We review our recent analysis of pion-pion scattering data in terms of Roy equations and Forward Dispersion Relations, and present some preliminary results in terms of a new set of once-subtracted coupled equations for partial waves. The first analysis consists of independent fits to the different pion-pion channels that satisfies rather well the dispersive representation. In the second analysis we constrain the fit with the dispersion relations. The latter provides a very precise and model independent description of data using just analyticity, causality and crossing.
Roy's equations are used to check if the scalar-isoscalar pion-pion scattering amplitudes fitted to experimental data fulfill crossing symmetry conditions. It is shown that the amplitudes describing the ``down-flat'' phase shift solution satisfy crossing symmetry below 1 GeV while the amplitudes fitted to the "up-flat'' data do not. In this way the long standing "up-down" ambiguity in the phenomenological determination of the scalar-isoscalar pion-pion amplitudes has been resolved confirming the independent result of the recent joint analysis of the pi+pi- and pi0pi0 data.
We complete and improve the fits to experimental $ππ$ 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$\bar{K}K$ 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 ampl…
The Roy equations determine the S- and P-wave pion-pion phase shifts on a low energy interval. They allow the derivation of threshold parameters from experimental data. We examine the solutions of these equations that are in the neighborhood of a given solution by means of a linearization procedure. An updated survey of known results on the dimension of the manifold of solutions is presented. The solution is unique for a low energy interval with upper end at 800 MeV. We determine its response to small variations of the input: S-wave scattering lengths and absorptive parts above 800 MeV. We con…
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 ampl…
The scalar-isoscalar, scalar-isotensor and vector-isovector pion-pion partial wave amplitudes are analyzed. Preliminary results indicate that only the scalar-isoscalar amplitude fitted to the "down-flat" data satisfies Roy's equations and consequently crossing symmetry.
Roy's equations, which incorporate crossing symmetry of the pion-pion scattering amplitudes, are used to resolve the present ambiguity between two solutions for the scalar-isoscalar phase shifts below 1 GeV. It is shown that the "down-flat" solution satisfies well Roy's equations and consequently crossing symmetry while the other solution called "up-flat" does not and thus should be eliminated.
We report our progress on the data analysis of $ππ$ scattering data in terms of Forward Dispersion Relations (FDR), as well as Roy equations (RE) and their once-subtracted counterpart, GKPY equations. The first part of the analysis consists of independent fits to the different $ππ$ channels. The GKPY equations provide a more stringent consistency check for the parametrizations of the S0-wave data in the region from 400 to 1100 MeV, In the second part we present our preliminary analysis where the fits are constrained to satisfy all dispersion relations within errors, including the new GKPY Eqs.…
We review results of an analysis of pipi interactions in S, P and D waves for two-pion effective mass from threshold to about 1.4 GeV. In particular we show a recent improvement of this analysis above the K anti-K threshold using more data for phase shifts and including the S0 wave inelasticity from pipi -> K anti-K. In addition, we have improved the fit to the f2(1270) resonance and used a more flexible P wave parametrization above the K anti-K threshold and included an estimation of the D2 wave inelasticity. The better accuracy thus achieved also required a refinement of the Regge analysis a…
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 $ππ$ 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.
