[Hybrid baryons] Description of electroproduction observables

[Viktor Mokeev]

Dear Colleagues,

Below I attach the comment on description of electroproduction observables we have discussed at our last meeting.

There are two completely identical ways to describe electroproduction observables: by connection them either to hadronic currents (a) or to electroproduction amplitudes (b). Two identical ways, however, results in different expressions for the terms which contains the virtual photon polarization parameter. It is demonstrated in the write up in e-mail below.

All electroproduction observables are determined by contraction between leptonic and hadronic tensors. This contraction is Lorentz invariant by design. This contraction produces the longitudinal contribution into cross section described by Eq (1). Note, that hadronic current is frame dependent but Lorentz covariant. Frame dependence and Lorentz covariance is experessed by factor \omega in Eq(1), which is the virtual photon energy in the chosen frame. In the lab frame and only in the lab frame \omega=\nu where \nu is invariant of electron scattering process. However, replacement \omega by \nu is valid ONLY in the lab frame. That is why leptonic and hadronic tensors are frequently contracted in lab frame. In general, \nu should be replaced by omega. The \epsilon_L parameter is coming from leptonic tensor. Yes, it is frame-dependent but reflect Lorentz covariance of the leptonic tensor dual to hadronic tensor by the factor 1/\omega.

As you may see from Eq. (3), after contraction between leptonic and hadronic tensors, frame dependence is canceled out. The result of the contraction is Lorentz invariant, as it should be.

Also, it is identical to describe electroproduction observables either through currents J_z with the factor \epsilon_L or through the products of amplitudes with the factor \epsilon. Since photoprtoduction cross sections are defined by the products of amplitudes, by convention, the longituidinal  (as well as interference term) contributions are defined through products of amplitudes. Under this convention, only \epsilon parameter should be used, as it was presented by Daniel. How epsilon_L is transformed into \epsilon under leptonic-hadronic tensor contraction is shown in my write up below.

Of course, we can use formalism where \epsilon_L directly enters into the expressions for the observables. It is a case when we describe obsevables through the current J_z.

Conclusion: Both ways describing electroproduction observables by employing epsilon_L and J_z current or by employing epsilon only with amplitudes instead of J_z are identical and Lorentz invariant.

My two cents and I appreciate your comments

Best Regards,
                              Victor







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From: Viktor Mokeev <mokeev at jlab.org>
Sent: Saturday, July 31, 2021 6:37 AM
To: Viktor Mokeev <mokeev at jlab.org>
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