Multi - legs, Superfluids & Semiclassics Riccardo Rattazzi, - - PowerPoint PPT Presentation

Multi-legs, Superfluids & Semiclassics

Riccardo Rattazzi, EPFL

with Gil Badel, Gabriel Cuomo, Alexander Monin, arXiv:1909.01269, arXiv:1911.08505

Extremal Correlators and Random Matrix Theory

Alba Grassi, Zohar Komargodski, Luigi Tizzano. Aug 27, 2019. 49 pp. e-Print: arXiv:1908.10306

The large charge limit of scalar field theories and the Wilson-Fisher fixed point at 𝜗=0 ϵ = 0

• G. Arias-Tamargo, D. Rodriguez-Gomez, J.G. Russo. Aug 29, 2019. 14 pp.

e-Print: arXiv:1908.11347

Accessing Large Global Charge via the 𝜗 ϵ -Expansion

Masataka Watanabe. Sep 3, 2019. 15 pp. e-Print: arXiv:1909.01337

[W eak vs Strong] & [Classical vs Quantum]

W eak coupling: loop expansion around leading trajectory

e−W = e−[S0+S1+S2+... ]

can further distinguish semi-classical and quantum observables semi-class quantum

Ex: around Ex: every day life

hφφφφi

Strong coupling: PI cannot be described by leading trajectory

hOi = O(γc`) + δqO

δqO > ∼ O(γc`)

δqO ⌧ O(γc`)

φ(γc`) = 0

γc`

Common practice: few legs in weakly coupled QFT

= small fluctuations around trivial trajectory

However when the number of legs grows expansion breaks down

How do we describe physics in this regime?

n-legs in φ4

✏ = E − nm n

a mess ?!

see old review by Rubakov, arXiv:9511236, 1995

1→n ∝ n! ✓ 8 ◆n−1 ⇣ ✏ m ⌘ 3n−1

2

A1→n ∝ n! ✓ λ 8m2 ◆ n−1

2

Aloop = Atree

• 1 + Bλn2 + Cλ2n4 + . . .
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Indeed .. but not completely all large effects can be proven to resum into

σ1→n ∼ enF (n,✏)

Exponential form suggest existence of non-trivial semiclassical trajectory describing the process Something indeed proven by Son, back in the 90’s…still a reasonable mess to work out quantitatively

Libanov, Rubakov, Son, T roitsky 1994 see Khoze, Reiness, 1810.01722

Charged

φ4

L = ∂µφ ∂µφ + λ 4 (φφ)2

λ

λ2 16π2

λ3 (16π2)2

Few Legs

Many Legs

φn

n 1

• Ex. anomalous dimension of

λn(n − 1)

λ2n(n − 1)(n − 2)

λ3n(n − 1)(n − 2)(n − 3)

λ2n(n − 1)

= n ⇥ λn + (λn)2 + (λn)3 + . . . ⇤

leading perturbation theory breaks down at

λn 16π2 > ∼ 1

n(n − 1) λ L

n(n − 1)(n − 2) λ2 (L2 + L)

n(n − 1)(n − 2)(n − 3) λ3 (L3 + L2 + L)

1 2n(n − 1)(n − 2)(n − 3) λ2 L2

Zφn = enL(

P

κ=0 λκPκ(λn)+O(L))

λ`n2`, . . . , λ`n`+2, λ`n`+1, . . . , λ`n

exponentiate

γn = d log Zφn dL

series can be organized as a double expansion

F0(λ Log) + λF1(λ Log) + . . .

similar to RG

• r to ’t Hooft large-N expansion

γn n = P0(λn) + 1 n ¯ P1(λn) + 1 n2 ¯ P2(λn) + . . .

γn n = 1 n d log Zn dL = P0(λn) + λ P1(λn) + λ2P2(λn) + . . .

Can one derive exponentiation and structure of the series without detailed diagrammatics? Can one resum the series?

