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Register a new userStringent constraints on the scalar K pi form factor from analyticity, unitarity and low-energy theorems
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Balasubramanian Ananthanarayan
Publication date
2009
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English
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We investigate the scalar K pi form factor at low energies by the method of unitarity bounds adapted so as to include information on the phase and modulus along the elastic region of the unitarity cut. Using at input the values of the form factor at t=0 and the Callan-Treiman point, we obtain stringent constraints on the slope and curvature parameters of the Taylor expansion at the origin. Also, we predict a quite narrow range for the higher order ChPT corrections at the second Callan-Treiman point.
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Motivated by the discrepancies noted recently between the theoretical calculations of the electromagnetic $omegapi$ form factor and certain experimental data, we investigate this form factor using analyticity and unitarity in a framework known as the method of unitarity bounds.We use a QCD correlator computed on the spacelike axis by operator product expansion and perturbative QCD as input, and exploit unitarity and the positivity of its spectral function, including the two-pion contribution that can be reliably calculated using high-precision data on the pion form factor. From this information, we derive upper and lower bounds on the modulus of the $omegapi$ form factor in the elastic region. The results provide a significant check on those obtained with standard dispersion relations, confirming the existence of a disagreement with experimental data in the region around 0.6 GeV.
Recent experimental data for the differential decay distribution of the decay $tau^-to u_tau K_Spi^-$ by the Belle collaboration are described by a theoretical model which is composed of the contributing vector and scalar form factors $F_+^{Kpi}(s)$ and $F_0^{Kpi}(s)$. Both form factors are constructed such that they fulfil constraints posed by analyticity and unitarity. A good description of the experimental measurement is achieved by incorporating two vector resonances and working with a three-times subtracted dispersion relation in order to suppress higher-energy contributions. The resonance parameters of the charged $K^*(892)$ meson, defined as the pole of $F_+^{Kpi}(s)$ in the complex $s$-plane, can be extracted, with the result $M_{K^*}=892.0 pm 0.9 $MeV and $Gamma_{K^*}=46.2 pm 0.4 $MeV. Finally, employing the three-subtracted dispersion relation allows to determine the slope and curvature parameters $lambda_+^{}=(24.7pm 0.8)cdot 10^{-3}$ and $lambda_+^{}=(12.0pm 0.2)cdot 10^{-4}$ of the vector form factor $F_+^{Kpi}(s)$ directly from the data.
We perform a dispersive analysis of the $omegapi$ electromagnetic transition form factor, using as input the discontinuity provided by unitarity below the $omegapi$ threshold and including for the first time experimental data on the modulus measured from $e^+e^-toomegapi^0$ at higher energies. The input leads to stringent parameterization-free constraints on the modulus of the form factor below the $omegapi$ threshold, which are in disagreement with some experimental values measured from $omegato pi^0gamma^*$ decay. We discuss the dependence on the input parameters in the unitarity relation, using for illustration an $N/D$ formalism for the P partial wave of the scattering process $omegapi to pipi$, improved by a simple prescription which simulates the rescattering in the crossed channels. Our results confirm the existence of a conflict between experimental data and theoretical calculations of the $omegapi$ form factor in the region around 0.6 GeV and bring further arguments in support of renewed experimental efforts to measure more precisely the $omegatopi^0gamma^*$ decay.
Dispersive representations of the Kpi vector and scalar form factors are used to fit the spectrum of tau ---> K pi nu_tau obtained by the Belle collaboration incorporating constraints from results for K_l3 decays. The slope and curvature of the vector form factor are obtained directly from the data through the use of a three-times-subtracted dispersion relation. We find $lambda_+=(25.49 pm 0.31) times 10^{-3}$ and $lambda_+= (12.22 pm 0.14) times 10^{-4}$. From the pole position on the second Riemann sheet the mass and width of the $K^*(892)^{pm}$ are found to be $m_{K^*(892)^pm}=892.0pm 0.5$~MeV and $Gamma_{K^*(892)^pm}=46.5pm 1.1$~MeV. The phase-space integrals needed for K_l3 decays are calculated as well. Furthermore, the Kpi isospin-1/2 P-wave threshold parameters are derived from the phase of the vector form factor. For the scattering length and the effective range we find respectively $a_{1}^{1/2},= ( 0.166pm 0.004),m_pi^{-3}$ and $b_{1}^{1/2},=( 0.258pm 0.009),m_pi^{-5}$.
The two-pion contribution from low energies to the muon magnetic moment anomaly, although small, has a large relative uncertainty since in this region the experimental data on the cross sections are neither sufficient nor precise enough. It is therefore of interest to see whether the precision can be improved by means of additional theoretical information on the pion electromagnetic form factor, which controls the leading order contribution. In the present paper we address this problem by exploiting analyticity and unitarity of the form factor in a parametrization-free approach that uses the phase in the elastic region, known with high precision from the Fermi-Watson theorem and Roy equations for $pipi$ elastic scattering as input. The formalism also includes experimental measurements on the modulus in the region 0.65-0.70 GeV, taken from the most recent $e^+e^-to pi^+pi^-$ experiments, and recent measurements of the form factor on the spacelike axis. By combining the results obtained with inputs from CMD2, SND, BABAR and KLOE, we make the predictions $a_mu^{pipi, LO},[2 m_pi,, 0.30 gev]=(0.553 pm 0.004) times 10^{-10}$ and $a_mu^{pipi, LO},[0.30 gev,, 0.63 gev]=(133. 083 pm 0.837)times 10^{-10}$. These are consistent with the other recent determinations, and have slightly smaller errors.
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