Momentum space spinning correlators and higher spin equations in three dimensions
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Springer
Received: July 27, 2020 Accepted: October 8, 2020 Published: November 11, 2020
Sachin Jain,a Renjan Rajan Johnb,c and Vinay Malvimata a
Indian Institute of Science Education and Research, Homi Bhabha Rd, Pashan, Pune 411 008, India b Università del Piemonte Orientale, Dipartimento di Scienze e Innovazione Tecnologica, Viale T. Michel 11, I-15121 Alessandria, Italy c I.N.F.N. — sezione di Torino, Via P. Giuria 1, I-10125 Torino, Italy
E-mail: [email protected], [email protected], [email protected] Abstract: In this article, we explicitly compute in momentum space the three and fourpoint correlation functions involving scalar and spinning operators in the free bosonic and the free fermionic theory in three dimensions. We also evaluate the five-point function of the scalar operator in the free bosonic theory. We discuss techniques which are more efficient than the usual PV reduction to evaluate one loop integrals. Our techniques can be easily generalised to momentum space correlators of complicated spinning operators and to higher point functions. The three dimensional fermionic theory has the interesting feature that ¯ is odd under parity. To account for this, we develop a parity odd the scalar operator ψψ basis which is useful to write correlation functions involving spinning operators and an odd ¯ operators. We further study higher spin (HS) equations in momentum space number of ψψ which are algebraic in nature and hence simpler than their position space counterparts. We use them to solve for three-point functions involving spinning operators without invoking conformal invariance. However, at the level of four-point functions, solving the HS equation requires additional constraints that come from conformal invariance and we could only verify that our explicit results solve the HS equation. Keywords: Conformal Field Theory, Higher Spin Symmetry ArXiv ePrint: 2005.07212
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP11(2020)049
JHEP11(2020)049
Momentum space spinning correlators and higher spin equations in three dimensions
Contents 1
2 List of theories and operators 2.1 Free bosonic theory 2.2 Free fermionic theory
2 2 3
3 Conformal Ward identities 3.1 Dilatation and special conformal Ward identities 3.2 Diffeomorphism and Weyl Ward identities: local and transverse parts of the correlator 3.2.1 Diffeomorphism/transverse Ward identity 3.2.2 Weyl/trace Ward identities 3.3 Reconstruction formula
4 4
4 Momentum basis for the transverse part 4.1 Projector basis for parity even correlators 4.1.1 A basis involving the Levi-Civita tensor 4.2 Projector basis for parity odd correlators
8 8 9 9
5 5 6 7
5 Results of explicit computations 5.1 Free bosonic theory in d = 3 5.1.1 Two and three-point functions 5.1.2 Four-point functions 5.1.3 Five-point functions 5.2 Free fermionic theory in d = 3 5.2.1 Two and three-point functions 5.2.2 Four-point functions
10 10 10 12 15 16 16 17
6 Correlators from higher spin Ward identities 6.1 Three-point sp
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