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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 (P1) - 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 J1. - 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 (gt99) 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.

- 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

- Start with the vector current term
- The wavefunction of the final - state leptons,

are solutions of the Dirac equation

This factor has the correct order of magnitude to

explain the smallness of

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.

- 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 , 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

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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 (2I1) instead of with

its isospin (I). - Thus the above relationship becomes
- 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 (

) are defined, - The relationship between the two bases is
- Here the quantities

are the Clebsch - Gordan coefficients.

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

- Consider the charge states
- Then apply C and the rotation to these

expressions

Thank you !

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