Scattering Amplitudes at Strong Coupling Beyond the Area Paradigm - - PowerPoint PPT Presentation

Benjamin Basso ENS Paris

based on work with Amit Sever and Pedro Vieira Strings 14 Princeton Scattering Amplitudes at Strong Coupling Beyond the Area Paradigm

Wednesday, 25 June, 14

1+1d background : flux tube sourced by two parallel null lines

Wilson loops at finite coupling in N=4 SYM

[Alday,Gaiotto,Maldacena,Sever,Vieira’10]

Sum over all flux-tube eigenstates bottom&top cap excite the flux tube out of its ground state

 

 

W = X

states ψ

Cbot(ψ) × e−E(ψ)τ+ip(ψ)σ+im(ψ)φ × Ctop(ψ)

Wednesday, 25 June, 14

[BB,Sever,Vieira’13]

vac vac ψ1 ψ2 ψ3 = X

ψi

"Y

i

e−Eiτi+ipiσi+imiφi # × P(0|ψ1)P(ψ1|ψ2)P(ψ2|ψ3)P(ψ3|0)

Valid at any coupling

Refinement : the pentagon way

Wednesday, 25 June, 14

[BB,Sever,Vieira’13]

vac vac ψ1 ψ2 ψ3 = X

ψi

"Y

i

e−Eiτi+ipiσi+imiφi # × P(0|ψ1)P(ψ1|ψ2)P(ψ2|ψ3)P(ψ3|0)

To compute amplitudes we need

The spectrum of flux-tube states All the pentagon transitions

Valid at any coupling

Refinement : the pentagon way

Wednesday, 25 June, 14

f6 = 1.04 (σ2 + τ 2)1/72 + O(e−

√ 2τ)

Simplest case : hexagon (n = 6) WL Pre-factor

minimal area in

[Alday,Gaiotto,Maldacena’09] [Alday,Maldacena,Sever,Vieira’10]

Wn=6 = f6 λ−

7 288 e √ λ 144 − √ λ 2π An=6(1 + O(1/

√ λ)) AdS5

[BB,Sever,Vieira’14]

classical quantum

Beyond the area paradigm

Wednesday, 25 June, 14

The flux-tube eigenstates

ψ = N particles state

(Adjoint) field insertions along a light-ray : create/annihilate state on the flux tube

p(u) = 2u + g2...

rapidity

p = p(u1) + · · · + p(uN)

Spectral data E(u) = twist + g2 . . .

can be found using integrability

Wednesday, 25 June, 14

Whex = = X

a

Z du P a(0|u) e−E(u)τ+ip(u)σ+imφP a(¯ u|0)

Lightest states dominate at large

i.e. in collinear limit

What are they?

Pentagon/OPE series for hexagon

Wednesday, 25 June, 14

Decoupling limit

Scalar mass is exponentially small at strong coupling

[Alday,Maldacena’07]

m = 21/4 Γ(5/4)λ1/8e−

√ λ 4 (1 + O(1/

p λ)) ⌧ 1

Low energy effective theory : (relativistic) O(6) sigma model

τ 1 For all heavy flux tube excitations decouple

Leff = √ λ 4π ∂X · ∂X with X2 =

6

X

i=1

X2

i = 1

E(p = 0)

gluons fermions scalars coupling mass

→

Wednesday, 25 June, 14

SNG = − 1 2πα0 q −det

• ∂αxµ∂βxνgAdS

µν

• − det
• ∂αyµ∂βyνgS5

µν

• square

pentagon hexagon

S5

AdS5

The pentagon/twist operator

Wednesday, 25 June, 14

Hexagon as a correlator of twist operators

5 5 4 4

3 3 3 2 2

1 1

6 6 6

=

W6 = h0| φD(τ, σ)φD(0, 0) |0i + O(e−

√ 2τ)

corrections from heavy modes irrelevant in collinear limit

WO(6)(z) z = m p σ2 + τ 2

Probes the physics of the O(6) sigma model :

Large distance z 1 Short distance

z ⌧ 1 WO(6) = 1 + O(e−2z)

WO(6) = ?

Wednesday, 25 June, 14

Insert complete basis of states

P(0|θ1, . . . , θN) = hθ1, . . . , θN| φD |0i

Pentagon transition = form factor of twist operator Normalization

h0| φD |0i = 1

which enforces that z → ∞ We found all these transitions so we can plot

WO(6) = X

N

1 N! h0| φD |θ1, . . . , θNi hθ1, . . . , θN| φD |0i e

−z P

i

cosh θi

See [Cardy,Castro-Alvaredo,Doyon’07] for similar considerations for computing entanglement entropy in integrable QFT

WO(6) → 1

OPE as form factor expansion

Wednesday, 25 June, 14

nmax=2 nmax=4 nmax=6 nmax=8

6 8 10 12 14 logH1êzL 0.10 0.15 0.20 0.25 log W

Plot of the truncated OPE/form factor series representation for log WO(6)

Numerical analysis

Wednesday, 25 June, 14

with the scaling dimension of the twist operator φk ∆k ∆k = c 12(k − 1 k )

[Knizhnik’87] [Lunin,Mathur’00] [Calabrese,Cardy’04]

c = central charge

2π(k − 1) = excess angle for φk

Short distance analysis

Short distance OPE (valid for )

z ⌧ 1

3-point function

Critical exponent A φD(τ, σ)φD(0, 0) ∼ log (1/z)B zA φ7(0, 0) A = 2∆D − ∆7 = 2∆5/4 − ∆3/2

