Infinite Symmetryin

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• 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 limitWeak 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

,2,1,0),1(22 nnM

] , [ 0nmnm m

kkBkAkkd ,0)()()( 11126

Spectrum0.5 1 1.5 2 2.5 3

-1

-0.5

0.5

1

0.5 1 1.5 2 2.5 3

-1

-0.5

0.5

1

0.5 1 1.5 2 2.5 3

-1

-0.5

0.5

1

-1-0.500.51

-1-0.500.51

0

10

20

30

k

kDkD

kCkCkC

kBkBkAk

kd ,0

)()(

)()()(

)()()()(

2111111

321111

2111

26

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

pmn

pmn kkxA

111

111 2211)(

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 hid

den 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 .

xiknm eXXkAkAV )();(

4321],[

4321 VVVVeDXDgVVVV gXS

Polarizations

• A natural basis of polarization:

xiknm eXXkAkAV )();(

xikAmAmBA eXXkV )(

)0,,0,0,0(

)0,,1,0,0(

/)0,,0,,(

/)0,,0,,(

iT

T

L

P

e

e

mEpe

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

k1k2

k3

k4

T

Infinitely many linear relations among 4-pt fx’s are obtained, and their ratios can be uniquely determined at the leading order.

What kind of relations?• Compare 4-pt. fx’s in a Family.

• 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.)

1143214321 :,,; MVmassVVVVVVVMFamily

1st covariant quantization

• Hilbert space: creation op’s -n acting on the vacuum. (-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

0,0 nL non

0, nL n

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 [ (+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:

_________________________________________________

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

kqmn

q

LL

m

LL

qmn

TT ,0,, 2211

2

11

xikqLmLqmnTqmnqmn eXXXNV 22,,,,

)!1()!1()!22(

1),,(

qmqmnN qmn

Spurious states

121121

1 P

nnn mL

Ζ

P

nn mL 2112

121

2 Z

n-2

kqmn

q

LL

m

LL

qmn

TT ,0,, 2211

2

11

What are the ratios?

oddmT

evenmTmM

T

qmn

nqmqm

qmn

,0

,!!12

11

),,(

)0,0,(2/

),,(

)()()()( 4433221),,(),,( kVkVkVkVT qmnqmn

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 .

)0,0,0(2/

),,,()0,0,( Tu

stTT

nTTTn

)logloglog(2/34)0,0,0( )(2 uuttssestueT

2D String

• W symmetry generated by discrete states

)0()1()0(2exp)2,,( JiMXXiMJJM

))(1(2112 21212211)(

2 MMJJMJMJ MJMJi

dz

01

!2

)12(

exp

)0(

),,1,()()2,,(

k

kik

k

kk

kki

kk

jiJ

xaSxa

XSS

MJjiSDetXiMJ

)0()()1()()1(2exp

)),(2,,()!1()(

)1(2exp)2,,()!(

12

2

XzJzXMi

jzXiMJDMJ

JiMXXiMJMJGMJ

ji

dzJ

JM

Zero norm states:

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

2)(,,2)(,21)( MJzJzJz jjj

Remarks• 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 obt

ained 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