Watson's Theorem Research Articles

Single-energy partial-wave analysis for pion photoproduction with fixed- t analyticity

Experimental data for pion photoproduction including differential cross sections and various polarization observables from four reaction channels, $\gamma p \to \pi^0 p$, $\gamma p \to \pi^+n$, $\gamma n \to \pi^- p$ and $\gamma n \to \pi^0 n$ from threshold up to $W=2.2$ GeV have been used in order to perform a single-energy partial wave analysis with minimal model dependence by imposing constraints from unitarity and fixed-$t$ analyticity in an iterative procedure. Reaction models were only used as starting point in the very first iteration. We demonstrate that with this procedure partial wave amplitudes can be obtained which show only a minimal dependence on the initial model assumptions. The analysis has been obtained in full isospin, and the Watson theorem is enforced for energies below $W=1.3$ GeV but is even fulfilled up to $W\approx 1.6$ GeV in many partial waves. Electromagnetic multipoles $E_{\ell\pm}$ and $M_{\ell\pm}$ are presented and discussed for $S,P,D$ and $F$-waves.

Weak kaon production off the nucleon and Watson's theorem

We have improved the tree-level model of Ref. [1] for weak production of kaons off nucleons by partially restoring unitarity. This is achieved by imposing Watson's theorem to the dominant vector and axial-vector contributions in appropriate angular momentum and isospin quantum number sectors. The observable consequences of this procedure are investigated.

Electronic width of the ψ(3770) resonance interfering with the background

Methods for extracting the $\ensuremath{\psi}(3770)\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}$ decay width from the data on the reaction cross section ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}D\overline{D}$ are discussed. Attention is drawn to the absence of the generally accepted method for determining ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\psi}(3770){e}^{+}{e}^{\ensuremath{-}}}$ in the presence of interference between the contributions of the $\ensuremath{\psi}(3770)$ resonance and background. It is shown that the model for the experimentally measured $D$ meson form factor, which satisfies the requirement of the Watson theorem and takes into account the contribution of the complex of the mixed $\ensuremath{\psi}(3770)$ and $\ensuremath{\psi}(2S)$ resonances, allows us to uniquely determine the value of ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\psi}(3770){e}^{+}{e}^{\ensuremath{-}}}$ by fitting. The ${\mathrm{\ensuremath{\Gamma}}}_{\ensuremath{\psi}(3770){e}^{+}{e}^{\ensuremath{-}}}$ values found from the data processing are compared with the estimates in the potential models.

Sensitivity of multipoles to phase variations in pion photo- and electro-production analyses

We use the Athens Model Independent Analysis Scheme (AMIAS) to examine the validity of using the Fermi–Watson theorem in the multipole analyses of pion photoproduction and electroproduction data. A standard practice in this field is to fix the multipoles’ phases from $$\pi N$$ scattering data, making use of the Fermi–Watson theorem. However, these phases are known with limited accuracy and the effect of this uncertainty on the obtained multipole extraction has not been fully explored yet. Using AMIAS we constrain the phases within their experimentally determined uncertainty. We first analyze sets of photoproduction pseudodata of increasing statistical precision and subsequently we apply the methodology for a re-analysis of the Bates/Mainz electroproduction data. It is found that the uncertainty induced by the $$\pi N$$ phases uncertainty to the extracted solutions would be significant only in the analysis of data with much higher precision than the current available experimental data.

A new class of double integrals involving Generalized Hypergeometric Function 3F2

The aim of this research paper is to evaluate fifty double integrals invoving generalized hypergeometric function (25 each) in the form of\begin{align*}\int_{0}^{1}\int_{0}^{1} & x^{c-1}y^{c+\al-1} (1-x)^{\al- 1}(1-y)^{\be-1}\, (1-xy)^{c+\ell-\al-\be+1}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; xy\right]\,dxdy\end{align*}and\begin{align*}\int_{0}^{1}\int_{0}^{1} & x^{c+\ell}y^{c+\ell+\al} (1-x)^{\al-1}(1-y)^{\be-1}\, (1-xy)^{c- \al-\be}\;\\ &\times \;{}_3F_2 \left[\begin{array}{c}a,\,\,\,\,\,b,\,\,\,\,\,2c+\ell+ 1 \\ \frac{1}{2}(a+b+i+1),\,\,2c+j \end{array}; 1-xy\right]\,dxdy\end{align*}in the most general form for any $\ell \in \mathbb{Z}$ and $i, j = 0, \pm 1, \pm2$.The results are derived with the help of generalization of Edwards's well known double integral due to Kim, {\it et al.} and generalized classical Watson's summation theorem obtained earlier by Lavoie, {\it et al}.More than one hundred ineteresting special cases have also been obtained.

