Potential Toolkit to Attack Nonperturbative Aspects of QFT - - PowerPoint PPT Presentation

Tatsuhiro MISUMI Masaru Hongo, Shigeki Sugimoto, Yuya Tanizaki, Hidetoshi Taya

Potential Toolkit to Attack Nonperturbative Aspects of QFT

• Resurgence and related topics-

Sep 7-25, 2020

@YITP , Kyoto U. & Online 09/07/20

YITP-iTHEMS molecule-type international workshop

Opening remarks & Brief introduction to Resurgence

Goals of this workshop

• 1. Review the novel techniques for nonperturbative effects
• f QFT, focusing on resurgence and the related techniques.
• 2. Study and summarize the very recent results in the

techniques.

• 3. Raise and consider questions in the techniques and their

physical results.

• 4. Propose their applications to physical problems other

than pure QFT.

• 5. Discuss the questions and applications, and produce new

works by collaborating with the participants.

• 1. Review the novel techniques for nonperturbative effects
• f QFT, focusing on resurgence and the related techniques.
• 2. Study and summarize the very recent results in the

techniques.

• 3. Raise and consider questions in the techniques and their

physical results.

• 4. Propose their applications to physical problems other

than pure QFT.

• 5. Discuss the questions and applications, and produce new

works by collaborating with the participants.

Three Lectures

Goals of this workshop

Six Talks & Poster Session

• 1. Review the novel techniques for nonperturbative effects
• f QFT, focusing on resurgence and the related techniques.
• 2. Study and summarize the very recent results in the

techniques.

• 3. Raise and consider questions in the techniques and their

physical results.

• 4. Propose their applications to physical problems other

than pure QFT.

• 5. Discuss the questions and applications, and produce new

works by collaborating with the participants.

Goals of this workshop

• 1. Review the novel techniques for nonperturbative effects
• f QFT, focusing on resurgence and the related techniques.
• 2. Study and summarize the very recent results in the

techniques.

• 3. Raise and consider questions in the techniques and their

physical results.

• 4. Propose their applications to physical problems other

than pure QFT.

• 5. Discuss the questions and applications, and produce new

works by collaborating with the participants.

Free Discussion Time

Goals of this workshop

It may be a new manner of holding academic workshops in Covid-19 and Post-Covid-19 eras.

How to join the workshop

• 1. Lectures, Invited talks, and Short talks will be held in

Zoom, whose URL has been emailed to all participants.

• 2. Poster sessions will be held in Mozilla hubs. Its URL and

how-to-use are shown in the email.

• 3. Discussion sessions will be also held in Mozilla hubs.

All of the important information are shown in SLACK, whose URL has been sent to you. We strongly recommend you to join SLACK ASAP

Since some of participants may not be familiar to resurgence theory in quantum physics, we are giving its very brief review and refer to problems to be considered !

See also Prof. Dunne’s lecture on 8th

• 1. Perturbative series and Borel resummation

Perturbation : quantum fluctuation is calculated as series of g^2 based

• n eigenstates of H0

Nonperturbative analysis : it is required to diagonalize whole hamiltonian.

Perturbative v.s. Non-perturbative analyses in QT

H = H0 + g2 H g2 1 H = H0 + g2 H g2 ∼ 1

E =

∞

X

n=0

cqg2q

E ≈ e− A

g2

cf.)

Trivial (perturbative) saddle Perturbative series

Path integral and Saddle points

Nontrivial saddles Non-perturbative contribution

δS δφ = 0

e−Ssol ∼ e− A

g2

Z = Z Dφ exp(−S[φ]) = X

σ∈saddles

Zσ

Z0 =

∞

X

q=0

aqg2q

Zσ ∝

: stationary points

Relation between Pert. and Non-pert.

X

q=0

aqg2q

exp  − A g2

• ❌

"They are not connected ? We just have independent contributions ?"

Perturbative series Non-perturbative contribution

No, it is not correct !

P(g2) =

∞

• q=0

aqg2q,

• H0 + g2Hpert
• ψ(x) = Eψ(x)

Borel resummation：Analytic function which has original perturbative series as asymptotic series Perturbative series is often divergent factorially

aq ∝ q!

