= = Cambridge, Dec 10th, 2007 Factorized World-Sheet Scatteringin near-flat AdS5xS5 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
Talk overview ► Factorization of three-particle world-sheet S-matrix in near-flat AdS5 x S5 to one loop in string σ-model ► Integrability Conserved charges Factorized scattering ► The 3-particle S-matrix has (163)2 compoments, but only 4 independent ones ! ► Mechanism of factorization from Feynman graphs
in flat space ►Simple Fock spectrum World-sheet scattering
in curved space ? ? ? ? ►Spectrum unknown World-sheet scattering
in AdS5 x S5 ►Spectrum from Bethe equations World-sheet scattering
Integrability of AdS string theory ► Classical Integrability • Trace of monodromy matrix is conserved and generates higher charges [Mandal, Suryanarayana, Wadia ‘02] [Bena, Polchinski, Roiban ‘03] • Algebraic curve describing classical strings [Kazakov, Marshakov, Minahan, Zarembo ‘04] ► Quantum Integrability • Bethe equations for quantum strings [Arutyunov, Frolov, Staudacher ‘04] • Check energies of multi-excitation states against Bethe equations (at tree-level) [Callan, McLoughlin, Swanson ‘04] [Hentschel, Plefka, Sundin ‘07] • Quantum consistency of AdS strings, and existence of higher charges in pure spinor formulation [Berkovits ‘05] Absence of particle production in bosonic sector in semiclassical limit [TK, McLoughlin, Roiban, Zarembo ‘06] Quantum consistency of monodromy matrix [Mikhailov, Schäfer-Nameki ‘07]
Integrability in 1+1d QFTs Existence of local higher rank conserved charges (in Lorentz invarint theory) [Shankar, Witten ‘78] [Zamolodchikov, Zamolodchikov ‘79] [Parke ‘80] No particle production or annihilation Conservation of the set of momenta -particle S-Matrix factorizes into 2-particle S-Matrices “Two mutually commuting local charges of other rank than scalar and tensor are sufficient for S-matrix factorization !” [Parke ‘80]
Superstrings on AdS5xS5 Sigma model on [Metsaev, Tseytlin ‘98] LC gauge [Frolov, Plefka, Zamaklar ‘06] Central extension [Beisert ‘06] [Arutyunov, Frolov, Plefka, Zamaklar ‘06] Two-particle S-Matrix Note: multi-particle factorization Group factorization
Symmetry constraints on the S-Matrix [Beisert ‘06] ► 2-particle S-Matrix: for one S-Matrix factor: of centrally extended algreba relate the two irreps of fixed up to one function ► 3-particle S-matrix: fixed up to four functions
3-particle S-matrix ► Eigenstates Extract coefficient functions from: ► Cross check using mixed processes, e.g.
Near-flat-space limit Decompactification limit built in ! Highly interacting Non-Lorentz invariant interactions ! giant magnons [Hofman, Maldacena ‘06] Coupling strength dependent on particle momenta ! Only quartic interactions ! near-flat-space Decoupling of right-movers ! [Maldacena, Swanson ‘06] ! UV-finiteness plane-wave [Berenstein, Maldacena, Nastase ‘02] ! quantum mechanically consistent reduction at least to two-loops Free massive theory
S-Matrix from Feynman diagrams ► 2-particle S-matrix ► 4-point amplitude compare for
S-Matrix from Feynman diagrams ► 3-particle S-matrix ► 6-point amplitude First non-triviality !?
Factorization Second non-triviality !? Factorization YBE
Emergence of factorization ► Tree-level diagrams ►One-loop diagrams ►Typical partial amplitude ►Phase space Divergences from internal propagators going on-shell
Emergence of factorization Cutting rule in 2d for arbitrary 1-loop diagrams [Källén, Toll ‘64] Here:
Emergence of factorization disconnected pieces probe the 2-particle S-Matrix [TK, McLoughlin, Minahan, Zarembo ‘07] Looks like, but cannot be identified separately!
Summary and open questions ! Proven the factorization of the 3-particle world-sheet S-Matrix to 1-loop in near-flat AdS5xS5 effectively fixes 3-particle S-matrix 1-loop computation of the highest-weight amplitudes, amplitude of mixed processes ! checks supersymmetries Direct check of quantum integrability of AdS string theory (albeit in the NFS limit) ! Extenstions of the above: higher loops, more particles, full theory ? ? Finite size corrections Asymptotic states?