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Helicity and Nuclear Beta-Decay Correlations

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Helicity and Nuclear Beta-Decay Correlations Helicity and nuclear beta-decay correlations R. Hong1 and A. Garc´ıa1 1Physics Department and CENPA, University of Washington, Seattle, Washington 98195 (Dated: February 19, 2013) We present simple derivations of nuclear beta-decay correlations taking into account the special role of helicity. These present a good opportunity to teach students about helicity and chirality in particle physics, to practice with simple aspects of quantum mechanics and to learn practical aspects of nuclear beta decays correlations. This article can be also used to introduce students to on-going experiments to search for hints of new physics in the low-energy precision frontier. I. INTRODUCTION beta-decay correlations. We will begin with a simple derivation of the elec- Helicity is the projection of the spin onto the direction tron asymmetry with respect to the polarization of the of the momentum. It is well defined for free fermions. parent nucleus in Sect. II. In Sect. III we will present Helicity plays an important role in modern physics. In a brief description of the weak interaction hamiltonian particular, in the study of atomic, nuclear and subnu- with some associated history. Although we present here clear properties via electroweak probes, helicity rules al- the interactions using Dirac's γ matrices, we think that low for understanding many of the important characteris- what is needed to follow that section is a brief intro- tics of these experiments. Thus, development of intuition duction to the Dirac equation. In our experience many with respect to different aspects of helicity in quantum students who have had only superficial exposure to the mechanics is a worthwhile exercise for classes that are Dirac equation, while not ready to maneuver comfort- taught to advanced undergraduate or beginning gradu- ably with related calculations, can clearly understand the ate students. general arguments presented here. In Sect. IV we derive expressions for the so-called Fierz interference terms. In In this paper we concentrate on the correlations that − arise in nuclear beta decays due to the combination of Sect. V we present a derivation of the e ν¯ correlations conservation of angular momentum and the helicity of the and we discuss the historical confusion which led physi- leptons. Because we concentrate on nuclear beta decays cists to believe that the weak interactions were carried we will use `weak interaction' as a synonym of `charged by scalar and tensor currents for a few years. In Sect. VI weak interactions'. Measurements of correlations from we show calculations for the case of neutron beta decay. nuclear beta decay originally established the V − A na- ture of the weak interactions about 50 years ago. In present times precise measurements of these correlations II. HELICITY AND CHIRALITY PROPERTIES are being pursued to search for forms of new physics with OF THE WEAK INTERACTION helicity properties that differ from the prescriptions of the standard model of particle physics (SM). The main features of beta decay are described in many The correlations can be calculated using trace tech- textbooks2{4. Here we briefly discuss the aspects that are niques and are sometimes brought up in this context as relevant for the present discussion. Nuclear beta decay exercises for students learning relativistic quantum me- is driven principally by two types of current that cor- chanics or field theory. For students without this training respond to two kinds of transitions: `Fermi transitions' the calculated expressions are presented and some com- (F), for which ∆J = 0 ∆T = 0, where J and T represent ments are added to show their plausibility. However, as the quantum numbers for spin and isospin, respectively, we will show below, with the aid of tools learned in the and `Gamow-Teller transitions'5(GT) with ∆J = 0; ±1 elementary quantum mechanics classes, students should ∆T = 0; ±1 but not for J = 0 ! 