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Gluons • Color Transformations • Quark Confinement • Scaling Violations • the Ratio R • Deep Inelastic Scattering • Structure of the Proton 2 Introductionqcd

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Gluons • Color Transformations • Quark Confinement • Scaling Violations • the Ratio R • Deep Inelastic Scattering • Structure of the Proton 2 Introductionqcd Quantum Chromo Dynamics • Evidence for Fractional Charges • Color • Gluons • Color Transformations • Quark Confinement • Scaling Violations • The Ratio R • Deep Inelastic Scattering • Structure of the Proton 2 IntroductionQCD QUANTUM ELECTRODYNAMICS: is the quantum theory of the electromagnetic interaction. ★ mediated by massless photons ★ photon couples to electric charge, ★ Strength of interaction : . QUANTUM CHROMO-DYNAMICS: is the quantum theory of the strong interaction. ★ mediated by massless gluons, i.e. propagator ★ gluon couples to “strong” charge ★ Only quarks have non-zero “strong” charge, therefore only quarks feel strong interaction Basic QCD interaction looks like a stronger version of QED, QED q QCD q Q √α √αS q q q 2 2 α = e /4π ~ 1/137 γ αS = gS /4π ~ 1 g ( subscript em is sometimes used to distinguish the of electromagnetism from ). Dr M.A. Thomson Lent 2004 2 QCD QUANTUM ELECTRODYNAMICS: is the quantum theory of the electromagnetic interaction. ★ mediated by massless photons ★ photon couples to electric charge, ★ Strength of interaction : . QUANTUM CHROMO-DYNAMICS: is the quantum theory of the strong interaction. ★ mediated by massless gluons, i.e. propagator ★ gluon couples to “strong” charge ★ Only quarks have non-zero “strong” charge, therefore onlyIntroductionquarks feel strong interaction Basic QCD interaction looks like a stronger version of QED, QED q QCD q Q √α √αS q q q 2 2 α = e /4π ~ 1/137 γ αS = gS /4π ~ 1 g 3 ( subscript em isCOLOUR sometimes used to distinguish the of electromagnetism from ). Dr M.A.In Thomson QED: Lent 2004 ★ Charge of QED is electric charge. ★ Electric charge - conserved quantum number. In QCD: ★ Charge of QCD is called “COLOUR” ★ COLOUR is a conserved quantum number with 3 VALUES labelled “red”, “green” and “blue” Quarks carry “COLOUR” Anti-quarks carry “ANTI-COLOUR” Leptons, , , DO NOT carry colour, i.e. “have colour charge zero” DO NOT participate in STRONG interaction. Note: Colour is just a label for states in a non-examinable SU(3) representation Dr M.A. Thomson Lent 2004 Evidence for Fractional Charge Production of lepton pairs in Pion-CarBon scattering: Electromagnetic annihilation of a quark and an anti-quark in pion- nucleon collisions: a virtual photon converts into a muon pair qq annihilate → photon → μ– + μ+ The cross section σ ~ charge of q × charge of q Since the q and q have opposite charges: σ ~ [charge of q]2 Evidence for Fractional Charge Use an 'Isoscalar' target – same numBer of u and d quarks 12 6C has 6 protons (uud) and 6 neutrons (udd) → 18 u + 18 d For π– (ud) → uu annihilation to μ– μ+ For π+ (ud) → dd annihilation to μ– μ+ − + − 2 σ(π C → µ µ + anything) ∝18 Qu =18 × 4 /9 + + − 2 σ(π C → µ µ + anything) ∝18 Qd =18 ×1/9 € € Good agreement With eXperiment Also results from electron and neutrino scattering from nuclei Quark Colour Why do We need coloured quarks? P + ++ In the Baryon decuplet J = 3/2 , the Δ is u↑ u↑ u↑ With angular momentum ℓ = 0 i.e. 3 fermions in the same state → violates Pauli EXclusion Principle Wavefunction symmetric in space, spin and flavour not alloWed for fermions Introduce a neW Degree of Freedom → Colour 3 'colours' red r, Blue B, green g NoW the 3 fermions are in different states: u↑ (red) u↑ (blue) u↑ (green) Not real colour – just a neW quantum numBer that distinguishes them! Colour The corner states of the decuplet (uuu), (ddd) and (sss) are symmetric under flavor- interchange (therefore, also the other states are assumed to Behave the same). The decuplet members are indeed the 10 quark combinations (out of 27) symmetric under space, spin and flavor eXchange We also assume that each Baryon is a properly normalized comBination of permuted 1 states, e.g. : (ddu) = (ddu + udd + dud) 3 1 (dus) = (dsu + uds + sud + sdu + dus + usd) 6 Since the decuplet members are spin 3/2 Baryons of the loWest mass, We assume the three quarks are in the (spatially symmetric) ground state (l=0). This implies that the three spins are parallel: is the Pauli principle violated? Need another degree of freedom (quantum numBer) that can Be different for the three quarks: color(red, blue, green) such that the hadron net color is 0. Indeed, eXperimentally one only detects protons and not red or green protons…. The three colors are the source of the strong-force charge. The strong force is independent of the quark color, i.e. the SU(3)color symmetry is eXact. Quark Colour HoWever – they Work like colours on a computer red green Blue Red 1 0 0 Green 0 1 0 Blue 0 0 1 Cyan 0 1 1 = anti-red r magenta 1 0 1 = anti-green g YelloW 1 1 0 = anti-blue B White 1 1 1 Black 0 0 0 Baryons consist of r + g + b = 1 1 1 = White – colourless Mesons consist of r + r = 1 0 0 + 0 1 1 = 1 1 1 = White – colourless (or g + g or b + B) Gluons Expect 9 comBinations (r, g, B) × (r, g, B) These are: r B Colourless and symmetric r g under r → b → g colour g r change. Doesn't interact g B b g b r miX r r g g 8 interacting gluons b B Colour Charge The 'Color Charge' is the Strong Interaction equivalent of Electric Charge in Electromagnetism Quantum Chromodynamics (QCD) is the quantum field theory of the strong 'color' interaction BetWeen quarks The strong force is carried By 8 massless vector gluons (JP = 1– ). They are equivalent to qq pairs i.e. rB, gr, .. etc .. This is an non-Abelian theory Where the force carriers have the charge and can interact With each other – unlike electromagnetism Where the photons carry NO charge and don't interact With each other QCD Colour Transformations QCD is a gauge theory. Gauge means scale or aBsolute potential. Consider a proton made up of one red, one green and one Blue quark. Do a gloBal colour transformation red ⇌ green everyWhere Still 'colourless' – nothing has changed QCD ColourLocal Colour TransformationsTransformations NoW do a local colour transformation red ⇌ green at one quark only Can Be made colourless if A emits an rg gluon Which is aBsorBed at C. When the gluon is aBsorBed at C the green and anti-green No longer colourless annihilate to make C red. g + rg → r Colourless again Local colour transformation → eXchange of gluons → force Particle Physics Quantum Chromodynamics The Local Gauge The Local Gauge PrinciplePrinciple , All the interactions between fermions and spin-1 bosons in the SM are specified by the principle of LOCAL GAUGE INVARIANCE , To arrive at QED, require physics to be invariant under the local phase transformation of particle wave-functions , Note that the change of phase depends on the space-time coordinate: •Under this transformation the Dirac Equation transforms as •To make “physics”, i.e. the Dirac equation, invariant under this local phase transformation FORCED to introduce a massless gauge boson, . + The Dirac equation has to be modified to include this new field: •The modified Dirac equation is invariant under local phase transformations if: Gauge Invariance The Local Gauge Principle , For physics to remain unchanged – must have GAUGE INVARIANCE of the new field, i.e. physical predictions unchanged for ,Hence the principle of invariance under local phase transformations completely specifies the interaction between a fermion and the gauge boson (i.e. photon): interaction vertex: QED ! , The local phase transformation of QED is a unitary U(1) transformation i.e. with Now extend this idea… From QED to QCD , Suppose there is another fundamental symmetry of the universe, say “invariance under SU(3) local phase transformations” • i.e. require invariance under where are the eight 3x3 Gell-Mann matrices introduced in handout 7 are 8 functions taking different values at each point in space-time 8 spin-1 gauge bosons wave function is now a vector in COLOUR SPACE QCD ! , QCD is fully specified by require invariance under SU(3) local phase transformations Corresponds to rotating states in colour space about an axis whose direction is different at every space-time point interaction vertex: , Predicts 8 massless gauge bosons – the gluons (one for each ) , Also predicts exact form for interactions between gluons, i.e. the 3 and 4 gluon vertices – the details are beyond the level of this course , For physics to remain unchanged – must have GAUGE INVARIANCE of the new field, i.e. physical predictions unchanged for ,Hence the principle of invariance under local phase transformations completely specifies the interaction between a fermion and the gauge boson (i.e. photon): interaction vertex: QED ! , The local phase transformation of QED is a unitary U(1) transformation i.e. with Now extend this idea… From QED to QCDFrom QED to QCD , Suppose there is another fundamental symmetry of the universe, say “invariance under SU(3) local phase transformations” • i.e. require invariance under where are the eight 3x3 Gell-Mann matrices introduced in handout 7 are 8 functions taking different values at each point in space-time 8 spin-1 gauge bosons wave function is now a vector in COLOUR SPACE QCD ! , QCD is fully specified by require invariance under SU(3) local phase transformations Corresponds to rotating states in colour space about an axis whose direction is different at every space-time point interaction vertex: , Predicts 8 massless gauge bosons – the gluons (one for each ) , Also predicts exact form for interactions between gluons, i.e. the 3 and 4 gluon vertices – the details are beyond the level of this course ColourColourin QCD in QCD ,The theory of the strong interaction, Quantum Chromodynamics (QCD), is very similar to QED but with 3 conserved “colour” charges In QED: • the electron carries one unit of charge • the anti-electron carries one unit of anti-charge • the force is mediated by a massless “gauge boson”–the photon In QCD: • quarks carry colour charge: • anti-quarks carry anti-charge: • The force is mediated by massless gluons , In QCD, the strong interaction is invariant under rotations in colour space i.e.
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