towards a continuum definition of the \({\mathbb{c}{p}^{N-1}}\) model. Phys. Rev. D87, 025015 (2013). arXiv:1210.3646

  31. Stingl, M.: Field theory amplitudes as resurgent functions. arXiv:hep-ph/0207349

  32. Mariño, M.: Open string amplitudes and large-order behavior in topological string theory. JHEP 0803, 060 (2008). arXiv:hep-th/0612127

  33. Mariño, M., Schiappa, R., Weiss, M.: Nonperturbative effects and the large-order behavior of matrix models and topological strings. Commun. Number Theory Phys. 2, 349 (2008). arXiv:0711.1954

  34. Mariño, M., Schiappa, R., Weiss, M.: Multi-instantons and multi-cuts. J. Math. Phys. 50, 052301 (2009). arXiv:0809.2619

  35. Pasquetti, S., Schiappa, R.: Borel and Stokes nonperturbative phenomena in topological string theory and c = 1 matrix models. Ann. Henri Poincaré 11, 351 (2010). arXiv:0907.4082

  36. Klemm, A., Mariño, M., Rauch, M.: Direct integration and non-perturbative effects in matrix models. JHEP 1010, 004 (2010). arXiv:1002.3846

  37. Garoufalidis, S., Its, A., Kapaev, A., Mariño, M.: Asymptotics of the instantons of Painlevé I. Int. Math. Res. Not. 2012, 561 (2012). arXiv:1002.3634

  38. Schiappa, R., Vaz, R.: The resurgence of instantons: multi-cut stokes phases and the Painlevé II equation. arXiv:1302.5138

  39. Başar G., Dunne G.V., Ünsal M.: Resurgence theory, ghost-instantons, and analytic continuation of path integrals. J. High Energy Phys. 2013, 41 (2013)

    Google Scholar 

  40. Cherman A., Dorigoni D., Dunne G.V., Ünsal M.: Resurgence in QFT: unitons, fractons and renormalons in the principal chiral model. Phys. Rev. Lett. 112, 012601 (2014)

    Article  Google Scholar 

  41. Delabaere, E.: Effective resummation methods for an implicit resurgent function. arXiv:math-ph/0602026

  42. Sen A.: S-duality improved superstring perturbation theory. J. High Energy Phys. 2013, 29 (2013)

    Article  Google Scholar 

  43. Beem C., Rastelli L., Sen A., van Rees B.C.: Resummation and S-duality in \({{\mathcal{N}}=4}\) SYM. J. High Energy Phys. 2014, 122 (2014)

    Article  Google Scholar 

  44. Banks, T., Torres, T.: Two point Padé approximants and duality. arXiv:1307.3689

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Correspondence to Inês Aniceto.

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Communicated by H. Ooguri

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Aniceto, I., Schiappa, R. Nonperturbative Ambiguities and the Reality of Resurgent Transseries. Commun. Math. Phys. 335, 183–245 (2015). https://doi.org/10.1007/s00220-014-2165-z

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