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 component 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 √ √ − 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 direction. Then chirality can be defined as “the spin projection 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 |θ±⟩ 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 |θ+⟩ = cos θ/2|+⟩ − e sin θ/2|−⟩ concepts are equivalent. iϕ |θ−⟩ = sin θ/2|+⟩ + 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 |θ−⟩, the possibility to have the +1/2 spin projection onto the z axis two positive energy solutions, i.e. particle solutions, with is sin2 θ/2. This implies that the probability of emission well-defined energies, momenta and helicities. This is due of the electron at an angle θ with respect to the initial to the fact that both momentum and helicity operators nuclear spin direction is given by: commute with the free particle hamiltonian. The spinor parts of these two solutions u±(E, p) have opposite he- sin2 θ/2 = (1 − cos θ)/2. (2) licities, but both of them have left-handed chirality pro- 3 jections with norms We have shown that assuming the chiral properties of √ the electrons emitted in beta decay leads to good agree- |P u (E, p)| = 1 − p/E, L + √ ment with experiment. As we shall see in the next section, this does not completely determine the chiral prop- |PLu−(E, p)| = 1 + p/E, (4) erties of the weak interaction. where PL is the left-handed chirality projection operator. This is consistent with Eq. 3. Note that |PLu+(E, p)| goes to zero in the massless limit where E = p, which III. SCALAR, VECTOR, AND TENSOR means that only the negative helicity states have left- CURRENTS handed chirality projection. In general, for massive particles like electrons, both of them should be included. At the time that Lee and Yang proposed that parity Therefore, the calculation that led to Eq. 2 should be was violated little was known about the weak interaction. replaced by: Dirac had already shown how to solve problems involving ( ) the electromagnetic interaction within a quantum theory sin2 θ/2 (1 + p /E ) /2+ 13 ( ) e e that correctly takes into account relativity . For elec- 2 cos θ/2 (1 − pe/Ee) /2 = (1 − (pe/Ee) cos θ) /2. (5) tron scattering from a proton at low-momentum transfer so that internal nucleon excitations can be neglected, for In summary, given that the nuclei experience a change example, the interaction can be expressed as a product of projection of angular momentum ∆M along a quan- of a nuclear current, a propagator for the photon, and an tization direction z, the electron, due to its helicity, will electronic current14–16: have an angular distribution: e2 H = (ψ¯ γµψ ) (ψ¯ γ ψ ). (10) p EM p p 2 − 2 e µ e 1 − ∆M · e,z (6) q M E e Here the ψ′s are Dirac spinor operators which can anni- with ∆M = ±1. For zero polarization of the parent hilate a particle (with certain momentum, energy, etc.) clearly there is no angular correlation. Once students in the initial state or create an antiparticle from vacuum, understand the mechanism they can calculate transitions ψ¯ means ψ†γ0, and q2 is the square of the 4-vector cor- starting in the different M-substates of the parent state. responding to the momentum transfer in the scattering For an initial state in the M = 4 substate, for example, process. The γµ’s are Dirac’s gamma matrices with the there will be both a spin-flip part (M = 4 → M = 3) with properties: the same angular distribution as above and non-spin-flip { → µ µ I for µ = 0 part (M = 4 M = 4) with no angular dependence. γ γ = − Following these arguments one can check that for an en- I for µ = 1, 2, 3 semble with polarization P = ⟨J⟩/J the decay rate is γµ γν = −γν γµ for µ ≠ ν; µ, ν = 0, 1, 2, 3. (11) proportional to: M is the mass of the carrier and in the case of the photon p 17 1 + AP · e . (7) M = 0. Fermi proposed a similar structure for the Ee weak interactions: − 60 G ¯ µ ¯ with A = 1 for the decay of Co. HFermi = −√ (ψpγ ψn)(ψeγµψν ) + h.c., (12) Antiparticles produced in weak decays present helici- 2 ties opposite to those of particles, so the angular distri- where h.c. indicates