2012

Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model

Dunne, Gerald V., Unsal, Mithat

Understand

This work is a step towards a non-perturbative continuum definition of quantum field theory (QFT), beginning with asymptotically free two dimensional non-linear sigma-models, using recent ideas from mathematics and QFT.

The ideas from mathematics are resurgence theory, the trans-series framework, and Borel-Ecalle resummation.
The ideas from QFT use continuity on R^1 x S^1_L, i.e, the absence of any phase transition as N \to infinity, or rapid-crossovers for finite-N, and the small-L weak coupling limit to render the semi-classical sector well-defined and calculable.
We classify semi-classical configurations with actions 1/N (kink-instantons), 2/N (bions and bi-kinks), in units where the 2d instanton action is normalized to one.

Built on

R. B. Dingle, Asymptotic expansions:their derivation and interpretation
1973
Earlier work this paper cites.
C. M. Bender and T. T. Wu, “Anharmonic oscillator,” Phys. Rev. 184
1973
Earlier work this paper cites.
C. M. Bender and T. T. Wu, “Statistical Analysis of Feynman Diagrams,” Phys. Rev. Lett. 37
1976
Earlier work this paper cites.
E. B. Bogomolny and V. A. Fateev, “Large Orders Calculations in the Gauge Theories,” Phys. Lett. B 71
1977
Earlier work this paper cites.
E. Brézin, G. Parisi and J. Zinn-Justin, “Perturbation Theory at Large Orders for Potential with Degenerate Minima,” Phys. Rev. D 16
1977
Earlier work this paper cites.
L. N. Lipatov, “Divergence of the Perturbation Theory Series and the Quasiclassical Theory,” Sov. Phys. JETP 45
1977
Earlier work this paper cites.
M. Stone and J. Reeve, “Late Terms in the Asymptotic Expansion for the Energy Levels of a Periodic Potential,” Phys. Rev. D 18
1978
Earlier work this paper cites.
A. D’Adda, M. Luscher and P. Di Vecchia, “A 1/n Expandable Series of Nonlinear Sigma Models with Instantons,” Nucl. Phys. B 146
1978
Earlier work this paper cites.
R. Balian, G. Parisi and A. Voros, “Discrepancies from asymptotic series and their relation to complex classical trajectories”, Phys. Rev. Lett. 41
1979
Earlier work this paper cites.
E. Brezin and J. Zinn-Justin,“Expansion of the H2+ ground state energy in inverse powers of the distance between the two protons”, Journ. Physique Lett. 40
1979
Earlier work this paper cites.
G. ’t Hooft, “Can We Make Sense Out of Quantum Chromodynamics?,” Subnucl. Ser. 15
1979
Earlier work this paper cites.
E. Witten, “Instantons, the Quark Model, and the 1/n Expansion,” Nucl. Phys. B 149
1979
Earlier work this paper cites.
A. Jevicki, “Instantons And The 1/n Expansion In Nonlinear Sigma Models,” Phys. Rev. D 20
1979
Earlier work this paper cites.
E. B. Bogomolny, “Calculation Of Instanton - Anti-instanton Contributions In Quantum Mechanics,” Phys. Lett. B91
1980
Earlier work this paper cites.
I. Affleck, “Testing The Instanton Method,” Phys. Lett. B 92
1980
Earlier work this paper cites.
J. Écalle, Les Fonctions Resurgentes
1981
Earlier work this paper cites.
D. J. Gross, R. D. Pisarski and L. G. Yaffe, “QCD and Instantons at Finite Temperature,” Rev. Mod. Phys. 53
1981
Earlier work this paper cites.
A. D’Adda and A. C. Davis, “Chiral Symmetry Restoration In The Cp**(n-1) In Two-dimensions Model With Fermions,” Phys. Lett. B 101
1981
Earlier work this paper cites.
M. C. Ogilvie and G. S. Guralnik, “Instantons And Vortices In Two-dimensions,” Nucl. Phys. B 190
1981
Earlier work this paper cites.
G. Veneziano and S. Yankielowicz, “An Effective Lagrangian for the Pure N=1 Supersymmetric Yang-Mills Theory,” Phys. Lett. B 113
1982
Earlier work this paper cites.
T. Eguchi and H. Kawai, “Reduction of Dynamical Degrees of Freedom in the Large N Gauge Theory,” Phys. Rev. Lett. 48
1982
Earlier work this paper cites.
A. Voros, “The return of the quartic oscillator. The complex WKB method”, Ann. de l’I. H. Poincare, A 39
1983
Earlier work this paper cites.
G. Munster, “A Study Of Cp**(n-1) Models On The Sphere Within The 1/n Expansion,” Nucl. Phys. B 218
1983
Earlier work this paper cites.
R. J. Damburg, R. K. Propin, S. Graffi, V. Grecchi, E. M. Harrell, J. Cizek, J. Paldus, H. J. Silverstone, “1/R Expansion for H2+: Analyticity, Summability, Asymptotics, and Calculation of Exponentially Small Terms”, Phys. Rev. Lett. 52
1984
Earlier work this paper cites.
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
Earlier work this paper cites.
F. David, “Cancellations Of Infrared Divergences In Two-dimensional Chiral Models,” Phys. Lett. B 96
1984
Earlier work this paper cites.

