M. Cobal, PIF 2003

Electromagnetic Interactions

Dimensionless coupling constant specifying strenght of interaction between charged particles and photons

Fine structure costant:(it determines spin-orbit splitting in atomic spectra)

Em fields have vector transformation properties.Photon is a vector particle spin parity JP = 1-

In the example seen, the photoelectric cross section (or matrix elements squared) is proportional to a first order process

The Rutherford scattering is a second order process

137

1

4

2

c

e

t

Photoelectric effect: absorption (or emission)of a photon by an electron (for an electron bound in an atom to ensure momentum Conservation.

M. Cobal, PIF 2003

e-

e+

+

-

ssememem

3

4~

22

Other examples at higher orders:

μμee

M. Cobal, PIF 2003

Renormalization and gauge invarianceQED: quantum field theory to compute

cross sections for em processes

renormalisability

Gauge invariance

Electron line: represents a”bare” electron

Real observable particles: “bare” particles “dressed” by these virtual processes (“self-energy” terms) which contribute to the mass and charge.

No limitation on the momentum k of these virtual particles logarithmically divergent termAs a consequence the theoretically calculated “bare” mass or charge (m0, e0) becomes infinite

kdk

e

Divergent terms of this type are present in all QCD calculations.

M. Cobal, PIF 2003

RENORMALIZATION:

“Bare” mass or charge are replaced by physical values e,m as determined from the experiment.A consequence of renormalization procedure:Coupling constants (such ) are not constants: depends on log of measurements energy scale

1/EM

EM(0

) =1/1

37.0

359895(6

1)

M. Cobal, PIF 2003

To have renormalisability:theory must be gauge invariant.

In electrostatics, the interaction energy which can be measured, depends only on changes in the static potential and not on its absolute magnitude invariant under arbitrary changes in the potential scale or gauge

In quantum mechanics, the phase of an electron wavefunction is arbitrary: can be changed at any local point in space-time and physics does not change

This local gauge invariance leads to the conservation of currents andof the electric charge.

GAUGE INVARIANCE:

M. Cobal, PIF 2003

• Conserved quantum numbers associated with c are colour charges in strong interactions they play similar role to the electric charge in em interactions.

• A quark can carry one of the three colours (red, blue, green). An anti-quark one of the three anti-colours

• All the observable particles are “white” (they do not carry colour)

• Quarks have to be confined within the hadrons since non-zero colour states are forbidden.

• 3 independent colour wavefunctions are represented by colour spinor

Hadrons: neutral mix of r,g,b colours

Anti-hadrons: neutral mix of r,g,b anti-colours

Mesons: neutral mix of colours and anti-colours

1

0

0

,

0

1

0

,

0

0

1

bgr

M. Cobal, PIF 2003

• These spinors are acted on by 8 independent “colour operators ” which are represented by a set of 3-dimensional matrices (analogues of Pauli matrices)

• Colour charges Ic3 and Yc are eigenvalues of corresponding operators

• Colour hypercharge Yc and colour isospin Ic3 charge are additive quantum numbers, having opposite sign for quark and antiquark.

Confinement condition for the total colour charges of a hadron: Ic3 = Yc = 0

M. Cobal, PIF 2003

Colour• Experimental data confirm predictions based on the assumption of symmetric wave functions Problem: ++ is made out of 3 quarks, and has spin J=3/2 (= 3 quarks of s= ½ in same state?) This is forbidden by Fermi statistics (Pauli principle)! Solution: there is a new internal degree of freedom (colour) which differentiate the quarks: ++=urugub

• This means that apart of space and spin degrees of freedom, quarks have yet another attribute • In 1964-65, Greenberg and Nambu proposed the new property – the colour – with 3 possible states, and associated with the corresponding wavefunction

Cx )(

M. Cobal, PIF 2003

Strong Interactions

• Take place between quarks which make up the hadrons

• Magnitude of coupling can be estimated from decay probability (or width ) of unstable baryons.

• Consider: =36 MeV, = 10-23 s

If we compare this with the em decay: , = 10-

19 s We get for the coupling of the strong charge

opK 13850

11920

10010

10 2

1

23

19

s

14

2

ss

g

M. Cobal, PIF 2003

QCD, Jets and gluons

• Quantum Chromodynamics (QCD): theory of strong interactions

Interactions are carried out by a massless spin-1 particle- gauge boson

In quantum electrodynamics (QED) gauge bosons are photons, in QCD, gluons

Gauge bosons couple to conserved charges: photons in QED- to conserved charges, and gluons in QCD – to colour charges.