the hermitian conjugate. Now the bution of the antineutrino around the direction of polar- process described is an incoming neutrino (or an outgo- ization is: ing antineutrino) and an outgoing electron. The h.c. term p 1 + BP · ν . (8) describes an incoming electron (or an outgoing positron) Eν and an outgoing neutrino. Because the mass of the car- 60 rier for the weak interactions is much larger than the mo- wiht B = +1 for the decay of Co. menta involved in the nuclear transitions the propagator Note that the expressions above show that the decay is constant to a very good approximation. When Fermi rate varies under the operation of parity (which inverts proposed this form for the interaction it was unknown the sign of coordinates). The experimental determination what the mass of the carrier was, but nowadays we know of A by Wu et al.6 showed clearly that parity was violated MW ∼ 80 GeV while the momentum transfer in nuclear by the weak interactions. beta decay is of the order of a few MeV. In fact, the weak- Using Eq. 4, the polarization of emitted electrons is ness of the weak interaction in nuclei is due to the large (1 − p/E)(+1) + (1 + p/E)(−1) mass of the W compared to the beta decay release en- P = = −p/E. (9) e (1 − p/E) + (1 + p/E) ergies. Although Fermi focused on the hypothesis of the vector interaction he suggested other possibilities were This was later directly confirmed by several allowed. Von Weizsäcker18 stressed this point suggest- experiments10–12. ing that the issue should be resolved experimentally. de 4

19 L/R Groet and Tolhoek calculated the decay rates for a phe- where we have used the notation ψ = PL/R ψ. We nomenological hamiltonian including all possible Lorentz have ignored the pseudo-scalar currents because they invariants and Lee and Yang7 included parity violation: turn out to be very small in nuclear beta decays. In ∑ this article we will assume the constants C to be real. ¯ i ¯ i Hint = (ψpO ψn)(CiψeOiψν + Allowing for complex phases brings in time-reversal sym- i=S,P,V,A,T metry violation, which is very interesting,20 but subject ′ ¯ 5 CiψeOiγ ψν ) + h.c. (13) for another paper. Notice that while the vector and axial- vector currents couple incoming and outgoing particles The Ci’s are constants that could be determined experi- with identical chiralities, the scalar and tensor currents mentally and the operators Oi’s are do the opposite. This is a direct consequence of Eq. 15.

OS = 1 Students should be encouraged to show that in the O = γ5 non-relativistic limit, appropriate for nucleons in nuclear P beta decays, the currents can be simplifed as: O = γ (14) V µ † 5 uu¯ → ϕ ϕ OA = iγµγ √ √ uγ¯ u → ϕ†(1, O(v/c))ϕ − − µ OT = σµν / 2 = i (γµγν γν γµ) /(2 2). 5 † uγ¯ µγ u → ϕ (O(v/c), σ)ϕ (19) The corresponding currents are called, respectively, { O(v/c) for µ = 0 scalar, pseudo-scalar, vector, axial-vector (or pseudo- uσ¯ u → µν −ϵ ϕ†σ ϕ for µ, ν, ρ = 1, 2, 3 vector), and tensor. The additional gamma matrix in- µνρ ρ cluded here, γ5, can be expressed in terms of the other uγ¯ 5u → O(v/c) four, γ5 = iγ1γ2γ3γ0. Using Eq. 11 one can show that where on the left-hand-side the spinor u’s are Dirac γ5 γ5 = I and γµ γ5 = −γ5 γµ (15) Spinors, while on the right-hand-side the spinor ϕ’s are the correponding Pauli Spinors and the σ matrices are The property of the operators under parity becomes evi- Pauli matrices. The leading order of the vector and scalar dent if one considers what happens when the coordinates currents (the axial vector and tensor currents) are iden- are inverted. Clearly a Lorentz scalar can not be the tical and called Fermi (Gamow-Teller) current. This also inner product of two vectors with different parities. On µ explains the selection rules given in Sect. II. the other hand, given that γ pµ (the kinetic term in We now see that the experiment of Wu et al., which the Dirac Equation) is a Lorentz scalar when acting on a µ identified the electron from nuclear beta decays as a par- state, the spacial components of γ should be parity-odd, ticle with left chirality, could not determine the chiral- while its time-like component should be parity-even. Be- 5 ity of the emitted antineutrino. Additional experiments cause the γ matrix is the product of one time-like matrix were proposed to determine whether the currents were and 3 space-like ones, it is parity-odd and multiplication scalar, vector, axial-vector or tensor, or some combina- by it reverses the parity property of all operators. This 5 14 tion of these. As we will see, eventually they determined γ matrix is by definition the chirality operator . Con- that the weak interaction is mediated by vector and axial- sequently, the operators: vector currents. 