・Construct an analytic function from asymptotic series

Perturbation and Borel resummation

Note that the analytic function is not unique for one asymptotic series.

Perturbation and Borel resummation

• H0 + g2Hpert
• ψ(x) = Eψ(x)

BP(t) :=

∞

• q=0

aq q! tq.

B(g2) = Z ∞ dt g2 e−t/g2BP(t) Borel resummation

P(g2) =

∞

• q=0

aqg2q,

Borel transform

Perturbative series is often divergent factorially

aq ∝ q!

• H0 + g2Hpert
• ψ(x) = Eψ(x)

B(g2) = Z ∞ dt g2 e−t/g2BP(t)

P(g2) =

∞

• q=0

aqg2q,

In special cases, Borel resum may give exact results

cf.) x^4 unharmonic oscillator

Perturbation and Borel resummation

Perturbative series is often divergent factorially

aq ∝ q!

• H0 + g2Hpert
• ψ(x) = Eψ(x)

BP(t) :=

∞

• q=0

aq q! tq.

Singularities on positive real axis leads to ambiguity

P(g2) =

∞

• q=0

aqg2q,

Borel transform can have singularities on positive real axis

B(g2e⌥i✏) = Z 1e±i✏ dt g2 e t

g2 BP(t)

Perturbation and Borel resummation

Perturbative series is often divergent factorially

aq ∝ q!

• H0 + g2Hpert
• ψ(x) = Eψ(x)

This should be cancelled by that from non-perturbative contribution!

Non-perturbative effect reappears in perturbative calculation through imaginary ambiguity !

P(g2) =

∞

• q=0

aqg2q,

Im[B(g2)] ≈ e− A

g2

B(g2e⌥i✏) = Re[B(g2)] ± iIm[B(g2)]

Perturbation and Borel resummation

Perturbative series is often divergent factorially

aq ∝ q!

Possible questions to be asked

• A resummation method is not unique. Is there a better

resummation formula ?

• Resummation method to get nonperturbative results

directly ?

• Resummation method for the divergent series beyond

Gevrey-1?

• Higher-order perturbative series in QFT ?

Costin, Dunne (17)(20) cf.) Stochastic perturbation theory

• 2. Resurgent structure and Trans-series in

ODE and Integral

Ecalle (81)

Formal solutions around z=∞

Resurgent structure in ODE

• ϕ′(z) + ϕ(z) = 1

z

• σe−z
• Φ0 =

∞

X

q=0

n!z−n−1

&

perturbative nonperturbative

Formal solutions around z=∞

• σe−z
• Φ0 =

∞

X

q=0

n!z−n−1

&

perturbative nonperturbative

resurgent structure!…….. but why?

• S+Φ0(z) − S−Φ0(z) = 2πie−z
• Resurgent structure in ODE
• ϕ′(z) + ϕ(z) = 1

z

• t=1

Ecalle (81)

Formal solutions around z=∞

：Stokes constant

• ϕ(z; σ) = Φ0 +σe−z
• σe−z
• S±ϕ(z; σ) = S±Φ0(z) + σe−z.
• ±
• S+ϕ(z; σ) → S−ϕ(z; σ + s)
• S+ϕ(z; σ) = S−ϕ(z; σ + s)
• S+Φ0(z) − S−Φ0(z) = 2πie−z
• Φ0 =

∞

X

q=0

n!z−n−1

s = 2πi

Borel-resum

&

perturbative nonperturbative

• 1. Solution is expressed as their linear combination = Trans-series
• 2. At arg[z]=0, an appropriate value of σ jumps = Stokes phenomenon
• 3. Continuity of solution leads to resurgent relation between two sectors

Resurgent structure in ODE

• ϕ′(z) + ϕ(z) = 1

z

• Ecalle (81)

For review for physicists, Marino(07), Aniceto, Schiappa, Vonk(13) Cherman, Dorigoni, Unsal(14) Dorigoni(14), Aniceto, Basar, Schiappa(18)

arg[z]=0+ arg[z]=0-

◆Resurgence theory and Alien calculus

・Group action connecting ± Borel resums：Stokes automorphism ・Operator associated with each singularity：Alien derivative ・Equation bridging alien and standard calculus：Bridge equation ・Bridge eq. reveals relation between each sector！

