9. The Weak Particle and Nuclear

Dr. Tina Potter

Dr. Tina Potter 9. The Weak Force 1 In this section...

The weak Four- Massive and the strength of the C- and violation universality interactions and the CKM

Dr. Tina Potter 9. The Weak Force 2 The Weak Interaction The weak interaction accounts for many decays in , e.g. − − − − µ → e ν¯eνµ τ → e ν¯eντ + − − π → µ ν¯µ n → pe ν¯e Characterised by long lifetimes and small interaction cross-sections

Dr. Tina Potter 9. The Weak Force 3 The Weak Interaction

Two types of weak interaction Charged current (CC): W ± (NC): Z bosons See Chapter 10 The weak force is mediated by massive vector bosons: mW = 80 GeV mZ = 91 GeV

Examples: (The list below is not complete, will see more vertices later!) Weak interactions of and : − νe − e e νe

Z W − W + Z

+ ν¯e e+ e ν¯e

Dr. Tina Potter 9. The Weak Force 4 Self-Interactions In QCD the carrycolour . In the weak interaction the W ± and Z bosons carry the W ± also carry the ⇒ boson self-interactions W − W −

Z γ

W + W +

Z W + Z W + γ W + W + W +

W − W − Z W − γ W − γ W −

(The list above is complete as far as weak self-interactions are concerned, but we have still not seen all the weak vertices. Will see the rest later) Dr. Tina Potter 9. The Weak Force 5 Fermi Theory The old (“imperfect”) idea Weak interaction taken to be a “4-fermion contact interaction” No

Coupling strength given by the Fermi constant GF −5 −2 GF = 1.166 × 10 GeV β-decay in Fermi Theory e− e−

GF GF − n ν¯e µ ν¯e

p νµ scattering in Fermi Theory − e νe

GF

νµ µ− Dr. Tina Potter 9. The Weak Force 6 Why must Fermi Theory be “Wrong”? − νe + n → p + e n p 2 2 dN 2 Ee dσ = 2π|M | = 2π4G dΩ G fi dE F (2π)3 F G 2s σ = F See Appendix F νe e− π − √ where Ee is the of the e in the centre-of- system and s is the energy in the centre-of-mass system.

In the laboratory frame: s = 2Eνmn (fixed target collision, see Chapter 3) −43 −2 ⇒ σ ∼ (Eν/ MeV) × 10 cm ν’s only interact weakly ∴ have very small interaction cross-sections. Here weak implies that you need approximately 50 -years of water to stop a 1 MeV neutrino! However, as Eν → ∞ the cross-section can become very large.√ Violates maximum value allowed by conservation of probability at s ∼ 1 TeV (“unitarity limit”). This is a big problem. ⇒ Fermi theory breaks down at high . Dr. Tina Potter 9. The Weak Force 7 Weak Charged Current: W ± Boson

Fermi theory breaks down at high energy True interaction described by exchange of charged W ± bosons 2 2 Fermi theory is the low energy (q  mW ) effective theory of the weak interaction Old Fermi Theory e− d d nu up VudgW β decay GF d u n ν¯e − ν¯ W gW e p e−

− − e νe e gW νe

− νee ± GF W scattering

ν − g µ µ νµ W µ− Dr. Tina Potter 9. The Weak Force 8 Weak Charged Current: W ± Boson Weak QED − Propagator − − Propagator e gW νe e α e 1 1 ∼ ∼ q2 − m2 q2 W ± W γ 2 2 gW e g αW = α α = νµ W µ− 4π µ− µ− 4π Charged Current Weak Interaction 2 2 1 1 At low energies, q  mW , propagator 2 2 → 2 q −mW −mW i.e. appears as the point-like interaction of Fermi theory. Massive propagator → short range 1 mW = 80.4 GeV ⇒ Range ∼ ∼ 0.002 fm mW Exchanged boson carries electromagnetic charge. Flavour changing - only the CC weak interaction changes flavour Parity violating - only the CC weak interaction can violate parity conservation Dr. Tina Potter 9. The Weak Force 9 Weak Charged Current: W ± Boson Compare Fermi theory with a massive propagator − e νµ

GF gW − µ ν¯e µ− e− − W gW

νµ ν¯e 2 2 2 gW For q  mW compare elements 2 → GF GF is small mW 2 because mW is large The precise relationship is: gW GF 2 → √ 8mW 2 The numerical factors are partly of historical origin (see Perkins 4th ed., page 210). −5 −2 mW = 80.4 GeV and GF = 1.166 × 10 GeV measured in β decay 2 g 1 2 g = 0.65 and α = W ∼ Compare to EM α = e ∼ 1 W W 4π 30 4π 137 The intrinsic strength of the weak interaction is actually greater than that of the electromagnetic interaction. At low energies (low q2), it appears weak owing to the massive propagator. Dr. Tina Potter 9. The Weak Force 10 Weak Charged Current: W ± Boson

