sepis an off-shell e-p cross section, K is a kinematical factor, Q is the virtual photon four momentum ,w is the energy transfer, q is the 3-momentum transfer.
The standard model of nuclear physics [PAC14 Workshop, 1] assumes that nuclear structure can be described by nucleons interacting via nucleon-nucleon potentials. The potentials themselves may be based on some meson exchange models. The dynamics has been treated non-relativistically ( Arriaga, [18], Morita, [7]) or relativistically (Udias [15]). Modern approaches to the four-body problem have included the use of Monte-Carlo techniques (Pudliner, [19] ) . Experimental tests of our understanding of nuclear structure in the few body systems have fruitfully employed electron induced proton knockout reactions, such as 4He(e,e'p)t ([9], [10], [16]) and 4He(e,e'p)X ([10], [16]). However, the complexity of the calculations has limited ab initio theoretical predictions thus far to the two body final nuclear state, p + t. These reactions are defined by a missing energy in the undetected recoiling nuclear system which corresponds to the triton in its ground state. Nevertheless, there is intense interest in examining the reaction at higher missing energies, that is at large internal excitation energy of the undetected nuclear system. The reason for this is that we anticipate that large missing energy correlates with large missing momentum. This has been seen explicitly in the three body system 3He(e,e'p)d,X, both in experiment ( show e89044 large pmiss -emiss spectrum) fig. 1, and in calculations (degli Atti [2]), fig. 2. The large pmiss kinematics show a significant enhancement in strength for the unbound p-n final state as compared to the bound p-n final state. Calculations in the 4 body system( Efros [3] ) show a similarly large strength at high missing momentum for large missing energy, fig. 3, and this is a general feature of the (e,e'p)X reaction ( degli Atti [4] ).
In the Impulse Approximation large pmiss corresponds to large momentum of the detected nucleon while it was in the 4He nuclear ground state. The nuclear mean field can not supply very high momentum components, so as a general rule, large momenta arise from violent nucleon-nucleon collisions. It is just under these conditions that the finite size and internal structure of the the nucleon should manifest itself. It is under these conditions that the standard model of nuclear structure should fail. This regime of the nuclear many body system is ideally explored by CEBAF, indeed, it was one of the original motivations for construction of CEBAF. Results about the pmiss strength in this region will be of interest whether they agree or disagree with the standard nuclear structure model. If they disagree, then we are exploring a new region in the nuclear many body system where quark degrees of freedom can no longer be ignored or smoothed over as some sort of effective potential. If they agree with the standard models then we could well ask why the nucleon's internal structure is not playing a more significant role.
The only previous experiment ( Le Goff, [10] ) on 4He(e,e'p)X which investigated high missing momentum and high missing energy found evidence for virtual photon absorption on a pair of nucleons, fig. 4. When the momentum distribution extracted from this experiment is compared to those obtained from the deuteron and the 3He(e,e'p)X an impressive agreement is observed , fig. 5. The authors of reference [10] attribute this agreement to that part of the NN interactions which can not be described in a mean field calculation and originating in the nuclear ground state. All nuclei should exhibit a similar high momentum distribution. Unfortunately, the interpretation of this similarity of high momentum distributions among different nuclei with that found in the deuteron is clouded by final state interactions. Bianconi et al [11], have shown that final state interactions can produce similarly shaped momentum distributions because the basic ingredient in final state interactions is a nucleon-nucleon collision in the outgoing state. Whereas the authors of the previous study [10] were forced to used non-parallel kinematics, where FSI are expected to be large ( see below), to achieve the desired high missing energies and momenta, the ability to repeat these measurement in parallel kinematics at CEBAF is a definite attraction.
A clean interpretation of the experimental results as due to single nucleon currents is not possible without also considering the other currents present in the nucleus in addition to the alteration of the outgoing detected proton's momentum as it interacts with the undetected nuclear system. The electron interacts with all the currents and charges in the nucleus, for example, meson exchange currents, isobar configurations, etc.. The strength of various contributions depends on the conditions of the virtual photon, for example, its four-momentum transfer, Q2. Both of these effects have been successfully treated theoretically. Meson exchange currents and isobar configurations have been included in the calculations of Laget[12], Schiavilla[13] for example. Several authors ( Laget[12], Schiavilla[13], Udias[14], etc.) have included the effects of final state interactions by optical potentials affecting the outgoing proton's wave function. Another approach to final state interactions (FSI) has been the use of the generalized eikonal approximation (GEA). This approach has been successfully applied to proton-nucleus scattering [Debruyne, 5 ] for proton kinetic energies above about 1 GeV. GEA calculations exist for d(e,e'p)n (Jeschonnek et al) and for 4He(e,e'p)X (Morita , [7] ). CEBAF's large energy span and the spectrometers of Hall A allow the (e,e'p) reaction to be studied for large outgoing proton kinetic energies. The kinematics in this proposal will maintain a nearly constant proton kinetic energy of slightly more than 1GeV.
