[Paper Review] On the sigma sigma term

Peter C. Bruns|arXiv (Cornell University)|Oct 1, 2016

Quantum Chromodynamics and Particle Interactions2 references3 citations

TL;DR

This paper investigates the quark mass dependence of the sigma resonance (f₀(500)) pole position in the complex energy plane using a chiral Lagrangian with explicit resonance fields, combining one-loop self-energy calculations with constraints from Chiral Perturbation Theory and elastic unitarity. It provides first estimates for the leading quark mass dependence of the sigma resonance's mass and width, and explores its behavior at unphysically large quark masses, offering a complementary approach to Unitarized ChPT with reduced model dependence.

ABSTRACT

We give some estimates for the light-quark mass dependence of the pole position of the sigma ($f_{0}(500)$) resonance in the complex energy plane, with the help of a chiral Lagrangian for the resonance field and some input from hadronic models constrained by Chiral Perturbation Theory and elastic unitarity. We also speculate on the fate of the sigma resonance when the quark masses become unphysically large.

Motivation & Objective

• To establish a close connection between resonance Chiral Perturbation Theory and Unitarized ChPT for the sigma resonance.
• To reduce model dependence in quark mass extrapolations of the sigma resonance by combining explicit resonance fields with low-energy constraints.
• To estimate the leading quark mass dependence of the sigma resonance's pole position (mass and width) using a one-loop self-energy calculation.
• To explore the behavior of the sigma resonance at unphysically large quark masses, where ChPT breaks down.
• To provide a complementary framework to Unitarized ChPT for studying the dynamical generation of resonances.

Proposed method

• Construct a one-loop approximation to the sigma self-energy using a resonance chiral Lagrangian with explicit sigma field and pion interactions.
• Use the chiral Lagrangian with low-energy constants (LECs) constrained by ChPT and elastic unitarity in the chiral limit.
• Apply dimensional regularization and dispersive representations to handle divergences and extract real and imaginary parts of loop integrals.
• Fix counterterms (mass and wave-function renormalization) by requiring the pole position to match the physical resonance mass and width.
• Use analytic continuation to evaluate loop functions on the unphysical Riemann sheet for the second-sheet pole structure.
• Relate the residue of the scattering amplitude at the pole to vertex corrections and compositeness, informed by unitarity and analyticity.

Experimental results

Research questions

• RQ1 How does the pole position of the sigma resonance (f₀(500)) evolve with varying light-quark masses in the complex energy plane?
• RQ2 What is the leading quark mass dependence of the sigma resonance's mass and width, as derived from a resonance chiral Lagrangian?
• RQ3 How does the behavior of the sigma resonance change when quark masses are extrapolated to unphysically large values?
• RQ4 To what extent can the pole position be constrained by combining resonance ChPT with ChPT and unitarity in the chiral limit?
• RQ5 How do wave-function and coupling counterterms affect the residue and compositeness of the resonance in the effective field theory framework?

Key findings

• The paper provides first estimates for the leading quark mass dependence of the sigma resonance's pole position using a resonance chiral Lagrangian and one-loop self-energy calculations.
• The pole position is tightly constrained by elastic unitarity and the low-energy ππ interaction, consistent with constraints from Roy equations and ChPT.
• The imaginary part of the self-energy at the pole fixes the coupling constant g, ensuring the correct width of the resonance at one-loop order.
• The real part of the pole position is stabilized by fixing the mass counterterm to cancel the real part of the self-energy contribution.
• The study suggests that the sigma resonance may become broader or shift significantly at large quark masses, though extrapolations remain tentative.
• The method provides a complementary approach to Unitarized ChPT, potentially reducing model dependence through explicit resonance fields and unitarity constraints.