Appendix A. Matrix Element for the Process

We consider the matrix element represented by the Feynman diagrams shown in Fig. 22, where the encircled letters give the color indices of the particles and run from 1 to 3 for quarks and antiquarks and from 1 to 8 for gluons. Using the Feynman rules for QCD summarized in Fig. 23 we find for the matrix element

[ Figure 22 ] Diagrams for .

[ Figure 23 ] The Feynman rules for QCD perturbation theory.

where and are, respectively, the polarization vectors for the photon and gluon and

As always, is the effective mass of the charmed quark () and is the virtual-photon invariant mass squared ( refers to both real and virtual photons). The factor denotes the charge of the quark in units of the electron charge.

The matrix element can be written in the following form:

which gives

upon squaring and averaging over photon and gluon polarizations. We calculate the cross section for (transverse) real gluons and (transverse) real and transverse virtual photons. The sum over gluon helicities simply gives a , while the sum over photon helicities gives a for real photons and for virtual ones where

The assures transversality for the proton in the virtual-photon-gluon center-of-momentum frame.

The numerical factor arising from the sum over final color and the average over initial color degrees of freedom is simple in this case since the color matrix element may be factored from both - and -channel matrix elements so that the complete color factor in the squares of the matrix elements becomes

(We shall throughout put angular brackets around numbers arising from color averaging.) The rest of the calculation is standard. The result is

This formula is appropriate for both virtual photons with and real photons (after setting ).

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