5 PL/R = (1 ± γ )/2 (16) are the projectors onto left- and right-handed chiralities. IV. FIERZ INTERFERENCE Under parity transformation, PL/R turns into PR/L, so a left-handed state transforms to a right-handed state. If both vector and scalar currents existed an interfer- In the relativistic limit, when the masses of the particles 5 ence effect should be present in the differential decay rate are negligible compared to their energies, γ becomes the as long as they can lead to the same final states. To helicity operator, and the two projectors above become fix ideas we first consider Fermi transitions. We also the projectors onto the helicity states. The interaction assume neutrinos are massless so the antineutrino final of Eq. 13 can be re-written in terms of left- and right- state with definite helicity has only one chirality compo- handed lepton operators. For the vector and axial-vector nent and the chiralities of the neutrino spinors should be currents: ∑ ( identical in the interfering terms. Both of the first lines ¯ i ′ ¯L L of Eq. 17 and Eq. 18 involve left-handed neutrinos, but Hint = (ψpO ψn) (Ci + Ci) ψe Oiψν + (17) i=V,A one projects the electron final state onto the left-handed ) chirality state while the other projects it onto the right- (C − C′) ψ¯RO ψR i i e i ν handed chirality state. Therefore, the product of these ′ ′ while for the scalar and tensor currents: two terms is proportional to (CV + CV )(CS + CS) multi- ∑ ( plied by the product of the left-handed and right-handed ¯ i ′ ¯R L Hint = (ψpO ψn) (Ci + Ci) ψe Oiψν + (18) norms of the electron final state, which are described in i=S,T ) Eq. 4. The final states21are solutions of the Dirac equa- − ′ ¯L R (Ci Ci) ψe Oiψν tion with well-defined helicities. For either helicity state, 5 the√ product of√ the left-handed and right-handed norms other. Assuming vector and axial-vector currents only, is 1 + pe/Ee 1 − pe/Ee, which is equal to me/Ee. we can repeat the chirality arguments of Sect. II. The The term that involves right-handed neutrinos is the probability of finding the antineutrino spin along ±z is 22 same as the term described above except that the (1±pν /Eν )/2. Given the antineutrino is with spin down, − ′ − ′ coupling-constant factor is (CV CV )(CS CS). The for example, the probability of finding the electron along sum of these two terms accounts for the interference ef- a direction at an angle θ will be given by Eq. 5. Thus fect between vector and scalar components, called the the total probability of finding the electron at an angle θ Fierz interference23 with rate proportional to: with respect to the antineutrino is:

′ ′ me (C C + C C ) . (20) 1 − pν /Eν 1 − (pe/Ee) cos θ V S V S E + (22) e 2 2 The m/E factor is typical of situations like the present 1 + pν /Eν 1 + (pe/Ee) cos θ one, where there is a ‘helicity mismatch’. A similar sit- 2 2 uation occurs for the highly sought-after neutrino-less double-beta decay24: if the neutrino is a Majorana par- which is equivalent to: ( ) ticle it can annihilate itself and the rate depends on a 1 p p factor m /E . Another important example is the sup- 1 + e · ν for V,A non-spin-flip. (23) ν ν 2 E E pression of the decay of the negative pion into an elec- e ν For spin-flip transitions the same arguments lead to: tron and its antineutrino compared to the decay into a ( ) muon and its antineutrino. The pion has zero spin so the 1 pe pν spin of the lepton and antilepton have to be in opposite 1 − · for V,A spin-flip. (24) 2 Ee Eν directions and momentum conservation requires them to come in opposite