Quantum Chromo Dynamics

Transcription

1 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 Introduction QUANTUM ELECTRODYNAMICS: is the uantum theory of the electromagnetic interaction. mediated by massless photons photon couples to electric charge, Strength of interaction :. QUANTUM CHROMO-DYNAMICS: is the uantum theory of the strong interaction. mediated by massless gluons, i.e. propagator gluon couples to strong charge Only uarks have non-zero strong charge, therefore only uarks feel strong interaction

3 Introduction Basic QCD interaction looks like a stronger version of QED, QED Q α α = e 2 /4π ~ 1/137 γ QCD α S α S = g S 2/4π ~ 1 g ( subscript em is sometimes used to distinguish the of electromagnetism from ). In QED: Charge of QED is electric charge. Electric charge - conserved uantum number. In QCD: Charge of QCD is called COLOUR COLOUR is a conserved uantum number with 3 VALUES labelled red, green and blue

4 Evidence for Fractional Charge Production of lepton pairs in Pion-Carbon scattering: Electromagnetic annihilation of a uark and an anti-uark in pionnucleon collisions: a virtual photon converts into a muon pair annihilate photon μ + μ + The cross section σ ~ charge of charge of Since the and have opposite charges: σ ~ [charge of ] 2

5 Evidence for Fractional Charge Use an 'Isoscalar' target same number of u and d uarks 12 C has 6 protons (uud) and 6 neutrons (udd) 6 18 u + 18 d For π (ud) uu annihilation to μ μ + For π + (ud) dd annihilation to μ μ + σ(π C µ + µ + anything) 18 Q u 2 =18 4 /9 σ(π + C µ + µ + anything) 18 Q d 2 =18 1/9 Good agreement with experiment Also results from electron and neutrino scattering from nuclei

6 Quark Colour Why do we need coloured uarks? In the baryon decuplet J P = 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 uantum number that distinguishes them!

7 The corner states of the decuplet (uuu), (ddd) and (sss) are symmetric under flavorinterchange (therefore, also the other states are assumed to behave the same). The decuplet members are indeed the 10 uark combinations (out of 27) symmetric under space, spin and flavor exchange We also assume that each baryon is a properly normalized combination of permuted states, e.g. : Colour (ddu) = 1 (ddu + udd + dud) 3 (dus) = 1 6 (dsu + uds + sud + sdu + dus + usd) Since the decuplet members are spin 3/2 baryons of the lowest mass, we assume the three uarks 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 (uantum number) that can be different for the three uarks: 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 uark color, i.e. the SU(3) color symmetry is exact.

8 Quark Colour However they work like colours on a computer red green blue Red Green Blue Cyan = anti-red r Magenta = anti-green g Yellow = anti-blue b White Black Baryons consist of r + g + b = = white colourless Mesons consist of r + r = = = white colourless (or g + g or b + b)

9 Gluons These are: Expect 9 combinations (r, g, b) (r, g, b) r b r g g r g b b g b r r r g g b b mix Colourless and symmetric under r b g colour change. Doesn't interact 8 interacting gluons

10 Colour Charge The 'Color Charge' is the Strong Interaction euivalent of Electric Charge in Electromagnetism Quantum Chromodynamics (QCD) is the uantum field theory of the strong 'color' interaction between uarks The strong force is carried by 8 massless vector gluons (J P = 1 ). They are euivalent to 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

11 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 uark. Do a global colour transformation red green everywhere Still 'colourless' nothing has changed

12 QCD Colour Transformations Local Colour Transformations Now do a local colour transformation red green at one uark 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 annihilate to make C red. No longer colourless g + rg r Colourless again Local colour transformation exchange of gluons force

13 The Local Gauge Principle The Local Gauge Principle 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, reuire 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 Euation transforms as To make physics, i.e. the Dirac euation, invariant under this local phase transformation FORCED to introduce a massless gauge boson,. + The Dirac euation has to be modified to include this new field: The modified Dirac euation is invariant under local phase transformations if: Gauge Invariance

14 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

15 From QED to QCD Suppose there is another fundamental symmetry of the universe, say invariance under SU(3) local phase transformations i.e. reuire 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 reuire 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

16 Colour in QCD Colour in QCD The theory of the strong interaction, Quantum Chromodynamics (QCD), is very similar to QED but with 3 conserved colour charges In QED: In QCD: 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 uarks carry colour charge: anti-uarks carry anti-charge: The force is mediated by massless gluons In QCD, the strong interaction is invariant under rotations in colour space i.e. the same for all three colours SU(3) colour symmetry This is an exact symmetry, unlike the approximate uds flavour symmetry discussed previously.

17 Colour in QCD Represent SU(3) colour states by: Colour states can be labelled by two uantum numbers: colour isospin colour hypercharge Exactly analogous to labelling u,d,s flavour states by Each uark (anti-uark) can have the following colour uantum numbers: and uarks anti-uarks

18 Colour Confinement It is believed (although not yet proven) that all observed free particles are colourless i.e. never observe a free uark (which would carry colour charge) conseuently uarks are always found in bound states colourless hadrons Colour Confinement Hypothesis: only colour singlet states can exist as free particles All hadrons must be colourless i.e. colour singlets To construct colour wave-functions for hadrons can apply results for SU(3) flavour symmetry to SU(3) colour with replacement g r just as for uds flavour symmetry can define colour ladder operators b

19 Colour Singlets It is important to understand what is meant by a singlet state Consider spin states obtained from two spin 1/2 particles. Four spin combinations: Gives four eigenstates of spin-1 triplet spin-0 singlet The singlet state is spinless : it has zero angular momentum, is invariant under SU(2) spin transformations and spin ladder operators yield zero In the same way COLOUR SINGLETS are colourless combinations: they have zero colour uantum numbers invariant under SU(3) colour transformations ladder operators all yield zero NOT sufficient to have : does not mean that state is a singlet

20 Meson Colour Wave-function Wavefunction Consider colour wave-functions for The combination of colour with anti-colour is mathematically identical to construction of meson wave-functions with uds flavour symmetry Coloured octet and a colourless singlet Colour confinement implies that hadrons only exist in colour singlet states so the colour wave-function for mesons is: Can we have a state? i.e. by adding a uark to the above octet can we form a state with. The answer is clear no. bound states do not exist in nature.

21 Baryon Colour Wavefunction Do bound states exist? This is euivalent to asking whether it possible to form a colour singlet from two colour triplets? Following the discussion of construction of baryon wave-functions in SU(3) flavour symmetry obtain No colour singlet state Colour confinement bound states of do not exist BUT combination of three uarks (three colour triplets) gives a colour singlet state (pages )

22 Baryon Color Wavefunction The singlet colour wave-function is: Check this is a colour singlet It has : a necessary but not sufficient condition Apply ladder operators, e.g. (recall ) Similarly Colourless singlet - therefore bound states exist! Anti-symmetric colour wave-function Allowed Hadrons i.e. the possible colour singlet states Mesons and Baryons Exotic states, e.g. pentauarks To date all confirmed hadrons are either mesons or baryons. However, some recent evidence for pentauark states (LHCb experiment).

23 Gluons In QCD uarks interact by exchanging virtual massless gluons, e.g. b r b r b rb br r r b r Gluons carry colour and anti-colour, e.g. b r r r b r b br rb rr Gluon colour wave-functions (colour + anti-colour) are the same as those obtained for mesons (also colour + anti-colour) OCTET + COLOURLESS SINGLET

24 Gluons So we might expect 9 physical gluons: OCTET: SINGLET: BUT, colour confinement hypothesis: only colour singlet states can exist as free particles Colour singlet gluon would be unconfined. It would behave like a strongly interacting photon infinite range Strong force. Empirically, the strong force is short range and therefore know that the physical gluons are confined. The colour singlet state does not exist in nature! NOTE: this is not entirely ad hoc. In the context of gauge field theory (see minor option) the strong interaction arises from a fundamental SU(3) symmetry. The gluons arise from the generators of the symmetry group (the Gell-Mann matrices). There are 8 such matrices 8 gluons. Had nature chosen a U(3) symmetry, would have 9 gluons, the additional gluon would be the colour singlet state and QCD would be an unconfined long-range force. NOTE: the gauge symmetry determines the exact nature of the interaction FEYNMAN RULES

25 Self Interaction Gluons that carry colour charges can interact with each other (Non-Abelian) 16 gluons 8 gluons 4 gluons 'Strength' of the colour charge appears to decrease as distance decreases (energy increases)

26 Gluon-Gluon Interactions Gluon-Gluon Interactions In QED the photon does not carry the charge of the EM interaction (photons are electrically neutral) In contrast, in QCD the gluons do carry colour charge Gluon Self-Interactions Two new vertices (no QED analogues) triple-gluon vertex uartic-gluon vertex In addition to uark-uark scattering, therefore can have gluon-gluon scattering e.g. possible way of arranging the colour flow

27 Free Quarks? Many searches for 'free' uarks not bound into hadrons: Search for stable fractional charged particles 'exotic' atoms - Moon rock, deep sea sludge,... Search for fractional charges in Millikan oil drop type experiments Look in accelerators and cosmic rays for particles with 1/9 or 4/9 ionisation of normal particles No evidence for isolated free fractionally charged uarks Why?

