Proving the low energy theorem of hidden local symmetry.

Based on the Ward-Takahashi identity for the BRS symmetry, we prove to all orders of the loop expansion the low energy theorem of hidden local symmetry for the vector mesons (KSRF (I) relation) in the U(N)L × U(N)R / U(N)V nonlinear chiral Lagrangian. ∗Address after April 1, 1993 : Department of Physics, Kyoto University, Kyoto 606-01, Japan Hidden local symmetry (HLS) is a natural framework for describing the vector mesons in a manner consistent with the chiral symmetry of QCD[1]. The HLS Lagrangian yields at tree level a successful phenomenology for the pions and the ρ mesons. By choosing a parameter a = 2 in this HLS Lagrangian, we have[2]: 1) Universality of the ρ meson coupling[3] gρππ = g (g: HLS gauge coupling); 2) KSRF relation[4] (version II) mρ = 2f 2 πg 2 ρππ; 3) ρ meson dominance of the electromagnetic form factor of the pion gγππ = 0[3]. Most remarkably, we further obtain an a-independent relation[5] gρ/gρππ = 2f 2 π , with gρ being the strength of ρ-γ mixing. This is actually another version of the celebrated KSRF relation (KSRF I) and is a decisive test of the HLS in the hadron physics. Since it follows from the symmetry structure alone, it was conjectured[5] to be a low energy theorem valid at zero momenta for any Lagrangian possessing the HLS and was further proved [6] at tree level. If it is indeed a genuine low energy theorem surviving the loop corrections, analogues of such a relation would also be useful for strongly coupled Higgs models or models of the dynamical electroweak symmetry breaking. Actually this HLS Lagrangian can be straightforwardly applied to such models[7]. Quite recently, it was shown in Landau gauge[8] that the above “low energy theorem” as well as the a = 2 tree-level results at zero momenta is not altered by the one-loop corrections, thus strongly suggesting that it may indeed be a true low energy theorem[9]. These results are actually the relations coming from O(p) operators or dimension-2 operators (counting only dimensions of the gauge fields and the derivatives). In this paper we shall prove that the above “low energy theorem” of HLS actually holds at any loop order, based on the Ward-Takahashi (WT) identity for the BecchiRouet-Stora (BRS) symmetry. By restricting ourselves to the dimension-2 operators mentioned above, we can prove it by mathematical induction in quite the same way as the renormalizability proof for gauge theories[10] and two-dimensional nonlinear