[Paper Review] Contributions of loops with dynamical vector mesons to masses and decay constants of pseudoscalar mesons and their quark mass dependence

Carla Terschlüsen, Stefan Leupold|arXiv (Cornell University)|Apr 6, 2016

Relativity and Gravitational Theory2 references3 citations

TL;DR

This paper investigates the impact of one-loop diagrams involving dynamical vector mesons on the masses and decay constants of pseudoscalar mesons, using a relativistic Lagrangian for the pseudoscalar octet and vector meson nonet. It finds that dynamical vector mesons significantly alter meson properties—especially the pion mass and decay constant—already at bare pion masses above 250 MeV, challenging the validity of purely static or chiral perturbation theory approximations in lattice QCD extrapolations.

ABSTRACT

The contributions of one-loop diagrams with dynamical vector mesons to masses and decay constants of pseudoscalar mesons are determined. Hereby, a relativistic Lagrangian for both the pseudoscalar-meson octet and the vector-meson nonet is used. The vector mesons are given in the antisymmetric tensor representation. Both the differences between static and dynamical vector mesons and the differences between calculations with and without vector mesons are studied as functions of the light quark mass.

Motivation & Objective

• To assess the quantitative impact of dynamical vector mesons on pseudoscalar meson masses and decay constants in chiral perturbation theory.
• To compare results from loops with static versus dynamical vector mesons, focusing on quark mass dependence.
• To evaluate how vector meson loops affect the quark mass dependence of meson observables, especially in lattice QCD extrapolation schemes.
• To determine whether standard chiral perturbation theory underestimates vector meson contributions in low-energy QCD.

Proposed method

• Uses a relativistic Lagrangian with pseudoscalar mesons (octet) and vector mesons (nonet) in the antisymmetric tensor representation.
• Calculates one-loop contributions to pseudoscalar two-point functions with dynamical vector mesons, employing renormalized loop integrals from prior work (Terschlüsen & Leupold, 2016).
• Compares results with those from standard chiral perturbation theory (χPT) and with static vector meson approximations at next-to-next-to-leading order (N2LO).
• Fixes the kaon mass at its physical value and varies the bare pion mass to study quark mass dependence.
• Focuses on the V–P–P vertex interaction as the dominant vector meson coupling, excluding mass splittings and V–V–P terms for simplicity.
• Analyzes finite parts of loop diagrams after renormalization, using physical input from ρ–π–π and ρ–K–K* couplings.

Experimental results

Research questions

• RQ1 How do dynamical vector mesons alter the masses and decay constants of pseudoscalar mesons compared to static vector meson approximations?
• RQ2 To what extent do vector meson loops modify the quark mass dependence of pseudoscalar meson observables in chiral perturbation theory?
• RQ3 Are the effects of vector mesons in one-loop diagrams negligible at low pion masses, as suggested by power-counting estimates?
• RQ4 How do results from a model with dynamical vector mesons compare to those from pure χPT in the context of lattice QCD extrapolation?
• RQ5 What is the quantitative significance of vector meson loops for the pion mass and decay constant in the two-flavor chiral limit?

Key findings

• The difference between static and dynamical vector mesons becomes significant for bare pion masses above approximately 250 MeV.
• Dynamical vector mesons induce a 10% correction to the pion mass ratio (Mπ / M̄π)² in the two-flavor chiral limit, primarily due to K*–pion loops.
• The deviation of the physical pion mass from the bare pion mass is substantial even at low quark masses, contradicting naive power-counting expectations.
• Vector meson loops generate effects scaling as Q⁴ / (m_V² (4πF)²), which are not negligible when Q ≈ M_K, explaining the observed 10% correction.
• The pion decay constant is also significantly modified by dynamical vector meson loops, with effects comparable in size to those from kaon and η tadpole diagrams.
• Pure χPT calculations grossly underestimate the importance of vector mesons in quark mass dependence studies, especially when extrapolating lattice QCD results to the physical point.