Forward Dispersion Relations Research Articles

F_{0}(1370) Controversy from Dispersive Meson-Meson Scattering Data Analyses.

We establish the existence of the long-debated f_{0}(1370) resonance in the dispersive analyses of meson-meson scattering data. For this, we present a novel approach using forward dispersion relations, valid for generic inelastic resonances. We find its pole at (1245±40)-i(300_{-70}^{+30}) MeV in ππ scattering. We also provide the couplings as well as further checks extrapolating partial-wave dispersion relations or with other continuation methods. A pole at (1380_{-60}^{+70})-i(220_{-70}^{+80}) MeV also appears in the ππ→KK[over ¯] data analysis with partial-wave dispersion relations. Despite settling its existence, our model-independent dispersive and analytic methods still show a lingering tension between pole parameters from the ππ and KK[over ¯] channels that should be attributed to data.

Analytic Approach to $\pi K$ Scattering and Strange Resonances

We review our analysis of $\pi K$ scattering using forward dispersion relations. The method yields a set of simple parameterizations that are compatible with forward dispersion relations up to 1.6 GeV while still describing the data. Once the partial waves are obtained, we calculate the poles in the complex plane by means of Pad\'e approximants, thus avoiding a particular model for the pole parameterization. The resonances calculated below 1.8 GeV are the much debated scalar $\kappa$-meson, nowadays known as $K_0^*(800)$, the scalar $K_0^*(1430)$, the $K^*(892)$ and $K_1^*(1410)$ vectors, the spin-two $K_2^*(1430)$ as well as the spin-three $K^*_3(1780)$.

Pion-kaon scattering amplitude constrained with forward dispersion relations up to 1.6 GeV

In this work we provide simple and precise parameterizations of the existing $\pi K$ scattering data from threshold up to 1.6 GeV, which are constrained to satisfy forward dispersion relations as well as three additional threshold sum rules. We also provide phenomenological values of the threshold parameters and of the resonance poles that appear in elastic scattering.

Measurement of the absolute differential cross section of proton–proton elastic scattering at small angles

The differential cross section for proton-proton elastic scattering has been measured at a beam energy of 1.0 GeV and in 200 MeV steps from 1.6 to 2.8 GeV for centre-of-mass angles in the range from 12-16 degrees to 25-30 degrees, depending on the energy. Absolute normalisations of typically 3% were achieved by studying the energy losses of the circulating beam of the COSY storage ring as it passed repeatedly through the windowless hydrogen target of the ANKE magnetic spectrometer. It is shown that the data have a significant impact upon a partial wave analysis. After extrapolating the differential cross sections to the forward direction, the results are broadly compatible with the predictions of forward dispersion relations.

Forward Dispersion Relations for $\pi K$ Scattering and the $K^*_0(800)$ Resonance

We review the recent analysis of π K scattering data in terms of forward dispersion relations, and also present the parameters of the strange resonances. This work consists of fits to the data that are constrained to satisfy analyticity requirements. The method yields a set of simple and consistent parameterizations that are compatible with forward dispersion relations up to 1.6 GeV while still describing the data. We also obtain the pole parameters of the K_(0)^(*)(800) and the K^(*)(892) resonances.

Nuclear-structure contribution to the Lamb shift in muonic deuterium

We consider the two-photon exchange contribution to the $2P-2S$ Lamb shift in muonic deuterium in the framework of forward dispersion relations. The dispersion integrals are evaluated using experimental data on elastic deuteron form factors and inelastic electron-deuteron scattering, both in the quasielastic and hadronic range. The subtraction constant that is required to ensure convergence of the dispersion relation for the forward Compton amplitude $T_1(\nu,Q^2)$ is related to the deuteron magnetic polarizability $\beta(Q^2)$. Based on phenomenological information, we obtain for the Lamb shift $\Delta E_{2P-2S}=2.01\pm0.74$ meV. The main source of the uncertainty of the dispersion analysis is due to lack of quasielastic data at low energies and forward angles. We show that a targeted measurement of the deuteron electrodesintegration in the kinematics of upcoming experiments A1 and MESA at Mainz can help quenching this uncertainty significantly.

Muonic-hydrogen Lamb shift: Dispersing the nucleon-excitation uncertainty with a finite-energy sum rule

We assess the two-photon exchange contribution to the Lamb shift in muonic hydrogen with forward dispersion relations. The subtraction constant $\bar T(0,Q^2)$ that is necessary for a dispersive evaluation of the forward doubly-virtual Compton amplitude, through a finite energy sum rule, is related to the fixed J=0 pole generalized to the case of virtual photons. We evaluated this sum rule using excellent virtual photoabsorption data that are available. We find that the "proton polarizability correction" to the Lamb shift in muonic hydrogen is $-(40\pm5)\mu$eV. We conclude that nucleon structure-dependent uncertainty by itself is unlikely to resolve the large (300$\mu$eV) discrepancy between direct measurement of the Lamb shift in $\mu H$ and expectations based on conventional Hydrogen measurements.

