People who have no weaknesses are terrible; there is no way of taking advantage of them. Anatole France (1844–1924) 27 Weak Interactions The decay of nuclei proceeds via α-, β-, γ-emission, or by fission. The first is governed by nuclear forces, the third by electromagnetic interactions. The β-emission has an entirely different origin. It is caused by weak interactions. Initially, such processes posed a serious puzzle: The observed emerging particles, electrons or positrons, did not come out with a definite energy equal to the mass difference between initial and final nuclei. Instead, they had an energy distribution with a maximal energy at the expected value. This led Nils Bohr to speculate that energy conservation could possibly be violated in quantum physics. In 1939, however, Wolfgang Pauli suggested in a historic letter to colleagues [1] meeting in T¨ubingen, that the missing energy was carried off by a neutral particle, now called neutrino. Its spin had to be 1/2, to preserve angular momentum. The simplest nuclear particle which shows β-decay is the neutron itself which decays, after about 900 seconds, as follows: n p + e− +¯ν , (27.1) → e whereν ¯e denotes a right-handed version of Pauli’s particle, which is nowadays re- ferred to as an antineutrino associated with the electron. As time went on, many more particles were discovered that arose from similar interactions. All of them are now called leptons. Recall the classification in Section 24.1. 27.1 Fermi Theory In 1934, Fermi made a first theoretical attempt to describe β-decay in lowest-order perturbation theory [2]. He suggested that the interaction is due to a local current- current interaction with a Lagrangian density of the form = C p¯(x)γµn(x)¯e(x)γ ν(x) + c.c.. (27.2) L µ In accordance with the standard short notation for the fields, we have denoted ψe(x) by e(x) and ψp(x) by p(x), etc. This interaction looks similar to the electric 1457 1458 27 Weak Interactions interaction in Eq. (12.85), except that the Coulomb potential describing the electric long-range interaction is replaced by a short-range δ-function potential. Fermi also realized that Lorentz invariance and locality allow a more general Lagrangian density of the form = CS p¯(x)n(x)¯e(x)ν(x) + CP p¯(x)iγ5n(x)¯e(x)iγ5ν(x) L µ µ + CV p¯(x)γ n(x)¯e(x)γµν(x) + CA p¯(x)γ γ5n(x)¯e(x)γµγ5ν(x) 1 λκ + 2 CT p¯(x)σ n(x)¯e(x)σλκν(x) + c.c.. (27.3) Fermi postulated the validity of time-reversal invariance, which to our present knowl- edge is fulfilled to a high degree of accuracy. A tiny violation of this is observable only in very few particularly sensitive processes. Hence the coefficients Ci in (27.3) have to be real, with only very small imaginary parts. If Γi denotes the five Dirac µ µ µν matrices 1, iγ5, γ , γ γ5, and σ /√2, one may write the interaction (27.3) short as = C p¯(x)Γ n(x)¯e(x)Γ ν(x) + c.c.. (27.4) L i i i i=S,P,V,A,TX The decay of an initial nucleus at rest Ψi into a final nucleus Ψf is governed by the matrix elements | i | i Ψ p¯(x)Γ n(x) Ψ . (27.5) h f | i | ii Due to the small energy differences involved, the final state will be almost at rest (since the decaying nucleus is usually big enough), and one may use the nonrelativis- tic limit for the matrix elements between nucleons. Taking the Dirac matrices in Dirac’s original representations, one can use the direct-product expressions (27.6), 0 3 2 5 1 γ = σ 1, ­ = iσ , γ = σ 1, (27.6) D × D × D × to rewrite the matrix elements ofp ¯(x)Γin(x) as follows: S :p ¯(x)n(x) p†(x)n(x), ≈ 5 2 P :p ¯(x)iγ n(x) p†(x) σ 1 n(x) 0, D ≈ − × ≈ 0 V :p ¯(x)γDn(x) p†(x)n(x), i ≈ 1 i p¯(x)γ n(x) p†(x) σ σ n(x) 0, D ≈ − × ≈ 0 5 1 A :p ¯(x)γDγDn(x) p†(x) σ 1 n(x) 0, i 5 ≈ × i ≈ p¯(x)γ γ n(x) p†(x)1 σ n(x), D D ≈ × 0i 2 i T :p ¯(x)σ n(x) p†(x) σ σ n(x) 0, ij ≈ − × k ≈ p¯(x)σ n(x) ǫ p†(x)1 σ n(x). (27.7) ≈ ijk × The lines containing mixtures of upper and lower components are very small for small momenta, and can be omitted. Thus we are left with only two types of nuclear matrix elements: Ψ p†(x)n(x) Ψ 1 , h f | | ii≡h i Ψ p†(x) n(x) Ψ , (27.8) h f | | ii≡h i 27.1 Fermi Theory 1459 where in front of a Dirac spinor acts only upon the spin label. The S-matrix elements are then approximately given by S =1 2πiδ(E + E + E E )T, (27.9) − f e ν¯ − i with the T -matrix [recall (9.293)] √2m T = ν 1 C u¯(p )v(p )+ C u¯(p )γ0v(p ) V h i S e ν¯ V e ν¯ n h 1 i ­ + [ C u¯(p ) v(p )+ C u¯(p ) γ v(p )] . (27.10) h i 2 T e ν¯ A e 5 ν¯ } The spin labels s3 for the electron and antineutrino have been omitted, for brevity. We have also normalized the antineutrino states in the same way as photon states like 0 (3) p′ p =2p δ (p′ p ), (27.11) h ν¯| ν¯i ν¯ ν¯ − ν¯ to allow a smooth limit as mν 0. When calculating decay rates,→ we may sum over the antineutrino polarizations which are hard to measure. The polarization sum yields √2m v¯(p )v(p )= /p m , (27.12) ν ν¯ ν¯ ν¯ − ν spinX pol and mν is set equal to zero at the end. For unpolarized nuclei, we have i j 1 2 i σ σ = , σ 1 =0. (27.13) h i 3| | h i If also the electrons are unpolarized, we obtain 1 2 2 1 2 2 1 2 T = 2 CS tr(/p ν¯ /p e)+ CV tr(γ0 /p ν¯γ0 /p e) | | 2meV |h i| | | |h i| spinX pol h 1 2 2 i i 1 2 2 i i + C tr(γ γ /p γ γ /p )+ C tr(σ /p γ σ /p ) 3 A|h i| 5 ν¯ 5 e 12 T |h i| ν¯ 5 e 2 1 2 i + 2C C 1 m tr(/p γ )+ C C m tr(σ /p γ /p ) , (27.14) S V |h i| e ν¯ 0 3 S V |h i| e ν¯ 0 e i or more explicitly 4E E p p p p 2 ν ν¯ 2 1 2 e ν¯ 2 1 2 e ν¯ T = 2 CS 1 + CV 1+ | | 2meV |h i| − EeEν¯ | | |h i| EeEν¯ spinX pol 2 2 1 pepν¯ 1 2 2 1 pepν¯ + CA 1 + CT 1+ |h i| − 3 EeEν¯ 4 |h i| 3 EeEν¯ 2 2me 2 me + 2CSCV 1 me CACT . (27.15) |h i| Ee − |h i| Ee The interference terms CSCV and CT CA show a strong threshold-dependence pro- portional to 1/Ee. Since this is not observed experimentally, one concludes that C C 0, C C 0. (27.16) S V ≈ T A ≈ 1460 27 Weak Interactions 2 One distinguishes now Gamow-Teller transitions,by for which = 0, and Fermi transitions, for which 1 2 = 0 (both without nuclear parity|h change).i| 6 For tran- |h i| 6 2 2 sitions in which initial and final nuclear spins are the same, only CS and CV con- tribute. These are distinguishable by the electron-antineutrino correlation in mo- mentum space. In the first case, electron and antineutrino come out preferably in opposite direction (pepν¯ < 0), in the second case in the same direction (pepν¯ > 0). Experiments done for the decay Ne19 F19 + e +¯ν, (27.17) → in which the nuclear spin is 1/2 before and after the decay show that the latter is true. Thus we can approximate CS 0. ≈ 2 Transitions which change the nuclear spin by one unit are sensitive to CA and 2 CT , with similar momentum correlations. The decay He6 Li6 + e +¯ν (27.18) → involves a change from nuclear spin 0 to 1. The electron and the neutrino emerge antiparallel, from which one deduces that C2 0. Thus only C and C are T ≈ V A appreciable. The antineutrinos which appear in these decays have been found to interact with matter extremely weakly. Only many years later, in 1953, was it possible to detect them directly via the inverse reaction ν¯ + p n + e+, (27.19) e → using a prolific antineutrino source of a large nuclear reactor filled with a high density of thermal neutrons. After many precise measurements of angular correlations and electron and nu- clear polarizations, the coefficients of the general interaction (27.3) have been quite well determined and led to GW λ gA = pγ¯ 1 γ5 n eγ¯ λ (1 γ5) ν + c.c., (27.20) L − √2 " − gV ! # − h i with the weak-interaction coupling constant 5 2 G = (1.14730 0.000641) 10− GeV− , (27.21) W ± × and the ratio g /g = 1.255 0.006.