A Beginners' Guide to Resurgence and Trans-Series In

Home , Asymptotic expansion

A Beginners’ Guide to Resurgence and Trans-series in Quantum Theories

Gerald Dunne

University of Connecticut

Recent Developments in Semiclassical Probes of Quantum Field Theories UMass Amherst ACFI, March 17-19, 2016

GD & Mithat Ünsal, reviews: 1511.05977, 1601.03414

GD, lectures at CERN 2014 Winter School

GD, lectures at Schladming 2015 Winter School Lecture 1

I motivation: physical and mathematical

I trans-series and resurgence

I divergence of perturbation theory in QM

I basics of Borel summation

I the Bogomolny/Zinn-Justin cancellation mechanism

I towards resurgence in QFT

I effective field theory: Euler-Heisenberg effective action Physical Motivation

infrared renormalon puzzle in asymptotically free QFT • non-perturbative physics without instantons: physical meaning• of non-BPS saddles "sign problem" in ﬁnite density QFT • exponentially improved asymptotics • Bigger Picture

non-perturbative deﬁnition of non-trivial QFT, in the continuum• analytic continuation of path integrals • dynamical and non-equilibrium physics from path integrals• uncover hidden ‘magic’ in perturbation theory • 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 → −∞   Physical Motivation

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

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− 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): closed set of functions:| {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 Perturbation theory

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 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 the least term (x dependent!) Asymptotic Series: optimal truncation & exponential precision ∞ n n 1 1 1 ( 1) n! x e x E − ∼ x 1 x n=0 X optimal truncation: N 1 exponentially small error opt ≈ x ⇒ −1/x N −N −N e RN (x) N! x N!N √Ne | |N≈1/x ≈ N≈1/x ≈ ≈ ≈ √x

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(x = 0.1) (x = 0.2) 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

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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

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x 0.5 1.0 1.5 2.0 2.5 3.0

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∞ 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+: directional 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 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 " # 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” ⇒

(question: to what extent is this true/useful in physics?) connection formula: K (e±iπ z ) = K ( z ) i π I ( z ) • 0 | | 0 | | ∓ 0 | | ±iπ − 1 Z (e λ) = Z (λ) i e 2λ Z (λ) ⇒ 1 2 ∓ 1

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 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 directional 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 (Im F 1 , 1 , 1; t iε = F 1 , 1 , 1; 1 t ) 2 1 2 2 − 2 1 2 2 − ±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 ﬁxes this

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 resurgence ﬁxes this Transseries Example: Painlevé II (matrix models, ﬂuids ... )

w00 = 2w3(x) + x w(x) , w 0 as x + → → ∞ x + asymptotics: w σ Ai(x) • → ∞ ∼ σ = real transseries parameter (ﬂucs Borel summable) n ∞ − 2 x3/2 2 +1 e 3 w(x) σ w(n)(x) ∼ 2√π x1/4 n=0 ! X x asymptotics: w x • → −∞ ∼ −√2 transseries exponentials: exp 2 2 ( x)3/2 p− 3 − imag. part of transseries parameter ﬁxed by cancellations

Hastings-McLeod: σ = 1 unique real solution on R • 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 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 ⇒ −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

QM: divergence of perturbation theory due to factorial growth of number of Feynman diagrams QFT: new physical eﬀects occur, due to running of couplings with momentum asymptotically free QFT • faster source of divergence: “renormalons” (IR & UV) ⇒ QFT requires a path integral interpretation resurgence: ‘generic’ feature of steepest-descents approx. • saddles, real and complex, BPS and non-BPS • Divergence of perturbation theory in QFT

C. A. Hurst (1952); W. Thirring (1953): • φ4 perturbation theory divergent (i) factorial growth of number of diagrams (ii) explicit lower bounds on diagrams F. J. Dyson (1952): •physical argument for divergence in QED pert. 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 • the birth of effective field theory • ~ 2 ~ 2 2 2 E B α 1 ~ 2 ~ 2 ~ ~ L = − + 2 E B + 7 E B + ... 2 90π Ec − · encodes nonlinear properties of QED/QCD vacuum • 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 QFT Application: Euler-Heisenberg 1935

Borel transform of a (doubly) asymptotic series • resurgent trans-series: analytic continuation B E • ←→ EH effective action Barnes function ln Γ(x) • ∼ ∼ R Euler-Heisenberg Effective Action: 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 characteristic factorial divergence • ∞ ( 1)n+1 Γ(2n + 2) cn = − 8 (k π)2n+4 k X=1 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) − 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 • 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 & beyond complex instantons (saddles) • ⇒ physics: optimization and quantum control • 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” 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 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 ⇒ 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 (saddles): x¨µ = Fµν(x)x ˙ ν • proper-time integral: ∂S(T ) = m2 • ∂T − 2 Im Γ e−Ssaddle(m ) ≈ saddlesX multiple turning point pairs complex saddles • ⇒ 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 • Lecture 2

