Abstract
Measurements of a permanent neutron electric dipole moment (EDM) potentially probe Beyond-the-Standard Model (BSM) sources of CP-violation. At low energy the CP-violating BSM interactions are parametrized by flavor-conserving CP-violating operators of dimension higher than four. QCD calculations of the nucleon matrix elements of these operators are required to fully reconstruct the sources and magnitudes of the different CP-violating contributions to the nucleon EDM. Herein we study the quark-chromo electric dipole moment (qCEDM) operator and the three-gluon Weinberg operator. The non-perturbative determination, using lattice QCD, of the nucleon matrix elements of these CP-violating operators is hampered by their short-distance behavior. Under renormalization these operators mix with lower dimensional operators, which induces power divergences in the lattice spacing, as the continuum limit is approached. We study the short-distance behavior of the qCEDM and the Weinberg operators using the gradient flow. We perform a short flow time expansion and determine, in perturbation theory, the expansion coefficients of the linearly-divergent terms stemming from the mixing with the pseudoscalar density and the topological charge, confirming the expectations of the operator product expansion. We introduce a new method to perform calculations at non-zero flow-time for arbitrary values of the external momenta. This method allows us to work in four dimensions for most of the calculations described in this paper, avoiding the complications associated with defining $\gamma_5$ in generic d dimensions. We show that leading contributions in the external momenta can be reproduced by defining $\gamma_5$ using the 't Hooft-Veltman-Breitenlohner-Maison scheme.
Highlights
The nucleon electric dipole moment (EDM) is a physical quantity that, once measured, will provide a unique opportunity to detect and investigate beyond-the-standard model (BSM) sources of charge and parity (CP) violation
The nucleon electric dipole moment (EDM) provides a unique opportunity to probe of sources of charge and parity (CP) violation in the Standard Model and beyond (BSM)
BSM theories that contain complex CP-violating couplings can induce a nonvanishing EDM, and at low energies one can parametrize the effects of the BSM degrees of freedom through effective, higher-dimensional CP-violating operators
Summary
The nucleon electric dipole moment (EDM) is a physical quantity that, once measured, will provide a unique opportunity to detect and investigate beyond-the-standard model (BSM) sources of charge and parity (CP) violation. Lattice QCD provides the most systematic method to calculate individual contributions from different CPviolating sources to the nucleon EDM in terms of the QCD fundamental degrees of freedom, quark and gluons. [21] we proposed using the gradient flow [22,23,24,25] to renormalize the θ term and the BSM CP-violating operators. Renormalization schemes based on the gradient flow include nonperturbative step-scaling approaches [56,57], removing power divergences in nonlocal operators relevant to hadron structure [58,59], and defining regularizationindependent quark-bilinear currents [60,61]. First results appeared in [66,67,68], and presently we determine the leading contribution to the short flow-time expansion (SFTE) coefficients of the CP-violating operators defined using the gradient flow. In Appendix C we use the calculation of the quark propagator as an example to elucidate the computational techniques for finite flow time
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