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Infinite Symmetry in the high energy limit Pei-Ming Ho 賀培銘 Physics, NTU Mar. 2006

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Collaborators Chuan-Tsung Chan (NCTS) 詹傳宗 Jen-Chi Lee (NCTU) 李仁吉 Shunsuke Teraguchi (NCTS/TPE) 寺口俊介 Yi Yang (NCTU) 楊毅

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References Ward identities and high-energy scattering amplitudes in string theory, Chan, Ho, Lee [hep-th/0410194] Nucl. Phys. B Solving all 4-point correlation functions for bosonic open string theory in the high energy limit, Chan, Ho, Lee, Teraguchi, Yang [hep-th/0504138] Nucl. Phys. B High-energy zero-norm states and symmetries of string theory, Chan, Ho, Lee, Teraguchi, Yang [hep-th/0505035] Phys. Rev. Lett. Comments on the high energy limit of bosonic open string theory, Chan, Ho, Lee, Teraguchi, Yang [hep- th/0509009] submitted to Nucl. Phys. B High energy scattering amplitudes of superstring theory, Chan, Lee, Yang [hep-th/0510247] Nucl. Phys. B

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To understand various aspects of a theory, we take various limits: Weak coupling limit strong coupling limit Weak field limit (strong field limit?) Low energy limit High energy limit ________________________________________ High energy limit: ( ) Yang-Mills theory Gross, Wilczek (1973); Politzer (1973) Closed string theory Gross, Mende (1987,88); Gross (1988,89) Open string theory Gross, Manes (1989)

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SSB in string theory? Spectrum of bosonic open strings in string units. Creation/annih. op’s massive higher spin gauge theory

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Spectrum

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A most generic spacetime field in the bosonic open string field theory is of the form:

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Why high energy limit? By high energy limit we mean we focus our attention on the leading order terms in the 1/E expansion. Theory is simplified in its high energy limit. Recall spontaneous symmetry breaking. We want to find the (legendary) huge hidden symmetry in string theory. [Gross, Mende, Manes]

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What to compute? Vertex operators: 4-point functions in the center of mass frame. It has 2 parameters E and f.

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Polarizations A natural basis of polarization: Note that components of e P and e L scale like E 1, e T scales like E 0, and components of (e P -e L ) scale like E -1.

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k1k1 k2k2 k3k3 k4k4 T

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Infinitely many linear relations among 4-pt fx’s are obtained, and their ratios can be uniquely determined at the leading order.

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What kind of relations? Compare 4-pt. fx’s in a Family. Focus on leading order terms in a Family. i.e., ignore 4-pt. fx’s subleading to a sibling. Do not try to mix families. ( Families with larger M dominate.)

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1 st covariant quantization Hilbert space: creation op’s a -n acting on the vacuum. (a -n are the annihilation op’s.) Virasoro constraint: physical states Spurious states are created by L -n and so they are (decoupled from) physical states. Physical spurious states are zero norm states, corresponding to gauge transformations

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How to get the relations? 1. Decouple spurious states OR 1’. Impose Virasoro constraints. 2. Count naïve dimension of a 4-pt. fx. (how it scales with E when E ) 3. Assumption: If the naïve dim. of a 4-pt. fx. is smaller than the leading naïve dim. (n) of the one with the highest spin, then it is subleading to it.

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Decouple spurious states at high energies States V 1, V 2 should have the same scattering ampl. w. other states in the high energy limit if ( V 1 – V 2 ) a spurious state. Polarization P L. The state is no longer spurious after the replacement. Otherwise it is impossible to obtain relations among physically inequivalent particles.

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m 2 = 2 At the lowest mass levels (m 2 = -2, 0), there are no more than one independent physical states. The lowest mass level as a nontrivial example is m 2 = 2. _________________________________________ Type I: [ k -1 -1 + -2 ] 0,k ; k = 0. = e L or e T Type 2: ½ [ ( +3k k ) -1 -1 + 5k - 2 ] 0,k = ½ [ 5 P -1 P -1 + L -1 L -1 + ] 0,k

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Decoupling of zero norm states: _________________________________________________ Count naïve order of E and replace P L: _________________________________________________ Solve the linear rel’s: _________________________________________________ Leading order result:

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Why can we derive relations this way? Consistency conditions for overlapping gauge transformations in a “smooth” high energy limit. A generic field theory (e.g. a naive massive vector/tensor field theory) [Fronsdal] does not have a smooth high energy limit.

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States at the leading order

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Spurious states

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What are the ratios? These relations are new. Gross and his collaborators’ computation was wrong.

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Scattering amplitudes s, t, u = Mandelstam variables: s = 4E 2, t -4E 2 sin 2 , u -4E 2 cos 2 .

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2D String W symmetry generated by discrete states

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Zero norm states: D(…, j) is almost the same as (…), but with the j-th row replaced by

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Remarks We can do similar things for n-pt. fx’s. But the relations will be incomplete. Ratios of 4pt. fx’s for superstring are also obtained this way. [Chan, Lee, Yang] Can all symmetries/linear relations be obtained from decoupling spurious states? Linear relations for subleading corr. fx’s? Linear relations at higher loops? We still do not know what the hidden symmetry is. Orz

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