I uniform WKB and some magic

I resurgence from all-orders steepest descents

I towards a path integral interpretation of resurgence

I large N

I connecting weak and strong coupling

I complex saddles and quantum interference 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 } n

m The Bigger Picture: Decoding the Path Integral

what is the origin of resurgent behavior in QM and QFT ?

to what extent are (all?) multi-instanton eﬀects encoded in perturbation theory? And if so, why?

QM & QFT: basic property of all-orders steepest descents integrals• Lefschetz thimbles: analytic continuation of path integrals• Towards 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 (Inﬁnite Dim.) 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 (Inﬁnite Dim.) 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 and Hydrodynamics (Heller/Spalinski 2015; Başar/GD, 2015)

resurgence: generic feature of diﬀerential equations • boost invariant conformal hydrodynamics • second-order hydrodynamics: T µν = uµ uν + T µν • E ⊥ d 4 τ E = + Φ dτ −3E dΦ 4 η 4 τII 1 λ1 2 τII = Φ Φ Φ dτ 3 τ − − 3 τ − 2 η2

asymptotic hydro expansion: 1 1 2η0 + ... • E ∼ τ 4/3 − τ 2/3 formal series trans-series • → 2/3 2/3 +e−Sτ (ﬂuc) + e−2Sτ (ﬂuc) + ... E ∼ Epert × × non-hydro modes clearly visible in the asymptotic hydro •series Resurgence and Hydrodynamics (Başar, GD, 1509.05046)

study large-order behavior (Aniceto/Schiappa, 2013) • 2 Γ(k + β) S c1,1 S c1,2 c ,k S c , + + + ... 0 ∼ 1 2πi Sk+β 1 0 k + β 1 (k + β 1)(k + β 2) − − − c (large order) 1.0 0,n ◆ 1-term ● 3-terms Hydro. expansion Borel plane c0,n (exact) ● 0.5 1.01 ●

● ● ● ●●●● ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●◆◆◆◆◆◆◆◆●●●●●●●●●●●●●●●●◆◆◆◆◆◆◆◆●●●●●●●●●●●●●●●●◆◆◆◆◆◆◆◆●●●●●●●●●●◆◆◆◆◆ 1.00 ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●●● ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆◆◆◆◆◆ 20 10 10 20 ◆◆◆◆◆◆◆◆◆◆◆◆ - - ◆◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆◆ ◆◆◆◆◆◆◆ 3 ◆◆◆◆◆◆ ◆◆◆◆◆ ◆◆◆◆◆ 0.99 ◆◆◆◆ ◆◆◆◆ ◆◆◆◆ ◆◆◆ 2 Cτ ◆◆◆ ◆◆◆ ◆◆◆ -0.5 ◆◆◆ ◆◆ ◆◆ ◆◆ ◆◆ 0.98 ◆◆ ◆◆ ◆◆ ◆◆ ◆◆ ◆◆ n 100 200 300 400 500 -1.0

resurgent large-order behavior and Borel structure veriﬁed to 4-instanton• level trans-series for metric coeﬃcients in AdS • ⇒ some magic: there is even more resurgent structure ... Uniform WKB & Resurgent Trans-series (GD/MÜ:1306.4405, 1401.5202)

~2 d2 ψ + V (x)ψ = E ψ − 2 dx2

weak coupling: degenerate harmonic classical vacua • 1 Dν √ ϕ(x) uniform WKB: ψ(x) = ~ ⇒ √ϕ0(x) S non-perturbative eﬀects: g2 ~ exp • ↔ ⇒ − ~ trans-series structure follows from exact quantization • condition E(N, ~) = trans-series → Zinn-Justin, Voros, Pham, Delabaere, Aoki, Takei, Kawai, Koike, ... • in fact ... (GD, Ünsal, 1306.4405, 1401.5202) ﬂuctuation factor:

~ 1 2 ∂Epert d~ ∂Epert(~,N) N + 2 ~ inst(~,N) = exp S ~ + P ∂N 3 ∂N − S " Z0 ~ !#

perturbation theory Epert(~,N) encodes everything ! ⇒

Connecting Perturbative and Non-Perturbative Sector

Zinn-Justin/Jentschura conjecture: generate entire trans-series from just two functions:

(i) perturbative expansion E = Epert(~,N) (ii) single-instanton ﬂuctuation function inst(~,N) (iii) rule connecting neighbouring vacua (parity,P Bloch, ...)

