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?
Z
Z
i
1
DA exp
S[A]
versus
DA exp − S[A]
~
~
1
2π
Z
∞
−∞
e i(
1 3
t +x t
3
) dt ∼
 2 3/2
−3 x

e√


 2 π x1/4

2
3/2 π


 sin(√3 (−x) 1/4+ 4 )
π (−x)
,
x → +∞
,
x → −∞
Physical Motivation
• what does a Minkowski path integral mean?
Z
Z
i
1
DA exp
S[A]
versus
DA exp − S[A]
~
~
1.0
0.5
-10
5
-5
10
-0.5
-1.0
1
2π
Z
∞
−∞
e
i( 13 t3 +x t)
dt ∼
 2 3/2
−3 x

e√


 2 π x1/4

2
3/2 π


 sin(√3 (−x) + 4 )
π (−x)1/4
,
x → +∞
,
x → −∞
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 l
1
c
f (g ) =
ck,l,p g
ln ± 2
exp − 2
g
g
| {z }
p=0 k=0 l=1
{z
}|
{z
}
perturbative fluctuations |
2
∞ X
∞ X
k−1
X
2p
k−instantons
quasi-zero-modes
• J. Écalle (1980): closed set of functions:
(Borel transform) + (analytic continuation) + (Laplace transform)
• trans-monomial elements: g 2 , e
−
1
g2
, ln(g 2 ), 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
n
m
Perturbation theory
• perturbation theory generally → divergent series
e.g. QM ground state energy: E =
I
Zeeman: cn ∼ (−1)n (2n)!
I
Stark: cn ∼ (2n)!
I
P∞
n
n=0 cn (coupling)
cubic oscillator: cn ∼ Γ(n + 12 )
I
quartic oscillator: cn ∼ (−1)n Γ(n + 12 )
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
f (x) =
N
−1
X
n=0
cn (x − x0 )n + RN (x)
convergent series:
|RN (x)| → 0
,
N →∞ ,
x fixed
asymptotic series:
|RN (x)| |x − x0 |N
−→
,
x → x0
,
N
fixed
“optimal truncation”:
truncate just before the least term (x dependent!)
Asymptotic Series: optimal truncation & exponential
precision
∞
X
1 1
(−1) n! x ∼ e x E1
x
n
n
n=0
optimal truncation: Nopt ≈
1
x
1
x
⇒ exponentially small error
√
e−1/x
|RN (x)|N ≈1/x ≈ N ! xN N ≈1/x ≈ N !N −N ≈ N e−N ≈ √
x
æ
0.920
æ
æ
0.90
æ
æ
æ
0.918
æ
0.85
æ
æ
0.916
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
æ
0.914
æ
0.80
æ
æ
æ
0.912
0.75
0
æ
5
10
15
(x = 0.1)
ææ
20
N
2
4
(x = 0.2)
6
8
æ
N
Borel summation: basic idea
write n! =
R∞
0
dt e−t tn
alternating factorially divergent series:
∞
X
(−1)n n! g n =
n=0
Z
0
∞
dt e−t
1
1+gt
(?)
integral convergent for all g > 0: “Borel sum” of the series
Borel Summation: basic idea
∞
X
(−1)n n! xn =
0
n=0
1.2
Z
∞
dt e−t
1
1 + xt
1.1
1.0
0.9
0.8
0.7
0.0
0.1
0.2
0.3
0.4
x
Borel summation: basic idea
write n! =
R∞
0
dt e−t tn
non-alternating factorially divergent series:
∞
X
n=0
n
n! g =
Z
∞
dt e−t
0
pole on the Borel axis!
1
1−gt
(??)
Borel summation: basic idea
write n! =
R∞
0
dt e−t tn
non-alternating factorially divergent series:
∞
X
n=0
n
n! g =
Z
∞
dt e−t
0
1
1−gt
(??)
pole on the Borel axis!
⇒ non-perturbative imaginary part
±
i π − g1
e
g
but every term in the series is real !?!
Borel Summation: basic idea
⇒
Borel
Re
"
∞
X
n! x
n
n=0
#
=P
Z
∞
−t
dt e
0
1
1
1
= e− x Ei
1 − xt
x
2.0
1.5
1.0
0.5
0.5
-0.5
1.0
1.5
2.0
2.5
3.0
x
1
x
Borel summation
Borel transform of series f (g) ∼
B[f ](t) =
P∞
n=0 cn g
∞
X
cn
n=0
n!
n:
tn
new series typically has finite radius of convergence.
Borel resummation of original asymptotic series:
Z
1 ∞
Sf (g) =
B[f ](t)e−t/g dt
g 0
warning: B[f ](t) may have singularities in (Borel) t plane
Borel singularities
avoid singularities on R+ : directional Borel sums:
1
Sθ f (g) =
g
Z
0
eiθ ∞
B[f ](t)e−t/g dt
C+
C-
go above/below the singularity: θ = 0±
−→
non-perturbative ambiguity: ±Im[S0 f (g)]
challenge: use physical input to resolve ambiguity
Borel summation: existence theorem (Nevanlinna & Sokal)
f (z) analytic in circle CR = {z : z −
f (z) =
N
−1
X
an z n + RN (z)
n=0
,
R
2
<
R
2}
|RN (z)| ≤ A σ N N ! |z|N
Borel transform
B(t) =
∞
X
an
n=0
n!
