Overview of High Energy String Scattering Amplitudes and Symmetries of String Theory

We review high energy symmetries of string theory at both the fixed angle or Gross regime (GR) and the fixed momentum transfer or Regge regime (RR). We calculated in details high energy string scattering amplitudes at arbitrary mass levels for both regimes. We discovered infinite linear relations among fixed angle string amplitudes conjectured by Gross in 1988 from decoupling of high energy zero-norm states (ZNS), and infinite recurrence relations among Regge string amplitudes from Kummer function U and Appell function F_1. However, the linear relations we obtained in the GR corrected [27-32] the saddle point calculations of Gross, Gross and Mende and Gross and Manes [1-5]. Our results were consistent with the decoupling of high energy ZNS or unitarity of the theory while those of them were not. In addition, for the case of high energy closed string scatterings, our results [36] differ from theirs by an oscillating prefactor which was crucial to recover the KLT relation valid for all energies. In the GR/RR regime, all high energy string amplitudes can be solved by these linear/recurrence relations so that all GR/RR string amplitudes can be expressed in terms of one single GR/RR string amplitude. In addition, we found an interesting link between string amplitudes of the two regimes, and discovered that at each mass level the ratios among fixed angle amplitudes can be extracted from Regge string scattering amplitudes. This result enables us to argue that the known SL(5,C) dynamical symmetry of the Appell function F_1 is crucial to probe high energy spacetime symmetry of string theory.

[1]  Edward Witten,et al.  Non-commutative geometry and string field theory , 1986 .

[2]  W. Miller.  Lie Theory and the Lauricella Functions FD , 1972 .

[3]  Yi Yang,et al.  High-energy zero-norm states and symmetries of string theory. , 2005, Physical review letters.

[4]  D. Gross,et al.  Gravitational effects in ultrahigh-energy string scattering , 2007, 0705.1816.

[5]  R. Brower,et al.  Analyticity for multi-Regge limits of the Bern-Dixon-Smirnov amplitudes , 2008, 0809.1632.

[6]  C. Yan,et al.  High-energy String Scattering Amplitudes and Signless Stirling Number Identity , 2010, 1012.5225.

[7]  P. West.  PHYSICAL STATES AND STRING SYMMETRIES , 1995 .

[8]  Yi Yang,et al.  Higher Spin String States Scattered from D-Particle in the Regge Regime and Factorized Ratios of Fixed Angle Scatterings , 2011, 1101.1228.

[9]  D. Amati,et al.  Superstring collisions at planckian energies , 1987 .

[10]  G. Bonelli.  On the tensionless limit of bosonic strings, infinite symmetries and higher spins , 2003, hep-th/0305155.

[11]  D. Lewellen,et al.  A Relation Between Tree Amplitudes of Closed and Open Strings , 1986 .

[12]  D. Gross,et al.  String Theory Beyond the Planck Scale , 1988 .

[13]  Quantized tension: Stringy amplitudes with Regge poles and parton behavior , 2004, hep-th/0410131.

[14]  The Hagedorn Transition, Deconfinement and N=4 SYM Theory , 1999, hep-th/9908001.

[15]  Jen-Chi Lee,et al.  Decoupling of degenerate positive-norm states in Witten's string field theory , 2002, hep-th/0212196.

[16]  Symmetries of the Bosonic String S-Matrix , 1993, hep-th/9310026.

[17]  Jen-Chi Lee.  New symmetries of higher spin states in string theory , 1990 .

[18]  D. Amati,et al.  CLASSICAL AND QUANTUM GRAVITY EFFECTS FROM PLANCKIAN ENERGY SUPERSTRING COLLISIONS , 1988 .

[19]  Massless higher spins and holography , 2002, hep-th/0205131.

[20]  Yi Yang.  Symmetry of String Scattering Amplitudes , 2017 .

[21]  Yi Yang,et al.  High-energy string scatterings of compactified open string , 2008, 0805.3168.

[22]  Z. Bern,et al.  New Relations for Gauge-Theory Amplitudes , 2008, 0805.3993.

[23]  Lee.  Decoupling of degenerate positive-norm states in string theory. , 1990, Physical review letters.

[24]  D. Gross,et al.  The High-Energy Behavior of String Scattering Amplitudes , 1987 .

[25]  C. R.O..  Massive Superstring Scatterings in the Regge Regime , 2018 .

[26]  X. Yin,et al.  ABJ triality: from higher spin fields to strings , 2012, 1207.4485.

[27]  Soldate,et al.  High-energy unitarity of gravitation and strings. , 1988, Physical review. D, Particles and fields.

[28]  A. Sagnotti,et al.  String Lessons for Higher-Spin Interactions , 2010, 1006.5242.

[29]  Bogdan Stefan Ski.  Gravitational couplings of D-branes and O-planes , 1999 .

[30]  Notes on high-energy limit of bosonic closed string scattering amplitudes , 2006, hep-th/0604122.

[31]  Yi Yang,et al.  Kummer function and High energy String Scatterings , 2008, 0811.4502.

[32]  Scatterings of massive string states from D-brane and their linear relations at high energies , 2006, hep-th/0610062.

