Broken Scale Invariance, Current Algebra, and Massive "Gravitation." I. General Formulation

A general analysis is given of the interaction of mesons of J^P=0^±, 1^±, and 2⁺ obeying the principles of broken scale invariance in the tree and seagull approximations. In analogy with current algebra, where one assumes that the vector and axial-vector currents are dominated by J^P=1^± and 0^± mesons in a field-current identity, we assume that the stress tensor ⊖^μν is dominated mainly by the 2⁺ and 0⁺ f, f^′, σ, and σ^′ mesons in a field-stress-tensor identity. A consistent formalism is seen to require also certain nonpole f-meson mass terms in ⊖^μν. With the usual smoothness conditions, the dynamics can be conveniently characterized by introducing an effective Lagrangian. The conservation law ∂_ν⊖^μν=0 and the Poincaré-group conditions then imply that (i) the f_μνⁱ(x) fields (i=1, 2, …) of the f, f^′, etc., mesons couple with all other "matter fields" (e.g., J^P=0^±, 1^± mesons) by making the usual matter Lagrangian of current algebra "generally covariant" by replacing the Lorentz metric η_μν by a "metric" formed from g_μνa≡η_μν+Σiλ_aif_μνⁱ (a=1, 2, …). (ii) The kinetic-energy part of the f-meson self-couplings must have the form of Einstein Lagrangians formed using the g_μνa, e.g., √-g_ag_a^μνR_μνa where R_μν is the contracted curvature tensor. (iii) Improvement for the spin-zero parts of the stress tensor is obtained by including "curvature" couplings, e.g., π²√-g_aR_a, where R_a is the curvature scalar formed from g_μνa. In general, then, the f-meson couplings are analogous to very strong gravitational couplings, with the f-meson mass terms breaking the gravitational gauge invariance. For the situation where one has only one f meson present our "metric space" is analogous to that of Zumino. However, the "metric space" considered here is considerably more complicated than such a Riemannian space as more than one metric, g_μνa, a=1, 2, …, is defined on it, and hence by algebraic combinations an infinite number of "metrics" exist. (We note that in general these metrics will depend nonlinearly on the f-meson fields.)

Broken scale invariance is introduced through a new postulate which requires that the improved Belinfante stress tensor and its trace play a fundamental role as sources of the J^P=2⁺, 0⁺ mesons with a universal coupling strength. The universality also leads to new relations of the type g_f=F_σm_f², etc., between the f-and σ-meson interpolating constants which resemble the Kawarabayashi-Suzuki-Riazuddin-Fayyazuddin-type relations in current algebra. The form of the vector current in the presence of broken scale invariance is derived. The condition of scale breaking implies that the vector current has canonical scale dimension 3, and the apparent conflict of conservation of vector current with the f couplings is resolved. Experimental tests of the present formalism are indicated here and will be examined in detail in a subsequent paper.