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Register a new userIsospin breaking in the phases of the Ke4 form factors
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V. Bernard
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Isospin breaking in the Kl4 form factors induced by the difference between charged and neutral pion masses is studied. Starting from suitably subtracted dispersion representations, the form factors are constructed in an iterative way up to two loops in the low-energy expansion by implementing analyticity, crossing, and unitarity due to two-meson intermediate states. Analytical expressions for the phases of the two-loop form factors of the Kpm -> pi^+ pi^- e^pm nu_e channel are given, allowing one to connect the difference of form-factor phase shifts measured experimentally (out of the isospin limit) and the difference of S- and P-wave pi-pi phase shifts studied theoretically (in the isospin limit). The isospin-breaking correction consists of the sum of a universal part, involving only pi-pi rescattering, and a process-dependent contribution, involving the form factors in the coupled channels. The dependence on the two S-wave scattering lengths a_0^0 and a_0^2 in the isospin limit is worked out in a general way, in contrast to previous analyses based on one-loop chiral perturbation theory. The latter is used only to assess the subtraction constants involved in the dispersive approach. The two-loop universal and process-dependent contributions are estimated and cancel partially to yield an isospin-breaking correction close to the one-loop case. The recent results on the phases of K^pm -> pi^+ pi^- e^pm nu_e form factors obtained by the NA48/2 collaboration at the CERN SPS are reanalysed including this isospin-breaking correction to extract values for the scattering lengths a_0^0 and a_0^2, as well as for low-energy constants and order parameters of two-flavour ChPT.
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Data on Ke4 decays allow one to extract experimental information on the elastic pi pi scattering amplitude near threshold, and to confront the outcome of the analysis with predictions made in the framework of QCD. These predictions concern an isospin symmetric world, while experiments are carried out in the real world, where isospin breaking effects - generated by electromagnetic interactions and by the mass difference of the up and down quarks - are always present. We discuss the corrections required to account for these, so that a meaningful comparison with the predictions becomes possible. In particular, we note that there is a spectacular isospin breaking effect in Ke4 decays. Once it is taken into account, the previous discrepancy between NA48/2 data on Ke4 decays and the prediction of pi pi scattering lengths disappears.
For direct CP-violation in $Ktopipi$ decays, the usual isospin-breaking effects at the percent level are amplified by the dynamics behind the $Delta I=1/2$ rule and conventionally encoded in $Omega_{rm IB}$ parameters. The updated prediction $Omega_{rm IB}^{(8)}=(15.9pm 8.2)times 10^{-2}$ of the Chiral Perturbation Theory for the strong isospin-breaking due to $pi_3-eta_8$ mixing confirms such a tendency but is quite sensitive to the theoretical input value of the low-energy constant corresponding to the flavour-singlet $eta_0$ exchange contribution in this truncated octet scheme. We rather exploit the phenomenological $eta_8-eta_0$ mixing as a probe for the non-negligible flavour-singlet component of the physical $eta$ pole to find $Omega_{rm IB}^{(9)}=(35pm7)times 10^{-2}$ in a complete nonet scheme. A large central value in the nonet scheme is thus substituted for a large uncertainty in the octet one. Including the experimental $pi^+-pi^0$ mass difference as the dominant electromagnetic isospin-breaking, we obtain for the effective parameter entering the ratio $epsilon/epsilon$ an improved result $hatOmega_{rm eff}^{(9)}=(29pm7)times 10^{-2}$ to be compared with $hatOmega_{rm eff}^{(8)}=(17pm9)times 10^{-2}$ used in recent analyses of $epsilon/epsilon$. Accordingly, we get a reduction from $(epsilon/epsilon)_{rm SM}^{(8)}=(17.4pm 6.1)times 10^{-4}$ to $(epsilon/epsilon)_{rm SM}^{(9)}=(13.9pm 5.2)times 10^{-4}$ and thereby an effective suppression of $(epsilon/epsilon)_{rm SM}$ by isospin-breaking corrections as large as $40%$ relative to the recent RBC-UKQCD value.
The axial-vector form factors and axial-vector constants of the baryon decuplet are investigated within a pion mean-field approach, which is also known as the chiral quark-soliton model. Given an axial-vector current with a specified flavor, there are four different form factors of a decuplet baryon. When we consider the singlet, triplet, and octet axial-vector currents, we have twelve different form factors for each member of the baryon decuplet. We compute all these axial-vector form factors of the baryon decuplet, taking into account the rotational $1/N_c$ corrections and effects of flavor SU(3) symmetry breaking. We find that, for a given flavor, two of the form factors for a decuplet baryon are only independent within the present approach. We first examine properties of the axial-vector form factors of the $Delta^+$ isobar and $Omega^-$ hyperon. We also compare the results of the triplet axial-vector form factors of $Delta^+$ with those from lattice QCD and those of the present work for the axial-vector constants of the baryon decuplet with the lattice data. All the results for other members of the baryon decuplet are then presented. The results of the axial charges are compared with those of other works. The axial masses and axial radii are also discussed.
One of the open issues in evaluations of the contribution from hadronic light-by-light scattering to the anomalous magnetic moment of the muon $(g-2)_mu$ concerns the role of heavier scalar, axial-vector, and tensor-meson intermediate states. The coupling of axial vectors to virtual photons is suppressed for small virtualities by the Landau-Yang theorem, but otherwise there are few rigorous constraints on the corresponding form factors. In this paper, we first derive the Lorentz decomposition of the two-photon matrix elements into scalar functions following the general recipe by Bardeen, Tung, and Tarrach. Based on this decomposition, we then calculate the asymptotic behavior of the meson transition form factors from a light-cone expansion in analogy to the asymptotic limits for the pseudoscalar transition form factor derived by Brodsky and Lepage. Finally, we compare our results to existing data as well as previous models employed in the literature.
We consider the evaluation of the $etapi$ isospin-violating vector and scalar form factors relying on a systematic application of analyticity and unitarity, combined with chiral expansion results. It is argued that the usual analyticity properties do hold (i.e. no anomalous thresholds are present) in spite of the instability of the $eta$ meson in QCD. Unitarity relates the vector form factor to the $etapi to pipi$ amplitude: we exploit progress in formulating and solving the Khuri-Treiman equations for $etato 3pi$ and in experimental measurements of the Dalitz plot parameters to evaluate the shape of the $rho$-meson peak. Observing this peak in the energy distribution of the $tauto eta pi u$ decay would be a background-free signature of a second-class amplitude. The scalar form factor is also estimated from a phase dispersive representation using a plausible model for the $etapi$ elastic scattering $S$-wave phase shift and a sum rule constraint in the inelastic region. We indicate how a possibly exotic nature of the $a_0(980)$ scalar meson manifests itself in a dispersive approach. A remark is finally made on a second-class amplitude in the $tautopipi u$ decay.
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