Zero-norm States and High-energy Scattering Amplitudes of

Superstring Theory

Jen-Chi Lee

Dept. of Electrophysics, National Chiao-Tung Univ. Hsin-Chu, Taiwan

Outline

1. Introduction & Overview

2. A simple example

3. Three calculations of string symmetry

(a) High-energy zero-norm states (HZNS)

(b) Virasoro constraints

(c) Saddle-point Method

4. Compares with Gross’s

5. (a) 2D string (b) Superstring

(c) Closed string

6. Conclusion

1. Introduction & Overview

• Q.F.T : “Symmetry dictates interaction”

e.g : Y. M. , G. R.

• String: Interaction Symmetry ?

Prescribed by self-consistent conditions of quantum string

e.g : 1-loop modular invariance of 10D Heterotic string.

massless Y.M.2E8 SO(32), aθ

...,(ab)θ(ab)θ [μν]μ

e.g :

Massive symmetries

……………?

How to identify string symmetry?

Lee, (1994)

1st Key

• Q.F.T :

• String:

hidden at low-energy

evident at high-energy!!

suggested by

• UV finiteness of quantum string!

• states with No free parameter.

: ' or E

fixedCM. ( Gross 1988)

PRL

•R. G., Asymptotic freedom, O.P.E. etc.•Spontaneous broken Symmetries

• High-energy limit. Saddle-point Approx.

• Infinite symmetry are

Huge symmetry Group!?

Conjectures

1. Existence of linear relations among High-energy Scattering Amplitudes (HSA) of different string states.　　

2. All HSA can be expressed in terms of that of tachyons.

However

1. Origin of symmetry charges were not understood.

2. Proportionality constants among HSA of different string states were Not calculated.

~4321 VVVV

( Gross 1988 P.R.L.)

4321 TTTT...

Tachyon

2nd Key

Zero-norm states (ZNS) in the old covariant first quantized (OCFR) spectrum

(Lee 1990)

There are two types of ZNS:

　0x|L 0,x|Lx|L , x|L 0211-

0,x|Lx|L , x|)L23

(L~

2

~

1

~21-2-

where

where

D=26 only

Type I: 　

Type II:

0x|1)(L~

0

2. A simple example (Chan & Lee 2003)

Combine 1st key and 2nd key

)'( (ZNS)

2nd key : Decoupling of ZNS

Ward Identity, 0VVVZNSV

(Lee 1994)M2=4

1st key : Taking high energy limit

fixed

E

CM

4M 2

CM2Te

3Te

1Te

2k 3k-

1k4k-

4Te

Index of the 2nd vertex

Can be measured!?

For Tachyon, tensor

(-1):(-1):1:8[LT] TTTT ::: (LT)LLTTTT

(3)TT CM39

TTT θsin-8E(tree)

1,2,3v 2v

E as ee LP

,0)2k,(EM1

e 22

P

(0,0,1)eT

,0)E, 22k (M1

e2

L

Sample calculation

(1)

!1)!-(2m21

M1

-VVVV i

qm4

1i

q2m

4321

iiii

......T...T...TTnnnn

4321

4321T

(valids to all loops order!)

,k0,|)(α)(α)(αqn,2m,|V qL2-

2mL1-

2q-2m-nT1-

(2)

Algebraically!!

1).-2(nM 2

Generalized to higher mass level

(tree only)(3)

)n]sinθ[-2E in

CM3......T...T...TT

nnnni

4321

4321T(T

with

2t)t)ln(s(s-tlntslns

-exp

2θ

cos2

θsinE2(-)π(n)

2n-5

CM

-3

CM2n-1-n-1-nT

is the only HSA at level )n ,n ,n ,(n4321

3. Three calculations of string symmetry

a. High-energy zero-norm state

(1),

1q2,-n,2m|1)-(2mqn,2m,|Mq1,-1,2m-n|L1-

(2)

(1)

(2)

m)q(n,0,q)(n,2m,

M1

-M3

-...M

1-2m-

TT

1qn,0,|Mqn,0,|21

q2,0,-n|L 2-

(n,0,0)TTq

q)(n,0,

2M1

-

(n,0,0)TT !1)!-(2m21

M1

-qmq2m

q)(n,2m,

(b) Virasoro constraints

“dual”0ψ|L1

Virasoro constraints are

Normalization factor & symmetry factors

Type I

'

E

Type II

By Type I

0ψ|L2

0x|L1-

0x|) (L~

2- 1-2L

2

3

0Ex. 4M 2

(c) Saddle-point Method

ee LP (s-t channel)

Saddle point

!1)!-(2m21

M1

-qmq2m

...T

n,0,00,

2

Tn

2q-2m-n3

T0

2q2mn-0

q2m

210

(2m) )k)(-e''(f(2m)!x-1Mkk

)(xu

Gross conjectures (1988)

are explicitly proved !

4. Compares with Gross’sEx.

Ex.

Note!

(1) s are missing (=0) in Gross & Manes (N.B.1989)

inconsistent with the decoupling of ZNS

(or Ward identities)

Violates unitarity!!

(2) A corrected saddle-point calculation was given in Chan, Ho & Lee (N.B.2005)

4M 2

6M 2

32

:96

-:964

-:3

62-:

31

:34

:16

(LL)(LLL)(TTL)LTT,(LLLL)(TTLL)(TTTT) ::::: TTTTTTT :

(-1):(-1):1:8[LT] TTTT ::: (LT)LLTTTT

One can construct a set of ZNS with discrete Polyakov momenta such that

• 2D discrete ZNS carry symmetry charges.

Space time symmetry algebra of 2D string was known to be

J,mG

a algebra!

}{Q mJ,

..(0))ψmJ-m(Jz2

..(0)(2)ψψ21212211 mm 1,-JJ1221mJmJ

( Chung ＆ Lee 1994)

( Klebanov ＆ Polyakov 1991)

( Witten 1992)

( Chung ＆ Lee 1994)

(0))GmJ-m(J(0)(z)GG2πidz

21212211 mm 1,-JJ2112,mJ,mJ

“Ground ring”

21212211 m,mJJ,mJ,mJ QQQ

.

.

.

5 (a) 2D string

• NS-sector, GSO even, polarization on the scattering plane

• GSO odd

5 (b) Superstring

L3/2-

T1/2- b,b|qn,2m,|

All are proportional to each other at each fixed mass level.

L3/2-

L1/2- b,b|q1,n,2m|

(Chan, Lee & Yang 2005)

• Polarizations orthogonal to the scattering plane.

• New high-energy scattering amplitudes due to the fermion exchange in the correction functions.

• Needs to consider high-energy massive fermion scattering amplitudes in the R-sector.

T1/2-b

E.g.

iT1/2-b Same answer ( up to a sign)

i i

-

Closed String

KLT formula :

High energy 4-tachyons :

‧Can be generalized to arbitrary mass levels.

5 (C)

(Chan, Lee & Yang 2005)

Veneziano (1968)

Gross & Mende (1987)

+ Are inconsistent with KLT formula !

symmetry in 2D string

6. Conclusion The importance of zero-norm state (ZNS) in string theory has been largely underestimated for decades!

(Kao& Lee 2002, Chan, Lee& Yang 2005)

High-energy symmetry

(Gross)

Gauge symmetry In WSFT

Zero-norm state (ZNS)

Discrete duality symmetries

T-duality (Lee 2000)…

(Chung & Lee 1994)

(CHLTY 2005)