[Paper Review] Minimally doubled fermions and their renormalization

Stefano Capitani, Michael Creutz+2 more|arXiv (Cornell University)|Oct 1, 2010

Quantum Chromodynamics and Particle Interactions12 references3 citations

TL;DR

This paper presents a one-loop lattice perturbative analysis of Bori{ç}i-Creutz and Karsten-Wilczek fermions—minimally doubled lattice fermions that preserve exact chiral symmetry and locality. It derives a conserved, ultralocal axial current and identifies three necessary counterterms to remove power divergences, showing both actions are renormalizable with consistent mixing patterns under hyper-cubic symmetry breaking.

ABSTRACT

Minimally doubled fermions have been proposed as a strictly local discretization of the QCD quark action, which also preserves chiral symmetry at finite cut-off. We study the renormalization and mixing properties of two particular realizations of minimally doubled fermions in lattice perturbation theory at one loop, and we construct conserved axial currents which have a simple form involving only nearest-neighbours sites. We also introduce a notation which allows a unified description of the renormalization properties of both actions.

Motivation & Objective

• To establish the renormalizability of minimally doubled fermions—specifically Bori{ç}i-Creutz and Karsten-Wilczek fermions—within lattice QCD.
• To identify and compute the necessary counterterms required to remove power divergences in the self-energy and vacuum polarization at one-loop order.
• To construct a conserved axial current with a simple, nearest-neighbor-only form, enabling efficient simulations.
• To unify the renormalization structure of both fermion actions using a common notation, revealing shared algebraic patterns despite different symmetries.
• To demonstrate that hyper-cubic symmetry breaking does not obstruct renormalization or Monte Carlo simulations, provided counterterms are consistently applied.

Proposed method

• Perform one-loop lattice perturbation theory on the vacuum polarization and self-energy functions for both Bori{ç}i-Creutz and Karsten-Wilczek fermions.
• Use the chiral symmetry of the actions to constrain the form of allowed counterterms, reducing the number of independent operators.
• Introduce a notation using $Λ_{\mu}$ to unify the algebraic structure of hyper-cubic-breaking terms in the vacuum polarization.
• Derive the explicit form of the conserved axial current using the Noether procedure, ensuring it depends only on nearest-neighbor sites.
• Compute the coefficients of dimension-four counterterms in the fermionic and gauge sectors via perturbative matching.
• Verify that power divergences are removed after counterterm subtraction, and no new divergences arise in the quantities studied.

Experimental results

Research questions

• RQ1 Can minimally doubled fermions such as Bori{ç}i-Creutz and Karsten-Wilczek be consistently renormalized at one-loop order in lattice perturbation theory?
• RQ2 What are the necessary counterterms to remove power divergences in the self-energy and vacuum polarization, and how do they depend on the action's symmetry breaking?
• RQ3 Can a conserved axial current be constructed with a simple, ultralocal form (involving only nearest neighbors) for these fermions?
• RQ4 Do the hyper-cubic-breaking structures in the vacuum polarization of both actions exhibit a common algebraic form when expressed in a unified notation?
• RQ5 Is the renormalization structure of these fermions sufficiently stable and predictable to allow reliable Monte Carlo simulations after counterterm subtraction?

Key findings

• Three counterterms are required in the fermionic action and one in the gauge sector to remove power divergences, with coefficients calculable in one-loop perturbation theory.
• The vacuum polarization for both actions can be expressed in a unified algebraic form using the $Λ_{\mu}$ notation, revealing a structural equivalence despite different symmetry breaking patterns.
• The self-energy’s logarithmic divergence is removed by counterterm subtraction, and no additional power divergences appear in the quantities computed.
• The axial current is conserved and has a simple, ultralocal form involving only nearest-neighbor sites, a rare feature among lattice fermions.
• The coefficient of the $Σ_{3}$ term in the vacuum polarization is $\tilde{b}\left(-\frac{8}{3}L + 19.99468\right)$ for Karsten-Wilczek and $\tilde{b}\left(-\frac{8}{3}L + 23.6793\right)$ for Bori{ç}i-Creutz, with $\tilde{b} = \frac{g_{0}^{2}C_{2}}{16\pi^{2}}$.
• The $d_{g}$ coefficient for the gauge counterterm is $-12.69766$ for Karsten-Wilczek and $-3.6376$ for Bori{ç}i-Creutz, confirming distinct but consistent renormalization patterns.