Resurgence theory, ghost-instantons, and analytic continuation of path integrals

Article

Abstract

A general quantum mechanical or quantum field theoretical system in the path integral formulation has both real and complex saddles (instantons and ghost-instantons). Resurgent asymptotic analysis implies that both types of saddles contribute to physical observables, even if the complex saddles are not on the integration path i.e., the associated Stokes multipliers are zero. We show explicitly that instanton-anti-instanton and ghost-anti-ghost saddles both affect the expansion around the perturbative vacuum. We study a self-dual model in which the analytic continuation of the partition function to negative values of coupling constant gives a pathological exponential growth, but a homotopically independent combination of integration cycles (Lefschetz thimbles) results in a sensible theory. These two choices of the integration cycles are tied with a quantum phase transition. The general set of ideas in our construction may provide new insights into non-perturbative QFT, string theory, quantum gravity, and the theory of quantum phase transitions.

Keywords

Solitons Monopoles and Instantons Nonperturbative Effects 

References

  1. [1]
    J. Zinn-Justin, Quantum field theory and critical phenomena, Oxford University Press, Oxford U.K. (2002).CrossRefGoogle Scholar
  2. [2]
    T. Schäfer and E.V. Shuryak, Instantons in QCD, Rev. Mod. Phys. 70 (1998) 323 [hep-ph/9610451] [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    S. Vandoren and P. van Nieuwenhuizen, Lectures on instantons, arXiv:0802.1862 [INSPIRE].
  4. [4]
    P. Argyres and M. Ünsal, A semiclassical realization of infrared renormalons, Phys. Rev. Lett. 109 (2012) 121601 [arXiv:1204.1661] [INSPIRE].ADSCrossRefGoogle Scholar
  5. [5]
    P.C. Argyres and M. Ünsal, The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion and renormalon effects, JHEP 08 (2012) 063 [arXiv:1206.1890] [INSPIRE].ADSCrossRefGoogle Scholar
  6. [6]
    G.V. Dunne and M. Ünsal, Resurgence and trans-series in quantum field theory: the CP N−1 model, JHEP 11 (2012) 170 [arXiv:1210.2423] [INSPIRE].ADSCrossRefGoogle Scholar
  7. [7]
    G.V. Dunne and M. Ünsal, Continuity and resurgence: towards a continuum definition of the CP N−1 model, Phys. Rev. D 87 (2013) 025015 [arXiv:1210.3646] [INSPIRE].ADSGoogle Scholar
  8. [8]
    F. Pham, Vanishing homologies and the n variable saddlepoint method, Proc. Symp. Pure Math. 40 (1983) 319.MathSciNetCrossRefGoogle Scholar
  9. [9]
    E. Witten, Analytic continuation of Chern-Simons theory, in Chern-Simons gauge theory: 20 years after, J.E. Andersen ed., AMS/IP studies in advanced mathematics 50, American Mathematical Society, U.S.A. (2011), arXiv:1001.2933 [INSPIRE].
  10. [10]
    M.V. Berry and C.J. Howls, Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. Lond. A 434 (1991) 657.MathSciNetADSCrossRefGoogle Scholar
  11. [11]
    S. Garoufalidis, A. Its, A. Kapaev and M. Mariño, Asymptotics of the instantons of Painleve I, Int. Math. Res. Not. 3 (2012) 561, arXiv:1002.3634 [INSPIRE].
  12. [12]
    M. Mariño, R. Schiappa and M. Weiss, Nonperturbative effects and the large-order behavior of matrix models and topological strings, Commun. Num. Theor. Phys. 2 (2008) 349 [arXiv:0711.1954] [INSPIRE].CrossRefMATHGoogle Scholar
  13. [13]
    M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [INSPIRE].ADSCrossRefGoogle Scholar
  14. [14]
    M. Mariño, R. Schiappa and M. Weiss, Multi-instantons and multi-cuts, J. Math. Phys. 50 (2009) 052301 [arXiv:0809.2619] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  15. [15]
    S. Pasquetti and R. Schiappa, Borel and Stokes nonperturbative phenomena in topological string theory and c = 1 matrix models, Annales Henri Poincaré 11 (2010) 351 [arXiv:0907.4082] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  16. [16]
    A. Klemm, M. Mariño and M. Rauch, Direct integration and non-perturbative effects in matrix models, JHEP 10 (2010) 004 [arXiv:1002.3846] [INSPIRE].ADSCrossRefGoogle Scholar
