Resurgence and Trans-Series in Quantum Theories

Resurgence and Trans-series in Quantum Theories

Gerald Dunne

University of Connecticut

Schladming Winter School: Intersections between QCD and Condensed Matter, March 1-6, 2015

GD & Mithat Ünsal, 1210.2423, 1210.3646, 1306.4405, 1401.5202

GD, lectures at CERN 2014 Winter School also with: G. Başar, A. Cherman, D. Dorigoni, R. Dabrowski: 1306.0921, 1308.0127, 1308.1108, 1405.0302, 1501.05671 Lecture 1

I motivation: physical and mathematical

I deﬁnition of resurgent trans-series

I divergence of perturbation theory in QM

I basics of Borel summation

I the Bogomolny/Zinn-Justin cancellation mechanism Physical Motivation

infrared renormalon puzzle in asymptotically free QFT • non-perturbative physics without instantons: physical meaning• of non-BPS saddles

Bigger Picture

I strongly interacting/correlated systems

I non-perturbative deﬁnition of non-trivial QFT in continuum

I analytic continuation of path integrals

I dynamical and non-equilibrium physics from path integrals

I uncover hidden ‘magic’ in perturbation theory

I “exact” asymptotics in QM, QFT and string theory − 2 x3/2 e√3 , x + ∞ 2 π x1/4 → ∞ 1 i 1 t3+x t e ( 3 ) dt  2π −∞ ∼  2 3/2 π Z  sin( 3 (−x) + 4 )  √ , x π (−x)1/4 → −∞  

Physical Motivation

what does a Minkowski path integral mean? • i 1 A exp S[A] versus A exp S[A] D D − Z ~ Z ~ Physical Motivation

what does a Minkowski path integral mean? • i 1 A exp S[A] versus A exp S[A] D D − Z ~ Z ~

− 2 x3/2 e√3 , x + ∞ 2 π x1/4 → ∞ 1 i 1 t3+x t e ( 3 ) dt  2π −∞ ∼  2 3/2 π Z  sin( 3 (−x) + 4 )  √ , x π (−x)1/4 → −∞   Mathematical Motivation

Resurgence: ‘new’ idea in mathematics (Écalle, 1980; Stokes, 1850) resurgence = unification of perturbation theory and non-perturbative physics perturbation theory generally divergent series • ⇒ series expansion trans-series expansion • −→ trans-series ‘well-defined under analytic continuation’ • perturbative and non-perturbative physics entwined • applications: ODEs, PDEs, fluids, QM, Matrix Models, QFT, String• Theory, ... philosophical shift: • view semiclassical expansions as potentially exact Resurgent Trans-Series

trans-series expansion in QM and QFT applications: • ∞ ∞ k−1 k l 2 2p c 1 f(g ) = ck,l,p g exp ln −g2 ±g2 p=0 k=0 l=1 X X X perturbative ﬂuctuations k−instantons quasi-zero-modes | {z } J. Écalle (1980): set of functions closed| under:{z } | {z } • (Borel transform) + (analytic continuation) + (Laplace transform)

− 1 trans-monomial elements: g2, e g2 , ln(g2), are familiar • “multi-instanton calculus” in QFT • new: analytic continuation encoded in trans-series • new: trans-series coeﬃcients ck,l,p highly correlated • new: exponentially improved asymptotics • 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 J. Écalle, 1980

m n I Zeeman: cn ( 1) (2n)! ∼ − I Stark: cn (2n)! ∼ 1 I cubic oscillator: cn Γ(n + ) ∼ 2 n 1 I quartic oscillator: cn ( 1) Γ(n + ) ∼ − 2 I periodic Sine-Gordon (Mathieu) potential: cn n! ∼ I double-well: cn n! ∼ note generic factorial growth of perturbative coeﬃcients

Perturbation theory

hard problem = easy problem + “small” correction • perturbation theory generally divergent series • →

∞ n e.g. QM ground state energy: E = n=0 cn (coupling) P Perturbation theory

hard problem = easy problem + “small” correction • perturbation theory generally divergent series • →

∞ n e.g. QM ground state energy: E = n=0 cn (coupling) P n I Zeeman: cn ( 1) (2n)! ∼ − I Stark: cn (2n)! ∼ 1 I cubic oscillator: cn Γ(n + ) ∼ 2 n 1 I quartic oscillator: cn ( 1) Γ(n + ) ∼ − 2 I periodic Sine-Gordon (Mathieu) potential: cn n! ∼ I double-well: cn n! ∼ note generic factorial growth of perturbative coeﬃcients Perturbation theory

but it works ... Perturbation theory works

QED perturbation theory:

1 1 α α 2 α 3 α 4 (g − 2) = − (0.32848...) + (1.18124...) − (1.7283(35)) + ... 2 2 π π π π

1 (g 2) = 0.001 159 652 180 73(28) 2 − exper 1 (g 2) = 0.001 159 652 184 42 2 − theory

QCD: asymptotic freedom

g3 11 4 N (g )= s N F s 16⇥2 3 C 3 2 ⇥ nobelprize.org

The left-hand panel shows a collection of different measurements by S. Bethke from High- Energy International Conference in Quantum Chromodynamics, Montpellier 2002 (hep- ex/0211012). The right-hand panel shows a collection by P. Zerwas, Eur. Phys. J. C34(2004)41. JADE was one of the experiments at PETRA at DESY. NNLO means Next-to- Next-to-Leading Order computation in QCD.

Although there are limits to the kind of calculations that can be performed to compare QCD with experiments, there is still overwhelming evidence that it is the correct theory. Very ingenious ways have been devised to test it and the data obtained, above all at the CERN LEP accelerator, are bounteous. Wherever it can be checked, the agreement is better than 1%, often much better, and the discrepancy is wholly due to the incomplete way in which the calculations can be made.

The Standard Model for Particle Physics

QCD complemented the electro-weak theory in a natural way. This theory already contained the quarks and it was natural to put all three interactions together into one model, a non- abelian gauge field theory with the gauge group SU(3) x SU(2) x U(1). This model has been called ‘The Standard Model for Particle Physics’. The theory explained the SLAC experiments and also contained a possible explanation why quarks could not be seen as free particles (quark confinement). The force between quarks grows with distance because of ‘infrared slavery’, and it is easy to believe that they are permanently bound together. There are many indications in the theory that this is indeed the case, but no definite mathematical proof has so far been advanced.

The Standard Model is also the natural starting point for more general theories that unify the three different interactions into a model with one gauge group. Through spontaneous symmetry breaking of some of the symmetries, the Standard Model can then emerge. Such

12 Perturbation theory

but it is divergent ... The series is divergent; therefore we may be able to do something with it

O. Heaviside, 1850 – 1925

Perturbation theory: divergent series

Divergent series are the invention of the devil, and it is shameful to base on them any demonstration whatsoever ... That most of these things [summation of divergent series] are correct, in spite of that, is extraordinarily surprising. I am trying to ﬁnd a reason for this; it N. Abel, 1802 – 1829 is an exceedingly interesting question. Perturbation theory: divergent series

The series is divergent; therefore we may be able to do something with it

O. Heaviside, 1850 – 1925 Asymptotic Series vs Convergent Series

N−1 n f(x) = cn (x x ) + RN (x) − 0 n=0 X convergent series:

RN (x) 0 ,N , x ﬁxed | | → → ∞

asymptotic series:

N RN (x) x x , x x ,N ﬁxed | | | − 0| → 0

“optimal truncation”: −→ truncate just before least term (x dependent!) Asymptotic Series vs Convergent Series

∞ n n 1 1 1 ( 1) n! x e x E − ∼ x 1 x n=1 X

æ 1.10 æ æ 0.920 æ 1.05 0.918 1.00

æ æ æ 0.95 0.916 æ æ æ æ æ æ æ æ æ æ æ 0.90 æ æ 0.914 æ æ æ 0.85 æ æ æ 0.912 0.80 æ 0.75 æ ææ N æ N 0 5 10 15 20 2 4 6 8 10

(x = 0.1) (x = 0.2)

optimal truncation order depends on x: N 1 opt ≈ x Asymptotic Series vs Convergent Series

contrast with behavior of a convergent series: more terms always improves the answer, independent of x

∞ ( 1)n − xn = PolyLog(2, x) n2 − n=0 X

-0.75 æ æ -0.62

æ æ -0.80 æ æ -0.64 æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ æ -0.66 -0.85 æ -0.68 -0.90 -0.70 -0.95 -0.72 -0.74 æ N æ N 5 10 15 20 5 10 15 20

(x = 1) (x = 0.75) Asymptotic Series: exponential precision

∞ n n 1 1 1 ( 1) n! x e x E − ∼ x 1 x n=0 X optimal truncation: error term is exponentially small

−1/x N −N −N e RN (x) N! x N!N √Ne | |N≈1/x ≈ N≈1/x ≈ ≈ ≈ √x

æ æ æ 0.920 æ 0.90

æ æ 0.918 æ æ 0.85 æ æ 0.916 æ æ æ æ æ æ æ æ æ æ æ æ æ 0.914 æ 0.80 æ æ

0.912 0.75

æ ææ N æ N 0 5 10 15 20 2 4 6 8

(x = 0.1) (x = 0.2) Asymptotic Series vs Convergent Series

Divergent series converge faster than convergent series because they don’t have to converge

G. F. Carrier, 1918 – 2002 Borel summation: basic idea

∞ −t n write n! = 0 dt e t alternatingR factorially divergent series:

∞ ∞ 1 ( 1)n n! gn = dt e−t (?) − 1 + g t n=0 0 X Z integral convergent for all g > 0: “Borel sum” of the series Borel Summation: basic idea

∞ ∞ 1 ( 1)n n! xn = dt e−t − 1 + x t n=0 0 X Z 1.2

1.1

1.0

0.9

0.8

0.7

x 0.0 0.1 0.2 0.3 0.4 non-perturbative imaginary part ⇒ i π − 1 e g ± g

but every term in the series is real !?!

Borel summation: basic idea

∞ −t n write n! = 0 dt e t non-alternatingR factorially divergent series:

∞ ∞ 1 n! gn = dt e−t (??) 1 g t n=0 0 X Z − pole on the Borel axis! Borel summation: basic idea

∞ −t n write n! = 0 dt e t non-alternatingR factorially divergent series:

∞ ∞ 1 n! gn = dt e−t (??) 1 g t n=0 0 X Z − pole on the Borel axis!

non-perturbative imaginary part ⇒ i π − 1 e g ± g

but every term in the series is real !?! Borel Summation: basic idea

∞ ∞ n −t 1 1 − 1 1 Borel e n! x = dt e = e x Ei ⇒ R P 1 x t x x "n=0 # 0 X Z − 2.0

1.5

1.0

0.5

x 0.5 1.0 1.5 2.0 2.5 3.0

-0.5 Borel summation

∞ n Borel transform of series f(g) cn g : ∼ n=0 ∞P cn [f](t) = tn B n! n=0 X new series typically has ﬁnite radius of convergence.

