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Critical points at infinity, non-Gaussian saddles, and bions
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  • Regular Article - Theoretical Physics
  • Open Access
  • Published: 13 June 2018

Critical points at infinity, non-Gaussian saddles, and bions

  • Alireza Behtash1,2,
  • Gerald V. Dunne1,3,
  • Thomas Schäfer2,
  • Tin Sulejmanpasic1,4 &
  • …
  • Mithat Ünsal1,2 

Journal of High Energy Physics volume 2018, Article number: 68 (2018) Cite this article

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A preprint version of the article is available at arXiv.

Abstract

It has been argued that many non-perturbative phenomena in quantum mechanics (QM) and quantum field theory (QFT) are determined by complex field configurations, and that these contributions should be understood in terms of Picard-Lefschetz theory. In this work we compute the contribution from non-BPS multi-instanton configurations, such as instanton-anti-instanton \( \left[\mathrm{\mathcal{I}}\overline{\mathrm{\mathcal{I}}}\right] \) pairs, and argue that these contributions should be interpreted as exact critical points at infinity. The Lefschetz thimbles associated with such critical points have a specific structure arising from the presence of non-Gaussian, quasi-zero mode (QZM), directions. When fermion degrees of freedom are present, as in supersymmetric theories, the effective bosonic potential can be written as the sum of a classical and a quantum potential. We show that in this case the semi-classical contribution of the critical point at infinity vanishes, but there is a non-trivial contribution that arises from its associated non-Gaussian QZM-thimble. This approach resolves several puzzles in the literature concerning the semi-classical contribution of correlated \( \left[\mathrm{\mathcal{I}}\overline{\mathrm{\mathcal{I}}}\right] \) pairs. It has the surprising consequence that the configurations dominating the expansion of observables, and the critical points defining the Lefschetz thimble decomposition need not be the same, a feature not present in the traditional Picard-Lefschetz approach.

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References

  1. S.R. Coleman, Aspects of Symmetry, Cambridge University Press (1979).

  2. J. Zinn-Justin, Quantum field theory and critical phenomena, Int. Ser. Monogr. Phys. 113 (2002) 1 [INSPIRE].

    Google Scholar 

  3. E.B. Bogomolny, Calculation of instanton-anti-instanton contributions in quantum mechanics, Phys. Lett. B 91 (1980) 431 [INSPIRE].

    Article  ADS  Google Scholar 

  4. J. Zinn-Justin, Multi-Instanton Contributions in Quantum Mechanics, Nucl. Phys. B 192 (1981) 125 [INSPIRE].

  5. A. Cherman, D. Dorigoni and M. Ünsal, Decoding perturbation theory using resurgence: Stokes phenomena, new saddle points and Lefschetz thimbles, JHEP 10 (2015) 056 [arXiv:1403.1277] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  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].

    Article  ADS  MathSciNet  Google Scholar 

  7. 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].

  8. T. Misumi, M. Nitta and N. Sakai, Resurgence in sine-Gordon quantum mechanics: Exact agreement between multi-instantons and uniform WKB, JHEP 09 (2015) 157 [arXiv:1507.00408] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  9. G. Basar, G.V. Dunne and M. Ünsal, Resurgence theory, ghost-instantons and analytic continuation of path integrals, JHEP 10 (2013) 041 [arXiv:1308.1108] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  10. 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].

    Article  ADS  MathSciNet  Google Scholar 

  11. M.M. Anber and T. Sulejmanpasic, The renormalon diagram in gauge theories on \( \mathrm{\mathbb{R}}3 \times \mathbb{S}1 \), JHEP 01 (2015) 139 [arXiv:1410.0121] [INSPIRE].

  12. A. Behtash, G.V. Dunne, T. Schäfer, T. Sulejmanpasic and M. Ünsal, Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence, arXiv:1510.03435 [INSPIRE].

  13. G.V. Dunne and M. Ünsal, New Nonperturbative Methods in Quantum Field Theory: From Large-N Orbifold Equivalence to Bions and Resurgence, Ann. Rev. Nucl. Part. Sci. 66 (2016) 245 [arXiv:1601.03414] [INSPIRE].

    Article  ADS  Google Scholar 

  14. T. Sulejmanpasic and M. Ünsal, Aspects of perturbation theory in quantum mechanics: The BenderWu Mathematica ® package, Comput. Phys. Commun. 228 (2018) 273 [arXiv:1608.08256] [INSPIRE].

  15. M. Kontsevich, On non-perturbative quantization, fukaya categories and resurgence, talk at Simons Center, (2015) [http://scgp.stonybrook.edu/video_portal/video.php?id=2183 ].

  16. M. Kontsevich, Resurgence from the path integral perspective, talk at Perimeter Institute, (2012) [https://www.perimeterinstitute.ca/videos/resurgence-path-integral-perspective].

