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 J P = 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 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- e+e+ ++ -- Other examples at higher orders:

M. Cobal, PIF 2003 Renormalization and gauge invariance QED: 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 term As a consequence the theoretically calculated “bare” mass or charge (m 0, e 0 ) becomes infinite 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/ (61)

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 and of 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

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 I c 3 and Y c are eigenvalues of corresponding operators Colour hypercharge Y c and colour isospin I c 3 charge are additive quantum numbers, having opposite sign for quark and antiquark. Confinement condition for the total colour charges of a hadron: I c 3 = Y c = 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:  ++ =u r u g u b This means that apart of space and spin degrees of freedom, quarks have yet another attribute In , Greenberg and Nambu proposed the new property – the colour – with 3 possible states, and associated with the corresponding wavefunction 

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,  = s If we compare this with the em decay:,  = s We get for the coupling of the strong charge

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 qrqr qbqb g rb Neutral combinations: rgb, rgb, rr, …

M. Cobal, PIF 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 q 2 ), a s < 1 and single gluon exchange is a good approximation. At low q 2 (= 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

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:

M. Cobal, PIF In the leading order of QCD,  s is given by: N f = number of allowed quark flavours  ~ 0.2 GeV is the QCD scale parameter which has to be defined experimentally

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.

M. Cobal, PIF 2003 e-e- e+e+ q q Colliding e + and e - can give 2 quarks in final state. Then, they fragment in hadrons 2 collimated jets of hadrons travelling in Opposite direction and following the momentum vectors of the original quarks

M. Cobal, PIF 2003 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 frame For a quark-antiquark pair: Where the fractional charge of a quark e q is taken into account and factor 3 arises from number of colours. If quarks have spin ½, angular distribution goes like (1+cos 2  ); if they have spin 0, like (1- cos 2  )

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+cos 2  )  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 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 E CM = 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 = 3  q e 2 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

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,

M. Cobal, PIF 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, G E (0) = 1 (low momentum transfer) - For q 2 , G E (q 2 )  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.

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 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 >> M p and a very large proton momentum P >>M p, x is fraction of the proton momentum carried by the struck quark; 0  x  1 -Energy E’ and angle  of scattered lepton are independent variables, describing inelastic process. This is a generalization of the elastic scattering formula

M. Cobal, PIF Structure functions F1 and F2 parameterize the interaction at the quark-photon vertex (just like G1 and G2 parameterize the elastic scattering) -Bjorken scaling: F 1,2 (x,Q 2 ) ~F 1,2 (x) - At Q >> M p, structure functions are approx. independent on Q 2 -If all particle masses, energies and momenta are multiplied by a scale factor, structure functions at any given x remain unchanged F2F2

M. Cobal, PIF 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; F 2 (x,Q 2 ) =  a e 2 a xf a (x) With f a (x) dx = probability of finding a parton a with fractional momentum between x and x+dx

M. Cobal, PIF 2003  Parton distributions f a (x) are not known theoretically  F 2 (x) has to be measured experimentally -Predictions for F 1 depend on the spin of a parton: F 1 (x,Q 2 ) = 0 (spin-0) 2x F 2 (x,Q 2 ) = F 2 (x,Q 2 ) (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, e 2 a can be evaluated: it appears to be consistent with square charges of quarks.