Borel resummation of original asymptotic series:

1 ∞ f(g) = [f](t)e−t/gdt S g B Z0

warning: [f](t) may have singularities in (Borel) t plane B Borel singularities

avoid singularities on R+: lateral Borel sums:

eiθ∞ 1 −t/g θf(g) = [f](t)e dt S g B Z0

C+

C-

go above/below the singularity: θ = 0± non-perturbative ambiguity: Im[ f(g)] −→ ± S0 challenge: use physical input to resolve ambiguity Borel summation: existence theorem (Nevanlinna & Sokal)

R R f(z) analytic in circle CR = z : z < { − 2 2 } N−1 n N N f(z) = an z + RN (z) , RN (z) A σ N! z | | ≤ | | n=0 X Borel transform ∞ R/2 an B(t) = tn n! n=0 X analytic continuation to Im(t) + Sσ = t : t R < 1/σ { | − | } 1/σ 1 ∞ Re(t) f(z) = e−t/z B(t) dt z Z0 Resurgence and Analytic Continuation

another view of resurgence:

resurgence can be viewed as a method for making formal asymptotic expansions consistent with global analytic continuation properties

the trans-series really IS the function ⇒ Resurgence: Preserving Analytic Continuation

zero-dimensional partition functions • ∞ √ − 1 sinh2( λ x) 1 1 1 Z1(λ) = dx e 2λ = e 4λ K0 −∞ √λ 4λ Z ∞ π Γ(n + 1 )2 ( 1)n(2λ)n 2 Borel-summable ∼ 2 − 1 2 r n=0 n!Γ 2 X √ π/ λ √ − 1 sin2( λ x) π − 1 1 Z2(λ) = dx e 2λ = e 4λ I0 0 √λ 4λ Z ∞ π Γ(n + 1 )2 (2λ)n 2 non-Borel-summable ∼ 2 1 2 r n=0 n!Γ 2 X naively: Z ( λ) = Z (λ) • 1 − 2 connection formula: K (e±iπ z ) = K ( z ) i π I ( z ) • 0 | | 0 | | ∓ 0 | | Resurgence: Preserving Analytic Continuation

Borel summation • ∞ π 1 − t 1 1 Z (λ) = dt e 2λ F , , 1; t 1 2 2λ 2 1 2 2 − r Z0 lateral Borel summation • iπ −iπ Z1(e λ) Z1(e λ) − ∞ π 1 − t 1 1 1 1 = dt e 2λ F , , 1; t iε F , , 1; t + iε 2 2λ 2 1 2 2 − − 2 1 2 2 r 1 Z ∞ π 1 − 1 − t 1 1 = (2i) e 2λ dt e 2λ F , , 1; t − 2 2λ 2 1 2 2 − r Z0 − 1 = 2 i e 2λ Z (λ) − 1 ±iπ − 1 connection formula: Z (e λ) = Z (λ) i e 2λ Z (λ) • 1 2 ∓ 1 reﬂection formula: ψ(1 + z) ψ(1 z) = 1 π cot(π z) • − − z −

∞ 1 π Im ψ(1 + iy) + + π e−2π k y ⇒ ∼ −2y 2 k X=1 “raw” asymptotics inconsistent with analytic continuation

Resurgence: Preserving Analytic Continuation

d Stirling expansion for ψ(x) = dx ln Γ(x) is divergent 1 1 1 1 174611 ψ(1 + z) ln z + + + + ... ∼ 2z − 12z2 120z4 − 252z6 ··· 6600z20 −

functional relation: ψ(1 + z) = ψ(z) + 1 • z

formal series Im ψ(1 + iy) 1 + π ⇒ ∼ − 2y 2 Resurgence: Preserving Analytic Continuation

d Stirling expansion for ψ(x) = dx ln Γ(x) is divergent 1 1 1 1 174611 ψ(1 + z) ln z + + + + ... ∼ 2z − 12z2 120z4 − 252z6 ··· 6600z20 −

functional relation: ψ(1 + z) = ψ(z) + 1 • z

formal series Im ψ(1 + iy) 1 + π ⇒ ∼ − 2y 2 reﬂection formula: ψ(1 + z) ψ(1 z) = 1 π cot(π z) • − − z −

∞ 1 π Im ψ(1 + iy) + + π e−2π k y ⇒ ∼ −2y 2 k X=1 “raw” asymptotics inconsistent with analytic continuation WKB Im E(z) √a e−b/z , z 0 ⇒ ∼ z → ∞ −b/z 1 a e a Γ(n + 2 ) cn dz = X ⇒ ∼ π zn+3/2 π bn+1/2 Z0

Borel Summation and Dispersion Relations

2 3 cubic oscillator: V = x + λ x A. Vainshtein, 1964

2 z=

1 E(z) z E(z ) = dz . o 0 2πi C z z0 R I − 1 R Im E(z) = dz π z z Z0 − 0 ∞ C 1 R Im E(z) = zn dz 0 π zn+1 n=0 0 X Z Borel Summation and Dispersion Relations

2 3 cubic oscillator: V = x + λ x A. Vainshtein, 1964

2 z=

1 E(z) z E(z ) = dz . o 0 2πi C z z0 R I − 1 R Im E(z) = dz π z z Z0 − 0 ∞ C 1 R Im E(z) = zn dz 0 π zn+1 n=0 0 X Z WKB Im E(z) √a e−b/z , z 0 ⇒ ∼ z → ∞ −b/z 1 a e a Γ(n + 2 ) cn dz = X ⇒ ∼ π zn+3/2 π bn+1/2 Z0 Instability and Divergence of Perturbation Theory

x2 x4 quartic AHO: V (x) = 4 + λ 4 Bender/Wu, 1969 Divergence of perturbation theory

an important part of the story ...

The majority of nontrivial theories are seemingly unstable at some phase of the coupling constant, which leads to the asymptotic nature of the perturbative series

A. Vainshtein (1964) Borel summation in practice

∞ n n f(g) cn g , cn β Γ(γ n + δ) ∼ ∼ n=0 X alternating series: real Borel sum • 1 ∞ dt 1 t δ/γ t 1/γ f(g) exp ∼ γ t 1 + t βg − βg Z0 " # nonalternating series: ambiguous imaginary part • 1 ∞ dt 1 t δ/γ t 1/γ Re f( g) exp − ∼ γ P t 1 t βg − βg Z0 − " # π 1 δ/γ 1 1/γ Im f( g) exp − ∼ ±γ βg − βg " # recall: divergence of perturbation theory in QM

∞ n e.g. ground state energy: E = n=0 cn (coupling) P n Zeeman: cn ( 1) (2n)! • ∼ −

Stark: cn (2n)! • ∼ n 1 quartic oscillator: cn ( 1) Γ(n + ) • ∼ − 2 1 cubic oscillator: cn Γ(n + ) • ∼ 2

periodic Sine-Gordon potential: cn n! • ∼

double-well: cn n! • ∼ recall: divergence of perturbation theory in QM

∞ n e.g. ground state energy: E = n=0 cn (coupling) P n Zeeman: cn ( 1) (2n)! stable • ∼ −

Stark: cn (2n)! unstable • ∼ n 1 quartic oscillator: cn ( 1) Γ(n + ) stable • ∼ − 2 1 cubic oscillator: cn Γ(n + ) unstable • ∼ 2

periodic Sine-Gordon potential: cn n! stable ??? • ∼

double-well: cn n! stable ??? • ∼ n! surprise: pert. theory non-Borel summable: cn n ∼ (2S) I stable systems

I ambiguous imaginary part

2S − 2 I i e g , a 2-instanton eﬀect ±

Bogomolny/Zinn-Justin mechanism in QM

......

degenerate vacua: double-well, Sine-Gordon, ... • − S splitting of levels: a real one-instanton eﬀect: ∆E e g2 ∼ Bogomolny/Zinn-Justin mechanism in QM

......

degenerate vacua: double-well, Sine-Gordon, ... • − S splitting of levels: a real one-instanton eﬀect: ∆E e g2 ∼ n! surprise: pert. theory non-Borel summable: cn n ∼ (2S) I stable systems

I ambiguous imaginary part

2S − 2 I i e g , a 2-instanton eﬀect ± Bogomolny/Zinn-Justin mechanism in QM

......

degenerate vacua: double-well, Sine-Gordon, ... • 1. perturbation theory non-Borel summable: ill-deﬁned/incomplete 2. instanton gas picture ill-deﬁned/incomplete: and ¯ attract I I regularize both by analytic continuation of coupling • ambiguous, imaginary non-perturbative terms cancel ! ⇒ Bogomolny/Zinn-Justin mechanism in QM

e.g., double-well: V (x) = x2(1 g x)2 −

2n E0 cn g ∼ n perturbation theory: X • 1 n − 2 cn 3 n!: Borel Im E π e 3g ∼ − ⇒ 0 ∼ ∓

1 non-perturbative analysis: instanton: g x0(t) = 1+e−t • 1 classical Eucidean action: S0 = 6g2 • non-perturbative instanton gas: • −2 1 Im E π e 6g2 0 ∼ ± BZJ cancellation E is real and unambiguous • ⇒ 0 “resurgence” cancellation to all orders ⇒ Bogomolny/Zinn-Justin mechanism in QM

double-well potential: V (x) = 1 x2 (1 g x)2 • 2 −

x0(R + t) , t > 0 approximate ¯ soln. : xcl(t) = II (x0(R t) , t < 0 − eﬀective interaction potential: U (t , t ) = 2 e−|t1−t2| int 1 2 − g2

1 − 1 Z = a2 dt dt e−Uint(t1,t2) a e 6g2 int 1 2 ≡ g √π Z Z ∞ T →∞ 1 2 T 2 a2 + T a2 dt exp e−t 1 + ... ∼ 2 g2 − Z0 instability: as g2 0, dominated by t 0 ??? • → → Bogomolny/Zinn-Justin mechanism in QM

∞ T →∞ 1 2 Z T 2 a2 + T a2 dt exp e−t 1 + ... int ∼ 2 g2 − Z0

BZJ idea: analytically continue g2 g2 → − dominated by ﬁnite t stable instanton gas ⇒ ⇒ ∞ 2 2 −t g 2 dt exp e 1 γE + ln + Ei −g2 − ∼ − 2 −g2 Z0

ambiguous imaginary part (from log) when g2 g2 • − → recall Z e−E0 T imaginary E from instanton gas • ∼ ⇒ 0 BZJ cancellation: cancels against ambiguous imaginary part from analytic continuation of Borel summation of perturbation theory −S/g2 2 4 pert ﬂucs about instanton: e 1 + a1g + a2g + ... divergent: n! −3S/g2 1 an n (a ln n + b) i π e a ln 2 + b ∼ (2S) ⇒ ± g 2 2 3-instanton: e−3S/g a ln 1 + b ln 1 + c • 2 − g2 − g2 resurgence: ad inﬁnitum, also sub-leading large-order terms

Decoding of Trans-series

∞ ∞ k−1 k q 2 2n S 1 f(g ) = cn,k,q g exp ln −g2 −g2 n=0 k q=0 X X=0 X ∞ 2n perturbative ﬂuctuations about vacuum: n=0 cn,0,0 g • n! divergent (non-Borel-summable): cn,0,0 α (2S)n • ∼P 2 ambiguous imaginary non-pert energy i π α e−2S/g ⇒ ∼ ± but c , , = α: BZJ cancellation ! • 0 2 1 − −S/g2 2 4 pert ﬂucs about instanton: e 1 + a1g + a2g + ... divergent: n! −3S/g2 1 an n (a ln n + b) i π e a ln 2 + b ∼ (2S) ⇒ ± g 2 2 3-instanton: e−3S/g a ln 1 + b ln 1 + c • 2 − g2 − g2 Decoding of Trans-series

