[Paper Review] The Advantage of Rightmost Ordering for gamma5 in Dimensional Regularization

Er-Cheng Tsai|ArXiv.org|May 10, 2009

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper proposes a rightmost $γ_5$ ordering scheme in dimensional regularization that preserves Ward-Takahashi identities for amplitudes without fermion loops by analytically continuing the Dirac matrices only after moving all $γ_5$ matrices to the rightmost position. This eliminates the need for finite counter-term corrections required in the Breitenlohner-Maison scheme, offering a simpler, gauge-invariant alternative for renormalized amplitudes in the Standard Model.

ABSTRACT

We propose a gamma5 scheme in dimensional regularization by analytically continuing the dimension after all the gamma5 matrices have been moved to the rightmost position. All Feynman amplitudes corresponding to diagrams with no fermion loops regulated in this manner automatically satisfy the Ward-Takahashi identities. This is in contrast to the scheme of Breitenlohner and Maison, in which finite counter-terms are needed to restore gauge invariance. This rightmost gamma5 scheme also has an advantage over the naive dimensional regularization scheme which does not have a definitive prescription consistent with gauge symmetry. Diagrams with fermion loops can be handled by selecting a proper cut point on each fermion loop to play the role of the point of the rightmost position.

Motivation & Objective

• To resolve the challenge of consistently defining $γ_5$ in dimensional regularization while preserving gauge symmetry.
• To eliminate the need for finite counter-term renormalization required in the Breitenlohner-Maison scheme.
• To provide a simpler, covariant alternative to existing $γ_5$ schemes that avoids splitting $n$-dimensional space into 4D and $(n-4)$D components.
• To establish a consistent framework for handling $γ_5$ in fermion loop diagrams via proper cut-point selection.

Proposed method

• Move all $γ_5$ matrices to the rightmost position in a Feynman amplitude before analytic continuation to $n \neq 4$ dimensions.
• Define $γ_5$ as $i\gamma^0\gamma^1\gamma^2\gamma^3$ in 4D, which anti-commutes only with $γ^\mu$ for $\mu = 0,1,2,3$.
• Use the decomposition $\gamma^\mu = \underline{\gamma}^\mu + \gamma_\Delta^\mu$, where $\underline{\gamma}^\mu$ acts in 4D and $\gamma_\Delta^\mu$ in the extra dimensions.
• Ensure that divergent sub-diagrams do not contain $γ_5$ by positioning it outside any 1PI sub-diagram, thus avoiding ambiguity in the $n \to 4$ limit.
• For fermion loops, define a proper cut point that avoids divergent sub-diagrams, enabling consistent continuation of the amplitude.
• Apply minimal subtraction renormalization after ensuring $γ_5$ is at the rightmost position, guaranteeing gauge invariance without additional finite counter-terms.

Experimental results

Research questions

• RQ1 Can a $γ_5$ scheme in dimensional regularization be constructed that preserves Ward-Takahashi identities without requiring finite counter-term corrections?
• RQ2 How can $γ_5$ be consistently continued to $n \neq 4$ dimensions while maintaining anti-commutation with $γ^\mu$ for $\mu = 0,1,2,3$?
• RQ3 What is the impact of $γ_5$ positioning on the $n \to 4$ limit of divergent amplitudes, particularly in the presence of divergent sub-diagrams?
• RQ4 Can fermion loop diagrams be consistently regulated in this scheme, and what conditions ensure gauge invariance?

Key findings

• Amplitudes with no fermion loops regulated using the rightmost $γ_5$ scheme automatically satisfy Ward-Takahashi identities without additional finite counter-terms.
• The rightmost $γ_5$ scheme avoids the need for finite counter-term renormalization required in the Breitenlohner-Maison scheme, simplifying calculations.
• The scheme ensures that the matrix product before $γ_5$ is fully $n$-dimensionally covariant, eliminating the need to split $n$-dimensional space into 4D and $(n-4)$D components.
• For superficially convergent diagrams, different proper $γ_5$ positions yield the same $n \to 4$ limit after renormalization, ensuring consistency.
• In superficially divergent diagrams, finite differences may arise from overall pole terms, but these are resolved by relying on Ward identities when using proper cut points.
• The method is consistent with the standard model and enables gauge-invariant regularization of amplitudes up to 2-loop order, as shown in a companion paper.