Starting from hyperbolic dispersion relations, we derive a closed system of Roy-Steiner equations for pion-nucleon scattering that respects analyticity, unitarity, and crossing symmetry. We work out analytically all kernel functions and unitarity relations required for the lowest partial waves. In order to suppress the dependence on the high-energy regime we also consider once- and twice-subtracted versions of the equations, where we identify the subtraction constants with subthreshold parameters. Assuming Mandelstam analyticity we determine the maximal range of validity of these equations. As…
We match the results for the subthreshold parameters of pion-nucleon scattering obtained from a solution of Roy-Steiner equations to chiral perturbation theory up to next-to-next-to-next-to-leading order, to extract the pertinent low-energy constants including a comprehensive analysis of systematic uncertainties and correlations. We study the convergence of the chiral series by investigating the chiral expansion of threshold parameters up to the same order and discuss the role of the Δ(1232) resonance in this context. Results for the low-energy constants are also presented in the counting sche…
We present a determination of the pion-nucleon ($πN$) $σ$-term $σ_{πN}$ based on the Cheng-Dashen low-energy theorem (LET), taking advantage of the recent high-precision data from pionic atoms to pin down the $πN$ scattering lengths as well as of constraints from analyticity, unitarity, and crossing symmetry in the form of Roy-Steiner equations to perform the extrapolation to the Cheng-Dashen point in a reliable manner. With isospin-violating corrections included both in the scattering lengths and the LET, we obtain $σ_{πN}=(59.1\pm 1.9\pm 3.0)$ MeV $=(59.1\pm 3.5)$ MeV, where the first error …
We review the structure of Roy-Steiner equations for pion-nucleon scattering, the solution for the partial waves of the t-channel process $ππ\to \bar N N$, as well as the high-accuracy extraction of the pion-nucleon S-wave scattering lengths from data on pionic hydrogen and deuterium. We then proceed to construct solutions for the lowest partial waves of the s-channel process $πN\to πN$ and demonstrate that accurate solutions can be found if the scattering lengths are imposed as constraints. Detailed error estimates of all input quantities in the solution procedure are performed and explicit p…
We give a brief review of the theoretical description of low energy pion-pion scattering by the combined use of Chiral Perturbation Theory and Roy equations, an update of the Regge parametrization of $ππ$ cross sections at high energies, and a short discussion of the scalar radius of the pion.
Starting from hyperbolic dispersion relations, we present a system of Roy--Steiner equations for pion Compton scattering that respects analyticity and unitarity requirements, gauge invariance, as well as crossing symmetry, and thus all symmetries of the underlying quantum field theory. To suppress the dependence on the high-energy region, we also consider once- and twice-subtracted versions of the equations, where the subtraction constants are identified with dipole and quadrupole pion polarizabilities. We consider the resolution of the $γγ\toππ$ partial waves by a Muskhelishvili-Omnès represe…
We present a set of once subtracted dispersion relations which implement crossing symmetry conditions for the $ππ$ scattering amplitudes below 1 GeV. We compare and discuss the results obtained for the once and twice subtracted dispersion relations, known as Roy's equations, for three $ππ$ partial JI waves, S0, P and S2. We also show that once subtracted dispersion relations provide a stringent test of crossing and analyticity for $ππ$ partial wave amplitudes, remarkably precise in the 400 to 1.1 GeV region, where the resulting uncertainties are significantly smaller than those coming from sta…
We have evaluated forward dispersion relations for scattering amplitudes that follow from direct fits to several sets of pion-pion scattering experiments, together with the precise K decay results, and high to energy data. We find that some of the most commonly used experimental sets, as well as some recent theoretical analyses based on Roy equations, do not satisfy these constraints by several standard deviations. Finally, we provide a consistent pion-pion amplitude by improving a global fit to data with these dispersion relations.
Assuming that axiomatic local field theory results hold for hadron scattering, André Martin and S. M. Roy recently obtained absolute bounds on the D-wave below threshold for pion-pion scattering and thereby determined the scale of the logarithm in the Froissart bound on total cross sections in terms of pion mass only. Previously, Martin proved a rigorous upper bound on the inelastic cross-section $σ_{inel}$ which is one-fourth of the corresponding upper bound on $σ_{tot}$, and Wu, Martin,Roy and Singh improved the bound by adding the constraint of a given $σ_{tot}$. Here we use unitarity and a…
We present a data-driven analysis of the resonant S-wave $ππ\to ππ$ and $πK \to πK$ reactions using the partial-wave dispersion relation. The contributions from the left-hand cuts are accounted for using the Taylor expansion in a suitably constructed conformal variable. The fits are performed to experimental and lattice data as well as Roy analyses. For the $ππ$ scattering we present both a single- and coupled-channel analysis by including additionally the $K\bar{K}$ channel. For the latter the central result is the Omnès matrix, which is consistent with the most recent Roy and Roy-Steiner res…
Starting from hyperbolic dispersion relations, we derive a system of Roy--Steiner equations for pion Compton scattering that respects analyticity, unitarity, gauge invariance, and crossing symmetry. It thus maintains all symmetries of the underlying quantum field theory. To suppress the dependence of observables on high-energy input, we also consider once- and twice-subtracted versions of the equations, and identify the subtraction constants with dipole and quadrupole pion polarizabilities. Based on the assumption of Mandelstam analyticity, we determine the kinematic range in which the equatio…
We present a data-driven analysis of the S-wave $ππ\to ππ\,(I=0,2)$ and $πK \to πK\,(I=1/2, 3/2)$ reactions using the partial-wave dispersion relation. The contributions from the left-hand cuts are parametrized using the expansion in a suitably constructed conformal variable, which accounts for its analytical structure. The partial-wave dispersion relation is solved numerically using the $N/D$ method. The fits to the experimental data supplemented with the constraints from chiral perturbation theory at threshold and Adler zero give the results consistent with Roy-like (Roy-Steiner) analyses. F…
We review our evaluation of forward dispersion relations for direct fits to the different, and often conflicting, pion-pion scattering experimental analyses. We find that some of the most commonly used data sets do not satisfy these constraints by several standard deviations. We also provide a consistent pion-pion amplitude by improving a global fit to data with these dispersion relations.
