What are hidden symmetries in QCD? Are introducing them natural in context of QCD, or we need to modify QCD?

**An edit.** It seems that I've understood applications of hidden symmetries (HS) conception in QCD, and the question about them is changed. First I'll briefly explain applications of HS in QCD, and after that I''ll reformulate the question.

**Description of vector mesons in QCD sigma model through hidden symmetries**

Suppose we have QCD sigma model (in other words - chiral perturbation theory), in which the basic object is matrix of pseudoscalar goldstone bosons $U$. It is transformed as $U \to e^{i\epsilon_{L}}Ue^{-i\epsilon_{R}}$ under global chiral transformations generated by $\left[U_{L}(3)\times U_{R}(3)\right]_{\text{global}}$ QCD group (here I don't write about anomaly effects), and sigma model action is invariant under this transformations. Improved by Wess-Zumino term (which violates some extra symmetries which are present in sigma model, but are absent in QCD fundamental theory; in fact WZ term generates all anomaly effects of QCD), this model describes correctly all processes involved only pseudoscalar mesons, SM gauge bosons and bounded states with half-integer spin (like nucleons and Delta-isobars), which arise as sigma model topological solutions called skyrmions.

But there isn't description of vector and axial-vector mesons in this model, so for being the correct low-energy theory of hadronic processes it has to be expanded. In early 80th people've realized how to include systematic description of vector mesons in sigma model. They have seen that $U$ can be realized through product

$$

\tag 1 U \equiv \zeta_{L}^{\dagger}\zeta_{M}\zeta_{R},

$$

where transformation laws of matrices $\zeta_{L/R/M}$ are

$$

\zeta_{L} \to G_{L/R}(x)g_{L/R}\zeta_{L/R}, \quad \zeta_{M} \to G_{L}(x)\zeta_{M}G_{R}^{\dagger}(x),

$$

and $g_{L/R} \in U_{L/R}(3), G_{L/R} \in \hat{U}_{L/R}(3)$; $\hat{U}_{L}(3)\times \hat{U}_{R}(3)$ is local symmetry. In fact, $(1)$ isn't transformed under $\hat{U}_{L}(3)\times \hat{U}_{R}(3)$, so that this symmetry is called **"hidden"**. We can now write down the most general action which contains $\hat{A}_{L/R}, \zeta_{L/R/M}$, where $\hat{A}_{L/R}$ are gauge fields associated with hidden symmetry.

The next step is the primary idea of hidden local symmetry approach in QCD. We fix "hidden" symmetry gauge so that $\zeta_{L/R} = e^{\pm i\frac{\Pi}{f_{\pi}}}, \zeta_{M} = 1$, where $\Pi$ denotes the matrix of pseudoscalar mesons, and say that hidden symmetry gauge fields $a_{L/R}$ in this gauge have quantum numbers of vector and axial-vector mesons. This breaks down the "hidden" symmetry, and the lagrangian becomes to be functional only of $U, a_{L/R}$. We then say that mass terms for $a_{L/R}$ are generated by Higgs mechanism and kinetic terms arise due to quantum fluctuations. Finally, if we want to integrate out $a_{L/R}$ fieds in the limit of their large masses, then we turn back to the initial sigma model. This provides the following statement: **"linear" model with "hidden" symmetry **$[U_{L}(3)\times U_{R}(3)]_{\text{global}}\times[\hat{U}_{L}(3)\times \hat{U}_{R}(3)]_{\text{local}}$ **is gauge equivalent to nonlinear sigma model based on spontaneously broken symmetry** $[U_{L}(3)\times U_{R}(3)]_{\text{global}}/U_{V}(3)$.

**The question**

It can be shown that introducing the vector mesons via hidden symmetry approach "almost" coincide with introducing them as background gauge fields with quantum numbers of vector and axial vector mesons in terms of minimally broken chiral gauge symmetries (masses of background gauge fields break this local symmetry); the last method is called "Massive Yang-Mills". Thus we may write down the full chiral invariant lagrangian which contains information about vector, pseudovector and pseudoscalar mesons, SM gauge bosons and half-integer spin hadrons.

The question is about the description of some anomalous processes in this model. The topic of question can be demonstrated on case of famous anomalous $\pi^{0} \to 2\gamma$ process. The term in lagrangian $\frac{N_{c}e^{2}}{48 \pi^{2} f_{\pi}}\pi^{0} F_{EM} \wedge F_{EM}$ which describes this process is generated by $U_{EM}(1)$ gauging of Wess-Zumino term. Corresponding matrix element gives amplitude of $\pi^{0} \to 2\gamma$ process with high accuracy in compare with experimental data. However, if we include vector mesons, then process $\pi^{0} \to \rho^{0}\omega^{0}\to 2\gamma$ will also give contribution, and it seems that the theory will give incorrect predictions (full decay width will be larger than experimental decay width). So, in general, is it true that theory has problems with description of such anomalous processes, or I'm wrong?