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

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Infinite Symmetry in the high energy limit

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Infinite symmetry in the high energy limit

Infinite Symmetryinthe high energy limit

Pei-Ming Ho 賀培銘

Physics, NTU

Mar. 2006



  • 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



  • 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

Infinite symmetry in the high energy limit

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)

Ssb in string theory

SSB in string theory?

  • Spectrum of bosonic open strings

    in string units. Creation/annih. op’s

  • massive higher spin gauge theory



Infinite symmetry in the high energy limit

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

Why high energy limit

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]

What to compute

What to compute?

  • Vertex operators:

  • 4-point functions in the center of mass frame.

  • It has 2 parameters E and f.



  • A natural basis of polarization:

Note that components of eP and eL scale like E1, eT scales like E0, and components of (eP-eL) scale like E-1.

Infinite symmetry in the high energy limit






Infinite symmetry in the high energy limit

Infinitely manylinear relations among 4-pt fx’s are obtained, and theirratios can be uniquely determined at the leading order.

What kind of relations

What kind of relations?

  • Compare 4-pt. fx’s in aFamily.

  • 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.)

1 st covariant quantization

1st 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

How to get the relations

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.

Decouple spurious states at high energies

Decouple spurious statesat high energies

  • States V1, V2 should have the same scattering ampl. w. other states in the high energy limit if (V1 – V2)  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.

M 2 2

m2 = 2

At the lowest mass levels (m2 = -2, 0), there are no more than one independent physical states.

The lowest mass level as a nontrivial example is

m2 = 2.


Type I: [k-1 -1+ -2]0,k; k = 0.

= eL or eT

Type 2: ½[(+3kk)-1 -1+ 5k-2]0,k

= ½[5P-1P -1+ L-1L -1+  ]0,k

Infinite symmetry in the high energy limit

Decoupling of

zero norm states:


Count naïve order of E

and replace P  L:


Solve the linear rel’s:


Leading order result:

Why can we derive relations this way

Why can we derive relationsthis 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.

States at the leading order

States at the leading order

Spurious states

Spurious states

What are the ratios

What are the ratios?

These relations are new.

Gross and his collaborators’ computation was wrong.

Scattering amplitudes

Scattering amplitudes

s, t, u = Mandelstam variables:

s = 4E2, t  -4E2 sin2, u  -4E2 cos2 .

2d string

2D String

  • W symmetry generated by discrete states

Infinite symmetry in the high energy limit

Zero norm states:

D(…, j) is almost the same as (…), but with the j-th row replaced by



  • 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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