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3.4 Parity conservation and intrinsic parity 63

A spherically symmetric potential has the property that V (−r) = V (r),so that states bound by such a potential—as is usually the case in atoms—can be parity eigenstates. For the hydrogen atom, the wavefunction in terms of the radial coordinate r and the polar and azimuthal angular coordinates θ and φ of the with respect to the is

χ ( θ φ) = η ( ) m (θ φ) r, , r Yl , where Y is the spherical harmonic function, with l the orbital angular and m its z-component. Under inversion, r → −r, θ → (π − θ), while φ → (π + φ) with the result that

m (π − θ π + φ) = (− )l m (θ φ) Yl , 1 Yl ,

Hence in this case, Pχ (r, θ, φ) = (−1)l χ (r, θ, φ) (3.3)

3.4 Parity conservation and intrinsic parity

In strong and electromagnetic interactions, parity is found to be conserved: the parity in the final state of a reaction is equal to that in the initial state. For example, for an electric (E1) transition in an atom, the change in l is governed by the l =±1. Thus from (3.3) the parity of the atomic state must change in such transitions, which are accompanied by the emission of of negative parity, so that the parity of the whole system (atom + ) is conserved. For a (less probable) magnetic dipole (M 1) transition, or for an electric quadrupole (E2) transition, the selection rules are l = 0 and 2 respectively, and in either case the radiation is emitted in a positive parity state. In high- physics, one is generally dealing with pointlike or nearly pointlike interactions and electromagnetic transitions involving small changes in (J =±1), in which case photons are emitted with negative parity. The of a pair of identical particles under interchange, which was described in Section 1.3, can be extended to include both spatial and functions of the particles. If the particles are non-relativistic, the overall wavefunction can be written as a simple product of space and spin functions: ψ = χ space α spin

Consider two identical , each of spin s = 1/2, described by a spin function α(S, Sz) where S is the total spin and Sz = 0or±1 is its component along the z-() axis. Using up and down arrows to denote z-components of sz =+1/2 and −1/2, we can write down the (2s + 1)2 = 4 possible states as follows: ⎫ α(1, +1) =↑↑ ⎬ α( − ) =↓↓ = 1, 1 √ ⎭ S 1, symmetric α(1, 0) = (↑↓+↓↑)/√2 α(0, 0) = (↑↓−↓↑)/ 2 S = 0, antisymmetric (3.4) 64 Conservation rules, symmetries, and of

The first three functions are seen to be symmetric under interchange, that is, α does not change sign, while for the fourth one it does. It is seen that the sign of the spin function under interchange is (−1)S+1 while that for the space wavefunction from (3.3) is (−1)L, where L is the total orbital angular momentum. Hence the overall sign change of the wavefunction under interchange of both space and spin coordinates of the two particles is

+ + ψ → (−1)L S 1 ψ (3.5)

As an example of the application of this rule, let us consider the determination of the intrinsic parity of the . This follows from the existence of the S-state capture of a negative pion in , with the emission of two :

− π + d → n + n (3.6)

The deuteron has spin 1, the pion spin 0, so that in the initial state and therefore in the final state also, the total angular momentum must be J = 1. If the total spin of the neutrons is S and their orbital angular momentum is L, then J = L + S. If J = 1 this allows L = 0, S = 1; or L = 1, S = 0or1;orL = 2, S = 1. Since the neutrons are identical particles it follows that their wavefunction ψ is antisymmetric, so that from (3.5) L + S must be even and L = S = 1isthe 3 L only possibility. Thus the neutrons are in a P1 state with parity (−1) =−1. The parities cancel on the two sides of (3.6), so that the pion must be assigned an intrinsic parity Pπ =−1, so that parity be conserved in this . The assignation of an intrinsic parity to a particle follows if the particle can be created or destroyed singly in a parity-conserving interaction, in just the same way that has been assigned in the same interaction to obey charge conservation. Clearly, in the above reaction, the number of is conserved and so the nucleon parity itself is conventional. It is assigned Pn =+1. However, in an interaction it is possible, if the energy is sufficient, to create a nucleon–antinucleon pair, and hence determine its parity by experiment. So while the parity of a nucleon is fixed by convention, the relative parity of nucleon and antinucleon—or any other –antifermion pair—is not. In the Dirac theory of fermions, particles and have opposite intrinsic parity. This prediction was verified in an experiment by Wu and Shaknov, shown in Fig. 3.1, using a 64Cu positron source. Positrons from this source came to rest in the surrounding absorber and formed positronium, an ‘atomic’ bound state of electron and positron, which has energy levels akin to those of the hydrogen atom, but with half the spacing because of the factor 2 in the reduced . The ground level of positronium occurs in two closely 3 spaced substates with different mean lifetimes: the spin-triplet ( S1) decaying −7 1 to three photons (lifetime 1.4×10 s), and the spin-singlet state ( S0) decaying to two photons (lifetime 1.25 × 10−10 s). We consider here the singlet decay:

+ − e e → 2γ (3.7)

The simplest wavefunctions describing the two-photon system, linear in the momentum vector k and in the polarization vectors (E-vectors) ε1 and ε2 of 3.5 Parity violation in weak interactions 65

Positron source (Cu64) S1

Aluminium scatterer

Lead shield Fig. 3.1 Sketch of the method used by Wu and Shaknov (1950) to measure the relative orientation of the polarization vectors of the 1 two photons emitted in the decay of S0 positronium. S1 and S2 are two anthracene counters recording the gamma rays after S2 scattering by aluminium cylinders. Their results proved that fermion and antifermion have opposite parity, as predicted by the Dirac theory of the electron. the photons will be

ψ1(2γ) = A (ε1 · ε2) ∝ cos φ (3.8a)

ψ2(2γ) = B (ε1 × ε2) · k ∝ sin φ (3.8b) where A and B are constants and φ is the angle between the planes of polarization. The first quantity ψ1 is a and therefore even under space inversion (φ →−φ), thus requires positive parity for the positronium system. The quantity ψ2 is the product of an axial vector with a polar vector, that is, a quantity which is odd under inversion. It corresponds to a positronium system of negative parity, with a sin2 φ distribution of the angle between the polarization vectors. In the experiment, the decays of singlet positronium were selected by observing the two photons emerging in opposite directions from a lead block. The photon polarization was determined indirectly by observing the Compton scattering off aluminium cubes, recorded in anthracene counters as shown in Fig. 3.1. The ratio of the scattering rates for φ = 90◦ and φ = 0◦ was 2.04±0.08, consistent with the ratio of 2.00 expected for positronium of negative parity. Since the ground states of positronium are S-states, the parity measured is the same as that of the electron–positron pair. This experiment therefore confirms that fermions and antifermions have opposite intrinsic parity, as predicted by the Dirac theory.

3.5 Parity violation in weak interactions

While parity is conserved in the strong and electromagnetic interactions, it is violated—what is more, maximally violated—in the weak interactions. This is manifested in the observation that fermions participating in the weak interactions are longitudinally polarized. Let σ represent the spin vector of a particle of energy E, momentum p, and v travelling along the z- axis, with σ 2 = 1. The longitudinal polarization P is the difference divided by the sum, of the numbers of particles N + and N − with σ parallel and