  17. E. Witten, A New Look At The Path Integral Of Quantum Mechanics, arXiv:1009.6032 [INSPIRE].

  18. E. Witten, Analytic Continuation Of Chern-Simons Theory, AMS/IP Stud. Adv. Math. 50 (2011) 347 [arXiv:1001.2933] [INSPIRE].

    Article  MathSciNet  MATH  Google Scholar 

  19. D. Harlow, J. Maltz and E. Witten, Analytic Continuation of Liouville Theory, JHEP 12 (2011) 071 [arXiv:1108.4417] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  20. A. Behtash, E. Poppitz, T. Sulejmanpasic and M. Ünsal, The curious incident of multi-instantons and the necessity of Lefschetz thimbles, JHEP 11 (2015) 175 [arXiv:1507.04063] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  21. A. Behtash, More on Homological Supersymmetric Quantum Mechanics, Phys. Rev. D 97 (2018) 065002 [arXiv:1703.00511] [INSPIRE].

  22. A. Behtash, T. Sulejmanpasic, T. Schäfer and M. Ünsal, Hidden topological angles and Lefschetz thimbles, Phys. Rev. Lett. 115 (2015) 041601 [arXiv:1502.06624] [INSPIRE].

  23. A. Behtash, G.V. Dunne, T. Schäfer, T. Sulejmanpasic and M. Ünsal, Complexified path integrals, exact saddles and supersymmetry, Phys. Rev. Lett. 116 (2016) 011601 [arXiv:1510.00978] [INSPIRE].

  24. C. Kozçaz, T. Sulejmanpasic, Y. Tanizaki and M. Ünsal, Cheshire Cat resurgence, Self-resurgence and Quasi-Exact Solvable Systems, arXiv:1609.06198 [INSPIRE].

  25. T. Fujimori, S. Kamata, T. Misumi, M. Nitta and N. Sakai, Nonperturbative contributions from complexified solutions in ℂP N −1 models, Phys. Rev. D 94 (2016) 105002 [arXiv:1607.04205] [INSPIRE].

  26. D. Dorigoni and P. Glass, The grin of Cheshire cat resurgence from supersymmetric localization, SciPost Phys. 4 (2018) 012 [arXiv:1711.04802] [INSPIRE].

    Article  Google Scholar 

  27. N. Nekrasov, Tying up instantons with anti-instantons, arXiv:1802.04202.

  28. M.V. Fedoryuk, The saddle-point method, Izdat. Nauka, Moscow, MR 58:22580 (1977).

  29. T. Schäfer and E.V. Shuryak, Instantons in QCD, Rev. Mod. Phys. 70 (1998) 323 [hep-ph/9610451] [INSPIRE].

  30. F. Pham, Vanishing homologies and the n variable saddlepoint method, Proc. Symp. Pure Math 2 (1983) 319.

    Article  MathSciNet  MATH  Google Scholar 

  31. V.I. Arnold, S.M. Gusein-Zade and A.N. Varchenko, Singularities of Differentiable Maps, Volume 1, Birkhäuser Basel (2012).

  32. E. Witten, Constraints on Supersymmetry Breaking, Nucl. Phys. B 202 (1982) 253 [INSPIRE].

  33. M.A. Shifman, Beginning supersymmetry (supersymmetry in quantum mechanics), in ITEP lectures on particle physics and field theory, vol. 1, pp. 301-344, World Scientific, Singapore (1995) [INSPIRE].

  34. I.I. Balitsky and A.V. Yung, Instanton Molecular Vacuum in N = 1 Supersymmetric Quantum Mechanics, Nucl. Phys. B 274 (1986) 475 [INSPIRE].

  35. P.V. Buividovich, G.V. Dunne and S.N. Valgushev, Complex Path Integrals and Saddles in Two-Dimensional Gauge Theory, Phys. Rev. Lett. 116 (2016) 132001 [arXiv:1512.09021] [INSPIRE].

    Article  ADS  MathSciNet  Google Scholar 

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

  37. 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 

  38. G.V. Dunne, T. Sulejmanpasic and M. Ünsal, Thimbles and Resurgence in the Triple-well System, work in progress.

  39. E. Brézin, J.C. Le Guillou and J. Zinn-Justin, Perturbation Theory at Large Order. 2. Role of the Vacuum Instability, Phys. Rev. D 15 (1977) 1558 [INSPIRE].

  40. R. Balian, G. Parisi and A. Voros, Quartic Oscillator, in Feynman Path Integrals. Proceedings of the International Colloquium held in Marseille, May 1978, pp. 337-360 (1978).

  41. J.L. Richard and A. Rouet, Complex Saddle Points Versus Dilute Gas Approximation in the Double Well Anharmonic Oscillator, Nucl. Phys. B 185 (1981) 47 [INSPIRE].

