a. bay beijing october 20051 summary standard model of particles (sm) - particles and interactions -...

A. Bay Beijing October 2005 1

Summary

Standard Model of Particles (SM) - particles and interactions - the electro-weak model


The situation in the sixties

Chaotic

similar to chemistry of 1800


The periodic tableThe periodic table

Mendeleev (1869) introduced the periodic table


Atomic model explains theMendeleev table

Rutherford (1912)showed that atomscontain a centralnucleus10

-10 m


e

e

u c t

d s bQuarks

Strong (gluons)

Electromagetic (photon)

Weak (W+, W-, Z)

Interactions (quanta)MatterElectriccharge [e]

0

1

2/3

1/3

The Standard Model of Particles

Gravity is absent: hopefully its effects are too weak...

each particle has an associated anti-particle:e and e+ , u and u , and , ...

each particle has an associated anti-particle:e and e+ , u and u , and , ...

__

how to distinguishthese two ?

spin 1/2 spin 1


Elementary particles in interaction:e. m.

q

q

e

eExchange of photons

Affects all the electrically chargedparticles: quarks + e,

Feynman graph:


W

q

q

e

Elementary particles in interaction:Weak

Exchange of W and Z

Affects the full set of particles

Feynman graph:


g

q

q

q

q

Elementary particles in interaction:Strong

Exchange of gluons

Affects only quarks

Feynman graph:


Quark model

A few examples: protons and neutrons are made of 3 quarks:

u

dNEUTRON: dd

uuPROTON:

d

NEUTRON:d

du

PROTON:d

uu

€

23 + 23 − 13 = +1

€

−23 − 23 + 13 = −1

Q=+1

Q=-1Q=0

Q=0

Charge mirror

Quarks hold together by "strong interaction" to form "hadrons": baryons (half-integer spin): p, n, , ... mesons (integer spin) : , , K, B, ...


Quark model .2

Easier to indicate the quark content of the hadrons with avector. Proton and neutrons and their antiparticles are:

€

p = uud C ⏐ → ⏐ p = u u d

n = udd C ⏐ → ⏐ n = u d d

C represents the "charge mirror"

More precisely, the "wave function" of a proton must contain the information of the movement of the 3 quarks, of their spin orientation, of the quark "flavour", and of of

an entity called the "colour" of the quark:

€

n = space spin flavour coulour

similar to atomic wave functions

new concept


Quark model .3

The mesons are built with one quark and one antiquark.

Lightest meson system are the 3 "pions":

€

+ = ud

π 0 =1

2uu − dd

π − = u d

the 1/sqrt(2) is to "QM average" the 2possible configurations uubar and ddbar

the minus sign is to respect aspecial condition of symmetry!


Quark model .4Theory must calculate the masses, spin, magnetic moments,decay probabilities, ... , of the hadrons. Quite a difficult task:

M(proton) ~ 1 GeV (remember: c=1) 3 quarks in the proton: 1 GeV/3~330 MeV/quarkM(pion) ~ 140 MeV 2 quarks in the pion: 140 MeV/2 = 70 MeV

This shows that the strong interaction dynamics definingthe binding energy is very important, very "strong".The theory of strong interactions is the colour theory:"chromodynamics".If one "switch of" strong interactions, the mass of the quarksshould be M(u)<M(d)<10 MeV, M(s)~100 MeV, M(c)~ 1.2 GeV, M(b)~4.5 GeV, M(t)~174 GeV


* pions are unstable, for instance:

Hadron decay

* p is (seems) stable, lifetime is at least 1029 years

* n is unstable, lifetime ~15min

W is the vector of theweak interactions

this is an e.m.interaction


The leptons

Neutrinos are neutral and have masses ~0.The electron is the lightest (known) charged particle (511 keV).

The muon (106 MeV) and the tau (1780 MeV) are unstable.

Seen by the theory of weakinteractions the processlook like this:

time


Measurement of masses

Mass of a particle of momentum p and energy E:M = sqrt(E2 p2)

Example of measurement: 0

- measure the 2 photons 4-vectors: (E1, p1), (E2, p2)- compute the parent (0) four-vector: (E, p)=(E1 +E2, p1 +p2)- compute M

E1

E2

detector(e.m. calorimeter)

0

M=135 MeV


The measurement of masses .2

The production threshold method was used by BES to measure

with very high precision the tau mass M :

Productionrate

Ebeam

e+ e- collider withbeam energy Ebeam.Minimal E neededto produce 2 taus is

Emin = 2 M c2

Each beam must have at least Ebeam = M c2

M= 1776.9 ±0.2 MeV+0.2

0.3


The BES method

In order to optimize the search, the energy scan was doneby the method of signal appearance / disappearance


Summary on SM elementary particles

The "elementary particles" of the Standard Model there are the quarks and the leptons (spin 1/2, they are "Fermions")Quarks have fractional charge (units of e), and they are the building blocks of hadrons (p, n,..., pions, kaons,...).Lepton have charge 0 or 1 (+1 the anti-leptons).

How can we explainthis mass spectrum ?


The interactions

€

α =α em =e2

4πε0hc

€

α s =g2

4πhcby analogy:

interaction gravitation e.m. weak strong

manifestation weight light beta decay nuclear forces

Couplingw/o dimension

lifetime

rangeweak


The interactions .2

QuickTime™ and aTIFF (LZW) decompressor

are needed to see this picture.



Coulomb scattering of anelectron by the field of a nucleus

Decay of a muonin electron and 2 neutrini


The interactions .3





annihilation of e+ e

into photon/Z decayinginto a pair of particles

when the 2 particles arequarks, they "hadronize"(i.e. they become hadrons)producing jets of particles

... we have not seen individual quarks !


Jets of hadrons

TASSO eventat PETRA

1+cos2distribution proving that the quark is a 1/2 spin particle

Open a parenthesis: jets of particle is an evidence of the existence of quarks


The interactions .4

We have also eventswith 3 or more jets

quarks andgluons hadronizeinto one jet each

HCAL

ECAL


How individual quarks (or gluons) transform into jets of hadrons ?This phenomenon is difficult to treat analytically because the intensityof the force is too strong (cannot do a "perturbative": calculation)

Phenomenology of hadronization

Potential model of qq interaction:

~ Coulomb at very short r (< 1fm)

E grows fastwith r(but not as much as elastic: E = k2/2)


Phenomenology of hadronization .2

Field lines at verysmall r

Field lines stay concentratedwhen you pull the two quarksapart.They form a string.


Phenomenology of hadronization .3

Energy accumulates in the string

when enough energy/fm couplesquark-antiquark can be produced

mesons fly apart

jet 1 jet 2


The interactions .5Quantum mechanics: interactions are mediated by quanta

Interaction quanta mass typical rangeStrong 8 gluons 0 1 fmE.m. photon 0 infiniteWeak W, Z ~100 GeV 10-3 fmGravity graviton 0 infinite

later we will try to understand why only Weak forces havequanta which are massive...


The interactions .5

* A photon travelling from a source to your eye has mass=0.If you measure p and E of this photon, you will find thatit has mass=0 : E2 p2 = 0. This is a "real" photon.

In a "collision", two charged particles exchange some p and E.QM says that this exchange is mediated by a photon.

Ex: electron and muon are charged particles, they can exchangethe E and p transported by a photon, the quantum of e.m. interaction

e


The interactions .6

pa = (Ea, pa), etc...

the transferred momentum is:

q = pa pc = (pd pb)

If you try to compute the massof this photon, you will finda value different from zero:this is a "virtual" photon.

This is possible within some restrictions imposed by theHeisenberg uncertainty principle...

chargedparticles

photon

with


The interactions .7

Heisenberg: measurements of position and momentum can only bedone with a finite precision because the microscopic processesare controlled by

€

xΔpx = h /2

A virtual photon violating momentumconservation by some p, can travela length x ~1/p.

Real photons do not violate anythingand they can travel as much as theywant.

This photon can travel ~1/mass = 1/|q|


The interactions .8

We are interested to determine the probability for a particle ato interact with particle b giving momenta pc and pd.Consider a and b like 2 wires carrying electric currents Ia=Qava

and Ib=Qbvb. The force acting between the 2 is given by:

€

F∝IaIb

r~ Qava

1

rQbvb

The QM result is similar:

€

Qava

1

q2Qbvb

2

~ α emva

1

q2vb

2

1/mass of the photon ~ distance

Ia Ib


The interactions .9

The QM theory of e.m. is called Quantum ElectroDynamics (QED)From the idea seen before we can infer a theory to computethe probability that a given process take place.The typical behaviour of a QED process, for instance of

e+ e

Probability of the QED process ~ (αem / E)2

expressed as a function of the total energy E is:

Indeed we have the cross section:

which has the nice behaviour 0 when E infinity,no "ultraviolet catastrophe".


The interactions .10

All these calculations are possible at the "perturbative level",which means that the "higher order corrections" must become smaller and smaller (expansion must converge). Diagrammatically something like:

> > > ...

Mathematically: ααCαDα

where A, B, C, D,... come from (often complex) calculations.

One sees that the coupling constant has better to be < 1 !

contribution fromone quantum 2 quanta 3 quanta


The interactions .11The technique has been generalized tothe other interactions.

"Currents J" of particles with charges ginteract via their specific quanta.

* e.m.: charge is e (or a fraction of e for the quarks) andthe quanta are the photons (or use α= αem e2)

* weak interaction: charge is gW (or simply g), quanta are W and Z

* strong interactions: charge is gs (or αs gs2) with 8 gluons

While Coulomb needs only 1 kind of charge, + and , stronginteractions have 3 kinds (r,g,b and -r,-g,-b) !!!

€

gJ1 × (propagator) × gJ2

J1 J2

g g



While Coulomb needs only 1 kind of charge, + and , stronginteractions have 3 kinds (r,g,b and -r,-g,-b) !!!

Consider the Coulomb force between 2 particlesAn electron has charge minus e, its anti-particle has charge plus e.Quarks have electric charge (2/3)e or (-1/3)e , and oppositesign for antiparticles.

Consider the strong force now.* The electron does not have strong interaction: its strongcharge is 0.* Quarks strongly interacts with a much more complicated algebra.They behaves like if they could be of 3 kind (SU(3) group)For instance, in a proton they must be of the 3 different coloursto give a white particle (r+g+b = white).



For a quark u, there are 3 possibilities u, u, u, etc.

During e.m. interaction, the electric charge stays on theparticle, because the photon is neutral.

During strong interaction, the charge can be transferredbecause the gluons carry the colour charge. Example:

red and blue quarks

blue-antired gluon exchange

blue and red quarks

time



g4

(r,g,b is an arbitrary index !)

€

3⊗ 3 = 8⊕1

Group theory: SU(3)couleur , basis 3 and 3:

Coloured gluons belongs to the 8 of

€

colour =

r

g

b

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ colour =

r

g

b

⎡

⎣

⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥


Interactions: some results

QED is capable to predict the Landé factor g for electron andmuon at the level of 10-9 precision:

€

=gqh

2mcs

dipolar magnetic moment of a particle of spin s, charge q, masse m

g = Landé factor g=2 for the Diracelectron

QED: (g2)/2 = ( 1'159'652.2 ± 0.2 ) 10-9

measured: ( 1'159'652.188 ± 0.004 ) 10-9

for the electron

€

g − 2

2= 0.5

α em

π− 0,32848

α em

π

⎛

⎝ ⎜

⎞

⎠ ⎟2

+1.19α em

π

⎛

⎝ ⎜

⎞

⎠ ⎟3

+ ...

with αem= 1/137.0339... from "static" measurement of e


Positronium gives αem at low energy

Coulomb potential

=>

E2

E1


αem at ~ 100 GeV

The relevant parameter isαem giving theinteraction strength

jet2

jet1

jet1

jet2

at 100 GeV αem~1/128


The relevant parameter isαs (alpha strong)giving the interactionstrength.

at 100 GeV αs~ 0.11

jet1

jet2

jet3g

Some results .3


αs at low(er) energy

Use "quarkonia"bound states of cc ()and bb (Y). Potentialis now

@ ~3 GeV

@ ~10 GeV


Running of the alphas

It is found that both the e.m. coupling constant αem

and αs vary with the energy of the process:

E of the process GeV

At 1 GeV (proton mass)αs>1, while at LEPenergy (~100 GeV)we have αs~ 0.1.

The oppositehappens for αem.At low energy itsvalue is ~1/137, and~1/128 at LEP.

running of αs


Running of the alphas

Hint of an unification of forces at high E ?

1 GeV

α

strong

e.m.

E (GeV)

Energy of(Grand) unification ?

Nice, but why do we whish some sort of "unification" ?


Unification of forces αstrong

e.m.

Why we whish some sort of "unification" ?

Unification means the reduction the number of entities in thetheory => more internal constraints => less free parameters =>the theory becomes more "predictive"

First example of (successful) unification of forces is the Maxwelltheory of electromagnetism.

A second example of unification is the electro-weak theory,which is part Standard Model.


The electro-weak theory

Historical background:

The e.m. theory was translated into a QM formalism atthe beginning of 1900, giving the Quantum Electro Dynamics.We have seen that this theory is very successful.

In 1934 E. Fermi wrote a model for the Weak Interactions (WI)inspired to QED. Because he didn't know the existence of theW and Z, he reduced the calculation to a "point-like theory"

QED Fermi model


The electro-weak theory .2

The Fermi model works well at very low energy (beta decay,...),but it cannot work at high energy:

Probability of a Fermi weak process: (E) ~ (GF E)2

This grows to infinity quite fast !