Wednesday, 25 June, 14

Critical exponent from one-loop anomalous dimensions

since in our case c = 5

Short distance OPE (valid for )

z ⌧ 1

3-point function

Critical exponent A = 1 36 B = −3 2A = − 1 24 B A φD(τ, σ)φD(0, 0) ∼ log (1/z)B zA φ7(0, 0)

Short distance analysis

Wednesday, 25 June, 14

z ⌧ 1

For

include subleading RG logs

and is thus non perturbative WO(6) = C z1/36 log (1/z)1/24 + . . . WO(6) → 1 when z → ∞ Constant is fixed in the IR by C

Short distance analysis

Wednesday, 25 June, 14

6 8 10 12 14 a

• 0.0130
• 0.0125
• 0.0120
• 0.0115
• 0.0110

log W + 1ê36 log z + 1ê24 log a

running coupling

α = log (1/z) + . . .

log C = −0.01

Numerical analysis

Wednesday, 25 June, 14

Short distance analysis

z ⌧ 1

For f6 = 1.04 (σ2 + τ 2)1/72 + O(e−

√ 2τ)

Pre-factor

WO(6) = C z1/36 log (1/z)1/24 + . . .

m ' 21/4 Γ(5/4)λ1/8e−

√ λ 4

z = m p σ2 + τ 2

Wn=6 = f6 λ−

7 288 e √ λ 144 − √ λ 2π An=6(1 + O(1/

√ λ))

An=6 = O(e−

√ 2τ)

controlled by the gluons

(i.e. 1 ⌧ τ ⌧ e

√ λ/4 )

Wednesday, 25 June, 14

Deep (infrared) collinear limit

O(6) σ model

α0 expansion 1/τ

m 1

Infrared/non-perturbative regime

Completely non perturbative

τ e

√ λ/4

z 1 equivalently

Wednesday, 25 June, 14

z ⌧ 1 1 ⌧ τ ⌧ e

√ λ/4

equivalently

O(6) σ model

α0 expansion 1/τ

m 1

Cross over

UV regime of O(6) model : perturbative collinear limit

Wednesday, 25 June, 14

O(6) σ model

α0 expansion 1/τ

m 1

Cross over

here we could match O(6) analysis with string perturbative expansion

Wednesday, 25 June, 14

O(6) σ model

α0 expansion 1/τ

m 1

Full stringy pre-factor

full thing : include all heavy modes gluons, fermions, ... f6 = 1.04 (σ2 + τ 2)1/72 + O(e−

√ 2τ) + O(e−2τ)

Next Strings maybe .... :)

Wednesday, 25 June, 14

At strong coupling SA develop a non-perturbative regime in the near collinear limit That’s because flux tube mass gap becomes extremely small m One should think in terms of correlators of twist operators This fixes the collinear limit of SA at strong coupling The string expansion breaks down for extremely large values of α0 τ ∼ − log u2 ∼ e

√ λ/4

Conclusions

Wednesday, 25 June, 14

Next-to-MHV amplitudes? Full one-loop pre-factor? Higher multiplicity (heptagon, ....)? One-loop Thermodynamical-Bubble-Ansatz equations? ... and many other questions...

Outlook

Wednesday, 25 June, 14

THANK YOU!

Wednesday, 25 June, 14

BACK UP

Wednesday, 25 June, 14

Asympotically a pentagon = 5 quadrants glued together excess angle = π 2

φD

1 1

2 2

3 3 4 4 4

5 5

=

twist operator P(ψedge 2|ψ0

edge 5) = hψ0| φD |ψi

Pentagon as twist operator

Wednesday, 25 June, 14

Monodromy

θ

θ5γ θ4γ

= =

θ + 5iπ 2 θ + 4iπ 2

One can go around the pentagon with 5 mirror rotations This is one more than for a square E − → ip − → −E − → −ip − → E γ

Wednesday, 25 June, 14

Hexagon as a correlator of twist operators

5 5

4 4 3 3 3

2 2

1 1 6 6 6

=

distance =

p σ2 + τ 2

W6 = h0| φD(τ, σ)φD(0, 0) |0i

computed in O(6) sigma model

Wednesday, 25 June, 14

Hexagon beyond 2pt approximation

W6 = 1 + 1 2 Z dθ1dθ2 (2π)2 |P(0|θ1, θ2)|2e−mτ(cosh θ1+cosh θ2)+imσ(sinh θ1+sinh θ2) + . . .

Multi-particle transitions

θ1 θ2 θ3 θ4

=

+ + π1 π2 π3

multi-particle states

Understood!

integrand = Y

i<j

1 P(θi|θj)P(θj|θi) × rational

θ w1 w2 w3

Wednesday, 25 June, 14

Higher multiplicity

Wn = h0| φD(τn−4, σn−4) . . . φD(τ1, σ1) |0i

Higher-point amplitudes correspond to higher-points correlators Overall short-distance scaling is controlled by OPE

φD . . . φD | {z }

n−4

∼ m−(n−4)∆( 5

4 )+∆( n 4 )φϕ

ϕ = 2π × n − 4 4

with final excess angle This leads to the addition

Wn = e−

√ λ 2π An+ √ λ(n−4)(n−5) 48n

+o( √ λ)

Wednesday, 25 June, 14