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Resurgence: “Exact” WKB in QM and QFT Gerald Dunne University Of

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Resurgence: “Exact” WKB in QM and QFT Gerald Dunne University Of Resurgence: “Exact” WKB in QM and QFT Gerald Dunne University of Connecticut CAQCD 2013: Minneapolis with Mithat Ünsal: arXiv:1210.2423 (JHEP), 1210.3646 (PRD), 1306.xxxx also: Gökçe Başar, Robert Dabrowski, 1306.0921 related: Argyres & Ünsal: arXiv:1204.1661 (PRL), 1206.1890 (JHEP) Physical Motivation SU(N) Yang-Mills on ℝ4 and ℂℙN-1 on ℝ2 • asymptotically free • instantons, theta vacua, ... two serious (unsolved) problems: • perturbative sector: infrared (IR) renormalons ⇒ perturbation theory ill-defined • non-perturbative sector: instanton scale moduli ⇒ instanton gas picture ill-defined ‘t Hooft, 1979; Affleck, 1980; David, 1981 analogous problem in QM: Bogomolny/Zinn-Justin (BZJ) mechanism degenerate classical vacua: double-well or Sine-Gordon ... ... S single-instanton sector: (i) level or band splitting e− instanton (ii) real and unambiguous ⇠ perturbation theory is non-Borel-summable: (i) ambiguous imaginary contrib. to real energy 2 S (ii) ie− instanton ⇠ ± non-perturbative sector: ¯ attractive II BZJ idea: rotate g2 → - g2; interaction repulsive; rotate back again produces ambiguous imaginary non-perturbative contribution which exactly cancels the term from perturbation theory ``resurgence’’ Bogomolny, 1980; Zinn-Justin, 1981; Balitsky/Yung 1986 recap: Borel summation and the BZJ mechanism t 1 e ( 1)nn!(g2)n = dt − (i) divergent, alternating: − 1+g2 t n Z0 X t 1 e n!(g2)n = dt − (ii) divergent, non-alternating: 1 g2 t n 0 X Z − i⇡ 1/g2 ⇒ ambiguous imaginary non-perturbative term: e− ±g2 t e− C+ “lateral Borel sums”: dt 2 C 1 g t C 1/g2 Z ± − − often identified with vacuum instability non-Borel-summable positive Borel pole ambig. imag. NP term 2 n g c i⇡ c c/g2 n! t = e− c 2 2 n , g , ± g X ✓ ◆ BZJ mechanism for SU(N) YM or ℂℙN-1 • for SU(N) YM and ℂℙN-1 QFT there is a new source of divergence, renormalons, producing dominant non-perturbative terms • asymptotically free theories have IR renormalons: positive Borel poles 2 S /β ie− instanton 0 instanton/anti-instanton poles ± β N 0 ⇠ UV renormalon poles IR renormalon poles neutral bion poles • but there are no such non-perturbative objects on ℝ4 or ℝ2, so perturbation theory remains incomplete and inconsistent ‘t Hooft, 1979; David, 1981 • spatially compactified ℂℙN-1 generates fractionalized instantons and bions, cancelling perturbative IR renormalon ambiguities against non- perturbative ambiguities in instanton/bion gas picture GD & Ünsal, 2012 the bigger picture: resurgence and non-perturbative QFT goal: fully consistent non-perturbative definition of asymptotically free QFT, in the continuum idea: “resurgence” unifies perturbative and non-perturbative sectors in such a way that the combination is unambiguous and well-defined under analytic continuation of the coupling question: is this a practically useful new idea in QFT beyond just a formalization of what is already done with condensates and OPE’s, multi-instanton-calculus, etc... ? What is Resurgence? Jean Écalle, 1980; Dingle 1960s; Stokes 1850s; Zinn-Justin 1980s; Pham 1980s; Olver, 1990s; Berry 1990s; ... • resurgence is very general: special functions, nonlinear DEs, difference eqs, iteration maps, functional eqs, holomorphic vector fields, Lie groups, steepest descents, ... , presumably also path integrals and QFT • leads to improved asymptotic expansions, incorporating exponentially small terms: “trans-series” • both formal and numerical improvement on Poincaré asymptotics, using Borel resummation analysis • philosophical shift: view semi-classical asymptotic expansions as `exact encoding’ of the function Resurgence and Trans-Series k 1 k q 1 1 − S 1 f(g2)= c g2n exp ln n,k,q −g2 −g2 n=0 q=0 X kX=0 X ✓ ◆ ✓ ◆ Écalle: set of functions based on these trans-monomial elements is closed: “any reasonable function’’ has a trans-series expansion (Borel transform) + (analytic continuation) + (Laplace transform) must not hit boundaries dramatic consequence: expansion coefficients extremely constrained (cf. BZJ cancellation mechanism) Decoding a Trans-Series k 1 k q 1 1 − S 1 f(g2)= c g2n exp ln n,k,q −g2 −g2 n=0 q=0 X kX=0 X ✓ ◆ ✓ ◆ 1 2n i ⇡ terms pert. theory: cn,0,0 g divergent, asymptotic ± n=0 X degenerate harmonic vacua: typically non Borel-summable n! 2S/g2 cn,0,0 ↵ i⇡↵e− ⇠ (2S)n ) ± but we find that c = ↵ so imaginary parts cancel! 