Symmetries

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Symmetries

• By Dong Xue
• Physics Astronomy
• University of South Carolina

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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

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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

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• 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.

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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?

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Parity of two-particle system

• The relationship between two bases
• The inversion of the axes in polar coordinates is
  

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Thus the parity of two-particle system is given
by

• Parity of two mesons with the same intrinsic
  parity

• Parity of Fermion - antifermion pair

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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

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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

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• 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

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• Meson and antimeson with non-zero spins
• The above relationship also holds for fermion
  - antifermion system.

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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.

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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

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

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• 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

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• Start with the vector current term
• The wavefunction of the final - state leptons,
  are solutions of the Dirac equation

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This factor has the correct order of magnitude to
explain the smallness of
Also start with the axial vector current term
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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.

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• Baryon number of the quarks is B 1/3
• Definition of quantum numbers of quark flavours

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Leptonic flavours and lepton number

• The lepton number is defined as
• Similarly, the lepton flavor numbers are given as
  

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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.

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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

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• 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.

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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

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• Consider the charge states
• Then apply C and the rotation to these
  expressions

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Thank you !