• S+ = S− ◦ S,

S = exp

• e−z∆
• [e−z∆, ∂σ] = 0
• e−z∆ϕ(z; σ) = s ∂σϕ(z; σ)
• SΦ0 = Φ0 + se−z,
• Sϕ(z; σ) = ϕ(z; σ + s)
• S−
• S+ϕ(z; σ) = S−ϕ(z; σ +s)
• S
• z∆,
• Resurgent structure in ODE
• ϕ(z; σ) = Φ0 +σe−z
• [e−z∆, ∂z] = 0

ϕ00 − zϕ = 0

Ex.) Airy equation

ϕ = Ai(z) ≈ e− 2

3 z 3 2 S±

X anz− 3

2 n + σ e 2 3 z 3 2 S±

X bnz− 3

2 n

(Irregular singularity on z = ∞)

0 ≤ arg[z] ≤ 2π Re[Ai(z)]

· Airy integral

Ai(g−2) = Z ∞

−∞

dφ exp  −i ✓φ3 3 + φ g2 ◆ ≈ r g 4π exp ✓ − 2 3g2 ◆

In integral, original contour decomposes into steepest decent contours (Lefschetz thimbles) associated with complex saddles Thimbles associated with distinct saddles have nontrivial relation via Stokes phenomena

Resurgent structure in integral

For review for physicists, Cherman, Dorigoni, Unsal(14) Tanizaki (14)

arg[g2] = 0+

Steepest descent method :

• riginal contour is decomposed into

thimbles associated with saddle points.

C = X

σ

nσJσ

Steepest descent contour = Thimble

· Airy integral

complex saddle points Ai(g−2) = Z ∞

−∞

dφ exp  −i ✓φ3 3 + φ g2 ◆

φ = ± i g

Complex saddle contributions in thimble decomposition (Steepest descent method)

Resurgent structure in integral

arg[g2] = 0+

+ i g

− i g

arg[g2] = 0+

Re[S] ≤ Re[S0]

Jσ

Thimble

nσ = hKσ, Ci

Intersection number

• f dual thimble K

and original contour

C = X

σ

nσJσ

・ ・

Im[S] = Im[S0]

· Airy integral

Ai(g−2) = Z ∞

−∞

dφ exp  −i ✓φ3 3 + φ g2 ◆

Complex saddle contributions in thimble decomposition (Steepest descent method)

Resurgent structure in integral

J−

+ i g

− i g

n+ = hK+, Ci = 0 n− = hK−, Ci = 1

arg[g2] = 0+

C = J−

C = X

σ

nσJσ

arg[g2] = 0+

K− J+ K+ J−

valid decomposition till arg[g2] = 2π

3 −

· Airy integral

Complex saddle contributions in thimble decomposition (Steepest descent method)

Resurgent structure in integral

C = X

σ

nσJσ

Stokes phenomenon : at special arg[g2], thimble decomposition discretely changes

arg[g2] = 2π 3 +

arg[g2] = 2π 3 +

n+ = hK+, Ci = 1 n− = hK−, Ci = 1

K− J+ K+ J−

· Airy integral

Complex saddle contributions in thimble decomposition (Steepest descent method)

C = J− + J+

Resurgent structure in integral

arg[g2] = −2π 3 → π

C = J−

arg[g2] = 2π 3 + arg[g2] = 2π 3 −

Complex saddle contributions in thimble decomposition (Steepest descent method)

· Airy integral

Thimble decomposition is discretely changed at Stokes line. Airy function is continuous even at the Stokes line.

C = J− + J+

J− 2π 3

−

= J− 2π 3

+

+ J+

Resurgent structure in integral

arg[g2] = −2π 3 → π

Two thimbles have resurgent relation via ambiguity due to Stokes phenomena !

C = J−

arg[g2] = 2π 3 + arg[g2] = 2π 3 −

Complex saddle contributions in thimble decomposition (Steepest descent method)

· Airy integral

Thimble decomposition is discretely changed at Stokes line. Airy function is continuous even at the Stokes line.

