Beta decay: helicity

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1 What are the consequences of parity viola:on in beta decay? h =! σ p! p helicity The eigenvalue of h is v/c. For a massless par:cle, the eigenvalues of h can be only +1 or -1. In general, the par:cle with h>0 is called right-handed h<0 is called lem-handed Experimentally, h(ν e ) 1, h(ν e ) +1 Beta decay: helicity (helicity flips under parity) hup://journals.aps.org/pr/abstract/ /physrev left-handed or right-handed? hup://hyperphysics.phy-astr.gsu.edu The global characteriza:on in terms of handedness is not meaningful for other par:cles, like electrons. An electron could have spin to the right and be traveling right and therefore be classified as righthanded. But from the reference frame of someone traveling faster than the electron, its velocity would be to the lem, while its spin would be unchanged. This would mean that the electron is a lemhanded par:cle with respect to that reference frame.

2 Beta decay: axial vector coupling constant The operators that are scalars, pseudoscalars and tensors produce leptons of both helici:es under a parity transforma:on. Only vector operators V and axial vector operators A can accommodate the observed result. Furthermore, since V and A are of different parity, they must appear in a linear combina:on. This leads to the V-A theory of beta decay. In principle, both V and A parts should be characterized by different coupling constants, G V and G A, respec:vely. In this theory, the weak interac:on acts only on lem-handed par:cles (and righthanded an:par:cles). Since the mirror reflec:on of a lem-handed par:cle is right-handed, this explains the maximal viola:on of parity. The vector current is known to be a conserved quan:ty (CVC hypothesis) How to relate G V and G A? pion decay constant pion-nucleon coupling constant g A G A G V = f π g πn M N c 2 Goldberger-Treiman relation Experimentally, g A =1.267(4). This value is very close, up to 3%, to the Goldberger- Treiman es:mate. This rela:on can be obtained by assuming the so-called par:ally conserved axial-vector current (PCAC) hypothesis. Now we are ready to es:mate the nuclear operator!

3 Beta decay: nuclear matrix elements V int gδ ( r! n r! p )δ ( r! n r! ) δ! e ( r n r! )Ô(n p) υ zero-range The nuclear operator transforming a neutron into a proton must be one body in nature. Hence it must involve the isospin raising or lowering operators. In the non-rela:vis:c limit, the vector part may be represented by the unity operator :mes τ ± and the axial-vector part by a product of τ ± and σ. (A proper deriva:on requires manipula:on with Dirac 4-component func:ons and γ matrices!) A j=1 [ ] V int G V τ ± ( j)+ g A! σ ( j)! τ ( j) Fermi decay, carries zero angular momentum Gamow-Teller decay, carries one unit of angular momentum G V determined from superallowed Fermi beta decays!

4 From the expression for ft, it is possible to determine the strength g of the beta-decay process, if one knows how to determine the reduced matrix element. For superallowed Fermi transi:ons, the matrix element is 2 so the ft values should be iden:cal. Superallowed beta decay between T=1 analog states corrected for nuclear structure effects (isospin mixing) and radia:ve correc:ons g = MeV fm 3 or, introducing the dimensionless constant G: G = g m 2 e c! =

5 Superallowed Fermi β-decay studies NUMBER OF PROTONS Impressive experimental effort worldwide Superallowed! emitters 10 0, NUMBER OF NEUTRONS Hardy and Towner survey (Feb. 2015) BR 0,1 + t 1/2 Q EC Kobayashi and Maskawa (2008): for "the discovery of the origin of broken symmetry, which predicts the existence of at least three families of quarks in nature." Vud 2 + Vus 2 + Vub 2 $"!$! $"!!!!"##!!"#$%&'()*'(+$,-&.'('-& nuclear meson decay -. -/ -0 $#%& $#%% $##' $##( '!!! '!!& '!!% )*+,

6 Beta decay: allowed transitions Fermi transitions J f M f T f T 0 f T J i M i T i T 0i = T i (T i +1) T 0i (T 0i 1)δ Ji J f δ Mi M f δ Ti T f δ T0i 1T 0 f In reality, isospin is violated by the electromagne:c force, but the viola:on is weak. J f = J i ( ΔJ = 0) T f = T i 0 T 0 f = T 0i 1 ΔT 0 =1 Δπ = 0 ( ΔT = 0, but T i = 0 T f = 0 forbidden) ( ) no parity change Gamow-Teller transitions ft = The matrix element strongly depends on the structure of the wave func:on! const F 2 + g A 2 GT 2 ΔJ = 0,1 ΔT = 0,1 T 0 f = T 0i 1 ( ΔT 0 =1) Δπ = 0 but J i = 0 J f = 0 forbidden but T i = 0 T f = 0 forbidden no parity change The absolute values of GT matrix elements are generally smaller than those for Fermi transi:ons.

7 Superallowed Gamow-Teller decay of the doubly magic nucleus 100 Sn Hinke et al., Nature 486, 341 (2012) Number distribution of log(ft) values for allowed β-transitions (obeying the selection rules).

8 Forbidden transitions Forbidden transi:ons involve parity change and a spin change of more than one unit. They come from the higher-order terms in the expansion of electron and neutrino plane waves into spherical harmonics. Forbidden decays are classified into different groups by the L-value of the spherical harmonics involved. Systema:c Uncertain:es in the Analysis of the Reactor Neutrino Anomaly A. Hayes et al., PRL 112, (2014) "...the correc:ons are nuclear-operator dependent and that an undetermined combina:on of matrix elements contributes to non-unique forbidden transi:ons. Also: r-process simula:ons.

9 Beta decay: electron capture Electron capture leads to a vacancy being created in one of the strongest bound atomic states, and secondary processes are observed such as the emission of X-rays and Auger electrons. Auger electrons are electrons emiued from one of the outer electron shells and take away some of the remaining energy. Capture is most likely for a 1s-state electron. The K-electron wave func:on at the origin is maximal and is given by ψ e (0) = 1 Zm e e 2 3/2 " % $ ' π #! 2 2 & g 2 The electron capture probability is thus given by: Example W EC = E v 2 M ' fi! Zm e e 2 $ # & π 2! 4 c 3 "! 2 % hup:// Manipula:ng life:mes in storage rings: hup:// 3