Method for Expressing Dirac Spinor Amplitudes in Terms of Invariants and Application to the Calculation of Cross Sections

Abstract

We describe here a method for expressing spinor amplitudes M =Z (p', s ' ) r u ( p , s ) in terms of invariants. The method also provides a simple formula for the square of M summed on spins. When r contains more than five t e r m s o r has more than two y-matrices per t e r m this formula involves the evaluation of fewer and shorter traces than the usual trace technique.Thus this should be a useful computational device for calculating cross sections. The methods presented here a r e independent of the spinor representation, but can be easily specialized to the important helicity case. A number of other possible applications of the amplitude formula a r e also briefly discussed.

New explicit expressions for Dirac bilinears

“…A strategy, developed about 50 years ago [14][15][16][17][18][19][20][21], consists in multiplying M Γ by the unity in the form 1 = M * Γ ′ /M * Γ ′ , where M Γ ′ is of the form (1) with an arbitrary matrix Γ ′ . One can then write…”

Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry

“…Such methods were first explored in refs. [93][94][95][96], using four-component spinors (see also refs. [97][98][99][100][101]).…”

P-odd observables at the $\Upsilon$ peak

“…To do this, we follow the method of reference [18], which permits the calculation of these amplitudes using standard trace techniques. Then, the complete results for the vector and axial currents, in the CM frame and with the momenta along the z-axis are, where we have applied a rotation to the leptonic current of the τ with respect to Eq.…”