0. As we will show be equipped for a derivation. Moreover, those students below, in the standard model the former come from the capable of doing the calculations via trace techniques of vector part of the current and don't flip the nuclear spin, gamma matrices may not realize that all that is involved while the latter come from the axial vector part of the is the conservation of angular momentum and the left- current and can flip the spin. handedness of the particles emitted in beta decay. In We start by considering the correlations between the this paper we will present simple derivations of the cor- polarization of the parent nucleus and the direction of the relation expressions. We have found that students benefit electrons for the famous experiment of Wu et al.6 which from following the arguments presented here even if they was one of the first experiments to confirm the hypothesis are able to do the calculation using trace techniques. A of parity violation by the weak interactions put forward review of Dirac, Majorana, and Weyl fermions with some by Lee and Yang7. Wu and collaborators polarized a overlap with the subject of this paper has recently been sample of radioactive 60Co atoms and observed the dis- published1. The present paper is aimed at students with tribution of electrons around the direction of the initial less background in field theory and focuses on nuclear- polarization. The corresponding decay scheme is shown 2 in Fig. 1. If we assume maximum polarization only the We have assumed here that the helicity of the electron is −1, but this is only correct if the mass of the electron is J=5 negligible compared to its energy. Note that the helicity 60Co of a particle is not a Lorentz invariant for a massive par- J=4 ticle. An observer moving faster than the particle will see its helicity in the opposite direction. According to 60Ni the standard model, only the left-handed chirality8 com- ponent of a particle is involved in the weak interaction. FIG. 1. Decay scheme for 60Co. In the relativistic limit these two concepts are equivalent but in general they are not. Konopinski2 gives an intu- M = 5 state is populated. In the beta decay transition itive description for chirality, which may otherwise seem a one unit of angular momentum is lost and the leptons rather abstract concept. Consider an electron moving in have to carry this angular momentum in the direction of the +z direction with momentum p, energy E, and spin the initial polarization (orbital angular momentum can along the +z direction. If we want to measure the veloc- be neglected in nuclear beta decays for which any other ity of this electron, we need to take an infinitesimal time possibility is allowed). Thus, the spin projections of the period. According to Heisenberg's uncertainty principle, two leptons onto the z axis have to be both +1=2, as the uncertainty of its energy goes to infinity. However, shown in Fig. 2. particles with infinite energy move at the speed of light, c, so the results of such a measurement are ±c. The phys- z ical state moving at velocity9 v = p=E can be described as a back-and-forth motion along the path at the speed c, pn with the \backward-motion" probability (1−p=E)=2 and qn \forward-motion" probability (1 + p=E)=2. The physical velocity v is just the mean velocity of this back-and-forth Sn motion. Therefore, the state we are considering can be p expressed as a linear combination of two internal velocity qe e states r r − 1 + p=E 1 p=E − S u+(E; p) = ϕ+(+c) + ϕ+( c); e 2 2 (3) where the subscript ± describes the spin along z direc- tion. Then chirality can be defined as \the spin projec- tion onto the internal velocity direction". In this sense, FIG. 2. Correlation between initial polarization and electron − 60 the state ϕ+(+c) has right-handed chirality and ϕ+( c) direction: in the decay of Co one unit of angular momentum has left-handed chirality. Therefore, a state with well- is lost by the nucleus and has to be carried out by the leptons. defined helicity, momentum and energy, like u (E; p), is Since the latter are spin-1/2 objects they have to both align + their spins in the direction of the initial polarization. This, a linear combination of two states with opposite chirality in conjunction with the helicities of the particles, determines and relative norms as in Eq. 3. If we build up a state the emission probabilities. like ϕ+(+c) + ϕ−(−c) which has a definite chirality, it is not a physical state because the spin is flipping when the For the electron emitted in the direction at an angle particle moves back-and-forth. If the particle is mass- θ relative to the z axis, the positive or negative helicity less, then one intrinsic velocity state is good enough to states jθ±i can be expressed as linear combinations of describe it, so the physical state also contains only one spin-up and spin-down states |±⟩ along the z axis: chirality component. In this case helicity and chirality describe the same property of the particle so these two iϕ jθ+i = cos θ=2j+i − e sin θ=2|−⟩ concepts are equivalent. iϕ jθ−i = sin θ=2j+i + e cos θ=2|−⟩ (1) A formal description of free fermions is shown in Ap- Now we add the assumption that the emitted electron pendix 1 using Dirac spinors. The motion of a free fermion is governed by the Dirac equation, which has has negative helicity. For this final state jθ−i, the pos- sibility to have the +1=2 spin projection onto the z axis two positive energy solutions, i.e.
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