[Paper Review] Off-shell renormalization in the presence of dimension 6 derivative operators. II. UV coefficients

Daniele Binosi, Andrea Quadri|arXiv (Cornell University)|Apr 14, 2019

Particle physics theoretical and experimental studies4 citations

TL;DR

This paper presents a systematic off-shell one-loop renormalization procedure for the Abelian Higgs-Kibble model with a dimension-6 derivative operator, using an auxiliary X-theory framework to disentangle UV divergences from generalized non-polynomial field redefinitions. It computes the complete set of UV coefficients for all radiatively generated dimension-6 gauge-invariant operators, including their dependence on the non-renormalizable coupling $ g $, and establishes gauge independence through cohomological techniques and contractible pairs.

ABSTRACT

The full off-shell one loop renormalization for all divergent amplitudes up to dimension 6 in the Abelian Higgs-Kibble model, supplemented with a maximally power counting violating higher-dimensional gauge-invariant derivative interaction $\sim g ~ ϕ^\dagger ϕ(D^μϕ)^\dagger D_μϕ$, is presented. This allows one to perform the complete renormalization of radiatively generated dimension 6 operators in the model at hand. We describe in details the technical tools required in order to disentangle the contribution to UV divergences parameterized by (generalized) non-polynomial field redefinitions. We also discuss how to extract the dependence of the $β$-function coefficients on the non-renormalizable coupling $g$ in one loop approximation, as well as the cohomological techniques (contractible pairs) required to efficiently separate the mixing of contributions associated to different higher-dimensional operators in a spontaneously broken effective field theory.

Motivation & Objective

• To develop a consistent off-shell renormalization framework for effective field theories with dimension-6 derivative operators in spontaneously broken gauge theories.
• To disentangle UV divergences arising from generalized non-polynomial field redefinitions from physical renormalization of gauge-invariant operators.
• To compute the one-loop UV coefficients of all dimension-6 operators radiatively generated in the Abelian Higgs-Kibble model with a power-counting-violating $\sim g\,\phi^\dagger\phi(D^\mu\phi)^\dagger D_\mu\phi$ interaction.
• To extract the $\beta$-function coefficients for the non-renormalizable coupling $g$ in one-loop approximation using cohomological methods.
• To ensure gauge independence of physical results by systematically handling gauge-dependent contributions from unphysical fields and sources.

Proposed method

• Utilizes an auxiliary X-theory formulation where the physical Higgs degree of freedom is described via gauge-invariant field coordinates $X_2 \sim \phi^\dagger\phi - v^2/2$, enforcing constraints via Lagrange multipliers $X_1$ and sources $\bar{c}^*$, $T_1$.
• Applies the Algebraic Renormalization approach with BRST cohomology to classify and isolate cohomologically trivial invariants associated with unphysical field redefinitions.
• Employs contractible pairs and the $R_\xi$-gauge formalism to systematically separate contributions from different higher-dimensional operators and ensure gauge independence.
• Performs one-loop calculations of 1-PI amplitudes in the X-theory, focusing on weak power-counting behavior that limits divergent amplitudes to a finite set per loop order.
• Maps the results from the X-theory back to the original target theory by on-shell reduction of $X_1$ and $X_2$, translating sources into physical operators in terms of $\phi$ and its covariant derivatives.
• Computes UV coefficients in dimensional regularization with $\epsilon$-poles, explicitly evaluating divergent parts of 2-, 3-, and 4-point functions involving $\sigma$, $\chi$, $A_\mu$, $\bar{c}^*$, $T_1$, and $\bar{c}$ fields.

Experimental results

Research questions

• RQ1 How can UV divergences from generalized non-polynomial field redefinitions be systematically separated from physical renormalization of gauge-invariant operators in effective field theories with higher-dimensional interactions?
• RQ2 What is the complete set of one-loop UV coefficients for all dimension-6 operators radiatively generated in the Abelian Higgs-Kibble model with a power-counting-violating $\sim g\,\phi^\dagger\phi D^\mu\phi D_\mu\phi$ interaction?
• RQ3 How does the $\beta$-function coefficient for the non-renormalizable coupling $g$ depend on the energy scale $\Lambda$ and the Higgs vacuum expectation value $v$ at one-loop order?
• RQ4 How can cohomological techniques, such as contractible pairs, be used to disentangle mixing between different higher-dimensional operators in a spontaneously broken gauge theory?
• RQ5 What is the role of gauge-dependent coefficients in the $X$-theory, and how do they ensure gauge independence of physical UV coefficients in the target theory?

Key findings

• The UV coefficient of the $\sigma\sigma\sigma$ vertex at zero momentum is $\frac{3}{8\pi^2 v^3}(m^4 + 2m^2M^2 + 2M^4 - m^2M_A^2(1 - \delta_{\xi 0}) + 6M_A^4)\frac{1}{\epsilon}$, with explicit $\delta_{\xi 0}$ dependence for non-Feynman gauges.
• The $\bar{c}^*\sigma\sigma$ vertex has zero one-loop divergence: $\overline{\Gamma}^{(1)}_{\bar{c}^*\sigma\sigma} = 0$, indicating no UV divergence from this source-field coupling.
• The $\chi\chi\chi\chi$ four-point function has a divergent part $\frac{3}{8\pi^2 v^4}(m^4 + 2m^2M^2 + 2M^4 - 2m^2M_A^2 + 6M_A^4)\frac{1}{\epsilon}$, with additional $\delta_{\xi 0}$-dependent terms and momentum-dependent corrections.
• The $\sigma\sigma\chi\chi$ vertex at zero momentum is $\frac{1}{8\pi^2 v^4}(m^4 + 2m^2M^2 + 2M^4 - 2m^2M_A^2(1 - \delta_{\xi 0}) + 6M_A^4)\frac{1}{\epsilon}$, showing explicit dependence on $M_A^2$ and $\delta_{\xi 0}$.
• The $T_1\sigma\sigma$ vertex at zero momentum is $\frac{1}{8\pi^2 v}\big(m^2(3M^2 + 2(1 - \delta_{\xi 0})M_A^2) + 6(M^4 + M_A^4)\big)\frac{1}{\epsilon}$, confirming non-trivial gauge dependence.
• The $\sigma\chi\chi$ vertex includes momentum-dependent divergent terms proportional to $\frac{g}{\Lambda v}$, with coefficients involving $m^2$, $M^2$, and $M_A^2$, demonstrating non-trivial mixing in the UV structure.