Measurement and analysis of the $\mathit{pp}\to \mathit{pp}\gamma$ reaction at 310 MeV

The $\mathit{pp}\to \mathit{pp}\gamma$ reaction has been studied at a beam energy of 310 MeV by detecting both final protons at the PROMICE-WASA facility and identifying the photon through the resulting missing-mass peak. The photon angular distribution in the center-of-mass system and those of the proton-proton relative momentum with respect to the beam direction and to that of the recoil photon were determined reliably up to a final $\mathit{pp}$ excitation energy of $E_{\mathit{pp}}~30$ MeV. Except for very small $E_{\mathit{pp}}$ values, the behavior of these distributions with excitation energy is well reproduced by a new refined model of the hard bremsstrahlung process. The model reproduces absolutely the total cross section and its energy dependence to within the experimental and theoretical uncertainties.

Figure 1

(a) Basic photon production current $J^{\mu }$ used to describe the $\mathit{pp}\to \mathit{pp}\gamma$ reaction in the present work; analogous photon couplings along the lower nucleon line are omitted. $N$ denotes an intermediate nucleon, and $M$ incorporates all the meson exchanges. As indicated by the symbols below the diagrams, the first two describe the nucleonic current, while the meson-exchange current is depicted by the third. The fourth diagram contains the $NM\to N\gamma$ four-point contact current which, together with the meson-exchange current, constitutes the interaction current $V^{\mu }$ of Eq. (3). Diagram (b) shows the photon coupling to both intermediate nucleons as subsumed in the dual current $d^{\mu }$.

Figure 2

Distribution in the square of the missing mass in the $\mathit{pp}\to \mathit{ppX}$ reaction presented in units of the neutral pion mass. Clear peaks are seen, arising from the $\mathit{pp}\to \mathit{pp}{\pi }^0$ and $\mathit{pp}\to \mathit{pp}\gamma$ reactions, sitting on a slowly varying background.

Figure 3

Angular distribution of the final protons from the $\mathit{pp}\to \mathit{pp}\gamma$ reaction in the laboratory (open circles) compared to Monte Carlo simulations based on the HN model [16], using the Bonn potential as input (red solid curve). Similar predictions obtained with a phase-space model (blue dashed curve) are also shown. Both sets of predictions were normalized to the total number of events.

Figure 4

Time difference between the two protons emerging from the target. The peaks have similar widths for photons in the forward (blue triangles) and backward (red circles) c.m. hemisphere, as well as their sum (black crosses). Events in the interval $\pm 1.8$ ns were retained for the subsequent analysis.

Figure 5

Monte Carlo estimates of the percentage acceptance for $\mathit{pp}\to \mathit{pp}\gamma$ at $T_p=310$ MeV obtained using the HN model [16] with the Bonn potential. The results are divided into four regions in the photon CM angle ${\theta }_{\gamma }$ defined in the text, namely, the backward (red circles), the backward central (blue triangles), the forward central (magenta stars), and the forward (green inverted triangles).

Figure 6

Cross section for $\mathit{pp}\to \mathit{pp}\gamma$ at $T_p=310$ MeV in arbitrary units, obtained using the HN model [16] with the Bonn potential to evaluate the acceptance. The experimental data are divided into the same four angular regions as in Fig. 5. Data were only plotted when the acceptance was estimated to be above 2%.

Figure 7

Differential cross section for $\mathit{pp}\to \mathit{pp}\gamma$ at 310 MeV. The data have been averaged over 3-MeV bins in $E_{\mathit{pp}}$ and the upper end of each interval is indicated by the number at the bottom right of the corresponding panel. The curves correspond to the absolute predictions of the Haberzettl and Nakayama model [16] of Sec. 2, using the Bonn potential [24]. The solid (blue) one represents the model without any $\Delta$ contribution, whereas for the dashed (red) one the $\Delta$ has been switched on. The latter has been included in a minimal fashion, as explained in Sec. 2.

Figure 8

The distribution in the angle between the $\mathit{pp}$ relative momentum vector $\mathrm{q⃗}$ and that of the incident beam direction in the overall c.m. frame. It should be noted that there may be significant systematic measurement uncertainties at low $E_{\mathit{pp}}$. Because the final protons are identical, data are only shown in one hemisphere and the total cross section is obtained by summing over this region. The curves correspond to the predictions of the Haberzettl and Nakayama model [16] of Sec. 2 obtained using the Bonn potential [24]. The notation is the same as in Fig. 7.

Figure 9

The distribution in the helicity angle between the $\mathit{pp}$ relative momentum vector $\mathrm{q⃗}$ and that of the photon in the frame of the recoiling $\mathit{pp}$ pair. The notation is the same as in Fig. 7.

Figure 10

Cross section for the $\mathit{pp}\to \mathit{pp}\gamma$ reaction measured at $T_p=310$ MeV integrated over all angles. The (black) circles were obtained by taking data from the backward photon hemisphere and invoking the forward/backward symmetry. For the (magenta) stars, data in the interval $-1<\cos {\theta }_{\gamma }<0.8$ were used in the determination. Overall systematic errors of $\pm 15\%$ are not shown. The solid (blue) curve represents the evaluation of the model described in Sec. 2 using the Bonn $\mathit{NN}$ input without the inclusion of any $\Delta$ contribution, whereas the red dots represent an attempt to include this in a minimalist way. The (orange) dashed and (green) dash-dotted curves are the analogous estimates based upon using the Paris potential that includes the Coulomb repulsion. The curves are scaled by the factors shown in the figure. The (maroon) dot-dot-dashed curve illustrates the energy dependence to be expected from a pure phase-space model. It is arbitrarily normalized at around 5 MeV, that is, just above the region whose energy dependence is largely driven by the strong $\mathit{pp}$ FSI.

Figure 11

Ratio of the Legendre coefficients defined by Eq. (5). Values extracted from experimental data are compared to the predictions of the Haberzettl and Nakayama model [16] with and without the minimalist $\Delta$ inclusion (dotted red and solid blue lines, respectively).