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

Resurgence: “Exact” WKB in QM and QFT

Gerald Dunne University of Connecticut

CAQCD 2013: Minneapolis

with Mithat Ünsal: arXiv:1210.2423 (JHEP), 1210.3646 (PRD), 1306.xxxx also: Gökçe Başar, Robert Dabrowski, 1306.0921 related: Argyres & Ünsal: arXiv:1204.1661 (PRL), 1206.1890 (JHEP) Physical Motivation

SU(N) Yang-Mills on ℝ4 and ℂℙN-1 on ℝ2

• asymptotically free • instantons, theta vacua, ... two serious (unsolved) problems:

• perturbative sector: infrared (IR) renormalons ⇒ perturbation theory ill-deﬁned • non-perturbative sector: instanton scale moduli ⇒ instanton gas picture ill-deﬁned

‘t Hooft, 1979; Afﬂeck, 1980; David, 1981 analogous problem in QM: Bogomolny/Zinn-Justin (BZJ) mechanism degenerate classical vacua: double-well or Sine-Gordon

...... S single-instanton sector: (i) level or band splitting e instanton (ii) real and unambiguous ⇠ perturbation theory is non-Borel-summable: (i) ambiguous imaginary contrib. to real energy 2 S (ii) ie instanton ⇠ ± non-perturbative sector: ¯ attractive II BZJ idea: rotate g2 → - g2; interaction repulsive; rotate back again produces ambiguous imaginary non-perturbative contribution which exactly cancels the term from perturbation theory

``resurgence’’ Bogomolny, 1980; Zinn-Justin, 1981; Balitsky/Yung 1986 recap: Borel summation and the BZJ mechanism

t 1 e ( 1)nn!(g2)n = dt (i) divergent, alternating: 1+g2 t n Z0 X t 1 e n!(g2)n = dt (ii) divergent, non-alternating: 1 g2 t n 0 X Z i⇡ 1/g2 ⇒ ambiguous imaginary non-perturbative term: e ±g2 t e C+ “lateral Borel sums”: dt 2 C 1 g t C 1/g2 Z ± often identiﬁed with vacuum instability

non-Borel-summable positive Borel pole ambig. imag. NP term

2 n g c i⇡ c c/g2 n! t = e c 2 2 n , g , ± g X ✓ ◆ BZJ mechanism for SU(N) YM or ℂℙN-1 • for SU(N) YM and ℂℙN-1 QFT there is a new source of divergence, renormalons, producing dominant non-perturbative terms • asymptotically free theories have IR renormalons: positive Borel poles 2 S / ie instanton 0 instanton/anti-instanton poles ± N 0 ⇠

UV renormalon poles IR renormalon poles neutral bion poles • but there are no such non-perturbative objects on ℝ4 or ℝ2, so perturbation theory remains incomplete and inconsistent ‘t Hooft, 1979; David, 1981 • spatially compactiﬁed ℂℙN-1 generates fractionalized instantons and bions, cancelling perturbative IR renormalon ambiguities against non-

perturbative ambiguities in instanton/bion gas picture GD & Ünsal, 2012 the bigger picture: resurgence and non-perturbative QFT

goal: fully consistent non-perturbative deﬁnition of asymptotically free QFT, in the continuum

idea: “resurgence” uniﬁes perturbative and non-perturbative sectors in such a way that the combination is unambiguous and well-deﬁned under analytic continuation of the coupling

question: is this a practically useful new idea in QFT beyond just a formalization of what is already done with condensates and OPE’s, multi-instanton-calculus, etc... ? What is Resurgence? Jean Écalle, 1980; Dingle 1960s; Stokes 1850s; Zinn-Justin 1980s; Pham 1980s; Olver, 1990s; Berry 1990s; ...

• resurgence is very general: special functions, nonlinear DEs, difference eqs, iteration maps, functional eqs, holomorphic vector ﬁelds, Lie groups, steepest descents, ... , presumably also path integrals and QFT

• leads to improved asymptotic expansions, incorporating exponentially small terms: “trans-series”

• both formal and numerical improvement on Poincaré asymptotics, using Borel resummation analysis

• philosophical shift: view semi-classical asymptotic expansions as `exact encoding’ of the function Resurgence and Trans-Series

k 1 k q 1 1 S 1 f(g2)= c g2n exp ln n,k,q g2 g2 n=0 q=0 X kX=0 X  ✓ ◆  ✓ ◆

Écalle: set of functions based on these trans-monomial elements is closed: “any reasonable function’’ has a trans-series expansion

(Borel transform) + (analytic continuation) + (Laplace transform)

must not hit boundaries dramatic consequence: expansion coefﬁcients extremely constrained (cf. BZJ cancellation mechanism) Decoding a Trans-Series k 1 k q 1 1 S 1 f(g2)= c g2n exp ln n,k,q g2 g2 n=0 q=0 X kX=0 X  ✓ ◆  ✓ ◆ 1 2n i ⇡ terms pert. theory: cn,0,0 g divergent, asymptotic ± n=0 X degenerate harmonic vacua: typically non Borel-summable

n! 2S/g2 cn,0,0 ↵ i⇡↵e ⇠ (2S)n ) ± but we ﬁnd that c = ↵ so imaginary parts cancel! 0,2,1 leading large order of pert. theory leading low order of ﬂuctuations

n! (n 1)! (n 2)! beyond leading: cn,0,0 ↵ n + n 1 + n 2 + ... ⇠ (2S) (2S) (2S)

2S/g2 1 2 4 ¯ e ln ↵ + g + g + ... II g2 ) ✓ ◆ Decoding a Trans-Series

k 1 k q 1 1 S 1 f(g2)= c g2n exp ln n,k,q g2 g2 n=0 q=0 X kX=0 X  ✓ ◆  ✓ ◆ ﬂuctuations around single-instanton sector (k=1):

S/g2 2 4 e 1+a1g + a2g + ... n! non Borel-summable: a (a ln n + b) n ⇠ (2S)n

3S/g2 1 i ⇡ e a ln + b ) ± g2 ✓ ◆ leading low orders of ﬂuctuations in k=3 (IIIbar, IIbarIbar) sector:

2 3S/g2 a 1 1 e ln + b ln + c 2 g2 g2 " ✓ ✓ ◆◆ ✓ ◆ #

... ad inﬁnitum II¯ “map” of all saddle points

vacuum

I II¯I,II¯ I¯

network of correspondences within trans-series lead to the “required” cancellations

2 4 S/g2 2 4 (1 + a1g + a2g + + ...) +e (1 + b g + b g + + ...) ··· 1 2 ···

2S/g2 2 4 3S/g2 2 4 +e (1 + c1g + c2g + + ...)+e (1 + d g + d g + + ...) ··· 1 2 ··· + ... +(log terms) graded resurgence triangle GD & Ünsal, 2012

resurgence triangle”:

f(0f(0,0),0) pert. theory around pert. vacuum instanton x (pert. flucs.) S S 2 +Ai⇥ ⇥ A i⇥⇥ e g +fi (1k,1) g2 i k e N f(1,1) e e Nf(1f(1, ,1)1) bion x (pert. flucs.) e e bi-instanton 2S 2S +2i2⇥A ⇥k 2S2A 2A ⇥2ki⇥ g2 +2i g2 g22i x (pert. flucs.) e e f(2,N2)f(2,2) ee f(2f(2,0),0) ee N f(2f(2, ,2) 2) e e

3S 3A ⇥ 3S 3A ⇥ 3S 3A ⇥ 33SA ⇥ +3i⇥ +3i k +i⇥+i k 2 i⇥ i k 33ii⇥k g2 N g2 N g N g2 N e e f(3,3) f(3,3) e e f(3,1)f(3,1) e e f(3, f1)(3, 1) ee f(3f(3, , 3)3) e e e e

4S 4A ⇥k 4S 4A ⇥k 4S4A 44SA ⇥k 4A4S ⇥k 2 +4i⇥+4i +2+2i⇥i 2 22ii⇥ 4i4i⇥ g N g2 N g g2 N g2 N e e f(4,4)f(4,4) ee f(4f,2)(4,2) e e f(4f(4,0),0) ee ff(4(4, , 2)2) ee ff(4(4, , 4)4) e e e e ...... (1.2)

which represents a general expansion of some observable. The rows correspond to a given in- stantonsectors number withn, with different associated ϴ perturbative dependence loop expansions cannot mix times or an cancel instanton prefactor k f () () n 1 c () , and with the topological phases speciﬁed. Only columns (n,k) ⌘ k=0 (n,k) of this triangle with matching ⇥ dependence can possibly mix via resurgence2. For example, P in the Bogomolny-Zinn-Justin (BZJ) approach to the periodic potential problem [24, 35], which has degenerate vacua and a topological theta angle, the ambiguity in the perturbative

contribution f(0,0)() to the ground state energy is cured by an ambiguity in the instanton- 2A anti-instanton amplitude, at order e f(2,0)(). This is in fact a general phenomenon that extends throughout the triangle: the ⇥-sectors are correlated with instanton sectors, which gives another tool for probing the mixing of the di↵erent terms in the trans-series represen- tation. Our main conjecture is that each column is a resurgent function of . This graded resurgence structure provides an interesting new perspective on instanton calculus and is born in a natural implementation of the theory of resurgence in the path integral formalism. Our long-term goal in applying resurgence to QFT is rather ambitious: We aim to give a non-perturbative continuum definition of quantum field theory, and provide a mathematically rigorous foundation. We also would like that such such a definition should be of practical value (not a formal tool) whose results can be compared with the numerical analysis of lattice field theory. In other words, generalizing the title of ’t Hooft’s seminal Erice lectures [38], we want to make sense out of general QFTs in the continuum. We emphasize that our immediate goal is not to provide theorems; rather we would like to reveal structure underlying QFT, a framework in which we can define QFT in a self- consistent manner without running into internal inconsistencies. We hope that whatever framework emerges along these lines may form the foundation of a rigorous definition. This point of view is close in spirit to Refs. [15, 39]. However, we ultimately hope that we will be able to use resurgence theory to provide exact and rigorous results for general QFTs, at least

2For notational simplicity we suppress log terms that generally also appear in the prefactor sums.

