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Basic Radiative Corrections

This summarizes my conversation with Andrei Afanasev.

Tree level

There are two diagrams that contribute at tree level. See Figs. 1a and 1b. For discussion purposes we will only talk about the straight diagram although all comments apply equally well to the crossed diagram. The cross section calculation is well known, goes by the name of the Klein-Nishina formula for obvious reasons and can be integrated over all solid angle to give a total cross section.

Fig. 1a. Tree-level, straight

Fig. 1b. Tree-level, crossed

Note that integration over all solid angle is not possible for electron scattering off a nucleus (for example). Only by introduction of atomic screening can that integral be made to converge. In the Compton case, the integral is finite without having to introduce any "artificial" cut-off.

Virtual corrections

At the next order in perturbation theory the diagrams including virtual photons (Fig. 2, for example) are introduced. The order α correction to tree level comes from the interference between the tree level diagrams and one-loop virtual photon diagrams. This prescription runs into difficulty. As the virual photon energy approaches 0, there is a term in the cross section integral over any finite solid angle does not converge, and is negative; this is the so-called infrared divergence problem.

Fig. 2. Radiative diagram with a virtual photon. (Representative; one of a few.)

Real corrections

The problem is solved by simultaneously considering higher order diagrams where two real photons are in the final state, as in Fig. 3. The cross section of this diagram has a term which is also infrared divergent, is positive, and exactly cancels the divergence in the real-virtual interfence term. One must therefore consider the sum of the processes to get a finite answer, non-radiative and radiative scattering together. Note that the "double Compton" scattering described in Fig. 3 does not interfere with the tree level diagrams; it has a different final state. So here "together" means the sum of two non-interfering processes.

Fig. 3. Radiative diagram with a real photon. (Representative; one of a few.)

The functional form of the divergent terms can be studied by imposing an artificial cut-off into the calculation, so that the behavior as the cut-off approaches zero can be characterized. This can be done as a finite mass for the photon, as a low energy limit on the photon energy, or even as a perturbation to four space-time dimensions (dimensional regularization). In all schemes the cancellation proceeds as described above. Further, the cancellation obtains order-by-order in perturbation theory.

Role of the detector

Although in principle the two processes (radiative and non-radiative) can be distinguished experimentally, in any real experiment, the detection of both in the same "channel" is unavoidable. At very low "extra" photon energy, any real-world detector cuts will include both radiative and non-radiative processes. There are several ways that this can happen:

1. The radiated photon is of such low energy that it is below threshold in all areas of the detector.
2. The radiated photon strikes the detector in the same place as one of the other particles and is not distinquished as a separate particle (particularly probable for Compton scattering with no magnetic field). Note
3. The radiated photon goes outside of the detector geometric acceptance and is not seen and does not disturb the kinematics of the other particles enough to cause the event to fail cuts.

So any count rate which includes the non-radiative process will also necessarily include some portion of the radiative process as well. Still, if the ability of the detector to distinguish radiative and non-radiative processes is understood the cross section for scattering into that detector can be calculated. Note that even if extended to include all solid angle, the Compton cross section remains finite and well-defined, as long as there is a full understanding of how radiative and non-radiative events are "combined" over the full solid angle. This means a measurement of the "total cross section including radiation" in a detector-independent way is problematic since the prescription for co-detecting radiative and non-radiative events is necessarly detector dependent.

Comparing different experiments

Non-radiative cross section

Since the measured cross section and the theoretical prediction described above depend on the details of the detector, comparison between different experiments is often done on the basis of a non-radiative cross section. For simplicity, let us consider the total cross section. At a given order in perturbation theory, the observed cross section σ_observed can be written as

σ_observed = σ₀(1 + δ_v + δ_r)

where δ_v represents the difference between the observed cross section calculated at lowest order (σ₀) and the observed cross section including after inclusion of virtual diagrams and δ_r represents the difference between σ₀ and the observed cross section including real radiation, and where the infrared divergences of both of these contributions have cancelled each other. Once δ_v and δ_r have been calculated, comparison between different experiments can be made on the basis of σ₀ which is independent of the detectors.

Compton scattering short-cut

For Compton scattering the radiative correction is relatively small, a few percent. Therefore, if we were to compare different experiments assuming that δ_vand δ_r are zero (since we do not have the mechanism for calculating them yet), the comparison would be valid as long as we believe that the radiative corrections for each experiment are similar. The non-radiative cross section thus obtained but the comparison (as a check of systematics) would be fine. Fine at what level? If radiative corrections for the different analyses vary by 10% and the total radiative correction is 2%, then the systematic difference in the comparison is only 0.2%.

This method has the advantage that varying geometric and analysis cuts are allowed among the different "experiments". Note that the variation in geometric acceptance is an essential part of the differences among the analysis. Allowing such variation makes the comparison exercise more useful. When we do have a good prescription for calculating radiative corrections, then we can pursue the non-radiative total cross section in the standard way.