November 11, 2004

Physics 4213/5213 Lecture 12

1 Introduction

This lecture takes a look at two phenomena, that are connected to the phenomena of mixing in the K^{0} and K̄^{0} system. Both these phenomena involve directly observing the mixture of the two states. In the first case, the phenomena of regeneration, a beam with no K_{S} can be converted to a beam with K_{S} by introducing a target. The second case looks at strangeness oscillations, were the net strangeness of the beam varies with time.

2 Regeneration

A phenomena that occurs due to mixing, is regeneration of the K_{S} component of the beam long after it has decayed away. This is done by placing a block of material in the path of the remaining beam. Since the K_{L} beam is composed of both K^{0} and K̄^{0} and the probability to interact with the material in the block is different for the two, the ratio of K^{0} and K̄^{0} will be different after the beam leaves the block. The possible strong interactions for the K^{0} and K̄^{0} are:

K̄^{0} + p → Λ + π^{+}
→ Σ^{+} + π^{0}
K̄^{0} + n → K^{−} + p
K^{0} + p → K^{+} + n  (1)

Notice, the K^{0} can only produce anti-baryons containing strange anti-quarks. This requires considerably more energy in order to conserve baryon number in the reaction, since two additional baryons need to be created.

After the K_{L} beam has traversed the target the wave-function is given by:

|ψ⟩ = [f|K^{0}⟩ + f̄|K̄^{0}⟩]  (2)

where f and f̄ are the fraction of the components that did not interact. This can be rewritten in terms of the K_{S} and K_{L} as:

|ψ⟩ = \frac{1}{\sqrt{2}} [f (|K_{S}⟩ + |K_{L}⟩) + f̄(|K_{L}⟩ − |K_{S}⟩)]
= \frac{1}{\sqrt{2}} [(f − f̄) |K_{S}⟩ + (f + f̄) |K_{L}⟩]  (3)

Therefore, as long as the reaction rates are different the K_{S} can be regenerated.

3 CPT

The combined symmetry CP T is satisfied for all interactions. This being proved in Quantum Field Theory. The assumptions that go into the proof (in simplified form) are:

1. The Lagrangian must be a Lorentz scalar;

2. Integral spin particles must obey Bose statistics and half integral spin particles must obey Fermi statistics.

Two consequence of this theorem, stated without proof, are that a particle and its anti-particle must have the same mass and the same lifetime—partial decay rates do not have to be the same and can be different if CP is violated.

4 Strangeness Oscillations

The first thing to notice is that the states K_{S} and K_{L} are not anti-particles of each other. Therefore, their decay rates and their masses can be different. Secondly and more important, is that in the production experiment, because of the larger strength of the strong interaction, the particle that is produced is the K^{0} or the K̄^{0} . These are a mixture of the weak interaction states:

|K^{0}⟩ = \frac{1}{\sqrt{2}} [|K_{S}⟩ + |K_{L}⟩]  |K̄^{0}⟩ = \frac{1}{\sqrt{2}} [|K_{L}⟩ − |K_{S}⟩]  (4)

Given that the K_{S} and K_{L} have different lifetimes, after some distance (time) the will be predominately K_{L} .

But more interesting than this, is the question of whether the K^{0} can transform itself into a K̄^{0} . This can be explored by looking at the time dependence of the wave-functions. This time dependence is given by the normal phase factor from the Schrodinger equation plus a factor to account for the lifetime:

a_{α}(t) = e^{−im_{α}t}e^{−t/2τ_{α}}  (5)

The particles are taken in their rest frame and again the masses and lifetimes are assumed different since the K_{S} and K_{L} are not anti-particles of each other. Assuming that the initially produced particle is a K^{0} , the phase factors are added to the wave-function:

|ψ(t)⟩ = \frac{1}{\sqrt{2}} [e^{−im_{1}t}e^{−t/2τ_{1}}|K_{S}⟩ + e^{−im_{2}t}e^{−t/2τ_{2}}|K_{L}⟩]

= \frac{1}{2} [e^{−im_{1}t}e^{−t/2τ_{1}} (|K^{0}⟩ − |K̄^{0}⟩) + e^{−im_{2}t}e^{−t/2τ_{2}} (|K^{0}⟩ + |K̄^{0}⟩)]

= \frac{1}{2} [(e^{−im_{1}t}e^{−t/2τ_{1}} + e^{−im_{2}t}e^{−t/2τ_{2}}) |K^{0}⟩ − (e^{−im_{1}t}e^{−t/2τ_{1}} − e^{−im_{2}t}e^{−t/2τ_{2}}) |K̄^{0}⟩] (6)

Notice that at t = 0 the state is pure K^{0} , but at later times there is a mixture of the two. The probability of finding either particle is given by the following expression:

|⟨K^{0}|ψ(t)⟩|^{2} = \frac{1}{4} [e^{−t/τ_{1}} + e^{−t/τ_{2}} + 2e^{−(t/2)(1/τ_{1}+1/τ_{2})} cos Δmt]
|⟨K̄^{0}|ψ(t)⟩|^{2} = \frac{1}{4} [e^{−t/τ_{1}} + e^{−t/τ_{2}} − 2e^{−(t/2)(1/τ_{1}+1/τ_{2})} cos Δmt]  (7)

Figure 1 shows the time dependence of the probability of finding either the K^{0} or the K̄^{0} .

From equation 7, unless there is a mass difference there will be no oscillation, the two would simply decay away. Experimentally an oscillation is observed. The experiment is done by placing a target at varying distances from the point of production and looking for reactions that indicate a K^{0} or a K̄^{0} interacted. The typical method of doing the experiment, is to look for evidence of the K̄^{0} through the reaction:

K̄^{0} + p → Λ + π^{+}
→ Σ^{+} + π^{0}  (8)

By looking at the period of oscillation, the mass difference between the K_{S} and K_{L} can be determined and is found to be 3.5 × 10^{−6} eV.

Figure 1: Plot of equations 7 for a value of Δmτ = 0.5. This shows how the amount of K^{0} and K̄^{0} vary with time.