–4– What is Resurgence?

resurgent functions display at each of their singular points a behaviour closely related to their behaviour at the “origin”. Loosely speaking, these functions resurrect, or surge up - in a slightly different guise, as it were - at their singularities

Écalle, 1980

question: to what extent can we reconstruct the full function from the perturbative series? What is Resurgence?

resurgence in saddle-point integrals: Berry & Howls (``hyperasymptotics’’)

2 (n) 2 f(z)/g (n) 2 1 f /g2 (n) 2 I (g )= dz e I (g ) e n T (g ) ) ⇠ g Z n (n) 2 1 2r (n) T (g )= g Tr Darboux’s theorem: large orders of expansion r=0 around one critical point governed by nhd. of X nearest singularity = other critical point n Resurgence: large orders of ﬂuctuations around an instanton governed by low m orders about “nearby” instanton(s)

2r 1 g (r 1)! (m) fn fm (m) T (n) ( 1)mn T + | |T + ... r ⇠ 2⇡i ( f f )r 0 g2(r 1) 1 m n m X | | ✓ ◆ zero dimensional QFT “path integrals”

2 (n) 2 f(z)/g (n) 2 1 f /g2 (n) 2 I (g )= dz e I (g ) e n T (g ) ) ⇠ g Z n 2 f(z)=sin (gz) min. saddle min.

1 2 vacuum vacuum 1 p⇡ (n + ) ¯ T (0) = g 2 g2n II (n + 1) n=0 X 1 9 75 cn (n 1)! (n 2)! + (n 3)! (n 4)! + ... ) ⇠ 4 32 128 1 2 2 1 (1) 1/g n p⇡ (n + 2 ) 2n I = ge ( 1) g (n + 1) n=0 X 1 9 75 T (1)(g2)=1 g2 + g4 g6 + ... ) 4 32 128 low orders in ﬂuctuations around “IIbar” saddle determine large-order behavior of ﬂuctuations around the “vacuum” cf. Darboux’s theorem resurgent trans-series structure is a basic property of all-orders saddle-point expansions of ordinary integrals

deeply embedded in perturbation theory and semi-classical analysis in QM and QFT, but its origin is very basic

aim: extend this to path integrals ... “functional Darboux theorem” one dimensional QFT : QM

V (x)=x2(1 + gx)2

2n E = cng n X n 53 1 1 1277 1 1 pert. theory: cn 3 n! 1 ... ⇠ 6 · 3 · n 72 · 32 · n(n 1) ✓ ◆ 1 2 53 2 1277 4 IIbar sector: ImE ⇡ e 6g2 1 g g ... ⇠ 6 72 ✓ ◆ 1 V (x)= sin2(gx) g2

5 1 13 1 c n! 1 ...... n ⇠ 2 · n 8 · n(n 1) ✓ ◆ 1 2 5 2 13 4 ImE ⇡ e 2g2 1 g g ... ⇠ 2 8 ✓ ◆ ﬂucs. around IIbar saddle determine large-order of ﬂucs. around vacuum 1 2 Sine-Gordon (Mathieu equation) spectrum V2(x)= sin (gx) 2g2

2 4 6 2 g g 3g 8 2 weak coupling: E± = 1 ... e g [1 + ...]+... 0 4 16 64 ± p⇡g  divergent 2 1 1 g + 4g2 128g6 + ... strong coupling: E± = 0 1 1 + 7 ... convergent ( 2g2 32g6 215g14

E0 1.4

1.2 energy band

1.0

0.8

0.6

0.4

0.2

g 0.0 0.2 0.4 0.6 0.8 1.0 Resurgence in QM wavefunctions

“Serious” WKB: Stokes, Dingle, Pham, Delabaere, Voros, Berry, Howls, ... 2 00 + Q (x) =0 1 x Q x Q = e (1 + ...)+e (1 + ...) pQ R R h i Stokes: reconstruct single-valued analytic function? Stokes sectors: keep both dominant and sub-dominant exponentials

Dingle: universal divergence ofx prefactor series singulant: F (x)=2 Q Zx0

1 F/2 k! F/2 k! e k + e k ⇠ pQ " F ( F ) # Xk Xk Darboux’s theorem: large orders of expansion around one critical point governed by nhd. of nearest singularity = other critical point resurgence “enforces” compatibility of asymptotic expansion with global analytic continuation properties

1 2 1 sin2(gx) ⇡ 1 1 1 p⇡ (n + ) 2 g2 2g2 2 2n Z+(g )= dx e = e I0 g g 2g2 ⇠ (n + 1) Z ✓ ◆ n=0 X 1 2 1 1 2 2 2 sinh (gx) 1 2 1 1 p⇡ (n + ) Z (g )= dx e g = e 2g K0 ( 1)n 2 g2n g 2g2 ⇠ (n + 1) Z ✓ ◆ n=0 X

2 ??? 2 Z ( g )=Z+(g )

i ⇡ but: K (e± z)=K (z) i ⇡ I (z) 0 0 ⌥ 0

i ⇡ 2 2 1/g2 2 Z (e± g )=Z+(g ) i ⇡ e Z (g ) ) ⌥

‘usual’ asymptotic expansion is incompatible with global analytic continuation properties, but resurgent expansion is compatible Resurgence prototype: Gamma function and Stirling’s Formula 1 1 1 1 174611 (1 + z) ln z + + + + ... ⇠ 2z 12z2 120z4 252z6 ··· 6600z20 leading (Stirling) (divergent!) correction 1 functional relation: (1 + z)= (z)+ z 1 reﬂection formula: (1 + z) (1 z)= ⇡ cot ⇡ z z `perturbative’ asymptotic expansion incompatible with reﬂection formula 1 ⇡ Im (1 + iz) + ⇠2z 2

1 ⇡ 1 ⇡ 1 2⇡kz Im (1 + iz)= + coth ⇡ z = + + ⇡ e 2z 2 2z 2 kX=1 non-perturbative terms generated from a resurgent analysis of the perturbative asymptotic expansion unlike the ``perturbative’’ asymptotic series, a resurgent trans-series expansion is fully compatible with global analyticity properties precisely this gamma function example appears in many QFT computations:

• Euler-Heisenberg effective actions • de Sitter/AdS effective actions • exact S-matrices • Chern-Simons partition functions • matrix models • ... but for non-trivial QFT we need to understand trans-series in more detail constructing the trans-series

• how does the trans-series structure arise? • where do the log terms come from? • what information is needed to construct the full trans-series? exact quantization condition conjecture: multi-instanton series 2 B A = A(E,g ) 1 1 2 A/2 B e = i p2⇡ 2 g2 ± ✓ ◆✓ ◆ Zinn-Justin, Jentschura

perturbation series B = B(E,g2) new result: A is determined by B GD, Ünsal, 2013 3

We are interested in cases where the potential V (x) has degenerate vacua, which are locally harmonic: V (x) x2 +.... The two paradigmatic cases we study in detail are the double-well (DW) and Sine-Gordon (SG) potentials:⇡

