[Paper Review] Perturbative renormalization of $\Delta F = 2$ four-fermion operators with the chirally rotated Schr\"odinger functional

Mattia Dalla Brida, Mauro Papinutto+1 more|arXiv (Cornell University)|May 29, 2016

Quantum Chromodynamics and Particle Interactions1 references4 citations

TL;DR

This paper presents a perturbative one-loop renormalization of parity-odd ∆F = 2 four-fermion operators in the chirally rotated Schrödinger functional (χSF) framework, leveraging automatic O(a) improvement to eliminate bulk O(a) lattice artifacts. The authors define a family of renormalization schemes via 3-point correlation functions, compute one-loop mixing coefficients and NLO anomalous dimensions, and demonstrate that scale-dependent renormalization constants approach the continuum limit with O(a²) corrections, enabling high-precision non-perturbative studies.

ABSTRACT

The chirally rotated Schr\"odinger functional ($\chi$SF) renders the mechanism of automatic $O(a)$ improvement compatible with Schr\"odinger functional (SF) renormalization schemes. Here we define a family of renormalization schemes based on the $\chi$SF for a complete basis of $\Delta F = 2$ parity-odd four-fermion operators. We compute the corresponding scale-dependent renormalization constants to one-loop order in perturbation theory and obtain their NLO anomalous dimensions by matching to the $\overline{ extrm{MS}}$ scheme. Due to automatic $O(a)$ improvement, once the $\chi$SF is renormalized and improved at the boundaries, the step scaling functions (SSF) of these operators approach their continuum limit with $O(a^{2})$ corrections without the need of operator improvement.

Motivation & Objective

• To establish a renormalization scheme for parity-odd ∆F = 2 four-fermion operators in lattice QCD using the chirally rotated Schrödinger functional (χSF).
• To enable automatic O(a) improvement in the χSF framework, eliminating bulk O(a) lattice artifacts without operator or action improvement.
• To compute one-loop mixing coefficients and NLO anomalous dimensions for these operators, ensuring compatibility with the MS scheme.
• To lay the groundwork for future non-perturbative studies by constructing flexible renormalization conditions that minimize NLO contributions.

Proposed method

• Define 3-point correlation functions in the χSF using chiral rotations of standard SF boundary operators to map parity-even to parity-odd four-fermion operators.
• Implement boundary conditions that preserve the rotated parity symmetry P5, enabling automatic O(a) improvement.
• Expand the renormalization conditions to O(g²) in perturbation theory and compute Feynman diagrams numerically to extract one-loop mixing coefficients Z_ij^(1).
• Use matching to the MS scheme to determine NLO anomalous dimensions and verify consistency with known leading-order results.
• Apply criteria to identify renormalization schemes with minimal NLO contributions by minimizing the norm of the J-matrix, ensuring stable and controlled perturbative evolution.
• Perform numerical checks on convergence to the continuum limit by subtracting finite and logarithmic divergences, confirming O(a²) scaling behavior.

Experimental results

Research questions

• RQ1 Can a consistent family of renormalization schemes be defined for parity-odd ∆F = 2 four-fermion operators in the χSF framework?
• RQ2 To what extent does automatic O(a) improvement in the χSF suppress lattice artifacts in the step scaling functions of these operators?
• RQ3 How do the one-loop mixing coefficients and NLO anomalous dimensions behave under perturbative matching to the MS scheme?
• RQ4 Can renormalization schemes be identified that minimize the size of NLO contributions in the running of the operators?
• RQ5 What is the scaling behavior of the renormalization factors, and does it confirm O(a²) convergence to the continuum limit?

Key findings

• The one-loop renormalization factors Z_ij^(1) for the parity-odd ∆F = 2 four-fermion operators are computed explicitly in terms of the lattice spacing and scheme parameters.
• The NLO anomalous dimensions are successfully obtained by matching to the MS scheme, confirming consistency with known leading-order results.
• All r1,ij and s1,ij coefficients are found to be consistent with zero, confirming the absence of O(a) lattice effects due to automatic O(a) improvement.
• The renormalization factors converge to the continuum limit with O(a²) scaling, as confirmed by numerical fits excluding the coarsest lattices.
• Schemes minimizing the norm of the J-matrix are identified, showing that NLO contributions can be kept small and stable, enabling reliable perturbative evolution.
• The results demonstrate that bulk operator improvement is unnecessary, and the step scaling functions of these operators can be extrapolated as O(a²), significantly improving precision in future non-perturbative studies.