[Paper Review] Induced QCD with two auxiliary bosonic fields

Bastian B. Brandt, Tilo Wettig|arXiv (Cornell University)|Nov 12, 2014

Quantum Chromodynamics and Particle Interactions7 references3 citations

TL;DR

This paper proposes a novel lattice discretization of SU(Nc) Yang-Mills theory using N_b ≥ N_c−1 auxiliary bosonic fields to induce gauge field self-interactions, avoiding the need for infinite auxiliary fields. Numerical evidence in 3D and 4D confirms the theory reproduces continuum Yang-Mills behavior at critical coupling, enabling dual formulations of QCD via gauge field integration.

ABSTRACT

Following a proposal of Budczies and Zirnbauer, we investigate an alternative lattice discretization of continuum ${ m SU}(N_c)$ Yang-Mills theory in which the self-interactions of the gauge field are induced by a path integral over $N_b\ge N_c-1$ auxiliary bosonic fields which are coupled linearly to the gauge field. In two dimensions there exists an analytic proof that the new discretization reproduces Yang-Mills theory in its non-perturbative continuum limit. We provide numerical evidence that this is also the case in three and four dimensions and that, after a suitable matching of the free parameters, the results of the induced theory agree with results from the ordinary plaquette action up to lattice artifacts. The new discretization is ideally suited to change the order of integration in the QCD path integral to arrive at formulations in which the gauge fields have been integrated out. The resulting theories might be amenable to methods previously used in the infinite-coupling limit, and we briefly discuss possibilities to arrive at dual representations of lattice QCD.

Motivation & Objective

• To develop a lattice formulation of Yang-Mills theory that induces gluon self-interactions via a finite number of auxiliary bosonic fields.
• To extend the Budczies-Zirnbauer 'designer action' from U(Nc) to SU(Nc) gauge theories.
• To provide numerical evidence that the induced theory reproduces continuum Yang-Mills theory in 3D and 4D.
• To explore the feasibility of deriving dual representations of QCD by integrating out gauge fields using this framework.
• To address sign problems in dual formulations by reformulating the theory with auxiliary fields.

Proposed method

• The theory uses a path integral over N_b auxiliary bosonic fields coupled linearly to SU(Nc) gauge fields via a modified BZ action.
• The gauge field path integral is evaluated exactly, yielding a weight factor proportional to |det(1 - α_BZ U_p)|^{-2N_b} for each plaquette.
• The continuum limit is approached as α_BZ → 1, with the weight factor converging to a δ-function on the identity group element.
• Numerical simulations evaluate the one-link integral ⟨Tr(U_p)/N_c⟩ in SU(3) for varying N_b and α_BZ to test convergence to 1.
• Dual representations are derived by expanding the fermionic action in Grassmann variables and integrating out gauge fields using the auxiliary field structure.
• Constraints on dual variables are derived to ensure gauge invariance and nonzero contributions after averaging over gauge fields.

Experimental results

Research questions

• RQ1 Does the induced Yang-Mills theory with a finite number of auxiliary bosonic fields reproduce continuum SU(Nc) Yang-Mills theory in 3D and 4D?
• RQ2 Can the critical coupling α_BZ → 1 limit be numerically confirmed to yield the correct continuum behavior for SU(3) gauge theory?
• RQ3 Is the proposed framework suitable for deriving dual representations of full QCD by integrating out gauge fields?
• RQ4 Can the sign problem in dual formulations be mitigated through this auxiliary field approach?
• RQ5 What is the minimal number of auxiliary fields N_b required for SU(Nc) gauge theories to reproduce Yang-Mills theory?

Key findings

• Numerical results show that the expectation value ⟨Tr(U_p)/N_c⟩ approaches 1 as α_BZ → 1, confirming convergence to the identity in the continuum limit for SU(3).
• The theory reproduces Yang-Mills physics at finite lattice spacing, with results matching those from the standard Wilson plaquette action after parameter matching.
• For SU(Nc), the critical number of auxiliary fields is N_b ≥ N_c−1, consistent with the analytic proof in 2D and universality arguments in higher dimensions.
• The dual formulation of QCD is derived by expanding the fermionic action and integrating out gauge fields, resulting in a partition function with dual variables and constraints.
• Additional constraints on dual variables are being derived to ensure gauge invariance and nonzero contributions, improving the efficiency of potential numerical simulations.
• The framework enables the use of methods previously applicable only in the infinite-coupling limit, offering a path to dual representations of full QCD with auxiliary field-induced dynamics.