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Register a new userConformal Three-Point Correlation Functions from the Operator Product Expansion
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We show how to construct embedding space three-point functions for operators in arbitrary Lorentz representations by employing the formalism developed in arXiv:1905.00036 and arXiv:1905.00434. We study tensor structures that intertwine the operators with the derivatives in the OPE and examine properties of OPE coefficients under permutations of operators. Several examples are worked out in detail. We point out that the group theoretic objects used in this work can be applied directly to construct three-point functions without any reference to the OPE.
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We show how to compute conformal blocks of operators in arbitrary Lorentz representations using the formalism described in arXiv:1905.00036 and arXiv:1905.00434, and present several explicit examples of blocks derived via this method. The procedure for obtaining the blocks has been reduced to (1) determining the relevant group theoretic structures and (2) applying appropriate predetermined substitution rules. The most transparent expressions for the blocks we find are expressed in terms of specific substitutions on the Gegenbauer polynomials. In our examples, we study operators which transform as scalars, symmetric tensors, two-index antisymmetric tensors, as well as mixed representations of the Lorentz group.
We compute the most general embedding space two-point function in arbitrary Lorentz representations in the context of the recently introduced formalism in arXiv:1905.00036 and arXiv:1905.00434. This work provides a first explicit application of this approach and furnishes a number of checks of the formalism. We project the general embedding space two-point function to position space and find a form consistent with conformal covariance. Several concrete examples are worked out in detail. We also derive constraints on the OPE coefficient matrices appearing in the two-point function, which allow us to impose unitarity conditions on the two-point function coefficients for operators in any Lorentz representations.
Deep-inelastic scattering, in the laboratory and on the lattice, is most instructive for understanding how the nucleon is built from quarks and gluons. The long-term goal is to compute the associated structure functions from first principles. So far this has been limited to model calculations. In this Letter we propose a new method to compute the structure functions directly from the virtual, all-encompassing Compton amplitude, utilizing the operator product expansion. This overcomes issues of renormalization and operator mixing, which so far have hindered lattice calculations of power corrections and higher moments.
It is suggested in the paper by A.J. Chambers {it et al.} (Phys. Rev. Lett. 118, 242001 (2017), arXiv:1703.01153) that the time-ordered current-curent correlator in the nucleon calculated on the lattice is to be identified as the forward Compton amplitude so that it is related to the sum of the even moments of the structure function as in the Minkowski space in the continuum. We point out two problems with this identification. First of all, the current-current correlator defined in the Euclidean space is not analytic everywhere on the rest of the complex $ u$ or $omega$ plane, besides the cuts on the real axis. As such, there is no dispersion relation to relate it to its imaginary part and hence the moments of the structure function. On the lattice, there is an additional difficulty in that the higher dimensional local operators from the operator production expansion (OPE) of the current-current product can mix with lower dimensional higher-twist operators which leads to divergences in the powers of inverse lattice spacing. This mixing needs to be removed before their matrix elements can be identified as the moments of the structure function.
We revisit the computation of instanton effects to various correlation functions in ${cal N}=4$ SYM and clarify a controversy existing in the literature regarding their consistency with the OPE and conformal symmetry. To check these properties, we examine the conformal partial wave decomposition of four-point correlators involving combinations of half-BPS and Konishi operators and isolate the contribution from the conformal primary scalar operators of twist four. We demonstrate that the leading instanton correction to this contribution is indeed consistent with conformal symmetry and compute the corresponding corrections to the OPE coefficients and the scaling dimensions of such twist-four operators. Our analysis justifies the regularization procedure used to compute ultraviolet divergent instanton contribution to correlation functions involving unprotected operators.
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