directions as well, so both leptons are For Fermi transitions, where there is no angular momen- forced into the same helicity state. The left-handedness tum difference between the parent and daughter nucleus, of the weak interaction only allows positive helicity mass- the expected correlation is just the same as for non-spin- less antineutrinos, but hinders the positive helicity mas- flip transitions. However, for GT decays we have to con- sive negative-charged leptons, so the decay is suppressed sider the non-spin-flip as well as spin-flip transitions. To by the m/E factor, where m is the mass of the negative- fix ideas we consider the case of the decay of 6He, whose charged lepton. Because the mass of the pion is much decay scheme is shown in Fig. 3. The daughter nucleus larger than the mass of the electron this factor is very small for the electron while for the muon it is of order J=0 unity and the decay proceeds mainly by π− → µ−ν¯ . µ 6He The decay of the positive pion is similar. J=1 For GT transitions the arguments above lead to a Fierz interference term: 6Li ′ ′ me (CA CT + CACT ) . (21) FIG. 3. Decay scheme for 6He. Ee Because of the 1/E dependence this interference effects e can have J = 1 and M = −1, 0, +1 so there are two can be identified measuring the electron energy distri- spin-flip transitions and one non-spin-flip one. It can be butions. In the 1950’s measurements had already deter- checked that the Clebsh-Gordan coefficients correspond- mined that these contributions had to be small so that ing to these three transitions with the operator repre- Fermi transitions were known to be driven by either S or sented by the Pauli matrices (an operator of rank 1 and V currents, but not by both, while GT transitions had projections equal to the M values) yields the same prob- to be driven by either A or T currents, but not by both. ability for all three transitions. Thus the correlation is ( ) 3 − 1 pe · pν V. e − ν¯ CORRELATION 1 . (25) 2 3 Ee Eν 6 Consider the directional correlation between the elec- Although we focussed on the decay of He it can be shown tron and the antineutrino in beta decays from non- that the same result holds for any pure GT transition. oriented nuclei. Again, we classify transitions into non- Students that can maneuver comfortably with Clebsh- spin-flip transitions (no nucleon spin flipping, ∆M = 0) Gordan-ery will figure out the way for the general proof. and spin-flip transitions (some nucleon will have its spin For the other students who are beginners we recommend flipped, ∆M = 1). trying to work out at least one other case. − We start considering non-spin-flip transitions. To fix To summarize, the general formula for the e ν¯ corre- ideas we can take one of the particles, say, the antineu- lation (up to a normalization) is trino, to be emitted along the +z direction. The equiva- p p 1 + a e · ν (26) lent of Fig. 2 will now have the two spins opposite to each Ee Eν 6 where a = +1 for Fermi transitions and a = −1/3 for ual’ to check that the simple arguments presented here pure GT transitions. If, on the other hand, we consider lead to the correct expressions for the correlation coeffi- scalar or tensor currents one of the leptons has chirality cients. For brevity we work within the standard model: ′ ′ opposite to those of the vector currents and the signs of no scalar or tensor currents (CS = CS = CT = CT = 0) ′ ′ the correlations are exactly opposite to those above. and no right-handed currents (CV = CV and CA = CA). A series of experiments were carried out in the 1950’s Because both the neutron and proton are spin-1/2 par- to determine the e − ν¯ correlation in nuclear beta de- ticles both kinds of transitions, Fermi and GT, are al- cays. Because neutrinos are hard to detect experiments lowed. Moreover, the GT part of the transition has a determined the recoiling-nucleus energy distribution in non-spin-flip part that interferes with the Fermi part and coincidence with electrons at different angles. But the affects the lepton correlations. We start by considering effect is bigger for lighter nuclei and larger energy re- the matrix elements