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Zero-norm States and High-energy Scattering Amplitudes of Superstring Theory

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. Introduction & Overview A simple example Three calculations of string symmetry (a) High-energy zero-norm states (HZNS)

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Zero-norm States and High-energy Scattering Amplitudes of Superstring Theory

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  1. Zero-norm StatesandHigh-energyScattering Amplitudes of Superstring Theory Jen-Chi Lee Dept. of Electrophysics, National Chiao-Tung Univ. Hsin-Chu, Taiwan

  2. Outline • Introduction & Overview • A simple example • 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

  3. 1. Introduction & Overview Prescribed by self-consistent conditions of quantum string e.g : 1-loop modular invariance of 10D Heterotic string. Q.F.T :“Symmetry dictates interaction” e.g : Y. M. , G. R. String: Interaction Symmetry ? massless Y.M. e.g : Massive symmetries Lee, (1994) ……………? How to identify string symmetry?

  4. : ( Gross 1988) PRL 1st Key • R. G., Asymptotic freedom, O.P.E. etc. • Spontaneous broken Symmetries hidden at low-energy evident at high-energy!! Q.F.T : String: • High-energy limit. Saddle-point Approx. • Infinite symmetry are • suggested by • UV finiteness of quantum string! • states with No free parameter. Huge symmetry Group!?

  5. ( Gross 1988 P.R.L.) Conjectures Tachyon 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 Origin of symmetry charges were not understood. Proportionality constants among HSA of different string states were Not calculated.

  6. (Lee 1990) Zero-norm states (ZNS) in the old covariant first quantized (OCFR) spectrum 2nd Key

  7. There are two types of ZNS: Type I: where Type II: where D=26 only

  8. 2. A simple example (Chan & Lee 2003) (ZNS) Decoupling of ZNS 2nd key : Ward Identity, M2=4 Combine 1st key and 2nd key (Lee 1994)

  9. 1st key : Taking high energy limit

  10. Sample calculation Index of the 2nd vertex Can be measured!? For Tachyon, tensor (tree)

  11. Generalized to higher mass level (1) (2) Algebraically!! (valids to all loops order!) (3) (tree only) with is the only HSA at level

  12. 3. Three calculations of string symmetrya. High-energy zero-norm state (1) (2) (1), (2)

  13. “dual” Type I Type II (b) Virasoro constraints 0 Ex. By Type I Virasoro constraints are Normalization factor & symmetry factors

  14. (c) Saddle-point Method (s-t channel) n Saddle point

  15. Gross conjectures (1988) are explicitly proved !

  16. Ex. 4. Compares with Gross’s Ex. Note! • 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)

  17. 5 (a) 2D string ( Chung ＆ Lee 1994) 2D discrete ZNS carry symmetry charges. . . ( Klebanov ＆ Polyakov 1991) Space time symmetry algebra of 2D string was known to be “Ground ring” ( Witten 1992) One can construct a set of ZNS with discrete Polyakov momenta such that . a algebra! ( Chung ＆ Lee 1994)

  18. 5 (b) Superstring (Chan, Lee & Yang 2005) • NS-sector, GSO even, polarization on the scattering plane • GSO odd All are proportional to each other at each fixed mass level.

  19. i i Same answer ( up to a sign) • 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. E.g.

  20. 5 (C) Closed String (Chan, Lee & Yang 2005) KLT formula : High energy 4-tachyons : Veneziano (1968) Gross & Mende (1987) Are inconsistent with KLT formula ! + ‧Can be generalized to arbitrary mass levels.

  21. 6. Conclusion The importance of zero-norm state (ZNS) in string theory has been largely underestimated for decades! High-energy symmetry (Gross) Gauge symmetry In WSFT (Kao& Lee 2002, Chan, Lee& Yang 2005) (CHLTY 2005) Zero-norm state (ZNS) symmetry in 2D string Discrete duality symmetries (Chung & Lee 1994) T-duality (Lee 2000) …

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