Electromagnetic Interactions

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

e+ - e- + Other examples at higher orders:

renormalisability Gauge invariance e Renormalization and gauge invariance QED: quantum field theory to compute cross sections for em processes a a 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 Divergent terms of this type are present in all QCD calculations.

EM(0) =1/137.0359895(61) 1/EM 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 a) are not constants: depends on log of measurements energy scale

GAUGE INVARIANCE: 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 and of the electric charge.

Conserved quantum numbers associated with cc 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

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

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 c

Strong Interactions • Take place between quarks which make up the hadrons • Magnitude of coupling can be estimated from decay probability (or • width G) of unstable baryons. • Consider: • G=36 MeV, t = 10-23 s • If we compare this with the em decay:, t = 10-19 s • We get for the coupling of the strong charge

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

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 qb grb Neutral combinations: rgb, rgb, rr, … qr

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

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 Single gluon exchange Confinment

Running of as • The asconstant is the QCD analogue of aem and is a measure of the • interaction strenght. • However as 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) ~ l r (r > 1 fm) • Short distance interactions are associated with the large • momentum transfer • Lorentz-invariant momentum transfer Q is defined as:

In the leading order of QCD, as is given by: • Nf = number of allowed quark flavours • L ~ 0.2 GeV is the QCD scale parameter which • has to be defined experimentally

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.

q Colliding e+ and e- can give 2 quarks in final state. Then, they fragment in hadrons e+ q e- 2 collimated jets of hadrons travelling in Opposite direction and following the momentum vectors of the original quarks

Comparison of the process with the reaction must show the same angular distribution both for muons and jets where q is the production angle with respect to the initial electron direction in CM frame For 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+cos2q); if they have spin 0, like (1-cos2q)

Angular distribution of the quark jet in e+e- annihilation, compared • with models • Experimentally measured angular dependence is clearly proportional • to (1+cos2q) jets are aligned with spin-1/2 quarks

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

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 as (probability for a quark to emit a gluon determined by as) • as= 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.

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 = 3Sq e2q

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

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,

If the electric charge is not point-like, but is spread with a • spherically symmetric density distribution e er(r), where r(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 ds/dW 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.

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

Deep inelastic • lepton-proton scattering • 1)First step: elastic scattering of the lepton from one of the quarks: • l-+q l-+q • 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:

Lorentz-invariant generalization for the transferred energy n, • defined by: • where W is the invariant mass of the final hadron state; in the rest • frame of the proton n = 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; 0x 1 • Energy E’ and angle q of scattered lepton are independent variables, • describing inelastic process. • This is a generalization of the elastic scattering formula

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

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) = Sae2axfa(x) • With fa(x) dx = probability of finding a parton a with fractional • momentum between x and x+dx

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, e2a can be evaluated: it appears to be consistent with square charges of quarks.

Electromagnetic Interactions