  42. A. Lapedes and E. Mottola, Complex Path Integrals and Finite Temperature, Nucl. Phys. B 203 (1982) 58 [INSPIRE].

  43. P.A. Millard, Complex Classical Paths and the One-dimensional sine-Gordon System, Nucl. Phys. B 259 (1985) 266 [INSPIRE].

  44. I.M. Gelfand and A.M. Yaglom, Integration in functional spaces and it applications in quantum physics, J. Math. Phys. 1 (1960) 48 [INSPIRE].

  45. S.R. Coleman, The Fate of the False Vacuum. 1. Semiclassical Theory, Phys. Rev. D 15 (1977) 2929 [Erratum ibid. D 16 (1977) 1248] [INSPIRE].

  46. C.G. Callan Jr. and S.R. Coleman, The Fate of the False Vacuum. 2. First Quantum Corrections, Phys. Rev. D 16 (1977) 1762 [INSPIRE].

  47. K. Kirsten, Spectral functions in mathematics and physics, Chapman and Hall/CRC (2001).

  48. H. Kleinert, Path Integrals in Quantum Mechanics, Statistics, Polymer Physics, and Financial Markets, World Scientific (2006).

  49. G.V. Dunne, Functional determinants in quantum field theory, J. Phys. A 41 (2008) 304006 [arXiv:0711.1178] [INSPIRE].

  50. M. Marino, Instantons and Large N, Cambridge University Press (2015).

  51. A.J. McKane and M.B. Tarlie, Regularization of functional determinants using boundary perturbations, J. Phys. A 28 (1995) 6931 [cond-mat/9509126] [INSPIRE].

  52. G.V. Dunne and H. Min, Beyond the thin-wall approximation: Precise numerical computation of prefactors in false vacuum decay, Phys. Rev. D 72 (2005) 125004 [hep-th/0511156] [INSPIRE].

  53. S. Friedli and Y. Velenik, Statistical Mechanics of Lattice Systems: A Concrete Mathematical Introduction, Cambridge University Press (2017),

  54. G.V. Dunne and M. Ünsal, Deconstructing zero: resurgence, supersymmetry and complex saddles, JHEP 12 (2016) 002 [arXiv:1609.05770] [INSPIRE].

    Article  ADS  MathSciNet  MATH  Google Scholar 

  55. M. Ünsal, Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions, Phys. Rev. D 80 (2009) 065001 [arXiv:0709.3269] [INSPIRE].

  56. E. Poppitz, T. Schäfer and M. Ünsal, Universal mechanism of (semi-classical) deconfinement and theta-dependence for all simple groups, JHEP 03 (2013) 087 [arXiv:1212.1238] [INSPIRE].

    Article  ADS  MATH  Google Scholar 

  57. 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].

  58. M. Nitta, Fractional instantons and bions in the O(N) model with twisted boundary conditions, JHEP 03 (2015) 108 [arXiv:1412.7681] [INSPIRE].

  59. Y. Liu, E. Shuryak and I. Zahed, Confining dyon-antidyon Coulomb liquid model. I., Phys. Rev. D 92 (2015) 085006 [arXiv:1503.03058] [INSPIRE].

  60. M. Nitta, Fractional instantons and bions in the principal chiral model on ℝ2 × S 1 with twisted boundary conditions, JHEP 08 (2015) 063 [arXiv:1503.06336] [INSPIRE].

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Authors and Affiliations

  1. Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA, 93106, U.S.A.

    Alireza Behtash, Gerald V. Dunne, Tin Sulejmanpasic & Mithat Ünsal

  2. Department of Physics, North Carolina State University, Raleigh, NC, 27695, U.S.A.

    Alireza Behtash, Thomas Schäfer & Mithat Ünsal

  3. Department of Physics, University of Connecticut, Storrs, CT, 06269-3046, U.S.A.

    Gerald V. Dunne

  4. Philippe Meyer Institute, Physics Department, École Normale Supérieure, PSL Research University, 24 rue Lhomond, F-75231, Paris Cedex 05, France

    Tin Sulejmanpasic

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  1. Alireza Behtash
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  2. Gerald V. Dunne
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Corresponding author

Correspondence to Thomas Schäfer.

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ArXiv ePrint: 1803.11533

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Behtash, A., Dunne, G.V., Schäfer, T. et al. Critical points at infinity, non-Gaussian saddles, and bions. J. High Energ. Phys. 2018, 68 (2018). https://doi.org/10.1007/JHEP06(2018)068

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  • Received: 27 April 2018

  • Accepted: 31 May 2018

  • Published: 13 June 2018

  • DOI: https://doi.org/10.1007/JHEP06(2018)068

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Keywords

  • Nonperturbative Effects
  • Differential and Algebraic Geometry
  • Resummation
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