Constraining the low-energy pion electromagnetic form factor with space-like and phase of time-like data

The Taylor coefficients c and d of the EM form factor of the pion are constrained using analyticity, knowledge of the phase of the form factor in the time-like region, 4m(pi)(2) <= t <= t(in) and its value at one space-like point, using as input the (g - 2) of the muon. This is achieved using the technique of Lagrange multipliers, which gives a transparent expression for the corresponding bounds. We present a detailed study of the sensitivity of the bounds to the choice of time-like phase and errors present in the space-like data, taken from recent experiments. We find that our results constrain c stringently. We compare our results with those in the literature and find agreement with the chiral perturbation-theory results for c. We obtain d similar to O(10) GeV-6 when c is set to the chiral perturbation-theory values.

New constraints on the pion EM form factor using Pi `(-Q(2))

We study the constraints arising on the expansion parameters c and d of the pion electromagnetic form factor from the inclusion of pure spacelike data and the phase of timelike data along with one spacelike datum, using as input the first derivative of the QCD polarization amplitude Pi'(-Q(2)). These constraints when combined with other analyses, provide a valuable check on a determination of c due to Guo et al. and on our previous work where pionic contribution to the (g - 2) of the muon was used as the input. This work further illustrates the power of analyticity techniques in form factor analysis.

Higher twist and higher order contributions to the pion electromagnetic form factor

The O(m(pi)4/(m(u) + (d))2Q2) and O(alpha(S)2) corrections to the leading term of the perturbative QCD calculation of the pion electromagnetic form factor are examined numerically. Both sets of terms provide significant corrections for values of Q2 between 1 and 15 GeV2/c2.

Implications of the recent high statistics determination of the pion electromagnetic form factor in the timelike region

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 pi pi scattering by Fermi-Watson theorem, are exploited by analytic techniques for finding correlations between the coefficients of the Taylor expansion at t = 0 and the values of the form factor at several points in the spacelike region. We do not use specific parametrizations, and the results are fully independent of the unknown phase in the inelastic region. Using for instance, from recent determinations, < r(pi)(2)> = (0.435 +/- 0.005) fm(2) and F(-1.6 GeV2) = 0.243(-0.014)(+0.022), we obtain the allowed ranges 3.75 GeV-4 less than or similar to c less than or similar to 3.98 GeV-4 and 9.91 GeV-6 less than or similar to d less than or similar to 10.46 GeV-6 for the curvature and the next Taylor coefficient, with a strong correlation between them. We also predict a large region in the complex plane where the form factor cannot have zeros.

Spacelike pion form factor from analytic continuation and the onset of perturbative QCD

The factorization theorem for exclusive processes in perturbative QCD predicts the behavior of the pion electromagnetic form factor F(t) at asymptotic spacelike momenta t(= -Q(2)) < 0. We address the question of the onset energy using a suitable mathematical framework of analytic continuation, which uses as input the phase of the form factor below the first inelastic threshold, known with great precision through the Fermi-Watson theorem from pi pi elastic scattering, and the modulus measured from threshold up to 3 GeV by the BABAR Collaboration. The method leads to almost model-independent upper and lower bounds on the spacelike form factor. Further inclusion of the value of the charge radius and the experimental value at -2.45 GeV2 measured at JLab considerably increases the strength of the bounds in the region Q(2) less than or similar to 10 GeV2, excluding the onset of the asymptotic perturbative QCD regime for Q(2) < 7 GeV2. We also compare the bounds with available experimental data and with several theoretical models proposed for the low and intermediate spacelike region.

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

We calculate upper and lower bounds on the modulus of the pion electromagnetic form factor on the unitarity cut below the omega pi inelastic threshold, using as input the phase in the elastic region known via the Fermi-Watson theorem from the pi pi 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 omega pi threshold from e(+)e(-) annihilation and tau-decay experiments. In particular, at low energies the calculated bounds offer a more precise description of the modulus than the experimental data.

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 -aEuro parts per thousand 2) and the phase of the pion electromagnetic form factor in the elastic region, known from pi pi 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.

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 pi pi 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 < r(pi)(2)> is an element of (0.42, 0.44) 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.

Bounds on the spacelike pion electromagnetic form factor from analyticity and unitarity

We use the recently measured accurate BaBaR data on the modulus of the pion electromagnetic form factor,Fπ(t), up to an energy of 3 GeV, the I=1P-wave phase of the π π scattering ampli-tude up to the ω−π threshold, the pion charge radius known from Chiral Perturbation Theory,and the recently measured JLAB value of Fπ in the spacelike region at t=−2.45GeV2 as inputs in a formalism that leads to bounds on Fπ in the intermediate spacelike region. We compare our constraints with experimental data and with perturbative QCD along with the results of several theoretical models for the non-perturbative contribution s proposed in the literature.

Two-pion low-energy contribution to the muon g-2 with improved precision from analyticity and unitarity

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 pi pi 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(-) ->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)(pi pi,LO)2m(pi), 0.30 GeV] = (0.553 +/- 0.004) x 10(-10) and a(mu)(pi pi,LO)0.30 GeV; 0.63 GeV] = (133.083 +/- 0.837) x 10(-10). These are consistent with the other recent determinations and have slightly smaller errors.

Threshold photoproduction of charged pions on14N

Pion photoproduction processes14Ngs(gamma, pgr +)14C and14Ngs(gamma, pgr –)14O have been studied in the threshold region. These processes provide an excellent tool to study the corrections to soft pion theorems and Kroll-Ruderman limit as applied to nuclear processes. The agreement with the available experimental data for these processes is better with the empirical wave functions while the shell-model wave functions predict a much higher value. Detailed experimental studies of these reactions at threshold, it is shown, are expected to lead to a better understanding of the shell-model inputs and radial distributions in the 1p state. We thank Dr. S.C.K. Nair for a helpful discussion during the initial stages of this work. One of us (MVN) thanks Dr. J.M. Laget for sending some unpublished data on pion photoproduction. He is also thankful to Dr. J. Pasupathy and Dr. R. Rajaraman for their interest and encouragement.