10.1 Basic Concepts and Formulae 543
e=gsinθw (10.10)
sin^2 θw= 0. 2319 (10.11)
Mw/Mz= 0. 88 (10.12)Feynman diagramsare short-hand for writing down individual terms in the calcula-
tion of transition matrix elements in various processes pictorially. As an example,
consider the diagram in Fig. 10.1. The solid lines are the fermion lines. The con-
vention used here is that time runs from left to right. The top solid lines represent
electron and bottom linesμ−. The diagram represents the scattering of a muon with
electron,μ−+e−→μ−+e−by electromagnetic interaction. The dots correspond-
ing to vertices are points where interactions occur. The wriggly line is the photon
line. At the vertex, the electron emits a photon which is absorbed by the muon or
viceversa. This corresponds to the lowest order of perturbation theory and is known
as the first order Feynman diagram or leading Feynman diagram.
Fig. 10.1Muon-electron
scattering
The arrows on the solid lines towards the vertices indicate the direction of
Fermions in time, the antifermions are indicated by reversed arrows, moving back-
ward in time. The photon being an antiparticle of itself does not need any arrow on
the photon line. The lines which begin and end within the diagrams are the internal
lines correspond to virtual particles that is those which are not observed. The lines
which enter or leave the diagram are the external lines which represent the observed
or real particles. The external lines show the physical process of an event, while the
internal lines indicate its mechanism. The direction of fermions is such that charge
is conserved on each vertex. Also, the four-momentum is conserved at each vertex.
The virtual particles which are exchanged (photon in Fig. 10.1) are not present in
the initial or final state. They exist briefly during the interaction. In this short time
τenergy can be violated compatible with the uncertainty principle,τ.ΔE ≈.
Consequently, a virtual particle is not required to satisfy the relativistic relation,
E^2 =p^2 +m^2. A virtual particle can be endowed with any mass which is different
from that of a free particle. In Fig. 10.1, the exchanged photon couples to the charge
of one electron at the top vertex and the second one at the bottom vertex. For each
vertex the transition amplitude carries a factor which is proportional to e that is
√
α
(square-root of fine structure constant). The transition matrix will be proportional
to
√
α√
αorα. The exchanged particle also introduces a propagation term in the