C = J− + J+

Resurgent structure in integral

Resurgent structure is expected to be in quantum theory, thus perturbative series could include nonpert. information !

Perturbative imaginary ambiguity Non-perturbative effect

S+Φ0(z) − S−Φ0(z) ≈ se−AzSΦ1(z)

In a certain class of quantum theories, we may be able to derive non-perturbative result from perturbation

Resurgent structure in ODE and integral

Possible questions to be asked

• Thimble decomposition is possible in QFT path integral ?
• Resurgent functions in resurgence theory for ODE can be

extended ?

• How can we figure out the intersection number ?
• Resurgent structure in partial differential equations ?

• 3. Resurgent structure in quantum theory

Example in Double-well QM

・Perturbation in Double-well QM

SI = 1 6g2

Singularity on positive real axis

Bogomolny(77) Zinn-Justin(81) Bender-Wu(73)

BPpert(t) = 1 π 1 t − 1/3

Im[Bpert(g2e⌥i✏)] = Im "Z 1e±i✏ dt g2 e t

g2

π 1 t 1/3 # = ⌥e

1 3g2

g2

H = p2 2 + x2(1 − gx)2 2

x(τ) = 1 2g ✓ 1 + tanh τ − τI 2 ◆

Instanton solution twice

aq = −3q+1 π q!

E0,pert = X

q=0

aqg2q

q 1

( )

Instanton-antiinstanton configuration = Bion

    

• [I ¯

I]

4

2

4

2

τ

Bion

x

V (x)

xI ¯

I = 1

2g  tanh τ − τI 2 − tanh τ − τ¯

I

2

• cancels the imaginary ambiguity in perturbation

⌥e−2SI g2

Eb ≈ − lim

β→∞

1 β Zb Z0 = e−2SI πg2  γ + log 2 g2 ± iπ

Complex bion solution and its contribution

d2z dτ 2 = ∂V ∂z x → z = x + iy

zcb(τ) = z1 − (z1 − zT ) 2 coth ωτ0 2  tanh ω(τ + τ0) 2 − tanh ω(τ − τ0) 2

• zT , τ0 ∈ C

The imaginary ambiguity cancels that from perturbative series ・Contribution from complex bion ・Complex bion solutions

Ecb = e−

1 3g2

⇡g2 ✓g2 2 ◆✏  − cos(✏⇡)Γ(✏) ± i⇡ Γ(1 − ✏)

• Behtash, Dunne, Schafer, Sulejmanpasic, Unsal(15)

For other models including CPN-1 QM, See Fujimori, et.al. (16)(17) For precise summation of bion contributions, see Sueishi (19).

Possible questions to be asked

• Non-trivial Borel singularities always correspond to

complex solutions ?

• How can we treat systems with fermions ?

By projection? By integrating out?

• Can we clarify relation of Stokes phenomena in semi-

classical calculation and Exact WKB analysis ?

• The same structure exists in quantum field theory ?

Sueishi-san’s talk on 18th

Examples in QFT and Matrix model

• 2D sigma model with compactification
• 3D Chern-Simons theory (with matter)
• 4D N=2 Super

Yang-Mills on S^4

• Matrix models & Topological String Theories

Schiappa, Marino, Aniceto(13~) Honda(16) Gukov, Marino, Putrov(16-) Honda(16) Fujimori, Honda, Kamata, TM, Sakai(18) Dunne, Unsal(12) TM, Nitta, Sakai(14)(15) Fujimori, et.al.(18) Ishikawa, et.al. (19) Marino(07~) Marino, Schiappa, Weiss(09), Hatsuda, Dorigoni(15)

• Prof. Dunne’s lecture on 8th

Hatsuda-san’s talk on 18th Honda-san’s talk on 25th Yoda-san’s poster on 11th

• 4. Resurgent structure in gauge theory

Infrared renormalon in QCD

‘t Hooft(79)

BP(t) = αs(µ)

• n
• −αs(µ)β0t

2 n = αs(µ) 1 + αs(µ)β0t/2

t = − 2 αs(µ)β0

Singularity on real axis ( )

C+ C− t

could be related to QCD scale and low-energy physics → IR renormalon

◆Adler function and renormalon

D(Q2) = 4π2 dΠ(Q2) dQ2 .