Neutrino Scattering with a Massive W Boson Replace contact interaction by massive boson exchange diagram: 2 Fermi theory dσ 2 Ee − ν = 2πGF e gW e dΩ (2π)3

W ± Standard Model dσ E 2  m2 2 = 2πG 2 e W g F 3 2 2 νµ W µ− dΩ (2π) mW − q

2 2 2 with |q~ | = 4Ee sin θ/2, where θ is the scattering angle. G 2s Integrate to give σ = F s  m2 π W G 2m2 σ = F W s  m2 see Appendix G π W Cross-section is now well behaved at high energies.

Dr. Tina Potter 9. The Weak Force 11 and helicity Consider a free particle of constant momentum, p~ Total angular momentum, J~ = ~L + S~ is always conserved The orbital angular momentum, ~L = ~r × p~ is perpendicular to p~ The spin angular momentum, S~ can be in any direction relative to p~ The value of spin S~ along p~ is always constant The sign of the component of spin along the direction of motion is known as the “helicity”, S~.~p h = |p~| Taking spin 1/2 as an example: S⃗ S⃗

⃗p ⃗p 1 1 h=+ h=− 2 2 “Right-handed” “Left-handed”

Dr. Tina Potter 9. The Weak Force 12 The

60 60 − β decay of Co → Ni + e + ν¯e

Align cooled 60Co nuclei with B~ field and look at direction of emission of electrons

e− always observed in direction opposite to spin – left-handed. p~ conservation: ν¯ must be emitted in opposite direction – right-handed.

Right-handed e− not observed here ⇒ Parity Violation

Dr. Tina Potter 9. The Weak Force 13 The Weak Interaction and Helicity

The weak interaction distinguishes between left- and right-handed states. This is an experimental observation, which we need to build into the Standard Model. The weak interaction couples preferentially to left-handed particles and right-handed antiparticles

To be precise, the probability for weak to the ± helicity state is 1  v  2 1 ∓ c for a lepton → coupling to RH particles vanishes 1  v  2 1 ± c for an antilepton → coupling to LH antiparticles vanishes

⇒ right-handed ν’s do not exist left-handedν ¯’s do not exist Even if they did exist, they would be unobservable.

Dr. Tina Potter 9. The Weak Force 14 Charge Conjugation

C-symmetry: the physics for +Q should be the same as for −Q. This is true for QED and QCD, but not the Weak force...

Charge Conjugation LH e− −−−−−−−−−−→ LH e+ EM, Weak EM, Weak

Charge Conjugation RH e− −−−−−−−−−−→ RH e+ EM, Weak EM, Weak

Charge Conjugation LH νe −−−−−−−−−−→ LH ν¯e Weak Weak

C-symmetry is maximally violated in the weak interaction.

Dr. Tina Potter 9. The Weak Force 15 Parity Violation Parity is always conserved in the strong and EM interactions η → π0π0π0 η → π+π−

Dr. Tina Potter 9. The Weak Force 16 Parity Violation Parity is often conserved in the weak interaction, but not always The weak interaction treatsLH andRH states differently and therefore can violate parity (because the interaction Hamiltonian does not commute with Pˆ).

K + → π+π−π+ K + → π+π0

Dr. Tina Potter 9. The Weak Force 17 Weak interactions of All weak charged current lepton interactions can be described by the W boson propagator and the weak vertex: W − The Standard Model Weak CC Lepton e−, µ−, τ − gW Vertex

νe, νµ, ντ + antiparticles

W bosons only “couple” to the (left-handed) lepton and neutrino within the same generation  e−  µ−  τ − 

νe νµ ντ ± − e.g. no W e νµ coupling g 2 g α = W W W 4π

Dr. Tina Potter 9. The Weak Force 18 Weak interactions of leptons Examples W − → e−ν¯ , µ−ν¯ , τ −ν¯ + + + + e µ τ W → e νe, µ νµ, τ ντ e−, µ−, τ − e+, µ+, τ +

W − W +

ν¯e, ν¯µ, ν¯τ νe, νµ, ντ

− − ν µ → e ν¯eνµ µ − − ντ τ → e ν¯eντ gW gW − µ e− τ − e− W − − gW W gW

ν¯e ν¯e

− − − n → pe ν¯e Bc → J/ψe ν¯e d d − Bc c¯ c¯ J/ψ nu up V g b VbcgW c d ud W u − − W ν¯ gW ν¯e W gW e e− e− Dr. Tina Potter 9. The Weak Force 19 µ Decay

Muons are fundamental leptons (mµ ∼ 206me) Electromagnetic decay µ− → e−γ is not observed (branching ratio < 2.4 × 10−12) ⇒ the EM interaction does not change flavour. Only the weak CC interaction changes lepton type, but only within a generation. “ conservation” for each lepton generation. − − decay weakly: µ → e ν¯eνµ − νµ e

gW GF − µ− e− µ ν¯e − W gW

ν¯e νµ

As mµ  mW can safely use Fermi theory to calculate decay width (analogous to nuclear β decay).

