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, Ringberg

Introduction

Goal: Interpolation of scattering amplitudes from weak to strong coupling

N=4 SYM: find remainder function R = R (u) cross ratios

From successful interpolation of anomalous dimensions

→ String theory in AdS can provide decisive input integrability at weak coupling not enough

Introduction: High Energy limit

Main Message: HE limit of remainder R at a=∞ is

determined 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]

Main Result and Plan1. Multi-Regge kinematics and regions

2. 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 TBA

• Mandelstam cuts & excited state TBA

• Formulas for MRL of Rn ,n=6,7 at a=∞

Cross ratios, MRL and regions

Kinematics

1.1 Kinematical invariants

t1

t2

t4 s4

s

s12

s123

2 → n – 2 = 5 production amplitude

t3 s3

s2

s1

½ (n2 -3n)

Mandelstam

invariants

1.1 Kinematical invariants

1.2 Kinematics: Cross Ratios

u3

1

u3

2u1

1 u1

2 u2

2

u2

1

u½ (n2 -5n)

basic cross

ratios (tiles) 3(n-5)

fundamental

cross ratios from

Gram det

1.3 Kinematics: Multi-Regge Limit

-ti << si xij ≈ si-1..sj-3

small

large

larger

1.4 Multi-Regge Regions2n-4 regions depending on the sign of ki0 = Ei

u2σ > 0 u3σ > 0 u2σ < 0 u3σ < 0

s1 < 0 s12 > 0 s123 < 0

s4 < 0 s34 > 0 s234 < 0

s1 > 0 s12 > 0 s123 > 0

s4 > 0 s34 > 0 s234 > 0

Weak Coupling

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.]

leading log

discontinuity

Continue cross ratios along

MHV

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] all loop LLA proposal using SVHP [Pennington]

[Bartels, Lipatov,Sabio Vera]

[Fadin,Lipatov]

LLA: [Bartels et al.]

([Lipatov,Prygarin])

H2 and its multi-site extension ↔ BKP Hamiltonian

are integrable

LLA and integrability

[Faddeev, Korchemsky]

ω(ν,n) eigenvalues of `color octet’ BFKL Hamiltonian

BFKL Greens fct in s2 discontinuity

← wave fcts of 2 reggeized gluons

[Lipatov]

↔ integrability in color singlet case = XXX spin chain

H2 = h + h*

Beyond 6 gluons - LLAn=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 regions

Including one that involves the

Eigenvalues of 3-site spin chain

?

paths

Strong Coupling

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 [KAB ↔ SAB] complex masses chemical potentials

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

R

Wall crossing & cluster algebras

3.2 TBA: Continution & Excitations [Dorey, Tateo]

Continue m along a curve in complex plane to m’ R

Solutions of = poles in integrand sign

contribution from excitations

Excitations created through change of parameters

3.3 TBA: Low Temperature LimitIn limit m → ∞ the integrals can be ignored:

Bethe Ansatz equations

energy of bare excitations

In low temperature limit, all energy is carried by

bare excitations whose rapidities θ satisfy BAEs.

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

,

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

Casimir energy vanishes

at infinite volume

[Bartels,Kotanski, VS]n=6 gluons:

u1→ 1u2,u3 → 0

∞

while keeping Cs and fixed

4D MRL = 2D IR

using check

6-gluon casesystem parameters solutions of Y3(θ) = -1 as function of ϕ

6-gluon case (contd)

solutions of Y1(θ) = -1solutions of Y2(θ) = -1

Solution of BA equations with 4 roots θ(2) = 0, θ3 = ± i π/4

n > 6 - gluons

[Bartels,VS, Sprenger ]

in prep.

Same identities at in LLA at weak coupling

n=7 gluons:

n = 7 gluons (contd)

n > 6 - gluons

[Bartels,VS, Sprenger ]

in prep.

Same identities as in LLA at weak coupling

n=7 gluons:

is under investigation….

General algorithm exists to compute remainder fct.

for all regions & any number of gluons at ∞ coupling

involves same number e2 ?

Conclusions and Outlook

Multi-Regge limit is low temperature limit of TBA natural kinematical regime Simplifications: TBA Bethe Ansatz

Mandelstam 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: