Chiral Properties of Baryon Interpolating Fields

TL;DR

This paper analyzes the chiral transformation properties of baryon interpolating fields, deriving identities that restrict their allowed multiplets and revealing their relation to Lorentz representations and axial charges.

Contribution

It provides a comprehensive classification of baryon interpolating fields based on chiral and Lorentz symmetries, including new restrictions from Fierz identities.

Findings

Only specific chiral multiplets are allowed for nucleons and Deltas.

Baryons come in two varieties with distinct Abelian axial charges.

Relations among operators reduce the number of independent baryon fields.

Abstract

We study the chiral transformation properties of all possible local (non-derivative) interpolating field operators for baryons consisting of three quarks with two flavors, assuming good isospin symmetry. We derive and use the relations/identities among the baryon operators with identical quantum numbers that follow from the combined colour, Dirac and isospin Fierz transformations. These relations reduce the number of independent baryon operators with any given spin and isospin. The Fierz identities also effectively restrict allowed baryon chiral multiplets. It turns out that the chiral multiplets of the baryons are equivalent to their Lorentz representation. For the two independent nucleon operators the only permissible chiral multiplet is the fundamental one $(\frac12,0)\oplus(0,\frac12)$. For the $\Delta$, admissible Lorentz representations are $(1,\frac12)\oplus (\frac12,1)$ and $(\frac32,0)\oplus(0,\frac32)$. In the case of the $(1,\frac12)\oplus (\frac12,1)$ chiral multiplet the $I(J)=\frac32(\frac32)$ $\Delta$ field has one $I(J)=\frac12(\frac32)$ chiral partner; otherwise it has none. We also consider the Abelian ($U_A(1)$) chiral transformation properties of fields and show that each baryon comes in two varieties: 1) with Abelian axial charge +3; and 2) with Abelian axial charge -1. In case of the nucleon these are the two Ioffe's fields; in case of the $\Delta$, the $(1,\frac12)\oplus (\frac12,1)$ multiplet has Abelian axial charge -1 and the $(\frac32,0)\oplus(0,\frac32)$ multiplet has Abelian axial charge +3.