[Paper Review] Towards a direct lattice calculation of m_d - m_u

André Walker-Loud|arXiv (Cornell University)|Nov 17, 2010

Particle physics theoretical and experimental studies7 references3 citations

TL;DR

This paper presents a novel lattice QCD method to directly compute the strong isospin-breaking parameter $ m_d - m_u $ using the baryon spectrum, employing a prudent partially quenched setup with symmetric valence quark mass splitting to reduce artifacts. The key result is a preliminary lattice determination of $ \delta_{\text{phys}} = 0.000216(15)(13) $ in lattice units, with a predicted strong contribution to the neutron-proton mass splitting of $ 3.40 \pm 0.20 \pm 0.74 \, \text{MeV} $, pending resolution of electromagnetic self-energy uncertainties.

ABSTRACT

We describe an independent method for determining the strong-isospin breaking mass parameter, 2delta = m_d - m_u, which utilizes the baryon spectrum. We use a prudent partially quenched choice of splitting the valence quark masses symmetrically about the light sea quark mass. This choice has the consequence of mitigating the most severe partial quenching artifacts. We also discuss the most significant hurdle to this method which is determining the electromagnetic self-energy of the neutron-proton mass splitting, a challenge which lacks a satisfactory answer. Despite these issues, the phenomenologically interesting dependence of m_n - m_p on delta can be determined.

Motivation & Objective

• To develop an independent lattice QCD method for determining the strong isospin-breaking parameter $ 2\delta = m_d - m_u $, which is fundamental to the QCD Lagrangian.
• To address the challenge of electromagnetic self-energy corrections in nucleon mass splitting, a long-standing systematic uncertainty in lattice calculations.
• To enable phenomenologically relevant predictions of the neutron-proton mass splitting dependence on $ \delta $, crucial for processes like neutron decay and time-reversal violation.
• To provide a consistency check via comparison with meson spectrum determinations of $ \delta $, using baryonic observables to indirectly constrain electromagnetic self-energy corrections.

Proposed method

• Utilizes a partially quenched lattice QCD framework where valence quark masses are symmetrically split about the degenerate sea quark mass: $ m_u^{\text{val}} = \hat{m} - \delta $, $ m_d^{\text{val}} = \hat{m} + \delta $, minimizing partial quenching artifacts.
• Employs the $ SU(4|2) $ heavy baryon chiral perturbation theory Lagrangian to model nucleon and cascade mass splittings at next-to-leading order, including both strong and partial quenching contributions.
• Performs finite-volume and chiral extrapolations using an NNLO formula that includes finite-volume corrections to extract the physical value of $ \delta $.
• Uses the ratio of $ \Xi^{-} $ and $ \Xi^{0} $ correlation functions to extract the $ \Xi^{-} - \Xi^{0} $ mass splitting with high precision on multiple ensembles.
• Relies on the Gasser and Leutwyler electromagnetic self-energy estimates for the nucleon, acknowledging the unresolved issue of the required subtraction constant in the dispersion relation.
• Applies the Cottingham formula with renormalized structure functions to compute electromagnetic self-energies, using modern knowledge of nucleon electromagnetic form factors to improve on prior estimates.

Experimental results

Research questions

• RQ1 Can the strong isospin-breaking parameter $ m_d - m_u $ be directly computed from the baryon spectrum in lattice QCD with controlled systematic errors?
• RQ2 How does the neutron-proton mass splitting depend on $ m_d - m_u $, and can this dependence be quantified using lattice QCD?
• RQ3 What is the impact of the electromagnetic self-energy correction—particularly the missing subtraction constant in the dispersion relation—on the determination of $ m_d - m_u $?
• RQ4 To what extent do finite-volume effects and pion mass dependence affect the extraction of $ \delta $ from $ \Xi^{-} - \Xi^{0} $ splittings?
• RQ5 Can the baryonic $ \Xi^{-} - \Xi^{0} $ splitting serve as a consistency check for the electromagnetic self-energy corrections used in nucleon mass splitting calculations?

Key findings

• The lattice calculation yields a preliminary determination of the physical isospin-breaking parameter as $ \delta_{\text{phys}} = 0.000216(15)(13) $ in lattice units, with statistical and systematic uncertainties.
• The strong contribution to the neutron-proton mass splitting is predicted to be $ 3.40 \pm 0.20 \pm 0.74 \, \text{MeV} $, consistent with previous lattice results.
• A significant volume dependence is observed in the $ \Xi^{-} - \Xi^{0} $ splitting, with data showing dependence on $ \exp(-m_\pi L) $, indicating strong finite-size effects.
• The pion mass dependence of the splitting is also substantial, requiring careful extrapolation using an NNLO formula with finite-volume corrections.
• The dominant uncertainty in the final result stems from the electromagnetic self-energy correction, particularly the uncomputed subtraction constant in the dispersion relation.
• The method enables a phenomenologically interesting determination of the $ m_n - m_p $ dependence on $ \delta $, even in the absence of a complete solution to the EM self-energy problem.