We show that a suitable setting for comparison of the low-energy representation for pion-pion scattering amplitudes, with dispersive representation for these amplitudes, is provided by certain manifestly crossing symmetric dispersion relations. We begin with a discussion of fixed-t dispersion relations and discuss the origin of crossing constraints that arise in this context when we consider resonance saturation with certain $l\geq 2$ states. We demonstrate that the approach advocated here does not require us to enforce such constraints. Our results are contrasted with those from fixed-t dispe…
We analyze the Roy equations for the lowest partial waves of elastic pi pi scattering and demonstrate that the two S-wave scattering lengths a_0^0 and a_0^2 are the essential parameters in the low energy region: Once these are known, the available experimental information determines the behaviour near threshold to within remarkably small uncertainties. An explicit numerical representation for the energy dependence of the S- and P-waves is given and it is shown that the threshold parameters of the D- and F-waves are also fixed very sharply in terms of a_0^0 and a^2_0. In agreement with earlier …
Starting from hyperbolic dispersion relations for the invariant amplitudes of pion-nucleon scattering together with crossing symmetry and unitarity, one can derive a closed system of integral equations for the partial waves of both the s-channel (pi N --> pi N) and the t-channel (pi pi --> Nbar N) reaction, called Roy-Steiner equations. After giving a brief overview of the Roy-Steiner system for pi N scattering, we demonstrate that the solution of the t-channel subsystem, which represents the first step in solving the full system, can be achieved by means of Muskhelishvili-Omnès techniques. In…
The Vector Pion form factor below 1 GeV is analyzed using experimental data on its modulus, the P-wave pion pion phase shifts and dispersion relation. It is found that causality is satisfied. Using dispersion relation, terms proportional to s squared and s cubed are calculated using the experimental data, where s is the momentum transfer. They are much larger than the one-loop and two-loop Chiral Perturbation Theory calculations. Unitarized model calculations agree very well with dispersion relation results.
In a series of papers we have applied several sum rules and forward dispersion relations, to pion-pion 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, 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 …
We investigate the temperature dependence of the shear and bulk viscosities and their ratios to the entropy density via a continuum QCD approach. We calculate the pion mass and decay constant in the framework of Dyson-Schwinger equations of QCD and the pion thermal width by combining with Roy equations. We obtain then the variation behaviors of the viscosities, especially a novel feature of the bulk viscosity, with respect to temperature.
The first three coefficients of the Taylor's series expansion of the vector pion form factor as a function of the momentum transfer are evaluated using experimental data on the pion form factor and the P-wave pi pi phase shifts. The real part of the form factor as a function of energy is also calculated by dispersion relation. Comparisons these results with Chiral Perturbation Theory and unitarized models are given.