2 2 2 3 2 4 VDW(x)=x (1 + gx) = x +2gx + g x (4) 1 1 V (x)= sin2(gx)=x2 g2 x4 + ... (5) SG g2 3 The Sine-Gordon case can be directly related to the Mathieu equation by simple changes of variables, given explicitly 6 in Appendix A. This permits detailed comparison with known results for Mathieu functions [6]. It is convenient to rescale the coordinate variable to y = gx: 1 An important observation is that if we replace ⌫ by an integer quantum number N, so that B = N + 2 ,then 3 the expansions (25, 62, 27) coincide2 precisely with the corresponding expansion obtained from standard Rayleigh- 4 d 2 6 Schrödinger perturbationWe theory areg interested about (y in the)+ casesNV (thy where)harmonic (y)= the potentialg oscillatorE (yV) (x level:) has degenerate vacua, which are locally(6) harmonic: V (x) x2 +.... dy2 The two paradigmatic cases we study in detail are the double-well (DW) and Sine-Gordon (SG) potentials:⇡ An important observation is that if we replace ⌫ by an integer quantum number1N,2 so that(NB) = N +21 ,then where E B = N + ,g = Epert. theory2 (g 2) 2 2 3 2 4 (28) the expansions (25, 62, 27) coincide precisely with the corresponding expansion obtained2 VDW from(x)= standardx (1 + Rayleigh-gx) = x +2gx + g x (4) ✓ ◆ th 2 2 1 1 Schrödinger perturbation theory about the N harmonic oscillator level:VDW(y)=y (1 + y) 2 2 2 2 4 (7) In particular, note that for each B, the perturbative expansionVSG(x)= in g ,2 assin in(gx (25)=, 62x, 27),g isx a divergent+ ... non-alternating (5) 2 g 3 series, which is not1 Borel summable.(N) V This(y)=sin fact will( bey) crucial below when we come to discuss the global(8) boundary E B = N + ,g2 = E SG(g2) (28) conditions that connect2 The one Sine-Gordon perturbativepert. theory case vacuum can be to directly another: related see to Section the Mathieu ... . equation by simple changes of variables, given explicitly It is well known that✓ in both thesein◆ cases Appendix the perturbative A. This permits energy detailed levels comparison are split by with non-perturbative known results for instanton Mathieu functions e↵ects. [6]. The corresponding perturbative2 expansion for u(y) [the function that appears in the argument of the parabolic In particular, note thatThis for level each splittingcylinderB, the perturbative isfunction (at leading in expansion the order) uniformIt is convenienta in single-instanton WKBg , as in ansatz to (25 rescale, 62 (10, e the↵)]27ect,), is coordinate is of and a the divergent is form: textbook variable non-alternating to materialy = gx: [7]. From (6) we see that series, which is not Borel4 summable. This2 fact will be crucial below when we come to discuss the global boundary g plays the role of ~ , and so we expect these non-perturbative e↵ects to be2 characterized by exponential factors of 4 d 2 conditions that connectthe one form perturbative vacuum to another: see Section ... . 2 g1 2k (y)+V (y) (y)=g E (y) (6) The corresponding perturbative expansion for u(y) [the function thatu(y appears)=u(y, in B, the g )= argument dyg 2 u ofk( they, B parabolic) (29) c k=0 cylinder function in the uniform WKB ansatz (10)] is ofwhere the form: exp X (9) With respect to its dependence on B,thecoecientg2 function u (y, B) is a polynomial of degree k in B,withdefinite 1 ✓ ◆ k 2 2 2 k 2k VDW(y)=y (1 + y) (7) parity:uu(yk()=y, uB(y,)=( B, g )=1) uk(y,g B).uk For(y, B the) DW and SG cases: (29) for some constant c>0. 2 k=0 VSG(y)=sin(y) (8) More interestingly, the perturbativeX series for these spectral problems2y isln non-Borel-summable,1+ 2y (1 + y)2 and therefore formally p 2 3 With respect to its dependenceit induces a on non-perturbativeB,thecoecient imaginary functionIt is welluDWu part,k( known(y,y)= B even) is that though a2 polynomial iny both1+ both these potentials of+ cases degreeg B thek are perturbativein completelyB,withdefinite energy stable levels+ and... are the split energy by non-perturbative should (30) instanton e↵ects. k 3 p 2y parity: uk(y, B)=(be1) purelyuk(y, real. B). For The the resolution DW and of SGThis this cases: puzzlelevel splitting is that is the (atr leading non-perturbative order) a single-instanton⇥ imaginary2 2 y 1+ part e3↵ect, is⇤ and in fact is textbook a two-instanton material [7]. From (6) we see that g4 plays the role of 2, and so we expect these non-perturbative e↵ects to be characterized by exponential factors of e↵ect, and is canceled by a corresponding non-perturbative2y ~ y2 imaginaryln cos contributiony q coming from the instanton/anti- 2y the2 formln 1+ 3 p(1 + y) 2 2 instantonuDW interaction(y)=p2 [y8–111+]. We+ referg Bu toSG this(y)=2 leading2 cancelation sin ++g...B as the Bogomolny-Zinn-Justiny + ... (30) (BZJ) mechanism. (31) 2y 2 p2 sin The resurgent trans-seriesr expression3 (2) for⇥2p the2 y energy1+ eigenvalue⇤ encodes2 the fact thatc there is in fact an infinite 3 exp (9) tower of suchHigher cancelations, order terms thereby are straightforward relatingy properties to generate of the but perturbative cumbersome sector to write. andg the2 non-perturbative sector. y ln cos q ✓ ◆ The BZJu cancelation(yWhile)=2p the2 is sin the perturbative first+ g2 ofB this expansion tower.2 + ... A (18 new) of observation the energy we yields, make with here the is that identification we(31) do not⌫ need toN, compute exactly the same SG 2 forp some constanty c>0. ! separately the perturbative and non-perturbative2 sin 2 sectors: the perturbative series contains all information about perturbative series for theMore energy interestingly, eigenvalue the perturbative as Rayleigh-Schrödinger series for these spectral perturbation problems theory is non-Borel-summable, [see (28)], the situation and therefore formally Higher order terms arethe straightforward non-perturbativeis quite to di↵ generate sector,erent for to but the allit cumbersome non-perturbativewave-function induces a non-perturbative to write. expansion orders. imaginary in This (19). provides To part, recover even a simple though the Rayleigh-Schrödinger and both explicit potentials illustration are completely perturbation of the stable and theory the energy should th 2 While the perturbativesurprising expansion powerwave-function ( of18 resurgent) of the for theenergy analysis.Nbe yields, purelylevel, we real.with identify The the resolutionidentification⌫ N,rewrite of this⌫ puzzley N=,gx is exactly, that and the expand the non-perturbative same in g : imaginary part is in fact a two-instanton ! ! perturbative series for the energy eigenvalue as Rayleigh-Schrödinger1 e↵ect, and is2 canceled perturbation by a4 corresponding theory [see non-perturbative (28)], the situation imaginary contribution coming from the instanton/anti- DN u0(gx)+g u1(gx)+g u2(gx)+... is quite di↵erent for the wave-function(N expansion) in (19g instanton). To recover interaction the Rayleigh-Schrödinger [8–11]. We refer to this perturbation leadingDN ( cancelationp2x theory) 2 as(N the) Bogomolny-Zinn-Justin4 (N) (BZJ) mechanism. th B. Strategy(x)= of the Uniform WKB Approach:2 Origin of the Trans-series+ g Expansion 1 (x)+g 2 (x)+... (32) wave-function for the N level, we identify ⌫ N(d/dx,rewrite⇣ )[The⇥ u ( resurgentygx=)+gx,g2 and trans-seriesu (gx expand)+g4 expression inu g(gx: )+ (...2)⇤ for]⌘/g the⌘ energyp2 eigenvalue encodes the fact that there is in fact an infinite ! tower0 of such cancelations,1 thereby2 relating properties of the perturbative sector and the non-perturbative sector. 1 2 4 DN u0(gx)+g u1(gx)+g u2(gx)+The... BZJ cancelation is the first of this tower. A new observationth we make here is that we do not need to compute (N) gBefore gettingThe leading into details, termp we is the first familiar state our harmonic strategy,DN (p2x oscillator and) the2 ( basicN wave) result, function4 ( whichN) for explains the N alreadylevel. why Interestingly, the expression if we truncate (x)= separately the perturbative+ g and 2k non-perturbative(x)+g (x)+ sectors:... the(32) perturbative series contains all information about for⇣ the energythe eigenvalues perturbative2 has4 expansion the trans-series of⌘u(gx) form at some in (2 order). g1 , and use2 this inside the uniform expression (10), we obtain a (d/dx)[⇥ u0(gx)+g u1(gx)+g u2(gx)+the... non-perturbative⇤] /g ⌘ p2 sector, to all non-perturbative orders. This provides a simple and explicit illustration of the Since themuch potentials better we approximation consider have to degenerate the wave-function harmonic than vacua, the in truncation the g2 of0 the limit Rayleigh-Schrödinger each classical vacuum perturbation has the theory surprising power of resurgentth analysis. The leading termp isform the familiar of a harmonicwave-function harmonic oscillator oscillator at the well. same wave Therefore order functiong it2k is for. natural The the uniformN to uselevel. approximation a parabolic Interestingly, uniform e!↵ ifectively we WKB truncate gives ansatz a resummation for the wave-function of many orders of 2k the perturbative expansion[3, 4]: of u(Rayleigh-Schrödingergx) at some order g perturbation, and use this theory. inside the uniform expression (10), we obtain a much better approximation to the wave-function than the truncationB. of the Strategy Rayleigh-Schrödinger of the Uniform perturbation WKB Approach: theory Origin of the Trans-series Expansion wave-function at the same order g2k. The uniform approximation e↵ectively gives1 a resummation of many orders of D⌫ g u(y) GD, Ünsal, 2013 Rayleigh-Schrödinger perturbation theory. UniformIII. (y)= GLOBAL WKB, Resurgence BOUNDARY and CONDITIONS Trans-Series (10) Before getting into details,⇣ we first⌘ state our strategy, and the basic result, which explains already why the expression for the energy eigenvalues hasu0( they) trans-series form in (2). 2 2 e.g. double-well potential:A. RelatingV (x)= onex vacuum(1 + gx) to another 2 Since the potentials wep consider have degenerate harmonic vacua, in the g 0 limit each classical vacuum has the Here D⌫ is aIII. parabolic GLOBAL cylinder BOUNDARY functionform of a [ harmonic6], CONDITIONS and ⌫ oscillatoris an ansatz well. Therefore parameter it is that natural is to to use be a determined. parabolic uniform! Substituting WKB ansatz for the wave-function this uniform WKB ansatz form of the wave-function into4 the Schrödinger2 2 equation2 produces a nonlinear equation for So far the entireSchrödinger discussion[3, 4]: has equation: been local, g in 00 the+ y neighborhood(1 + y) = ofg theE minimum of one of the classical vacua. To the argument functionA. Relatingu(y) that one can vacuum be solved to perturbatively. another Purely local analysis in the immediate vicinity of the4 potential minimum,proceed, we where need the to potential specify how is harmonic, one classical leads vacuum to a perturbative relates to another. expansion Here1 of the the energy details (explained of the double-well in and D⌫ g u(y) detail in SectionSine-Gordon ... below): cases di↵er slightly in details, but in each case we impose (y)= a global boundary condition at the midpoint (10) So far the entire discussionThe has coe beencient local,Ek in(⌫) the is in neighborhood fact a polynomial,uniform of the of minimumWKB degree approximation: (k of+ one 1), in of the the as-yet-undetermined classical vacua.⇣ To⌘ ansatz parameter ⌫.In of the barrier between two neighboring classical vacua. The result illustratesu the0(y) physics of level splitting (DW) and proceed, we need to specifythe howg2 one0 classical limit, the vacuum ansatz parameter relates to⌫ another.tends to an Here integer theN details, labelling of the the double-well unperturbed and harmonic oscillator energy band! spectra (SG), respectively. 1 Sine-Gordon cases di↵er slightlylevel. in Indeed, details, when but in⌫ = eachN, theHere case expansion weD⌫ imposeis a parabolic (11 a) global coincides2 cylinder boundary precisely2 functionk condition with [6], standard and at⌫ theis Rayleigh-Schrödinger an midpointp ansatz parameter perturbation that is to be determined. Substituting Consider first thelocal DW analysis: case. EachE = E level(⌫,g labeled)= byg theEk index(⌫) N splits into two levels due to tunneling(11) between the theory: this uniform WKB ansatz form of the wave-function into the Schrödinger equation produces a nonlinear equation for of the barrier between two neighboringtwo classical classical vacua. vacua. To see The how result this illustrates arises, consider thek physics=0N = of 0 and level note splitting that the (DW) ground and state wave-function is a node-less band spectra (SG), respectively. the argument function2 u(y)X that(N) can be solved2 perturbatively. Purely local analysis in the1 immediate vicinity of the function, which is therefore an evenE functionN,g about= E the midpoint(g ) between(non-Borel-summable) the two wells (ymidpoint =(12) ), while the Consider first the DW case. Each level labeled by thepotential index N minimum,splits into where two levels thepert potential. duetheory to is tunneling harmonic, between leads to the a perturbative expansion 2 of the energy (explained in first excited state wave-functiondetail in Section (which ... also below): has N = 0) has one node and is therefore an odd function about this two classical vacua. To see howThis this perturbative arises, consider seriesN expression= 0 and note is incomplete, that the ground and indeed state ill-defined, wave-function because is a the node-less series is notth Borel summable. midpoint. Thus, the global boundaryRayleigh-Schrödinger condition to be imposed perturbation at ymidpoint1 theoryis: for N level function, which is therefore anThe even fact function that it is about incomplete the midpoint should not between be too surprising the two wells because (ymidpoint so far the= analysis), while has the been purely local, making no reference to the existence of neighboring degenerate classical vacua. To fully determine2 2 1 the2k energy we must impose first excited state wave-function (which also has Nglobal= 0) hasanalysis: one node Neumann and is therefore or Dirichlet an odd b.c.E function= atE1 barrier(⌫,g about)= midpoint thisg Ek(⌫ ) (11) a global boundary condition that relates one classicalground vacuum state : to another. DW0 When=0 we do this we learn that ⌫ is only (33) midpoint. Thus, the global boundary condition to be imposed at ymidpoint is: 2 k=0 exponentially close to the integer N, with a small correction being a function✓ ◆ of both NXand g2: 1 2 2 1 ground state : 0 ⌫ =0first(N,g excited)=N state+ ⌫( :N,g ) DW =0(33) (13) (34) DW 2 global 2 ✓ ◆ 2 ✓ ◆ The explicit form of the correction term ⌫(N,g1 ) is derived and discussed below in Section ... . For now we state that genericallyfirst it excited has a trans-series state : form: =0 1 (34) DW u 2 e.g.2 odd state: D⌫ =0 ⌫ = N ? k 1✓ ◆ g k l 1 1 1 c ! 1 ) ⌫(N,g2)= d exp ln g2p (14) k,l,p g2N+1 g2 g2 p=0 kX=1 Xl=1 X ✓  ◆ ✓  ◆ This form follows directly from properties of the parabolic cylinder functions, and so it is generic to problems having degenerate vacua that are harmonic. Having solved the global boundary condition to determine the parameter ⌫ as a function of N and g2, as in (13) 2 and (14), to obtain the corresponding energy eigenvalue we insert this value ⌫global(N,g ) back into the perturbative expansion (11) for the energy, leading to the final exact expression for the energy eigenvalue:

