Invariance Principles and Conservation Laws

Recall from last lecture:

Parity

Parity is a multiplicative quantity.

Parity is conserved in strong and electromagnetic interactions.

Parity of the Spherical Harmonics

For any system bound by a central potential, V(r), the wave function can be decomposed into radial and angular parts, with the angular parts described by spherical harmonics:

ψ(r,θ,φ)=χ(r)Y_l^m(θ,φ)

The spherical harmonics are given by:

Sqrt{(2l+1)(l-m)!/4π(l+m)!} P_l^m(cosθ)exp(i mφ)

The parity operation on spherical coordinates changes r to -r, θ to π-θ, and φ to φ+π. Thus:

exp(i mφ) goes to exp(i mφ + i mπ) = exp(i mπ)exp(i mφ) = (-1)^mexp(i mφ)

and

P_l^m(cosθ) goes to P_l^m(cos(π-θ)) = (-1)^l+mp_l^m(cosθ)

Assembling this yields:

P Y_l^m = (-1)^lY_l^m

Parity will be conserved if the Hamiltonian is invariant under the parity operation, that is, if the Hamiltonian is invariant under a reversal of sign of all the coordinates. This is true for the strong and electromagnetic forces, but, as we'll learn, doesn't hold for the weak force. Therefore, parity is conserved in strong and electromagnetic interactions, but violated in weak interactions. Conveniently, the binding of quarks into hadrons is dominated by the strong interaction with some small influence from the electromagnetic interaction, and a vanishingly small effect from the weak interaction. Therefore, parity is another quantum number useful in defining a particle.

Parity of the photon

The basic atomic transition (E1) is characterized by a change of orbital angular momentum by one unit, Δl=±1. Thus, the parity of the photon is P_γ=-1.

Parity of baryons

The conservation of baryon number makes the parity of baryons irrelevant. Ian any interaction, the same number of baryons must appear in the initial and final states. There absolute parity relative to mesons is therefore irrelevant. By convention we choose the parity of the proton P_p=+1. The parity of the neutron is the same as that of the proton, P_n=+1.

Pion Spin and Parity

The application of parity becomes non-trivial for pions. The determination of the parity is intertwined with the determination of the pion spin, so the both are discussed here. We'll begin our discussion here with a discussion of the decisive experiments from which the spin and parity was originally determined.

Spin of the Pion

Charged pions

The spin of the charged pions was determined by measurement of the forward and backward cross sections for the reaction:

p p <==> π⁺ d

Parity of the neutral pion

The parity of the neutral pion was determined by looking at the so called double Dalitz decay π⁰-->e⁺e^-e⁺e^-. The determination is that the π⁰ also has negative parity.

At the time, it was not clear if the particles π^± were somehow related, or distinct. The fact that they have the same spin and parity is an indication that they are related, and is why they are grouped as pions. That they are not identical particles is clear from the difference in their masses, almost 5MeV, and the tremendous difference in their lifetimes.

Parity of particle and anti-particle

The parity of an anti-fermion is opposite the parity of the fermion, while the parity of an anti-boson is the same as the parity of the boson. Thus the parity of e⁺e^- or p-pbar is -1, not including the orbital angular momentum factor. The parity of π⁺π^- is +1, not including the orbital angular momentum factor.

As an example, the ρ meson of mass 770 MeV and J=1 (spin 1) decays (strongly) to π⁺π^-. The spin 0 pions must be in an L=1 orbital state in order that angular momentum is conserved. Therefore, the parity of the ρ is (-1)^L = -1. We write that the ρ is a state with J^P = 1^-.

Charge Conjugation

The operation of charge conjugation in the classical sense is to change every positive charge with a negative charge, and vice versa. The directions of electric fields reverse under charge conjugation. The directions of magnetic fields also reverse since magnetic fields are created by moving charges -- the directions of the charges don't change, but the signs do, resulting in an overall change of sign. These changes leave Maxwell's equations unchanged since they are linear in the electric and magnetic fields.

In quantum field theory the operation of charge conjugation exchanges particle and anti-particle. For charged particles, this is the same thing as changing the sign. The difference occurs only for neutral particles which are not their own anti-particle, and examples abound: K⁰, D⁰, B⁰, neutrinos, Σ⁰, and Ξ⁰ are a few.

The charge conjugation or C-pairty eigenstates must be neutral, since C|π⁺> --> |π^-> Beyond that, the C-parity operation is like parity.

C-parity of the photon

Both E and B fields change sign under C-parity. Therefore, photons have negative C-parity, C=-1.

C-parity of the π⁰

The π⁰ decays electromagnetically to two photons π⁰ --> γγ and therefore has even C-parity.

The decay of a π⁰ to one photon is ruled out by energy and momentum conservation. The decay of a π⁰ to three photons violates C-parity. Of course, C-parity is not a truly fundamental symmetry, and even seems to be violated since the world around us is matter dominated, and certainly not symmetric in particles and anti-particles. Therefore we'd like to know how well C-parity works. An indication of this is from limits to forbidden processes such as the 3γ decay of the pion. The limit, expressed as a fraction of the 2γ branching ratio is: < 3×10^-8

CPT Invariance

An important result from field theory is that all interactions are invariant under the simultaneous operations of C (charge conjugation), P (parity), and T (time reversal). A consequence is that particles and anti-particles must have the same mass and lifetime, and equal but opposite electric charges and magnetic moments. Up to now, the consequences of the CPT theorem have agreed with experimental tests.

CP violation and T violation

While the combination CPT is thought to be absolutely invariant, no other combination seems to have that property. C and P are violated in the weak interaction.

Discuss C and P violation in the weak interaction using neutrinos as an example.

The combination CP was believed to be conserved by the weak interaction, until:

it was observed to be violated in K_L decays, and
Sakharov pointed out that CP violation was required in order to explain the evolution of the universe from an initial state with equal amounts o matter and anti-matter, to our present matter-dominated universe.