Dr. Tina Potter 9. The Weak Force 20 µ Decay

5 Fermi theory gives decay width ∝ mµ (Sargent Rule) However, more complicated phase space integration (previously neglected kinetic energy of recoiling nucleus) and taking account of helicity/spin gives different constants 2 1 GF 5 Γµ = = 3mµ τµ 192π Muon mass and lifetime known with high precision.

mµ = 105.6583715 ± 0.0000035 MeV

−6 τµ = (2.1969811 ± 0.0000022) × 10 s

Use muon decay to fix strength of weak interaction GF −5 −2 GF = (1.16632 ± 0.00002) × 10 GeV

GF is one of the best determined fundamental quantities in particle physics.

Dr. Tina Potter 9. The Weak Force 21 τ Decay

The τ mass is relatively large mτ = 1.77686 ± 0.00012 GeV

Since mτ > mµ, mπ, mp, ... there are a number of possible decay modes ν ν τ τ ντ g gW W gW − − τ − e− τ µ τ − d, s W − W − − gW gW W gW

ν¯e ν¯µ u¯ Measure the τ branching fractions as:

− − τ → e ν¯eντ 17.83 ± 0.04%

− − τ → µ ν¯µντ 17.41 ± 0.04%

τ − → 64.76 ± 0.06%

Dr. Tina Potter 9. The Weak Force 22 Lepton Universality Do all leptons have the same weak coupling? Look at measurements of the decay rates and branching fractions. ντ νµ gW gW − − − τ e µ e− W − − gW W gW ν¯e ν¯e 1 G 2 2 F 5 1 Γτ→e 1 GF 5 = Γµ→e = m 3 µ = = 3mτ τµ 192π ττ B(τ → e) 0.178 192π 5 ττ mµ If weak interaction strength is universal, expect: = 0.178 5 τµ mτ

mµ = 105.6583715 ± 0.0000035 MeV

Measure mµ, mτ , τµ to high precision: mτ = 1.77686 ± 0.00012 GeV −6 τµ = (2.1969811 ± 0.0000022) × 10 s

−13 −13 Predict ττ = (2.903 ± 0.005) × 10 s Measure ττ = (2.903 ± 0.005) × 10 s ⇒ same weak CC coupling for µ and τ Dr. Tina Potter 9. The Weak Force 23 Lepton Universality

We can also compare ντ ντ

gW gW τ − e− τ − µ− − − W gW W gW

ν¯e ν¯µ B(τ − → µ−ν¯ ν ) If the couplings are the same, expect: µ τ − − = 0.9726 B(τ → e ν¯eντ ) (the small difference is due to the slight reduction in phase space due to the non-negligible muon mass). − − Measured B(τ → µ ν¯µντ ) − − = 0.974 ± 0.005 consistent with prediction. B(τ → e ν¯eντ )

⇒ same weak CC coupling for e, µ and τ ⇒ Lepton Universality

Dr. Tina Potter 9. The Weak Force 24 Universality of Weak Coupling

− Compare GF measured from µ decay with that from nuclear β decay

νµ d d nu up gW V g d ud W u µ− e− − W gW − ν¯ W gW e − ν¯e e

µ −5 −2 β −5 −2 GF = (1.16632±0.00002)×10 GeV GF = (1.136 ± 0.003) × 10 GeV

β Ratio GF µ = 0.974 ± 0.003 GF Conclude that the strength of the weak interaction is almost the same for leptons as for . But the difference is significant, and has to be explained.