R/2
tn
analytic continuation to
Sσ = {t : |t − R+ | < 1/σ}
Z
1 ∞ −t/z
f (z) =
e
B(t) dt
z 0
Im(t)
1/σ
Re(t)
Borel summation in practice
f (g) ∼
∞
X
cn g n
cn ∼ β n Γ(γ n + δ)
,
n=0
• alternating series: real Borel sum
1
f (g) ∼
γ
Z
0
∞
dt
t
1
1+t
t
βg
δ/γ
" #
t 1/γ
exp −
βg
• nonalternating series: ambiguous imaginary part
" #
δ/γ
Z ∞ dt
1
t
t 1/γ
1
Re f (−g) ∼
P
exp −
γ
t 1−t
βg
βg
0
" #
δ/γ
π 1
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?)
Resurgence: Preserving Analytic Continuation
• zero-dimensional partition functions
Z ∞
√
1
1
1
1
− 2λ
sinh2 ( λ x)
4λ
dx e
Z1 (λ) =
= √ e K0
4λ
λ
−∞
r ∞
1 2
X
Γ(n + 2 )
π
∼
(−1)n (2λ)n
Borel-summable
2
2
n! Γ 1
n=0
Z2 (λ) =
Z
√
π/ λ
2
√
1
− 2λ
sin2 ( λ x)
dx e
0
∼
r
∞
Γ(n + 21 )2
πX
(2λ)n
2
2
n! Γ 12
n=0
• naively: Z1 (−λ) = Z2 (λ)
1
π
= √ e− 4λ I0
λ
1
4λ
non-Borel-summable
Resurgence: Preserving Analytic Continuation
• zero-dimensional partition functions
Z ∞
√
1
1
1
1
− 2λ
sinh2 ( λ x)
4λ
dx e
Z1 (λ) =
= √ e K0
4λ
λ
−∞
r ∞
1 2
X
Γ(n + 2 )
π
∼
(−1)n (2λ)n
Borel-summable
2
2
n! Γ 1
n=0
Z2 (λ) =
Z
√
π/ λ
2
√
1
− 2λ
sin2 ( λ x)
dx e
0
∼
r
∞
Γ(n + 21 )2
πX
(2λ)n
2
2
n! Γ 12
n=0
1
π
= √ e− 4λ I0
λ
1
4λ
non-Borel-summable
• naively: Z1 (−λ) = Z2 (λ)
• connection formula: K0 (e±iπ |z|) = K0 (|z|) ∓ i π I0 (|z|)
⇒
1
Z1 (e±iπ λ) = Z2 (λ) ∓ i e− 2λ Z1 (λ)
Resurgence: Preserving Analytic Continuation
• Borel summation
r
Z ∞
t
1 1
π 1
− 2λ
Z1 (λ) =
dt e
, , 1; −t
2 F1
2 2λ 0
2 2
• directional Borel summation:
Z1 (eiπ λ) − Z1 (e−iπ λ)
r
Z ∞
t
π 1
1 1
1 1
=
dt e− 2λ 2 F1
, , 1; t − iε − 2 F1
, , 1; t + iε
2 2λ 1
2 2
2 2
r
Z ∞
t
1 1
π 1 −1
−
e 2λ
, , 1; −t
= −(2i)
dt e 2λ 2 F1
2 2λ
2 2
0
1
= −2 i e− 2λ Z1 (λ)
(Im 2 F1 21 , 21 , 1; t − iε =
2 F1
1 1
2 , 2 , 1; 1
−t )
1
• connection formula: Z1 (e±iπ λ) = Z2 (λ) ∓ i e− 2λ Z1 (λ)
Resurgence: Preserving Analytic Continuation
Stirling expansion for ψ(x) =
ψ(1 + z) ∼ ln z +
d
dx
ln Γ(x) is divergent
1
1
1
174611
1
−
+
−
+ ··· +
− ...
2
4
6
2z 12z
120z
252z
6600z 20
• functional relation: ψ(1 + z) = ψ(z) +
formal series
⇒
1
z
1
Im ψ(1 + iy) ∼ − 2y
+
π
2
Resurgence: Preserving Analytic Continuation
Stirling expansion for ψ(x) =
ψ(1 + z) ∼ ln z +
d
dx
ln Γ(x) is divergent
1
1
1
174611
1
−
+
−
+ ··· +
− ...
2
4
6
2z 12z
120z
252z
6600z 20
• functional relation: ψ(1 + z) = ψ(z) +
formal series
⇒
1
z
1
Im ψ(1 + iy) ∼ − 2y
+
• reflection formula: ψ(1 + z) − ψ(1 − z) =
π
2
1
z
− π cot(π z)
∞
⇒
X
π
1
e−2π k y
Im ψ(1 + iy) ∼ − + + π
2y
2
k=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)
!2n+1
2 3/2
∞
X
e− 3 x
w(x) ∼
σ √ 1/4
w(n) (x)
2 πx
n=0
p
• x → −∞ asymptotics: w ∼ − x2
√
transseries exponentials: exp − 2 3 2 (−x)3/2
imag. part of transseries parameter fixed by cancellations
• Hastings-McLeod: σ = 1 unique real solution on R
Borel Summation and Dispersion Relations
cubic oscillator: V = x2 + λ x3
z=
2
I
1
E(z)
dz
2πi C
z − z0
Z
Im E(z)
1 R
=
dz
π 0
z − z0
Z
∞
X
1 R
Im E(z)
n
=
z0
dz
π 0
z n+1
E(z0 ) =
.zo
R
C
A. Vainshtein, 1964
n=0
Borel Summation and Dispersion Relations
cubic oscillator: V = x2 + λ x3
z=
A. Vainshtein, 1964
2
I
1
E(z)
dz
2πi C
z − z0
Z
Im E(z)
1 R
=
dz
π 0
z − z0
Z
∞
X
1 R
Im E(z)
n
=
z0
dz
π 0
z n+1
E(z0 ) =
.zo
R
C
n=0
WKB ⇒ Im E(z) ∼
⇒
cn ∼
e−b/z
√a
z
a
π
Z
0
∞
dz
,
z→0
e−b/z
a Γ(n + 21 )
=
π bn+1/2
z n+3/2
X
Instability and Divergence of Perturbation Theory
quartic AHO:
V (x) =
x2
4
4
+ λ x4
Bender/Wu, 1969
recall: divergence of perturbation theory in QM
e.g. ground state energy: E =
P∞
n=0 cn (coupling)
• Zeeman: cn ∼ (−1)n (2n)!