D(Q2) =

∞

X

n=0

αs Z ∞ dˆ k2 F(ˆ k2) ˆ k2 " β0αs log ˆ k2Q2 µ2 #n ≈ αs X

n

✓ −αsβ0 2 ◆2 n! + UV contr.

k

≈ ✓ΛQCD Q ◆4

β0 = −11Nc − 2Nf 12π

In asymptotically free QFT, another source of Im ambiguity exists.

B(αs) = ReB ± iπ β0 e

2 αsβ0 t = − 2 αs(µ)β0

・Non-trivial Polyakov-loop holonomy for small S1

QCD(adj.) on R3 × S1

1/N fractional instantons (Q=1/N, S=SI/N)

BPS KK BPS KK (1, 1/2) (−1, 1/2) (−1, −1/2) (1, −1/2)

cf.)SU(2) Neutral bion = IR-renormalon

Argyres-Unsal (12)

Conjecture : Neutral bion is identified as IR-renormalon

???

P = diag[1, e2πi/N, e4πi/N, · · ·, e2π(N−1)i/N]

P N = 1

( )

• Prof. Unsal’s lecture on 10th
• Prof. Cherman’s lecture on 22th

Morikawa-san’s talk on 24th Yamazaki-san’s talk on 25th

For YM on R3×S1 with ’t Hooft twist see also Yamazaki, Yonekura(17), Itou(19). For confinement via magnetic bions, see Unsal(07).

・ZN-twisted boundary condition on S1 direction

CPN-1 model on R1 × S1

1/N fractional instantons (Q=1/N, S=SI/N) Bion imaginary ambiguity seems consistent with IR-renormalon. But, is it related to renormalon on R2 ? There have been arguments.

Dunne-Unsal (12)

Conjecture : Complex bion is identified as IR-renormalon

s(x1, x2)

x

y

2 4 4 2 1 −1 −2 −4

x1 x2

s(x1,y1)

~ z(x1, x2 + L) = Ω~ z(x1, x2),

Ω = diag(1, !, . . . , !N1).

Fujimori-san’s & Morikawa-san’s talks on 24th Yamazaki-san’s talk on 25th

Fujimori,et.al. (16)(17)(18)

Bion

Ishikawa, et.al. (19) Yamazaki, Yonekura (19) etc….

Adiabatic continuity in compactified QFT

Adiabatic continuity conjecture: Vacuum structure & ZN symmetry persist during ZN-twisted compactification If adiabatic continuity exists, the resurgent structure on compactified spacetime has implications on decompactified spacetime.

Cherman, Schafer, Unsal (16), (See also Iritani, Itou,TM(15))

• 2D ZN-twisted model
• QCD(adj.) + ZN-twisted quarks
• YM with ’t Hooft twist

Sulejmanpasic (16) Tanizaki, TM, Sakai (17) Hongo, Tanizaki, TM(18) TM, Tanizaki, Unsal(19) Fujimori, et.al. (20)

• Prof. Unsal’s lecture on 10th
• Prof. Cherman’s lecture on 22th

Fujimori-san’s talk on 24th Chen-san’s, Itou-san’s, Tanizaki-san’s posters on 11th

Yamazaki, Yonekura (17), Itou (19)

• 5. Further applications of resurgence theory

Further applications

• Condensed matter physics, High-Tc superconductor
• Hydrodynamics, Fluid dynamics
• Schwinger mechanism
• Quantization conditions via exact-WKB, TBA equations
• Cosmology & Astrophysics
• Phase transition
• Prof. Dunne’s lecture on 8th

Hatsuda-san’s talk on 18th Honda-san’s talk on 25th Posters by Arraut, Du, Harris, Imaizumi, Shimazaki, Yoda

Goals of this workshop

• 1. Review the novel techniques for nonperturbative effects
• f QFT, focusing on resurgence and the related techniques.
• 2. Study and summarize the very recent results in the

techniques.

• 3. Raise and consider questions in the techniques and their

physical results.

• 4. Propose their applications to physical problems other

than pure QFT.

• 5. Discuss the questions and applications, and produce new

works by collaborating with the participants.