(N) 2 2 2 1 2k 2 E (g )=E N + ⌫(N,g ),g = g Ek(N + ⌫(N,g )) (15) k=0 X 2 2 Re-expanding the polynomial coecients Ek(N + ⌫(N,g )) in powers of the coupling g , we obtain an expression for the N th energy level, E(N)(g2), that has the trans-series form in (2). We stress that this uniform WKB approach makes it clear why the trans-series form of the energy is generic for problems with degenerate harmonic classical vacua: all properties of the g2 0 limit reduce to properties of the parabolic cylinder functions, which lead directly to the trans-series form for!⌫(N,g2)in(13). In particular, all analytic continuations needed to analyze questions of resurgence and cancellation of ambiguities can be expressed in terms of the analytic continuation properties of the parabolic cylinder functions, and all these analytic continuation properties are rigorously known [6]. [Comment: for example, the square double-well problem has a trans-series form, but with no log terms, because the individual wells are not described by parabolic cylinder functions. Correspondingly, even though this is a double-well problem with degenerate minima, the perturbative expansion in g2 is unambiguously summable.]

C. Perturbative Expansion of the Uniform WKB Ansatz

Recalling that the parabolic cylinder function D⌫ (z) satisﬁes the di↵erential equation d2 1 z2 D (z)+ ⌫ + D (z)=0 (16) dz2 ⌫ 2 4 ⌫ ✓ ◆ we see that the uniform WKB ansatz (10) converts the Schr¨odinger equation (6) to the following non-linear equation for the argument function u(y) appearing in (10):

4 1 2 2 2 2 1 2 g u00 0 V (y) u (u0) g E + g ⌫ + (u0) + pu =0 (17) 4 2 2 0 (u )3/2 ✓ ◆ ✓ 0 ◆ At ﬁrst sight, it looks like (17) is more dicult to solve than (6), but we will see that the perturbative solution of (17) has some advantages over the the perturbative solution of (6). We solve (17) for u(y) and E by making simultaneous perturbative expansions: 2 4 E = E0 + g E1 + g E2 + ... (18) 2 4 u(y)=u0(y)+g u1(y)+g u2(y)+... (19) 3

