[Paper Review] Renormalization of tensor self-energy in Resonance Chiral Theory

Karol Kampf, Jiří Novotný+1 more|ArXiv.org|Mar 12, 2008

Advanced NMR Techniques and Applications3 citations

TL;DR

This paper investigates the renormalization of tensor self-energy in Resonance Chiral Theory using the antisymmetric tensor formalism for $1^{--}$ vector resonances. It shows that one-loop quantum corrections require new kinetic counterterms with two derivatives—absent in the original leading-order Lagrangian—to cancel ultraviolet divergences, which can lead to additional poles in the propagator corresponding to states with opposite parity, potentially including negative-norm ghosts or tachyons.

ABSTRACT

We study the problems related to the renormalization of propagators in Resonance Chiral Theory, concentrating on the case of vector $1^{--}$ resonances in the antisymmetric tensor formalism. We have found that renormalization of the divergences of the self-energy graphs needs new type of kinetic counterterms with two derivatives which are not present in the original leading order Lagrangian. The general form of the propagator for antisymmetric tensor fields could then contain not only poles corresponding to the original $1^{--}$ resonance states but also to the additional states with opposite parity which decouple in the free field limit. In some cases, these dynamically generated additional states might be negative norm ghosts or tachyons.

Motivation & Objective

• To analyze the renormalization of propagators for $1^{--}$ vector resonances in the antisymmetric tensor formulation of Resonance Chiral Theory.
• To identify inconsistencies arising in one-loop quantum corrections when using the antisymmetric tensor formalism.
• To determine whether new types of counterterms are required to cancel ultraviolet divergences in the self-energy graphs.
• To investigate the physical interpretation of additional poles in the propagator, including their potential to correspond to negative-norm states or tachyons.
• To extend the analysis to alternative formulations (vector and first-order formalisms) and assess the universality of the observed issues.

Proposed method

• Formalism is based on the antisymmetric tensor field $ R_{\mu\nu} $ to describe $1^{--}$ vector resonances in Resonance Chiral Theory.
• The two-point 1PI Green function is decomposed into transverse and longitudinal projectors $ \Pi^{T} $ and $ \Pi^{L} $, with the propagator derived via inversion.
• One-loop self-energy diagrams are computed using dimensional regularization, with vertices derived from the interaction Lagrangian involving derivatives and field strengths.
• Counterterms are introduced to cancel ultraviolet divergences, including a new kinetic term with two derivatives not present in the original leading-order Lagrangian.
• The structure of the full propagator is analyzed to identify poles beyond the physical $1^{--}$ resonance, including those from opposite-parity states.
• The results are compared across the antisymmetric tensor, vector, and first-order formalisms to assess generality of the findings.

Experimental results

Research questions

• RQ1 What counterterms are required to renormalize the self-energy of $1^{--}$ resonances in the antisymmetric tensor formulation of Resonance Chiral Theory at one-loop order?
• RQ2 Do the one-loop corrections generate additional poles in the propagator that correspond to unphysical states such as ghosts or tachyons?
• RQ3 Is the need for new kinetic counterterms with two derivatives a generic feature of spin-1 resonance formulations in effective field theories?
• RQ4 Can the presence of additional poles with opposite parity be linked to the absence of gauge symmetry in massive spin-1 theories?
• RQ5 How do the results in the antisymmetric tensor formalism compare to those in the vector and first-order formalisms in terms of consistency and unitarity?

Key findings

• The one-loop self-energy corrections in the antisymmetric tensor formulation require a new type of kinetic counterterm with two derivatives, not present in the original leading-order Lagrangian.
• The divergent part of the self-energy is canceled by counterterms with coefficients $ \delta M^2 \sim -\frac{40}{3}d_1^2(M/F)^2\lambda_\infty $ and $ \alpha \sim -\frac{40}{3}d_1^2(M/F)^2\lambda_\infty $, where $ \lambda_\infty $ is the divergent part of the loop integral.
• The full propagator contains two types of poles: one corresponding to the physical $1^{--}$ resonance and another corresponding to a state with opposite parity, which may be a negative-norm ghost or tachyon.
• The presence of such unphysical states is not limited to the antisymmetric tensor formalism but also appears in the vector and first-order formalisms, indicating a generic issue in massive spin-1 effective field theories.
• The results suggest that Resonance Chiral Theory, as currently formulated, may suffer from internal inconsistencies at the quantum level unless these additional degrees of freedom are properly accounted for.
• The phenomenon is linked to the known problem in massive vector field theories without gauge symmetry, where unphysical states can emerge unless carefully regulated.