[33]  Yi Yang,et al.  The Exact SL(K+3,C) Symmetry of String Scattering Amplitudes , 2016, 1603.00396.

[34]  Jen-Chi Lee.  Reduction of stringy scattering amplitudes through massive ward identities , 1994 .

[35]  On stringy AdS5 × S5 and higher spin holography , 2003, hep-th/0305052.

[36]  Yi Yang,et al.  Linear relations and their breakdown in high energy massive string scatterings in compact spaces , 2007, 0705.1872.

[37]  D. Gross,et al.  The High-energy Behavior of Open String Scattering , 1989 .

[38]  P. Vanhove,et al.  Minimal basis for gauge theory amplitudes. , 2009, Physical review letters.

[39]  Yi Yang,et al.  Anatomy of zero-norm states in string theory , 2005, hep-th/0501020.

[40]  Yi Yang,et al.  Recurrence relations of higher spin BPST vertex operators for open strings , 2013, 1304.6948.

[41]  Yi Yang,et al.  Stirling number Identities and High energy String Scatterings , 2009, 0909.3894.

[42]  Ward identities and high-energy scattering amplitudes in string theory , 2004, hep-th/0410194.

[43]  On higher spins and the tensionless limit of string theory , 2003, hep-th/0311257.

[44]  Jen-Chi Lee.  Stringy Symmetries and Their High-energy Limit , 2003, hep-th/0303012.

[45]  Yi Yang,et al.  Patterns of high energy massive string scatterings in the Regge regime , 2008, 0812.4190.

[46]  Yi Yang,et al.  The Appell Function F1 and Regge String Scattering Amplitudes , 2014, 1406.1285.

[47]  Taejin Lee,et al.  Solving Lauricella string scattering amplitudes through recurrence relations , 2017, 1707.01281.

[48]  Jen-Chi Lee.  Spontaneously broken symmetry in string theory , 1994, hep-th/0503056.

[49]  An orientifold of AdS5 × T11 with D7-branes, the associated α′2-corrections and their role in the dual = 1 Sp(2N+2M) × Sp(2N) gauge theory , 2002, hep-th/0206071.

[50]  D. Amati,et al.  Can spacetime be probed below the string size , 1989 .

[51]  D. Gross,et al.  High-energy symmetries of string theory. , 1988, Physical review letters.

[52]  Finite in All Directions , 1993, hep-th/9305139.

[53]  Stringy gravity, interacting tensionless strings and massless higher spins , 2001, hep-th/0103247.

[54]  Jen-Chi Lee,et al.  Recurrence relations of Kummer functions and Regge string scattering amplitudes , 2012, 1212.6915.

[55]  J. Avan,et al.  Classical integrability and higher symmetries of collective string field theory , 1991 .

[56]  Yi Yang,et al.  Comments on the high energy limit of bosonic open string theory , 2005, hep-th/0509009.

[57]  The Pomeron and gauge/string duality , 2006, hep-th/0603115.

[58]  E. Lee,et al.  Generalized On-Shell Ward Identities in String Theory , 2000 .

[59]  W. N. Bailey.  Confluent Hypergeometric Functions , 1960, Nature.

[60]  A. Polyakov,et al.  INTERACTION OF DISCRETE STATES IN TWO-DIMENSIONAL STRING THEORY , 1991, hep-th/9109032.

[61]  Xiaoxia Wang,et al.  Recursion formulas for Appell functions , 2012 .

[62]  G. Veneziano.  Construction of a crossing-simmetric, Regge-behaved amplitude for linearly rising trajectories , 1968 .

[63]  S. Kawamoto,et al.  To see Symmetry in a Forest of Trees , 2013, 1309.3443.

[64]  J. Avan,et al.  Quantum integrability and exact eigenstates of the collective string field theory , 1991 .

[65]  Zero-norm states and high-energy symmetries of string theory , 2004, hep-th/0401133.

[66]  田中 正,et al.  SUPERSTRING THEORY , 1989, The Lancet.

[67]  D. Gross.  Strings at superplanckian energies: in search of the string symmetry , 1989, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[68]  Yi Yang,et al.  The string BCJ relations revisited and extended recurrence relations of nonrelativistic string scattering amplitudes , 2016, 1601.03813.

[69]  Yi Yang,et al.  High-energy massive string scatterings from orientifold planes , 2007, 0712.4245.

[70]  J. Atick,et al.  The Hagedorn Transition and the Number of Degrees of Freedom of String Theory , 1988 .

[71]  Song He,et al.  Exponential Fall-Off Behavior of Regge Scatterings in Compactified Open String Theory , 2010, 1012.3158.

[72]  M. Soldate.  Partial-wave unitarity and closed-string amplitudes , 1987 .

[73]  R. Hagedorn,et al.  Statistical thermodynamics of strong interactions at high-energies. 2. Momentum spectra of particles produced in pp-collisions , 1965 .

[74]  B. Ovrut,et al.  Zero-norm states and enlarged gauge symmetries of the closed bosonic string with massive background fields , 1990 .

[75]  R. Hagedorn.  Hadronic matter near the boiling point , 1968 .

[76]  Superstring Scattering from D-Branes , 1996, hep-th/0611173.