Symmetries
By Dong Xue
Physics & Astronomy
University of South Carolina
Outline
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Symmetries
Parity(P)
Particle-antiparticle conjugation(C)
Time reversal(T)
Pion decay
Quark flavours and baryonic number
Leptonic flavours and lepton number
Isospin
Sum of two isospins
G - parity
Symmetries
• The conservation laws limit the possibility of an initial
state transforming into another state in a quantum
process (collision or decay) and are expressed in terms
of the quantum numbers.
• Noether's theorem : Symmetries
Conservation Laws
Symmetry
Translation in time
Translation in space
Rotation
Gauge transformation
Conversation Law
Energy
Momentum
Angular momentum
Charge
• Discrete additive quantum numbers.
"Charges" of all fundermental interactions.
Quark flavours, baryon number, lepton flavours and
lapton numbers.
• Discrete multiplicative quantum numbers.
Parity, particle - antiparticle conjugation and time
reversal.
Parity
• The parity operation is the inversion of the three
coordinate axes.
• Definition of parity for different particles:
Proton: positive parity (P=+1)
Fermions
Other fermions : relative to the proton
QFT requires fermions and antifermions to have opposite
parities while bosons and antibosons to have the same
parity.
At the quark level, all quarks have positive parity and
antiquarks have negative parity.
The parity of the photon is negative.
How about the parity for strange hyperons?
Parity of two-particle system
• The relationship between two bases :
• The inversion of the axes in polar coordinates is :
Thus the parity of two-particle system is given by:
• Parity of two mesons with
the same intrinsic parity
• Parity of Fermion antifermion pair
The parity of the pion
• Consider the following process:
The initial angular momentum of the reaction is J=1.
The deuterium nucleus contains two nucleons, of
positive intrinsic parity, in an S wave.
Final state contains 2 identical fermions, there is one
choice for this state:
Particle-antiparticle conjugation
• The particle-antiparticle conjugation operator C changes
the particle into its antiparticle, leaving space
coordinates, time and spin unchanged, but the sign of all
the additive quantum numbers is changed.
• The charge conjugation of the photon
For a state of n photons:
• The charge conjugation of the pions :
• The charge conjugation of the meson :
• The charge conjugation of the particle - antiparticle pair :
Meson and antimeson with zero spin :
Meson and antimeson with non-zero spins :
The above relationship also holds for fermion antifermion system.
Time reversal and CPT
• Time reversal operator inverts time leaving the
coordinates unchanged.
• The invariance of the theories under the combined
operations P, C and T is called CPT.
• A sequence of CPT is that the mass and lifetime of a
particle and its antiparticle must be identical.
Pion decay
• Charged pions decay predominantly (>99%) in the
channel :
• The second most probable channel is :
• The ratio of decay width between the two channels is :
2
1
 if 
M fi   E 
2E
E is the total energy,   E  is the phase - space volume,
M is the matrix element.
 a ,cd 
pf
8 m
2
M a ,cd
2
• The matrix element contains their wavefunctions
combined in a covariant quantity.
• Following are the possible combinations:
• Another three factors of matrix element :
 : the wavefunction of the pion in its initial state. (PS)
f : the pion decay constant. (S)
p  : the four - momentum of the pion. (V)
• Construct the possibe matrix elements with the above
elements:
Pseudoscalar term
Scalar term
Axial vector current term
Vector current term
• Start with the vector current term:
• The wavefunction of the final - state leptons, are
solutions of the Dirac equation:
  M  ml
2
2
This factor has the correct order of magnitude to explain
the smallness of    e  /     
Also start with the axial vector current term :
Quark flavours and baryonic number
• Definition of the baryon number :
• Within the limits of experiments, all known interactions
conserve the baryon number.
• Consider the proton decay :
• The present limit is almost 1034 years.
• Baryon number of the quarks is B = 1/3
• Definition of quantum numbers of quark flavours :
Leptonic flavours and lepton number
• The lepton number is defined as :
• Similarly, the lepton flavor numbers are given as :
Isospin
• Symmetry property of nuclear forces :
two nuclear states with the same spin and the same
parity differing by the exchange of a proton with a
neutron have approximately the same energy.
• Proton and neutron are considered two states of the
nucleon, which has isospin I = 1/2.
• For isospin I , the dimensionality 2I +1 is the number of
different particles or nuclear levels, they differ by the
third component I z , the group is called an isotopic
multiplet.
Next introduce the flavour hypercharge :
The third component of the isospin is defined by Gell - Mann
and Nishijima relationship :
The sum of two isospins
• The rules for isospin composition are the same as for
angular momentum.
• Consider a system of two particles, one of isospin 1 and
one of isospin 1/2. The total isospin can be 1/2 or 3/2.
This statement can be written as : 1  1/ 2  1/ 2  3/ 2
• Alternative is to label the representation with the number
of its states (2I+1) instead of with its isospin (I).
Thus the above relationship becomes : 3  2  2  4
• Oberserve the following reaction :
• Consider two bases:
• The isospins and their third components of each
particle are defined, which are given as
• The total isospin (I) and its third component (I z ) are
defined,
• The relationship between the two bases is :
• Here the quantities
Clebsch - Gordan coefficients.
are the
G-Parity
• G-parity is convenient when dealing with non - strange
states with zero baryonic number.
0
• Start with the  , which is an eigenstate of the charge
conjugation C.
• G is defined as C followed by a 180 rotation around the
y - axis in isotopic space, namely :
• Consider the charge states:
• Then apply C and the rotation to these expressions :
Thank you !