Complex phase structure of the meson-baryon T -matrix

The full complex phase structure of the meson-baryon reaction amplitude in coupled channels approach is investigated, including also the photon-baryon channel. The result may be viewed as a generalization of the well-known Watson's theorem. Furthermore, the complex phase structure is exhibited for the pole and nonpole parts of the reaction amplitude in such a way that it will serve as a convenient common starting point for constructing models with different levels of approximation, in particular, for building isobar models where the basic properties of the $S$-matrix can be maintained. Such models should be useful, especially, in coupled multichannel calculations, where a large amount of experimental data are considered in resonance analyses, a situation encountered in modern baryon spectroscopy. In particular, it is shown that the unitarity of the pole part of the $T$-matrix arises automatically from the dressing mechanism inherent in the basic scattering equation. This implies that no separate conditions are required for making this part of the amplitude unitary as it has been done in some of the existing isobar models.

Model-independent constraint on the pion scalar form factor and light quark masses

We investigate the pion scalar form factor in the Meiman-Okubo framework, implementing the phase below the inelastic $K\overline{K}$ threshold, where it is known from the $\ensuremath{\pi}\ensuremath{\pi}$ scalar isoscalar phase shift ${\ensuremath{\delta}}_{0}^{0}$ by Watson theorem. State-of-the-art knowledge of the perturbative QCD expansion of the scalar correlator and the phase shift ${\ensuremath{\delta}}_{0}^{0}$ is used as input. No assumptions about the phase above the inelastic threshold or the possible zeros of the form factor in the complex plane are necessary. We obtain a model-independent constraint relating the sum of the light quark masses to the slope and the curvature of the pion scalar form factor at the origin. The recent lattice results for the light quark masses and the pion scalar radius are found to satisfy this constraint. We obtain also a strong correlation between the pion scalar radius and the curvature of the form factor, with rather high values predicted for the curvature.

Applications of extended Watson's summation theorem

Applications of extended Watson's summation theorem

Watson’s theorem and theNΔ(1232)axial transition

We present a new determination of the $N\Delta$ axial form factors from neutrino induced pion production data. For this purpose, the model of Hernandez {\it et al.} [Phys. Rev. D76, 033005 (2007)] is improved by partially restoring unitarity. This is accomplished by imposing Watson's theorem on the dominant vector and axial multipoles. As a consequence, a larger $C_5^A(0)$, in good agreement with the prediction from the off-diagonal Goldberger-Treiman relation, is now obtained.

EXTRACTION OF RESONANCE PARAMETERS AND ROLE OF THE FINAL STATE INTERACTIONS

We present an overview of the SAID group effort to analyze new γn → π-p cross sections vs. the world database to get new multipoles and determine neutron electromagnetic couplings. The differential cross section for the processes γn → π-p was extracted from new measurements at CLAS and MAMI-B accounting for Fermi motion effects in the impulse approximation (IA) as well as NN- and πN-FSI effects beyond the IA. We evaluated results of several pion photoproduction analyses and compared πN PWA results as a constraint for analyses of pion photoproduction data (Watson's theorem).

NOTE ON THE CLASSICAL WATSON'S THEOREM FOR THE SERIES3F2

Summation theorems for hypergeometric series <TEX>$_2F_1$</TEX> and generalized hypergeometric series <TEX>$_pF_q$</TEX> play important roles in themselves and their diverse applications. Some summation theorems for <TEX>$_2F_1$</TEX> and <TEX>$_pF_q$</TEX> have been established in several or many ways. Here we give a proof of Watson's classical summation theorem for the series <TEX>$_3F_2$</TEX>(1) by following the same lines used by Rakha [7] except for the last step in which we applied an integral formula introduced by Choi et al. [3].

Ambiguities in the rate of oxygen formation during stellar helium burning in the12C(α,γ) reaction

The rate of oxygen formation determines the C/O ratio during stellar helium burning. It is the single most important nuclear input of stellar evolution theory including the evolution of Type II and Type Ia supernova. Yet the low energy cross section of the fusion of 4He + 12C denoted as the 12C(a,g)16O reaction still remains uncertain after forty years of extensive work. We analyze and critically review the most recent measurements of complete angular distributions of the outgoing gamma-rays at very low energies (Ecm > 1.0 MeV). Our analysis of the angular distribution measured with the EUROGAM/GANDI arrays lead us to conclude considerably larger error bars than published hence they are excluded from the current sample of "world data". We show that the current sample of "world data" of the measured E2 cross section factors below 1.7 MeV cluster to two distinct groups that lead to two distinct extrapolations of SE2(300) ~ 60 or ~ 154 keVb. We point to a much neglected discrepancy between the measured E1-E2 phase difference (phi_12) and unitarity as required by the Watson theorem, suggesting systematic problem(s) of some of the measured gamma-ray angular distributions. The ambiguity of the extrapolated SE2(300) together with a previously observed ambiguity of SE1(300) represent the current state of the art of the field and they must be resolved by future measurements of complete and detailed angular distributions of the 12C(a,g) reaction at very low energies (Ecm < 1.0 MeV).

