Abstract
We study the crossing equations in d = 3 for the four point function of two U(1) currents and two scalars including the presence of a parity violating term for the s-channel stress tensor exchange. We show the existence of a new tower of double trace operators in the t-channel whose presence is necessary for the crossing equation to be satisfied and determine the corresponding large spin parity violating OPE coefficients. Contrary to the parity even situation, we find that the parity odd s-channel light cone stress tensor block do not have logarithmic singularities. This implies that the parity odd term does not contribute to anomalous dimensions in the crossed channel at this order light cone expansion. We then study the constraints imposed by reflection positivity and crossing symmetry on such a four point function. We reproduce the previously known parity odd collider bounds through this analysis. The contribution of the parity violating term in the collider bound results from a square root branch cut present in the light cone block as opposed to a logarithmic cut in the parity even case, together with the application of the Cauchy-Schwarz inequality.
Highlights
Numerical coefficients njs, njf correspond to the parity invariant sector, while pj is the parity violating coefficient
We study the crossing equations in d = 3 for the four point function of two U(1) currents and two scalars including the presence of a parity violating term for the s-channel stress tensor exchange
In this paper we have studied the modifications to the crossing equation for a four point function of two U(1) currents and two scalars due to the presence of a parity violating term in the stress tensor exchange in the s-channel
Summary
Following [27], the differential operator required to obtain the parity odd JJT correlator from φφT is given by, j(P1; Z1)j(P2; Z2)T (P3; Z3) = αDl(e3f)t + βDl(e4f)t φ(P1; Z1)φ(P2; Z2)T (P3; Z3) , (2.1). Where we have expressed the correlators in the embedding space formalism. Note that Pis and Zis are defined in a five dimensional embedding space. To project into the real 3 dimensional space, one uses the following projection formulae. In order to compare it against the known structure of parity odd three point functions in literature [7, 10] we re-express the building blocks of the three point functions in the embedding space i.e. Hij, Vi,jk and ij, in terms of the tensor structures used in [7].
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