The rates and spectra of the anomalous $etatopi^+pi^-gamma$ and $eta topi^+pi^-gamma$ decays are calculated. The approach is based on the effective meson Lagrangian obtained in the Nambu-Jona-Lasinio model with vector and axial-vector mesons by integ
rating out quark fields. The resulting action is affected by mixing between members of pseudoscalar $J^{PC}=0^{-+}$ and axial-vector $1^{++}$ nonets that violates some low-energy theorems. In this note we point out that a gauge covariant procedure to diagonalize this mixing allows for consistent description of the $etatopi^+pi^-gamma$ and $eta topi^+pi^-gamma$ decays.
We reconsider the contribution due to $pi a_1$-mixing to the anomalous $gammatopi^+pi^0pi^-$ amplitude from the standpoint of the low-energy theorem $F^{pi}=e f_pi^2 F^{3pi}$, which relates the electromagnetic form factor $F_{pi^0togammagamma}=F^pi$
with the form factor $F_{gammatopi^+pi^0pi^-}=F^{3pi}$ both taken at vanishing momenta of mesons. Our approach is based on a recently proposed covariant diagonalization of $pi a_1$-mixing within a standard effective QCD-inspired meson Lagrangian obtained in the framework of the Nambu-Jona-Lasinio model. We show that the two surface terms appearing in the calculation of the anomalous triangle quark diagrams or AVV- and AAA-type amplitudes are uniquely fixed by this theorem. As a result, both form factors $F^pi$ and $F^{3pi}$ are not affected by the $pi a_1$-mixing, but the concept of vector meson dominance (VMD) fails for $gammatopi^+pi^0pi^-$.
A realization of the composite Higgs scenario in the context of the effective model with the $SU(2)_Ltimes U(1)_R$ symmetric four-Fermi interactions proposed by Miransky, Tanabashi and Yamawaki is studied. The model implements Nambus mechanism of dyn
amical electroweak symmetry breaking leading to the formation of $bar tt$ and $bar bb$ quark condensates. We explore the vacuum structure and spectrum of the model by using the Schwinger proper-time method. As a direct consequence of this mechanism, the Higgs acquires a mass in accord with its experimental value. The present prediction essentially differs from the known overestimated value, $m_chi= 2m_t$, making more favourable the top condensation scenario presented here. The mass formulas for the members of the second Higgs doublet are also obtained. The Nambu sum rule is discussed. It is shown that the anomalous $U(1)_A$ symmetry breaking modifies this rule at next to leading order in $1/N_c$.
Within the context of an extended Nambu - Jona-Lasinio model, we analyze the role of the axial-vector $a_1(1260)$ and $a_1(1640)$ mesons in the decay $tauto u_tau rho^0pi^-$. The contributions of pseudoscalar $pi$ and $pi (1300)$ states are also cons
idered. The form factors for the decay amplitude are determined in terms of the masses and widths of these states. To describe the radial excited states $pi (1300)$ and $a_1(1640)$ we introduce two additional parameters which can be estimated theoretically, or fixed from experiment. The decay rate and $rhopi$ mass spectrum are calculated.
An extended Nambu-Jona-Lasinio (NJL) model with chiral group $U(2)times U(2)$ and spin-0 and spin-1 four quark interactions is used to develop the gauge covariant approach to the diagonalization of the $pi-a_1$ mixing in the presence of electroweak f
orces. This allows for manifestly gauge covariant description of both the non-anomalous and anomalous parts of the effective Lagrangian. It is shown that in the non-anomalous sector the theory is equivalent to the standard non-covariant approach.
It is found that in presence of electroweak interactions the gauge covariant diagonalization of the axial-vector -- pseudoscalar mixing in the effective meson Lagrangian leads to a deviation from the vector meson and the axial-vector meson dominance
of the entire hadronic electroweak current. The essential features of such a modification of the theory are investigated in the framework of the extended Nambu-Jona-Lasinio model with explicit breaking of chiral $U(2) times U(2)$ symmetry. The Schwinger-DeWitt method is used as a major tool in our study of the real part of the relevant effective action. Some straightforward applications are considered.
The anomalous decays $f_1(1285)torho^0pi^+pi^-$ and $a_1(1260)toomegapi^+pi^-$ violating natural parity for vectors and axial-vectors are studied in the framework of the Nambu -- Jona-Lasinio model. We consider the Lagrangian with $U(2)_Ltimes U(2)_R
$ chiral symmetric four quark interactions. The theory is bosonized and corresponding effective meson vertices are obtained in the leading order of $1/N_c$ and derivative expansions. The uncertainties related with the surface terms of anomalous quark triangle diagrams are fixed by the corresponding symmetry requirements. We make a numerical estimate of the decay widths $Gamma (f_1(1285)torho^0pi^+pi^-)=2.78, mbox{MeV}$ and $Gamma (a_1(1260)toomegapi^+pi^-)=87, mbox{keV}$. Our result on the $f_1(1285)torho^0pi^+pi^-$ decay rate is in a good agreement with experiment. It is shown that a strong suppression of the $a_1(1260)toomega pipi$ decay is a direct consequence of destructive interference between box and triangle anomalies.
Using the Nambu--Jona-Lasinio model with the $U(2)times U(2)$ chiral symmetric effective four-quark interactions, we derive the amplitude of the radiative decay $f_1(1285) topi^+pi^-gamma$, find the decay width $Gamma (f_1topi^+pi^-gamma)=346,mbox{ke
V}$ and obtain the spectral di-pion effective mass distribution. It is shown that in contrast to the majority of theoretical estimates (which consider the $a_1(1260)$ meson exchange as the dominant one), the most relevant contribution to this process comes out from the $rho^0$-resonance exchange related with the triangle $f_1rho^0gamma$ anomaly. The spectral function is obtained to be confronted with the future empirical data.
Within the framework of the extended Nambu -- Jona-Lasinio model, we calculate the matrix element of the $tau to f_1(1285) pi^{-} u_{tau}$ decay, obtain the invariant mass distribution of the $f_1pi$ -system and estimate the branching ratio Br$(tau
to f_1 pi^{-} u_{tau})=4.0times 10^{-4}$. The two types of contributions are considered: the contact interaction, and the axial-vector $I^G(J^{PC})=1^-(1^{++})$ resonance exchange. The latter includes the ground $a_1(1260)$ state, and its first radially excited state, $a_1(1640)$. The corrections caused by the $pi -a_1$ transitions are taken into account. Our estimate is in a good agreement with the latest empirical result Br$(tau to f_1 pi^{-} u_{tau})=(3.9pm 0.5)times 10^{-4}$. The distribution function obtained for the decay $tau to f_1(1285) pi^{-} u_{tau}$ shows a clear signal of $a_1(1640)$ resonance which should be compared with future experimental data including our estimate of the decay width $Gamma (a_1(1640) to f_1 pi )=14.1,mbox{MeV}$.
