Lecture 11 - Weak Interactions

Weak Charged Currents and β Decay • Weak Neutral Currents • - • and Decays • Lepton Universality • Lepton Flavour Conservation •

1 Weak Charged Currents The exchange of a heavy W + or W − describes weak interactions with charged currents The currents are either: A charged lepton changing into a neutrino (or vice-versa) • These must be of the same flavour! An up-type changing into a down-type quark •

Weak interaction strength is related to the Fermi constant GF : 2 GF g −5 −2 = 2 GF = 1.16 10 GeV √2 8MW × where g is a dimensionless constant

2 Description of β Decay Quark level: + − u de νe d ue ν¯e → →

Hadron level (mp = 938MeV, mn = 940MeV): − n pe ν¯e is allowed → Free is unstable and decays with τn = 886s + p ne νe is forbidden → Free is stable; decay forbidden by mp < mn Nuclear level:

14 14 ∗ + + O N + e + νe β decay → 60 60 ∗ − − Co Ni + e +ν ¯e β decay → Which type is allowed depends on available between nuclei

3 for β Decay

ν¯e

W − d e−

¡ u

2 2 for W boson is igµν /(M q ) • − W − Dimensionless coupling constant is g for • modified by CKM factors for (Vud) The operator is γµ 1 (1 γ5) instead of γµ • 2 − Vector minus Axial-vector (V-A)

4 Amplitude for β Decay

The matrix element “factorizes” into lepton and quark currents:

g µ 1 5 1 g µ 1 5 = u¯dγ (1 γ )uu u¯ν γ (1 γ )ue M √2 2 −  M 2 q2 √2 e 2 −  W − Neglecting lepton the lepton current gives:

µ 1 5 2 u¯ν γ (1 γ )ue = 8EeEν (1 + cos θ) | e 2 − | where θ is the opening angle between e and ν µ 1 5 2 The quark current squared u¯dγ (1 γ )uu is parametrised by a | 2 − | “hadronic form factor”, e.g. F (n p), which is a function of q2 → In nuclear β decay use “matrix elements”, e.g. 60Co 60 Ni∗ → 2 2 2 At low q can replace g /MW with Fermi constant GF

5 β decay (Kurie plot)

2 dΓ GF 2 2 Spectrum: = 3 E (E0 Ee) dEe π e 2 5 − 1 GF E0 Total rate: Γ = τ = 30π3 neglecting lepton and nuclear matrix element

Shape near endpoint E0 determines m(νe) < 2eV

6 Weak Neutral Currents Described by the exchange of a heavy Z0 boson Z0 couples to all types of fermion but there are no flavour-changing neutral currents! amplitude for νe νe scattering: → G µ 1 5 µ e e 5 = u¯ν γ (1 γ )uν u¯eγ (cV cAγ )ue M √2  2 −  −  cV and cA are vector and axial-vector couplings which depend on fermion type:

Lepton 2cV 2cA Quark 2cV 2cA

νe, νµ, ντ 1 1 u,c,t 0.38 1 e, µ, τ -0.06 -1 d,s,b -0.68 -1

7 Neutrino Scattering Neutrino- ν e− ν e−: − − e e •e e → − e νe

Z W

¡ νe νe ¡ νe e− Antineutrino-Electron scatteringν ¯ e− ν¯ e−: − − e e • →− − e e e e

Z W

¡

¡ ν¯e ν¯e ν¯e ν¯e

8 Scattering − − Muon neutrino-Electron scattering νµe νµe : • − − → e e e− e−

Z Z

¡ νµ νµ ¡ ν¯µ ν¯µ

− − Inverse muon decay νµe νeµ : − • e → νe

W

¡ νµ µ−

9 νµ scattering cross-sections − − Charged current exchange diagram νµe νeµ : • → dσ G2 s G2 s 2 = 16G2 s2 = F σ = F |M| F dΩ 4π2 π Neutral current scattering of muon : • 2 dσ GF s 2 2 2 = (cV + cA) +(cV cA) (1 y) dy 4π − −   2 − − GF s 2 2 σ(νµe νµe )= (c + cV cA + c ) → 3π V A Neutral current scattering of muon antineutrinos: • 2 − − GF s 2 2 σ(¯νµe ν¯µe )= (c cV cA + c ) → 3π V − A

10 νe scattering cross-sections

νe scattering is superposition of charged and neutral currents Scattering of electron neutrinos: • 2 dσ − − GF s 2 2 2 (νee νee )= g + g (1 y) dy → π L −   Scattering of electron antineutrinos: • 2 dσ − − GF s 2 2 2 (¯νee ν¯ee )= g + g (1 y) dy → π R L −   where gL =1+(cV + cA)/2 and gR =(cV cA)/2 − Total cross-sections: • 2 2 − G s − G s σ(ν e )= F (g2 + g2 /3) σ(¯ν e )= F (g2 + g2 /3) e π L R e π R L − −41 2 In the lab frame σ(νee ) 10 Eν cm where Eν is in GeV ≈

11 Muon Decay ν¯e

W − g µ− e g

¡ νµ Matrix element:

GF µ 1 5 µ 1 5 = u¯ν γ (1 γ )uµ u¯eγ (1 γ )uν M √2  µ 2 −  2 − e  Michel spectrum: 2 dΓ GF 2 2 4Ee = 3 mµEe 3 dEe 12π  − mµ  Total decay rate: 2 5 GF mµ 1 Γµ = 3 τµ = = 2.197µs 192π Γµ

12 Tau Decays The τ lepton is heavy enough to decay to many final states

(mτ = 1.777 GeV)

− − − − τ e ντ ν¯e τ µ ντ ν¯µ → → − − τ duν¯ τ τ suν¯ τ → → Naive branching fractions (with factor 3 for quark color): × (τ e)= (τ µ) = 20% (τ ) = 60% B → B → B → The decay to an electron can be related to muon decay: 5 Γ → m τ e = τ Γµ mµ 

This agrees with the measured τ lifetime ττ = 291fs

13 Lepton Universality

Couplings of W and Z are the same for all families of leptons This can be tested in τ decays: − − (τ e ν¯eντ ) = (17.84 0.06)% B → ± − − (τ µ ν¯µντ ) = (17.36 0.06)% B → ± small difference is due to the muon mass

Other tests include: Michel spectrum shape in µ and τ decays W ℓν and Z ℓ+ℓ− at high energy → → Heavy quark decays b cℓν and c sℓν → →

14 50 years of µ e flavour violation searches →

15 25 years of τ flavour violation searches

16