Integrability - PowerPoint Presentation

Slide1

Integrability

in the

Multi-Regge Regime

Volker SchomerusDESY Hamburg

Based on work w. Jochen Bartels, Jan Kotanski , Martin Sprenger, Andrej Kormilitzin, 1009.3938, 1207.4204 & in preparation

Amplitudes 2013,

RingbergSlide2

Introduction

Goal: Interpolation of scattering amplitudes from weak to strong coupling

N=4 SYM: find remainder function R = R (u) cross ratiosFrom successful interpolation of anomalous dimensions→ String theory in

AdS can provide decisive input integrability

at weak coupling not enoughSlide3

Introduction: High Energy limit

Main Message

: HE limit of remainder R at a=∞ isdetermined by IR limit of 1D q-integrable system Weak coupl: HE limit computable ← integrabilityBFKL,BKP

TBA integral eqs

 algebraic BA eqse.g.

Useful to consider kinematical limits: here HE limit

[↔ Sever’s talk]Slide4

Main Result and Plan

1. Multi-Regge

kinematics and regions2. Multi-Regge limit at weak coupling (N)LLA and (BFKL) integrability, n=6,7,8…

3. Multi-Regge limit at strong coupling

MRL as low temperature limit of TBAMandelstam cuts & excited state TBAFormulas for MRL of Rn ,n=6,7 at a=∞

Cross ratios, MRL and regionsSlide5

Kinematics Slide6

1.1 Kinematical invariants

t

1t2

t4

s4 s

s12s

1232 → n – 2 = 5 production amplitude

t

3

s

3

s

2

s

1

½ (n

2

-3n)

Mandelstam

i

nvariants Slide7

1.1 Kinematical invariantsSlide8

1.2 Kinematics: Cross Ratios

u

31

u32

u11u12

u22

u2

1

u

½ (n

2

-5n)

b

asic cross

r

atios (tiles)

3(n-5)

fundamental

cross ratios

f

rom

Gram

detSlide9

1.3 Kinematics: Multi-Regge

Limit

-

t

i

<< si

xij ≈ si-1

..s

j-3

small

large

largerSlide10

1.4 Multi-Regge

Regions

2n-4 regions depending on the sign of ki0 = Ei

u

2σ > 0 u3σ

> 0 u

2σ < 0 u3σ

< 0

s

1

< 0

s

12

>

0

s

123

< 0

s

4

< 0

s

34

>

0

s

234

< 0

s

1

>

0

s

12

>

0

s

123

> 0

s

4

>

0

s

34

>

0

s

234

> 0Slide11

Weak Coupling Slide12

Weak Coupling: 6-gluon 2-loop

[Lipatov,Prygarin]

2-loop n=6 remainder function R(2)(u1,u2,u3) known [Del Duca et al.] [

Goncharov et al.]

l

eading log

discontinuity

Continue cross ratios along

MHVSlide13

Leading log approximation LLA

The (N)LLA for can be obtained from

Impact factor

Φ

& BFKL eigenvalue

ω

known in (N)LLA

Explicit formulas for R in (N)LLA derived to 14(9) loops

[

Dixon,Duhr,Pennington

]

a

ll loop LLA proposal

using SVHP

[Pennington]

[Bartels,

Lipatov,Sabio

Vera]

[

Fadin,Lipatov

]

LLA: [Bartels et al.]

([

Lipatov,Prygarin

])Slide14

H2 and its multi-site extension

↔ BKP Hamiltonianare

integrable LLA and integrability[Faddeev

, Korchemsky]ω(ν,n)

eigenvalues of `color octet’ BFKL Hamiltonian BFKL Greens fct in s

2 discontinuity ← wave fcts of 2

reggeized gluons

[Lipatov]

↔

integrability

in

color

singlet

case

= XXX

spin

chain

H

2

= h + h

*Slide15

Beyond 6 gluons - LLA

n

=7: Four interesting regions(N)LLA remainder involves the same BFKL ω(

ν,n) as for n = 6

[Bartels,

Kormilitzin,Lipatov,Prygarin]

n=8: Eleven interesting regionsIncluding one that involves the Eigenvalues of 3-site spin chain

?

 pathsSlide16

Strong CouplingSlide17

3.1 Strong Coupling: Y-System

Scattering amplitude → Area of minimal surface 

[

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

A=(a,s) a=1,2,3; s = 1, …, n-5

`particle densities’ rapidity

R = free energy of 1D quantum system involving 3n-15 particles [

mA,CA]

with

integrable

interaction

[K

AB

↔ S

AB

]

c

omplex masses

c

hemical potentials

R = R(u) = R(m(u),C(u)) by inverting

R

 Wall crossing & cluster algebras Slide18

3.2 TBA:

Continution

& Excitations

[

Dorey, Tateo]

Continue m along a curve in complex plane to m’

RSolutions of = poles in integrand

sign

c

ontribution from excitations

Excitations created through change of parameters Slide19

3.3 TBA: Low

T

emperature Limit

In limit m → ∞ the integrals can be ignored: Bethe

Ansatz equations energy of bare excitationsIn low temperature limit, all energy is carried by bare excitations whose

rapidities θ satisfy BAEs.

= large volume L => large m = ML ; IR limit

,Slide20

3.4 The Multi-Regge

Regime

[Bartels, VS, Sprenger] Multi-Regge regime reached when

Casimir

energy vanishesat infinite volume[

Bartels,Kotanski, VS]n

=6 gluons:

u

1

→ 1

u

2

,u

3

→

0

 ∞

w

hile keeping C

s

and

f

ixed

4D MRL = 2D IR

using

checkSlide21

6-gluon case

s

ystem parameters solutions of Y3(θ) = -1 as function of

ϕSlide22

6-gluon case (contd

)

solutions of Y

1(θ) = -1

solutions of Y2(θ) = -1 Solution of BA equations with 4 roots θ(2) = 0, θ3 = ± i

π/4 Slide23

n > 6 - gluons

[

Bartels,VS

,

Sprenger ] in prep.

Same identities at in LLA at weak couplingn=7 gluons: Slide24

n = 7 gluons (contd

)Slide25

n > 6 - gluons

[

Bartels,VS

, Sprenger

] in prep.Same identities as in LLA at weak couplingn=7 gluons:

i

s under investigation….

General algorithm exists to compute remainder

fct

.

for all regions & any number of gluons at ∞ coupling

i

nvolves same

n

umber e

2

?Slide26

Conclusions and Outlook

Multi-Regge limit is low temperature limit of TBA

natural kinematical regime Simplifications: TBA  Bethe AnsatzMandelstam cut contributions ↔ excit. energies

Regge regime is the only known kinematic limit in which amplitudes simplify at weak

and strong coupling Regge Bethe Ansatz provides qualitative and quantitative predictions for Regge-limit of amplitudes at strong coupling

Interpolation between weak and strong coupling ?

Two new entries in AdS/CFT dictionary: