Integrability in the Multi- Regge Regime

1. Integrability in the Multi-Regge Regime Amplitudes 2013, Ringberg Volker Schomerus DESY Hamburg Based on work w. Jochen Bartels, Jan Kotanski, Martin Sprenger, Andrej Kormilitzin, 1009.3938, 1207.4204 & in preparation

2. Introduction Goal: Interpolation of scattering amplitudes from weak to strong coupling cross ratios N=4 SYM: find remainder function R = R (u) From successful interpolation of anomalous dimensions → String theory in AdS can provide decisive input integrability at weak coupling not enough

3. Introduction: High Energy limit Useful to consider kinematical limits: here HE limit [↔ Sever’s talk] BFKL,BKP Weak coupl: HE limit computable ← integrability Main Message: HE limit of remainder R at a=∞ is determined by IR limit of 1D q-integrable system TBA integral eqs algebraic BA eqs e.g.

4. Main Result and Plan 1. Multi-Reggekinematics and regions Cross ratios, MRL 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=∞

5. Kinematics

6. 1.1 Kinematical invariants s1 t1 s12 ½ (n2 -3n) Mandelstam invariants s123 t2 s2 s s3 t3 t4 s4 2 → n – 2 = 5 production amplitude

7. 1.1 Kinematical invariants

8. 1.2 Kinematics: Cross Ratios ½ (n2 -5n) basic cross ratios (tiles) u31 u32 u11 u u12 u21 3(n-5) fundamental cross ratios u22 from Gram det

9. 1.3 Kinematics: Multi-Regge Limit small xij≈ si-1..sj-3 -ti<< si larger large

10. 1.4 Multi-Regge Regions 2n-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

11. Weak Coupling

12. Weak Coupling: 6-gluon 2-loop MHV 2-loop n=6 remainder function R(2)(u1,u2,u3) known [Del Duca et al.] [Goncharov et al.] Continue cross ratios along [Lipatov,Prygarin] leading log discontinuity

13. Leading log approximation LLA The (N)LLA for can be obtained from [Bartels, Lipatov,Sabio Vera] [Fadin,Lipatov] Impact factor Φ & BFKL eigenvalue ω known in (N)LLA LLA: [Bartels et al.] ([Lipatov,Prygarin]) Explicit formulas for R in (N)LLA derived to 14(9) loops using SVHP [Dixon,Duhr,Pennington] [Pennington] all loop LLA proposal

14. LLA and integrability ω(ν,n) eigenvalues of `color octet’ BFKL Hamiltonian H2 = h + h* BFKL Greens fct in s2 discontinuity ← wave fcts of 2 reggeized gluons H2 and its multi-site extension ↔BKP Hamiltonian are integrable [Lipatov] [Faddeev, Korchemsky] ↔ integrability in colorsingletcase = XXX spinchain

15. Beyond 6 gluons - LLA  paths 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

16. Strong Coupling

17. 3.1 Strong Coupling: Y-System [Alday,Gaiotto, Maldacena][Alday,Maldacena,Sever,Vieira] Scattering amplitude → Area of minimal surface  R = free energy of 1D quantum system involving 3n-15 particles [mA,CA]with integrable interaction [KAB ↔ SAB] rapidity complex masses chemical potentials A=(a,s) a=1,2,3; s = 1, …, n-5 `particle densities’ R R = R(u) = R(m(u),C(u)) by inverting  Wall crossing & cluster algebras

18. 3.2 TBA: Continution& Excitations [Dorey, Tateo] R Continue m along a curve in complex plane to m’ Solutions of = poles in integrand sign contribution from excitations Excitations created through change of parameters

19. 3.3 TBA: Low Temperature Limit = large volume L => large m = ML ; IR 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 rapiditiesθ satisfy BAEs.

20. 3.4 The Multi-Regge Regime Multi-Regge regime reached when [Bartels, VS, Sprenger]  ∞ 4D MRL = 2D IR fixed while keeping Cs and u1→ 1 u2,u3 →0 check using Casimir energy vanishes at infinite volume n=6 gluons: [Bartels,Kotanski, VS]

21. 6-gluon case system parameters solutions of Y3(θ) = -1 as function of ϕ

22. 6-gluon case (contd) solutions of Y1(θ) = -1 solutions of Y2(θ) = -1 Solution of BA equations with 4 roots θ(2) = 0, θ3 = ± iπ/4

23. n > 6 - gluons n=7 gluons: [Bartels,VS, Sprenger ] in prep. Same identities at in LLA at weak coupling

24. n = 7 gluons (contd)

25. n > 6 - gluons n=7 gluons: [Bartels,VS, Sprenger ] in prep. Same identities as in LLA at weak coupling is under investigation…. involves same number e2 ? General algorithm exists to compute remainder fct. for all regions & any number of gluons at ∞ coupling

26. Conclusions and Outlook ReggeBethe Ansatz provides qualitative and quantitative predictions for Regge-limit of amplitudes at strong coupling Multi-Regge limit is low temperature limit of TBA Two new entries in AdS/CFT dictionary: Simplifications: TBA  Bethe Ansatz natural kinematical regime Mandelstam cut contributions ↔ excit. energies Regge regime is the only known kinematic limit in which amplitudes simplify at weak and strong coupling Interpolation between weak and strong coupling ?