Parametrisation-free determination of the shape parameters for the pion electromagnetic form factor

Recent data from high-statistics experiments that have measured the modulus of the pion electromagnetic form factor from threshold to relatively high energies are used as input in a suitable mathematical framework of analytic continuation to find stringent constraints on the shape parameters of the form factor at t=0. The method uses also as input a precise description of the phase of the form factor in the elastic region based on Fermi–Watson theorem and the analysis of the ππ scattering amplitude with dispersive Roy equations, and some information on the spacelike region coming from recent high precision experiments. Our analysis confirms the inconsistencies of several data on the modulus, especially from low energies, with analyticity and the input phase, noted in our earlier work. Using the data on the modulus from energies above 0.65 GeV, we obtain, with no specific parametrisation, the prediction \(\langle r_{\pi}^{2} \rangle\in(0.42,\,0.44)\ \text{fm}^{2}\) for the charge radius. The same formalism leads also to very narrow allowed ranges for the higher-order shape parameters at t=0, with a strong correlation among them.

The pion form factor from analyticity and unitarity

Analyticity and unitarity techniques are employed to estimate Taylor coefficients of the pion electromagnetic form factor at t = 0 by exploiting the recently evaluated two-pion contribution to the muon (g − 2) and the phase of the pion electromagnetic form factor in the elastic region, known from ππ scattering by Fermi–Watson theorem and the values of the form factor at several points in the space-like region. Regions in the complex t-plane are isolated where the form factor cannot have zeros.

Model independent bounds on the modulus of the pion form factor on the unitarity cut below the ωπ threshold

We calculate upper and lower bounds on the modulus of the pion electromagnetic form factor on the unitarity cut below the ωπ inelastic threshold, using as input the phase in the elastic region known via the Fermi–Watson theorem from the ππ P-wave phase shift, and a suitably weighted integral of the modulus squared above the inelastic threshold. The normalization at t=0, the pion charge radius and experimental values at spacelike momenta are used as additional input information. The bounds are model independent, in the sense that they do not rely on specific parametrizations and do not require assumptions on the phase of the form factor above the inelastic threshold. The results provide nontrivial consistency checks on the recent experimental data on the modulus available below the ωπ threshold from e + e − annihilation and τ-decay experiments. In particular, at low energies the calculated bounds offer a more precise description of the modulus than the experimental data.

On a new hypergeometric transformation

The aim of this article is to establish a general transformation for generalized hypergeometric function involving hypergeometric polynomials, by the method of elementary manipulation of series representation and to derive certain Chaundy's formulae by another method. Two applications are presented; Watson's theorem on the sum of $_3F_{2}$ and their contiguous summation formulae are deduced by means of the generalized Gauss' second summation theorem. Also several earlier results by Driver - Johnston and Coffey - Johnston follow as special cases of our main findings.

Near-threshold pion production in diproton reactions with polarised beams and target at ANKE-COSY

An extensive experimental study of near-threshold pion production in dipro- ton reactions is underway at the ANKE-COSY spectrometer (JThe programme is aimed at isolating the four-nucleon-pion contact interaction term appearin g in thePT ex- pansions of these processes. This will establish links between pion production and other low energy phenomena within thePT approach. The first step in the programme was to measure the differential cross-section and the proton analysing power in the~!f ppg s� 0 and~n!f ppg s� − reactions over the full angular range. Heref ppg s denotes a diproton, i.e.,a two-proton system in a 1 S 0 state. These data allow a partial wave analysis to be carried out provided that simplifying assumptions are made when applying the Watson theorem. To make the analysis more robust, and independent of the uncertainties in the relative normalization, the spin-correlation coeffi cients Ax; x and Ay㭹 in the~ n~! f ppg s� − reac- tion were measured in a follow-up experiment. The first results of the data analysis are presented and future developments of the programme are outlined.

Model dependence of single-energy fits to pion photoproduction data

Model dependence of multipole analysis has been explored through energy-dependent and single-energy fits to pion photoproduction data. The MAID energy-dependent solution has been used as input for an event generator producing realistic pseudo data. These were fitted using the SAID parametrization approach to determine single-energy and energy-dependent solutions over a range of lab photon energies from 200 to 1200 MeV. The resulting solutions were found to be consistent with the input amplitudes from MAID. Fits with a $\chi$-squared per datum of unity or less were generally achieved. We discuss energy regions where consistent results are expected, and explore the sensitivity of fits to the number of included single- and double-polarization observables. The influence of Watson's theorem is examined in detail.

GENERALIZED DOUBLE INTEGRAL INVOLVING KAMPÉ DE FÉRIET FUNCTION

The aim of this paper is to obtain twenty five Eulerian type double integrals in the form of a general double integral involving Kamp<TEX>$\'{e}$</TEX> de F<TEX>$\'{e}$</TEX>riet function. The results are derived with the help of the generalized classical Watson's theorem obtained earlier by Lavoie, Grondin and Rathie. A few interesting special cases of our main result are also given.

Single-energy amplitudes for pion photoproduction in the first resonance region

We consider multipole amplitudes for low-energy pion photoproduction, constructed with minimal model dependence, at single energies. A previous fit making minimal use of Watson's theorem has been reexamined in light of more recent measurements. Problems associated with the choice of a unique solution are discussed. Comparisons with more recent fits to the full resonance region are made. Explanations are suggested for the discrepancies and more precise measurements of the recoil polarization, $P$, are motivated.