Weak Interaction
Outline
Outline
Introduction to weak interactions
Charged current (CC) interactions
Neutral current (NC)
Weak vector bosons W± and Z0
Weak charged interactions
Beta decay
Flavour changing charged current
W± boson propagator
Fermi coupling constant
Parity violation
Muon decay
Decay rate /lifetime
Lepton Universality
W± boson couplings for leptons
Tau decays
Weak quark decays
W± boson couplings for quarks
Cabibbo angle, CKM mechanism
Spectator model
Nuclear and Particle Physics
Franz Muheim
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Introduction
Weak Interactions
Account for large variety of physical processes
Muon and Tau decays, Neutrino interactions
Decays of lightest mesons and baryons
Z0 and W± boson production at √s ~ O(100 GeV)
Natural radioactivity, fission, fusion (sun)
Major Characteristics
Long lifetimes
Small cross sections (not always)
“Quantum Flavour Dynamics”
Charged Current (CC)
Neutral Current (NC)
mediated by exchange of
W± boson
Z0 boson
Intermediate vector bosons MW = 80.4 GeV
W± and Z0 have mass
MZ = 91.2 GeV
Self Interactions
of W± and Z0
also W± and γ
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Beta Decay
Weak Nuclear Decays
See also Nuclear Physics
Recall β+ = e+
β- = eContinuous energy spectrum of e- or e+
Î 3-body decay, Pauli postulates neutrino, 1930
Interpretation
Fermi, 1932
Bound n or p decay
n → pe −ν e
⎛−t⎞
⎟⎟
N(t) = N (0) exp⎜⎜
τ
⎝ n⎠
τ n = 885.7 ± 0.8 s
τ 1 = τ n ln 2 = 613.9 ± 0.6 s
2
p → ne +ν e
(bound p)
τ p > 10 32 y
(p stable)
Modern quark level picture
Weak charged current mediated
by exchange of virtual W± boson
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Weak Charged Interactions
Fundamental Vertex
Weak Charged Current
QED
Flavour Changing
Weak charged current changes
lepton and quark flavours
W± boson transforms νe into e-, u into d-quark
Only weak interaction is flavour changing
Weak Charged Current
W± boson couples to weak charge gW of
neutrino/electron or up/down quark pair
Coupling strength ∝ gW
Î For each vertex add factor gW to
matrix element
Probability for weak vertex
cross section or decay rate ∝ gW2
W± Boson Propagator
W boson has mass MW
Recall photons and gluons are massless
Î Add propagator term 1/(q2 – MW2)
to matrix element/amplitude
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Fermi Theory of
Weak Interaction
W± Boson Propagator
for small
q2 << MW2 (q2 → 0)
1/(q2 – MW2) → 1/MW2
Propagator is constant
Fermi Coupling Constant GF
Weak coupling constant
includes propagator
gW2
GF ∝ 2
MW
exact
GF
gW2
=
2
2 8 MW
Fermi
different from QED & QCD Perkins, p 151&210
gW dimensionless → units of GF are GeV-2
GF/(ħc)3 = 1.16637(1) ·10-5 GeV-2
Range of Weak Interaction
Massive exchange boson ↔ short range
Rweak =
hc
≈ 2 ⋅ 10 −18 m = 0.002 fm << 0.1 fm
2
MW c
Analogous to Yukawa interaction
Strength of Weak Interaction
GF & MW
⇒
1
1
gW2
=
> α em =
gW = 0.66 ⇒ αW =
4π
29
137
Not intrinsically weak
at low q2 weak due to large MW
αS ≈ 0.2 > αW ≈ 0.03 > αem ≈ 0.008 all running
Will these meet at high energy? – unification?
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Parity Violation
Parity P
Intrinsic quantum number of particles
Conserved in strong and electromagnetic
interactions
Kaon Decays
Observe K+ → π+ π0 and K+ → π+ π- π+ decays
Parity of K+ and π+
P (K + ) = P (π + ) = −1
Parity of final states P (π +π 0 ) = Pπ Pπ (− 1)L= 0 = 1
L= 0
P (π +π −π + ) = Pπ Pπ Pπ (− 1) = −1
Î Puzzle
Parity Violation in Weak Interactions
1956 proposed by Lee and Yang
1957 experimentally confirmed by Wu et al
Measure e- from 60Co beta decay 60Co→60Ni*e−νe
Polarised 60Co spin J aligned to magnetic field B
at 0.01 K B,J
B,J
If parity conserved expect equal e- rates
parallel and antiparallel to B-field and spin
Observe asymmetry Î Parity violated
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Muon Decay
Muon
Fundamental lepton with mass mµ = 105.7 MeV
does not interact strongly
Electromagnetic decay µ+ → e+γ is forbidden
Decays weakly – flavour changing
µ − → e− ν e ν µ
µ + → e+ ν e ν µ
Amplitude and Decay Rate
Matrix element / amplitude ∝ gWgW 1/(q2 – MW2)
q2 << MW2 → decay rate Γµ ∝ GF2
dimensions → Γµ ∝ GF2 mµ5
Total decay rate 1
GF2 m µ5
= Γµ =
(h = c = 1)
or lifetime τµ
τµ
192π 3
Measurements
Lifetime τµ = 2.19703(4) 10-6 s
Mass
mµ = 105.658369(9) MeV
Î
GF = 1.16637(1) ·10-5 GeV-2
weak coupling constant GF is measured
in muon decay
see 4th year projects for measuring τµ and mµ
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Lepton Universality in
Weak Interaction
Tau Decays
mτ = 1.777 GeV > mµ, mπ, mρ, …
Several weak decay modes possible
Branching
Fractions
Tau Decay Rates
BF (τ − → X ) = Γ(τ − → X ) / Γ(τ − → all )
τ − → e −ν eν τ
BF = 17.8 ± 0.1%
τ − → µ −ν µν τ
BF = 17.4 ± 0.1%
τ − → hadronsν τ
BF = 64.7 ± 0.2%
Investigate decay τ − → e −ν eν τ
1
ττ
= Γτ →all
Γτ →all
1
1 GF2 mτ5
Γτ → eν eν υ =
Γτ →eν eν υ =
=
BFτ →eν ν
0.178 192π 3
Γτ → eν eν υ
e υ
compare with muon decay
Expect lifetime τ τ = τ µ
Measure
BFτ → eν eν τ
m µ5
5
mτ
= (2.91 ± 0.01) × 10 −13 s
τ τ = (2.906 ± 0.011) × 10−13 s
Î Weak coupling of τ and µ identical
Lepton Universality in Standard Model
W± boson couples identically to all leptons
e − → W −ν e or ν e → e −W +
µ − → W −ν µ or ν µ → µ −W + τ − → W −ν τ or ν τ → τ −W +
Charged weak current
Couples within lepton doublets
Nuclear and Particle Physics
Franz Muheim
⎛ν e ⎞
⎜⎜ − ⎟⎟
⎝e ⎠
⎛ν µ ⎞
⎜⎜ − ⎟⎟
⎝µ ⎠
⎛ν τ ⎞
⎜⎜ − ⎟⎟
⎝τ ⎠
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Weak Interaction
of Quarks
Weak charged quark currents
Are weak charged quark currents also universal?