Amplitudes and observables in pp elastic scattering at $\sqrt{s}=7\ \mbox{TeV}$

A precise analysis of the pp elastic scattering data at 7 TeV in terms of its amplitudes is performed as an extension of previous studies for lower energies. Slopes B R and B I of the real and imaginary amplitudes are independent quantities, and a proper expression for the Coulomb phase is used. The real and imaginary amplitudes are fully disentangled, consistently with forward dispersion relations for amplitudes and for slopes. We present analytic expressions for the amplitudes that cover all t range completely, while values of total cross section σ, ratio ρ, B I , and B R enter consistently to describe forward scattering. It is stressed that the identification of the amplitudes is an essential step for the description of elastic scattering, and pointed out the importance of the experimental investigation of the transition range from non-perturbative to perturbative dynamics, which may confirm the three gluon exchange mechanism observed at lower energies.

Pion-pion scattering amplitude. IV. Improved analysis with once subtracted Roy-like equations up to 1100 MeV

We improve our description of pion-pion scattering data by imposing additional requirements to our previous fits, in the form of once-subtracted Roy-like equations, while extending our analysis up to 1100 MeV. We provide simple and ready to use parametrizations of the amplitude. In addition, we present a detailed description and derivation of these once-subtracted dispersion relations that, in the 450 to 1100 MeV region, provide an additional constraint which is much stronger than our previous requirements of Forward Dispersion Relations and standard Roy equations. The ensuing constrained amplitudes describe the existing data with rather small uncertainties in the whole region from threshold up to 1100 MeV, while satisfying very stringent dispersive constraints. For the S0 wave, this requires an improved matching of the low and high energy parametrizations. Also for this wave we have considered the latest low energy Kl4 decay results, including their isospin violation correction, and we have removed some controversial data points. These changes on the data translate into better determinations of threshold and subthreshold parameters which remove almost all disagreement with previous Chiral Perturbation Theory and Roy equation calculations below 800 MeV. Finally, our results favor the dip structure of the S0 inelasticity around the controversial 1000 MeV region.

Precise dispersive analysis of the f0(600) and f0(980) resonances from ππ scattering

We show how the new precise data on kaon decays together with forward dispersion relations, sum rules and once- and twice-subtracted Roy equations allow for a precise analysis of ππ scattering. The once subtracted Roy equations provide a more stringent consistency check for the parametrizations of the S0-wave data in the region from 400 to 1100 MeV that allows us to present a preliminary determination of the f0(600) and f0(980) poles from the constrained dispersive data analysis.

PRECISE DISPERSIVE DATA ANALYSIS OF THE f0(600) POLE

We review how the use of recent precise data on kaon decays together with forward dispersion relations (FDR) and Roy's equations allow us to determine the sigma resonance pole position very precisely, by using only experimental input. In addition, we present preliminary results for a modified set of Roy-like equations with only one subtraction, that show a remarkable improvement in the precision around the σ region. We also improve the matching between the parametrizations at low and intermediate energy of the S0 wave, and show that the effect of this on the sigma pole position is negligible.

Sigma pole position and errors of a once and twice subtracted dispersive analysis of pi-pi scattering data

We show how the new precise data on kaon decays together with forward dispersion relations, sum rules and once- and twice-subtracted Roy’s equations allow for a precise determination of the sigma meson pole position. We present a comparison and a study of the different sources of uncertainties when using either once- or twicesubtracted Roy’s equations to analyze the data. Finally we present a preliminary determination of the σ pole from the constrained dispersive data analysis.

In memory of Paco Ynduráin: A precise determination of ππ scattering from experiment and dispersion relations

This talk is dedicated to the memory of Paco Yndurain, the original speaker in the conference. After a short account of his scientific career, we briefly review our ongoing collaboration to determine precisely the pi pi scattering amplitude including the most recent data by means of Forward Dispersion Relations and Roy Equations. A remarkable improvement in precision over the intermediate energy region is obtained by using once-subtracted Roy Equations in addition to the standard twice-subtracted ones.

Pion-pion scattering amplitude. III. Improving the analysis with forward dispersion relations and Roy equations

We complete and improve the fits to experimental π π scattering amplitudes, both at low and high energies, that we performed in the previous papers of this series. We then verify that the corresponding amplitudes satisfy analyticity requirements, in the form of partial wave analyticity at low energies, forward dispersion relations (FDR) at all energies, and Roy equations below K K threshold; the first by construction, the last two, inside experimental errors. Then we repeat the fits including as constraints FDR and Roy equations. The ensuing central values of the various scattering amplitudes verify very accurately FDR and, especially, Roy equations, and change very little from what we found by just fitting data, with the exception of the D2 wave phase shift, for which one parameter moves by 1.5 σ. These improved parametrizations therefore provide a reliable representation of pion- pion amplitudes with which one can test various physical relations. We also present a list of low energy parameters and other observables. In particular, we find a_(0)^((0)) = 0.223 ± 0.009M_(π)^(-1), a_(0) ^(2) = -0.0444 ± 0.0045M -(π)^(-1), and δ_((0))^(0) (m_(K)^(2)) – δ_((2))^(0)(m_(K)^(2)) = 50.9 ± 1.2o

Forward analysis of πN scattering with an expansion method

The π N forward-scattering data are analyzed using an expansion method, where the invariant amplitudes are represented by expansions satisfying the forward dispersion relations. The experimental errors of the data are taken into account through the covariance matrix of the coefficients of the expansions in a careful error analysis. From the results, some coefficients, c n0 ± , of the subthreshold expansions have been calculated with proper error bars.

