Proton-proton bremsstrahlung: Consequences of different on-shell-point conditions

Abstract

Background: Proton-proton bremsstrahlung ($pp\ensuremath{\gamma}$) is investigated both theoretically and experimentally. The $pp\ensuremath{\gamma}$ amplitudes can be classified according to the number and location of on-shell points at which they are evaluated. Quantitative understanding of the effect on the $pp\ensuremath{\gamma}$ cross section of using different on-shell-point conditions is lacking, but it is essential to identifying the photon emission mechanism governing the $pp\ensuremath{\gamma}$ process. Method: Four different $pp\ensuremath{\gamma}$ amplitudes, which include four-on-shell-point amplitudes and one-on-shell-point amplitude, are generated from a realistic one-boson-exchange (ROBE) model for $pp$ scattering. These ROBE amplitudes are used to investigate the consequence of using different on-shell-point conditions in calculating the $pp\ensuremath{\gamma}$ cross sections. Purpose: We verify the validity of the ROBE $pp\ensuremath{\gamma}$ amplitudes. We explore similarities between the four-on-shell-point ROBE and two-$u$-two-$t$ special (TuTts) soft-photon $pp\ensuremath{\gamma}$ amplitudes and important differences with the one-on-shell-point ROBE and Low $pp\ensuremath{\gamma}$ amplitudes. We demonstrate that the precision Kernfysisch Versneller Instituut (KVI) data can only be described by the four-on-shell-point (ROBE or TuTts) amplitude. We use the ROBE four-on-shell-point amplitude and one-on-shell-point amplitude to investigate systematically the effect of using different on-shell-point conditions to calculate the $pp\ensuremath{\gamma}$ cross section. Furthermore, we identify a general principle that governs the process. This general principle is also applicable to other bremsstrahlung processes involving the scattering of two identical nucleons. Results: (i) The four-on-shell-point ROBE (or TuTts) amplitude describes the high-precision KVI data much better than does the one-on-shell-point ROBE (or Low) amplitude. Although the contribution from the anomalous magnetic moment of the proton is very significant, it does not completely dominate the KVI $pp\ensuremath{\gamma}$ cross sections. (ii) The four-on-shell-point ROBE (or TuTts) amplitude describes the TRIUMF data better than does the one-on-shell-point ROBE (or Low) amplitude. (iii) The effect on the $pp\ensuremath{\gamma}$ cross section of using different on-shell-point conditions is significant in the hard-photon region, i.e., for small proton scattering angles $\ensuremath{\theta}$ ($={\ensuremath{\theta}}_{3}={\ensuremath{\theta}}_{4}$, symmetric scattering angles) far from the elastic limit ($\ensuremath{\theta}\ensuremath{\rightarrow}{45}^{\ensuremath{\circ}}$); in contrast, the effect becomes insignificant in the vicinity of the elastic limit. Near the limit as $\ensuremath{\theta}$ tends to ${45}^{\ensuremath{\circ}}$ (or as the photon energy $K$ approaches zero), the four-on-shell-point and one-on-shell-point amplitudes approach one another, a general principle applicable to all bremsstrahlung processes because only kinematics is involved. Conclusion: The four-on-shell-point ROBE amplitude provides a quantitative description of $pp\ensuremath{\gamma}$ cross sections. The anomalous magnetic moment is an important component of the photon emission mechanism. The four-on-shell-point property of the full ROBE amplitude is essential to properly describing the complete range of the precision KVI data and the TRIUMF data, although the one-on-shell-point ROBE amplitude is adequate in the region near the elastic limit.