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Intrinsic parities of fermions and bosons
Intrinsic parity can be defined if
a particle is at rest.
For scalars (spin 0), vectors (spin 1) and tensors,
parity is equivalent to a rotation by 2π
P2=1 so P= ±1
p.91
Subtle point: Bosons have well-defined parity but fermions
(spin ½) are spinors and produced in pairs. So one can define
relative parity of fermions and anti-fermions that satisfy the
Dirac equation. In addition, the parity of Dirac fermions can be
real or imaginary. By convention, we choose it to be real.
Pf Pf = -1
Intrinsic Parities of fermions
and anti-fermions are opposite
Intrinsic parities of fermions and bosons (continued)
P
J (spectroscopic notation)
J=total spin and P is the intrinsic parity
Question: What is the JP for the photon ?
Answer: 1Question: What is the JP for a pion (and an anti-pion) ?
Answer: O-
Example of parity of a two-body system with
relative orbital angular momentum l
P = P1P2 (-1)
l
Here P1 and P2 are the
intrinsic parities of the
two particles.
The derivation closely follows the example that we
did for the hydrogen atom last time.
Question: What is the parity of two pions in a pwave (l=1) state ?
Answer: P(pion)P(pion)(-1) = -1
Question: What is the parity of a proton and anti-proton
in an orbital angular momentum state l?
Ans: These are fermions (spin ½) so they have opposite
intrinsic parities. (1)(-1)l = (-1)l+1
Question: What are the possible values of JP for a spin ½
particle and its anti-particle if they are in a S-wave state
or a P-wave state (an example in atomic physics is
positronium)
Hint: addition of angular momentum in QM
S - wave J P = 0- or 1-
P - wave J P = 1+ or 0 + ;1+ ;2+
Do you understand the
spectroscopic notation ? What is
the left superscript ?
Charge conjugation (continued)
C eigenvalues of the photon and π0
Cg =-g
C ng = (-1) ng
n
Question: What is the charge conjugation of the π0 ?
Ans: it is the charge conjugation of two photons ?
(-1)(-1) = +1
Question: What does the charge conjugation operator
do to a charged pion ?
C p+ = + p- ; C p- = + p+
Charge conjugation (continued)
Question: What is the charge conjugation of a
charged meson-antimeson pair with relative orbital
angular momentum l ? (Do two cases: when the
mesons are spin zero and when they have non-zero
spin)
Hint: M+  M- and M- and M+ under C.
spin 0:
C m + , m - = (-1)l m + , m Now let’s try spin 1 mesons (spin 0, 1, 2).
What is the symmetry of spin 0, 2 ?
spin s:
C m + , m - = (-1)l+s m + , m -
So there is an
extra factor of
(-1)s
fermions with spin 1/2:
C f f = (-1)l+s f f
Time reversal and CPT
There is a theorem from QFT (Quantum Field Theory)
called the CPT theorem, which states if a local theory of
interacting fields is invariant under the proper Lorentz
group, it will also be invariant under the combination of C
(particle-anti particle conjugation), space inversion (P)
and time reversal (T).
Consequences: if CP is violated then T
is violated (and the theory is not
invariant under the reversal of the
direction of time
particle
Anti-particle
ASACUSA (low energy anti-proton
experiment at CERN)
| m p - m p | /m p £ 10
-8
2003
Mystery of charged pion decay
G(p ® en )
= 1.2 ´10 -4
G(p ® mn )
But the ratio of phase space volume goes like
p(electron)/p(muon) as discussed in Chapter 1.
p
mp - m
140 - 0.5
= 2
=
= 2.3
2
2
pm mp - mm 140 - 106
*
e
*
2
2
e
2
2
2
using the result
mp - m
p =
2mp
2
*
l
2
l
Two body decay kinematics
Question: How can we the
four order of magnitude
discrepancy ?
Question: But why are muons and
electrons different ?
Answer: the ratio of the masses 106 MeV
(muon) versus 0.511 MeV (electron)
M ∼ ml fpfp l(1- g 5 )p g mn l
m
The V-A nature of the weak interaction explains the
ml2 (mass-squared) dependence of the decay rate.