I divergence of perturbation theory in QFT

I Euler-Heisenberg eﬀective actions & Schwinger eﬀect

I complex instantons and quantum interference

I IR renormalon puzzle in asymptotically free QFT Resurgence: recall from lecture 1

what does a Minkowski path integral mean? • i 1 A exp S[A] versus A exp S[A] D D − Z ~ Z ~ perturbation theory is generically asymptotic •

resurgent trans-series • ∞ ∞ k−1 k l 2 2p c 1 f(g ) = ck,l,p g exp ln −g2 ±g2 p=0 k=0 l=1 X X X perturbative ﬂuctuations k−instantons quasi-zero-modes | {z } | {z } | {z } Towards Resurgence in QFT

resurgence analytic continuation of trans-series • ≡ eﬀective actions, partition functions, ..., have natural integral •representations with resurgent asymptotic expansions analytic continuation of external parameters: temperature, chemical• potential, external ﬁelds, ... e.g., magnetic electric; de Sitter anti de Sitter, . . . • ↔ ↔ matrix models, large N, strings, ... (Mariño, Schiappa, ...) • soluble QFT: Chern-Simons, ABJM, matrix integrals • →

asymptotically free QFT ? . . . “renormalons” • Divergence from combinatorics

n n typical leading growth: cn ( 1) β Γ(γ n + δ) • ∼ ± factorial growth of number of Feynman diagrams • + 1 ∞ 1 x2 g x4 ∞ n n e− 2 − dx = J g J ( 1) (n 1)! √ n ⇒ n ∼ − − 2π n=0 Z−∞ X 4 3 n φ and φ : J c n! (Hurst, 1952; Thirring, 1953) • n ∼ QED: (Riddell, 1953) • (n!)2 n! J(n, , ρ) = (!)2(n )! · ρ! 1 (n ρ) ! 2(n ρ)/2 − 2 − − = # ext. electron lines , ρ = # ext. photon lines comment: large N limit in YM/QCD: number• of planar diagrams grows as a power law!

planar n J c (Koplik, Neveu, Nussinov, 1977) n ∼ Divergence of perturbation theory in QFT

C. A. Hurst (1952); W. Thirring (1953) φ4 pert. theory divergent (i) factorial growth # diagrams (ii) explicit lower bounds on diagrams

If it be granted that the perturbation expansion does not lead to a convergent series in the coupling constant for all theories which can be renormalized, at least, then a reconciliation is needed between this and the excellent agreement found in electrodynamics between experimental results and low-order calculations. It is suggested that this agreement is due to the fact that the S-matrix expansion is to be interpreted as an asymptotic expansion in the ﬁne-structure constant ... Dyson’s argument (QED)

F. J. Dyson (1952): •physical argument for divergence of QED perturbation theory

2 2 4 F (e ) = c0 + c2e + c4e + ...

Thus [for e2 < 0] every physical state is unstable against the spontaneous creation of large numbers of particles. Further, a system once in a pathological state will not remain steady; there will be a rapid creation of more and more particles, an explosive disintegration of the vacuum by spontaneous polarization.

suggests perturbative expansion cannot be convergent • Euler-Heisenberg Eﬀective Action (1935) review: hep-th/0406216

. . .

1-loop QED effective action in uniform emag field • e.g., constant B field: • B2 ∞ ds 1 s m2s S = coth s exp −8π2 s2 − s − 3 − B Z0

2 ∞ 2n+2 B n 2B S = B2 +4 −2π2 (2n + 4)(2n + 3)(2n + 2) m2 n=0 X Euler-Heisenberg Eﬀective Action

e.g., constant B ﬁeld: characteristic factorial divergence • ∞ ( 1)n+1 1 cn = − Γ(2n + 2) 8 (k π)2n+4 k X=1 recall Borel summation: • ∞ n n f(g) cn g , cn β Γ(γ n + δ) ∼ ∼ n=0 X

1 ∞ ds 1 s δ/γ s 1/γ f(g) exp → ∼ γ s 1 + s βg − βg Z0 " # reconstruct correct Borel transform: • ∞ s 1 1 s = coth s k2π2(s2 + k2π2) −2s2 − s − 3 k X=1 328 The European Physical Journal D Schwinger effect: recent~ technological advances in laser science, and also in 2eE 2mc2 theoreticalmc ∼ refinements of the Heisenberg-Euler computation, suggest that lasers such as those planned for ELI may be⇒ able to reach this elusive nonperturbative regime. 2 3 Thism c has the potential16 to open up an entirely new domain Ec 10 V/cm Fig. 1. PairWKB production tunneling as the from separation Dirac sea of a virtual vacuum ∼of experiments,e~ ≈ with the prospect of fundamental discov- dipole pairIm underS thephysical influence pair of anproduction external electric rate field. eries and practical applications, as are described in many → talks in this conference. Euler-Heisenberg series must be divergent • + asymptotic e e− pairs if they gain the binding energy of 2mc2 from the external field, as depicted in Figure 1.This is a non-perturbative process, and the leading exponential 2TheQEDeffective action part of the probability, assuming a constant electric field, was computed by Heisenberg and Euler [2,3]: In quantum field theory, the key object that encodes vacuum polarization corrections to classical Maxwell electro- π m2 c3 dynamics is the “effective action” Γ [A], which is a func- PHE exp , (3) tional of the applied classical gauge field A (x)[22–24]. ∼ − e ! µ & E ' The effective action is the relativistic quantum field the- building on earlier work of Sauter [18]. This result sets a ory analogue of the grand potential of statistical physics, basic scale of a critical field strength and intensity near in the sense that it contains a wealth of information about which we expect to observe such nonperturbative effects: the quantum system: here, thenonlinearpropertiesofthe quantum vacuum. For example, the polarization tensor 2 3 δ2Γ m c 16 Πµν = contains the electric permittivity %ij and c = 10 V/cm δAµδAν E e ! ≈ the magnetic permeability µij of the quantum vacuum, c 2 29 2 and is obtained by varying the effective action Γ [A]with Ic = c 4 10 W/cm . (4) 8π E ≈ × respect to the external probe Aµ(x). The general formal- As a useful guiding analogy, recall Oppenheimer’s compu- ism for the QED effective action was developed in a se- tation [19]oftheprobabilityofionizationofanatomof ries of papers by Schwinger in the 1950’s [22,23]. Γ [A]is binding energy E in such a uniform electric field: defined [23]intermsofthevacuum-vacuumpersistence b amplitude 3/2 4 √2mEb Pionization exp . (5) i 3 e 0out 0in =exp Re(Γ )+i Im(Γ ) . (8) ∼ (− ! ) & | ' { } E &! ' Taking as a representative atomic energy scale the binding Note that Γ [A]hasarealpartthatdescribesdispersiveef- me4 energy of hydrogen, Eb = 2 13.6eV,wefind fects such as vacuum birefringence, and an imaginary part 2! ≈ that describes absorptive effects, such as vacuum pair pro- 2 m2 e5 P hydrogen exp . (6) duction. Dispersive effects are discussed in detail in Gies’s ∼ −3 !4 contribution to this volume [25]. The imaginary part en- & E ' codes the probability of vacuum pair production as This result sets a basic scale of field strength and inten- 2 sity near which we expect to observe such nonperturbative Pproduction =1 0out 0in ionization effects in atomic systems: − |& | '| 2 =1 exp Im Γ m2e5 − −! ionization = = α3 4 109 V/cm & ' c 4 c 2 E ! E ≈ × Im Γ (9) Iionization = α6I 6 1016 W/cm2. (7) ≈ ! c c ≈ × These, indeed, are the familiar scales of atomic ioniza- here, in the last (approximate) step we use the fact that ionization Im(Γ )/ is typically very small. The expression (9)canbe tion experiments. Note that c differs from c ! 3 7E E viewed as the relativistic quantum field theoretic analogue by a factor of α 4 10− .Thesesimpleestimates explain why vacuum∼ pair× production has not yet been of the well-known quantum mechanical fact that the ion- observed – it is an astonishingly weak effect with con- ization probability is determined by the imaginary part ventional lasers [20,21]. This is because it is primarily a of the energy of an atomic electron in an applied electric non-perturbative effect, that depends exponentially on the field. (inverse) electric field strength, and there is a factor of From a computational perspective, the effective action 107 difference between the critical field scales in the atomic∼ is defined as [22–24] regime and in the vacuum pair production regime. Thus, Γ [A]=! ln det [iD/ m] with standard lasers that can routinely probe ionization, − there is no hope to see vacuum pair production. However, = ! tr ln [iD/ m] . (10) −

Euler-Heisenberg Eﬀective Action and Schwinger Eﬀect

B field: QFT analogue of Zeeman effect E field: QFT analogue of Stark effect B2 E2: series becomes non-alternating → − 2 2 2 Borel summation Im S = e E ∞ 1 exp k m π ⇒ 8π3 k=1 k2 − eE P h i Euler-Heisenberg Effective Action and Schwinger Effect

B field: QFT analogue of Zeeman effect E field: QFT analogue of Stark effect B2 E2: series becomes non-alternating → − 2 2 2 Borel summation Im S = e E ∞ 1 exp k m π ⇒ 8π3 k=1 k2 − eE 328 TheP European Physicalh Journali D Schwinger effect: recent~ technological advances in laser science, and also in 2eE 2mc2 theoreticalmc ∼ refinements of the Heisenberg-Euler computation, suggest that lasers such as those planned for ELI may be⇒ able to reach this elusive nonperturbative regime. 2 3 Thism c has the potential16 to open up an entirely new domain Ec 10 V/cm Fig. 1. PairWKB production tunneling as the from separation Dirac sea of a virtual vacuum ∼of experiments,e~ ≈ with the prospect of fundamental discov- dipole pairIm underS thephysical influence pair of anproduction external electric rate field. eries and practical applications, as are described in many → talks in this conference. Euler-Heisenberg series must be divergent • + asymptotic e e− pairs if they gain the binding energy of 2mc2 from the external field, as depicted in Figure 1.This is a non-perturbative process, and the leading exponential 2TheQEDeffective action part of the probability, assuming a constant electric field, was computed by Heisenberg and Euler [2,3]: In quantum field theory, the key object that encodes vacuum polarization corrections to classical Maxwell electro- π m2 c3 dynamics is the “effective action” Γ [A], which is a func- PHE exp , (3) tional of the applied classical gauge field A (x)[22–24]. ∼ − e ! µ & E ' The effective action is the relativistic quantum field the- building on earlier work of Sauter [18]. This result sets a ory analogue of the grand potential of statistical physics, basic scale of a critical field strength and intensity near in the sense that it contains a wealth of information about which we expect to observe such nonperturbative effects: the quantum system: here, thenonlinearpropertiesofthe quantum vacuum. For example, the polarization tensor 2 3 δ2Γ m c 16 Πµν = contains the electric permittivity %ij and c = 10 V/cm δAµδAν E e ! ≈ the magnetic permeability µij of the quantum vacuum, c 2 29 2 and is obtained by varying the effective action Γ [A]with Ic = c 4 10 W/cm . (4) 8π E ≈ × respect to the external probe Aµ(x). The general formal- As a useful guiding analogy, recall Oppenheimer’s compu- ism for the QED effective action was developed in a se- tation [19]oftheprobabilityofionizationofanatomof ries of papers by Schwinger in the 1950’s [22,23]. Γ [A]is binding energy E in such a uniform electric field: defined [23]intermsofthevacuum-vacuumpersistence b amplitude 3/2 4 √2mEb Pionization exp . (5) i 3 e 0out 0in =exp Re(Γ )+i Im(Γ ) . (8) ∼ (− ! ) & | ' { } E &! ' Taking as a representative atomic energy scale the binding Note that Γ [A]hasarealpartthatdescribesdispersiveef- me4 energy of hydrogen, Eb = 2 13.6eV,wefind fects such as vacuum birefringence, and an imaginary part 2! ≈ that describes absorptive effects, such as vacuum pair pro- 2 m2 e5 P hydrogen exp . (6) duction. Dispersive effects are discussed in detail in Gies’s ∼ −3 !4 contribution to this volume [25]. The imaginary part en- & E ' codes the probability of vacuum pair production as This result sets a basic scale of field strength and inten- 2 sity near which we expect to observe such nonperturbative Pproduction =1 0out 0in ionization effects in atomic systems: − |& | '| 2 =1 exp Im Γ m2e5 − −! ionization = = α3 4 109 V/cm & ' c 4 c 2 E ! E ≈ × Im Γ (9) Iionization = α6I 6 1016 W/cm2. (7) ≈ ! c c ≈ × These, indeed, are the familiar scales of atomic ioniza- here, in the last (approximate) step we use the fact that ionization Im(Γ )/ is typically very small. The expression (9)canbe tion experiments. Note that c differs from c ! 3 7E E viewed as the relativistic quantum field theoretic analogue by a factor of α 4 10− .Thesesimpleestimates explain why vacuum∼ pair× production has not yet been of the well-known quantum mechanical fact that the ion- observed – it is an astonishingly weak effect with con- ization probability is determined by the imaginary part ventional lasers [20,21]. This is because it is primarily a of the energy of an atomic electron in an applied electric non-perturbative effect, that depends exponentially on the field. (inverse) electric field strength, and there is a factor of From a computational perspective, the effective action 107 difference between the critical field scales in the atomic∼ is defined as [22–24] regime and in the vacuum pair production regime. Thus, Γ [A]=! ln det [iD/ m] with standard lasers that can routinely probe ionization, − there is no hope to see vacuum pair production. However, = ! tr ln [iD/ m] . (10) − Euler-Heisenberg and Matrix Models, Large N, Strings, ...