• Stark: cn ∼ (2n)!
• quartic oscillator: cn ∼ (−1)n Γ(n + 21 )
• cubic oscillator: cn ∼ Γ(n + 12 )
• periodic Sine-Gordon potential: cn ∼ n!
• double-well: cn ∼ n!
n
recall: divergence of perturbation theory in QM
e.g. ground state energy: E =
P∞
n=0 cn (coupling)
n
• Zeeman: cn ∼ (−1)n (2n)!
stable
• Stark: cn ∼ (2n)!
unstable
• quartic oscillator: cn ∼ (−1)n Γ(n + 21 )
stable
• cubic oscillator: cn ∼ Γ(n + 12 )
unstable
• periodic Sine-Gordon potential: cn ∼ n!
stable ???
• double-well: cn ∼ n!
stable ???
Bogomolny/Zinn-Justin mechanism in QM
...
...
• degenerate vacua: double-well, Sine-Gordon, ...
splitting of levels: a real one-instanton effect: ∆E ∼ e
−
S
g2
Bogomolny/Zinn-Justin mechanism in QM
...
...
• degenerate vacua: double-well, Sine-Gordon, ...
splitting of levels: a real one-instanton effect: ∆E ∼ e
surprise: pert. theory non-Borel summable: cn ∼
I
stable systems
I
ambiguous imaginary part
I
±i e
− 2S
2
g
, a 2-instanton effect
−
n!
(2S)n
S
g2
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:
I and Ī attract
• 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
E0 ∼
• perturbation theory:
cn ∼ −3n n!
:
X
Borel
cn g 2n
n
−
⇒
Im E0 ∼ ∓ π e
• non-perturbative analysis: instanton: g x0 (t) =
• classical Eucidean action: S0 = 6g12
• non-perturbative instanton gas:
−2
Im E0 ∼ ± π e
1
3g 2
1
1+e−t
1
6g 2
• BZJ cancellation ⇒ E0 is real and unambiguous
“resurgence” ⇒ cancellation to all orders
Decoding of Trans-series
2
f (g ) =
∞ X
∞ X
k−1
X
n=0 k=0 q=0
cn,k,q g
2n
k q
S
1
exp − 2
ln − 2
g
g
P
2n
• perturbative fluctuations about vacuum: ∞
n=0 cn,0,0 g
n!
• divergent (non-Borel-summable): cn,0,0 ∼ α (2S)n
⇒ ambiguous imaginary non-pert energy ∼ ±i π α e−2S/g
• but c0,2,1 = −α: BZJ cancellation !
2
Decoding of Trans-series
2
f (g ) =
∞ X
∞ X
k−1
X
n=0 k=0 q=0
cn,k,q g
2n
k q
S
1
exp − 2
ln − 2
g
g
P
2n
• perturbative fluctuations about vacuum: ∞
n=0 cn,0,0 g
n!
• divergent (non-Borel-summable): cn,0,0 ∼ α (2S)n
⇒ ambiguous imaginary non-pert energy ∼ ±i π α e−2S/g
• but c0,2,1 = −α: BZJ cancellation !
pert flucs about instanton: e−S/g
2
1 + a1 g 2 + a2 g 4 + . . .
divergent:
n!
−3S/g 2 a ln 1 + b
an ∼ (2S)
(a
ln
n
+
b)
⇒
±i
π
e
n
g2
2
2
a
1
1
−3S/g
• 3-instanton: e
+ b ln − g2 + c
2 ln − g 2
2
Decoding of Trans-series
2
f (g ) =
∞ X
∞ X
k−1
X
n=0 k=0 q=0
cn,k,q g
2n
k q
S
1
exp − 2
ln − 2
g
g
P
2n
• perturbative fluctuations about vacuum: ∞
n=0 cn,0,0 g
n!
• divergent (non-Borel-summable): cn,0,0 ∼ α (2S)n
⇒ ambiguous imaginary non-pert energy ∼ ±i π α e−2S/g
• but c0,2,1 = −α: BZJ cancellation !
pert flucs about instanton: e−S/g
2
1 + a1 g 2 + a2 g 4 + . . .
divergent:
n!
−3S/g 2 a ln 1 + b
an ∼ (2S)
(a
ln
n
+
b)
⇒
±i
π
e
n
g2
2
2
a
1
1
−3S/g
• 3-instanton: e
+ b ln − g2 + c
2 ln − g 2
resurgence: ad infinitum, also sub-leading large-order terms
2
Towards Resurgence in QFT
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
F (e2 ) = c0 + c2 e2 + c4 e4 + . . .