2 2 VDW(y)=y (1 + y) (7) 2 VSG(y)=sin(y) (8) It is well known that in both these cases the perturbative energy levels are split by non-perturbative instanton e↵ects. This level splitting is (at leading order) a single-instanton e↵ect, and is textbook material [7]. From (6) we see that g4 plays the role of ~2, and so we expect these non-perturbative e↵ects to be characterized by exponential factors of the form c exp (9) g2 ✓ ◆ for some constant c>0. More interestingly, the perturbative series for these spectral problems is non-Borel-summable, and therefore formally 8 it induces a non-perturbative imaginary part, even though both potentials are completely stable and the energy should C. Global Boundary Condition for the Double-Well System be purely real. The resolution of this puzzle is that the non-perturbative imaginary part is in fact a two-instanton e↵ect, and is canceled byTo a derivecorresponding the form of the non-perturbative global boundary condition, imaginary recall contribution that the global coming boundary from conditions the instanton/anti- (33, 34) are imposed 1 2 instanton interaction [8at–11 the]. barrier We refer midpoint to thisymidpoint leading= 2 . cancelation When we analytically as the continue Bogomolny-Zinn-Justing o↵ the positive real (BZJ) axis, this mechanism. renders Borel 1 1 The resurgent trans-seriessummable expression the argument (2) foru the energyof the parabolic eigenvalue cylinder encodes function theD⌫ factappearing that there in the uniform is in fact WKB an ansatz infinite (10). g 2 tower of such cancelations,But now thereby this argument relating1 u properties1 is a complex of the number, perturbative o↵ the real sector positive and axis. the Thus non-perturbative in the limit where the sector. modulus g 2 The BZJ cancelation isof theg2 firstapproaches of this zero, tower. the appropriate A new observation asymptotic behavior we make of the here parabolic is that cylinder we do function not need is not to just compute given by ⌫ z2/4 D (z) z e , (z + ), as used in (24). We now need to use the (resurgent) asymptotic behavior of the separately the perturbative⌫ and⇠ non-perturbative! 1 sectors: the perturbative series contains all information about the non-perturbative sector,parabolic to cylinder all non-perturbative functions throughout orders. the relevant This region provides of the a complex simple plane, and given explicit by [6 illustration]: of the 8 surprising power of resurgent analysis. 8 ⌫ z2/4 1 i⇡⌫ p2⇡ 1 ⌫ z2/4 1 ⇡ D (z) z e F + e± z e F , C.< Globalarg(z) < Boundary⇡ Condition(37)8 for the Double-Well System ⌫ ⇠ 1 z2 ( ⌫) 2 z2 2 ± ✓ ◆ C. Global✓ Boundary◆ Condition for the Double-Well System C. Global Boundary Condition for theTo Double-Well derive the form System of the global boundary condition, recall that the global boundary conditions (33, 34) are imposed B. Strategywhere of the Uniform WKB Approach: Origin of the Trans-series Expansion1 2 To derive the form of the globalat the boundary barrier condition, midpoint y recallmidpoint that= the2 global. When boundary we analytically conditions continue (33, 34g) areo↵ imposedthe positive real axis, this renders Borel ⌫ 1 1 ⌫ k 2 at the barrier midpoint1 ymidpoint = . When we analytically1 continue1 g o↵ the positive real axis, this renders Borel 1 k 2 summablek + 2 2 2 the1 argument2 u of the parabolic cylinder function D⌫ appearing in the uniform WKB ansatz (10). To derive the form of the global boundaryF condition,= recall that1 the1 global boundary conditionsg 2 (33, 34) are imposed(38) summable1 2 the argument ⌫u 1 of⌫ the parabolic2 cylinder1 function1 D appearing in the uniform WKB ansatz (10). Before getting into details, we first state our1 strategy,z and the basic g result,But2 which now2 this explainsk argument! z alreadyu whyis a complexthe⌫ expression number, o↵ the real positive axis. Thus in the limit where the modulus at the barrier midpoint ymidpoint = . When✓ we◆ analyticallyk=0 1 continue2 1 2 g 2 o↵the✓ positive◆ g real axis,2 this renders Borel 2 But now this argumentX u is2 a complex number, o↵ the real positive axis. Thus in the limit where the modulus for the energy eigenvalues has the1 trans-series1 form in (2). g 1 ⌫2of g approaches⌫ zero, thek appropriate asymptotic behavior of the parabolic cylinder function is not just given by summable the argument u of the parabolic2 1 cylinder1 k function+ + D ⌫kappearing+1+ 2 1 in the2 uniform WKB ansatz (10). g 2 ofF g approaches= zero, the2 appropriate2 2 asymptotic⌫ 2 z /4 behavior of the parabolic cylinder(39) function is not just given by Since the potentials we consider have degenerate2 harmonic2 2 vacua, in1 the⌫D⌫ (gz) ⌫z0e limit each, (z2 classical+ ), as vacuum used in ( has24). the We now need to use the (resurgent) asymptotic behavior of the 1 1 z ⌫ z /4 k! z But now this argument u is a complexD (z) number,z e o↵, (thez real2++ positive2), as1+ used!⇠ axis.2 in ( Thus24). We in! the now limit1 need where to use the the modulus (resurgent) asymptotic behavior of the form of a harmonic oscillator well.g Therefore 2 it is natural⌫ ✓ ◆ to usek=0 a parabolicparabolic uniform cylinder WKB functions✓ ansatz◆ throughout for the wave-function the relevant region of the complex plane, given by [6]: 2 parabolic⇠ cylinderX functions! 1 throughout the relevant region of the complex plane, given by [6]: of g approaches zero, the appropriate asymptotic behavior of2 the parabolic cylinder function is not just given by [3, 4]: 2 z /4 ⌫ Noticez /4 that there are two di↵erent exponential terms e± in (37). Normally on the real2 axis one1 or other is dominantp2⇡ 2 1 ⇡ D⌫ (z) z e , (z + ), as used in (24). We2 now need to use the (resurgent)⌫ asymptoticz /4 behaviori⇡⌫ of the 1 ⌫ z /4 ⇠ or sub-dominant,! 1 but for certain rays of z in the complex⌫ z2/4 plane1 they mayDi⌫⇡⌫(z be)p equally2⇡z e1 important.⌫ Fz21/4 2 This1+ e± is the Stokes⇡z e F2 2 , < arg(z) < ⇡ (37) parabolic cylinder functions throughout the relevantD ( regionz) z ofe theF complex+ plane,e± given⇠ z by [6]:e Fz , ( ⌫) < arg(z) < ⇡z (37) 2 ± phenomenon. D ⌫ 1 u⇠(y) 1 z2 ( ⌫) ✓ 2 ◆z2 2 ± ✓ ◆ ⌫ g ✓ ◆ ✓ ◆ Consider the global boundary (y condition)= with Dirichlet boundarywhere condition at the midpoint, as in (34). Using(10) the 2 p 2 full analytic⌫ expressionz /4 (137), thewhere globali⇡⌫ boundary2⇡⇣ 1 condition⌫ ⌘z /4 (34)1 can be written⇡ as D (z) z e F + e± uz0(y) e F , < arg(z) < ⇡ (37) 2 ⌫ 1 2 2 2 ⌫ 1 ⌫ k ⇠ z ( ⌫) z 2 ± 1 k k + ⌫ ⌫ 1 1 ⌫ k 2 2 2 1 2 ✓ ◆ i⇡ ✓ 1◆ 1 k kF+1 =1 2 (38) Here D is a parabolic cylinder function [6], and ⌫ is anp1 ansatze± parameter2 that2 is to be determined.2 2 2 2 Substituting ⌫ 1 ⌫ 2 ⌫ =F1 ⇠ H2 (⌫,g= ) ⌫ 1 ⌫z 2 (40) k! z (38) where presentation is in terms of two important quantum mechanical2 examples,z 0 the double-well and✓ Sine-Gordon◆ k! k=0z potentials,2 2 2 ✓ ◆ ( ⌫) g ✓ ◆ k=0 2 2 2 ✓X ◆ this uniform WKB ansatz form of the wave-function into the ✓ Schrödinger◆ equationX produces a nonlinear equation for 1 ⌫ ⌫ k since these contain already much of the physics relevant for the discussion of non-perturbative1 ⌫ 1 e↵ects⌫ due1 tok degenerate+k + k +1+ 1 2 the argument function u(y) that can be solved perturbatively.N 1 ⌫ Purely1 local⌫ analysis1 1 ink thek + immediate+ F k +1+ vicinity= 1 of2 the 2 2 2 (39) minimawhere in gauge theories and1 1models. k But wek + stress that1 the basic2 idea of2 resurgent2 2 2 trans-series2 analysis1 ⌫ is ⌫ 2 CP 2 2 F22 2 = 1 ⌫ z ⌫ 2 + 1+ k! (39)z potential minimum, where the potentialF1 is harmonic,= leads to⌫ a perturbative1 ⌫ z expansion of the+ energy ✓1+◆ (explainedk=0k! z(38) in 2 2 2 ✓ ◆ much more general, applyingz2 to both linear and nonlinear problems,✓ k◆! andz2k=0 to functional 2 2 problems 2 likeX QFT.✓ ◆ ✓ ◆u2 1k=0 u2 2 1 2 21 ✓ 1X◆ detail in Section ... below): 0 2X 0 2 z2/4 ⇠ exp 1 ⌫ = exp⌫ k z2/4 (41) UniformNotice2 1 that WKB, there are2 Resurgence two di↵erentNotice2 exponential and that thereTrans-Series2 terms are twoe di↵inerent (37). exponential Normally on terms thee real± axisin one (37 or). otherNormally is dominant on the real axis one or other is dominant ⌘ 1s ⇡ g "k+ 22g+ 2# k3+1+⇡g 2 16g 2 ± 2 F = or sub-dominant, 2 but for certain rays of z in the(39) complex plane they may be equally important. This is the Stokes 2 2 or sub-dominant,11 but⌫ for certain⌫ rays of z in the2 complex plane they may be equally important. This is the Stokes B.z What are2 trans-series, +2k and1+u2 where1 dok! theyz come from? k=0 2 21 p phenomenon.