1 N+ 2 ~ 1 32 −S/~ E(~,N) = Epert(~,N) e inst(~,N) + ... ± √2π N! P ~ Connecting Perturbative and Non-Perturbative Sector

Zinn-Justin/Jentschura conjecture: generate entire trans-series from just two functions:

(i) perturbative expansion E = Epert(~,N) (ii) single-instanton ﬂuctuation function inst(~,N) (iii) rule connecting neighbouring vacua (parity,P Bloch, ...)

1 N+ 2 ~ 1 32 −S/~ E(~,N) = Epert(~,N) e inst(~,N) + ... ± √2π N! P ~

in fact ... (GD, Ünsal, 1306.4405, 1401.5202) ﬂuctuation factor:

~ 1 2 ∂Epert d~ ∂Epert(~,N) N + 2 ~ inst(~,N) = exp S ~ + P ∂N 3 ∂N − S " Z0 ~ !#

perturbation theory Epert(~,N) encodes everything ! ⇒ S − 0 1 2 resurgence : e ~ 1 + ~ 102N 174N 71 72 − − − 1 2 4 3 2 + ~ 10404N + 17496N 2112N 14172N 6299 + ... 10368 − − − known for all N and to essentially any loop order, directly from• perturbation theory ! diagramatically mysterious ... •

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

1 1 1 − 8 48 16

a1 b 11 b 21

1 1 − "QFT computation": 3-loop ﬂuctuation about 24 12

b12 b 22 • 1 for double-well and Sine-Gordon: 8

b I 23 1 − Escobar-Ruiz/Shuryak/Turbiner 1501.03993, 1505.05115 8

b 24

1 1 1 − S 16 16 8 − 0 71 2 DW : e ~ 1 ~ 0.607535 ~ ... 72 d e f − − − 1 1 − 16 48 g h

Figure 2: Diagrams contributing to the coeﬃcient B2. The signs of contributions and symmetry factors are indicated.

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

1 1 1 − 8 48 16

a1 b 11 b 21

1 1 − "QFT computation": 3-loop ﬂuctuation about 24 12

b12 b 22 • 1 for double-well and Sine-Gordon: 8

b I 23 1 − Escobar-Ruiz/Shuryak/Turbiner 1501.03993, 1505.05115 8

b 24

1 1 1 − S 16 16 8 − 0 71 2 DW : e ~ 1 ~ 0.607535 ~ ... 72 d e f − − − 1 1 − 16 48 g h S0 1 − 2 Figure 2: Diagrams contributing to the coeﬃcient B2. The signs of contributions and symmetry resurgence ~ : e 1 + ~ 102N 174N 71factors are indicated. 72 − − − 1 2 4 3 2 + ~ 10404N + 17496N 2112N 14172N 6299 + ... 10368 − − − known for all N and to essentially any loop order, directly11 from• perturbation theory ! diagramatically mysterious ... • Connecting Perturbative and Non-Perturbative Sector

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

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

phase transitions? Resurgence and Matrix Models, Topological Strings

Mariño, Schiappa, Weiss: Nonperturbative Eﬀects and the Large-Order Behavior of Matrix

Models and Topological Strings 0711.1954; Mariño, Nonperturbative eﬀects and

nonperturbative deﬁnitions in matrix models and topological strings 0805.3033 resurgent Borel-Écalle analysis of partition functions etc in •matrix models 1 Z(gs,N) = dU exp tr V (U) g Z s

two variables: gs and N (’t Hooft coupling: λ = gsN) • e.g. Gross-Witten-Wadia: V = U + U −1 • double-scaling limit: Painlevé II • 3rd order phase transition at λ = 2: condensation of instantons• similar in 2d Yang-Mills on Riemann surface • Resurgence in the Gross-Witten-Wadia Model

Buividovich, GD, Valgushev 1512.09021 → PRL

unitary matrix model 2d U(N) lattice gauge theory • ≡ third order phase transition at λ = 2 • N Z = U exp Tr(U + U †) D λ Z in terms of eigenvalues eizi of U • N π −S(zi) Z = dzi e i Y=1−Zπ 2N 2 zi zj S(zi) cos(zi) ln sin − ≡ − λ − 2 i i

phase transition driven by complex saddles • (a) Im(z) (b) Im(z) (c) Im(z) (d) Im(z) λ = 1.5, m = 0 λ = 1.5, m = 1 λ = 1.5, m = 2 λ = 1.5, m = 17

