[Paper Review] Beyond-the-Standard-Model matrix elements with the gradient flow

Andrea Shindler, Jordy de Vries+1 more|arXiv (Cornell University)|Sep 9, 2014

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper proposes a novel lattice QCD method using the gradient flow to compute beyond-the-Standard-Model (BSM) matrix elements, specifically nucleon electric dipole moments (EDMs) from the QCD θ-term and the s̄s content of the nucleon, which are critical for dark matter and CP violation searches. By defining the topological charge via the gradient flow and using small flow-time expansions, the method enables unitary, O(a)-improved calculations without fermion mixing, providing a robust path to first-principles determinations of BSM observables.

ABSTRACT

At the Forschungszentrum Juelich (FZJ) we have started a long-term program that aims to determine beyond-the-Standard-Model (BSM) matrix elements using the gradient flow, and to understand the impact of BSM physics in nucleon and nuclear observables. Using the gradient flow, we propose to calculate the QCD component of key beyond the Standard Model (BSM) matrix elements related to quark and strong theta CP violation and the strange content within the nucleon. The former set of matrix elements impacts our understanding of Electric Dipole Moments (EDMs) of nucleons and nuclei (a key signature of BSM physics), while the latter contributes to elastic recoil of Dark Matter particles off nucleons and nuclei. If successful, these results will lay the foundation for extraction of BSM observables from future low-energy, high-intensity and high-accuracy experimental measurements.

Motivation & Objective

• To compute nucleon electric dipole moments (EDMs) induced by the QCD θ-term and higher-dimensional CP-odd operators, key signatures of new physics beyond the Standard Model.
• To determine the s̄s quark content of the nucleon, which directly affects elastic dark matter scattering cross sections.
• To develop a method that avoids fermion mixing and renormalization complications in lattice calculations of BSM matrix elements.
• To enable accurate, first-principles lattice determinations of BSM observables using the gradient flow for topological charge definition.
• To lay the foundation for extracting BSM parameters from future high-precision low-energy experiments on EDMs and dark matter.

Proposed method

• The gradient flow is used to define the topological charge density q(x,t), enabling a safe continuum limit and avoiding renormalization issues in correlators involving the θ-term.
• The matrix element for the nucleon EDM is extracted from the three-point function ⟨N|J_em^μ|N⟩_θ using a linear expansion in θ, with the topological charge Q computed via the gradient flow.
• For the s̄s content, a small flow-time expansion relates the matrix element of the scalar density at non-zero flow-time to the physical matrix element at t=0, eliminating mixing through the coefficient c₃(t).
• The physical matrix element is reconstructed via C^sub(0,x) = (G_π / G_π,t) × [⟨N O_s(t,x) N†⟩ − ⟨O_s(t,x)⟩⟨N N†⟩], where G_π,t is the pseudoscalar two-point function at flow-time t.
• The method ensures automatic O(a) improvement when using twisted mass fermions at maximal twist, and the final result is independent of flow-time t in the continuum limit.
• The approach allows the use of any fermion action without requiring Ginsparg-Wilson fermions, increasing flexibility and reducing systematic uncertainties.

Experimental results

Research questions

• RQ1 Can the nucleon EDM induced by the QCD θ-term be computed accurately using the gradient flow to define the topological charge?
• RQ2 How can the s̄s content of the nucleon be extracted from lattice QCD without operator mixing or complex renormalization?
• RQ3 What is the role of the small flow-time expansion in connecting physical matrix elements to those computed at non-zero flow-time?
• RQ4 How can the gradient flow method ensure unitarity and O(a) improvement in BSM matrix element calculations?
• RQ5 Can this method be generalized to other BSM matrix elements involving CP-odd or scalar operators in nucleons and nuclei?

Key findings

• The method enables the computation of the nucleon EDM from the θ-term via the gradient flow, avoiding the sign problem and allowing a linear expansion in θ.
• The topological susceptibility computed via the gradient flow shows stable behavior with flow-time, validating the method's consistency.
• The small flow-time expansion allows the physical matrix element of the scalar density to be extracted from correlators at non-zero flow-time, with the coefficient c₃(t) determined non-perturbatively from pseudoscalar two-point functions.
• The renormalization factors cancel between the scalar density and the pion two-point function, reducing systematic uncertainties.
• The matrix element is independent of flow-time t in the continuum limit, confirming the validity of the small flow-time expansion and the absence of higher-dimensional operator contamination.
• The approach is unitary, O(a)-improved with twisted mass fermions, and applicable to any fermion action, making it a robust tool for future BSM lattice studies.