for the nuclear part of the transition. lease. Thus, a particularly important case was the decay Since the energy scale of the decay is small compared to of 6He, which is one of the lightest nuclei that decay the nucleon mass, we can use non-relativistic forms of the by a pure GT transition and has a relatively large en- operators in Eq. 20 for the nuclear part. The operator for ergy release (about 3.5 MeV). The experiment is hard the Fermi part is just unity with a coupling constant CV , because one needs to detect the rather low energy recoil- while the operator for the GT part is the Pauli matrices, 6 ing Li ions and the electrons versus the angle at which σ, with a coupling constant CA. Thus, the total decay they were emitted with respect to each other. One mea- rate is simply proportional to (we will show in Appendix surement published25 in 1953 and confirmed26 in 1955 2 that the interferences cancel when integrating over the seemed to have clearly pinned down the interaction to leptons directions): 19 ( ) be of tensor type. In addition, measurements on Ne C2 1 + 3|λ|2 (27) confirmed27 that the interaction was of the S and T type. V But a few years later evidence from other experiments where we use the property of the Pauli matrices σ2 = mounted against this possibility. Feynman gives an in- 3I (with I the 2 × 2 identity matrix) and we define the teresting and amusing account of these times in an article quantity λ = CA/CV for the ratio of the axial to vector called ‘The 7 percent solution’28. Eventually there was coupling constants. a determination of the helicity of the neutrinos using a To calculate the beta asymmetry we consider a neutron very ingenious idea29 and two different groups30,31 per- with spin aligned in the +z direction. The GT operator formed different versions of the 6He experiment with very can be considered separately for the non-spin-flip part, convincing evidence that the interaction was of the vec- σz, and for the spin-flip part σ− with ratio tor and axial-vector type. In retrospect it is not hard to + √ ⟨p ↓ |σ−T |n ↑⟩ see the difficulties in the first experiment and guess that = 2, (28) ⟨ ↑ | +| ↑⟩ more should have been demanded. But for a few years p σzT n the world of physics believed weak interactions were me- where T + is the isospin rasing operator that turns a neu- diated by scalar and tensor currents. Nowadays it is clear tron into a proton. If we ignore the interference between that the weak interactions are carried principally by vec- the GT and Fermi operators, the decay rate is propor- tor and axial-vector currents. tional to a non-spin-flip part with probability propor- Nevertheless, tensor currents should not be considered tional to 1 + |λ|2 and a spin-flip part proportional to as completely strange objects: the nucleon-nucleon ef- 2|λ|2. Following the arguments of Sect. II and integrat- fective potential at low energies, for example, contains ing over all antineutrino directions, the differential decay tensor currents. With respect to the weak interaction, rate is proportional to: ( ) models that go beyond the standard model in search for ( ) p explanations of some facts that seem to beg for a deeper 1 + |λ|2 + 2|λ|2 1 − P · e . (29) theory, predict scalar or tensor currents. Experimental Ee searches for these new forms of physics with improved However, the non-spin-flip part of the axial current can sensitivity thanks to much ingenuity and developments interfere with the vector current because they both con- of new tools32–36 continues via low-energy precision ex- nect identical initial and final states. One may naively periments and at the LHC (see Bhattacharya et al.37 for think that this interference is independent on the neu- a comparison of the sensitivity of both). tron polarization or on the direction of the spin of the leptons, just as happened for the non-flip contributions considered above. However, the nuclear matrix element VI. NEUTRON BETA DECAY for the axial current depends on the spin of the neutron while the matrix element for the vector does not depend − on it. As a consequence this interference depends linearly The decay of the neutron (n → p + e +ν ¯) presents on the polarization and the z-projection of the spin of the some additional calculation challenges, but there is also electron, leading to an additional term much to be learned for students interested in deeper un- ( ) ( ) derstanding. The complete expressions for the correla- 2 pe 2 pe 38 1 + |λ| − 2λP · + 2|λ| 1 − P · . (30) tions are available and can be used as a ‘solutions man- Ee Ee 7

