[Paper Review] Renormalizable Expansion for Nonrenormalizable Theories: II. Gauge Higher Dimensional Theories

D. I. Kazakov, G. S. Vartanov|ArXiv.org|Feb 1, 2007

Cosmology and Gravitation Theories15 references3 citations

TL;DR

This paper extends the 1/Nf expansion to higher-dimensional gauge theories, demonstrating that a dimensionless effective coupling emerges through resummation of fermion loops, rendering the theory renormalizable despite initial nonrenormalizability. The method leads to a logarithmically divergent perturbation theory with a nonpolynomial beta function, while the original dimensionful gauge coupling acts as a mass parameter and is multiplicatively renormalized.

ABSTRACT

The previously developed renormalizable perturbative 1/N-expansion in higher dimensional scalar field theories is extended to gauge theories with fermions. It is based on the $1/N_f$-expansion and results in a logarithmically divergent perturbation theory in arbitrary high odd space-time dimension. Due to the self-interaction of non-Abelian fields the proposed recipe requires some modification which, however, does not change the main results. The new effective coupling is dimensionless and is running in accordance with the usual RG equations. The corresponding beta function is calculated in the leading order and is nonpolynomial in effective coupling. The original dimensionful gauge coupling plays a role of mass and is also logarithmically renormalized. Comments on the unitarity of the resulting theory are given.

Motivation & Objective

• To extend the 1/N expansion, previously developed for scalar theories, to gauge theories with fermions in arbitrary odd space-time dimensions.
• To address the challenge of gauge invariance and self-interactions in non-Abelian gauge theories within the 1/Nf framework.
• To demonstrate that the resulting perturbative expansion is renormalizable, with a dimensionless coupling that runs logarithmically via standard RG equations.
• To analyze the unitarity of the resulting theory, particularly concerning complex poles and spectral functions in the gauge propagator.
• To clarify the role of the original dimensionful coupling as a mass parameter, distinct from the expansion parameter.

Proposed method

• Introduce a 1/Nf expansion by coupling Nf fermion fields to a U(1) gauge field with interaction strength scaled by 1/√Nf to suppress loop factors.
• Resum all leading-order vacuum polarization diagrams (fermion bubbles) into a dressed photon propagator with a momentum-dependent polarization function.
• Derive the effective propagator in the form Dμν(p) = -i/(p²) × (gμν - pμpν/p²) × 1/(1 + e²f(D)(-p²)^{D/2-2}), which suppresses high-momentum behavior.
• Re-normalize the gauge field and redefine the coupling to introduce a dimensionless effective coupling h, replacing the original dimensionful e.
• Apply dimensional regularization in odd D to handle divergences and derive the running of h via the beta function in leading order.
• Analyze the analytic structure of the propagator, including cuts and possible complex poles, to assess unitarity and interpret potential ghost states.

Experimental results

Research questions

• RQ1 Can the 1/Nf expansion be consistently extended to non-Abelian gauge theories in higher odd-dimensional spacetimes?
• RQ2 How does the presence of triple and quartic gauge self-interactions affect the 1/Nf expansion and the resulting renormalizability?
• RQ3 What is the behavior of the effective coupling h in the renormalization group flow, and does it lead to a nonpolynomial beta function?
• RQ4 How does the original dimensionful coupling e, now acting as a mass parameter, transform under renormalization?
• RQ5 To what extent is the resulting theory unitary, given the appearance of complex poles and cuts in the gauge propagator?

Key findings

• The 1/Nf expansion leads to a renormalizable perturbative framework in higher-dimensional gauge theories, even when the original theory is formally nonrenormalizable.
• The effective coupling h is dimensionless and runs logarithmically according to the standard renormalization group equations, with a nonpolynomial beta function in the leading order.
• The original dimensionful coupling e is multiplicatively renormalized and plays the role of a mass parameter, not an expansion parameter.
• The resummed gauge propagator exhibits improved high-energy behavior, suppressing divergences and leading to logarithmic, rather than power-like, divergences.
• Despite the presence of cuts and possible complex poles in the propagator, the theory remains unitary at the level of physical amplitudes, as cuts correspond to standard physical states.
• The method is applicable in arbitrary odd space-time dimensions; even dimensions introduce logarithmic terms that complicate the regularization but are expected to be treatable in future work.