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 − B
~2
E
α 1 ~ 2 ~ 2 2
~
~
+
E −B
+7 E·B
+ ...
L=
2
90π Ec2
• 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
R
• EH effective action ∼ Barnes function ∼ ln Γ(x)
Euler-Heisenberg Effective Action: e.g., constant B field
B2
S=− 2
8π
Z
∞
0
ds
s2
m2 s
1 s
exp −
coth s − −
s 3
B
∞
B2 X
2B 2n+2
B2n+4
S=− 2
2π
(2n + 4)(2n + 3)(2n + 2) m2
n=0
• characteristic factorial divergence
cn =
∞
(−1)n+1 X Γ(2n + 2)
8
(k π)2n+4
k=1
• reconstruct correct Borel transform:
∞
X
s
1
1 s
= − 2 coth s − −
k 2 π 2 (s2 + k 2 π 2 )
2s
s 3
k=1
Euler-Heisenberg Effective Action and Schwinger Effect
B field: QFT analogue of Zeeman effect
E field: QFT analogue of Stark effect
B 2 → −E 2 : series becomes non-alternating
h
i
2 2 P∞
1
k m2 π
exp
−
Borel summation ⇒ Im S = e8πE3
k=1 k2
eE
Euler-Heisenberg Effective Action and Schwinger Effect
B field: QFT analogue of Zeeman effect
E field: QFT analogue of Stark effect
B 2 → −E 2 : series becomes non-alternating
h
i
2 2 P∞
1
k m2 π
exp
−
Borel summation ⇒ Im S = e8πE3
k=1 k2
eE
The European Physical Journal D
Schwinger effect:
recent~technological
advanc
2eE
∼ refinements
2mc2
theoretical
of t
mc
tion, suggest
that
lasers
su
⇒
may be able to reach this e
3
m2 chas
This
the potential to op
Ec ∼of experiments,
≈ 1016 V/cm
with the pr
WKB
tunneling
from
Dirac sea
g. 1. Pair
production
as the
separation
of a virtual vacuum
e~
eries
and
practical
applicat
ole pair
under
the
influence
of
an
external
electric
field.
Im S → physical pair production rate
talks in this conference.
• Euler-Heisenberg series must be divergent
mptotic e+ e− pairs if they gain the binding energy of
2
QED/QCD effective action and the “Schwinger effect”
• formal definition:
Γ[A] = ln det (i D
/ + m)
Dµ = ∂µ − i
e
Aµ
~c
• vacuum persistence amplitude
i
i
hOout | Oin i ≡ exp
Γ[A] = exp
{Re(Γ) + i Im(Γ)}
~
~
• encodes nonlinear properties of QED/QCD vacuum
• vacuum persistence probability
2
2
2
|hOout | Oin i| = exp − Im(Γ) ≈ 1 − Im(Γ)
~
~
• 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) = E cos(ωt)
• adiabaticity parameter: γ ≡ me cEω
h
i
2 3
• WKB ⇒ PQED ∼ exp −π me ~ cE g(γ)
PQED ∼
h


exp
−π





m2 c3
e~E
i
2
e E 4mc /~ω
ωmc
,
γ 1 (non-perturbative)
,
γ 1 (perturbative)
• 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
−φ̈ − (p3 − e A3 (t))2 φ = (m2 + p2⊥ )φ
+i m
bp⇥
ap⇥
“imaginary time method”
• pair production probability: P ≈
R
im
d3 p |bp |2
• imaginary time method
I
q
2
2
2
2
|bp | ≈ exp −2 Im dt m + p⊥ + (p3 − eA3 (t))
• more structured E(t) involve quantum interference
the envelope of the oscillations in the distribution function
g probis related to the temporal width of the slits. The width of the
e quanenvelope of oscillations thus also becomes a probe of the
ses no
Hebenstreit, Alkofer, GD, Gies, PRL 102, 2009
subcycle structure of the laser.
ter. We
To complete the physical picture, we consider the overmakes
all envelope of the longitudinal momentum distribution,
mentum
again for " ¼ 0, averaging over the rapid oscillations.
own in
When there are more than three cycles per pulse (! *
unction
3), the peak of the momentum
2 distribution is located near
hat, for
pk ð1Þ ¼ 0, whereas for ! & 3 the peak is shifted to a
illatory
nonzero value. Furthermore,
2 the Gaussian width of the
fset inemployed
e, howpffiffiffiffiffiffiffiffi WKB approximation Eq. (4), which scales
~ is obviously somewhat broader than the
with eE0 =$,
field is
true distribution, as is shown in Fig. 5. We can quantify
c distrithis discrepancy in the width, by extending the WKB result
e oscilbeyond the Gaussian approximation inherent in Eq. (4). We
a direct
PHYSICAL REVIEW LETTERS
mmetric PRL 102, 150404 (2009)
14
14
o is the
5 10
5 14
10
5 10
perfect
omenta,
produc14
14
3 14
10
3 10
ts from 3 10
r-phase
esting a
14
14
1 10
rch for 1 101 1410
Carrier Phase Effect
t
E(t) = E exp −
cos (ωt + ϕ)
τ
• sensitivity to carrier phase ϕ ?
week ending
17 APRIL 2009
k MeV
0.6
0.4
0.2
0
0.2
0.4
0.6 k MeV
ood in a
0.4
0.2
0
0.2
0.4
0.6
as first
FIG. 5. Comparison of the asymptotic distribution
~ 1Þ for k~?function
3.
distribution function fðk;
¼0
ioniza- FIG.
