Chiral symmetry breaking

In massless QCD, a quark condensate breaks

S U (N_{f})_{L} \times S U (N_{f})_{R}

to the diagonal vector subgroup. The associated Goldstone modes are the light pseudoscalar mesons.

Table of contents

Linear sigma model

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The linear sigma model packages scalar and pseudoscalar fields into a linearly transforming multiplet. A symmetry-breaking potential gives a massive radial sigma mode and massless pions before explicit breaking is added.

Chiral perturbation theory

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Chiral perturbation theory is the low-energy effective field theory of pseudo-Goldstone bosons from chiral symmetry breaking. It expands amplitudes in derivatives and light-quark masses over the chiral scale.

Leading pion mass from a two-flavour condensate ( $m_{π}^{2} = 2 v (m_{u} + m_{d}) / F^{2}$ )

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For condensate normalization

⟨ \overset{q}{ˉ}_{L j} q_{R i} ⟩ = - v δ_{ij}

, replacing the condensate matrix by

- vU

maps the real quark mass matrix

M = diag (m_{u}, m_{d})

v tr [M (U + U^{†})]

. Since

(π \cdot τ)^{2} = ∣ π ∣^{2} I

, expansion gives a constant minus

v (m_{u} + m_{d}) ∣ π ∣^{2} / F^{2}

. Comparison with

- m_{π}^{2} ∣ π ∣^{2} /2

gives the displayed common leading mass. Strong isospin-breaking mass splitting enters beyond this leading action; electromagnetic effects are also excluded.

Uniqueness of the two-derivative two-flavour chiral action ( $L_{2} = F^{2} tr (\partial_{μ} U^{†} \partial^{μ} U) /4$ )

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For

U \in S U (2)

with

U \mapsto A U B^{- 1}

, the chiral group acts transitively. A derivative-free invariant is therefore constant. At

U = I

, its diagonal stabilizer acts on tangent vectors by the irreducible three-dimensional rotation representation; the only invariant symmetric metric is a multiple of