We calculate the nucleonic and pionic dispersion relations at finite temperature T and non-vanishing chemical potentials $(μ_f)$ in the context of an effective chiral theory that describes the strong and electromagnetic interactions for nucleons and pions. The dispersion relations are calculated in the broken chiral symmetry phase, where the nucleons are massive and pions are taken as massless. The calculation is performed at lowest order in the energy expansion, working in the framework of the real time formalism of thermal field theory in the Feynman gauge. These one-loop dispersion relation…
Chiral perturbation theory, the low energy effective theory of the strong interactions for the light pseudoscalar degrees of freedom, is based on effective Lagrangian techniques and is an expansion in the powers of the external momenta and the powers of the quark masses, which correct the soft-pion theorems. Our primary emphasis will be on the problem of $ππ$ scattering. After briefly reviewing these features and some results, we review some features of $π-N$ scattering.
We study the behavior of the pion dispersion relation in a pion medium at finite density and temperature, introducing a chemical potential to describe the finite pion number density. Such description is particularly important during the hadronic phase of a relativistic heavy-ion collision, between chemical and thermal freeze-out, where the pion number changing processes, driven by the strong interaction, can be considered to be frozen. We make use of an effective Lagrangian that explicitly respects chiral symmetry through the enforcement of the chiral Ward identities. The pion dispersion relat…
The pion electromagnetic form factor is investigated by using the dispersion relation with the superconvergence condition, which was proposed to synthesize the vector meson dominance model and QCD. The absorptive part is given as an addition of the resonance and the QCD terms, the latter being parametrized so as to realize the prediction of perturbative QCD. The experimental data are analyzed and the parameters for the QCD part, the resonance part as well, are determined. Existing experiments for the form factor are reproduced very well both for the space-like and time-like momenta.
We consider a model with $ρππ$, $ρKK$, and gauge-invariant $ργ$ couplings and obtain the pion form factor $F_π$ at timelike momentum transfers by resummation of pion and kaon loops. We use a dispersion representation for the loop diagrams and analyse ambiguities related to subtraction constants. The resulting representation for $F_π$ is shown to have the form of the conventional vector meson dominance formula with one important distinction - the effective $ρ$-meson decay constant $f^{\rm eff}_ρ$ turns out to depend on the momentum transfer. For the electromagnetic pion form factor we include i…
In this work we provide simple and precise parameterizations of the existing $πK$ 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.
We establish the framework for the comparison of $πK$ scattering amplitudes from SU(3) chiral perturbation theory with suitable dispersive representations which result from the combination of certain fixed-t dispersion relations with dispersion relations on hyperbolic curves. This allows for predictions for some combinations of low energy constants appearing in higher order calculations of chiral perturbation theory. Using a simple parametrization for the lowest partial waves, first estimates for some combinations are presented.
The scattering of $γπ\to ππ$ is studied using the axial anomaly, elastic unitarity, analyticity and crossing symmetry. Using the technique to derive the Roy's equation, an integral equation for the P-wave amplitude is obtained in terms of the strong P-wave pion pion phase shifts. Its solution is obtained numerically by an iteration procedure using the starting point as the solution of the integral equation of the Muskelshsvilli-Omnes type. It is, however, ambiguous and depends sensitively on the second derivative of the P-wave amplitude at $s=m_π^2$ which cannot directly be measured.
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 sigma 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 negligib…
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 sigma resonance region. For practical applications, these results are shown to be very well approximated by a very simple conformal expansion.
We present a determination of the pion charge radius from high precision data on the pion vector form factor from both timelike and spacelike regions, using a novel formalism based on analyticity and unitarity. At low energies, instead of the poorly known modulus of the form factor, we use its phase, known with high accuracy from Roy equations for $ππ$ elastic scattering via the Fermi-Watson theorem. We use also the values of the modulus at several higher timelike energies, where the data from $e^+e^-$-annihilation and $τ$-decay are mutually consistent, as well as the most recent measurements …
We consider the lightcone sum-rule description of the pion-photon transition form factor, based on dispersion relations, in combination with the renormalization group of QCD, in terms of the formal solution of the Efremov-Radyushkin-Brodsky-Lepage evolution equation, and show that the emerging scheme amounts to a certain version of Fractional Analytic Perturbation Theory (FAPT). In order to ensure the correct asymptotic behavior of the considered physical quantity, this modified FAPT version has to be supplemented by process-specific boundary conditions---in contrast to the standard one. Howev…
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 twice-subtracted Roy's equations to analyze the data. Finally we present a preliminary determination of the sigma pole from the constrained dispersive data analysis.