z z More details, including the reason for the sign, are shown Both terms with pe and pν flip signs under the inversion z z in Appendix 2. of the polarization direction, but the pepν term does not. In summary, the decay rate from polarized neutron If we add these two rates, the term with C4 does not beta decay after integrating over neutrino momentum is vanish. However, for spin-1/2 particles like neutrons, the ( ) sum rate should be the same as that for non-polarized ( ) p 1 + 3|λ|2 1 + AP · e , (31) neutron decay, so it should not depend on any specific Ee direction. Therefore C4 must be zero for neutron decay. 60 where the beta asymmetry correlation coefficient A is However, for other cases, such as the decay of Co this term has to be included. λ + |λ|2 A = −2 . (32) 1 + 3|λ|2 VII. CONCLUSIONS Similarly, if the integration is performed over all electron directions, the differential decay rate is proportional We have presented simple derivations of the nuclear to: ( ) beta decay correlations for some particular cases. In the ( ) pν process we described some of the history of the discov- 1 + 3|λ|2 1 + BP · , (33) Eν eries that led to understanding the weak interaction. In our experience students develop intuition about the prop- where the antineutrino asymmetry correlation coefficient erties of chirality and helicity working through these ar- B is: guments. Alternative interesting questions to consider λ − |λ|2 are, for example, how do the answers given here vary if B = −2 . (34) 1 + 3|λ|2 instead of electron emission one considers positron emission; or, if instead of the V-A currents assumed for some Note that the sign of the non-interfering part of A and of the calculations, such as those for neutron beta-decay, B are opposite because they are in the spin-flip transi- one considers S and T currents. The arguments used can tion, while the interfering part of A and B are identical also be applied to predict the angular distribution ex- because they are in the non-spin-flip transition. pected for Mott scattering, neutrino-nucleus scattering The expression for the e−ν¯ correlation can be obtained and to neutrino-electron conversion in a nuclear target. by adding the non-spin-flip and spin-flip expressions of Eqs. 24 and ?? multiplied by the respective probabilities: ( ) ( ) 1 + |λ|2 p p 2|λ|2 p p VIII. ACKNOWLEDGMENTS 1 + e · ν + 1 − e · ν . 1 + 3|λ|2 E E 1 + 3|λ|2 E E e ν e ν We acknowledge the support of the U.S. Department (35) of Energy under Grant No. DE-FG02-97ER41020. We In summary, the decay rate for polarized neutrons is pro- thank Steve Ellis, Jerry Miller, Ann Nelson, Derek Storm portional to for helpful comments. ( ) p p p p 1 + a e · ν + P · A e + B ν (36) Ee Eν Ee Eν APPENDIX 1 with A and B given by Eqs. 32 and 34 and In order to reconcile quantum mechanics with special 1 − |λ|2 relativity, Dirac wrote down a hamiltonian that is linear a = . (37) 1 + 3|λ|2 in ∇ for free fermions: Because the ratio of the axial to vector coupling constants H = α · ∇ + βm (39) has the value λ ∼ −1.27 the beta asymmetry and e − ν¯ correlation coefficients end up being negative and small, where αi = γ0γi (i = 1, 2, 3) and β = γ0 are 4 × 4 while the antineutrino asymmetry is large and positive. matrices, and the γ matrices are The careful reader may note that while the e−ν¯ corre- ( ) ( ) ( ) lation coefficient was derived assuming non-oriented neu- 0 0 I i 0 σi 5 I 0 γ = , γ = − , γ = − . trons we ended up quoting it for the decay from polarized I 0 σi 0 0 I neutrons. Arguably, if both the electron and antineutrino (40) momenta are measured from beta decay of nuclei oriented along the +z direction, the differential decay rate should Here I stands for