Production of e+ e− pairs in proton-deuteron capture to He
Introduction
The electromagnetic probe is a well-established and powerful tool to investigate the structure of hadronic systems. In exclusive processes one can distinguish three different regimes depending on the four-momentum of the photon. First, in the space-like region (q2<0) in quasi-free kinematics the (e,e′p) reaction directly probes the single-particle structure of the nuclear vertex. Second, for real photons, the (γ,p) or (p,γ) reactions are more sensitive to the details of the reaction mechanism and to meson exchange currents (MEC). Third, the rather less well-known time-like region (q2>0), which can be explored using dilepton production, addresses additional aspects as compared to real photons: (i) the coupling of longitudinally polarized photons, (ii) the time-like form factor in the “unphysical” region (4m2e<q2<4m2p). Basically there are two alternatives to explore the time-like region: virtual Compton scattering and the bremsstrahlung processes with virtual photons, such as and capture reactions.
In this paper we study the capture reaction at intermediate energies (proton energies up to a few hundred MeV) for both real and virtual photons. The main motivation for the present work are experiments at TSL (Uppsala) which in the past have explored [1] only small photon invariant masses (q2<(10 MeV)) and KVI (Groningen) experiments which cover larger photon invariant masses [2].
Since we feel it is important to satisfy gauge invariance, we start from the covariant impulse approximation model of [3] and make it explicitly gauge invariant by the introduction of an additional internal amplitude. An important input in this approach is the pdHe vertex function, for which recent calculations 4, 5, 6 using a realistic NN interaction (Argonne v14 and v18) are used.
Section snippets
Description of the model
At small photon energies the photon production amplitude is dominated by radiation from the external legs (first three diagrams of Fig. 1), and this consideration led to the development of low-energy theorems (LET)s 7, 8, 9. In kinematical conditions where the photon energy is not small we may still assume the dominance of the above external amplitude, without applying an expansion in powers of the photon energy.
The 4-momenta of the proton, deuteron, He nucleus, and (virtual) photon are
Cross section and response functions for dilepton production
The c.m. cross section for the p+d He+e++e− reaction can be decomposed (in complete analogy to the spacelike (e,e′p) reaction) into the sum of products of kinematical factors and four response functions (RFs),Here is the invariant mass of the virtual photon, , Tp is the proton kinetic energy in the lab frame, pc and qc are the c.m. 3-momenta of the proton
Results of calculations and discussion
The model is first tested for the real photon reaction. Fig. 2 (upper panel) shows cross sections for the reaction He→pd at EγLAB = 245 MeV, related to the capture process at TpLAB = 358 MeV via time reversal. Note the unsatisfactory discrepancy between the disintegration and capture reaction measurements, known for a long time and recently pointed out again in Ref. [20]. As seen from the figure the agreement with the photodisintegration data 17, 18 is quite reasonable and considerably better
Acknowledgements
We thank Robert Wiringa for calculating the He–d and He–d overlap integrals. We would also like to thank Justus Koch, Ulla Tengblad, Jan Ryckebusch and Rob Timmermans for useful discussions, and Betsy Beise for sending data file with the deuteron form factors. This work is supported by the Fund for Scientific Research-Flanders (FWO-Vlaanderen).
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Permanent address: National Science Center `Kharkov Institute of Physics and Technology', 310108 Kharkov, Ukraine.