scalar QED EH in self-dual background (F = F˜): • ± F 2 ∞ dt 1 1 1 S = e−t/F + 16π2 t sinh2(t) − t2 3 Z0 Gaussian matrix model: λ = g N • 1 ∞ dt 1 1 1 = e−2λ t/g + F −4 t sinh2(t) − t2 3 Z0 c = 1 String: λ = g N • 1 ∞ dt 1 1 1 = e−2λ t/g F 4 t sin2(t) − t2 − 3 Z0 Chern-Simons matrix model: • 1 ∞ dt 1 1 1 = e−2(λ+2π i m) t/g + F −4 t 2 − t2 3 m∈Z 0 sinh (t) X Z de Sitter/ anti de Sitter eﬀective actions (Das & GD, hep-th/0607168)

explicit expressions (multiple gamma functions) • 2 d/2 n m (AdSd) K AdSd (K) an 2 L ∼ 4π n m X 2 d/2 n m (dSd) K dSd (K) an 2 L ∼ 4π n m X (AdSd) n (dSd) changing sign of curvature: an = ( 1) an • − odd dimensions: convergent • even dimensions: divergent • n d Γ(2n + d 1) a(AdSd) B2 + 2( 1)n − n ∼ n(2n + d) ∼ − (2π)2n+d

pair production in dSd with d even • QED/QCD effective action and the “Schwinger effect” formal definition: • e Γ[A] = ln det (i D/ + m) Dµ = ∂µ i Aµ − ~c vacuum persistence amplitude • i i O O exp Γ[A] = exp Re(Γ) + i Im(Γ) h out | ini ≡ { } ~ ~ encodes nonlinear properties of QED/QCD vacuum • vacuum persistence probability • 2 2 O O 2 = exp Im(Γ) 1 Im(Γ) |h out | ini| − ≈ − ~ ~ probability of vacuum pair production 2 Im(Γ) • ≈ ~ cf. Borel summation of perturbative series, & instantons • QED/QCD effective action

encodes nonlinear properties of QED/QCD vacuum • δ2Γ polarization tensor: Πµν • δAµδAν → Euler & Heisenberg (1935): • 4 e ~ 2 2 ik = δik + 2 E~ B~ δik + 7BiBk 45πm4c7 − h i 4 e ~ 2 2 µik = δik + 2 E~ B~ δik 7EiEk 45πm4c7 − − h i the electromagnetic properties of the vacuum can be described by a ﬁeld-dependent electric and magnetic polarisability of empty space, which leads, for example, to refraction of light in electric ﬁelds or to a scattering of light by light V. Weisskopf, 1936

PVLAS; ALPS, GammeV, BMV, OSQAR, ... • QFT in Extreme Background Fields: physical motivation

perturbation theory is not applicable • semiclassical/instanton/resurgence methods • non-perturbative lattice methods • I vacuum energy: mass generation; dark energy

I beyond standard model: axion, ALP, dark matter searches

I non-equilibrium QFT: e.g. quark-gluon-plasma

I astrophysics: neutron stars, magnetars, black holes

I cosmological particle production (Parker, Zeldovich)

I Hawking radiation

I back-reaction, cascading

I ultimate electric ﬁeld limit? Schwinger Eﬀect: Beyond Constant Background Fields

constant ﬁeld •

sinusoidal or single-pulse•

envelope pulse with sub-cycle• structure; carrier-phase eﬀect

chirped pulse; Gaussian beam• , ... envelopes and beyond require complex instantons • physics: optimization and quantum control • Beyond Constant Background Fields

Keldysh (1964): atomic ionization in E(t) = cos(ωt) • √ E adiabaticity parameter: γ ω 2mEb • ≡ eE √ 3/2 2mE WKB P exp 4 b g(γ) • ⇒ ionization ∼ − 3 e~E

√ 3/2 2mE exp 4 b , γ 1 (non-perturbative) − 3 e~E  Pionization  ∼   2Eb/~ω √eE , γ 1 (perturbative) 2ω 2mEb    semi-classical instanton interpolates between non-perturbative •‘tunneling ionization” and perturbative “multi-photon ionization” Keldysh Approach in QED Brézin/Itzykson, 1970; Popov, 1971

Schwinger eﬀect in E(t) = cos(ωt) • E adiabaticity parameter: γ m c ω • ≡ e E 2 3 WKB P exp π m c g(γ) • ⇒ QED ∼ − e ~ E h i 2 3 exp π m c , γ 1 (non-perturbative) − e ~ E PQED  h i ∼ 2  4mc /~ω  e E , γ 1 (perturbative) ω m c   semi-classical instanton interpolates between non-perturbative •‘tunneling pair-production” and perturbative “multi-photon pair production” we will come back to this later ... Scattering Picture of Particle Production

Feynman, Nambu, Fock, Brezin/Itzykson, Marinov/Popov, ...

over-the-barrier scattering: e.g. scalar QED • φ¨ (p e A (t))2 φ = (m2 + p2 )φ − − 3 − 3 ⊥

+i m

ap⇥ bp⇥ “imaginary time method” i m

3 2 pair production probability: P d p bp • ≈ | | imaginary time method R •

2 2 2 2 bp exp 2 Im dt m + p + (p eA (t)) | | ≈ − ⊥ 3 − 3 I q more structured E(t) involve quantum interference • week ending PRL 102, 150404 (2009) PHYSICALREVIEWLETTERS 17 APRIL 2009

