Lecture 15 – Next steps

●

The Higgs boson

●

Review of the Standard Model



Problems of the Standard Model



Proposed Solutions

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The Standard Model

Goal: a theory which describes all of the fundamental constituents of nature and their interactions with the minimum of assumptions and free parameters. Ultimately describe all interactions over small distance scales and cosmological observations.

The Standard Model is our best attempt at this - assess how successfult in this lecture.

6 quarks, 6 leptons, 3 exchange bosons + antiparticles. Two independent forces (electroweak and QCD).

19 free parameters: particle masses, mixing angles, CP-violating term, couplings....

Consistent method of introducing interactions via so-called gauge invariance and Feynam diagram formalism (next lecture course).

The Standard Model assumes massless neutrinos but this is easily fixed.

Barring neutrino oscillations, the Standard Model has never failed a single experimental test.

There is still one test left to pass - finding the Higgs boson.

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The Higgs boson

The missing particle in the Standard Model. Explains mass generation of the fundamental particles.

The Higgs mechanism is a way of explaining why, in an apparently unified electroweak theory, the

W

 an d

Z

0 Some consequences: A spin-0 massive boson, the Higgs particle

H

0 , is required.

A Higgs field pervades space: fermions interacting with the field acquire mass.

A ferm ion with mass

m f

can also couple to the Higgs boson with strength

g Hff

.

g Hff

 2

g W

 

m m W f

  (15.01) Couplings to other particles, with strength proportional to particle mass.

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How do we look for the Higgs ?

How is it produced and how does it decay ?

At LEP:

e

 

e

 

H

0 

Z

0

208 GeV centre-of-mass energy Sensitive to Higgs masses up to 120 GeV.

Production mechanism

b b b

H

0 FK7003 4

Have we already found it ?

Lots of excitement around 2000/2001 as LEP reached the end of its life.

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Observation of a Higgs ?

An excess of events was seen at mass  115 GeV but reanalysis of data and rigorous statistical calculation of significance means it is impossible (and stupid) to conclude a Higgs was seen.

Lower mass limit

M H

> 113.5 GeV (15.02) FK7003 6

Been here before - top quark non discovery…

●

1984 CERN

●

UA1 experiment

●

pp

(630 GeV cm energy)

●

Something they would rather forget

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Race for the Higgs

The Tevatron (

pp

at 2 TeV centre-of-mass energy) is now hunting the Higgs.

The LHC (

pp

at 14 TeV centre-of-mass energy) will take up the chase in 2009. Different production mechanisms compared with LE P and different decays sought.

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Where is the Higgs ?

FK7003 Excluded by direct search.

Most likely Higgs mass value from fits to measured electroweak quantities in the Standard Model.

The Higgs is either just around the corner or nature is more complicated than we suppose.

9

How good is the Standard Model ?

Criteria

Predictivity and testability Completeness* Compactness

U,G or VG

VG – the only ’failure’ is neutrino masses and we can patch that up by adding extra parameters.

Higgs yet to be found.

The SM can be killed but is still v. much alive!

U – no quantum theory of gravity ? Dark matter ? ….

G - Based on 19 free parameters – not bad for describing EM,weak and strong forces below



1TeV.

*

The focus of the rest of this lecture FK7003 10

Speculation strategy

We have few answers but that doesn't mean we can't ask sensible questions.

(1) At which energies can we expect that the Standard model will not describe subatomic particle interactions ? (2) In which areas is the Standard Model incomplete and which theories have been proposed address these problems ?

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How well can we localise a particle ?

To what precision can we know the position of a particle, eg electron ? In quantum mechanics the position can be known to infinite accuracy if we accept we have no knowledge of its momentum. Eg from b asic quantum mechanics: Heisenberg's microscope.

Resolution in position

x

 1

p

 (2.36) ;   probing photon wavelength.

p

 =photon momentum 

p x x p x

1 (2.12) Above picture assumes reaction:  

e

 Quantum field theory changes this picture. If

p



e

  2

m e

(

m e

=electron particle)  kinematically feasible reaction:  

e



e

 

e

 

e

 Two ident ical particles in final state. No longer possible to say anything about electron position for

p

  2

m e

.