the 2 × 2 identity matrix and σi for pz pz include a term proportional to e ν : the Pauli matrices. The matrices are such that squaring Ee Eν Eq. 39 yields a relation between momentum and energy z z z z 2 2 2 pe pν pe pν pe pν consistent with relativity, E = p + m . We choose the 1 + C1 + C2 + C3 · + C4 (38) Ee Eν Ee Eν Ee Eν Weyl representation to make the discussion on helicity 8 and chirality easier. This hamiltonian leads to the Dirac vector current, which yields Equation: 4C2 λ (i)2ψ¯Lγ3γ5ψLψ¯Lγ0ψL × (iγµ∂ − m)ψ = 0 (41) V e ν ν e µ + − ⟨p ↑ |σzT |n ↑⟩⟨n ↑ |T |p ↑⟩ + h.c, (44) which describes the motion of a free fermion with a − + − 4 component Dirac spinor ψ. where the nuclear current ⟨p ↑ |σzT |n ↑⟩⟨n ↑ |T |p ↑⟩ The Dirac equation has two particle solutions is equal to 1. Using the explicit expressions in Appendix −iEt+ip·x u±(E, p)e and two antiparticle solutions 1 one gets iEt−ip·x v±(E, p)e . Here ( ) ( √ ) ( √ ) ± ± † σ 0 † E ∓ pχ − E ∓ pχ (i)2ψ¯Lγ3γ5ψLψ¯Lγ0ψL = ψL z ψLψ¯L ψL. u± = √ , v∓ = √ , (42) e ν ν e e ν ν e E ± pχ± E ± pχ± 0 σz (45) where ( ) ( ) 1 − 0 This indicates that this interference term has a linear χ+ = , χ = (43) 0 1 dependence on the z component of the electron and antineutrino spins. One can obtain the dependence on lep- correspond to the two directions of spin. Here we sup- ton spins by working with the eigenstates of the σz oper- pose that z is the quantization axis and momentum is ator. Because every spinor in Eq. 45 is left-handed so we also in the +z direction. For antiparticle states, one can just work with the upper 2 components of the Dirac iEt−ip·x − can interpret v−(E, p)e as the absence of a par- spinors, where χ+( ) indicates spin-up(down) electrons, ticle with energy −E, momentum −p, spin along the or spin-down(up) anti-neutrinos. The definition of χ+(−) +z direction and thus negative helicity, or the presence and the reason for the opposite signs for particle and an- of an anti-particle with energy E, momentum p and tiparticle is explained in Appendix 1. If the electron spin spin along the −z direction and thus also negative he- and antineutrino spin are identical, Eq. 45 is zero because + − licity. The projectors PL/R in Eq. 16 project out the up- σz is diagonal and χ is normal to χ . This is consistent per/lower two components of the spinor. Therefore, the with the fact that the electron and antineutrino are emit- upper/lower components have left-handed/right-handed ted with total angular momentum zero along the z axis chiralities and the norms in Eq. 4 can be calculated. For in non-spin-flip transitions and they should be opposite massive particles the chirality eigenstates are not solu- to each other. For spin-up electron and spin-down anti- tions to the Dirac equation, so free massive particles neutrino, Eq. 45 is positive (and negative for the opposite cannot have a well defined chirality. One should note case). Therefore, the righthand side of Eq. 45 should be 5 e − ν e ν that the chiralities are the eigenvalues of γ : +1 for left- proportional to Sz Sz , where Sz and Sz are the values handed chirality and −1 for right-handed chirality ac- of the z component of electron and antineutrino spins. cording to the convention used in this paper. For mass- We have seen before that the spin of the electron has less particles, left-handed chirality corresponds to nega- a negative correlation with its direction, while the spin tive helicity, while for massless anti-particles, left-handed of antineutrino has a positive correlation with its direc- chirality corresponds to positive helicity. tion. Thus the interference term is −2λP · pe −2λP · pν , Ee Eν where the factor 2 comes from adding the hermitian conjugate term and the other factors are the same for the APPENDIX 2 non-interfering terms and have been taken out. After integrating over all electron directions and all anti-neutrino The interference term in neutron decay is the prod- directions, the interference term disappears so it does not uct of the non-spin-flip part of the axial current and the show up in the total decay rate.

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