  17. [17]
    I. Aniceto, R. Schiappa and M. Vonk, The resurgence of instantons in string theory, Commun. Num. Theor. Phys. 6 (2012) 339 [arXiv:1106.5922] [INSPIRE].MathSciNetCrossRefMATHGoogle Scholar
  18. [18]
    R. Schiappa and R. Vaz, The resurgence of instantons: multi-cuts stokes phases and the Painleve II equation, arXiv:1302.5138 [INSPIRE].
  19. [19]
    M. Mariño, Lectures on non-perturbative effects in large-N gauge theories, matrix models and strings, arXiv:1206.6272 [INSPIRE].
  20. [20]
    J. Écalle, Les fonctions resurgentes, volumes I–III, Publications mathématiques d’Orsay, France (1981).Google Scholar
  21. [21]
    E. Delabaere, Introduction to the Écalle theory, in Computer algebra and differential equations, E. Delabaere ed., London Math. Society Lecture Note Series 193, Cambridge University Press, Cambridge U.K. (1994).Google Scholar
  22. [22]
    O. Costin, Asymptotics and Borel summability, Chapman & Hall/CRC, U.S.A. (2009).MATHGoogle Scholar
  23. [23]
    G. Guralnik and Z. Guralnik, Complexified path integrals and the phases of quantum field theory, Annals Phys. 325 (2010) 2486 [arXiv:0710.1256] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  24. [24]
    D.D. Ferrante, G.S. Guralnik, Z. Guralnik and C. Pehlevan, Complex pathl integrals and the space of theories, arXiv:1301.4233 [INSPIRE].
  25. [25]
    I. Aniceto and R. Schiappa, Nonperturbative ambiguities and the reality of resurgent transseries, arXiv:1308.1115 [INSPIRE].
  26. [26]
    M. Stone and J. Reeve, Late terms in the asymptotic expansion for the energy levels of a periodic potential, Phys. Rev. D 18 (1978) 4746 [INSPIRE].ADSGoogle Scholar
  27. [27]
    NIST digital library of mathematical functions, http://dlmf.nist.gov/.
  28. [28]
    R.B. Dingle, Asymptotic expansions: their derivation and interpretation, Academic Press, U.S.A. (1973).MATHGoogle Scholar
  29. [29]
    R. Balian, G. Parisi and A. Voros, Quartic oscillator, in the proceedings of Feynman Path Integrals, Marseille, France (1979).Google Scholar
  30. [30]
    R. Balian, G. Parisi and A. Voros, Discrepancies from asymptotic series and their relation to complex classical trajectories, Phys. Rev. Lett. 41 (1978) 1141.ADSCrossRefGoogle Scholar
  31. [31]
    C.M. Bender and S.A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill, New York U.S.A. (1978).MATHGoogle Scholar
  32. [32]
    M. Stingl, Field theory amplitudes as resurgent functions, hep-ph/0207349 [INSPIRE].
  33. [33]
    E.B. Bogomolny, Calculation of instanton-anti-instanton contributions in quantum mechanics, Phys. Lett. B 91 (1980) 431 [INSPIRE].ADSCrossRefGoogle Scholar
  34. [34]
    J. Zinn-Justin, Multi-instanton contributions in quantum mechanics, Nucl. Phys. B 192 (1981) 125 [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    S. Banerjee et al., Topology of future infinity in dS/CFT, arXiv:1306.6629 [INSPIRE].
  36. [36]
    J. Zinn-Justin and U. Jentschura, Multi-instantons and exact results I: conjectures, WKB expansions and instanton interactions, Annals Phys. 313 (2004) 197 [quant-ph/0501136] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  37. [37]
    J. Zinn-Justin and U. Jentschura, Multi-instantons and exact results II: specific cases, higher-order effects and numerical calculations, Annals Phys. 313 (2004) 269 [quant-ph/0501137] [INSPIRE].MathSciNetADSCrossRefMATHGoogle Scholar
  38. [38]
    C.M. Bender and T.T. Wu, Anharmonic oscillator, Phys. Rev. 184 (1969) 1231 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  39. [39]
    C.M. Bender and T. Wu, Anharmonic oscillator. 2: a study of perturbation theory in large order, Phys. Rev. D 7 (1973) 1620 [INSPIRE].MathSciNetADSGoogle Scholar
  40. [40]
    G.V. Dunne and M. Ünsal, Uniform WKB and resurgent trans-series, to appear.Google Scholar
  41. [41]
    AuroraScience collaboration, M. Cristoforetti, F. Di Renzo and L. Scorzato, New approach to the sign problem in quantum field theories: High density QCD on a Lefschetz thimble, Phys. Rev. D 86 (2012) 074506 [arXiv:1205.3996] [INSPIRE].ADSGoogle Scholar
