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

Infinite Symmetryinthe high energy limit

Pei-Ming Ho 賀培銘

Physics, NTU

Mar. 2006


Collaborators
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
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


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




Why high energy limit
Why high energy limit? 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]


What to compute
What to compute? field theory is of the form:

  • Vertex operators:

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

  • It has 2 parameters E and f.


Polarizations
Polarizations field theory is of the form:

  • 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.


k field theory is of the form:3

T

k2

k1

k4


Infinitely many field theory is of the form:linear 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? field theory is of the form:

  • 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
1 field theory is of the form: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


How to get the relations
How to get the relations? field theory is of the form:

  • 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 states field theory is of the form:at 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
m field theory is of the form:2 = 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 field theory is of the form:

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 relations field theory is of the form: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.


States at the leading order
States at the leading order field theory is of the form:


Spurious states
Spurious states field theory is of the form:


What are the ratios
What are the ratios? field theory is of the form:

These relations are new.

Gross and his collaborators’ computation was wrong.


Scattering amplitudes
Scattering amplitudes field theory is of the form:

s, t, u = Mandelstam variables:

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


2d string
2D String field theory is of the form:

  • W symmetry generated by discrete states


Zero norm states: field theory is of the form:

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


Remarks
Remarks field theory is of the form:

  • 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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