5 10 14 5 10 14

3 10 14 3 10 14

14 1 10 14 1 10 k MeV k MeV 0.4 0.2 0 0.2 0.4 0.6 0.4 0.2 0 0.2 0.4 0.6

FIG. 3. Asymptotic distribution function f k;~ for k~ 0 FIG. 4. Asymptotic distribution function f k;~ for k~ 0 ð 1Þ ? ¼ ð 1Þ ? ¼ for 5, E 0:1E , and =4. The center of the for 5, E0 0:1Ecr, and =2. The center of the 0 cr ¼ ¼ ¼À distribution¼ is shifted¼ to p ¼À102 keV. distribution is shifted to p 137 keV. kð1Þ% kð1Þ% sensitive dependence on the other shape parameters, such creation events. The fringes are large for =2, since ¼À as . then the field strength has two peaks of equal size (though In fact, there is an even more distinctive dependence on opposite sign) which act as two temporally separated slits. the carrier phase upon which the form of the scattering Moving the carrier phase away from =2 corre- ¼À potential !2 k;~ t is extremely sensitive. The carrier- sponds to gradually opening or closing the slits, resulting in phase dependenceÀ ð Þ is difficult to discuss in the WKB ap- a varying degree of which-way information and thus a proach, because a nonzero carrier phase breaks the E t varying contrast of the interference fringes. A quantitative E t symmetry of the pulse shape, which in turn makesð Þ¼ consequence of this double-slit picture is that the width of theðÀ imaginaryÞ time treatment of the WKB scattering prob- the envelope of the oscillations in the distribution function lem significantly more complicated [21]. But in the quan- is related to the temporal width of the slits. The width of the envelope of oscillations thus also becomes a probe of the tum kinetic approach, the carrier phase causesCarrier no Phase Effect Hebenstreit, Alkofer, GD, Gies, PRL 102, 2009 computational problems; it is just another parameter. We subcycle structure of the laser. have found that the introduction of the carrier phase makes To complete the physical picture, we consider the over- the oscillatory behavior in the longitudinal momentum all envelope of the longitudinal momentum distribution, again for 0, averaging over the rapid oscillations. distribution even more pronounced. This is shown in ¼ Figs. 3 and 4, where the momentum distribution function When there are more than three cycles per pulse ( * is plotted for =4 and =2. We see that, for 3), the peak of the momentum2 distribution is located near ¼À ¼À p 0, whereas for t & 3 the peak is shifted to a the same values of the other parameters, the oscillatoryE(tk)ð1 =Þ¼ exp cos (ωt + ϕ) behavior becomes more distinct as the phase offset in- nonzeroE value. Furthermore,−τ 2 the Gaussian width of the creases. The most distinctive momentum signature, how- employed WKB approximation Eq. (4), which scales ever, is found for =2, when the electric field is with peE0= ~, is obviously somewhat broader than the ¼À true distribution, as is shown in Fig. 5. We can quantify totally antisymmetric. In this case, the asymptotic distri- ffiffiffiffiffiffiffiffi this discrepancy in the width, by extending the WKB result bution function f k;~ t vanishes at the minima of the oscil- ð Þ beyondsensitivity the Gaussian approximation to carrier inherent phase in Eq.ϕ (4).? We week ending lations, as shown in Fig. 4. This feature also has a direct PHYSICALREVIEWLETTERS analogue in the scattering picture: For an antisymmetric PRL• 102, 150404 (2009) 17 APRIL 2009 field, the gauge potential Eq. (2) is symmetric, and so is the 14 5 10 14 5 1410 scattering potential well !2 k;~ t . In this case, perfect 5 10 transmission is possible forÀ certainð Þ resonance momenta, corresponding to zero reflection and thus zero pair produc- 14 3 10 3 10 14 tion. Also note that the center of the distribution shifts from 3 10 14 p 0 to a nonzero value again. These carrier-phase kð1Þ¼ effects provide distinctive signatures, strongly suggesting a 14 1 10 14 new experimental strategy and probe in the search for 1 10 14 1 10 Schwinger pair production. k MeV 0.6 0.4 0.2 0 0.2 0.4 0.6 k MeV k MeV These momentum signatures can also be understood in a 0.4 0.2 0 0.2 0.4 0.6 0.4 0.2 0 0.2 0.4 0.6 quantum-mechanical double-slit picture, which has first FIG. 5. Comparison of the asymptotic distribution function ~ ~ been developed in the context of above-threshold ioniza- FIG. 3. Asymptotic distribution function f k;~ for k~ 0 FIG. 4. Asymptotic distribution function f k; for k 0 f k;~ for k~ 0 (oscillating solid line)ð with1Þ the prediction? ¼ ð 1Þ ? ¼ for 5, E 0:1E , and =4. The center of the for 5, E0 0:1Ecr, and =2. The center of the tion with few-cycle laser pulses [35]: In this picture, the ofð Eq.1Þ (4) (dotted0 ? ¼ line)cr and the improved WKB approximation ¼ ¼ ¼Àπ distribution¼ is shifted¼ to p ¼À102 keV. distribution is shifted to p 137 keV. oscillations are fringes in the momentum spectrum that based on an expansionϕ ofkð1 Eq.=Þ% (5 0) (dashed line) for 5, E0 ϕkð1=Þ% result from the interference of temporally separated pair 0:1E , and 0. ¼ ¼ 2 cr ¼ sensitive dependence on the other shape parameters, such creation events. The fringes are large for =2, since then the field strength has two peaks of equal¼À size (though 150404-3as . In fact, there is an even more distinctive dependence on opposite sign) which act as two temporally separated slits. the carrier phase upon which the form of the scattering Moving the carrier phase away from =2 corre- ¼À potential !2 k;~ t is extremely sensitive. The carrier- sponds to gradually opening or closing the slits, resulting in phase dependenceÀ ð Þ is difficult to discuss in the WKB ap- a varying degree of which-way information and thus a proach, because a nonzero carrier phase breaks the E t varying contrast of the interference fringes. A quantitative E t symmetry of the pulse shape, which in turn makesð Þ¼ consequence of this double-slit picture is that the width of theðÀ imaginaryÞ time treatment of the WKB scattering prob- the envelope of the oscillations in the distribution function lem significantly more complicated [21]. But in the quan- is related to the temporal width of the slits. The width of the tum kinetic approach, the carrier phase causes no envelope of oscillations thus also becomes a probe of the computational problems; it is just another parameter. We subcycle structure of the laser. have found that the introduction of the carrier phase makes To complete the physical picture, we consider the over- the oscillatory behavior in the longitudinal momentum all envelope of the longitudinal momentum distribution, again for 0, averaging over the rapid oscillations. distribution even more pronounced. This is shown in ¼ Figs. 3 and 4, where the momentum distribution function When there are more than three cycles per pulse ( * is plotted for =4 and =2. We see that, for 3), the peak of the momentum distribution is located near ¼À ¼À p 0, whereas for & 3 the peak is shifted to a the same values of the other parameters, the oscillatory kð1Þ¼ behavior becomes more distinct as the phase offset in- nonzero value. Furthermore, the Gaussian width of the creases. The most distinctive momentum signature, how- employed WKB approximation Eq. (4), which scales ever, is found for =2, when the electric field is with peE0= ~, is obviously somewhat broader than the ¼À true distribution, as is shown in Fig. 5. We can quantify totally antisymmetric. In this case, the asymptotic distri- ffiffiffiffiffiffiffiffi this discrepancy in the width, by extending the WKB result bution function f k;~ t vanishes at the minima of the oscil- beyond the Gaussian approximation inherent in Eq. (4). We lations, as shownð in Fig.Þ 4. This feature also has a direct analogue in the scattering picture: For an antisymmetric

ﬁeld, the gauge potential Eq. (2) is symmetric, and so is the 5 10 14 scattering potential well !2 k;~ t . In this case, perfect transmission is possible forÀ certainð Þ resonance momenta,

corresponding to zero reﬂection and thus zero pair produc- 14 3 10 tion. Also note that the center of the distribution shifts from p 0 to a nonzero value again. These carrier-phase kð1Þ¼ effects provide distinctive signatures, strongly suggesting a 14 new experimental strategy and probe in the search for 1 10 Schwinger pair production. k MeV These momentum signatures can also be understood in a 0.6 0.4 0.2 0 0.2 0.4 0.6 quantum-mechanical double-slit picture, which has ﬁrst FIG. 5. Comparison of the asymptotic distribution function been developed in the context of above-threshold ioniza- f k;~ for k~ 0 (oscillating solid line) with the prediction tion with few-cycle laser pulses [35]: In this picture, the ofð Eq.1Þ (4) (dotted? ¼ line) and the improved WKB approximation oscillations are fringes in the momentum spectrum that based on an expansion of Eq. (5) (dashed line) for 5, E0 result from the interference of temporally separated pair 0:1E , and 0. ¼ ¼ cr ¼

150404-3 Particle Production as the Stokes Phenomenon Dumlu, GD, 2010 anti-Stokes line Stokes line dom dom 2 00 + ~ ψ + Q ψ = 0 sub sub + Stokes line ↓ anti-Stokes line 1 i z dom ψ = exp Q1/2 + ± 1/4 Q ±~ sub sub Z + dom anti-Stokes line Stokes line exp(±i z3/2) suppose Q has simple zero at z = 0: ψ± • ∼ z1/4 WKB solutions deﬁned locally, inside Stokes wedges • propagating from t = to t = + necessarily crosses Stokes• lines −∞ ∞ “birth” of a new exponential = particle production • multiple sets of turning points quantum interference • ⇒ Carrier Phase Eﬀect from the Stokes Phenomenon

4 4

2 2

0 0

-2 -2

-4 -4 -4 -2 0 2 4 -4 -2 0 2 4

interference produces momentum spectrum structure • interference t = ⇥ t =+

P 4 sin2 (θ) e−2 ImW θ: interference phase ≈ double-slit interference, in time domain, from vacuum • Ramsey eﬀect: N alternating sign pulses N-slit system • 2 ⇒ coherent N enhancement Akkermans, GD, 2012 ⇒ Dynamically Assisted Schwinger Eﬀect Schützhold, GD, Gies, 2008

optical+X-ray laser pulse: E(t) = O(Ωt) + εX (ωt) • E exponential enhancement due to new turning points • 10 10

5 5 L L t t H 0 H 0 Im Im

-5 -5

-10 -10 -4 -2 0 2 4 -4 -2 0 2 4 Re t Re t

H L H L “multi-photon assisted tunneling” • lowers Schwinger critical ﬁeld from 1029W/cm2 to 1025W/cm2 ∼

(Di Piazza et al, 2009) Worldline Instantons GD, Schubert, 2005 To maintain the relativistic invariance we describe a trajectory in space-time by giving the four variables xµ(u) as functions of some ﬁfth parameter (somewhat analogous to the proper-time) Feynman, 1950

worldline representation of eﬀective action • ∞ T 4 dT −m2T 2 Γ = d x e x exp dτ x˙ + Aµ x˙ µ − T D − µ Z Z0 Ix Z0 double-steepest descents approximation: • worldline instantons: x¨µ = Fµν(x)x ˙ ν • proper-time integral: ∂S(T ) = m2 • ∂T − 2 Im Γ e−Sinstanton(m ) ≈ instantonsX multiple turning point pairs complex instantons • ⇒ Divergence of derivative expansion GD, T. Hall, hep-th/9902064

time-dependent E ﬁeld: E(t) = E sech2 (t/τ) • 4 ∞ j ∞ 2k m ( 1) 2E Γ(2k + j)Γ(2k + j 2) k j Γ = − ( 1)k − B2 +2 −8π3/2 (mλ)2j − m2 1 j k j!(2k)!Γ(2k + j + 2 ) X=0 X=2 Borel sum perturbative expansion: large k (j ﬁxed): • 1 j Γ(2k + 3j ) c( ) 2 − 2 k ∼ (2π)2j+2k+2

j m2π 1 m4π Im Γ(j) exp ∼ − E j! 4τ 2E3 resum derivative expansion • m2π 1 m 2 Im Γ exp 1 + ... ∼ − E − 4 Eτ Divergence of derivative expansion

Borel sum derivative expansion: large j (k ﬁxed): • 5 (k) 9 −2k Γ(2j + 4k 2 ) c 2 2 − j ∼ (2π)2j+2k

(2πEτ 2)2k Im Γ(k) e−2πmτ ∼ (2k)! resum perturbative expansion: • Eτ Im Γ exp 2πmτ 1 + ... ∼ − − m compare: • m2π 1 m 2 Im Γ exp 1 + ... ∼ − E − 4 Eτ 2 diﬀerent limits of full: Im Γ exp m π g m • ∼ − E E τ derivative expansion must be divergenth i • QFT: Renormalons reviews: Beneke, 1998; Shifman, 2014

QM: divergence of perturbation theory due to factorial growth of number of Feynman diagrams

n n! cn ( 1) ∼ ± (2S)n QFT: new physical eﬀects occur, due to running of couplings with momentum faster source of divergence: “renormalons” • n n β0 n! cn ( 1) ∼ ± (2S)n both positive and negative Borel poles • Renormalons

2 Adler function in QED: D(Q2) = 4π2Q2 dΠ(Q )

• − dQ2

q q k

bubble-chains, momentum k interpolating expression • → ∞ 2 2 2 k d(k ) αs(Q ) D(Q2) = Q2 2 2 3 β α (Q2) 0 (k + Q ) 1 0 s ln(Q2/k2) Z − 4π 2 running coupling αs(k ): • 2 2 αs(Q ) αs(k ) = 2 1 β0 αs(Q ) ln(Q2/k2) − 4π β = ﬁrst beta-function coeﬃcient • 0 2 αs(Q ) expansion has factorial divergences from both small & • large k2 Renormalons

split k2 integral: 0 k2 Q2 and Q2 k2 • ≤ ≤ ≤ ≤ ∞ 2 2 low-momentum: t = 2 ln Q ; high momentum: t = ln k • k2 Q2 Q2 2 2 ∞ 2 2 1 2 k αs(k ) 1 2 k αs(k ) D = 4 dk 2 2 3 + 4 dk 2 2 3 Q (1 + k /Q ) Q 2 (1 + k /Q ) Z0 ZQ 2 ∞ ∞ a a αs(Q ) ( 1) (1 + a) ( 1) (2 + a) = dt e−t − + − 2  β0αs t β0αs t  0 a=0 1 1 + Z X  − 4π (a+2) 4π (a+1)  in Borel form, with poles on both R± •   8π 12π 16π IR : tIR = , , ,..., a β α β α β α 0 s 0 s 0 s 4π 8π 12π UV : tUV = , , ,..., a −β α −β α −β α 0 s 0 s 0 s key physics question: does the weakly-coupled theory "know •enough" to extend into the strongly coupled region ? IR Renormalon Puzzle in Asymptotically Free QFT

2S − 2 perturbation theory: i e β0 g −→ ± − 2S instantons on R2 or R4: i e g2 −→ ±

instanton/anti-instanton poles

UV renormalon poles IR renormalon poles

appears that BZJ cancellation cannot occur asymptotically free theories remain inconsistent

’t Hooft, 1980; David, 1981 Lecture 3

N−1 I BZJ cancellation in 2d CP theory

I why resurgence?

