Is a q-number δ required in (8,8) chiral symmetry breaking?

Abstract

We investigate the assumption that δ (the chiral-invariant scale-breaking part of the Hamiltonian density) has a dimension in the (8,8) model of chiral symmetry breaking. Using a spectral function sum rule for the vacuum expectation value of the chiral symmetry breaking Hamiltonian density u( x), restrictions for the mass m ϵ and the width Γ ϵ of η 0+(700–1000) are derived from this assumption. They imply that a q-number δ with dimension l δ ⩾ 1 is incompatible with Γ ϵ ⩽ m ϵ and m ϵ ⩾ 600 MeV. On the other hand, if δ is a c-number ( l δ = 0) it is shown that Γ ϵ ⩽ m ϵ is incompatible with m ϵ ⩾ 690 MeV while if Γ ϵ is below m ϵ (e.g. Γ ϵ ⩽ 0.9 m ϵ ) only an ϵ with a lower mass ( m ϵ ⩽ 670 MeV) is allowed. The present phenomenological situation concerning ϵ makes impossible a definite conclusion on the dimensional content of δ in the (8,8) model. However, experiments favor values of m ϵ and Γ ϵ which imply that δ is a q-number consisting of at least two terms with different dimensions and nonvanishing vacuum expectation values. Quite generally, our results may be used to check if a particular theoretical prediction for m ϵ and for Γ ϵ is compatible with a c-number δ in the (8,8) model. We assume in the paper that u has a dimension and that conventional ϵ-saturation is valid. However, independent of the latter assumption it is shown here that in the (8,8) chiral symmetry breaking scheme the dimension of u is between 0 and 4 if δ is a c-number (this may also be seen in the ( 3,3) ⊗ (3, 3) model from the analogous sum rule).