Abstract
A new mathematical framework for the diagonalization of the nondiagonal vector–scalar and axial-vector–pseudoscalar mixing in the effective meson Lagrangian is described. This procedure has unexpected connections with the Hadamard product of n×n matrices describing the couplings, masses, and fields involved. The approach is argued to be much more efficient as compared with the standard methods employed in the literature. The difference is especially noticeable if the chiral and flavor symmetry is broken explicitly. The paper ends with an illustrative application to the chiral model with broken U(3)L×U(3)R symmetry.
Highlights
The QCD Lagrangian with n massless flavors is known to possess a large global symmetry, namely the symmetry under U (n)V × U (n)A chiral transformations of quark fields
Resorting to arguments pertaining to the Lie algebra associated with chiral transformations and to Chisholm’s theorem, we have shown that one may always use the most general linear shifts of
V μ and Aμ fields (22) for dealing with V –σ and A–φ mixing in chiral models without compromising the chiral symmetry properties of the Lagrangian, as long as one admits the corresponding new transformation laws (24)–(25) for the shifted fields. This result is independent of the number of flavors and works even when the U (n) × U (n) chiral symmetry is explicitly broken to U (1)n
Summary
The QCD Lagrangian with n massless flavors is known to possess a large global symmetry, namely the symmetry under U (n)V × U (n)A chiral transformations of quark fields. It follows from the gap equation that the constituent quark mass matrix M is diagonal but its eigenvalues are all unequal and nonzero M = diag(Mu, Md, Ms) This leads to a new mixing between the vector, V μ, and the scalar, σ , fields and, as a result, to the redefinition of the longitudinal component of the vector field: V μ = V μ + κ ∂μσ. The effects of flavor symmetry breaking, collected in the matrix M, do not spoil the U (3) × U (3) group transformation laws of the fields, M enters the transformations It shows that a linear replacement of variables that diagonalizes the free part of the meson Lagrangian is legitimate, unique, and does not ruin the pattern of explicit symmetry breaking of the theory. The model may have some interest for the readers who might wish to use it to take into account the explicit and flavor symmetry breaking effects in hot/dense/magnetized matter [32,33], or apply it in the study of hybrid stars [34], where eight-quark interactions seem to have an important impact
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