 Fundamental limitation on knowledge of position: 1 2

m

(15.03) FK7003 12

Compton Wavelength

Compton wavelength of a particle: 

c

 2 

m

(15.04) Introduced in lecture 12 as the distance below which the electromagnetic coupling constant starts to change i.e. the distance at which quantum field theory below important in describing particle behaviour.

Electron: 

c

 2 

m e

  10 m. (12.03) Different ways to think about this number but the poi nt is that 

c

that a quantum description of matter says that we can localise a particle of 

c

 2 

m

.

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Gravity

gravitational singularity (black hole): Scharzschild radius

r s G

 Gravitational constant.

 2

Gm

.

Quantum d escription of nature implies that a particle position be known to accuracy: 

C

 2 

m

.

However, for 

C



r c

the particle is contained within such a small size that a gravitational singularity occurs.

The qu antum prediction of a particle localised to a certain distance must be invalid if that localisation is taking place inside a black hole :).

 (naively) quantum gravity becomes important at:

r c

 

C

 2

Gm

 2 

m



G

(15.05) Formally define the Planck mass  1

G

  19 The Standard Model must fail for masses and energies > Planck mass and a theory of quantum gravity is needed.

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Question

Compare the values of the electromagnetic and gravitational attractive forces between two stationary massive particles with charges 

e

and -

e

if the particles have (a) mass=1 GeV and (b) Planck mass. The particles are separated by a macroscopic distance.

R



F em F grav



e

2 4  0

r

2 2

m G r

2  11 

e

2 4

m

2  0

G G



m

 1 GeV   

m



m p

 10 10  2 19 GeV  -1 -2  27 kg  0    19  2  12 Fm

e

  27  2  8.85 10  12   11  8  10 36  19 C  Gravity is extremely weak until we get to the Planck scale.

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Other possible energy scales

From lecture 12: The coupling constants vary with momentum transfer (or distance) Eg strong force becomes weak at short distances (<1fm)  asymptotic freedom.

E Weak Electromagnetic  GUT scale Strong Log(Momentum transfer,

Q

(GeV) ) 

s

 

s



M Z

   1   6 

N f

 

s



M Z

 ln  

Q M Z

     1 (1 2.05) Couplings appear to unify for

Q

 10 16 GeV.  Grand unified theories (GUTs) unify em, weak and strong forces (to come).

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Speculation strategy

We have few answers but that doesn't mean we can't ask sensible questions.

(1) At which energies can we expect that the Standard model will not describe subatomic particle interactions ? Quantum gravit y effects must play a role for masses and energies at and  above the Planck scale ( 10 19 GeV). The GUT scale ( 10 16 GeV) looks a promising energy for "new physics" to appear.

(2) In which areas is the Standard Model incomplete and which theories have been proposed address these problems ?

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Problems of the Standard Model

A subjective selection of three open areas in particle physics about which the Standard Model has nothing to say. (i) Cosmology: Dark matter. 22% of universe's energy budget in the form of "dark matter".

Current evidence suggests that WIMPs: electrically neutral and weakly interacting massive particles with masses 1  10 TeV may be responsible ( LHC energies) (ii) Forces: unification and gravity Is th ere hope for a theory which unifies all of the fundamental forces or at least the strong, em and weak forces ? Why is gravity weak until the Planck mass (the hierarchy problem) ? (iii) Properties of par ticles: electric charge quantisation Why do we never observe particles with charge, eg, 1.5234

e

? If the ultimate aim is a

theory of everything

which predicts particles, forces and cosmological measure ments from a single principle/equations then solutions to one of the above problems should address in some way the other problems.

*There's loads more, eg matter - antimatter asymmetry, the strong CP problem (why is there no observed CP violation in the strong processes), neutrino masses, dark energy etc. but we'll take (i), (ii) and (iii) as opportunities to show how a problem is defined and solutions proposed.

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Supersymmetry

Every Standard Model has a supersymmetry partner.

Symmetry between bosons and fermion Quarks (fermions)  Squarks (bosons) ; 

g

(bosons)  

g

(fermions) Symmetry is broken otherwise SM and SUSY particles (sparticleS) would have the same mass.

sparticles decaying to SM particles.

R=(-1) 3   2

S

  1 SM particl es = -1 SUSY partner particles.

B L S

(15.07) FK7003 19

Why look for SUSY ?

Many reasons for looking for SUSY, amongs them...

(1) It predicts a dark matter candidate: i.e. a WIMP with mass Neutralino:  0 TeV.

 .

(2) Unificati on of the couplings is more exact if SUSY sparticles exist.

Can develop SUSY grand unified theories (GUTs) which unify the electromagnetic, weak and strong forces.

Standard Model E Electromagnetic Weak Strong Standard Model+SUSY Weak Electromagnetic Strong Log(Momentum transfer,

Q

(GeV) ) (3) Solves the hierarchy problem (beyond this course) Log(Momentum transfer,

Q

(GeV) ) Lecture 17 - explore how to look for SUSY at a LHC experiment.

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Grand Unified Theories

Incorporate strong, electromagnetic and weak forces into a GUT.

Simplest model: SU(5) (Georgi-Glashow).

Prediction of proton decay.

Violation of lepton and baryon number.

Eg

p

  0 

e

 Predictions for lifetime  10 30 years.

Current limits (SuperK- lecture)  Other GUTS predict    10 33 years.

10 33 years. GUTs also predict heavy magnetic monopoles

m

 1 0 16 GeV and explain charge quantisation. FK7003 21

Extra spatial dimensions

Original ideas on extra dimensions from T. Kaluza and O. Klein (1921). Several different models incorporating extra dimensions on the market today.

Large Extra Dimensions.

Hierarchy problem  gravity is weak since it propagates in extra dimensions (bulk) and we see a diluted form of it in our 3+1 dimension world (brane). Gravitational potential

n

  number of extra dimensions.

1

r n

 1 (15.08) where

r



R R

 distance scale for interactions at which the effects of extra dimensions are observed.

n

 2 

R

 1 mm (15.09) In general, many extra dimensions theories often predict "new" heavy particles with masse s TeV and provide dark matter candidates.

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Micro Black Holes at the LHC

In general, when two particles pass each other with enough energy, a micro black hole can be formed.

For three spatial dimensions, gravity is too weak. With extra dimensions gravity becomes stronger, micr o black holes can be created.

"Normal" black hole: size mass m

sun

, temperature km, 0.01K,   "Micro" blackhole: size temperature  Hawking rad ition.) 1 TeV, 10  27 s (evaporate through The world won't end when we turn on the LHC.

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Electric charge quantisation

Maybe its better not to be too ambitious and just focus on one specific problem.

Electric charge quantisation. Why is electric charge always meaured in integer multiples of the elementary Why are the electron and proton charges the same (barring a sign) ? The best limits state:

q electron



q q electron proton

 10  20 (15.10) Is there any way to accommodate electric charge quantisation within q uantum mechanics ? For clarity - use practical units for following derivation.

Also, we'll derive from start to finish...

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Maxwell’s equations

Electric and magnetic fields from electric charges and currents 

q e

, 

e

, magnetic charges and currents 

q m

, 

m

,

j m



j e

 and 

e

 0 (15.11) ;   0 0 

E



t



j e

(15.14) Lorentz force law:

F



q e



E

  0

m

(15.12) ;

E



B



t

 0

j m

(15.13)

B

 

q m B

 1

c

2

v



E

(15.15)

v B E E B q e v q m v

No magnetic monopoles have ever been observed 

q m

  0

e

(15.11) ;

B

0 (15.16) ; 

B



t

(15.17) ;  

m

Lorentz force law:

F



qE B

(15.18)  0,

j m

 0   0 0 

E



t



j e

(15.14) FK7003 25

Monopoles and charge quantisation

Alternative version of Dirac's argument (1931)

q e q m z r



P q e

respectively, at point

P

:

E



q r e

4  0

r

3 (15.19) ;

B

  4 0 

q r m r

 3  (15.20)

q m

,

r

'

dz

ˆ ;

r

  

r

2 

d

2  2

rd

cos   1 2

d q m



r

 4 0

q m



q e r



dz

ˆ 

r

2 

d

2  2

rd

cos   3 2

x

(15.21) Momentum density in electromagnetic field : p   0     0

q q e m

  2

r

3 

r

2  

d

2  

z

ˆ  2

rd

cos   3 2 (15.22) Angular momentum density =

r

  0 FK7003

q q d e m

  2

r r

3 

r

2 

d

2 

z

ˆ   2

rd

cos   3 2 (15.23) 26

r



z

ˆ    2

r z

ˆ 

r

2 cos 

r

ˆ  2

r z

ˆ (15.24)  Angular momentum in the field: L Set

u x



y

ˆ cos   0

q q d e m

  2

z

ˆ    0

q q d e m

  2

z

ˆ 

r

2  1 1   0  

r

2

r

 1 

u

2 

dr



d

2  2

rdu

 3 2

z

 cos 2   cos   1 

r

2 (15.25) sin 

r

3 

r

2 

d

2 (15.27)  2

rd

c os   3 2   0 

rdr

 

ru



r

2 

d

2  2

rdu

 3 2

d

 1 

u

2 

r

2 0  0

q q e m

8 

d z

ˆ 1

d

 1 1    1  1  

d u u u

 1 2    

du

1

u

2     0

q q e

4 

m

Q

z

ˆ (15.29)   

d



d d

2  0

q q e

8 

m

 2

rdu

 1 1 

u

  1 2 

d

(15.28)

z

ˆ  1 1   1   1 =

u



u

2    0

q q e m

8 

z

ˆ 1  1 1 

u d u

2  

d u

2 2 FK7003 (15.26) 27

Dirac’s quantisation condition

z

Angular momentum in the field:   0 4 

m z

ˆ (15.29)

r



q m

Angular momentum is quantised:   0 4  

q e



n

4  2  0

q m



nh

 0

q m

(15.31)

m



n

2 (15.30)

d q e



r P x

 If there's one type of magn etic charge in the universe,

anywhere in the universe

, this "explains" why electric charge is quantised ; its a consequence of angular momentum quantisation. This is one reason why we look for them.

In addition they also turn up just about everywhere else in physics (except in experiments), eg GUTs (

m

 Possible monopole charge:

q e



q D



h

 0

e

= "Dirac monopole" charge. (15.32) Coupling constant for Dirac monopoles: 

m

 1 

m

  0

q

4 

D

2  34 (15.33)  (1) field theory/Feynman diagram formalism impossible ; (2) several thousand times greater ionisation energy loss than, eg, proton with same momentum (lecture 16).

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Speculation strategy

We have few answers but that doesn't mean we can't ask sensible questions.

(1) At which energies can we expect that the Standard model will not describe subatomic particle interactions ? Quantum gravit y effects must play a role for masses and energies at and  above the Planck scale ( 10 19 GeV). The GUT scale ( 10 16 GeV) looks a promising energy for "new physics" to appear.

(2) In which areas is the Standard Model incomplete and which theories have been proposed address these problems ?

Dark matter, hierarchy problem, force unification, charge quantisation  (to name but four) SUSY, extra dimensions, magnetic monopoles are just some of the things we've been speculating..But this is a game - we need data! FK7003 29

So how close are we to a unified theory of all the forces ?

At present string theory offers the best hope. It is the most promising candidate theory for quantum gravity.

However, its been the most promising theory for over 20 years now...

Lecture 9 - hadron masses can be calculated using a picture of hadrons as excitations of string. This formed part of the early ideas which led to string theory. Point-like particles are tiny quantised one-dimensional strings. Extra dimensions and supersymmetry accommodated within string theory.

Extremely challenging to come up with a quantitative prediction from string theory which can be tested. Time will tell.

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Summary

●

Higgs discovery would be confirmation of the Standard Model

●

Standard Model is incomplete

●

A range of proposed solutions exist which postulate the existence of ”new” particles which could be ”around the corner” at LHC energies.

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