Symmetries

By Dong Xue

Physics & Astronomy

University of South Carolina

Outline

• 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 Conversation Law Translation in time Energy Translation in space Momentum Rotation Angular momentum Gauge transformation 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 :

E is the total energy, is the phase - space volume, M is the matrix element.

E

21

2if fiM EE

2

, ,28f

a cd a cd

pM

m

• 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)

: the pion decay constant. (S)

: the four - momentum of the pion. (V)

• Construct the possibe matrix elements with the above

elements:

Pseudoscalar term Axial vector current term

Scalar term Vector current term

fp

• Start with the vector current term:

• The wavefunction of the final - state leptons, are solutions of the Dirac equation:

2 2lM m

This factor has the correct order of magnitude to explainthe 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 years.3410

• 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 nucl

eon, which has isospin I = 1/2.• For isospin I , the dimensionality 2I +1 is the number of d

ifferent particles or nuclear levels, they differ by the third component , the group is called an isotopic multiplet.zI

Next introduce the flavour hypercharge :

The third component of the isospin is defined by Gell - Mannand 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 : • 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 : • Oberserve the following reaction :

1 1/ 2 1/ 2 3/ 2

3 2 2 4

• Consider two bases:• The isospins and their third components of each parti

cle are defined, which are given as• The total isospin (I) and its third component ( ) are defin

ed,

• The relationship between the two bases is :

• Here the quantities are the Clebsch - Gordan coefficients.

zI

G-Parity

• G-parity is convenient when dealing with non - strange states with zero baryonic number.

• Start with the , which is an eigenstate of the charge conjugation C.

• G is defined as C followed by a rotation around the y - axis in isotopic space, namely :

0

180

• Consider the charge states:

• Then apply C and the rotation to these expressions :

Thank you !