Symmetry of string theory in the high energy limit via zero norm states ____________________________

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Symmetry of string theoryin the high energy
limitvia zero norm states_______________________
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• Chan, Lee hep-th/0312226
• Chan, Lee hep-th/0401133
• Chan, Lee, Ho hep-th/0410194
• Pei-Ming Ho
• National Taiwan University
• Nov. 2004

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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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• Linear relations among all scattering ampls
• ?
• All ampls expressed in terms of one ampl. (say,
  the tachyon ampl.), which may be determined by
  unitarity.
• ?
• Huge symmetry of string theory
• ?
• Spontaneously broken at low energies
• (cf SM at high energies.)

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• We consider bosonic open string theory.
• A salient feature is the big gauge symmetry,
  which corresponds to zero norm states in the
  covariant 1st quantized formulation.
• ___________________________________
• In Wittens string field theory,
• ?? ? Q? ?, ?.

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4 point fx.
? Euler number g closed string coupling
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• What is a zero norm state?
• In the covariant first quantized string theory
• Hilbert space ? ? ? ???? ??-m???-n?0, k?
• If
• (Ln ?n0 ) ? ? ? 0, ? n ? 0
• then
• ? ? ? physical state.
• If a physical state ? ? ? ?
• ? ? ? ? ? 0 , ? physical states ? ?gt
• then
• ? ? ? zero norm state

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• m2 0 as an example
• Physical states ? ? ? ????-1?0, k?
• Virasoro constraints
• (Ln ?n0 ) ? ? ? 0, ? n ? 0
• ? k? k? 0, ?? k? 0
• Zero norm states Physical states with
• ?? ak?
• Since ? ? ? ? ? 0 ? phys. states ? ?gt
• ? ? ? ? ? ? ? ? ? ?
• ? ?? ? ?? ak?

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Zero norm states
Type I
Type II
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• zero norm state ? gauge transformation
• ? ? ? ? ? ?? ? ? ? ? ? ? ?
• ________________________________________
• 2 states differ by a zero norm state
• 2 states related by a gauge transformation
• ? Zero norm states must decouple
• ?V1(k1) V2ZNS(k2) V3(k3) V4(k4)? 0
• Can we use this condition to derive linear
  relations among correl. fxs?

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• In terms of a basis ?Va? of the Hilbert space,
• V1ZNS(k1) ?a ca Va(k1)
• ?
• ?a ca ?Va(k1) V2a(k2) V3(k3) V4(k4)? 0
• However, physically different states
• (i.e. different particles)
• are not related by zero norm states.
• ?
• It is impossible to derive nontrivial relations
  among particles this way. (?)

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• Gauge symmetry ? Ward identities
• quantum version of charge conservation
• ________________________________________
• Example of a U(1) gauge field
• If we know how A couples to other fields
• SI ? A? J? d4x,
• we can use the gauge symmetry
• A? ? A? ???
• to derive the Ward identity
• ?? J? (?) ? 0.
• But the coupling is precisely the correl. fx.
• one needs to compute in string theory.

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• Remarkably, in the high energy limit,
• zero norm states do lead to nontrivial
• linear relations among scattering ampls.
• Key point
• We have spontaneous symmetry breaking (most gauge
  fields are massive).
• Assumption
• In the high energy limit, there is a consistent
  theory with massless gauge fields.
• ? constraints

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• Field theory analogue
• Smooth massless limit Fronsdal
  (1980)
• Massive gauge field with the Lagrangian
• has the wave eq.
• which is not smooth in the limit m2 ? 0.

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• Prescription (for 4-pt. fxs)
• Decouple all zero norm states in corr. fxs.
• T ?V1(k1) V2ZNS(k2) V3(k3) V4(k4)? 0
• Take high energy limit (fixed angle), assume
  that
• Solve the linear rels at the leading order in E.
• The 2nd step assumes a smooth massless limit.

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Take polarizations in the basis
Assign a naïve order of energy to every
quantity eP eL E eT E0 ?nx? ??-n k? E
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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?

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• Decoupling of
• zero norm states
• _________________________________________________
• Count naïve order of E
• and replace P ? L
• _________________________________________________
• Solve the linear rels
• _________________________________________________
• Leading order result

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The same can be done for m2 4.The decoupling
of zero norm states imply
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Keep terms in the highest naïve leading order
? Keep terms in the next order ?
These linear relations are uniquely solved by
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Conjectures

• The same for any given mass level?
• Linear relations fix the ratio uniquely among all
  ampls at the leading order in energy.
• All ampls at the leading order are included.
• All particles are included.
• These are explicitly verified to be true for
• m2 2, 4, 6.

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• General result for 4-pt. fxs

All other ampls in the leading order only differ
from this one by an overall numerical factor.
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• Another general result for 4 pt. fxs
• This can be obtained by saddle point calculation.
• (Perhaps it can also be obtained using SFT.)
• The contribution of the 2nd vertex, for example,
  is given by

_______________________________________ This
formula fails when the polarization of the factor
with la 1 is eL, i.e. there is (eL? ??-1).
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• The 1st relation includes all ampls at the
  leading order (true or naïve) for a given mass
  level.
• Subleading ampls are ignored (its naïve leading
  order may or may not vanish).
• ___________________________________
• The 2nd relation gives all ampls at the naïve
  leading order.
• When the naïve leading order vanishes, the
  amplitude is ignored.

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• Comparison with Gross Manes
• Saddle point approx. at the leading order
• Their claim In the high energy limit (?? ? ?),
• the path integral is dominated by the WS action,
• ? the saddle point is universal in the leading
  order.

The only modulus is the cross ratio at the
leading order.
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? 4 Tachyon ampl. correctly reproduced
The only modification for other ampls is the
additional factors given by the saddle pt.
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• However, their result do not agree with direct
  computation.
• Their mistake
• Subleading terms in the WS action give powers of
  E, which is needed to establish linear relations.
• As a result they did not really establish any
  linear relation among ampls.

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• Remarks
• The decoupling of zero norm states should also
  hold for diagrams with loops.
• At every mass level all leading ampls differ
  only by numerical factors.
• Questions
• What can we say about the symmetry of string
  theory?
• What is the high energy string field theory which
  can reproduce all the 4-pt fxs?