One of the great appeals of using the electromagnetic interaction is
its ability to study finer details of the reaction and nuclear structure
through the extraction of response functions, rather than relying only
on the differential cross section.
The form of the differential cross for (e,e'p) reactions in the single
photon ( Born approximation) is
{ put in equation here }
The kinematic variables are indicated in fig. 3.
All the currents present in the nucleus contribute to the response functions. The response functions can be separated by techniques described in the previous proposal, exp - 89044. By a suitable choice of the kinematic parameters, beam energy, electron and proton angles, for example, the RL, RT, and RLT response functions can be measured. These are the response functions accessible if the electron scattering plane and the plane formed by q and pp are the same. Because of final state interactions the missing momentum associated with the detected event may not be the momentum that the nucleon had before it was ejected by the virtual photon, as mentioned above. Calculations ( Morita[7], Bianconi[8] ) and a heuristic argument (Templon [9] ) show that final state interactions are most prominent in perpendicular kinematics. In this case the angle between the virtual photon's 3 momentum, q, and the missing momentum pX , is large ( about 70 degrees in the kinematics proposed here). Despite this effect, calculations for the deuteron target d(e,e'p)n (Jeschonnek [20]) show that the response function RLT is less sensitive to FSI than the cross section alone. The calculations of Jeschonnek and Donnelly [20] for the case of d(e,e'p)n show a sensitivity in the RTL response function to the choice of a realistic NN potential, i.e., the Bonn versus the Paris potential at high missing momentum. These calculations[20] include final state interactions, including the spin-orbit interaction, using Glauber theory. MEC and IC are not included in these calculations. One could hope that a good calculation of the 4He(e,e'p)t including FSI, MEC, and IC could use the RLT information to arrive at non ambiguous conclusions about high momentum components in the 4He ground state.
Interesting information is accessible even for lower missing momenta, say below 0.3 GeV/c because in other nuclei calculations [5 ] of RLT and the associated asymmetry ALT , where,
ALT = (s(f=0) - s(f=180))/(s(f=0)+s(f=180)), (1)
using a relativisitic mean field approach to obtain the nuclear ground state and the outgoing proton wave function show measurable differences from calculations using a non-relativistic model. The inclusion of the lower component of the bound state nucleon wave function and the outgoing proton wave function [5,15] is critical in fitting the observed value of ALT for the 16O(e.e'p) data of Gao, et al. [ 6]. Relativtistic effects do not change the calculated cross section by more than about 10% from the non-relativistic calculation[5], nor do relativisitic corrections in the calculation of the nuclear ground state for light nuclei have a significant effect in the calculated nucleon momentum distribution, n(k), up to missing momentum of 0.8 GeV/c [4]. However, the calculated ALT [5] changes from about -0.25(non-relativistic calculation) to -0.45(relativistic calculation) at missing momentum of 0.25 GeV/c for the p1/2 state in the data of Gao[6].
The RL and RT response functions are separated using parallel kinematics. In this case q and pX are either parallel or anti parallel. In both cases the proton is detected in the direction of q. These kinematic conditions show the smallest effect of FSI on the outgoing proton's momentum distribution. Here also, the high energy available at CEBAF allows large pX to be measured. The only way earlier experiments could get to large missing momentum was to use off-parallel kinematics, where FSI become large.
An interesting parameter to measure, referred to in the case of d(e,e'p)n (Jeschonnek et al) and in the case of 4He(e,e'p)X (Morita [7]), is the forward- backward asymmetry (qqX= 0) and ( qqX= 180) in the deduced distorted momentumdistribution. This is described in [7] , thusly:
sepis an off-shell
e-p cross section, K is a kinematical factor, Q is the virtual photon four
momentum ,w is the
energy transfer, q is the 3-momentum transfer.
The forward-backward asymmetry in the distorted nucleon momentum distribution is defined by
The case for 4He is shown in fig.6
. It should be noted that these calculations do not contain MEC or IC,
so they only show the effects of FSI on the reaction. The two curves refer
to the use of a realistic ground state wave function and a Jastrow wave
function. Below missing momentum of about 0.3 GeV/c the asymmetry is independent
of the wave function used. Above 0.3 GeV/c the asymmetries diverge, indicating
that different high momentum component distributions will yield different
asymmteries. Different nucleon-nucleon potentials will produce different
high momentum distributions. The foward-backward asymmetry is mainly sensitive
to the S-state component of the wave function. The effects of different
percentages of D-state components is shown in the figure. It would be interesting
to see if the inclusion of MEC and IC in the calculation wash out this
sensitivity to VNN. One of the goals of the proposal is
to provide data in the high missing momentum region where the asymmetry
could be tested against a complete calculation.