( 22 ) local analysis:✓E =◆ phenomenon.E(⌫,g )= g⌫+ 2Ek(⌫) (non-Borel-summable)✓ ◆ (11) X 2 1 F1 2g2 2 1 Consideru the2 global boundaryConsider condition the with1 global Dirichlet2 boundary1 boundary condition2 condition1 with at Dirichlet the midpoint, boundary as in condition (34). Using at the midpoint, as in (34). Using the H0(⌫,g ) k=0 2 ✓ ◆ exp u u (42) z /4 1 2 0 In this paper we concentrate⌘ 2full on trans-series analytic1 2 expressionX expressions (37),u the2full for1 global analytic energy boundary expression eigenvalues2g condition (372 in), the( certain34) global can be2 QM boundary written problems, as condition with (34 a ) can be written as Notice that there are two di↵erent exponentialu terms e± ! in (37u).( Normally 2 ) on the real axis one or other is dominant 2 0 2 F2 22  ✓ 2✓ ◆ ✓ ◆◆ or sub-dominant,coupling constant butglobal forg certain. analysis: Our rays notation of ze.g.2 isin chosenodd the complex state: to matchD plane⌫ the they coupling2g may=0 parameter be equallyg important.in certain This QFTs is such the Stokes as Yang-Mills g ⌫ i⇡ ⌫ N 1 q ✓ !◆ i⇡ 1 e 2 phenomenon.or CP models. One could also discuss trans-series representations1 of thee± wave-function;2 this2 is implicit± in our 2 2 = ⇠ H2 (⌫,g ) = ⇠ H0(⌫,g ) (40) (40) 2 p 2 2 ⇡ 0 2 Consideranalysis, theNote global but that since⇠ boundaryis we the are familiar⌫ condition motivatedz non-perturbative/4 with by attempts Dirichlet instantoni⇡⌫ to boundary2 compute⇡ factor,1 ⌫ condition while QFTz /4H quantities0((⌫,g at⌫)) the is perturbative suchg midpoint, as a mass as in g in, gap, ( and34).( to represents Using⌫ be) veryg the the concrete D⌫ (z) z e [1 + ...]+e± z e [1 + ...]✓ ◆ < arg(z) <⇡ ✓ ◆ we focusfluctuations on energy around eigenvalues. the instanton. For a complementary( ⌫) discussion of trans-series as solutions2 ± to di↵erential equations full analytic expression (37),⇠ the global boundary condition (34) can be written as 2 see [Pham,The ...]. expression (40) is an implicitwhere relation for ⌫ as a functionwhere of the coupling g . We solve by expanding ⌫ = N +⌫, where N is a non-negative integer, noting that ⌫ The general perturbative expansion of ani⇡ energy level has the1 form 2 1 1 1 1 e± 2 u2 2 u u12 1 u2 1 1 ⌫ =N ⇠ H0 0(⌫2,g ) 0 2 0 2 0 (40)2 i⇡ 2 i⇡ ⇠ exp i⇡ ⇠ = expexp = exp (41) (41) 1 e± 2( ⌫) g e± 2 ⌘ s ⇡ g2 e± 22g2 ⌘ 3⇡⇡2gg2 2 6g2 2 g2 2 6g2 = ✓N! (N) ◆ ⌫ 1+ln "(N) #hN s(⌫)+ ...  " (43)# 3⇡g ( ⌫) g2 E g2 (g2)= g2kE g2 (1)  pert. theory k 2 1 2 1 ✓ ◆ ✓ ◆ ⇢  ✓ ◆ 1 u ( ) u where ⌫+ p 2 1 p ( 2 ) k=0 2 F1 2 ⌫+ th 2 u2 1 2g 2 1 2 F1 2g2 where hNdeterminesis the N harmonic as a number function [6]. This of g2 implies: that1X (for the odd2 state) 1 u 1 2 1 2 1 1 1 1 H0(⌫,g ) 2 ✓ ◆ exp 2 u u (42)2 2 H0(⌫,g ) 2 1 2 ✓ 0 ◆ exp u u0 (42) 2 1 2 1 ⌘ 2 1 2 (Nu)( ) 2g 2 2 1 2 2 u u 1 u 1 ! ⌘ 2 2 1 2 u ( 2 ) 2g 2 2 where N is an integer0 labeling2 N the energy02 2 level, and thei⇡ perturbative0 2 coe2NcientsFE2 k u2 can2 2 be computed ✓! by✓ straight-◆ ✓ ◆◆ ⇠ 2 expH (N,g ) = e exp2 2 H (N,g 2)0g 2 (41)F2 2g2  ✓ ✓ ◆ ✓ ◆◆ 2 0 2 ± 2 2 0 2 3 forward iterative⌫⌘= procedures.Ns+ ⇡ g For the cases2 g⇠ we study+ln 3⇡ here,gq thish perturbativeN6g expansion✓ is not⇠◆ + BorelO(⇠ ) summable,(44) which g2 " N! # g2  g2 Nq! ✓ ◆ means that on its own✓ it is incomplete◆ and indeed inconsistent.✓ ◆ ✓ ◆ ✓ ◆ 2 2 Note that ⇠ is the familiar2 1 non-perturbative instanton factor, while H0(⌫,g ) is perturbative in g , and represents2 the 2 u ( ) Note that ⇠ is the familiar non-perturbative instanton factor, while H0(⌫,g ) is perturbative in g , and represents the This situation can be remedied by recognizing⌫+ 1 p that the2 full expansion of the energy at small coupling is in fact of If we repeat the computationfluctuations2 using1 the Neumann around2 F1 the boundary2g instanton.2 condition at the midpoint, then the only change is a feed back1 into perturbativeu expansion ⇒fluctuations trans-series1 around expression the1 instanton. 1 2 the “trans-series”Hchange(⌫,g2) in sign form: on the RHS of (40The),2 which expression leads (✓40 to) a is change an implicit◆ exp of sign relation of the foru odd2⌫ as powers a function ofu⇠2 in of ( the44). coupling Thus, tog leading.(42) We solve by expanding ⌫ = N 2+⌫, 0 1 The expression2 (40) is an implicit0 relation for ⌫ as a function of the coupling g . We solve by expanding ⌫ = N +⌫, ⌘ 2 1 where2 N is a non-negativeu2 integer, noting2g that 2 2 order in ⇠ the splittingu of the levels is! symmetric up( and 2 ) down, but this is not true at higher orders. This expansion 0 2 F2k 1 2 where N is a✓ non-negative✓ ◆k integer,✓ noting◆◆l that is the trans-series form ( ofN) the parameter ⌫1mentioned 12g already in1 (13)-(14) inc Section ... . This1 discussion makes (N) 2 2 ⌫ N 2p E q(g )=E (g )+ ✓ cik,l,p⇡ ◆ exp i⇡ ln g i⇡ (2) pert . theory 1 e± 2 2N+1 e± 2 2 i⇡ ⌫ 2 e± 2i⇡ N 2 i⇡ g 1g e± 2 g e± 2 e± 2 k=1 l=1 p=0 2 = 2N! 2 ⌫ 2 +ln 2 hN (⌫) + ... (43) 2 Note that ⇠ is the familiar non-perturbative instanton factor,( ⌫) whileg H✓ (⌫,g ) is perturbativeg ◆ ✓ ing , and=◆N represents! g the ⌫ +ln hN (⌫) + ... (43) X X X✓ ◆ 0 ✓ ( ⌫◆) ⇢g2  ✓ g◆2 g2 fluctuations around the instanton. ✓ ◆ 2 ✓ ◆ ⇢  ✓ ◆ The second part of the trans-seriesgenericwhere involves: propertyh is a the sum Nofth over parabolicharmonic non-perturbative number cylinder [6]. This factors functions implies exp[ thatk c/g (for], the multiplied odd state) by prefactors The expression (40) is an implicit relation forN⌫ as a function of the coupling g2. Weth solve by expanding ⌫ = N +⌫, that are themselves series in g2 and in ln(1/g2). The basicwhere buildinghN is blocks the N ofharmonic the trans-series, number [6g].2 This,exp[ impliesc/g2 that] and (for the odd state) where Nln(1is/g a2 non-negative), are called integer, “trans-monomials”, noting that and are all familiarN from2 QM and QFT.i⇡ Remarkably, the2 expansionN coe2 2 - 2 H0(N,g ) e± N2 2 2 H0(N,gi⇡ ) 2 3 2N 2 2 ⌫ = N + ⇠ +ln 2 H0(hN,gN ) e± 2 ⇠ + O(⇠ )2 (44)H0(N,g ) cients ck,l,p of the trans-series⌫ are inter-twinedN amongstg2 themselves,N! ⌫ and= N also+ withg2 the coe cients⇠g2 of the+ln perturbativeN! h ⇠2 + O(⇠3) (44) i⇡ i⇡ i⇡ 2 2 N 2 1 e± 2 e± 2 ✓ ◆ e± 2  ✓g 2 ◆ N!✓ ◆ ✓ g ◆ g N! expansion, in such a way that= theN! total trans-series⌫ is real +ln and unambiguous.hN✓ ( For⌫◆) example,+ ... a Borel analysis(43)✓ of◆ the ✓ ◆ ✓ ◆ ( ⌫) g2 If we repeatg2 the computation 2 using theg2 Neumann boundary condition at the midpoint, then the only change is a perturbative series✓ requires◆ an analytic✓ continuation◆ ⇢ in g ,If producing we✓ repeat◆ non-perturbativethe computation using imaginary the Neumann parts, boundary but these condition at the midpoint, then the only change is a are precisely cancelled by the imaginarychange in parts sign on associated the RHS of with (40), the which ln( leads1/g2 to) factors a change in of the sign non-perturbative of the odd powers portion of ⇠ in (44). Thus, to leading th order in ⇠ the splitting of the levelschange is in symmetric sign on the up and RHS down, of (40 but), which this is leads not true to a at change higher of orders. sign of This the expansion odd powers of ⇠ in (44). Thus, to leading where hN is the N harmonic number [6]. This implies that (for the odd state) of the trans-series. Ambiguitiesis only the arise trans-series if you form look of at the justorder parameter one in isolated⇠ the⌫ mentioned splitting portion of already the of the levels in trans-series (is13 symmetric)-(14) in expansion, Section up and ... down, for . This but discussion this is not makes true at higher orders. This expansion example just the perturbative part, or just some particular multi-instanton sector. When viewed as a whole, the N 2 i⇡ is the trans-series2N form2 of the2 parameter ⌫ mentioned already in (13)-(14) in Section ... . This discussion makes analytic continuation2 H of0( theN,g trans-series) expressione± 2 is unique and2 exact.H This0(N,g will) be2 explained3 and demonstrated ⌫ = N + ⇠ +ln hN ⇠ + O(⇠ ) (44) explicitly below.g2 N! g2 g2 N! ✓ ◆  ✓ ◆ ✓ ◆ ✓ ◆ We have three main goals in this paper: If we repeat the computation using the Neumann boundary condition at the midpoint, then the only change is a change in1. signExplain on the in RHS a simple of (40 manner), which how leads such to a a change trans-series of sign expansion of the odd (2) powers arises, and of ⇠ alsoin (44 in). what Thus, sense to leading it is generic. order in ⇠ the splitting of the levels is symmetric up and down, but this is not true at higher orders. This expansion is the trans-series2. Explain form the of origin the parameter of the inter-relations⌫ mentioned within already the trans-series, in (13)-(14) and in Section their physical ... . This consequences. discussion makes 3. In its strongest form, “resurgence” claims that complete knowledge of the perturbative series is sucient to generate the remainder of the trans-series, including all orders of the non-perturbative expansion. We demonstrate this very strong claim in simple and explicit detail for several QM spectral problems, including the double-well and Sine-Gordon models.

