Hidden Local Symmetry at One Loop

We show that one-loop corrections with the hidden gauge boson loop preserve in the low energy limit all the successful tree level predictions of the hidden local symmetry in the SU(2)L×SU(2)R/SU(2)V chiral Lagrangian. Most amazingly, the ρ meson dominance of the pion electromagnetic form factor survives the one-loop corrections at any momentum, if and only if we take the parameter choice a = 2. For the choice a = 1 (“vector limit”), a is not renormalized (Za = 1) and no deviation from a = 1 is induced by the loop effects of the hidden gauge bosons. Actually, a = 1 is a nontrivial ultraviolet fixed point. email:[email protected] email:[email protected] Work supported in part by the Takeda Science Foundation and the Ishida Foundation, and also by the International Collaboration Program of the Japan Society for the Promotion of Science. It is now a popular notion to identify the ρmeson with the dynamical gauge boson of the hidden local symmetry in the SU(2)L×SU(2)R/SU(2)V nonlinear chiral Lagrangian[1]. By setting a parameter choice a = 2 in this hidden local symmetry Lagrangian, we can successfully reproduce three phenomenological facts[2]: The ρ-coupling universality, gρππ = g, the KSRF relation (II) , m 2 ρ = 2f 2 πg 2 ρππ[3], and the ρ meson dominance of the electromagnetic form factor of the pion[4]. Most remarkably, we obtain the celebrated KSRF (I) relation, gρ = 2f 2 πgρππ, as an aindependent relation which is actually characteristic to the hidden local symmetry and hence may be regarded as a “low energy theorem” of the symmetry[5]. In fact it was proved to be a low energy theorem at tree level[6]. Moreover, this hidden local symmetry Lagrangian has been applied to the strongly coupled Higgs model and/or the effective theory of the technicolor including the techni-ρ meson (“BESS model”)[7]. Recently, one-loop corrections of the pion loop have been extensively studied in the chiral Lagrangian (chiral perturbation theory[8, 9]), which succeeded in reproducing systematically the low energy hadron physics slightly away from the low energy limit dictated by the chiral symmetry. However, this approach appears to fail in reproducing the higher energy region, even considerably lower than the ρ meson pole, and needs explicit degree of a new field, the ρ meson. Amazing fact is that the finite part of the one-loop counter terms in the chiral perturbation theory is saturated by the tree level effects of the ρ meson[10]. Thus the effective Lagrangian including the ρmeson, if it yields successful tree level results, should be a good starting point towards constructing a true “effective field theory” including the quantum corrections. As such we take the hidden local symmetry Lagrangian mentioned above. Actually, it was pointed out[10, 11] that loop effects of the vector mesons are crucial to the π−π mass difference[12]. Our goal is thus to promote the hidden local symmetry Lagrangian into an “effective field theory” valid up to beyond the ρ meson pole by including the full quantum effects.