Unitarity, crossing symmetry and duality of the S-matrix in large N Chern-Simons theories with fundamental matter (original) (raw)
2015, Journal of High Energy Physics
We present explicit computations and conjectures for 2 → 2 scattering matrices in large N U (N) Chern-Simons theories coupled to fundamental bosonic or fermionic matter to all orders in the 't Hooft coupling expansion. The bosonic and fermionic S-matrices map to each other under the recently conjectured Bose-Fermi duality after a level-rank transposition. The S-matrices presented in this paper may be regarded as relativistic generalization of Aharonov-Bohm scattering. They have unusual structural features: they include a non-analytic piece localized on forward scattering, and obey modified crossing symmetry rules. We conjecture that these unusual features are properties of S-matrices in all Chern-Simons matter theories. The S-matrix in one of the exchange channels in our paper has an anyonic character; the parameter map of the conjectured Bose-Fermi duality may be derived by equating the anyonic phase in the bosonic and fermionic theories. 4.8 The onshell limit 4.8.1 An infrared 'ambiguity' and its resolution 4.8.2 Covariantization of the amplitude 4.9 The S-matrix in the adjoint channel 4.10 The S-matrix for particle-particle scattering 5. The onshell one loop amplitude in Landau Gauge 6. Scattering in the fermionic theory 6.1 The offshell four point amplitude 6.2 The onshell limit 6.3 S-matrices 6.3.1 S-matrix for adjoint exchange in particle-antiparticle scattering 6.3.2 S-matrix for particle-particle scattering 7. Scattering in the identity channel and crossing symmetry 7.1 Crossing symmetry 7.2 A conjecture for the S-matrix in the singlet channel 7.3 Bose-Fermi duality in the S-channel 7.4 A heuristic explanation for modified crossing symmetry 7.5 Direct evaluation of the S-matrix in the identity channel 7.5.1 Double analytic continuation 7.5.2 Schrodinger equation in lightfront quantization? 8. Discussion A. The identity S-matrix as a function of s, t, u B. Tree level S-matrix B.1 Particle-particle scattering B.2 Particle-antiparticle scattering B.3 Explicit tree level computation C. Aharonov-Bohm scattering C.1 Derivation of the scattering wave function C.2 The scattering amplitude C.3 Physical interpretation of the δ function at forward scattering D. Details of the computation of the scalar S-matrix D.1 Computation of the effective one particle exchange interaction D.2 Euclidean rotation D.3 Solution of the Euclidean integral equations D.4 The one loop box diagram computed directly in Minkowski space D.4.1 Scalar poles D.4.2 Contributions of the gauge boson poles off shell-2-D.4.3 The onshell contribution of the gauge boson poles 88 E. Details of the one loop Landau gauge computation 91 E.1 Simplification of the integrand of the box graph E.2 Simplification of the remaining integrands 93 E.3 Absence of IR divergences 94 E.4 Absence of gauge boson cuts 95 E.5 Potential subtlety at special values of external momenta 98 F. Details of scattering in the fermionic theory 98 F.1 Off shell four point function 98 G. Preliminary analysis of the double analytic continuation 103 G.1 Analysis of the scalar integral equation after double analytic continuation 103 G.2 The oneloop box diagram after double analytic continuation 103 G.2.1 Setting up the computation 103 G.2.2 The contribution of the pole at zero 105 G.2.3 The contribution of the remaining four poles 105 G.3 Solutions of the Dirac equation at q ± = 0 after double analytic continuation. 107 G.4 Aharonov-Bohm in the non-relativistic limit 108 5 Readers familiar with the relationship between Chern-Simons theory and WZW theory may recognize this formula in another guise. C 2 (R) k is the holomorphic scaling dimension of a primary operator in the integrable representation R, and e 2πiνm is the monodromy of the four point function < R1, R2,R1,R2 > in the conformal block corresponding to the OPE R1R2 → Rm. 6 The additional-1 in the fermionic theory comes from Fermi statistics. We have used −1 = e ±iπ = e −iπsgn(λ F) .
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