I uniform WKB

I path integral interpretation: functional Darboux theorem

I thimbles and analytic continuation of path integrals IR Renormalon Puzzle in Asymptotically Free QFT

2S − 2 perturbation theory: i e β0 g −→ ± − 2S instantons on R2 or R4: i e g2 −→ ±

instanton/anti-instanton poles

UV renormalon poles IR renormalon poles

appears that BZJ cancellation cannot occur asymptotically free theories remain inconsistent

’t Hooft, 1980; David, 1981 IR Renormalon Puzzle in Asymptotically Free QFT

resolution: there is another problem with the non-perturbative instanton gas analysis (Argyres, Ünsal 1206.1890; GD, Ünsal, 1210.2423) scale modulus of instantons • spatial compactiﬁcation and principle of continuity • 2 dim. CPN−1 model: • instanton/anti-instanton poles

UV renormalon poles IR renormalon poles

neutral bion poles

cancellation occurs ! (GD, Ünsal, 1210.2423, 1210.3646) Topological Molecules in Spatially Compactiﬁed Theories 2 1 1 ℝ → SL x ℝ CPN−1: regulate scale modulus problem with (spatial) compactiﬁcation: R2 S1 R1 → L ×

x2 x2

x1 x1

Euclidean time

Sinst Sinst ZN twist: instantons fractionalize: Sinst = −→ N β0 Topological Molecules in Spatially Compactiﬁed Theories

temporal conpactiﬁcation: information only about deconﬁned phase

ℝ1 x Sᵦ1 ℝ1 ℝ2 high T low T

spatial compactiﬁcation: semi-classical small L regime continuously connected to large L: principle of continuity

1 1 SL x ℝ ℝ1 ℝ2 “continuity” Topological Molecules in Spatially Compactiﬁed Theories

weak-coupling semi-classical analysis • non-perturbative: kink-instantons: i, i = 1, 2,...,N • I bions: topological molecules of ¯ • II “orientation” dependence of ¯ interaction: • II charged bions ij = [ i ¯j]: repulsive bosonic interaction • B I I neutral bions ii = [ i ¯i]: attractive bosonic interaction • B I I instanton/anti-instanton amplitude is ambiguous: •

2 8π 8π g N 16 − 2 16 − 2 i ¯i = ln γ e g N iπ e g N I I ± 8π − g2 N ± g2 N Perturbative Analysis

weak-coupling semi-classical analysis • perturbative eﬀective QM problem (Mathieu) • → perturbation theory diverges & non-Borel summable • perturbative sector: lateral Borel summation •

2 8π 2 1 −t/g 2 16 − 2 B± (g ) = dt B (t) e = Re B (g ) iπ e g N E g2 E E ∓ g2 N ZC± compare: • 2 8π 8π g N 16 − 2 16 − 2 i ¯i = ln γ e g N iπ e g N I I ± 8π − g2 N ± g2 N exact ("BZJ") cancellation ! explicit application of resurgence to nontrivial QFT The Bigger Picture

Q: should we expect resurgent behavior in QM & QFT? QM uniform WKB ⇒ (i) trans-series structure is generic (ii) all multi-instanton eﬀects encoded in perturbation theory

(GD, Ünsal, 1306.4405, 1401.5202)

Q: what is behind this resurgent structure ? basic property of all-orders steepest descents integrals •

Q: could this extend to (path) functional integrals ? Uniform WKB & Resurgent Trans-Series ( 1306.4405, 1401.5202) d2 V (g x) d2 ψ + ψ = E ψ g4 ψ(y) + V (y)ψ(y) = g2 E ψ(y) −dx2 g2 →− dy2

weak coupling: degenerate harmonic classical vacua • 2 c non-perturbative eﬀects: g ~ exp 2 • ↔ ⇒ − g approximately harmonic • uniform WKB with parabolic cylinder functions ⇒ 1 Dν u(y) ansatz (with parameter ν): ψ(y) = g • √u0(y) “similar looking equations have similar looking solutions” Uniform WKB & Resurgent Trans-Series

perturbative expansion for E and u(y): • ∞ 2 2k E = E(ν, g ) = g Ek(ν) k X=0 ν = N: usual perturbation theory (not Borel summable) • global analysis boundary conditions: • ⇒

y -1 - 1 y 2 - 3 Π -Π - Π Π Π 3 Π 2 2 2 2

midpoint 1 ; non-Borel summability g2 e±i g2 • ∼ g ⇒ → trans-series encodes analytic properties of Dν • generic and universal ⇒ Uniform WKB & Resurgent Trans-Series

ν −z2/4 ±iπν √2π −1−ν z2/4 Dν(z) z e (1 + ... ) + e z e (1 + ... ) ∼ Γ( ν) − exact quantization condition −→ −ν 2 1 e±iπ 2 e−S/g = (ν, g2) Γ( ν) g2 π g2 P −

p −S/g2 ν is only exponentially close to N (here ξ e ): ⇒ ≡ √π g2 N 2 2 g2 (N, g ) ν = N + P ξ N! 2N 2 ±iπ g2 ∂ e 2 P + ln ψ(N + 1) 2 ξ2 + O(ξ3) −(N!)2 P ∂N g2 − P 2 ∞ 2k insert: E = E(ν, g ) = g Ek(ν) trans-series • k=0 ⇒ P in fact ... (GD, Ünsal, 1306.4405, 1401.5202)

2 g dg2 ∂E(N, g2) N + 1 g2 (N, g2) = exp S 1 + 2 P g4 ∂N − S " Z0 !#

perturbation theory E(N, g2) encodes everything ! ⇒

Connecting Perturbative and Non-Perturbative Sector

Zinn-Justin/Jentschura: generate entire trans-series from (i) perturbative expansion E = E(N, g2) (ii) single-instanton ﬂuctuation function (N, g2) (iii) rule connecting neighbouring vacua (parity,P Bloch, ...) Connecting Perturbative and Non-Perturbative Sector

in fact ... (GD, Ünsal, 1306.4405, 1401.5202)

2 g dg2 ∂E(N, g2) N + 1 g2 (N, g2) = exp S 1 + 2 P g4 ∂N − S " Z0 !#

perturbation theory E(N, g2) encodes everything ! ⇒ Connecting Perturbative and Non-Perturbative Sector

e.g. double-well potential: B N + 1 ≡ 2 1 19 E(N, g2) = B g2 3B2 + g4 17B3 + B − 4 − 4 375 459 131 g6 B4 + B2 + ... − 2 4 32 − non-perturbative function( (...) exp[ A/2]): • P ∼ − 1 19 153B A(N, g2) = + g2 17B2 + + g4 125B3 + 3g2 12 4 17815 23405 22709 +g6 B4 + B2 + + 12 24 576 simple relation: • ∂E ∂A = 3g2 2B g2 ∂B − − ∂g2 1 1 1 − 8 48 16

a1 b 11 b 21

1 1 − ﬂuctuation about for double-well: 24 12

b12 b 22 • I 1 2-loop (Shuryak/Wöhler, 1994); 3-loop 8

b 23

1 − (Escobar-Ruiz/Shuryak/Turbiner, arXiv:1501.03993) 8

b 24

1 1 1 − − S0 71 16 16 8 E/S/T : e g 1 g 0.607535 g2 ... 72 d e f − − − 1 1 − 16 48 g h − S0 1 D/Ü g 2 Figure 2: Diagrams contributing to the coeﬃcient B2. The signs of contributions and symmetry : e 1 + g 102N 174N 71 factors are indicated. 72 − − − 1 + g2 10404N 4 + 17496N 3 2112N 2 14172N 6299 + ... 10368 − − −

Resurgence at work

ﬂuctuations about (or ¯) saddle determined by those about •the vacuum saddle, toI all ﬂuctuationI orders Resurgence at work

ﬂuctuations about (or ¯) saddle determined by those about •the vacuum saddle, toI all ﬂuctuationI orders

1 1 1 − 8 48 16

a1 b 11 b 21

1 1 − ﬂuctuation about for double-well: 24 12

b12 b 22 • I 1 2-loop (Shuryak/Wöhler, 1994); 3-loop 8

b 23

1 − (Escobar-Ruiz/Shuryak/Turbiner, arXiv:1501.03993) 8

b 24

11 Connecting Perturbative and Non-Perturbative Sector

all orders of multi-instanton trans-series are encoded in perturbation theory of ﬂuctuations about perturbative vacuum

why ? turn to path integrals .... Analytic Continuation of Path Integrals

The shortest path between two truths in the real domain passes through the complex domain

Jacques Hadamard, 1865 - 1963 All-Orders Steepest Descents: Darboux Theorem

all-orders steepest descents for contour integrals: •

hyperasymptotics (Berry/Howls 1991, Howls 1992)

1 1 − f(z) 1 − fn I(n)(g2) = dz e g2 = e g2 T (n)(g2) C 1/g2 Z n p T (n)(g2): beyond the usual Gaussian approximation • asymptotic expansion of ﬂuctuations about the saddle n: • ∞ T (n)(g2) T (n) g2r ∼ r r=0 X All-Orders Steepest Descents: Darboux Theorem

universal resurgent relation between diﬀerent saddles: • ∞ −v (n) 2 1 γnm dv e (m) Fnm T (g ) = ( 1) 2 T 2π i m − 0 v 1 g v/(Fnm) v X Z −

exact resurgent relation between ﬂuctuations about nth saddle • and about neighboring saddles m

γnm 2 (n) (r 1)! ( 1) (m) Fnm (m) (Fnm) (m) Tr = − − r T0 + T1 + T2 + ... 2π i m (Fnm) " (r 1) (r 1)(r 2) # X − − − universal factorial divergence of fluctuations (Darboux) • fluctuations about different saddles explicitly related ! • All-Orders Steepest Descents: Darboux Theorem

d = 0 partition function for periodic potential V (z) = sin2(z)

π − 1 sin2(z) I(g2) = dz e g2 Z0 π two saddle points: z0 = 0 and z1 = 2 .

min. saddle min.

vacuum vacuum II¯ All-Orders Steepest Descents: Darboux Theorem

large order behavior about saddle z : • 0 1 2 (0) Γ r + 2 Tr = √πΓ(r + 1) (r 1)! 1 9 75 − 1 4 + 32 128 + ... ∼ √π − (r 1) (r 1)(r 2) − (r 1)(r 2)(r 3) − − − − − − !

low order coeﬃcients about saddle z : • 1

1 9 75 T (1)(g2) i √π 1 g2 + g4 g6 + ... ∼ − 4 32 − 128 ﬂuctuations about the two saddles are explicitly related • Resurgence in Path Integrals: “Functional Darboux Theorem”

could something like this work for path integrals?