Similar

S. Coleman, Aspects of Symmetry
1985
Cited alongside, same era.
I. I. Balitsky and A. V. Yung, “Instanton Molecular Vacuum In N=1 Supersymmetric Quantum Mechanics,” Nucl. Phys. B 274
1986
Cited alongside, same era.
E. Mottola and A. Wipf, “Unsuppressed Fermion Number Violation At High Temperature: An O(3) Model,” Phys. Rev. D 39
1989
Cited alongside, same era.
W. J. Zakrzewski, Low Dimensional Sigma Models
1989
Cited alongside, same era.
P. Achuthan, H. J. W. Muller-Kirsten and A. Wiedemann, “Perturbation Theory And Boundary Conditions: Analogous Treatments Of Anharmonic Oscillators And Double Wells And Similarly Related Potentials And The Calculation Of Exponentially Small Contributions To Eigenvalues,” Fortsch. Phys. 38
1990
Cited alongside, same era.
J. C. Le Guillou and J. Zinn-Justin, (eds.), Large order behavior of perturbation theory
1990
Cited alongside, same era.
M. V. Berry and C. J. Howls, “Hyperasymptotics”, Proc. R. Soc. A 430
1991
Cited alongside, same era.
E. Delabaere, “Spectre de l’opérateur de Schrödinger stationnaire unidimensionnel à potentiel polynôme trigonométrique”, C. R. Acad. Sci. Paris 314
1992
Cited alongside, same era.
M. Beneke, “Renormalons,” Phys. Rept. 317
1993
Cited alongside, same era.
P. M. Sutcliffe, “Instanton chains with soliton limits,” Phys. Lett. B 302
1993
Cited alongside, same era.
B. Candelpergher, J.C. Nosmas and F. Pham, “Premiers Pas en Calcul Étranger,” Ann. Inst. Fourier 43
1993
Cited alongside, same era.
E. Delabaere, “Introduction to the Ecalle theory”, In Computer Algebra and Differential Equations
1994
Cited alongside, same era.

