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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 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

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?

  • Spectrum of bosonic open strings

    in string units. Creation/annih. op’s

  • massive higher spin gauge theory


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

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?

  • 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.






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

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.)

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?

  • 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 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.

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

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 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

Spurious states

What are the ratios?

These relations are new.

Gross and his collaborators’ computation was wrong.

Scattering amplitudes

s, t, u = Mandelstam variables:

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

2D String

  • W symmetry generated by discrete states

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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