“functional Darboux theorem” ?

multi-dimensional case is already non-trivial and interesting • Pham (1965); Delabaere/Howls (2002) Picard-Lefschetz theory •

do a computation to see what happens ... • double-well potential: V (x) = x2(1 gx)2 • − vacuum saddle point • n 53 1 1 1277 1 1 cn 3 n! 1 ... ∼ − 6 · 3 · n − 72 · 32 · n(n 1) − − instanton/anti-instanton saddle point: • −2 1 53 1277 Im E π e 6g2 1 g2 g4 ... ∼ − 6 · − 72 · −

Resurgence in Path Integrals (GD, Ünsal, 1401.5202)

periodic potential: V (x) = 1 sin2(g x) • g2 vacuum saddle point • 5 1 13 1 cn n! 1 ... ∼ − 2 · n − 8 · n(n 1) − − instanton/anti-instanton saddle point: • −2 1 5 13 Im E π e 2g2 1 g2 g4 ... ∼ − 2 · − 8 · − Resurgence in Path Integrals (GD, Ünsal, 1401.5202)

periodic potential: V (x) = 1 sin2(g x) • g2 vacuum saddle point • 5 1 13 1 cn n! 1 ... ∼ − 2 · n − 8 · n(n 1) − − instanton/anti-instanton saddle point: • −2 1 5 13 Im E π e 2g2 1 g2 g4 ... ∼ − 2 · − 8 · − double-well potential: V (x) = x2(1 gx)2 • − vacuum saddle point • n 53 1 1 1277 1 1 cn 3 n! 1 ... ∼ − 6 · 3 · n − 72 · 32 · n(n 1) − − instanton/anti-instanton saddle point: • −2 1 53 1277 Im E π e 6g2 1 g2 g4 ... ∼ − 6 · − 72 · − resurgence: asymptotic expansions about diﬀerent saddles are closely related requires a deeper understanding of complex conﬁgurations and analytic continuation of path integrals ...

Stokes phenomenon: intersection numbers k can change with phase of parameters N

Analytic Continuation of Path Integrals: Lefschetz Thimbles

1 i 1 − 2 S[A] − 2 Simag[Ak] − 2 Sreal[A] A e g = k e g A e g D N D k Γk Z thimblesX Z Lefschetz thimble = “functional steepest descents contour” remaining path integral has real measure: (i) Monte Carlo (ii) semiclassical expansion (iii) exact resurgent analysis Analytic Continuation of Path Integrals: Lefschetz Thimbles

resurgence: asymptotic expansions about diﬀerent saddles are closely related requires a deeper understanding of complex conﬁgurations and analytic continuation of path integrals ...

Stokes phenomenon: intersection numbers k can change with phase of parameters N Complex Instantons

recall complex instantons in non-perturbative imaginary part of• QED eﬀective action worldline instantons are Lefschetz thimbles • how to compute them eﬃciently ? • Thimbles from Gradient Flow

gradient ﬂow to generate steepest descent thimble:

∂ δS A(x; τ) = ∂τ −δA(x; τ)

keeps Im[S] constant, and Re[S] is monotonic • ∂ S S¯ 1 δS ∂A δS ∂A − = = 0 ∂τ 2i −2i δA ∂τ − δA ∂τ Z ∂ S + S¯ δS 2 = ∂τ 2 − δA Z

Chern-Simons theory (Witten 2001) • comparison with complex Langevin (Aarts 2013, ...) • lattice (Tokyo/RIKEN, Aurora, 2013): Bose-gas • X Thimbles and Gradient Flow: an example Thimbles, Gradient Flow and Resurgence Quartic model ∞ σ x4 Z = dx exp x2 + relation to Lefschetz thimbles GA 13 − 2 4 Z−∞ critical points: (Aarts, 2013; GD, Unsal, ...)

2 4 stable thimble z0 = 0 unstable thimble σ = 1+i, λ = 1 z = i σ/λ 1 2 ± ±

y 0 p y 0 thimbles can be stable thimble unstable thimble computed not contributing analytically -1 -2

-2 -2 -1 0 1 2 -4 Im S(z0) = 0 -4 -2 0 2 4 x x Im S(z ) = AB/2λ ± − contributing thimbles change with phase of σ for A > 0: only 1 thimble contributes• need all three thimbles when Re[σ] < 0 integrating along thimble gives• correct result, with inclusion of complex Jacobianintegrals along thimbles are related (resurgence) • SIGN 2014 – p. 6 resurgence: preferred unique “ﬁeld” choice • Ghost Instantons: Analytic Continuation of Path Integrals

(Başar, GD, Ünsal, arXiv:1308.1108)

− R dτ 1 x˙ 2+ 1 sd2(g x|m) (g2 m) = x e−S[x] = x e 4 g2 Z | D D Z Z doubly periodic potential: real & complex instantons •

instanton actions:

2 arcsin(√m) SI (m) = m(1 m) − 2 arcsin(p √1 m) SG(m) = − − m(1 m) − p Ghost Instantons: Analytic Continuation of Path Integrals

large order growth of perturbation theory: • 16 1 ( 1)n+1 an(m) n! n − n ∼ − π (S ¯(m)) +1 − S ¯(m) +1 II | GG |

naive ratio Hd=1L ratio Hd=1L ò 3 ò ò ìò ò ò ò ìò ìò ìò ìò ìò ìò ìò ìò ìò ìò ìò ìò ìò ò ò 1.0 ìò ìò ìò ìò ìò ò ò ìò ìò à à à à à à à à à à à ì ò ìò à à à à æ æ æ æ æ æ æ ò à à à æ æ æ æ æ 2 ìò à à æ æ æ ò 0.8 æà ìò ìò à æ æ ò ì ì ì ì ì ì ì ìò à æ ì ì ì ì ì à æ æ ì ìò à æ 0.6 à æ à æ 1 à à à à à à à æà æà æà æà æà æà æà æà æà æà æà æ à à æà æà æ æ æ æ æ æ ìò æ à à à à æà æ æ à æ æ æ æ ì ì ì ì 0.4 æ æ æ æà ì ì ì ì ì ì ì ìò ì ì ò ò ò n 0.2 5 ò10ò 15 20 25 ò ò ò ò ò ò n -1 ò ò 0 5 10 15 20 25

without ghost instantons with ghost instantons complex instantons directly aﬀect perturbation theory, even •though they are not in the original path integral measure Non-perturbative Physics Without Instantons

Dabrowski, GD, 1306.0921, Cherman, Dorigoni, GD, Ünsal, 1308.0127, 1403.1277

O(N) & principal chiral model have no instantons ! • Yang-Mills, CPN−1, O(N), principal chiral model, ... all have •non-BPS solutions with ﬁnite action

(Din & Zakrzewski, 1980; Uhlenbeck 1985; Sibner, Sibner, Uhlenbeck, 1989) “unstable”: negative modes of ﬂuctuation operator • what do these mean physically ? • resurgence: ambiguous imaginary non-perturbative terms should cancel ambiguous imaginary terms coming from lateral Borel sums of perturbation theory

− 1 S[A] − 1 S[A ] A e g2 = e g2 saddle (ﬂuctuations) (qzm) D × × Z allX saddles Lecture 4

I connecting weak-coupling to strong-coupling

I resurgence and localization: some examples

I = 2 SUSY gauge theories and all-orders WKB N I quantum geometry Connecting weak and strong coupling

main physics question: does weak coupling analysis contain enough information to extrapolate to strong coupling ?

. . . even if the degrees of freedom re-organize themselves in a very non-trivial way?

classical asymptotics is clearly not enough: is resurgent asymptotics enough? Connecting weak and strong coupling

often, weak coupling expansions are divergent, but strong-coupling• expansions are convergent (generic behavior for special functions) e.g. Euler-Heisenberg • 4 ∞ 2n+4 m n 2eB Γ(B) B2 +4 ∼ −8π2 (2n + 4)(2n + 3)(2n + 2) m2 n=0 X

(eB)2 1 m2 3 m2 2 m2 Γ(B) = + ζ0( 1) + ln(2π) 2π2 −12 − − 4eB 4 2eB − 4eB ( 1 m2 1 m2 2 m2 γ m2 2 + + ln −12 4eB − 2 2eB 2eB − 2 2eB " # ∞ n m2 m2 ( 1)nζ(n) m2 +1 + 1 ln + − 2eB − 2eB n(n + 1) 2eB n=2 ) X Resurgence and Localization

(Drukker et al, 1007.3837; Mariño, 1104.0783; Aniceto, Russo, Schiappa, 1410.5834) certain protected quantities in especially symmetric QFTs can be• reduced to matrix models resurgent asymptotics ⇒ 3d Chern-Simons on S3 matrix model • → 1 1 1 2 ZCS(N, g) = dM exp tr (ln M) vol(U(N)) −g 2 Z

ABJM: = 6 SUSY CS, G = U(N)k U(N)−k • N × (σ) N ( 1) dxi 1 ZABJM (N, k) = − x −x N! 2πk N xi i σ(i) σ∈S i 2ch ch XN Z Y=1 i=1 2 2k Q = 4 SUSY Yang-Mills on S4 •N 2 1 1 2 ZSYM (N, g ) = dM exp trM vol(U(N)) −g2 Z Mathieu Equation Spectrum

~2 d2ψ + cos(x) ψ = u ψ − 2 dx2

non-Borel-summable perturbation theory: •

2 1 ~2 1 1 u(N, ~) 1 + ~ N + N + + ∼ − 2 − 16 2 4 " # 3 ~3 1 3 1 N + + N + ... −162 2 4 2 − " #

energy is really a function of two variables: • u = u(N, ~) Mathieu Equation Spectrum: (~ plays role of g)

~2 d2ψ + cos(x) ψ = u ψ − 2 dx2 u(ℏ)

2.5

2.0

1.5

1.0

0.5

ℏ 0.5 1.0 1.5 -0.5

-1.0 Mathieu Equation Spectrum

~2 d2ψ + cos(x) ψ = u ψ − 2 dx2 small N: divergent, non-Borel-summable: • 2 1 ~2 1 1 u(N, ~) 1 + ~ N + N + + ∼ − 2 − 16 2 4 " # 3 ~3 1 3 1 N + + N + ... −162 2 4 2 − " # large N: convergent expansion: • 2 4 2 8 ~ 2 1 2 5N + 7 2 u(N, ~) N + + ∼ 8 2(N 2 1) ~ 32(N 2 1)3(N 2 4) ~ − − − 9N 4 + 58N 2 + 29 2 12 + + ... 64(N 2 1)5(N 2 4)(N 2 9) ~ − − − ! diﬀerent expansions and diﬀerent degrees of freedom ! • uniform asymptotics: • N exp N 2 + 1 1 1 1 g2 N g ZN = IN N 1 g Ng ∼ hq i4  1  √ 2 1 1 + 1 + N g 2 2π N + g2 ( )   q analogue of Keldysh tunneling/multi-photon transition •

Small g and Large N

often we study theories with both g and N • ’t Hooft limit: λ N g fixed • ≡ planar n planar limit of QCD/YM: Jn n! but Jn c • ∼ ∼ e.g. Bessel functions: • g 1/g 1 1 2π e , g 0,N fixed ZN IN N N → g ≡ Ng ∼ q√ 1 e ,N , g fixed  2πN 2Ng → ∞ uniform asymptotics:  • N exp N 2 + 1 1 1 1 g2 N g ZN = IN N 1 g Ng ∼ hq i4  1  √ 2 1 1 + 1 + N g 2 2π N + g2 ( )   q analogue of Keldysh tunneling/multi-photon transition • Non-perturbative splittings

u(ℏ)

2.5

2.0

1.5

1.0

0.5

ℏ 0.5 1.0 1.5 -0.5

-1.0

N− 1 2 24(N+1) 2 2 8 narrow bands: ∆uband exp N ∼ π N! − r ~ ~ 2 2N gap N ~ e narrow gaps: ∆uN ∼ 2π N ~ band gap equal bands and gaps: ∆uN ∆u O(~) ∼ N ∼ recall Keldysh tunneling/multi-photon transition • Strong/weak coupling

what about a QFT in which the vacuum re-arranges itself in a non-trivial manner? Mathieu equation: • ~2 d2ψ N~ + Λ2 cos(x) ψ = u ψ , a − 2 dx2 ≡ 2

