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Register a new userLow-energy theorem for $gammato 3pi$: surface terms against $pi a_1$-mixing
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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 $gammatopi^+pi^0pi^-$.
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Using a recently proposed gauge covariant diagonalization of $pi a_1$-mixing we show that the low energy theorem $F^{pi}=e f_pi^2 F^{3pi}$ of current algebra, relating the anomalous form factor $F_{gamma to pi^+pi^0pi^-} = F^{3pi}$ and the anomalous neutral pion form factor $F_{pi^0 to gammagamma}=F^pi$, is fulfilled in the framework of the Nambu-Jona-Lasinio (NJL) model, solving a long standing problem encountered in the extension including vector and axial-vector mesons. At the heart of the solution is the presence of a $gamma pi {bar q} q $ vertex which is absent in the conventional treatment of diagonalization and leads to a deviation from the vector meson dominance (VMD) picture. It contributes to a gauge invariant anomalous tri-axial (AAA) vertex as a pure surface term.
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.
A reliable determination of the isospin breaking double quark mass ratio from precise experimental data on $etato 3pi$ decays should be based on the chiral expansion of the amplitude supplemented with a Khuri-Treiman type dispersive treatment of the final-state interactions. We discuss an extension of this formalism which allows to estimate the effects of the $a_0(980)$ and $f_0(980)$ resonances and their mixing on the $etato 3pi$ amplitudes. Matrix generalisations of the equations describing elastic $pipi$ rescattering with $I=0,,2$ are introduced which accomodate both $pipi/Kbar{K}$ and $etapi/Kbar{K}$ coupled-channel rescattering. Isospin violation induced by the physical $K^+-K^0$ mass difference and by direct $u-d$ mass difference effects are both accounted for in the dispersive integrals. Numerical solutions are constructed which illustrate how the large resonance effects at 1 GeV propagate down to low energies. They remain small in the physical region of the decay, due to the matching constraints with the NLO chiral amplitude, but they are not negligible and go in the sense of further improving the agreement with experiment for the Dalitz plot parameters.
We evaluate the $a_1(1260) to pi sigma (f_0(500))$ decay width from the perspective that the $a_1(1260)$ resonance is dynamically generated from the pseudoscalar-vector interaction and the $sigma$ arises from the pseudoscalar-pseudoscalar interaction. A triangle mechanism with $a_1(1260) to rho pi$ followed by $rho to pi pi$ and a fusion of two pions within the loop to produce the $sigma$ provides the mechanism for this decay under these assumptions for the nature of the two resonances. We obtain widths of the order of $13-22$ MeV. Present experimental results differ substantially from each other, suggesting that extra efforts should be devoted to the precise extraction of this important partial decay width, which should provide valuable information on the nature of the axial vector and scalar meson resonances and help clarify the role of the $pisigma$ channel in recent lattice QCD calculations of the $a_1$.
We show that the a_1-rho-pi Lagrangian is a decisive element for obtaining a good phenomenological description of the three-pion decays of the tau lepton. We choose it in a two-component form with a flexible mixing parameter sin(theta). In addition to the dominant a_1-> pho pi intermediate states, the a_1->pi sigma ones are included. When fitting the three-pion mass spectra, three data sets are explored: (1) ALEPH 2005 pi-pi-pi+ data, (2) ALEPH 2005 pi-pi0pi0 data, and (3) previous two sets combined and supplemented with the ARGUS 1993, OPAL 1997, and CLEO 2000 data. The corresponding confidence levels are (1) 28.3%, (2) 100%, and (3) 7.7%. After the inclusion of the a_1(1640) resonance, the agreement of the model with data greatly improves and the confidence level reaches 100% for each of the three data sets. From the fit to all five experiments [data set (3)] the following parameters of the a_1(1260) are obtained m_{a_1}=(1233+/-18) MeV, Gamma_{a_1}=(431+/-20) MeV. The optimal value of the Lagrangian mixing parameter sin(theta)= 0.459+/-0.004 agrees with the value obtained recently from the e+e- annihilation into four pions.
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