λn

Semiclassical expansion around non-trivial trajectory Common answer:

Z  ∂ ¯ φ∂φ + λ 4 ¯ φφ 2

• −

→ 1 λ Z  ∂ ¯ φ∂φ + 1 4 ¯ φφ 2

• ≡ S

λ

φ → φ √ λ

Compute semiclassically for

λ ⌧ 1

λn = fixed

h¯ φn(xf)φn(xi)i = 1 λn R D ¯ φDφ e− 1

λ[

R ∂ ¯ φ∂φ+ 1

4 ( ¯

φφ)2−λn(ln ¯ φ(xf )+ln φ(xi))]

R D ¯ φDφ e− 1

λ[

R ∂ ¯ φ∂φ+ 1

4 ( ¯

φφ)2]

φ(x)n → 1 λn/2 φn(x) ≡ 1 λn/2 e

nλ λ ln φ(x)

saddle point

φcl ≡ φcl(λn, xfi)

xfi ≡ xf − xi

1 λn R D ¯ φDφ e− 1

λ[

R ∂ ¯ φ∂φ+ 1

4 ( ¯

φφ)2−λn ln ¯ φf φi]

R D ¯ φDφ e− 1

λ[

R ∂ ¯ φ∂φ+ 1

4 ( ¯

φφ)2]

= 1 λn+ 1

2 e 1 λ [Γ−1(nλ,xfi) + λΓ0(nλ,xfi) + ... ]

= n! e

1 λ[˜

Γ−1(nλ,xfi) + λ˜ Γ0(nλ,xfi) + ...]

breaking zero mode of solution

U(1)

Stirling

γn n = P0(λn) + λP1(λn) + . . .

• proves ‘exponentiation’ to all orders
• semiclassical expansion valid for all as long as
• must match diagrammatic expansion at

λn

λ ⌧ 1

λn ⌧ 1

!

n! e

1 λ[˜

Γ−1(nλ,xfi) + λ˜ Γ0(nλ,xfi) + ...]

∂2φ(x) − 1 2φ2(x)¯ φ(x) = − λn ¯ φ(xf)δ(d)(x − xf), ∂2 ¯ φ(x) − 1 2φ(x)¯ φ2(x) = − λn φ(xi)δ(d)(x − xi)

main problem: finding classical solution can perform perturbation theory in not straightforward to find solution at finite

λn

λn

basic difficulty must regulate to where not scale invariant: radial dependence non-trivial

d = 4 − ✏

φ4

focus on Wilson-Fisher fixed point in

d = 4 − ✏

∗ (4⇡)2 = ✏ 5 + 3✏2 25 + O(✏3)

β(λ∗) = 0

λ =  −✏ + 5

• (4⇡)2 − 15 2

(4⇡)4 + O(3)

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Conformally map theory to cylinder R × Sd−1

W ay out

hO|

O

Mapping to the cylinder & operator state correspondence

r

τ = ln r

O

|Oi

h O(r) O(0) i = 1 r2∆

hO|e−Hτ|Oi = e−∆τ

• on cylinder

is conserved off-criticality

D = Hcy`

• expect solution to be stationary for

τf τi

• simple consistent ansatz at

φc` = ρei

χ = −iµτ

ρ = const

τf τ τi

this corresponds to a homogeneous superfluid

µ ≡

chemical potential

lowest dimension primary of charge n, dominates path integral at large times in charge n sector

Scyl = Z dτdΩd−1 ✓ |∂φ|2 + ξdR|φ|2 + λ 4 |φ|4 − i n Ωd−1 χf + i n Ωd−1 χi ◆

φn ≡

projects on charge n

‘conformal mass’ on sphere ξdR = ✓d − 2 2 ◆2 ≡ m2

d

φ ≡ ρ eiχ

Sd−1 = 2π

d 2

Γ( d

2)

boundary eom bulk eom

µ(µ2 − m2

d) =

λn 4Sd−1

m2

d ≡

✓d − 2 2 ◆2

Plug back into action and perform systematic loop expansion

2ρ2µ = λn Sd−1

−µ2 + m2

d + 1

2ρ2 = 0

Leading order

d → 4

= ∗ ∝ ✏ → 0

λn = fixed

x ≡ λ∗n 16π2

1 λ∗n∆−1 =

3 ⇥ 9x − √ 81x2 − 3 ⇤1/3 + 32/3 ⇥ 9x − √ 81x2 − 3 ⇤ h 9x − √ 81x2 − 3 2/3 + 31/3 i2

