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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

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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)

- 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:

- 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]

- 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.

k3

T

k2

k1

k4

- 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.)

- 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

- 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.

- States V1, V2 should have the same scattering ampl. w. other states in the high energy limit if (V1 – V2) a spurious state.
- Polarization PL.
- The state is no longer spurious after the replacement. Otherwise it is impossible to obtain relations among physically inequivalent particles.

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.

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Type I: [k-1 -1+ -2]0,k; k = 0.

= eL or eT

Type 2: ½[(+3kk)-1 -1+ 5k-2]0,k

= ½[5P-1P -1+ L-1L -1+ ]0,k

Decoupling of

zero norm states:

_________________________________________________

Count naïve order of E

and replace P L:

_________________________________________________

Solve the linear rel’s:

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Leading order result:

- 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.

These relations are new.

Gross and his collaborators’ computation was wrong.

s, t, u = Mandelstam variables:

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

- 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