  42. [42]
    M. Cristoforetti, F. Di Renzo, A. Mukherjee and L. Scorzato, Monte Carlo simulations on the Lefschetz thimble: taming the sign problem, Phys. Rev. D 88 (2013) 051501 [arXiv:1303.7204] [INSPIRE].ADSGoogle Scholar
  43. [43]
    M. Ünsal, Theta dependence, sign problems and topological interference, Phys. Rev. D 86 (2012) 105012 [arXiv:1201.6426] [INSPIRE].ADSGoogle Scholar
  44. [44]
    D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  45. [45]
    A. Jaffe, C.D. Jäkel and R.E. Martinez, Complex classical fields and partial Wick rotations, arXiv:1302.5935 [INSPIRE].
  46. [46]
    S. Garoufalidis, Chern-Simons theory, analytic continuation and arithmetic, Acta Math. Vietnam. 33 (2008) 335 [arXiv:0711.1716] [INSPIRE].MathSciNetMATHGoogle Scholar
  47. [47]
    G. ’t Hooft, Can we make sense out of quantum chromodynamics?, Subnucl. Ser. 15 (1979) 943.Google Scholar
  48. [48]
    M. Beneke, Renormalons, Phys. Rept. 317 (1999) 1 [hep-ph/9807443] [INSPIRE].
  49. [49]
    G.V. Lavrelashvili, V. Rubakov, M. Serebryakov and P. Tinyakov, Negative euclidean action: instantons and pair creation in strong background fields, Nucl. Phys. B 329 (1990) 98 [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  50. [50]
    C.K. Dumlu and G.V. Dunne, Complex worldline instantons and quantum interference in vacuum pair production, Phys. Rev. D 84 (2011) 125023 [arXiv:1110.1657] [INSPIRE].ADSGoogle Scholar
  51. [51]
    B. Tekin, K. Saririan and Y. Hosotani, Complex monopoles in the Georgi-Glashow-Chern-Simons model, Nucl. Phys. B 539 (1999) 720 [hep-th/9808045] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  52. [52]
    G. Alexanian, R. MacKenzie, M. Paranjape and J. Ruel, Path integration and perturbation theory with complex Euclidean actions, Phys. Rev. D 77 (2008) 105014 [arXiv:0802.0354] [INSPIRE].MathSciNetADSGoogle Scholar
  53. [53]
    A. Cherman, D. Dorigoni, G.V. Dunne and M. Ünsal, Resurgence in QFT: unitons, fractons and renormalons in the principal chiral model, arXiv:1308.0127 [INSPIRE].
  54. [54]
    R. Dabrowski and G.V. Dunne, Fractionalized non-self-dual solutions in the CP N−1 model, Phys. Rev. D 88 (2013) 025020 [arXiv:1306.0921] [INSPIRE].ADSGoogle Scholar
  55. [55]
    T. Okuda and T. Takayanagi, Ghost D-branes, JHEP 03 (2006) 062 [hep-th/0601024] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  56. [56]
    C. Bauer, G.S. Bali and A. Pineda, Compelling evidence of renormalons in QCD from high order perturbative expansions, Phys. Rev. Lett. 108 (2012) 242002 [arXiv:1111.3946] [INSPIRE].ADSCrossRefGoogle Scholar
  57. [57]
    G.S. Bali, C. Bauer, A. Pineda and C. Torrero, Perturbative expansion of the energy of static sources at large orders in four-dimensional SU(3) gauge theory, Phys. Rev. D 87 (2013) 094517 [arXiv:1303.3279] [INSPIRE].ADSGoogle Scholar
  58. [58]
    Z. Ambrozinski and J. Wosiek, Resummation of not summable series, arXiv:1210.3554.
  59. [59]
    Z. Ambrozinski, Tunneling with Tamm-Dancoff method, arXiv:1207.3898 [INSPIRE].
  60. [60]
    Z. Ambrozinski, Tunneling in cosine potential with periodic boundary conditions, Acta Phys. Polon. B 44 (2013) 1261 [arXiv:1303.0708] [INSPIRE].MathSciNetADSCrossRefGoogle Scholar
  61. [61]
    C. Beem, L. Rastelli, A. Sen and B.C. van Rees, Resummation and S-duality in N = 4 SYM, arXiv:1306.3228 [INSPIRE].
  62. [62]
    A. Sen, S-duality improved superstring perturbation theory, arXiv:1304.0458 [INSPIRE].
  63. [63]
    T. Banks and T. Torres, Two point Pade approximants and duality, arXiv:1307.3689 [INSPIRE].

Copyright information

© SISSA, Trieste, Italy 2013

Authors and Affiliations

  • Gökçe Basar
    • 1
  • Gerald V. Dunne
    • 2
  • Mithat Ünsal
    • 3
  1. 1.Department of Physics and AstronomyStony Brook UniversityStony BrookU.S.A.
  2. 2.Department of PhysicsUniversity of ConnecticutStorrsU.S.A.
  3. 3.Department of Physics and AstronomySFSUSan FranciscoU.S.A.

Personalised recommendations