[Paper Review] Off-shell renormalization of the abelian Higgs model in the unitary gauge

Hidenori Sonoda|arXiv (Cornell University)|Aug 29, 2001

Particle physics theoretical and experimental studies3 citations

TL;DR

This paper demonstrates that the abelian Higgs model in the unitary gauge is off-shell renormalizable when using properly chosen composite interpolating fields for the massive photon and Higgs boson. By identifying that extra UV divergences arise from improper field choices and compositeness, the authors show these singularities are removable via a modified renormalization scheme, restoring renormalizability equivalent to standard gauges.

ABSTRACT

We discuss the off-shell renormalization properties of the abelian Higgs model in the unitary gauge. The model is not renormalizable according to the usual power counting rules. In this paper, however, we show that with a proper choice of interpolating fields for the massive photon and the Higgs particle, their off-shell Green functions can be renormalized. An analysis of the nature of the extra singularities in the unitary gauge is given, and a recipe for the off-shell renormalization is provided.

Motivation & Objective

• To resolve the longstanding issue of apparent non-renormalizability in the unitary gauge of the abelian Higgs model.
• To clarify the origin of extra UV divergences unique to the unitary gauge, which are not present in renormalizable gauges.
• To establish that off-shell Green functions can be renormalized when using appropriate interpolating fields with correct scaling dimensions.
• To show that the unitary gauge is physically consistent and as renormalizable as standard gauges when field definitions are properly chosen.
• To provide a systematic recipe for off-shell renormalization applicable to S-matrix calculations and beyond.

Proposed method

• Use of dimensional regularization in $D = 4 - \epsilon$ to handle UV divergences while preserving gauge invariance.
• Identification of the massive photon and Higgs particle via composite interpolating fields with scale dimensions $\geq 2$, rather than elementary fields.
• Application of BRST symmetry and Ward identities to derive consistency conditions for the renormalization procedure.
• Derivation of a modified renormalization scheme that accounts for non-integrable singularities arising from coincident composite fields in coordinate space.
• Explicit construction of the relation between the $B_\mu B_\nu$ and $\mathcal{A}_\mu \mathcal{A}_\nu$ propagators via self-energy corrections and 1PI diagrams.
• Use of diagrammatic techniques to prove Ward identities for higher-point functions, ensuring consistency under renormalization.

Experimental results

Research questions

• RQ1 Why do extra UV divergences appear in the unitary gauge despite the model being physically equivalent to renormalizable gauges?
• RQ2 What is the origin of the non-renormalizable behavior in the unitary gauge under standard field choices?
• RQ3 Can off-shell Green functions be renormalized in the unitary gauge if the interpolating fields are redefined appropriately?
• RQ4 How do composite interpolating fields with higher scaling dimensions affect UV behavior and renormalization?
• RQ5 To what extent can the S-matrix remain finite and unitary despite the presence of extra UV singularities in off-shell amplitudes?

Key findings

• The unitary gauge is not fundamentally non-renormalizable; the apparent non-renormalizability stems from improper choices of interpolating fields.
• The extra UV divergences in the unitary gauge arise from the compositeness of the standard interpolating fields, which generate non-integrable singularities when fields coincide in space.
• Proper interpolating fields for the massive photon and Higgs boson are composite fields with scale dimension 2 or higher, ensuring finite off-shell Green functions after renormalization.
• The UV singularities introduced by compositeness do not affect the S-matrix, as they vanish in the asymptotic limit of infinite spatial separation.
• A consistent off-shell renormalization scheme is constructed, showing that the unitary gauge can be used for full Green function calculations with proper field definitions.
• The relation between the $B_\mu B_\nu$ and $\mathcal{A}_\mu \mathcal{A}_\nu$ propagators is derived explicitly, confirming the equivalence of the unitary and $R_\xi$ gauges at the quantum level.