Dr. Tina Potter 9. The Weak Force 25 Weak Interactions of Quarks Impose a symmetry between leptons and quarks, so weak CC couplings take place within one generation: e− µ− τ − Leptons − e−µ−τ − W − W − W

νe νµ ντ ν¯ ν¯e ν¯µ τ s b Quarks d     u c t W − W − W − d s b u¯ c¯ t¯ + + + + So π → µ νµ would be allowed but K → µ νµ would not νµ νµ

+ + π u VudgW gW K u VusgW gW

+ + d¯ W s¯ W

µ+ µ+ + + + But we have observed K → µ νµ ! (much smaller rate than π decay.) Dr. Tina Potter 9. The Weak Force 26 Quark Mixing

Instead, alter the lepton-quark symmetry to: (only considering 1st and 2nd gen. here) Leptons Quarks  − −    0 e µ u c where d = d cos θC + s sin θC 0 0 0 νe νµ d s s = −d sin θC + s cos θC

Now, the down type quarks in the weak interaction are actually linear superpositions of the down type quarks i.e. weak eigenstates (d 0,s0) are superpositions of the mass eigenstates (d,s)

 0     Weak Eigenstates d cos θC sin θC d = Mass Eigenstates s0 − sin θC cos θC s

◦ ⇒ Cabibbo angle θC ∼ 13 (from experiment)

Dr. Tina Potter 9. The Weak Force 27 Quark Mixing Now, the weak coupling to quarks is: d cos θC + s sin θC d0 d s

− − − W gW cos θC W gW sin θC W = + u¯ u¯ u¯

−d sin θC + s cos θC s0 d s

− − − W −gW sin θC W gW cos θC W = + c¯ c¯ c¯ + + + + Quark mixing explains the lower rate of K → µ νµ compared to π → µ νµ β and the ratio GF µ = 0.974 ± 0.003 GF 2 β 2 2 Difference in couplings affects |M| ∝ (GF ) ∝ (cos θC ) β Now expect GF ◦ µ = cos θC which holds for θC ∼ 13 GF Dr. Tina Potter 9. The Weak Force 28 CKM matrix Cabibbo-Kobayashi-Maskawa Matrix Extend quark mixing to three generations 0 d0 s0 b

− W − W − W

u¯ c¯ t¯  d 0   d  Weak Eigenstates 0 Mass Eigenstates  s  = VCKM  s  b0 b    3  Vud Vus Vub cos θC sin θC sin θC 2 VCKM =  Vcd Vcs Vcb  ∼  − sin θC cos θC sin θC  3 2 Vtd Vts Vtb sin θC − sin θC 1  0.975 0.220 0.01  ∼  −0.220 0.975 0.05  0.01 −0.05 1

Dr. Tina Potter 9. The Weak Force 29 Quark Mixing

Weak interactions between quarks of the same family are “Cabibbo Allowed” d s b

− − − g V W gW Vud W gW Vcs W W tb

u¯ c¯ t¯

between quarks differing by one family are “Cabibbo Suppressed” s d b s

− − − − g V W gW Vus W gW Vcd W gW Vcb W W ts

u¯ c¯ c¯ t¯

between quarks differing by two families are “Doubly Cabbibo Suppressed” b d

− − g V W gW Vub W W dt

u¯ t¯

Dr. Tina Potter 9. The Weak Force 30 Quark Mixing Examples

+ + K → µ νµ νµ us¯ coupling ⇒ Cabbibo suppressed

+ K u VusgW gW 2 4 2 4 2 + s¯ W |M| ∝ gW Vus = gW sin θC

µ+ D0 → K −π+ D0 → K +π− D0 u¯ u¯K − D0 u¯ u¯π − c gW Vcs s c gW Vcd d

W + u W + u gW Vud gW Vus d¯π + s¯K + 0 + − 2 2 4 Expect Γ(D → K π ) (gW VcdVus) sin θC 0 − + ∼ 2 2 = 4 ∼ 0.0028 Γ(D → K π ) (gW VcsVud) cos θC Measure0 .0038 ± 0.0008 D0 → K +π− is Doubly Cabibbo suppressed (two Cabibbo suppressed vertices)

Dr. Tina Potter 9. The Weak Force 31 Summary of the Weak CC Vertex

All weak charged current quark interactions can be described by the W boson propagator and the weak vertex: W − The Standard Model Weak CC Quark Vertex q = d, s, b gW Vqq0

+ antiparticles q0 = u, c, t W ± bosons always change quark flavour W ± prefers to couple to quarks in the same generation, but quark mixing means that cross-generation coupling can occur. two generations is less probable than one.

W -lepton coupling constant −→ gW W -quark coupling constant −→ gW VCKM

Dr. Tina Potter 9. The Weak Force 32 Summary

Weak interaction (charged current)

Weak force mediated by massive W bosons mW = 80.385 ± 0.015 GeV Weak force intrinsically stronger than EM interaction 1 1 α ∼ α ∼ W 30 EM 137 Universal coupling to quarks and leptons, but... Quarks take part in the interaction as mixtures of the mass eigenstates Parity & C-symmetry can be violated due to the helicity structure of the interaction Strength of the weak interaction given by µ −5 −2 GF = (1.16632 ± 0.00002) × 10 GeV from µ decay.

Up next... Section 10: Electroweak Unification Dr. Tina Potter 9. The Weak Force 33