Resurgence of = 2 SUSY SU(2) (Başar, GD, 1501.05671) N moduli parameter: u = tr Φ2 • h i electric: u 1; magnetic: u 1 ; dyonic: u 1 • ∼ ∼ − ∂F a = scalar , aD = dual scalar , aD = • h i h i ∂a Nekrasov prepotential: • class. pert. inst. NS(a, ~) = (a, ~) + (a, ~) + (a, ~) F F F F 2 4 8 4 4 8 inst ~ Λ 21Λ ~ Λ 219Λ + + ... + + + ... + ... F ∼ 2πi 16a4 256a8 2πi 64a6 2048a10 2 2 2 2 ∞ 2n class pert a a ~ a 2 ~ + log log + ~ d2n F F ∼ −2πi Λ2 − 48πi 2Λ2 a n=1 X Resurgence of = 2 SUSY SU(2) (Başar, GD, 1501.05671) N moduli parameter: u = tr Φ2 • h i electric: u 1; magnetic: u 1 ; dyonic: u 1 • ∼ ∼ − ∂F a = scalar , aD = dual scalar , aD = • h i h i ∂a Nekrasov prepotential: • class. pert. inst. NS(a, ~) = (a, ~) + (a, ~) + (a, ~) F F F F 2 4 8 4 4 8 inst ~ Λ 21Λ ~ Λ 219Λ + + ... + + + ... + ... F ∼ 2πi 16a4 256a8 2πi 64a6 2048a10 2 2 2 2 ∞ 2n class pert a a ~ a 2 ~ + log log + ~ d2n F F ∼ −2πi Λ2 − 48πi 2Λ2 a n=1 X Mathieu equation: • ~2 d2ψ N~ + Λ2 cos(x) ψ = u ψ , a − 2 dx2 ≡ 2 All-orders WKB and = 2 SUSY SU(2) Mironov-Morozov N

all-orders WKB action: (Dunham, 1932) • π π 0 √2 ~2 (V )2 a(u) = √u V dx dx ... 6 5/2 2π −π − − 2 −π (u V ) − Z Z − ∞ 2n a(u) = ~ an(u) ⇒ n=0 X Bohr-Sommerfeld in large u (electric) region: • N invert a(u) = ~ = u = u(N, ~) = u(a, ~) 2 ⇒

Matone relation: • 2 iπ ∂ NS(a, ~) ~ u(a, ~) = Λ F 2 ∂Λ − 48 ∗ Resurgence in = 2 and = 2 Theories (Başar, GD, 1501.05671) N N Mathieu & Lamé eqs encode Nekrasov prepotential • u(ℏ) electric sector ←− 2.5 (convergent)

2.0 1.5 magnetic sector ←− 1.0

0.5

ℏ 0.5 1.0 1.5 -0.5 dyonic sector ←− -1.0 (divergent)

resurgent WKB: u = u(N, ~) • ’t Hooft coupling: λ N ~ • ≡ very diﬀerent physics for λ 1, λ 1, λ 1 • ∼ Divergent versus Convergent

dyonic region (divergent, non-Borel-summable): • 2 1 ~2 1 1 u(N, ~) 1 + ~ N + N + + ∼ − 2 − 16 2 4 " # 3 ~3 1 3 1 N + + N + ... −162 2 4 2 − " # electric region (convergent, but coefficients have poles): • 2 4 2 8 ~ 2 1 2 5N + 7 2 u(N, ~) N + + ∼ 8 2(N 2 1) ~ 32(N 2 1)3(N 2 4) ~ − − − 9N 4 + 58N 2 + 29 2 12 + + ... 64(N 2 1)5(N 2 4)(N 2 9) ~ − − − ! different expansions and different degrees of freedom ! • Uniform WKB and = 2 SUSY SU(2) (Başar, GD, 1501.05671) N Bohr-Sommerfeld misses non-perturbative physics • misses band and gap splittings • smooth transition through magnetic region ? • u(ℏ)

2.5

2.0

1.5

1.0

0.5

ℏ 0.5 1.0 1.5 -0.5

-1.0 Non-perturbative splittings u(ℏ) electric sector 2.5 ←− (convergent) 2.0

1.5 magnetic sector 1.0 ←− 0.5

ℏ 0.5 1.0 1.5 dyonic sector -0.5 ←− (divergent) -1.0

N− 1 2 24(N+1) 2 2 8 dyonic: ∆uband exp N ∼ π N! − r ~ ~ 2 2N gap N ~ e electric: ∆uN ∼ 2π N ~ band gap magnetic: ∆uN ∆u O(~) ∼ N ∼ recall Keldysh tunneling/multi-photon transition • Uniform WKB and = 2 SUSY SU(2) (Başar, GD, 1501.05671) N

Bohr-Sommerfeld misses non-perturbative physics • universal band/gap splitting: (Landau, Dykhne, Keller, ...) • 2 ∂u 2π ∆u(N, ~) exp Im aD ∼ π ∂N − ~

~ Cb ~ Cg Cg

-2π 0 -2π 0 Cb

N− 1 dyonic sector: ∆u(N, ~) √64 32 2 exp 8 • ∼ π ~ − ~ N 2 e 2N electric sector: ∆u(N, ~) ~ • ∼ 2π ~N magnetic sector: bands & gaps O(~) (equal !) • ∼ Multi-instantons at strong coupling (!)

in this region instantons are large • bands and gaps are of equal width • u(Q) 1.15

1.10

1.05

1.00

0.95

0.90

4 Q= 0 10 20 30 40 50 60 ℏ2 degrees of freedom re-organize from tight-binding ‘atomic’ states• to ‘nearly-free’ scattering states Uniform WKB provides uniform analysis

all-orders WKB action: (Dunham, 1932) • ∞ 2n a = ~ an(u) n=0 X

1 (N+ 2 )~ dyonic: expand an(u) for u 1; invert with a = • ∼ − 2 N~ electric: expand an(u) for u 1, & invert with a = • 2 1 (N+ 2 )~ magnetic: expand an(u) for u 1, & invert with a = 2 or • N~ ∼ a = 2 Matone relation: • 2 iπ ∂ NS(a, ~) ~ u(a, ~) = Λ F 2 ∂Λ − 48 Perturbative/Non-perturbative connection (Başar, GD, 1501.05671)

Zinn-Justin: B(u, ~), A(u, ~) determine full trans-series • GD, Ünsal: u(B, ~) encodes A(B, ~): • ∂u ~ ∂A = 2B + ~ ∂B −16 ∂ ~ simple proof from Nekrasov and Matone relation • F ∂ ∂u ∂ ∂ ∂aD u Λ F Λ F = Λ ∼ ∂Λ ⇒ ∂a ∼ ∂Λ ∂a ∂Λ identiﬁcations: • ~ ~ 1 a B , aD A + shift , Λ ↔ 2 ↔ 4π ∼ ~

quantum geometry: a(u, ~) and aD(u, ~) related • uniform WKB spans electric/magnetic/dyonic sectors • Conclusions

Resurgence systematically uniﬁes perturbative and non-perturbative• analysis, via trans-series trans-series ‘encode’ analytic continuation information • expansions about diﬀerent saddles are intimately related • there is extra un-tapped ‘magic’ in perturbation theory • matrix models, large N, strings, SUSY QFT • IR renormalon puzzle in asymptotically free QFT • multi-instanton physics from perturbation theory • = 2 and = 2∗ SUSY gauge theory •N N fundamental property of steepest descents • moral: go complex and consider all saddles, not just minima • A Few References: books

I J.C. Le Guillou and J. Zinn-Justin (Eds.), Large-Order Behaviour of Perturbation Theory

I C.M. Bender and S.A. Orszag, Advanced Mathematical Methods for Scientists and Engineers

I R. B. Dingle, Asymptotic expansions: their derivation and interpretation

I O. Costin, Asymptotics and Borel Summability

I R. B. Paris and D. Kaminski, Asymptotics and Mellin-Barnes Integrals

I E. Delabaere, “Introduction to the Ecalle theory”, In Computer Algebra and Diﬀerential Equations 193, 59 (1994), London Math. Soc. Lecture Note Series A Few References: papers

I C.M. Bender and T.T. Wu, “Anharmonic oscillator”, Phys. Rev. 184, 1231 (1969); “Large-order behavior of perturbation theory”, Phys. Rev. Lett. 27, 461 (1971).

I E. B. Bogomolnyi, “Calculation of instanton–anti-instanton contributions in quantum mechanics”, Phys. Lett. B 91, 431 (1980).

I M. V. Berry and C. J. Howls, “Hyperasymptotics for integrals with saddles”, Proc. R. Soc. A 434, 657 (1991)

I J. Zinn-Justin & U. D. Jentschura, “Multi-instantons and exact results I: Conjectures, WKB expansions, and instanton interactions,” Annals Phys. 313, 197 (2004), quant-ph/0501136, “Multi-instantons and exact results II: Speciﬁc cases, higher-order eﬀects, and numerical calculations,” Annals Phys. 313, 269 (2004), quant-ph/0501137

I E. Delabaere and F. Pham, “Resurgent methods in semi-classical asymptotics”, Ann. Inst. H. Poincaré 71, 1 (1999) A Few References: papers

I 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, 349 (2008) arXiv:0711.1954

I M. Mariño, “Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories,” J. Phys. A 44, 463001 (2011), arXiv:1104.0783

I I. Aniceto, R. Schiappa and M. Vonk, “The Resurgence of Instantons in String Theory,” Commun. Num. Theor. Phys. 6, 339 (2012), arXiv:1106.5922

I M. Mariño, “Lectures on non-perturbative eﬀects in large N gauge theories, matrix models and strings,” arXiv:1206.6272

I E. Witten, “Analytic Continuation Of Chern-Simons Theory,” arXiv:1001.2933

I I. Aniceto, J. G. Russo and R. Schiappa, “Resurgent Analysis of Localizable Observables in Supersymmetric Gauge Theories”, arXiv:1410.5834 A Few References: papers

I P. C. Argyres & M. Ünsal, “The semi-classical expansion and resurgence in gauge theories: new perturbative, instanton, bion, renormalon eﬀects,” JHEP 1208, 063 (2012), arXiv:1206.1890

I G. V. Dunne & M. Ünsal, “Resurgence and Trans-series in Quantum Field Theory: The CP(N-1) Model,” JHEP 1211, 170 (2012), and arXiv:1210.2423

I G. V. Dunne & M. Ünsal, “Generating Non-perturbative Physics from Perturbation Theory,” arXiv:1306.4405; “Uniform WKB, Multi-instantons, and Resurgent Trans-Series,” arXiv:1401.5202.

I G. Basar, G. V. Dunne and M. Unsal, “Resurgence theory, ghost-instantons, and analytic continuation of path integrals,” JHEP 1310, 041 (2013), arXiv:1308.1108

I A. Cherman, D. Dorigoni, G. V. Dunne & M. Ünsal, “Resurgence in QFT: Nonperturbative Eﬀects in the Principal Chiral Model”, Phys. Rev. Lett. 112, 021601 (2014), arXiv:1308.0127

I G. Basar & G. V. Dunne, arXiv:1501.05671, “Resurgence and the Nekrasov-Shatashvili Limit: Connecting Weak and Strong Coupling in the Mathieu and Lamé Systems”