II. UNIFORM WKB FOR POTENTIALS WITH DEGENERATE CLASSICAL VACUA

A. The Spectral Problem

Consider the spectral problem

d2 (x)+V (x) (x)=E (x) (3) dx2 13 13 B. Explicit Relation Between Perturbative and Non-Perturbative Sectors

In this SectionB. we show Explicit that Relation in both the Between DW and Perturbative SG cases, all and aspects Non-Perturbative of the trans-series Sectors can in fact be deduced2 from the perturbative expansion. The global boundary conditions (i.e., exact eigenvalue conditions) in (47) and (65) presentationhaveIn this been Section expressed is in we terms show in of the that two same important in both notation the quantum DW used and by mechanical Zinn-Justin. SG cases, examples, all However, aspects the of our double-well the interpretation trans-series and Sine-Gordon can is quite in fact di↵ be potentials,erent, deduced and from the perturbative expansion. The global boundary conditions (i.e., exact eigenvalue conditions) in (47) and (65) sincethis leads these to contain a new already observation. much ofZinn-Justin the physics expressed relevant forthe the global discussion boundary of non-perturbative condition as an implicit e↵ects due expression to degenerate for the 8 have been expressed in the same2 notationN 1 used by Zinn-Justin. However, our2 interpretation is quite di↵erent, and minimaenergy E inas gauge a function theories of andg , by writing models. both B Butand weA stressas functions that the of basicE and ideag . of In resurgent the uniform trans-series approach analysis it is more is CP2 2 2 thismuchnatural leads more to aexpress general, new observation.E applying= E(B,g to Zinn-Justin) both as a linear function expressed and of nonlinearB and theg global problems,, and boundary therefore and to condition also functionalA = A as( problemsB,g an implicit) as like a function expression QFT. of forB and the energyg2. Moreover,E as a function ZJ computed of g2, byA(E,g writingC.2) separately Global both B Boundaryand fromA theas functionsperturbative Condition of forE computation theand Double-Wellg2. In of theB uniform(E,g System2). Theapproach computation it is more of naturalA(E,g to2) express was veryE complicated,= E(B,g2) asinvolving a function multi-instanton of B and g2 calculations,, and therefore both also analyticA = A and(B,g numerical:2) as a function of B and g2. Moreover,To ZJ derive computed the formA(E,g of the2) globalseparately boundary from condition, the perturbative recall that computation the global boundary of B(E,g2 conditions). The computation (33, 34) are of imposed B. What are trans-series,1 and where2 do they come from? 2quote: “The calculation of the ‘instanton function’ ADW(E,g ) is a little more2 dicult (see Appendices F.6 A(E,g ) wasat very the barrier complicated, midpoint involvingymidpoint multi-instanton= 2 . When we calculations, analytically continue both analyticg o↵ the and positive numerical: real axis, this renders Borel 1 1 andsummable F.7): The the calculation argument is basedu onof the the evaluation parabolic cylinder of successively function higherD⌫ appearing orders of in the the contour uniform integral WKB ansatz (10). g 2 2 Inquote: thisof the paper “The WKB we calculation expansion.” concentrate of the on1 ‘instanton trans-series1 function’ expressionsA for(E,g energy) is a eigenvalues little more di in certaincult (see QM Appendices problems, F.6 with a But now this2 argument u is a complex number,DW o↵ the real positive2 axis. Thus in the limit where the modulus couplingand constantF.7): Theg calculation. Our notation isg based is chosen2 on the to evaluation match the coupling of successively parameter higherg in orders certain of the QFTs contour such as integral Yang-Mills N 1 2 2 orWeCP show hereofmodels.g thatapproaches in One all could the zero, cases also the considered discuss appropriate trans-series in [ asymptotic20], the representations non-perturbative behavior of of the thefunction parabolic wave-function;A( cylinderB,g ) can this function be is trivially implicit is not deduced in just our given by of the WKB expansion.”⌫ z2/4 2 analysis,from the but perturbativeD⌫ ( sincez) wez e are function motivated, (z E(B,g+ by attempts).), asThis used remarkable to in compute (24). fact We QFT now is not quantities need at all to obvious use such the as when (resurgent) a mass these gap, functions asymptotic to be very are concretebehavior written of the ⇠ 2 ! 12 Wewein show focusterms here on ofparabolicE energy thatas B in( eigenvalues.cylinderE,g all the) and cases functionsA For( consideredE,g a throughout complementary), but in become [20 the], the relevant obvious discussion non-perturbative region when of written trans-series of the function complex in terms as solutionsA plane, of(B,gB. given2) tocan di by be↵erential [6 trivially]: equations deduced fromsee [Pham, the perturbative ...]. function E(B,g2). This remarkable fact is not at all obvious when these functions are written 2 2 in termsThe general of E as perturbativeB(E,g ) and expansionA⌫(E,gz2/),4 of but an1 energybecome level obviousi⇡⌫ p has2⇡ the when1 form⌫ writtenz2/4 in terms1 of B. ⇡ D (z) z e F + e± z e F , < arg(z) < ⇡ (37) ⌫ ⇠ 1 z2 1. Magic formula( ⌫) for DW 2 z2 2 ± ✓ ◆ 1 ✓ ◆ E(N) (g2)= g2kE(N) (1) Of course,where these are equivalent, becausepert we1.. theory can Magic use formula the Lagrange-Bürmann for DWk theorem to invert these expressions. k=0E 2 Thus, converting the results of [20] to our notation (EZJ X,⌫ and gZJ 1 g ⌫)wehave: k 1 1 !k 2 k + ! 1 2 2 2 2 (N) Of course, these are equivalent, becauseF1 we can= use the Lagrange-Bürmann ⌫ 1 ⌫ theorem to invert these expressions. (38) where N is an integer labelingE the energy3 level,z21 and the perturbative35 25 coecients1155kE! k z2can735 be computed175 by straight- 2 2 2✓ ◆ k4=0 E3 2 2 26 2 ✓ 4 ◆ 2 Thus,forward converting iterativeBDW the procedures.(E,g results)= of For [20+] theg to ourcasesE notation we+ study+ (EgX here,ZJ thisE2 ,+ perturbative and EgZJ+ gg expansion)wehave:E + is notE Borel+ summable, which 2 4 4 !8 1 4 ⌫ ! 32⌫ 16 k 32 1 means that on its own it is incomplete✓ and indeed1 ◆ inconsistent.✓ k + 2 + 2◆ k +1+✓ 2 1 2 ◆ E 450453 F2 1 45045= 35 3117525 1155 735 175 (39) This situationB ( canE,g be2)= remedied++g8g by2 recognizingE2E+5z+2 + thatg4 E the3 +E full3+ expansion1 +EE⌫ ++... of1+g6 the⌫ energyEk4!+ atz small2 E2 coupling+ is in fact(79) of DW 2 1284 ✓ 4 ◆ 64 k=0 8 1282 4 2 2 32 ✓ 16◆ 32 the “trans-series” form: ✓✓ Resurgence◆ X✓ and Trans-Series◆◆ ✓ ◆ z2/4 Notice that there are two8 45045 di↵erent5 exponential45045k 1 3 terms31175e± in (37). Normallyk on the reall axis one or other is dominant +g E +1 1E + 1E + ... c 1 (79) or sub-dominant,(N) 2 but(N) for certain1282 rays of64z2 in the complex128 plane they may be equally important.2p This is the Stokes E 2 (g )=1Epert.2theory17(g2)+19 4 227ck,l,p 3 187 exp 6 47431ln4 34121 g 2 28829 (2) ADW(E,g )= + g ✓ E + + g E +g2N+1◆E + g g2 E +g2 E + phenomenon. 3g2 4 12k=1 l=1 p=0 8 ✓ 8  ◆192✓  96◆ 576 Consider the global boundary✓ conditionX◆X X with✓ Dirichlet boundary◆ B condition✓ at the midpoint, as in (◆34). Using the 317629 264725 1 8429091 2 The secondfull part analytic2 of the expression trans-series1 8 2 (1737 involves),2 the5 19 global a sum boundary over4 3 227 non-perturbative condition3 187 (34) factors can6 A/ be474312 exp[ writtenk4 c/g asZinn-Justin,234121], multiplied2 Jentschura28829 by prefactors ADWexact(E,g quantization)= +g+ g condition:EE ++ + gE + E +BE +E2...+ ge = Ei + E + (80) 2 2 2 p2⇡ 2 g ± 2 2 that are themselves series3g in g 128and4 in ln(112/g48). The8 basic✓384 building◆✓8 ◆ blocks of192 the trans-series,96 g ,exp[576c/g ] and ✓ ✓ ◆ ✓ ◆⌫ ◆ ✓ ◆ ln(1/g2), are called “trans-monomials”, and are all1 familiare i⇡ from2 QM and QFT. Remarkably, the expansion 1 coe- The first expression can8 be317629 inverted to5 give:264725 3 842909± 2 uniform+g WKB condition:E + E + 2 E +=... ⇠ H0(⌫,g ) B = ⌫ + (80) (40) cients ck,l,p of the trans-series128 are inter-twined48 amongst( ⌫) themselves,384g and also with the coecients of the perturbative2 ✓ ✓ ◆◆ expansion, in such a way2 that the total2 trans-series2 1 is4 real and3 unambiguous.19 6 For375 example,4 459 a Borel2 131 analysis of the E (B,g )=2B 2g 3B + 2g 172 B + B 2g B + B + Theperturbative firstperturbative expressionwhere seriesDW can requires be function: inverted an analytic to give: continuation4 in g , producing4 non-perturbative 2 imaginary4 parts,32 but these are precisely cancelled by the imaginary✓ parts associated◆ ✓ with the ln( ◆1/g2) factors✓ in the non-perturbative◆ portion 8 106892 15 23405 2 3 1 22709 of the trans-series. Ambiguities2 2g only2 u arise0 2B2 if+1 you lookuB40 at+ just2 3 one19B isolated1 ... portion6 3751 of4 the459 trans-series2 131 expansion,(81) for EDW(B,g )=2B⇠ 2g 3B+ exp 2g 17B += B exp2g B + B + (41) example just the perturbative part,⌘ or4⇡ justg2 some4 8 particular2 g2 64 multi-instanton4 2 sector.6g22 When4 viewed32 as a whole, the ✓s✓ ◆ " ✓ # ◆3⇡◆g  ✓ ◆ analytic continuation of the trans-series expression is unique and exact. This will be explained and demonstrated When we insert this into the second8 10689 expression5 23405 we obtain3 22709A as a function2 1 of B: 1 pu ( 2 ) explicitlyinstanton below. function:2g B + B +⌫+ 2 F B ... (81) 4 u82 1 64 1 2g2 We have three main goals1 2 in this✓ paper:1 19 2 153◆ 178151 2234051 2227091 2 H0(⌫,g ) 2 2 4 3 ✓ ◆6exp 4 u 2 u0 (42) ADW(B,g )= + g 17B + + g 125B + B2 1+ g B2 + B + 2 ⌘ 2 1 2 u ( 2 ) 2g 2 2 When we insert this into3 theg second expressionu0 12 we obtain! A asF a4 function of B:12 ✓ ✓24 ◆ ✓576 ◆◆ 1. Explain in a simple manner✓ how such 2 a◆ trans-series✓ expansion2 2 (◆g22) arises,✓ and also in what sense it is generic.◆ 8 87549q 5 50715 3 217663 ✓ ◆ 2 1+g 2 B2 + 19 B 4 + 3 B153 ... 6 17815 4 23405 2 22709 (82) 2. ExplainA (B,g the)= origin of+ theg inter-relations17B + within+ g the125B trans-series,+ B and+ g their physical2 B + consequences.B + 2 DWNote that ⇠ is the2 familiar4 non-perturbative2 instanton64 factor, while H0(⌫,g ) is perturbative in g , and represents the 3g ✓ @EDW12 2 ◆24@ADW 12 24 576 fluctuationsamazingly: around the✓ instanton. ◆= 6g✓ 2B + g ◆ ✓ ◆ 3. In its strongest form, “resurgence” claims that complete knowledge2 of the perturbative series is sucient2 to gen- In this secondThe form expression we notice8 (4087549) a is remarkably an5 implicit@B50715 simple relation3 relation217663 for ⌫ as between a function@g the of perturbative the coupling g expression2. We solveE byDW expanding(B,g ) and⌫ the= N +⌫, erate the remainder+g of the trans-series,B +2 includingB + ✓ all ordersB of... the non-perturbative◆ expansion. We demonstrate(82) non-perturbativewhere N expressionis a non-negativeADW4 (B,g integer,): 2 noting that64 this very strong claim✓ in simple and explicit detail for several◆ QM spectral problems, including the double-well and Sine-Gordon models. i⇡ @⌫ EDW i⇡ 2 N 4 @ADW i⇡ GD, Ünsal, 20132 In this secondsimilar form wefor notice Sine-Gordon,1 a remarkablye± 2 0(d) simple =AHO, relatione12± Bg2 Fokker-Planck, between 6g the perturbative Stark,e± 2 expression ... EDW(B,g ) and(83) the 2 @=B N! ⌫ @g2 +ln h (⌫)2 + ... (43) non-perturbative expression( A⌫DW) (B,gg2 ): g2 g2 N ✓ ◆ ✓ ◆ ⇢  ✓ ◆ This meanstheII. thatperturbative UNIFORM the non-perturbative WKB expansion FOR expression POTENTIALS containsA (B,g WITHALL2) is completely information DEGENERATE determined of CLASSICAL the by thetrans-series perturbative VACUA expression th @EDW DW 2 4 @ADW where2 hN is the N harmonic number= [6].12 ThisBg implies6g that (for the odd state) (83) EDW(B,g ). Moreover, recall that the computation@B of ADW (E,g@)in[g2 20] was very complicated, involving multi- instanton computations, with both analyticA. and The numerical Spectral input. Problem The relation (83) implies that none of this is N 2 i⇡ 2N 2 2 2 H0(N,g ) 2 e± 2 2 H0(N,g ) 2 3 This means that the⌫ non-perturbative= N + expression ⇠ADW(B,g+ln) is completelyhN determined by the perturbative⇠ + O expression(⇠ ) (44) 2 g2 N! g2 g2 N! EDWConsider(B,g ). the Moreover, spectral problemrecall✓ that◆ the computation of ADW✓ (E,g◆)in[20✓] was very◆ ✓ complicated,◆ involving multi- instanton computations, with both analytic and numerical input. The relation (83) implies that none of this is If we repeat the computation usingd2 the Neumann boundary condition at the midpoint, then the only change is a change in sign on the RHS of (40), which leads to a change of sign of the odd powers of ⇠ in (44). Thus, to leading 2 (x)+V (x) (x)=E (x) (3) order in ⇠ the splitting of the levels dx is symmetric up and down, but this is not true at higher orders. This expansion is the trans-series form of the parameter ⌫ mentioned already in (13)-(14) in Section ... . This discussion makes physical meaning of non-self-dual classical solutions Dabrowski, GD, 2013 1 1 2 SYM[A] S [n] Ae g ne g2 CPN D D Z Z small g2 : dominated by critical points; i.e., finite action solutions of classical Euclidean equations