~ 1ÞAsymptotic
fðk;
for k~ ¼ 0 (oscillating solid line) with the prediction
¼ 5, E0 ¼? 0:1Ecr , and " ¼ !#=4. The center of the
ure, the for of! Eq.
(4) (dotted line) and the improved WKB approximation
ð1Þ
%
102
keV.
distribution
is
shifted
to
p
k
um that
based on an expansion of
Eq. (5) (dashed line) for ! ¼ 5, E0 ¼
0:1Ecr , and " ¼ 0.
ed pair
sensitive dependence on the other shape parameters, such
as !.
150404-3
ϕ=0
0.4
0.2
0
0.2
0.4
0.6
k MeV
~ 1Þ for k~? ¼ 0
FIG. 4. Asymptotic distribution function fðk;
for ! ¼ 5, E0 ¼ 0:1Ecr , and " ¼ !#=2. The center of the
distribution is shifted to pk ð1Þ % 137 keV.
ϕ=
π
2
creation events. The fringes are large for " ¼ !#=2, since
then the field strength has two peaks of equal size (though
Carrier Phase Effect from the Stokes Phenomenon
4
4
2
2
0
0
-2
-2
-4
-4
-4
-2
0
2
-4
4
-2
• interference produces momentum spectrum structure
interference
t=
⇥
P ≈ 4 sin2 (θ) e−2 ImW
t=+
θ: interference phase
• double-slit interference, in time domain, from vacuum
• Ramsey effect: N alternating sign pulses ⇒ N -slit system
⇒ coherent N 2 enhancement
Akkermans, GD, 2012
0
2
4
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
Z T
I
Z
Z ∞
dT −m2 T
2
4
e
Dx exp −
dτ ẋµ + Aµ ẋµ
Γ=− d x
T
0
x
0
• double-steepest descents approximation:
• worldline instantons (saddles): ẍµ = Fµν (x) ẋν
∂S(T )
∂T
= −m2
X
2
Im Γ ≈
e−Ssaddle (m )
• proper-time integral:
saddles
• 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 )
2k
∞
∞
Γ(2k + j)Γ(2k + j − 2)B2k+2j
m4 X (−1)j X
k 2E
Γ = − 3/2
(−1)
2j
2
(mλ)
m
8π
j!(2k)!Γ(2k + j + 21 )
j=0
k=2
• Borel sum perturbative expansion: large k (j fixed):
(j)
ck ∼ 2
(j)
Im Γ
Γ(2k + 3j − 12 )
(2π)2j+2k+2
4 j
m π
m2 π 1
∼ exp −
E
j! 4τ 2 E 3
• resum derivative expansion
m2 π
1 m 2
Im Γ ∼ exp −
1−
+ ...
E
4 Eτ
Divergence of derivative expansion
• Borel sum derivative expansion: large j (k fixed):
(k)
cj
∼2
9
−2k
2
Im Γ(k) ∼
Γ(2j + 4k − 25 )
(2π)2j+2k
(2πEτ 2 )2k −2πmτ
e
(2k)!
• resum perturbative expansion:
Eτ
Im Γ ∼ exp −2πmτ 1 −
+ ...
m
• compare:
m2 π
1 m 2
Im Γ ∼ exp −
1−
+ ...
E
4 Eτ
h
2
• different limits of full: Im Γ ∼ exp − mE π g
• derivative expansion must be divergent
m
Eτ
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?
Z
Z
i
1
S[A]
versus
DA exp
DA exp − S[A]
~
~
• perturbation theory is generically asymptotic
• resurgent trans-series
k l
1
c
ck,l,p g
f (g ) =
ln ± 2
exp − 2
g
g
| {z }
p=0 k=0 l=1
|
{z
}
|
{z
}
perturbative fluctuations
2
∞ X
∞ X
k−1
X
2p
k−instantons
n
m
quasi-zero-modes
The Bigger Picture: Decoding the Path Integral
what is the origin of resurgent behavior in QM and QFT ?
n
m
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
I
(n)
2
(g ) =
Z
−
dz e
1
g2
(Berry/Howls 1991, Howls 1992)
f (z)
Cn
1
− 1 f
=p
e g2 n T (n) (g 2 )
1/g 2
• T (n) (g 2 ): beyond the usual Gaussian approximation
• asymptotic expansion of fluctuations about the saddle n:
T (n) (g 2 ) ∼
∞
X
r=0
Tr(n) g 2r
All-Orders Steepest Descents: Darboux Theorem
• universal resurgent relation between different saddles:
Z ∞
1 X
e−v
dv
(n) 2
γnm
(m) Fnm
T (g ) =
(−1)
T
2π i m
v 1 − g 2 v/(Fnm )
v
0
• exact resurgent relation between fluctuations about nth saddle
and about neighboring saddles m
"
X (−1)γnm
(r
−
1)!
Fnm
(Fnm )2
(m)
(m)
(m)
Tr(n) =
T
+
T
+
T2 + . . .
r
0
1
2π i m (Fnm )
(r − 1)
(r − 1)(r − 2)
• 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)
Z π
− 1 sin2 (z)
2
I(g ) =
dz e g2
0
two saddle points: z0 = 0 and z1 = π2 .
min.
vacuum
saddle
I I¯
min.
vacuum
All-Orders Steepest Descents: Darboux Theorem
• large order behavior about saddle z0 :
Tr(0)
=
2
Γ r + 21
√
π Γ(r + 1)
∼
(r − 1)!
