Introduction to Cosmology

Electroweak Interactions 123

Weak leptonic reactions are

e±+

(−)

휈e→e±+

(−)

휈e, (6.34)

(−)

휈e+

(−)

휈e→

(−)

휈e+

(−)

휈e, (6.35)

where

(−)

휈estands for휈eor휈e. There is also the annihilation reaction

e−+e+→휈e+휈e, (6.36)

as well as the reverse pair production reaction.
Similar reactions apply to the two other lepton families, replacing e above by휇or
휏, respectively. Note that the휈ecan scatter against electrons by the W±exchange and
Z^0 exchange, respectively. In contrast,휈휇and휈휏can only scatter by the Z^0 exchange
diagram, because of the separate conservation of lepton-family numbers.

Spin and Statistics. The leptons and nucleons all have two spin states each. In the
following we shall refer to them asfermions,afterEnrico Fermi(1901–1954), whereas
the photon, theHiggs bosonand the W and Z arebosons, afterSatyendranath Bose
(1894–1974).
The difference between bosons and fermions is deep and fundamental. The number
of spin states is even for fermions, odd for bosons (except the photon). They behave
differently in a statistical ensemble. Fermions have antiparticles which most bosons
do not. Thefermion numberis conserved, indeed separately for leptons and baryons.
The number of bosons is not conserved; for instance, in pp collisions one can produce
any number of pions and photons.
Two identical fermions refuse to get close to one another. This is thePauli exclu-
sion forcefor which Wolfgang Pauli (1900–1958) received the Nobel prize, and which
is responsible for the electrondegeneracy pressurein white dwarfs and the neutron
degeneracy pressure in neutron stars. A gas of free electrons will exhibit pressure even
at a temperature of absolute zero. According to quantum mechanics, particles never
have exactly zero velocity: they always carry out random motions, causing pressure.
For electrons in a high-density medium such as a white dwarf with density 10^6 휌⊙,the
degeneracy pressure is much larger than the thermal pressure, and it is enough to
balance the pressure of gravity.
Bosons do not feel such a force, nothing inhibits them getting close to each other.
The massive vector bosons W±and Z^0 have three spin or polarization states: the
transversal(vertical and horizontal) states which the photons also have, and thelon-
gitudinal statealong the direction of motion, which the photon is lacking.
The number of distinct states ordegrees of freedom,푔, of photons in a statistical
ensemble (in a plasma, say) is two. In general, due to the intricacies of quantum statis-
tics, the degrees of freedom are the product of the number of spin states,푛spin,the
number푛antiwhich is 2 for particles with distinct antiparticles and otherwise 1, and
afactor푛Pauli=^78 , which only enters for fermions obeying Fermi–Dirac statistics. For
bosons this factor is unity. The degrees of freedom are then

푔=푛spin푛anti푛Pauli, (6.37)

tabulated in the fifth column of Table A.5.