1 S • minima: instantons (self-dual) ⇒ e g2 inst ⇠ • saddle points: non-self-dual but ﬁnite action - explicit for ℂℙN-1 (Din & Zakrzewski) - existence for SU(2) YM; ansätze for SU(2), SU(3); simple embedding for N>3 (Sibner, Sibner & Uhlenbeck; Sadun; Vandoren, ...) k S - “unstable”: instantons & anti-instantons ie g2 inst - subdominant and imaginary ⇠ ± - physics: resurgent non-perturbative contributions “instability” of non-self-dual solutions action charge

# zero modes: 2 x 6 = 12 far-separated count: 4 x 6 = 24

“instability”: (some) zero modes become negative modes fractionalized twisted non-self-dual solutions

action

charge Conclusions

moral: perturbative series expansions are typically divergent, and are not usually compatible with global analytic continuation properties

• non-Borel-summable perturbation theory is incomplete and inconsistent • reﬂected in incompleteness of corresponding non-perturbative instanton gas picture • together, a resurgent expansion is complete and consistent

resurgent analysis and trans-series provide a way to “enforce” proper analytic continuation properties by treating both perturbative and non-perturbative sectors on an equal footing, restoring consistency Conclusions

question: is this a practically useful new idea in QFT beyond just a formalization of what is already done with condensates and OPE’s, multi-instanton-calculus, etc... ?

• semi-classical non-perturbative interpretation of infrared renormalons in asymptotically free ℂℙN-1 and Yang-Mills • Chern-Simons partition functions and large N matrix models (Schiappa, Mariño, ...) • general strategy comes from steepest descents expansion of ordinary integrals • physical interpretation of “unstable” non-self-dual saddle point solutions

• O(N), Sp(N) and Grassmannian QFT models (Argyres, GD, Ünsal) • detailed relation to large N limit • full path integral formulation of resurgence