√
π
1−
1
4
(r − 1)
+
9
32
(r − 1)(r − 2)
−
75
128
(r − 1)(r − 2)(r − 3)
• low order coefficients about saddle z1 :
T (1) (g 2 ) ∼ i
√
1
9 4
75 6
π 1 − g2 +
g −
g + ...
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 ...
Resurgence in (Infinite Dim.) Path Integrals
• periodic potential: V (x) =
1
g2
(GD, Ünsal, 1401.5202)
sin2 (g x)
• vacuum saddle point
5 1 13
1
cn ∼ n! 1 − · −
·
− ...
2 n
8 n(n − 1)
• instanton/anti-instanton saddle point:
5
13
−2 12
2
4
Im E ∼ π e 2g 1 − · g −
· g − ...
2
8
Resurgence in (Infinite Dim.) Path Integrals
• periodic potential: V (x) =
1
g2
(GD, Ünsal, 1401.5202)
sin2 (g x)
• vacuum saddle point
5 1 13
1
cn ∼ n! 1 − · −
·
− ...
2 n
8 n(n − 1)
• instanton/anti-instanton saddle point:
5
13
−2 12
2
4
Im E ∼ π e 2g 1 − · g −
· g − ...
2
8
• double-well potential: V (x) = x2 (1 − gx)2
• vacuum saddle point
1
53 1 1 1277 1
n
·
cn ∼ 3 n! 1 −
· · −
·
− ...
6 3 n
72 32 n(n − 1)
• instanton/anti-instanton saddle point:
53 2 1277 4
−2 12
6g
1−
Im E ∼ π e
·g −
· g − ...
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 = E uµ uν + T⊥µν
4
= − E +Φ
3
4η
4 τII
1 λ1 2
τII
=
−Φ−
Φ−
Φ
3τ
3 τ
2 η2
1
0
• asymptotic hydro expansion: E ∼ τ 4/3
1 − τ2η
+
.
.
.
2/3
τ
dE
dτ
dΦ
dτ
• formal series → trans-series
E ∼ Epert +e−Sτ
2/3
× (fluc) + e−2Sτ
2/3
× (fluc) + . . .
• non-hydro modes clearly visible in the asymptotic hydro
series
Resurgence and Hydrodynamics
(Başar, GD, 1509.05046)
• study large-order behavior
(Aniceto/Schiappa, 2013)
S c1,1
S 2 c1,2
Γ(k + β)
c0,k ∼ S1
c
+
+
+
.
.
.
1,0
k + β − 1 (k + β − 1)(k + β − 2)
2πi S k+β
c0,n (large order)
◆ 1-term ● 3-terms
1.0
Hydro. expansion
Borel plane
c0,n (exact)
●
0.5
1.01 ●
1.00
0.99
0.98
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
◆
100
200
300
400
-20
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●
●●●●●●●●●●●●●●●●●●
-10
3
-0.5
n
500
10
2 Cτ
-1.0
• resurgent large-order behavior and Borel structure verified to
4-instanton level
• ⇒ trans-series for metric coefficients in AdS
20
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
⇒ uniform WKB:
ψ(x) =
• non-perturbative effects:
Dν
√1 ϕ(x)
~
ϕ0 (x)
√
g2 ↔ ~
⇒
exp − S~
• trans-series structure follows from exact quantization
condition → E(N, ~) = trans-series
•
Zinn-Justin, Voros, Pham, Delabaere, Aoki, Takei, Kawai, Koike, ...
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 Pinst (~, N )
(iii) rule connecting neighbouring vacua (parity, Bloch, ...)
~ 1
E(~, N ) = Epert (~, N ) ± √
2π N !
32
~
N + 1
2
e−S/~ Pinst (~, N ) + . . .
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 Pinst (~, N )
(iii) rule connecting neighbouring vacua (parity, Bloch, ...)
~ 1
E(~, N ) = Epert (~, N ) ± √
2π N !
in fact ...
(GD, Ünsal, 1306.4405, 1401.5202)
" Z
~
∂Epert
d~
Pinst (~, N ) =
exp S
3
∂N
0 ~
32
~
N + 1
2
e−S/~ Pinst (~, N ) + . . .
fluctuation factor:
!#
N + 12 ~2
∂Epert (~, N )
−~+
∂N
S
⇒ perturbation theory Epert (~, N ) encodes everything !
Resurgence at work
• fluctuations about I (or Ī) saddle are determined by those
about the vacuum saddle, to all fluctuation orders
−
1
1
1
8
48
16
a1
• "QFT computation": 3-loop fluctuation about
I for double-well and Sine-Gordon:
b 11
b 21
b 12
b 22
1
12
DW :
e
S0
~
71
2
1− ~−0.607535 ~ − . . .
72
1
8
b 23
Escobar-Ruiz/Shuryak/Turbiner 1501.03993, 1505.05115
−
1
−
24
−
1
8
b
24
1
1
1
16
16
8
d
e
f
1
−
1
−
16
48
g
h
Figure 2: Diagrams contributing to the coefficient B2 . The signs of contributions and symm
factors are indicated.
11
Resurgence at work
• fluctuations about I (or Ī) saddle are determined by those
about the vacuum saddle, to all fluctuation orders
−
1
1
1
8
48
16
a1
• "QFT computation": 3-loop fluctuation about
I for double-well and Sine-Gordon:
b 11
b 21
b 12
b 22
1
1
−
12
24
1
8
b 23
Escobar-Ruiz/Shuryak/Turbiner 1501.03993, 1505.05115
−
1
8
71
2
1− ~−0.607535 ~ − . . .