Then

M. Ünsal, “Abelian duality, confinement, and chiral symmetry breaking in QCD(adj),” Phys. Rev. Lett. 100
Original
2008
Later among the works it cites.
M. Ünsal, L. G. Yaffe, “Center-stabilized Yang-Mills theory: Confinement and large N volume independence,” Phys. Rev. D78
Original
2008
Later among the works it cites.
J. C. Myers and M. C. Ogilvie, “New phases of SU(3) and SU(4) at finite temperature,” Phys. Rev. D 77
Original
2008
Later among the works it cites.
M. Shifman and M. Ünsal, “QCD-like Theories on R(3) x S(1): A Smooth Journey from Small to Large r(S(1)) with Double-Trace Deformations,” Phys. Rev. D 78
Original
2008
Later among the works it cites.
F. Bruckmann, “Instanton constituents in the O(3) model at finite temperature,” Phys. Rev. Lett. 100
Original
2008
Later among the works it cites.
O. Costin, Asymptotics and Borel Summability
2009
Later among the works it cites.
M. Ünsal, “Magnetic bion condensation: A New mechanism of confinement and mass gap in four dimensions,” Phys. Rev. D 80
Original
2009
Later among the works it cites.
W. Brendel, F. Bruckmann, L. Janssen, A. Wipf and C. Wozar, “Instanton constituents and fermionic zero modes in twisted CP**n models,” Phys. Lett. B 676
Original
2009
Later among the works it cites.
D. Harland, “Kinks, chains, and loop groups in the CP**n sigma models,” J. Math. Phys. 50
Original
2009
Later among the works it cites.
E. Vicari and H. Panagopoulos, “Theta dependence of SU(N) gauge theories in the presence of a topological term,” Phys. Rept. 470
Original
2009
Later among the works it cites.
B. Collie and D. Tong, “The Partonic Nature of Instantons,” JHEP 0908
Original
2009
Later among the works it cites.
I. Aniceto, R. Schiappa and M. Vonk, “The Resurgence of Instantons in String Theory,” arXiv:1106.5922 ; S. Pasquetti and R. Schiappa, “Borel and Stokes Nonperturbative Phenomena in Topological String Theory and c=1 Matrix Models,” Annales Henri Poincaré 11
Original
2010
Later among the works it cites.
M. M. Anber and E. Poppitz, “Microscopic Structure of Magnetic Bions,” JHEP 1106
Original
2011
Later among the works it cites.
M. Douglas, “Foundations of Quantum Field Theory,” String-Math 2011
2011
Later among the works it cites.
J. Zinn-Justin, “Multi - Instanton Contributions In Quantum Mechanics,” Nucl. Phys. B 192
2011
Later among the works it cites.
M. Mariño, “Nonperturbative effects and nonperturbative definitions in matrix models and topological strings,” JHEP 0812
Original
2011
Later among the works it cites.
M. Aguado and M. Asorey, “Theta-vacuum and large N limit in C P N − 1 CP^{N-1} sigma models,” Nucl. Phys. B 844
Original
2011
Later among the works it cites.
E. Poppitz and M. Unsal, “Seiberg-Witten and ’Polyakov-like’ magnetic bion confinements are continuously connected,” JHEP 1107
Original
2011
Later among the works it cites.
P. A. Bolokhov, M. Shifman and A. Yung, “BPS Spectrum of Supersymmetric CP(N-1) Theory with ZN Twisted Masses,” Phys. Rev. D 84
Original
2011
Later among the works it cites.
T. Vachaspati, “Soliton Creation with a Twist,” Phys. Rev. D 84
Original
2011
Later among the works it cites.
P. Argyres and M. Ünsal, “A semiclassical realization of infrared renormalons,” Phys. Rev. Lett. 109
Original
2012
Closest in time.
P. C. Argyres and M. Ünsal, “The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, and renormalon effects,” JHEP 1208
Original
2012
Closest in time.
M. Kontsevich, “Resurgence from the path integral perspective,” Perimeter Institute Seminar, August 2012
2012
Closest in time.
J. G. Russo, “A Note on perturbation series in supersymmetric gauge theories,” JHEP 1206
Original
2012
Closest in time.
X. Cui and M. Shifman, “Perturbative Aspects of Heterotically Deformed CP(N-1) Sigma Model. I,” Phys. Rev. D 82
Original
2012
Closest in time.
E. Poppitz and M. Ünsal, “Index theorem for topological excitations on R 3 × S 1 R^{3}\times S^{1} and Chern-Simons theory,” JHEP 0903
Original
2085
Closest in time.

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