π π π π

Re(z) Re(z) Re(z) Re(z) 0 0 0 0

(e) Im(z) (f) Im(z) (g) Im(z) (i) Im(z) λ = 4.0, m = 0 λ = 4.0, m = 1 λ = 4.0, m = 2 λ = 4.0, m = 7

π π π π

Re(z) Re(z) Re(z) Re(z) 0 0 0 0

eigenvalue tunneling into the complex plane • weak-coupling: “instanton” is m = 1 conﬁguration • has negative mode resurgent trans-series • ⇒ strong-coupling: dominant saddle is m = 2, complex ! • Resurgence in the Gross-Witten-Wadia Model 1512.09021

weak-coupling “instanton” action from string eqn • weak S( ) = 4/λ 1 λ/2 arccosh ((4 λ)/λ) , λ < 2 I − − − strong-couplingp “instanton” action from string eqn • strong S( ) = 2arccosh (λ/2) 2 1 4/λ2, λ 2 I − − ≥ p 2.0

(1/N) |Re(S1 - S0)|, N = 400 (1/N) |Re(S2 - S0)|, N = 400 (w) λ 1.5 SI ( ) / N (s) λ SI ( ) / N

S| / N 1.0 ∆ |Re

0.5

0.0 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 λ

interpolated by Painlevé II (double-scaling limit) • 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 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 in = 2 and = 2 Theories (Başar, GD, 1501.05671) N N ~2 d2ψ + cos(x) ψ = u ψ − 2 dx2 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)

energy: u = u(N, ~); ’t Hooft coupling: λ N ~ • ≡ very diﬀerent physics for λ 1, λ 1, λ 1 • ∼ Mathieu & Lamé encode Nekrasov twisted superpotential • Resurgence of = 2 SUSY SU(2) N

moduli parameter: u = tr Φ2 • h i electric: u 1; magnetic: u 1 ; dyonic: u 1 • ∼ ∼ − ∂W a = scalar , aD = dual scalar , aD = • h i h i ∂a Nekrasov twisted superpotential (a, ~, Λ): • W Mathieu equation: (Mironov/Morozov) • ~2 d2ψ N~ + Λ2 cos(x) ψ = u ψ , a − 2 dx2 ≡ 2

Matone relation: • iπ ∂ (a, ~, Λ) ~2 u(a, ~) = Λ W 2 ∂Λ − 48 Mathieu Equation Spectrum: (~ plays role of g2)

~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 trans-series • → 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: ?? trans-series ?? • −→ 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) ~ − − − ! Resurgence of = 2 SUSY SU(2) (Başar, GD, 1501.05671) N N~ 1, deep inside wells: resurgent trans-series • ∞ N−1/2 ∞ (±) n 32 32 − 8 n u (N, ~) cn(N)~ e ~ dn(N)~ + ... ∼ ± √π N! n=0 ~ n=0 X X Borel poles at two-instanton location • N~ 1, far above barrier: convergent series • 2 2 N−1 2 2 2N N−1 (±) ~ N αn(N) ~ βn(N) u (N, ~) = ~ + ... 8 4n ± 8 (2N−1(N 1)!)2 4n n=0 ~ n=0 ~ X − X (Basar, GD, Ünsal, 2015) coeﬃcients have poles at O(two-(complex)-instanton) • N~ 8 , near barrier top: “instanton condensation” • ∼ π (±) π 2 u (N, ~) 1 ~ + O(~ ) ∼ ± 16 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 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 E. Delabaere and F. Pham, “Resurgent methods in semi-classical asymptotics”, Ann. Inst. H. Poincaré 71, 1 (1999)

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 non-perturbative eﬀects in large N gauge theories, matrix models and strings,” arXiv:1206.6272

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 A Few References: papers

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

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 I. Aniceto and R. Schiappa, “Nonperturbative Ambiguities and the Reality of Resurgent Transseries,” Commun. Math. Phys. 335, no. 1, 183 (2015) arXiv:1308.1115. A Few References: papers

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

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 and G. V. Dunne, “Hydrodynamics, resurgence, and transasymptotics,” arXiv:1509.05046.

I A. Behtash, G. V. Dunne, T. Schaefer, T. Sulejmanpasic and M. Ünsal, “Toward Picard-Lefschetz Theory of Path Integrals, Complex Saddles and Resurgence,” arXiv:1510.03435.

I G. V. Dunne and M. Ünsal, “What is QFT? Resurgent trans-series, Lefschetz thimbles, and new exact saddles", arXiv:1511.05977.

I P. V. Buividovich, G. V. Dunne and S. N. Valgushev, “Complex Saddles in Two-dimensional Gauge Theory,” arXiv:1512.09021.