Abstract
A method to find relations between the operators in the mesonic Lagrangian of Chiral Perturbation Theory at order p6 is presented. The procedure can be used to establish if the basis of operators in the Lagrangian is minimal. As an example, we apply the method to the two-flavor case in the absence of scalar and pseudo-scalar sources (s=p=0), and conclude that the minimal Lagrangian contains 27 independent operators.
Highlights
The global chiral symmetry of the QCD Lagrangian for vanishing quark masses, and its spontaneous breaking to the diagonal group, characterizes the strong interactions among the lightest hadronic degrees of freedom – the pseudoscalar mesons – at low energies
An additional relation among the operators in the basis of [5] for the n = 2 case was proven [6], where no additional manipulations but those already used in [5] were required. This showed that the derivation of an algorithm to exhaust all possible algebraic conditions among the L6 operators imposed by partial integration, equations of motion, Bianchi identities and, Cayley–Hamilton relations, is a nontrivial task
The large number of low-energy constants in the mesonic chiral Lagrangian of order p6 makes their determination by direct comparison with the experiment rather difficult
Summary
The global chiral symmetry of the QCD Lagrangian for vanishing quark masses, and its spontaneous breaking to the diagonal group, characterizes the strong interactions among the lightest hadronic degrees of freedom – the pseudoscalar mesons – at low energies. An additional relation among the operators in the basis of [5] for the n = 2 case was proven [6], where no additional manipulations but those already used in [5] were required This showed that the derivation of an algorithm to exhaust all possible algebraic conditions among the L6 operators imposed by partial integration, equations of motion, Bianchi identities and, Cayley–Hamilton relations, is a nontrivial task. The question about the minimality of the O(p6) chiral Lagrangian is proper and, to the best of our knowledge, remains unanswered It is the aim of the present work to describe a method that provides necessary conditions for the existence of additional relations between the operators of the L6 Lagrangian, and to show its application to the two-flavor case when massless quarks are considered.
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