Triplet Asymmetry Calculation

The goal is to establish a method for measuring the polarization of a GeV-scale photon beam in a plane-polarization basis. The method should be based upon a process with a large physics asymmetry, for which the theoretical uncertainties on the asymmetry are small. Atomic pair production has a large asymmetry but its theoretical calculation is complicated by the dominance of the low-q2 part of the cross section where the nuclear charge is partially screened by atomic electrons. This can be taken into account by using the atomic form-factor, but that assumes that the scattering is elastic and fails to take into account atomic excitations that take place during the scattering process. One way to avoid the complications of atomic physics is to select the kinematics of pair production from a free electron and then detect the recoil electron in the final state. If the recoil electron is detected with sufficient momentum then the final-state interactions will be weak, and the process can be calculated from QED.

To our knowledge, so far no complete QED calculation of this process that takes polarization into account has been published. We undertake to carry out a numerical calculation ourselves, at tree level QED, and publish the results. As a check, we want to compare our results with published results that should be good approximations for some range of kinematics.

Table of Contents

1. What kinematics should we use to describe the process?
2. Does the calculation lead to a practical result?
3. Do we agree on the shape of the unpolarized cross section?

From my notes taken during our conversation at YerPhI on November 14, I understand the kinematics you are using to describe triplet production as follows. A basic accounting of degrees of freedom shows that for a 3-particle final state, 9 momentum components and 4 constraints from conservation of momentum leaves 5 independent kinematical variables. Your choice is as follows.

1. Fix completely the momentum of the positron and the electron. This is 6 variables, so it is overdetermined if the initial state is given. Instead, you allow the incident photon energy to be a free variable. This leads to a unique solution for all remaining variables.
2. To simplify the problem, take symmetric kinematics for the pair. This means that the pair has equal momentum. In the frame where the incident gamma is along the z-axis, the specified polar angles of the electron and positron are fixed at the same value and their azimuthal angles sum to zero. Under these conditions it is seen that the recoil electron always lies in the zx plane.
3. As a test case, we consider the following values.

   E+ = E-	= 500 MeV
   θ+ = θ-	= 7.5 mr

   Asymmetry will be seen in the variation of the cross section with the azimuthal angle of the positron φ+ = -(φ-).
I coded these constraints into a program and solved the kinematics. Click the following links for plots of E_recoil, theta_recoil, and E_gamma vs φ+. I checked that these solutions satisfy energy and momentum conservation to within a precision of 1eV. Sergei, my results do not agree with your plots shown here. The agreement is good if you shift φ by 90° in your plots. Is it possible that your results are off by 90°? I checked mine carefully, and I think mine are correct. For example, at φ=0° the positron and electron momenta are parallel which means that the recoil momentum must be large to cancel the transverse part. But in your plot, the recoil energy is negligible at φ=0°. Can you please check this?

You are right I have an extra shift in the Phi axis by 90 degrees. The reason is that I used Delta Phi = Phi_e+ - Phi_e- in calculations and pass it as axis value. So, we can say that our kinematics results coincide.

I've seen your web site shared with Sergey and have two remarks:

1. Polarized calculation of the triplet using Borselino's plus 6 additional graphs were so far already calculated. Please see the CLAS-NOTE 98-018[1] and reference therein on Endo and Kabayashi's TPP code [2] of polarized triplet calculation. Of course for special kinematical configuration adopted to non-magnetic analysis of the forward pair.
2. In kinematical consideration of the triplet production in particular for the high enough energies, kinematical reconstruction of the forward pair may be too rough for kinematical reconstruction of fine details in recoil arm. For example a reasonable energy resolution like 1% for each of forward particles might be comparable with energy of the recoil with corresponding consequences.

It surprises me to hear you say that complete results for the triplet were presented at the Hall B workshop, because I attended that workshop and I distinctly remember Maximon saying that the triplet cross section had never been done properly. I have written him an email to try to find out if that situation has changed, and I will update you on what I learn. Meanwhile I have searched the web and found a couple more references [3,4] that also refer to complete calculations, although the authors themselves are more interested in experimental details than in the theoretical results.

With regard to experimental resolutions, we are not yet at the point of being able to propose specific kinematics for the measurement, but from what I have seen so far the cross section is not very sensitive to the exact energy of the forward pair. The same is not true of the recoil. The recoil is low-energy and so it sets the scale for the q² of the process, whereas the energy of the pair is weakly dependent on m². To be more quantitative we need to compare some specific kinematics.

To compare our differential cross section results, we need to agree on what we use for the differential measure. What I am using is shown in the formula below.



where p₊ and p_- refer to the forward e⁺ and e^- pair, while p and p' refer to the target and recoil system and k is the momentum of the incident photon. Using this for the differential measure, I plot my cross section in units of mb/MeV² as a function of φ₊ = φ_- in Fig. 1.