Integrability in the Multi-Regge Regime Volker Schomerus DESY Hamburg Based on work w. Jochen Bartels, Jan Kotanski, Martin Sprenger, Andrej Kormilitzin, , & 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 integrability BFKL,BKP TBA integral eqs algebraic BA eqs e.g. Useful to consider kinematical limits: here HE limit [ Severs talk]

Main Result and Plan 1. 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 R n,n=6,7 at a= Cross ratios, MRL and regions

Kinematics

1.1 Kinematical invariants t1t1 t2t2 t 4 s4s4 s s 12 s n – 2 = 5 production amplitude t 3 s3s3 s2s2 s1s1 ½ (n 2 -3n) Mandelstam invariants

1.1 Kinematical invariants

1.2 Kinematics: Cross Ratios u 31 u 32 u 11 u 12 u 22 u 21 u ½ (n 2 -5n) basic cross ratios (tiles) 3(n-5) fundamental cross ratios from Gram det

1.3 Kinematics: Multi-Regge Limit -t i << s i x ij s i-1..s j-3 small large larger

1.4 Multi-Regge Regions 2 n-4 regions depending on the sign of k i0 = E i u 2σ > 0 u 3σ > 0 u 2σ < 0 u 3σ < 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 > 0

Weak Coupling

Weak Coupling: 6-gluon 2-loop [Lipatov,Prygarin] 2-loop n=6 remainder function R (2) (u 1,u 2,u 3 ) 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])

H 2 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 s 2 discontinuity wave fcts of 2 reggeized gluons [Lipatov] integrability in color singlet case = XXX spin chain H 2 = h + h *

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 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 [m A,C A ] with integrable interaction [K AB S AB ] complex masseschemical 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 Limit In 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: u 1 1 u 2,u 3 0 while keeping C s and fixed 4D MRL = 2D IR using check

6-gluon case system parameters solutions of Y 3 (θ) = -1 as function of ϕ

6-gluon case (contd) solutions of Y 1 (θ) = -1 solutions of Y 2 (θ) = -1 Solution of BA equations with 4 roots θ (2) = 0, θ 3 = ± 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 e 2 ?

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: