Particle Phy

Particle Physics 5th Handout

Electroweak Theory •

Divergences: cancellation requires introduce W, introduce Z, introduce Higgs

•

Gauge theories: gauge symmetries bosons, introduction of W+Z; Problems with massive W+Zs  the Higgs.

http://ppewww.ph.gla.ac.uk/~parkes/teaching/PP/PP.html

Chris Parkes

Grand Unification ?

Unification • Emag, weak, strong, gravity – Distinct characteristics (conservation rules of interaction, coupling strength..)

• Different aspects of single universal interaction at v. high energy ? – Symmetry broken at low mass or energy scales

• e.g. Electricty & Magnetism – Single theory, electromagnetic field – One arbitrary constant, c S

ee- p Force from B-field

S`

(See Feynman Lect. Vol.2 13-8)

ep eForce from E field

2

Electroweak • Electromagnetic and Weak – Different aspects of single electroweak interaction – Same coupling e – Low energy broken symmetry – Massless γ, massive W+,W-,Z

3

Divergences

See Perkins, Intro to HEP, 3rd edition chapter 9

• Predicted amplitudes for physical processes finite at all energies, orders of coupling constants – QED arbitrary parameters h,e,m(electron)

• Fermi Weak Theory

G2s – point-like contact interaction σ = π • Elastic Scattering process

νe e−

νe

– Scattered intensity cannot exceed incident intensity π 2 – Unitarity limit σ max =

e−

2

• Cross-section exceeds wave theory limit – xsec grows as s, – i.e. at some point more particles out than you put in !

4

Divergences – Add W • Introduce W boson – Propagator – ‘spread’ interaction over finite range W± q 2 −1 ) Propagator term (1 + 2 MW • For q2 large, tends to • Cancels s dependence

MW q2

νe

2

σ E →∞

e

G 2MW = π

W

-

νe e−

−

Coupling strength, propagator

2

– i.e. well behaved at high energy

• BUT diverergences appear in other processes • Need to systematically cancel them

5

Divergences – Add Z

•Diagrams contribute to amplitude •Total xsec well behaved

• Divergences with QED diagrams as well as W – Adding Neutral currents solves divergences

6

Divergence in Electroweak 1) Electroweak - Cancel all divergences – well behaved theory – Photon and weak couplings related – unification – Same intrinsic coupling strength gW ≈ e (numerical factors neglected)

2) Works exactly if electron mass=0 – For finite electron mass need to add Higgs boson also 7

Unification Conditions

e = gW sin θW = g Z cos θW 2 2ε 0 1 where e = 4πα , (α ≈ 137 )

(gZ depends on particles at vertex, discuss form later)

MW cos θW = MZ

Predicts mass MW

At low energy W interaction strength given by GF Fermi constant 2

GF g = W2 2 MW

hence, M W =

πα

sin θW GF 2 8

Higher Orders ν

ν

ν

Z

µ W

• Measure neutrino – nucleon scattering • NUTEV expt

q q q sin2θW=0.2277±0.0013(stat)±0.0009(syst) →MW =78.1±0.2 GeV/c2 using unification formula

BUT measurements (LEP,TeVatron) give MW =80.39±0.03 GeV/c2 Why ? Higher order diagrams e.g. t H W

W

Z/W

Z/W

b Hence, MW sensitive to mass top quark, mass Higgs boson

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•

In Q.M. connection between

Global transformations and conserved quantities, e.g. • • •

Translational Invariance→ Linear momentum conservation Rotational Invariance→Angular momentum conservation Translations in time→Energy conservation Noether’s theorem – Symmetries (invariances) naturally lead to conserved quantities

Emmy Noether

Gauge Transformations

ψ ( x , t ) → ψ ' ( x , t ) = e iqθψ

Clearlyψ *ψ unchanged, e iqθ e -iqθ = 1

Schrodinger or Dirac eqn of form: ∂ψ ( x , t ) i = H ( x )ψ ( x , t ) ∂t ∂ψ ' ( x , t ) ∂ ∂ψ and = i (e iqθψ ) = e iqθ i ∂t ∂t ∂t ∂ψ ' or similiarly ∂x

So, ψ’ still satisfies eqn of motion→ no change in observables 10

Physics invariant under Global transformation of this form (known as U(1))

Local Gauge Transform - QED •

Now consider local transformation

ψ ( x , t ) → ψ ' ( x , t ) = e iqθ ( x ,t )ψ ( x , t ) Phase θ different at every point is space-time

•Ψ’ no longer a solution of eqn of motion for free particle •

Add Electromagnetism – –

Can now be made invariant ! i.e. invariance under U(1) local transformation → electromagnetic field •

• •

(Conserved quantity is electric charge)

Interpretation: i ( Et − p . x ) ψ ≈ e Change of phase ≡change in E,p

•

Exactly compensated by changes in emag. Field

9. Emag field carries changes away •

Virtual photons

10. To cancel over all space-time range must be ∞ •

so, photon massless

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Local Gauge Transform - QCD • This time use colour state of quark ∀ →3 component vector Λ in r,g,b, space