+ 9 × 31/3x ⇥ 9x − √ 81x2 − 3 ⇤2/3 2 h 9x − √ 81x2 − 3 2/3 + 31/3 i2

Supposed to resum leading powers of n at all loops!

and comparing with diagrams

γn = λn(n − 1) 32π2 − λ2n2(n − 1) 512π4 + . . .

they happily agree

∆φn = n + λn2 32π2 − λ2n3 512π4 + λ3n4 4096π6 + O

• λ4n5

expanding at small λn

Makes one want to check subleading terms To do so fixed point condition in is essential

d = 4 − ✏

n

• diag = ✏n(n − 1)

10 − ✏2 n(n2 − 4n) 50 + O

• ✏3n4

diagrammatic computation gives Semiclassically, subleading terms should come from Casimir energy on cylinder

Semiclassically on cylinder

e−∆nτ = R Dφ e−S+inχf −inχi R Dφ e−S

= e−Scl(n)− 1

2 ln det S(2) n

e− 1

2 ln det S(2)

1-loop

1 τ ✓1 2 ln det S(2)

n

− 1 2 ln det S(2) ◆

= 1 2 X

`

n`,d [!+(`, d) + !−(`, d) − 2!0(`, d)]

Casimir Energy on Sd-1, convergent for sufficiently low d in terms of the renormalized coupling

n = Rn(n − 1) 32⇡2 + R✏n2 64⇡2 − 2

R(2n3 + 3n2)

1024⇡4 + . . .

λR

✏n(n − 1) 10 − ✏2 n(n2 − 4n) 50 + O

• ✏3n4

fixed point

perfect match !

total match at 3-loops, partial match at 4-loops plus boosting of available computations

matching to available computations with n ≤ 4 up to 5-loops

Large charge limit for d → 4

= ∗ ∝ ✏ → 0

Interpretation:

m2

ρ ⇠ µ2 ⇠ (λn)2/3

1 R2 = 1

integrate out radial mode: ‘pure’ conformal superfluid EFT

L ∼ (∂χ)4 + R(∂χ)2 + R2 + . . .

known large charge behaviour

Hellerman, Orlando, Reffert, W atanabe ’15 Monin, Pirtskhalava, RR, Seibold ’16 Jafferis, Mukhametzhanov, Zhiboedov, ‘17

∆φn = π2 λ " 3 8 ✓λn π2 ◆4/3 + ✓λn π2 ◆2/3 − 2 3 + O ⇣ λn/π2−2/3⌘#

λn 1

For quantum corrections provide proper scaling

d = 4 − ✏

∆φn = 1 ✏ " c−1(✏) ✓2 5✏n ◆ 4−✏

3−✏

+ c0(✏) ✓2 5✏n ◆ 3−✏

2−✏

+ . . . #

1 λ ⇣ |∂φ|2 + |φ|6⌘

in

3 − ✏

() = ✏ + a3

in d=3 conformally invariant for any up to 1-loop

λ

∆n = 1 f0(n, ✏) + f1(n, ✏) + f2(n, ✏) + . . .

✏ → 0

Result non-trivially interpolates between universal quantum effects in genuine 3D CFT at large charge and Feynman diagram computations

Summary

Wilson-Fisher fixed points: simple but rich playground to get structural insight on regime in QFT

λn 1

Loop expansion for non-trivially and systematically encapsulated by semiclassical supefluid configuration

γφn

hφn1 . . . φnpi

can be studied by extension of method, providing dynamical information, akin to amplitudes Properties of nearby operators, ex , described by hydrodynamic modes

φn−2∂φ∂φ

…but it would be nice to get back to the SM… someone certainly will before the FCC begins