e.g. is M(d → uW-) = M(µ-→ νµW-) ?
Nature is not quite that simple
Comparison
Measure GF in muon and nuclear beta decay
Obtain ratio GF(beta)/ GF(muon) = 0.974(3)
Î Charged weak current almost equal
for leptons and up/down quarks
Leptonic Pion and Kaon Decays
Observe π+ → µ+ νµ and K+ → µ+ νµ
Evidence for flavour changing coupling u → sW+
between quark generations
Rate Γ( K+ → µ+ νµ ) ≈ 5% expected Γ( π+ → µ+ νµ )
Matrix element
M(u → sW+) ~ sinθC
M(u → dW+) ~ cosθC
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Cabibbo Mixing Angle
Quark Flavour Mixing
Two generations & four quark flavours (u, d, c, s)
weak eigenstates not equal to mass eigenstates
are linear combination of mass eigenstates
Cabibbo Hypothesis
W boson couples to
weak doublet (u,d’)
Cabbibo angle θC rotates weak eigenstates
Coupling strength to W boson within doublet
⎛ d ' ⎞ ⎛ cos θ C
⎜⎜ ⎟⎟ = ⎜⎜
⎝ s' ⎠ ⎝ − sin θ C
⎛ν e ⎞
⎜ ⎟ → gW
⎜ e− ⎟
⎝ ⎠
sin θ C ⎞
⎟
cos θ C ⎟⎠
⎛d ⎞
⎜⎜ ⎟⎟
⎝s ⎠
⎛u⎞
⎜⎜ ⎟⎟ → gW cos θ C = gW Vud
⎝d ⎠
⎛ u⎞
⎜⎜ ⎟⎟ → gW sin θ C = gW Vus
⎝s⎠
Experimental Results
Muon decay
Beta decay
Pion decay
Kaon decay
Cabibbo angle
(
)
Γ (d → uν ν ) ∝ G cos θ
Γ (π → µ ν ) ∝ G cos θ
Γ (K → µ ν ) ∝ G sin θ
Γ µ − → e −ν eν µ = GF2 m µ5 / 192π 3 = Γµ
e
+
+
+
+
µ
2
F
2
µ
2
F
2
µ
θ C = 12.96(9)0
2
F
= GF2 Vud
2
C
= GF2 Vud
2
C
= GF2 Vus
2
C
2
sin θ C = 0.2243(16)
Cabibbo-Kobayashi-Maskawa
Mechanism
Cabibbo-Kobayashi-Maskawa
Mechanism
Weak Charged Interaction
W couples quarks of different generations
For 3 generations ⎛ d ' ⎞ ⎛Vud Vus Vub ⎞ ⎛ d ⎞
⎟ ⎜ ⎟
⎜ ⎟ ⎜
s
V
V
V
'
=
⎜
⎜
⎟
⎜ s⎟
cd
cs
cb ⎟
expand to
Nuclear and Particle Physics
⎜ b' ⎟ ⎜ V
Vts
td
⎝ ⎠Franz
⎝ Muheim
Vtb ⎟⎠ ⎜⎝ b ⎟⎠
10
Weak Hadron Decays
Lightest Mesons and Baryons
only flavour changing weak decay possible for
Pions (π+ = u dbar), kaons (K+ = u sbar),
charmed (D) and B mesons, neutron, Λ, Σ, Ξ, Ω
Exceptions: zero net quantum numbers allow
electromagnetic decay, e.g. π0 →γγ
Spectator
Spectator Model
Model
Weak decay of heavy B or D meson (b or c quark)
Decay rate of B or D meson equal to decay rate
of heavy b or c quark, light quark is only spectator
Predictions of Spectator Model
2
Semileptonic decay rate
GF2 mb5
−
Γ (B → X c ,u e ν e ) =
V
cb ( ub )
192π 3
B → Xc,u e- νbar
Equal total decay rates/lifetimes for
D+, D0, Ds+ and B+, B0, Bs+
D meson lifetimes τ D ≈ 2.5 τ D ≈ 2.5 τ D
B mesons lifetimes are equal within 10%
Spectator model works well for B mesons
+
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s
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