3 Different schemes for inclusion of Vector Mesons in effective lagrangians

In the previous section, we have discussed the implicit role played by vector mesons in chiral lagrangians, through saturation of the counterterm contributions. Beyond the very low energy region, where a lowest-order power expansion in is sufficient, one may try to extend the validity of chiral lagrangians through the explicit introduction of vector meson fields. Two main lines have been developed, the massive Yang Mills (YM) approach proposed by Meissner [1] among other authors, and the Hidden Symmetry (HS) scheme proposed by Bando et al. [2]. In the former, spin-1 fields are introduced in the ungauged lagrangian through the usual covariant derivatives. Restricting, for simplicity of exposition, to the sole SU(2) fields, i.e. pions and mesons, one starts with the usual lowest order lagrangian

and then makes the substitution

where now contains only the pion field and is the meson field. This results in the lowest order lagrangian

where, for the sake of the present argument, we have shown only some terms involving interactions between neutral mesons and pions. The gauge coupling constant g is determined by neutral decay, , to have the value . In the massive Yang Mills scheme, photonic interactions are then introduced through conventional Vector Meson Dominance, i.e. through the lagrangian

which mediates all transitions between photons and matter fields. Applying this rule to the decay , one obtains the relation and . The experimental values MeV and KeV are then fitted with . The dots in eq.(17) refer to further quadratic interactions between and an even number of pion fields, but no mass terms for the vector mesons. This is to be contrasted with what happens in the lagrangian proposed by Bando et al.[2], in which the vector mesons enter explicitly as gauge fields of a ``hidden'' local symmetry of the chiral lagrangian, whereas the electro-weak gauge bosons are explicitly introduced through the usual covariant derivatives. A Higgs-like mechanism then generates the vector meson masses.

To wit, in the ``hidden symmetry approach'' of Bando et al.[2] the most general lagrangian containing pseudoscalar, vector and (external) electroweak gauge fields with the smallest number of derivatives, is given at the lowest order, by the linear combination , a being an arbitrary parameter, of the two SU(3) symmetric lagrangians

The matrices and contain the pseudoscalars fields, P, and the unphysical (or compensator) scalar fields, , that will be absorbed to give a mass to the vector mesons

The full covariant derivative is

where only the photon field, , (but not its weak partners, as before) has been explicitly shown, and P and V now stand for the SU(3) octet and nonet matrices

The lagrangian can be reduced to the chiral lagrangian (1) for any value of the parameter a [2]. In fact, working in the unitary gauge ( ) to eliminate the unphysical scalar fields and substituting the solution of the equation of motion for , the part vanishes and becomes identical to the non linear chiral lagrangian (1).

The ``hidden symmetry'' lagrangian (19) can be easily seen to contain, among other things, a vector meson mass term, the pseudoscalar weak decay constants, the vector-photon conversion factor and the couplings of both vectors and photons to pseudoscalar pairs. The latter can be eliminated fixing a = 2, thus incorporating conventional vector-dominance in the electromagnetic form-factors of pseudoscalars. Returning, as before, to the simpler SU(2) case, for one has

In the above equation, the constant a has been fixed to the value a=2 in order to recover the relation between and the couplings, previously discussed. This lagrangian obviously reproduces all the lowest order results, as the previous one does, however it does not contain couplings between two or more - mesons and an even number of pion fields.

It is not yet clear if these two lagrangians are fully equivalent or not. At first sight, processes involving one meson and four or more pion fields could be used to discriminate between the two [9], since these processes could proceed through multiple -pion interactions in the YM scheme, while not so in the HS approach. However, it has been pointed out that axial vector terms, not yet introduced in the above lagrangians, would probably play a role. This might render experimental discrimination rather difficult [10].

To the two schemes just discussed, one must also add the simplest and oldest of them all, i.e. conventional Vector Meson Dominance [11], in which all photon-pseudoscalar interactions proceed through vector meson fields with the same interaction lagrangian seen before and the vector meson fields are introduced through the covariant derivatives in the free pion lagrangian . The lagrangian thus obtained

contains the same vertices as the previous two, but there are of course no multipion interactions, which in conventional VMD are basically frozen into the role played by the vector mesons.

After this brief introduction, we shall now discuss some vector meson decay processes which can be measured at DANE , and in which the validity of the theoretical scenarios we have illustrated can be tested.