DW : e
72
S0
1
resurgence : e− ~ 1 +
~ −102N 2 − 174N −71
72
1
2
4
3
2
+
~ 10404N + 17496N − 2112N − 14172N −6299 + . . .
10368
b
−
S0
~
24
1
1
1
16
16
8
d
e
f
1
−
1
−
16
48
g
h
Figure 2: Diagrams contributing to the coefficient B2 . The signs of contributions and symm
factors are indicated.
• known for all N and to essentially any loop order, directly
from perturbation theory !
11
• 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
n
m
Z
−
DA e
1
S[A]
g2
=
X
all saddles
e
−
1
S[Asaddle ]
g2
× (fluctuations) × (qzm)
Analytic Continuation of Path Integrals: Lefschetz Thimbles
Z
−
DA e
1
S[A]
g2
=
X
thimbles k
Nk e
−
i
g2
Simag [Ak ]
Z
Γk
−
DA e
1
S
[A]
g 2 real
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
Z
−
DA e
1
S[A]
g2
=
X
thimbles k
Nk e
−
i
g2
Simag [Ak ]
Z
Γk
−
DA e
1
S
[A]
g 2 real
Lefschetz thimble = “functional steepest descents contour”
remaining path integral has real measure:
(i) Monte Carlo
(ii) semiclassical expansion
(iii) exact resurgent analysis
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 Nk can change with
phase of parameters
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
∂
∂τ
S − S̄
2i
∂
∂τ
1
=−
2i
• Chern-Simons theory
Z S + S̄
2
δS ∂A δS ∂A
−
δA ∂τ
δA ∂τ
Z 2
δS = − δA
=0
(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
mbles
2λ
σ 2 x4
Z=
dx exp −
x +
2
4
GA 13
−∞
Z
∞
(Aarts, 2013; GD, Unsal, ...)
2
4
stable thimble
unstable thimble
σ = 1+i, λ = 1
2
0
y
y
1
0
stable thimble
unstable thimble
not contributing
-1
-2
-2
-2
-1
0
x
1
2
-4
-4
-2
• contributing thimbles change with phase of σ
mble contributes
• need all three thimbles when Re[σ] < 0
imble gives correct result, with
• integrals along thimbles
are related (resurgence)
x Jacobian
SIGN 2014 – p. 6
• resurgence: preferred unique “field” choice
0
x
2
4
Ghost Instantons: Analytic Continuation of Path Integrals
(Başar, GD, Ünsal, arXiv:1308.1108)
2
Z(g |m) =
Z
−S[x]
Dx e
=
Z
−
Dx e
R
dτ
1 2
ẋ + 12
4
g
sd2 (g x|m)
• doubly periodic potential: real & complex instantons
instanton actions:
√
2 arcsin( m)
SI (m) = p
m(1 − m)
√
−2 arcsin( 1 − m)
p
SG (m) =
m(1 − m)
Ghost Instantons: Analytic Continuation of Path Integrals
• large order growth of perturbation theory:
16
1
(−1)n+1
an (m) ∼ − n!
−
π
(SI Ī (m))n+1 |SG Ḡ (m)|n+1
naive ratio Hd=1L
3
ò
ì
2
ò
ì
1
-1
ratio Hd=1L
ò
ì
ò
ì
ò
ì
ò
ò
ò
ì
ì
ì
ò
ò
ò
ò
ò
ò
òì
òì
òì
òì
òì
ò
òì
òì
òì
òì
òì
òì
òì
ì
òòì
òì
òì
ì
òì
ò ì
àààààààààà
àààààææææææææææ
ì
ò
à
à
æ
à
ì
ò
à
ææ
æ
à
æ
æ
ì
ò à
æ
à
ì
ò
ì
ò
à æææ
à
ì
ò à ææ
ààææ
ì
òææ
à
ææ
ì
ò
ì
ì
0.8
ì
ì
ì
ì
à
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àæ
àààæ
ì
àæææ
ææ
ì ì ì ì ì
ì
ì
ì
ì ì ì
ì
ò ì
ò ò ò
5
15
20
25
ò10ò
ò ò
ò ò
ò ò
ò ò
æ
à
ì
ò
1.0
0.6
à
without ghost instantons
0.4
n
0.2
0
5
10
15
20
25
with ghost instantons
• complex instantons directly affect perturbation theory, even
though they are not in the original path integral measure
n
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
Z
X
− 1 S[A]
− 1 S[A
]
DA e g2
=
e g2 saddle × (fluctuations) × (qzm)
all 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
Z
1
tr V (U )
Z(gs , N ) = dU exp
gs
• two variables: gs and N (’t Hooft coupling: λ = gs N )
• 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
Z
N
†
Z = DU exp
Tr(U + U )
λ
• in terms of eigenvalues eizi of U
N Z
Y
π
Z =
dzi e−S(zi )
i=1−π
X
2N X
2 zi − zj
cos(zi ) −
ln sin
S(zi ) ≡ −
λ
2
i
i<j
• at large N search numerically for saddles:
∂S
∂zi
=0
Resurgence in the Gross-Witten-Wadia Model
1512.09021
• phase transition driven by complex saddles
(a)
Im(z)
λ = 1.5, m = 0
(b)
Im(z)
π
(c)
λ = 4.0, m = 0
Im(z)
π
λ = 4.0, m = 1
(g)
Re(z)
Im(z)
0
λ = 4.0, m = 2
(i)
Im(z)
π
Re(z)
0
λ = 4.0, m = 7
π
Re(z)
0
λ = 1.5, m = 17
π
0
π
Re(z)
Im(z)
(d)
Re(z)
0
(f)
λ = 1.5, m = 2
π
Re(z)
0
Im(z)
Im(z)
π
Re(z)
(e)
λ = 1.5, m = 1
Re(z)
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 !
