Phenomenological study of the $pp\to {\pi }^{+}pn$ reaction

Fully constrained bubble chamber data on the $pp\to {\pi }^{+}pn$ and $pp\to {\pi }^{+}d$ reactions are used to investigate the ratio of the counting rates for the two processes as function of the $pn$ excitation energy $Q$. Though it is important to include effects associated with the $p$-wave nature of pion production, the data are insufficient to establish unambiguously the dependence on $Q$. The angular distributions show the presence of higher partial waves which seem to be anomalously large at small $Q$. The dispersion relation method to determine scattering lengths is extended to encompass cases where, as for the $pp\to {\pi }^{+}pn$ reaction, there is a bound state and, in a test example, it is shown that the values deduced for the low-energy neutron-proton scattering parameters are significantly influenced by the pion $p$-wave behavior.

Figure 1

Values of $F\left(x\right)$ extracted from the PNPI bubble chamber data at 900.2 MeV [1] (black stars), 940.7 MeV [2] (red circles), and 988.6 MeV [3] (blue squares) using Eq. (1), The errors shown do not include those arising from the numbers of $pp\to d{\pi }^{+}$ events measured since these affect all the data at a given beam energy. The laboratory beam energies quoted here are the nominal ones listed in Refs. [1, 2, 3].

Figure 2

Values of $F^{*}\left(x\right)$ extracted from the PNPI bubble chamber data at 900.2 MeV [1] (black stars), 940.7 MeV [2] (red circles), and 988.6 MeV [3] (blue squares) on the basis of Eqs. (1) and (2). The errors shown do not include those arising from the numbers of $pp\to d{\pi }^{+}$ events measured. Deviations between the values of $F^{*}\left(x\right)$ for different energies at large $x$ are due in part to the spread in the measured beam energy; the nominal beam energies are used in the evaluation of the momentum factor of Eq. (2).

Figure 3

Values of $F^{*}\left(x\right)$ averaged over the PNPI beam energies of 900.2, 940.7, and 988.6 MeV [1, 2, 3] obtained using Eqs. (1) and (2). The curves are fits made on the basis of Eq. (3) with $A=1$ (fixed), $B=-0.054$, and $C=0.0223$.

Figure 4

Distributions in the Gottfried-Jackson angle ${\theta }_p$ for all the $pp\to pn{\pi }^{+}$ PNPI bubble chamber events in the range 900–1000 MeV [1, 2, 3]. The (red) stars correspond to data chosen with $0<Q<20$ MeV, whereas the (blue) circles are those where $140<Q<160$ MeV. The lines represent the Legendre polynomial curves of Eq. (4), with the values of the fitted parameters $C_2$ and $C_4$ being shown in Fig. 5.

Figure 5

Variation of the parameters $C_2$ (red stars) and $C_4$ (blue circles) of the Legendre polynomial fit of Eq. (4) to the Gottfried-Jackson angular distribution observed in the PNPI bubble chamber measurements of the $pp\to pn{\pi }^{+}$ differential cross section [1, 2, 3].

Figure 6

Estimates of the spin-triplet $np$ scattering length ($a_1$) and effective range ($r_1$) deduced from Eqs. (5) and (12) as functions of the cutoff energy $Q_{\max }$ in the integration. The true values with no cutoff should be $a_1=5.414$ fm and $r_1=1.757$ fm. The solid lines were obtained with the bare Jost function of Eq. (13) as input but with $\alpha \to -\alpha$. The dashed lines were derived by using the modified Jost input of Eq. (15), where the effects of the $p$-wave nature of the pion production have been included.