We present an extraction of the pion-nucleon ($πN$) scattering lengths from low-energy $πN$ scattering, by fitting a representation based on Roy-Steiner equations to the low-energy data base. We show that the resulting values confirm the scattering-length determination from pionic atoms, and discuss the stability of the fit results regarding electromagnetic corrections and experimental normalization uncertainties in detail. Our results provide further evidence for a large $πN$ $σ$-term, $σ_{πN}=58(5)$ MeV, in agreement with, albeit less precise than, the determination from pionic atoms.
We reply to the Comment by Fil'kov and Kashevarov, arXiv:0805.4486 [hep-ph]. We show that the discrepancies between ChPT and dispersion theory, reported for the polarizability of the pion, result from applying dispersion theory to non-analytic functions.
Within fixed-t dispersion relations the results of new VPI partial-wave analysis for the multipole amplitudes $M_{1+}^{3/2},E_{1+}^{3/2}$ are successfuly described, and the resonance and nonresonance contributions into these amplitudes are separated in correspondence with the interpretation on the language of diagram approach, dynamical models and effective Lagrangian approach. The amplitudes $A_p^{3/2}$ and $A_p^{1/2}$ corresponding to the $Δ$ contributions into $M_{1+}^{3/2},E_{1+}^{3/2}$ are obtained. They are in better agreement with quark model predictions than the amplitudes extracted wi…
The multichannel S- and P-wave amplitudes for the pion-pion scattering, constructed requiring analyticity and unitarity of the S-matrix and using the uniformization procedure, are elaborated using the dispersion relations with imposed crossing symmetry condition. The amplitudes are modified in the low-energy region to improve their consistency with experimental data and the dispersion relations. Agreement with data is achieved for both amplitudes from the threshold up to 1.8 GeV and with dispersion relations up to 1.1 GeV. Consequences of the applied modifications, e.g. changes of the S-wave l…
By looking at the complex plane of the pion-mass squared we establish a dispersion relation which the static quantities, such as baryon masses, magnetic moments, polarizabilities, should obey. This dispersion relation yields insight into the differences between the heavy-baryon and relativistic calculations in the baryon sector of chiral perturbation theory.
We investigate the charged-neutral pion self-energy difference at finite temperature. Within Chiral Perturbation Theory we extend previous analysis in the chiral and soft pion limits. Our analysis with physical masses leads to additional contributions for temperatures typical of a meson gas, including a momentum-dependent function for the self energy. In addition, a nonzero imaginary part arises, which we define consistently in the Coulomb gauge and comes from an infrared enhanced contribution due to thermal bath photons. For distributions typical of a heavy-ion meson gas, the charged and neut…
A dispersion relation formalism for the virtual Compton scattering (VCS) reaction on the proton is presented, which for the first time allows a dispersive evaluation of 4 generalized polarizabilities at a four-momentum transfer $Q^2 \leq$ 0.5 GeV$^2$. The dispersive integrals are calculated using a state-of-the-art pion photo- and electroproduction analysis. The dispersion formalism provides a new tool to analyze VCS experiments above pion threshold, thus increasing the sensitivity to the generalized polarizabilities of the nucleon.
Recent attempts to determine the pion polarizability by dispersion relations yield values that disagree with the predictions of chiral perturbation theory. These dispersion relations are based on specific forms for the absorptive part of the Compton amplitudes. The analytic properties of these forms are examined, and the strong enhancement of intermediate-meson contributions is shown to be connected to non-analytic structures
Using the tadpole improved Wilson quark action on small, coarse and anisotropic lattices, $Kπ$ scattering length in the $I=3/2$ channel is calculated within quenched approximation. The results are extrapolated towards the chiral and physical kaon mass region. Finite volume and finite lattice spacing errors are also analyzed and a result in the infinite volume and continuum limit is obtained. Our result is compared with the results obtained using Roy equations, Chiral Perturbation Theory, dispersion relations and the experimental data.