0
Resurgence in the Gross-Witten-Wadia Model
1512.09021
• weak-coupling “instanton” action from string eqn
p
(weak)
SI
= 4/λ 1 − λ/2 − arccosh ((4 − λ)/λ) , λ < 2
• strong-coupling “instanton” action from string eqn
p
(strong)
SI
= 2arccosh (λ/2) − 2 1 − 4/λ2 , λ ≥ 2
2.0
(1/N) |Re(S1 - S0)|, N = 400
(1/N) |Re(S2 - S0)|, N = 400
SI(w)(λ) / N
SI(s)(λ) / N
|Re∆S| / N
1.5
1.0
0.5
0.0
1.0
1.5
2.0
2.5
3.0
λ
3.5
4.0
4.5
5.0
• 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
Z
1
1
1
2
dM exp − tr
(ln M )
ZCS (N, g) =
vol(U (N ))
g
2
• ABJM: N = 6 SUSY CS, G = U (N )k × U (N )−k
N
X (−1)(σ) Z Y
dxi
ZABJM (N, k) =
N!
2πk QN 2ch
i=1
σ∈SN
i=1
• N = 4 SUSY Yang-Mills on S4
ZSY M (N, g 2 ) =
1
vol(U (N ))
Z
xi
2
1
ch
1
dM exp − 2 trM 2
g
xi −xσ(i)
2k
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
∞
m4 X
B2n+4
2eB 2n+4
Γ(B) ∼ − 2
8π
(2n + 4)(2n + 3)(2n + 2) m2
n=0
Γ(B) =
(
2
1
m2
3 m2
m2
0
+
ln(2π)
− + ζ (−1) −
−
12
4eB 4 2eB
4eB
2 # 2 2
m2
1 m2
m
1
γ m2
−
+ − +
ln
−
12 4eB 2 2eB
2eB
2 2eB
n+1 )
∞
X
m2
m2
(−1)n ζ(n) m2
+
1 − ln
+
2eB
2eB
n(n + 1)
2eB
(eB)2
2π 2
"
n=2
Resurgence in N = 2 and N = 2∗ Theories
−
u(ℏ)
2.5
(Başar, GD, 1501.05671)
~2 d2 ψ
+ cos(x) ψ = u ψ
2 dx2
←− electric sector
(convergent)
2.0
1.5
←− magnetic sector
1.0
0.5
0.5
-0.5
-1.0
1.0
1.5
ℏ
←− dyonic sector
(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 N = 2 SUSY SU(2)
• moduli parameter: u = htr Φ2 i
• electric: u 1;
• a = hscalari ,
magnetic: u ∼ 1 ;
aD = hdual scalari ,
dyonic: u ∼ −1
aD =
∂W
∂a
• Nekrasov twisted superpotential W(a, ~, Λ):
• Mathieu equation:
−
(Mironov/Morozov)
~2 d2 ψ
+ Λ2 cos(x) ψ = u ψ
2 dx2
,
a≡
• Matone relation:
u(a, ~) =
iπ ∂W(a, ~, Λ) ~2
Λ
−
2
∂Λ
48
N~
2
Mathieu Equation Spectrum: (~ plays role of g 2 )
−
~2 d2 ψ
+ cos(x) ψ = u ψ
2 dx2
u(ℏ)
2.5
2.0
1.5
1.0
0.5
0.5
-0.5
-1.0
1.0
1.5
ℏ
Mathieu Equation Spectrum
−
~2 d2 ψ
+ cos(x) ψ = u ψ
2 dx2
• small N : divergent, non-Borel-summable → trans-series
"
#
~2
1 2 1
1
+
−
N+
u(N, ~) ∼ −1 + ~ N +
2
16
2
4
"
#
~3
1 3 3
1
− 2
N+
+
N+
− ...
16
2
4
2
• large N : convergent expansion: −→ ?? trans-series ??
4
8
~2
1
2
5N 2 + 7
2
2
u(N, ~) ∼
N +
+
8
2(N 2 − 1) ~
32(N 2 − 1)3 (N 2 − 4) ~
!
12
9N 4 + 58N 2 + 29
2
+
+ ...
64(N 2 − 1)5 (N 2 − 4)(N 2 − 9) ~
Resurgence of N = 2 SUSY SU(2)
(Başar, GD, 1501.05671)
• N ~ 1, deep inside wells: resurgent trans-series
N −1/2
∞
∞
X
8 X
32
32
(±)
n
u (N, ~) ∼
cn (N )~ ± √
e− ~
dn (N )~n + . . .
π N! ~
n=0
n=0
• Borel poles at two-instanton location
• N ~ 1, far above barrier: convergent series
N −1
N
−1
2 2N
X
~2 N 2 X αn (N ) ~2
βn (N )
(±)
~
u (N, ~) =
±
+ ...
4n
N
−1
2
8
~
8 (2
(N − 1)!)
~4n
n=0
n=0
(Basar, GD, Ünsal, 2015)
• coefficients have poles at O(two-(complex)-instanton)
• N ~ ∼ π8 , near barrier top: “instanton condensation”
u(±) (N, ~) ∼ 1 ±
π
~ + O(~2 )
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
• N = 2 and N = 2∗ SUSY gauge theory
• 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.