Forward dispersion relations and Roy equations in ππ scattering

We first review the results of an analysis of ππ interactions in S, P and D waves for the two-pion effective mass from threshold to about 1.4 GeV. In particular, we show a recent improvement of this analysis above the K \( \bar K \) threshold using more data for phase shifts and including the S0-wave inelasticity from ππ → K \( \bar K \). In addition, we have improved the fit to the f 2(1270)-resonance and used a more flexible P-wave parametrization above the K \( \bar K \) threshold and included an estimation of the D2-wave inelasticity. The better accuracy thus achieved also required a refinement of the Regge analysis above 1.42 GeV. Finally, in this work we check that the ππ scattering amplitudes obtained in this approach satisfy remarkably well forward dispersion relations and Roy’s equations.

Chiral-dispersive calculations of ππ scattering confront experiment

In a series of papers we have applied several sum rules and forward dispersion relations, to ππ scattering. We have found that some widely used data sets fail to satisfy these constraints, and we have provided an amplitude that describes data consistently with the dispersive tests. Furthermore, we noted that the input and precision claimed in a Roy equation analysis by Colangelo, Gasser and Leutwyler (CGL), lead to several mismatches with some sum rules. Subsequently, Caprini, Colangelo, Gasser and Leutwyler claimed that our Regge parametrization was incorrect. We collect here the answers to their various claims, and try to clarify the points of agreement and disagreement, showing experimental evidence that substantiates our results, and that, in addition, their representation fails to satisfy all three forward dispersion relations up to s ⩽ 800 MeV by several standard deviations.

Extended partial-wave analysis ofπNscattering data

We present results from a comprehensive partial-wave analysis of pi+-p elastic scattering and charge-exchange data, covering the region from threshold to 2.6 GeV in the lab pion kinetic energy, employing a coupled-channel formalism to simultaneously fit pi-p-->eta n data to 0.8 GeV. Our main result, solution SP06, utilizes a complete set of forward and fixed-t dispersion relation constraints applied to the piN elastic amplitude. The results of these analyses are compared with previous solutions in terms of their resonance spectra and preferred values for couplings and low-energy parameters.

Erratum: Pion-pion scattering amplitude. II. Improved analysis aboveK¯Kthreshold [Phys. Rev. D74, 014001 (2006)

We improve, in the energy region between $\bar{K}K$ threshold and $\sim~1.4$ GeV, the energy-dependent phase shift analysis of $\pi\pi$ scattering presented in a previous paper. For the S0 wave we have included more data above $\bar{K}K$ threshold and we have taken into account systematically the elasticity data on the reaction $\pi\pi\to\bar{K}K$. We here made a coupled channel fit. For the D0 wave we have considered information on low energy parameters, and imposed a better fit to the $f_2$ resonance. For both waves the expressions we now find are substantially more precise than the previous ones. We also provide slightly improved D2 and P waves, including the estimated inelasticity for the first, and a more flexible parametrization between 1 and 1.42 GeV for the second. The accuracy of our amplitudes is now such that it requires a refinement of the Regge analysis, for $s^{1/2}\geq1.42$ GeV, which we also carry out. We show that this more realistic input produces $\pi\pi$ scattering amplitudes that satisfy better forward dispersion relations, particularly for $\pi^0\pi^0$ scattering.

Pion-pion scattering amplitude. II. Improved analysis aboveK¯Kthreshold

We improve, in the energy region between the KK threshold and ~ 1.4 GeV, the energy-dependent phase shift analysis of π π scattering presented in a previous paper. For the S0 wave, we have included more data above the KK threshold and we have taken into account systematically the elasticity data on the reaction π π →KK. We here made a coupled channel fit. For the D0 wave, we have considered information on low energy parameters, and imposed a better fit to the f_2 resonance. For both waves the expressions we now find are substantially more precise than the previous ones. We also provide slightly improved D2 and P waves, including the estimated inelasticity for the first, and a more flexible parametrization between 1 and 1.42 GeV for the second. The accuracy of our amplitudes is now such that it requires a refinement of the Regge analysis, for^(1/2) ≥ 1.42 GeV, which we also carry out. We show that this more realistic input produces π π scattering amplitudes that satisfy better forward dispersion relations, particularly for π^0 π^0 scattering.