Renormalization and additional degrees of freedom within the chiral effective theory for spin-1 resonances

Abstract

We study in detail various aspects of the renormalization of the spin-1 resonance propagator in the effective field theory framework. First, we briefly review the formalisms for the description of spin-1 resonances in the path integral formulation with the stress on the issue of propagating degrees of freedom. Then we calculate the one-loop 1 −− meson self-energy within the Resonance chiral theory in the chiral limit using different methods for the description of spin-one particles, namely the Proca field, antisymmetric tensor field and the first order formalisms. We discuss in detail technical aspects of the renormalization procedure which are inherent to the power-counting non-renormalizable theory and give a formal prescription for the organization of both the counterterms and one-particle irreducible graphs. We also construct the corresponding propagators and investigate their properties. We show that the additional poles corresponding to the additional one-particle states are generated by loop corrections, some of which are negative norm ghosts or tachyons. We count the number of such additional poles and briefly discuss their physical meaning.

On the chiral expansion of vector meson masses

“…However, it is unclear how to order this plethora of terms and how to consider the loop effects implied [9,22]. In both approaches the challenge is caused by meson resonances that are close to the vector mesons in mass [23][24][25].…”

Chiral behavior of vector meson self-energies

“…Thus the scalar degrees of freedom of the four-vector fields decouple from the PGBs. For a review of the problems with additional degrees of freedom, and other general aspects in the description of spin-1 fields in an effective field theory framework, we refer to the recent study in [55]. We also note that we set the external source fieldsv µ ,â µ ,p, introduced in the general ChPT framework [11,17], to zero.…”

“…It is straightforward to construct the corresponding counterterm Lagrangian for such energy-dependent terms, see e.g. [49,55]. We will not need the explicit form of these terms here.…”

The vector form factor at the next-to-leading order in 1/N C : chiral couplings L9(μ) and C88(μ) − C90(μ)

“…[10] and [11], respectively. In spite of all the complexity associated with the still not so well understood renormalization of RχT [11,12,13,14,15,16], these pioneering calculations showed the potential predictability at the NLO in 1/N C .…”