0,2,1 − leading large order of pert. theory leading low order of fluctuations n! (n 1)! (n 2)! beyond leading: cn,0,0 ↵ n + β −n 1 + γ −n 2 + ... ⇠ (2S) (2S) − (2S) − 2S/g2 1 2 4 ¯ e− ln ↵ + βg + γg + ... II − −g2 ) ✓ ◆ Decoding a Trans-Series k 1 k q 1 1 − S 1 f(g2)= c g2n exp ln n,k,q −g2 −g2 n=0 q=0 X kX=0 X ✓ ◆ ✓ ◆ fluctuations around single-instanton sector (k=1): S/g2 2 4 e− 1+a1g + a2g + ... n! non Borel-summable: a (a ln n + b) n ⇠ (2S)n 3S/g2 1 i ⇡ e− a ln + b ) ± g2 ✓ ◆ leading low orders of fluctuations in k=3 (IIIbar, IIbarIbar) sector: 2 3S/g2 a 1 1 e− ln + b ln + c 2 −g2 −g2 " ✓ ✓ ◆◆ ✓ ◆ # ... ad infinitum II¯ “map” of all saddle points vacuum I II¯I,II¯ I¯ network of correspondences within trans-series lead to the “required” cancellations 2 4 S/g2 2 4 (1 + a1g + a2g + + ...) +e− (1 + b g + b g + + ...) ··· 1 2 ··· 2S/g2 2 4 3S/g2 2 4 +e− (1 + c1g + c2g + + ...)+e− (1 + d g + d g + + ...) ··· 1 2 ··· + ... +(log terms) graded resurgence triangle GD & Ünsal, 2012 resurgence triangle”: f(0f(0,0),0) pert. theory around pert. vacuum instanton x (pert. flucs.) S S 2 +Ai⇥ ⇥ A i⇥⇥ e− g +fi (1k,1) − g2 − i k e− λ N f(1,1) e e− λ − Nf(1f(1, ,1)1) bion x (pert. flucs.) e e −− bi-instanton 2S 2S +2i2⇥A ⇥k 2S2A 2A ⇥2ki⇥ g2 +2i g2 g22i x (pert. flucs.) e− e− λ f(2,N2)f(2,2) e−e− λf(2f(2,0),0) e−e−λ − −N f(2f(2, ,2) 2) e e − − 3S 3A ⇥ 3S 3A ⇥ 3S 3A ⇥ 33SA ⇥ +3i⇥ +3i k +i⇥+i k 2 i⇥ i k 33ii⇥k g2 λ N g2 λ N − g − λ N g2λ N e− e− f(3,3) f(3,3) e− e− f(3,1)f(3,1) e e− −f(3, f1)(3, 1) ee−− −− f(3f(3, , 3)3) e e e − − e −− 4S 4A ⇥k 4S 4A ⇥k 4S4A 44SA ⇥k 4A4S ⇥k 2 +4i⇥+4i +2+2i⇥i 2 22ii⇥ 4i4i⇥ − g λ N g2 λ N − g λ g2λ N λg2 N e e− f(4,4)f(4,4) e−e− f(4f,2)(4,2) e e− f(4f(4,0),0) ee−− −− ff(4(4, , 2)2) ee−− −− ff(4(4, , 4)4) e e e −− e −− . .. .. (1.2) which represents a general expansion of some observable. The rows correspond to a given in- stantonsectors number withn, with different associated ϴ perturbative dependence loop expansions cannot mix times or an cancel instanton prefactor β k f (λ) (λ) n 1 c (λ) , and with the topological phases specified. Only columns (n,k) ⌘ − k=0 (n,k) of this triangle with matching ⇥ dependence can possibly mix via resurgence2. For example, P in the Bogomolny-Zinn-Justin (BZJ) approach to the periodic potential problem [24, 35], which has degenerate vacua and a topological theta angle, the ambiguity in the perturbative contribution f(0,0)(λ) to the ground state energy is cured by an ambiguity in the instanton- 2A anti-instanton amplitude, at order e− λ f(2,0)(λ). This is in fact a general phenomenon that extends throughout the triangle: the ⇥-sectors are correlated with instanton sectors, which gives another tool for probing the mixing of the di↵erent terms in the trans-series represen- tation. Our main conjecture is that each column is a resurgent function of λ. This graded resurgence structure provides an interesting new perspective on instanton calculus and is born in a natural implementation of the theory of resurgence in the path integral formalism. Our long-term goal in applying resurgence to QFT is rather ambitious: We aim to give a non-perturbative continuum definition of quantum field theory, and provide a mathematically rigorous foundation. We also would like that such such a definition should be of practical value (not a formal tool) whose results can be compared with the numerical analysis of lattice field theory. In other words, generalizing the title of ’t Hooft’s seminal Erice lectures [38], we want to make sense out of general QFTs in the continuum. We emphasize that our immediate goal is not to provide theorems; rather we would like to reveal structure underlying QFT, a framework in which we can define QFT in a self- consistent manner without running into internal inconsistencies. We hope that whatever framework emerges along these lines may form the foundation of a rigorous definition. This point of view is close in spirit to Refs. [15, 39]. However, we ultimately hope that we will be able to use resurgence theory to provide exact and rigorous results for general QFTs, at least 2For notational simplicity we suppress log terms that generally also appear in the prefactor sums. –4– What is Resurgence? resurgent functions display at each of their singular points a behaviour closely related to their behaviour at the “origin”. Loosely speaking, these functions resurrect, or surge up - in a slightly different guise, as it were - at their singularities Écalle, 1980 n m question: to what extent can we reconstruct the full function from the perturbative series? What is Resurgence? resurgence in saddle-point integrals: Berry & Howls (``hyperasymptotics’’) 2 (n) 2 f(z)/g (n) 2 1 f /g2 (n) 2 I (g )= dz e− I (g ) e− n T (g ) Γ ) ⇠ g Z n (n) 2 1 2r (n) T (g )= g Tr Darboux’s theorem: large orders of expansion r=0 around one critical point governed by nhd.
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