Theory of spin dependence at very high energies. [Review, S matrix] Page: 3 of 13
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A. Spinology
The discussion that follows will be at the level of the textbook
of Bjorken and Drell3. The reader can refer to it for more details.
The process we shall consider is sketched in Fig. 1.
P 2
1
Fig. 1. One of the Feynman diagrams for 1/2+ 1/2+ elastic scattering
mediated by 1- exchange.
We shall be interested in the structure of this diagram, which
follows directly from an application of the Feynman rules:
M = u(P2 pu(pl)g (q2 vu(ql) (1)
is proportional to the invariant amplitude for Fig. 1. We have
dropped factors of i and e2 as well as the momentum dependence of
the vector propagator since we do not intend to use (1) to compute
absolute rates, but rather wish to highlight the spin dependence.
This spin dependence comes in through the fermion wave functions
u(p) = E+ X(2)
E+m X
where X is a two component spinor. The fermion is characterized by
both a momentum 4-vector pu and a polarization 4-vector wu: in
the rest frame pu = (m,0,0,0) and wu = (0,1) , w ,z) with w - oX =
X. For computations with Dirac wave functioiis, 1t is convenient to
recall the covariant projection operators
u(p) ii(p) = (+m)(1+y5y&) (3)
2
Using this fact we can evaluate the absolute square of M [Eq. (1)].
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Thomas, G. H. Theory of spin dependence at very high energies. [Review, S matrix], article, January 1, 1977; Illinois. (https://digital.library.unt.edu/ark:/67531/metadc1059264/m1/3/: accessed July 16, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.