Particle Physics

Chris Parkes

5th Handout

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

Electroweak Theory

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

2. Gauge theories: gauge symmetriesbosons, introduction of W+Z; Problems with massive W+Zs the Higgs.

2

Unification

e-

e-

Force from B-field

S S` e-

e-

Force from E fieldp p

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

Grand 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

3

Electroweak• Electromagnetic and Weak

– Different aspects of single electroweak interaction

– Same coupling e– Low energy broken symmetry– Massless , massive W+,W-,Z

4

Divergences• Predicted amplitudes for physical processes finite at

all energies, orders of coupling constants– QED arbitrary parameters h,e,m(electron)

• Fermi Weak Theory– point-like contact interaction

• Elastic Scattering process– Scattered intensity cannot exceed incident intensity– Unitarity limit

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

sG2

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

e

e e

e

2

2

max

5

Divergences – Add W• Introduce W boson

– Propagator – ‘spread’ interaction over finite range

• For q2 large, tends to

• Cancels s dependence

– i.e. well behaved at high energy

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

12

2

)1( WM

qW

2

2

q

MW

22W

E

MG

e

ee

e

Coupling strength,propagator

W-Propagator term

6• Divergences with QED diagrams as well as W

– Adding Neutral currents solves divergences

•Diagrams contribute to amplitude•Total xsec well behaved

Divergences – Add Z

7

Divergence in Electroweak

1) Electroweak - Cancel all divergences– well behaved theory– Photon and weak couplings related –

unification– Same intrinsic coupling strength

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

boson also

egW (numerical factors neglected)

8

Unification Conditions

Z

WW

WZWW

M

M

e

gge

cos

)( ,4 where

cossin22

1371

0

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

Predicts mass MWAt low energy W interaction strength given by GF Fermi constant

2sin hence,

2 2

2

FW

W

W

WF

GM

M

gG

9

Higher Orders• Measure neutrino –

nucleon scattering

• NUTEV expt

t

W Wb

H

Z/W Z/W

Z

q

W

q qsin2θW=0.22770.0013(stat)0.0009(syst)MW =78.10.2 GeV/c2 using unification formula

BUT measurements (LEP,TeVatron) give MW =80.390.03 GeV/c2

Why ? Higher order diagrams e.g.

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

10

• In Q.M. connection between

Global transformations and conserved quantities, e.g.1. Translational Invariance Linear momentum conservation

2. Rotational InvarianceAngular momentum conservation

3. Translations in timeEnergy conservation

iqetxtx ),('),(

Emmy Noether

Noether’s theorem – Symmetries (invariances) naturally lead to conserved quantities

Gauge Transformations1 unchanged, Clearly * -iqθiqθee

x

tiee

ti

t

tx

txxHt

txi

iqiq

' similiarlyor

)(),('

and

),()(),(

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

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

Schrodinger or Dirac eqn of form:

11

Local Gauge Transform - QED • Now consider local transformation

• Add Electromagnetism– Can now be made invariant !– i.e. invariance under U(1) local transformation

electromagnetic field• (Conserved quantity is electric charge)

• Interpretation:1. Change of phase change in E,p

• Exactly compensated by changes in emag. Field

2. Emag field carries changes away• Virtual photons

3. To cancel over all space-time range must be • so, photon massless

),(),('),( ),( txetxtx txiq Phase θ different at every point is space-time•Ψ’ no longer a solution of eqn of motion for free particle

).( xpEtie

12

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

• 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

),(),('),( ),( txetxtx txi

13

Weak Interactions• So QED local gauge

QCD

• What about weak ?• Need nature of particle also to change

• Transform

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

eeqgq

00 ,

,

WWee

WeWe

ee

ee

),(),('),( ),( txetxtx txig

14

Weak Transform

01

101

0

02 i

i

10

013

Generators of SU(2) T are Pauli matrices:

3 basis states W+,W-,W0

Arrange particles in pairs in generations:

Weak Isospin space – up and down componentse.g.

R

R

L

R

L

e

d

u

d

u

ee

'

'

R

R

L

R

L

s

c

s

c

'

'

'

'

R

R

L

R

L

b

t

b

t

Left-handed doublet

Right-handed singletWeak force acts on LH

operator lowering )(

operator raising )(

2121

2121

i

i

e

e

e

e

Caveat: RH neutrinos?

15

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:

1. W0 same form and strength as W

• But not true experimentally

2. W+,W-,W0 all predicted massless• But heavy, W ~ 80 GeV/c2 , Z ~91 GeV/c2

WZWW gg cossin

And gZ depends on particles at vertex

16

Problem 1: neutral bosons• Clearly ,Z0 related to W0,B0 but how ?• Mixtures

• 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 couplinge- - weak part & emag part for charge 1u - weak part & emag part for charge 2/3

WW

WW

WBZ

WB

cossin

sincos00

00

Mixtures give rise to unification condition relate ,W,Z couplings and explain gZ variation with particle type

17

Problem 2: Masses for W & Z• Gauge invariance leads to zero masses

– Need to cancel at infinite range– QED – massless – 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

18

Higgs Mechanism• Cocktail party

– People at party !– Higgs field is NOT empty

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

• 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

19

Higgs Field• Introduce doublet of scalar fields• Vacuum state

– Not zero

• Emag bowl shaped– Vacuum field 0

• Higgs field, “Mexican Hat”-like– Vacuum expectation value of field, v

• Ground state is degenerate– Spontaneous symmetry breaking

2

22 v

Redefine all fields wrt physical vacuumPotential Energy not symmetric about this pointSymmetry between W and B fields is broken

20

Higgs Mechanism Predictions

• 1) W and Z acquire masses– Masses from interaction of gauge fields with non-

zero vac. expectation value, v, of Higgs Field

• 2) Neutral spin-zero Higgs bosons H– 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– Coupling proportional to particle mass

WZ

W

M

M cos

2v

W gM

Standard Model does not predict Higgs mass, W/Z mass, fermion masses

ff

H

21

• 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 ?

Electroweak Summary