Compare to the nice behaviour of QED:

Probability of a QED process: (E) ~ (αem / E)2

To avoid the ultraviolet catastrophe the simplest solution isto introduce a particle playing a role analogous to the photonin QED. The main difference with QED is that the Wmust be massive...

Fermi constant

cross section


The electro-weak theory .3Why do we need a massive W ?

The Fermi model is OK at low energy. It starts to be wrongonly around 100 GeV.So the virtual particle must become "real" at this energy:

€

1

q2

€

1

q2 − MW2

The QED term becomes

This explains why wedo not have free W going aroundlike the photon. To produce themyou need a lot of energy.

It also explain why the weak interaction is "weak": in realityit is weak only at E<<MW. At high E, it is comparable to e.m.More of this in a moment.

q


The electro-weak theory .4Why do we need a neutral Z?

Because "second order" diagrams diverge, like in thecalculation of corrections to e+ e

this diagram givesa divergent result

contributions from theseprocesses allow to exactlyget rid of the infinities, if ...



The needed relations between Z, W and photon are incorporatedin the Glashow Weinberg Salam electroweak theory:

Weinberg angle

Here g is related to the electric charge g =e/2√2 e=1.60 10-19 C

We introduce here two "weak charges" gW (GF~(gW)2) and gZ.

gW, and W are free parameters of the theory (not predicted)

... if we impose the correct relations to link e.m. and weak sectors.


Observation of W and Z

In 1982 CERN commissioning of a p - antip collider withbeams of 270 GeV (E in the centre of mass = 540 GeV).W and Z can be created by the collisions of quarks fromthe two protons:

X and X' are the "spectators" (a lot of particles which are thereto complicate the life of the physicist).


Observation of W and Z .2Once produced the W and Z decay and we have to observethey decay products in "ad hoc" detectors.A (quite) simple case is Z e+ e :

e


Observation of W and Z .3

In reality it is not too bad.The 2 electrons can be selectedquite easily and the invariantmass of the mother computed

one event energyLego plot


Observation of W and Z .4

In the '90 the LEP and SLAC have produced Z by collisionof e+ and e beams at the correct energy to excite the Z resonance:

Example:

€

e+e− → Z → μ +μ−

E c.m.86 88 90 92 94 GeV


e+e- into quarks (jets)

1 10 100 GeV

€

ss €

cc

€

bb €

e+e− → Z → qq



Fighting against infinities is a constant source of inspiration.

Another example: the prediction of the existence of a c quarkby Glashow, Iliopoulos, Maiani (GIM) in 1970:

K0

diverges !

+ c

g2 sinC cosC g2 (sinC) cosC

stabilized !



We have more instabilitiesto explore !

In the SM we have 3 familiesof lepton and quarks"doublets". €

e

e

⎛

⎝ ⎜

⎞

⎠ ⎟L

ν μ

μ

⎛

⎝ ⎜

⎞

⎠ ⎟L

ν τ

τ

⎛

⎝ ⎜

⎞

⎠ ⎟L

€

u

dC

⎛

⎝ ⎜

⎞

⎠ ⎟L

c

sC

⎛

⎝ ⎜

⎞

⎠ ⎟L

t

bC

⎛

⎝ ⎜

⎞

⎠ ⎟L

C and L indiceswill be explainedlater

The symmetry of the families lepton-quark is very aesthetic.But do we have any (mathematical) reason to believe that shouldbe like that ? Yes: breaking this symmetry generates infinities inthe "triangle anomalies".

any chargedfermion

Each triangle gives + or infinity.The sum of the triangles gives zeroif and only if the sum of the charges isalso zero. Indeed for each family

€

3×2

3−

1

3

⎛

⎝ ⎜

⎞

⎠ ⎟+ (−1) = 0

quarkscomewith 3colours

symptom fora deeper reality?!



One more instability...

the scattering WW WWdiverges and violates "unitarity"around E~1 TeV

possible cure:add a spin 0 (scalar) particlewith ad hoc couplings.

This is one reason to believethat there must be somenew particle before 1 TeV.

(or e+e WW)


Summary

We have explored at a qualitative level the behaviourof particles communicating by the different kind ofinteractions.QM requires a quantification of the (E,p) exchange.This is obtained via quanta (photon, gluons, W,Z),playing the role of mediators of the force.The strength of an interaction in a process is parametrizedby the charges (e.m., weak, strong).

To avoid infinities in the calculations we have to assume that:* the quark and lepton families are linked* e.m. and weak charges are also linked* there is an hint for the existence of a new 0-spin particle with mass < 1 TeV.

a. bay beijing october 20051 summary standard model of particles (sm) - particles and interactions -...

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