[Paper Review] Infrared behavior of the running coupling in scalar field theory

Marco Frasca|arXiv (Cornell University)|Feb 8, 2008

Cosmology and Gravitation Theories4 references3 citations

TL;DR

This paper derives the infrared behavior of the running coupling in Yang-Mills theory using a strong coupling expansion of a massless scalar field theory, showing that the beta function is β(λ) = 4λ, leading to a running coupling α(p) ∝ p⁴ that vanishes in the infrared. This supports the MOM scheme as the correct definition of the running coupling, consistent with lattice results and suggesting a scenario dominated by instantons rather than a fixed point.

ABSTRACT

We compute the Green function of the massless scalar field theory in the infrared till the next-to-leading order, providing a fully covariant strong coupling expansion. Applying Callan-Symanzik equation we obtain the exact running coupling for this case by computing the beta function. This result is applied using a recently proved mapping theorem between a massless scalar field theory and Yang-Mills theory. This beta function gives a running coupling going to zero as $p^4$ in agreement with lattice results presented in Boucaud et al. [JHEP 0304 (2003) 005] and showing that the right definition of the running coupling for a Yang-Mills theory in the infrared is given in a MOM scheme. The emerging scenario is supporting a quantum field theory based on instantons.

Motivation & Objective

• To resolve the ambiguity in defining the running coupling in Yang-Mills theory in the infrared regime.
• To establish a fully covariant strong coupling expansion for the massless scalar field theory as a proxy for Yang-Mills theory.
• To derive the exact beta function for the running coupling using the Callan-Symanzik equation in the infrared limit.
• To demonstrate that the running coupling goes to zero as p⁴, consistent with lattice data and incompatible with a fixed point.
• To validate the MOM scheme as the physically correct definition of the running coupling in the infrared.

Proposed method

• Formal strong coupling expansion of the massless scalar field theory partition function in the limit λ → ∞.
• Derivation of the two-point function (propagator) in the infrared at next-to-leading order using a gradient expansion method.
• Application of the Callan-Symanzik equation to the derived propagator to extract the beta function.
• Mapping of the scalar theory to Yang-Mills theory via a recently proven theorem, transferring results to the gauge theory.
• Identification of the running coupling α(p) via the MOM scheme, defined as α_MOM(p²) = J²(p)Z(p), where Z and J are dressing functions.
• Use of the relation λ → Ng² to map the scalar theory result to Yang-Mills theory, yielding α(p) ∝ p⁴.

Experimental results

Research questions

• RQ1 What is the correct definition of the running coupling in the infrared for Yang-Mills theory?
• RQ2 How does the running coupling behave in the infrared when computed via a strong coupling expansion of a scalar field theory?
• RQ3 Does the Callan-Symanzik equation yield a consistent beta function in the strong coupling limit of the scalar theory?
• RQ4 Is the behavior α(p) ∝ p⁴ compatible with lattice results and phenomenological data?
• RQ5 Can the infrared dynamics of Yang-Mills theory be consistently described via instantons, given the coupling behavior?

Key findings

• The beta function for the scalar field theory in the infrared is β(λ) = 4λ, indicating a trivial fixed point.
• The running coupling in the infrared behaves as α(p) ∝ p⁴, vanishing at low momenta rather than approaching a fixed point.
• This behavior is consistent with lattice computations by Boucaud et al., which show α(p) ∝ p⁴ in the MOM scheme.
• The ghost propagator decouples in the infrared, supporting the use of the MOM scheme over definitions involving ghost dressing functions.
• The mapping from the scalar theory to Yang-Mills theory confirms that the running coupling in the latter also vanishes as p⁴ in the infrared.
• The result supports a quantum field theory scenario dominated by instantons, with the gluon propagator reaching a finite value and the coupling vanishing at low energies.