[Paper Review] Massive Yang-Mills Fields in Interaction with Gravity

D. R. Grigore, G. Scharf|ArXiv.org|Aug 26, 2008

Geomagnetism and Paleomagnetism Studies11 references3 citations

TL;DR

This paper determines the most general form of interaction between gravity and massive Yang-Mills fields using the causal perturbation theory framework. It shows that gauge invariance via the cohomological structure of the BRST operator uniquely fixes the interaction Lagrangian, which couples gravity only to the physical degrees of freedom through a conserved energy-momentum tensor, while ghost fields appear only to maintain gauge invariance and do not contribute to physical S-matrix elements.

ABSTRACT

We determine the most general form of the interaction between the gravitational field and an arbitrary Yang-Mills system of fields (massless and massive). We work in the perturbative quantum framework of the causal approach (of Epstein and Glaser) and use a cohomological definition of gauge invariance for both gauge fields. We also consider the case of massive gravity. We discuss the question whether gravity couples to the unphysical degrees of freedom in the Yang-Mills fields.

Motivation & Objective

• To derive the most general interaction Lagrangian between massive Yang-Mills fields and gravity in a perturbative quantum field theory framework.
• To clarify whether gravity couples to unphysical degrees of freedom (ghosts) in massive Yang-Mills theories.
• To extend the causal approach and BRST cohomology techniques to include massive Yang-Mills fields and massive gravity.
• To demonstrate that the resulting interaction Lagrangian is gauge-invariant and leads to a conserved energy-momentum tensor.
• To show that ghost fields do not contribute to physical S-matrix elements despite appearing in the interaction.

Proposed method

• Uses the causal perturbation theory of Epstein and Glaser to recursively construct chronological products with causal support.
• Applies a cohomological definition of gauge invariance via the BRST operator $ d_Q $, defined as the graded commutator with the gauge charge $ Q $.
• Solves the descent equations $ d_Q T^I = i abla_ u T^{I u} $ using the algebraic Poincaré lemma and cohomological techniques.
• Constructs the interaction Lagrangian $ T_{\text{int}} $ as a sum of the Yang-Mills interaction $ t^{\text{YM}}_{\text{int}} $, a BRST-exact term $ d_Q B $, and a divergence $ \partial_\mu B^\mu $, ensuring gauge invariance.
• Expresses the final interaction in terms of the physical part of the massive vector field $ A^\text{phys}_\mu = A_\mu + \frac{1}{m^2} \partial_\mu \partial_\nu A^\nu $, which contains only transverse physical modes.
• Derives the energy-momentum tensor $ \mathcal{T}^{\mu\nu} $ as a sum of the Yang-Mills and massive vector contributions, showing its conservation $ \partial_\nu \mathcal{T}^{\mu\nu} = 0 $.

Experimental results

Research questions

• RQ1 What is the most general form of interaction between massive Yang-Mills fields and gravity in a perturbative quantum field theory?
• RQ2 Does gravity couple to the unphysical degrees of freedom (ghosts) in the Yang-Mills sector?
• RQ3 How can gauge invariance be consistently implemented in the interaction Lagrangian when gravity is coupled to Yang-Mills fields?
• RQ4 Can the interaction be rewritten in terms of physical degrees of freedom only, and does this form preserve gauge invariance?
• RQ5 What is the role of ghost fields in the interaction if they do not contribute to physical S-matrix elements?

Key findings

• The most general interaction Lagrangian is uniquely determined by gauge invariance and is given by $ T_{\text{int}} = t^{\text{YM}}_{\text{int}} + d_Q B + i \partial_\mu B^\mu $, with no additional free parameters.
• Ghost fields appear in the interaction but do not contribute to physical S-matrix elements, resolving the paradox of coupling to non-energy-carrying degrees of freedom.
• The interaction can be rewritten in terms of the physical vector field $ A^\text{phys}_\mu $, which contains only the three transverse physical modes, showing that gravity couples exclusively to physical degrees of freedom.
• The energy-momentum tensor $ \mathcal{T}^{\mu\nu} $ is conserved, as shown by $ \partial_\nu \mathcal{T}^{\mu\nu} = 0 $, even though the naive free tensor is not conserved due to the constraint $ \partial_\mu A^\mu = 0 $.
• The final interaction takes the standard form $ t_{\text{int}} = \hat{h}_{\mu\nu} \mathcal{T}^{\mu\nu} $, where $ \mathcal{T}^{\mu\nu} $ includes the physical part of the massive vector field and the Yang-Mills field strength.
• The canonical dimension of the interaction is 7, which is non-renormalizable in the standard sense, but the structure is preserved through BRST invariance and the causal framework.