Gluons do not have electric charge and couple to colour charges strong nteractions are flavour-independent

M. Cobal, PIF 2003

• Colour simmetry is supposed to be exact quark-quark force is independent of the colours involved

• Colour quantum numbers of the gluon are: r,g,b

• Gluons carry 2 colours, and there are 8 possible combinations colour-anticolour

• Gluons can have self-interactions

qr

qb

grb Neutral combinations:rgb, rgb, rr, …

M. Cobal, PIF 2003

- Gluons can couple to other gluons

- Bound colourless states of gluons are called glueballs (not detected experimentally yet).

-Gluons are massless long-range interaction

Principle of asymptotic freedom

-At short distances, strong interactions are sufficiently weak (lowest order diagrams) quarks and gluons are essentially free particles

-At large distances, higher-order diagramsdominate interaction is very strong

M. Cobal, PIF 2003

• For violent collisions (high q2), as < 1 and single gluon exchange is a good approximation.

• At low q2 (= larger distances) the coupling becomes large and the theory is not calculable. This large-distance behavior is linked with confinement of quarks and gluons inside hadrons.

• Potential between two quarks often taken as:

• Attempts to free a quark from a hadron results in production of new mesons. In the limit of high quark energies the confining potential is responsible for the production of the so-called “jets

krr

V ss

3

4

Single gluon exchange Confinment

M. Cobal, PIF 2003

Running of s

The s constant is the QCD analogue of em and is a measure of the interaction strenght.

However s is a “running constant”, increases with increase of r, becoming divergent at very big distances.

- At large distances, quarks are subject to the “confining potential” which grows with r:

V(r) ~ r (r > 1 fm)

-Short distance interactions are associated with the large momentum transfer

Lorentz-invariant momentum transfer Q is defined as:222qEqQ

)( 1 rOq

M. Cobal, PIF 2003

-In the leading order of QCD, s is given by:

Nf = number of allowed quark flavours ~ 0.2 GeV is the QCD scale parameter which has to be defined experimentally

)/ln()233(

1222

QN f

s

M. Cobal, PIF 2003

QCD jets in e+e- collisions

- A clean laboratory to study QCD:

-At energies between 15 GeV and 40 GeV, e+e- annihilation produces a photon which converts into a quark-antiquark pair

- Quark and antiquark fragment into observable hadrons

- Since quark and antiquark momenta are equal and counterparallel, hadrons are produced in two opposite jets of equal energies

- Direction of a jet reflects direction of a corresponding quarks.

hadronsee *

M. Cobal, PIF 2003

e-

e+ q

EM Sq

Colliding e+ and e- can give 2 quarks in final state. Then, theyfragment in hadrons

2 collimated jets of hadrons travelling in Opposite direction and following the momentum vectors of the original quarks

M. Cobal, PIF 2003

*ee

Comparison of the process with the reaction

must show the same angular distribution both for muons and jets

where is the production angle with respect to the initial electron direction in CM frameFor a quark-antiquark pair:

Where the fractional charge of a quark eq is taken into account

and factor 3 arises from number of colours. If quarks have spin ½, angular distribution goes like (1+cos2); if they have spin 0, like (1-cos2)

)cos1(2

)(cos

22

2

Qee

d

d

)(cos

3)(cos

2

eed

deqqee

d

dq

M. Cobal, PIF 2003

Angular distribution of the quark jet in e+e- annihilation, compared with models

-Experimentally measured angular dependence is clearly proportional to (1+cos2) jets are aligned with spin-1/2 quarks

M. Cobal, PIF 2003

If a high momentum (hard) gluon is emitted by the quark or the anti-quark, it fragments to a jet, leading to a 3-jet events

A 3-jet event seen in a e+e- annihilation at the DELPHI experiment

M. Cobal, PIF 2003

-In 3-jet events it is difficult to understand which jet come from the quarks and which from the gluon

-Observed rate of 3-jet and 2-jet events can be used to determine value of s (probability for a quark to emit a gluon determined by s)

s= 0.15 0.03 for ECM = 30-40 GeV

Principal scheme of hadroproduction in e+e- hadronization begins at distances of 1 fm between partons.

M. Cobal, PIF 2003

The total cross section of e+e- hadrons is often expressed as:

Where:

Assuming that the main contribution comes from quark-antiquark two jet events:

And hence: R = 3q e2q

)(

)(

ee

hadronseeR

2

2

3

4

Qee

)(3

)()(

2

eee

qqeehadronsee

qq

q

M. Cobal, PIF 2003

R allows to check the number of colours in QCD and number of quark flavours allowed to be produced at a given Q

R(u,d,s)=2 R(u,d,s,c)=10/3 R(u,d,s,c,b)=11/3

If the radiation of hard gluons is taken into account, the extra factor proportional to s arises:

Measured R with theoretical predictions for five available flavours (u,d,s,c,b), using two different s calculations

))(

1(32

2

Q

eR s

qq

M. Cobal, PIF 2003

Elastic electron scattering- Beams of structureless leptons are a good tool for investigating properties of hadrons

- Elastic lepton hadron scattering can be used to measure sizes of hadrons

-Angular distribution of an electron momentum p scattered by a static electric charge e is given by the Rutherford formula,

)2

(sin4 42

22

p

m

d

d

R

M. Cobal, PIF 2003

-If the electric charge is not point-like, but is spread with a spherically symmetric density distribution e e(r), where (r) is normalized

Then the differential cross-section is replaced:

Where the electric form factor

- For q=0, GE(0) = 1 (low momentum transfer)- For q2, GE(q2) 0 (large mom. transfer)

Measurements of d/d determine the form-factor and hence the charge distribution of the proton. At high momentum transfers, the recoil energyof the proton is not negligible, and is replaced by the Lorentz-invariant Q

At high Q, static interpretation of charge distribution breaks down.

1)( 3 xdr

)( 22 qGd

d

d

dE

R

xderqG xqiE

32 )()(

q

222 )'()'( EEppQ

M. Cobal, PIF 2003

Inelastic lepton scattering-Historically, was first to give evidence of quark constituents of the proton. In what follows only one-photon exchange is considered

-The exchanged photon acts as a probe of the proton structure

-The momentum transfer corresponds to the photon wavelenght which must be small enough to probe a proton big momentum transfer is needed.

-When a photon resolves a quark within a proton,the total lepton-proton scattering is a two-step process

M. Cobal, PIF 2003

Deep inelastic lepton-proton scattering

1)First step: elastic scattering of the lepton from one of the quarks: l-+q l-+q

2) Fragmentationof the recoil quark and the proton remnant into observable hadrons

Angular distributions of recoil leptons reflect properties of quarks from which they are scattered

For further studies, some new variables have to be defined:

M. Cobal, PIF 2003

-Lorentz-invariant generalization for the transferred energy , defined by:

where W is the invariant mass of the final hadron state; in the rest frame of the proton = E-E’

-Dimensionless scaling variable x:

- For Q >> Mp and a very large proton momentum P >>Mp, x is fraction of the proton momentum carried by the struck quark; 0x 1

-Energy E’ and angle of scattered lepton are independent variables, describing inelastic process.

This is a generalization of the elastic scattering formula

2222 pp MQWM

pMQ

x2

2

),(2

sin),(2

[cos1

)2

(sin4''2

12

222

22

42

2

QxFxM

QQxF

EddE

d

p

M. Cobal, PIF 2003

-Structure functions F1 and F2 parameterize the interaction at the quark-photon vertex (just like G1 and G2 parameterize the elastic scattering)

-Bjorken scaling: F1,2(x,Q2) ~F1,2(x)

- At Q >> Mp, structure functions are approx. independent on Q2

-If all particle masses, energies and momenta are multiplied by a scale factor, structure functions at any given x remain unchanged

F2

M. Cobal, PIF 2003

-SLAC data from ’69 were first evidence of quarks

-Scaling is observed at very small and very big x Approximate scaling behaviour can be explained if protons are considered as composite objects

- The trivial parton model: proton consists of some partons. Interaction between partons are not taken into account

- The parton model can be valid if the target p has a sufficiently big momentum, so that: z = x

- Measured cross section at any given x is prop. to the probability of finding a parton with a fraction z = x of the proton momentum,

If there are several partons; F2(x,Q2) = ae2axfa(x)

With fa(x) dx = probability of finding a parton a with fractional momentum between x and x+dx

M. Cobal, PIF 2003

Parton distributions fa(x) are not known theoretically F2(x) has to be measured experimentally

-Predictions for F1 depend on the spin of a parton:

F1(x,Q2) = 0 (spin-0) 2x F2(x,Q2) = F2(x,Q2) (spin-1/2)

The expressionfor spin-1/2 is called Callan-Gross relation and is very well confirmed by experiments partons are quarks

Comparing proton and neutron structure functions and those from n scattering, e2

a can be evaluated: it appears to be consistent with square charges of quarks.