Subatomic Physics

11.7. Chirality versus Helicity 353
invariance arguments. An electron is described by its energy, its momentum p,
and its spin J. For spin 1/2, it is customary to use instead of the spin J the
dimensionless Pauli spin operator σ; it is connected to J by
σ = 2J
. (11.50)
�
The only axial vector available is J, orσ. The operator A must therefore be
proportional to σ. The axial charge operator, A0, changes sign under the parity
operation as indicated by Eq. (11.46); since σ · p has this property, we set
A =(A0, A), A0 =
σ · p
, A = σ. (11.51)
mc
The factor 1/mc in A0 is chosen to make the operator dimensionless.
The nonrelativistic operators, as given in Eqs. (11.40) and (11.51), cannot be
used for the evaluation of the muon and tau[on] decays because there all particles in
the final state must be treated relativistically. The generalization of the operators
V and A to relativistic leptons is well known. (17) Calculations with the relativistic
operators are, however, beyond our means here, and we therefore give the transition
rate for the muon decay without proof. The rate dwµ(Ee) for the emission of an
electron with energy between Ee and Ee + dEe becomes, for Ee ≫ mec 2 ,
dwµ(Ee) =G 2 F
m2 µ
4π3�7c2 E2 �
e 1 − 4
3
Ee
mµc 2
�
dEe. (11.52)
This expression, after replacing the electron energy by the electron momentum,
agrees very well with the spectrum shown in Fig. 11.7.
11.7 Chirality versus Helicity
• In Eq. (9.33) we gave a definition for the helicity of particles. For massive particles
this quantity is not frame independent as can be seen from the fact that the dot
product involves only the space-like components of the momentum so a Lorentztransformation
can clearly change it. In other words, an observer moving faster
than the particle would see the opposite helicity as one moving slower than the
particle.
In the relativistic treatment of quantum mechanics another observable emerges
which is called chirality. It plays a central role in the proper definition of the
currents. We don’t have the room to properly define it here, but we can make a
connection to limiting cases. In particular, for highly relativistic particles (p ≫ mc)
it can be shown that: (17)
chirality → helicity. (11.53)
17 See Halzen and Martin, Quarks and Leptons, John Wiley & Sons (1984); Chapter 5.