1. Cambridge, Dec 10th, 2007 Thomas Klose Princeton Center for Theoretical Physics based on work with Valentina Giangreco Puletti and Olof Ohlson Sax: hep-th/0707.2082 also thanks to T. McLoughlin, J. Minahan, R. Roiban, K. Zarembo for further collaborations Factorized World-Sheet Scattering in near-flat AdS 5 xS 5 ==

2. Talk overview ► Factorization of three-particle world-sheet S-matrix in near-flat AdS 5 x S 5 to one loop in string σ-model ► Integrability Conserved charges Factorized scattering ► The 3-particle S-matrix has (16 3 ) 2 compoments, but only 4 independent ones ! ► Mechanism of factorization from Feynman graphs

3. ► Simple Fock spectrum in flat space World-sheet scattering

4. ? ? ? ? ► Spectrum unknown in curved space World-sheet scattering

5. in AdS 5 x S 5 ► Spectrum from Bethe equations World-sheet scattering

6. Integrability of AdS string theory [Mandal, Suryanarayana, Wadia ‘02] [Bena, Polchinski, Roiban ‘03] ► Classical Integrability ► Quantum Integrability  Quantum consistency of monodromy matrix  Absence of particle production in bosonic sector in semiclassical limit  Quantum consistency of AdS strings, and existence of higher charges in pure spinor formulation [Berkovits ‘05] [Arutyunov, Frolov, Staudacher ‘04] [TK, McLoughlin, Roiban, Zarembo ‘06] [Mikhailov, Schäfer-Nameki ‘07]  Bethe equations for quantum strings  Trace of monodromy matrix is conserved and generates higher charges [Callan, McLoughlin, Swanson ‘04] [Hentschel, Plefka, Sundin ‘07]  Check energies of multi-excitation states against Bethe equations (at tree-level)  Algebraic curve describing classical strings [Kazakov, Marshakov, Minahan, Zarembo ‘04]

7. Integrability in 1+1d QFTs  No particle production or annihilation  Conservation of the set of momenta  -particle S-Matrix factorizes into 2-particle S-Matrices Existence of local higher rank conserved charges [Zamolodchikov, Zamolodchikov ‘79] [Shankar, Witten ‘78] [Parke ‘80] (in Lorentz invarint theory) “Two mutually commuting local charges of other rank than scalar and tensor are sufficient for S-matrix factorization !” [Parke ‘80]

8. LC gauge Sigma model on [Metsaev, Tseytlin ‘98] [Frolov, Plefka, Zamaklar ‘06] Superstrings on AdS 5 xS 5 Two-particle S-Matrix Group factorization [Beisert ‘06] Central extension [Arutyunov, Frolov, Plefka, Zamaklar ‘06] Note: multi-particle factorization

9. Symmetry constraints on the S-Matrix of centrally extended algreba relate the two irreps of fixed up to one function ► 2-particle S-Matrix: for one S-Matrix factor: fixed up to four functions ► 3-particle S-matrix: [Beisert ‘06]

10. 3-particle S-matrix ► Eigenstates Extract coefficient functions from: ► Cross check using mixed processes, e.g.

11. Near-flat-space limit Non-Lorentz invariant interactions Decoupling of right-movers ! ! UV-finiteness quantum mechanically consistent reduction at least to two-loops ! ! Decompactification limit built in ! Coupling strength dependent on particle momenta ! near-flat-space [Maldacena, Swanson ‘06] giant magnons plane-wave [Hofman, Maldacena ‘06] [Berenstein, Maldacena, Nastase ‘02] Highly interacting Only quartic interactions ! Free massive theory

12. S-Matrix from Feynman diagrams ► 2-particle S-matrix ► 4-point amplitude for compare

13. S-Matrix from Feynman diagrams ► 3-particle S-matrix ► 6-point amplitude !? First non-triviality

14. Factorization !? Second non-triviality YBE

15. Emergence of factorization ► Tree-level diagrams Divergences from internal propagators going on-shell ► One-loop diagrams ► Phase space ► Typical partial amplitude

16. [Källén, Toll ‘64] Cutting rule in 2d for arbitrary 1-loop diagrams Here: Emergence of factorization

17. [TK, McLoughlin, Minahan, Zarembo ‘07] disconnected pieces probe the 2-particle S-Matrix Looks like, but cannot be identified separately!

18. Summary and open questions 1-loop computation of  the highest-weight amplitudes,  amplitude of mixed processes Finite size corrections Proven the factorization of the 3-particle world-sheet S-Matrix to 1-loop in near-flat AdS 5 xS 5 ! ? Asymptotic states? ! ? Extenstions of the above: higher loops, more particles, full theory fixes 3-particle S-matrix checks supersymmetries Direct check of quantum integrability of AdS string theory (albeit in the NFS limit) ! effectively