ψ (x, t) →ψ ' (x, t) = e

iβ λ ⋅ Λ ( x , t )

ψ (x, t)

• Symmetry group is SU(3) • λ are matrices which transform the colour state – 8 basis states

• i.e. SU(3) gauge symmetry → 8 massless, coloured gluons 12

Weak Interactions • So QED local gauge QCD

e − → e −γ

q → qg

• What about weak ? • Need nature of particle also to change e − → υ eW − , υ e → e −W + e − → e −W 0 , υ e → υeW 0

• Transform

ψ (x, t) →ψ ' (x, t) = e

igτ ⋅ Λ ( x ,t )

ψ (x, t)

• Symmetry group SU(2) – Λ is a 2 component vector – Τ are the matrix states

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Weak Transform Generators of SU(2) T are Pauli matrices: 0 − i  τ 2 =  i 0 

0 1  τ 1 =  1 0

1 0   τ 3 =   0 − 1

→3 basis states W+,W-,W0

Arrange particles in pairs in generations: υe   −   e L u    d 'L

(e )

 υµ   −   µ L c    s'  L

−

R

(u) R ( d ') R

(µ ) −

R

( c) R ( s ') R

 υτ   −  τ  L t    b'  L

(τ ) −

R

(t) R ( b ') R

Left-handed doublet Right-handed singlet Weak force acts on LH Caveat: RH neutrinos?

Weak Isospin space – up and down components e.g. 1 (τ + iτ ) raising operator e − → υe 2

1 2

1

2

(τ 1 − iτ 2 ) lowering operator

υe → e −

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Electroweak Transform •

Combined Electroweak – –

Symmetry SU(2)xU(1) Triplet (W±,W0) and singlet (B0) of massless (∞ range) fields – Predicts W+,W-, neutral currents, photon – Explains Fermi theory, cancels divergences

• •

Two problems remain: W0 same form and strength as W± •

•

But not true experimentally gW sin θW = g Z cos θW And gZ depends on particles at vertex

W+,W-,W0 all predicted massless •

But heavy, W ~ 80 GeV/c2 , Z ~91 GeV/c2

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Problem 1: neutral bosons

• Clearly γ,Z0 related to W0,B0 but how ? 0 0 • Mixtures→ γ = B cos θW + W sin θW Z = − B 0 sin θW + W 0 cos θW • W± - weak force

– couples left-handed particle states discussed earlier

• Z – mixture weak & electromagnetic – Emag part couples to electric charge of particle • Same for LH,RH parts

– Weak part couples to weak isospin • i.e. only to RH part of particle

– e.g.

ν – only weak component of coupling e- - weak part & emag part for charge 1 u - weak part & emag part for charge 2/3 Mixtures give rise to unification condition → relate γ,W,Z couplings 16 and explain gZ variation with particle type

Problem 2: Masses for W & Z • Gauge invariance leads to zero masses – Need to cancel at infinite range – QED – massless γ – QCD – massless g

• BUT not for (Electro)Weak • Overcome by introducing Higgs Field Mechanism to: • give particles masses •make theory gauge invariant Higgs boson is the quanta of the Higgs field. Only particle in SM not discovered

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Higgs Mechanism

http://hepwww.ph.qmw.ac.uk/epp/higgs.html

• Cocktail party

– People at party ! – Higgs field is NOT empty

• An ex-PM arrives – People cluster around her – She acquires mass from the Higgs field

• Rumour passes through room – Cluster of people – Excitation of Higgs field – Higgs boson 18

Higgs Field

• Introduce doublet of scalar fields • Vacuum state – Not zero

• Emag bowl shaped – Vacuum field 0

• Higgs field, “Mexican Hat”-like

v2 – Vacuum expectation value of field, v φ = 2 • Ground state is degenerate 2

– Spontaneous symmetry breaking Redefine all fields wrt physical vacuum Potential Energy not symmetric about this point Symmetry between W and B fields is broken 19

Higgs Mechanism Predictions • 1) W and Z acquire masses

MW = cos θW MZ

– Masses from interaction of gauge fields with nonzero vac. expectation value, v, of Higgs Field M W = g

• 2) Neutral spin-zero Higgs bosons H

v 2

– Quanta of Higgs field from gauge invariance

• 3) Particle masses – Particles travel through Higgs field and acquire masses – Fermions/bosons also interact with Higgs boson H f – Coupling proportional to particle mass f 20 Standard Model does not predict Higgs mass, W/Z mass, fermion masses

Electroweak Summary • Electroweak theory provides well-behaved theory without divergences • Gauge invariance leads to introduction of weak force • Higgs Mechanism leads to particle masses • Tests of Theory: – Find Neutral Currents  – Discover W,Z bosons  – Measure W,Z couplings and masses  – Find Higgs Boson

?

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