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.
Similar content being viewed by others
References
J. Zinn-Justin, Quantum field theory and critical phenomena, Oxford University Press, Oxford U.K. (2002).
T. Schäfer and E.V. Shuryak, Instantons in QCD, Rev. Mod. Phys. 70 (1998) 323 [hep-ph/9610451] [INSPIRE].
S. Vandoren and P. van Nieuwenhuizen, Lectures on instantons, arXiv:0802.1862 [INSPIRE].
P. Argyres and M. Ünsal, A semiclassical realization of infrared renormalons, Phys. Rev. Lett. 109 (2012) 121601 [arXiv:1204.1661] [INSPIRE].
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].
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].
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].
F. Pham, Vanishing homologies and the n variable saddlepoint method, Proc. Symp. Pure Math. 40 (1983) 319.
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].
M.V. Berry and C.J. Howls, Hyperasymptotics for integrals with saddles, Proc. Roy. Soc. Lond. A 434 (1991) 657.
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].
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].
M. Mariño, Nonperturbative effects and nonperturbative definitions in matrix models and topological strings, JHEP 12 (2008) 114 [arXiv:0805.3033] [INSPIRE].
M. Mariño, R. Schiappa and M. Weiss, Multi-instantons and multi-cuts, J. Math. Phys. 50 (2009) 052301 [arXiv:0809.2619] [INSPIRE].
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].
A. Klemm, M. Mariño and M. Rauch, Direct integration and non-perturbative effects in matrix models, JHEP 10 (2010) 004 [arXiv:1002.3846] [INSPIRE].
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].
R. Schiappa and R. Vaz, The resurgence of instantons: multi-cuts stokes phases and the Painleve II equation, arXiv:1302.5138 [INSPIRE].
M. Mariño, Lectures on non-perturbative effects in large-N gauge theories, matrix models and strings, arXiv:1206.6272 [INSPIRE].
J. Écalle, Les fonctions resurgentes, volumes I–III, Publications mathématiques d’Orsay, France (1981).
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).
O. Costin, Asymptotics and Borel summability, Chapman & Hall/CRC, U.S.A. (2009).
G. Guralnik and Z. Guralnik, Complexified path integrals and the phases of quantum field theory, Annals Phys. 325 (2010) 2486 [arXiv:0710.1256] [INSPIRE].
D.D. Ferrante, G.S. Guralnik, Z. Guralnik and C. Pehlevan, Complex pathl integrals and the space of theories, arXiv:1301.4233 [INSPIRE].
I. Aniceto and R. Schiappa, Nonperturbative ambiguities and the reality of resurgent transseries, arXiv:1308.1115 [INSPIRE].
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].
NIST digital library of mathematical functions, http://dlmf.nist.gov/.
R.B. Dingle, Asymptotic expansions: their derivation and interpretation, Academic Press, U.S.A. (1973).
R. Balian, G. Parisi and A. Voros, Quartic oscillator, in the proceedings of Feynman Path Integrals, Marseille, France (1979).
R. Balian, G. Parisi and A. Voros, Discrepancies from asymptotic series and their relation to complex classical trajectories, Phys. Rev. Lett. 41 (1978) 1141.
C.M. Bender and S.A. Orszag, Advanced mathematical methods for scientists and engineers, McGraw-Hill, New York U.S.A. (1978).
M. Stingl, Field theory amplitudes as resurgent functions, hep-ph/0207349 [INSPIRE].
E.B. Bogomolny, Calculation of instanton-anti-instanton contributions in quantum mechanics, Phys. Lett. B 91 (1980) 431 [INSPIRE].
J. Zinn-Justin, Multi-instanton contributions in quantum mechanics, Nucl. Phys. B 192 (1981) 125 [INSPIRE].
S. Banerjee et al., Topology of future infinity in dS/CFT, arXiv:1306.6629 [INSPIRE].
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].
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].
C.M. Bender and T.T. Wu, Anharmonic oscillator, Phys. Rev. 184 (1969) 1231 [INSPIRE].
C.M. Bender and T. Wu, Anharmonic oscillator. 2: a study of perturbation theory in large order, Phys. Rev. D 7 (1973) 1620 [INSPIRE].
G.V. Dunne and M. Ünsal, Uniform WKB and resurgent trans-series, to appear.
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].
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].
M. Ünsal, Theta dependence, sign problems and topological interference, Phys. Rev. D 86 (2012) 105012 [arXiv:1201.6426] [INSPIRE].
D. Harlow, J. Maltz and E. Witten, Analytic continuation of Liouville theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].
A. Jaffe, C.D. Jäkel and R.E. Martinez, Complex classical fields and partial Wick rotations, arXiv:1302.5935 [INSPIRE].
S. Garoufalidis, Chern-Simons theory, analytic continuation and arithmetic, Acta Math. Vietnam. 33 (2008) 335 [arXiv:0711.1716] [INSPIRE].
G. ’t Hooft, Can we make sense out of quantum chromodynamics?, Subnucl. Ser. 15 (1979) 943.
M. Beneke, Renormalons, Phys. Rept. 317 (1999) 1 [hep-ph/9807443] [INSPIRE].
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].
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].
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].
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].
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].
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].
T. Okuda and T. Takayanagi, Ghost D-branes, JHEP 03 (2006) 062 [hep-th/0601024] [INSPIRE].
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].
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].
Z. Ambrozinski and J. Wosiek, Resummation of not summable series, arXiv:1210.3554.
Z. Ambrozinski, Tunneling with Tamm-Dancoff method, arXiv:1207.3898 [INSPIRE].
Z. Ambrozinski, Tunneling in cosine potential with periodic boundary conditions, Acta Phys. Polon. B 44 (2013) 1261 [arXiv:1303.0708] [INSPIRE].
C. Beem, L. Rastelli, A. Sen and B.C. van Rees, Resummation and S-duality in N = 4 SYM, arXiv:1306.3228 [INSPIRE].
A. Sen, S-duality improved superstring perturbation theory, arXiv:1304.0458 [INSPIRE].
T. Banks and T. Torres, Two point Pade approximants and duality, arXiv:1307.3689 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Basar, G., Dunne, G.V. & Ünsal, M. Resurgence theory, ghost-instantons, and analytic continuation of path integrals. J. High Energ. Phys. 2013, 41 (2013). https://doi.org/10.1007/JHEP10(2013)041
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2013)041