28 Quark Confinement The reason there are no free uarks, and the only stable configurations are and (and possible gg 'glueball' states) is due to the fact that α S decreases with increasing energy and increases with increasing distance At short distance, inside a proton, α S is small and the uarks are 'free' Try and pull one out and α S increases and the force tries to keep it in

29 Gluon Self-Interaction and Confinement Gluon self-interactions and Confinement Gluon self-interactions are believed to give rise to colour confinement Qualitative picture: Compare QED with QCD In QCD gluon self-interactions sueeze lines of force into a flux tube What happens when try to separate two coloured objects e.g. e + e - Form a flux tube of interacting gluons of approximately constant energy density Reuire infinite energy to separate coloured objects to infinity Coloured uarks and gluons are always confined within colourless states In this way QCD provides a plausible explanation of confinement but not yet proven (although there has been recent progress with Lattice QCD)

30 Hadronization and Jets Hadronisation and Jets Consider a uark and anti-uark produced in electron positron annihilation i) Initially Quarks separate at high velocity ii) Colour flux tube forms between uarks iii) Energy stored in the flux tube sufficient to produce pairs iv) Process continues until uarks pair up into jets of colourless hadrons This process is called hadronisation. It is not (yet) calculable. The main conseuence is that at collider experiments uarks and gluons observed as jets of particles e + e

31 Hadronization In high energy interactions you don't see the produced uarks but 'jets' of hadrons Analogy: Gluons are like bits of elastic and uarks are the ends There are no free ends If you pull hard enough you get two bits of elastic but still no free ends

32 Quark Gluon Interaction The Quark Gluon Interaction Representing the colour part of the fermion wave-functions by: Particle wave-functions The QCD g vertex is written: Only difference w.r.t. QED is the insertion of the 3x3 SU(3) Gell-Mann matrices Gluon a Isolating the colour part: Hence the fundamental uark - gluon QCD interaction can be written colour i! j

33 External Lines spin 1/2 spin 1 Feynman Rules of QCD Feynman Rules for QCD incoming uark outgoing uark incoming anti-uark outgoing anti-uark incoming gluon outgoing gluon Internal Lines (propagators) spin 1 gluon Vertex Factors spin 1/2 uark a, b = 1,2,,8 are gluon colour indices i, j = 1,2,3 are uark colours, a = 1,2,..8 are the Gell-Mann SU(3) matrices + 3 gluon and 4 gluon interaction vertices Matrix Element -im = product of all factors

34 Quark Quark Scattering Consider QCD scattering of an up and a down uark u d u d The incoming and out-going uark colours are labelled by In terms of colour this scattering is The 8 different gluons are accounted for by the colour indices NOTE: the -function in the propagator ensures a = b, i.e. the gluon emitted at a is the same as that absorbed at b Applying the Feynman rules: where summation over a and b (and and ) is implied. Summing over a and b using the -function gives: Sum over all 8 gluons (repeated indices)

35 QCD vs QED QCD vs QED QED e e QCD u u QCD Matrix Element = QED Matrix Element with: or euivalently d d + QCD Matrix Element includes an additional colour factor

36 Colour Factors QCD colour factors reflect the gluon states that are involved Gluons: Configurations involving a single colour r r Only matrices with non-zero entries in 11 position are involved r r Similarly find

37 r r Colour Factors Other configurations where uarks don t change colour e.g. Only matrices with non-zero entries in 11 and 33 position are involved b b Similarly Configurations where uarks swap colours r g e.g. Only matrices with non-zero entries in 12 and 21 position are involved g r Gluons Configurations involving 3 colours r b b g e.g. Only matrices with non-zero entries in the 13 and 32 position But none of the matrices have non-zero entries in the 13 and 32 positions. Hence the colour factor is zero colour is conserved

38 Colour Factors Recall the colour part of wave-function: The QCD g vertex was written: Now consider the anti-uark vertex The QCD g vertex is: Note that the incoming anti-particle now enters on the LHS of the expression For which the colour part is i.e indices ij are swapped with respect to the uark case Hence c.f. the uark - gluon QCD interaction

39 Colour Factors Finally we can consider the uark anti-uark annihilation QCD vertex: with Conseuently the colour factors for the different diagrams are: e.g. Colour index of adjoint spinor comes first

40 Quark-Quark Scattering Quark-Quark Scattering Consider the process which can occur in the high energy proton-proton scattering jet There are nine possible colour configurations of the colliding uarks which are all eually likely. Need to determine the average matrix element which is the sum over all possible colours divided by the number of possible initial colour states p jet d u d u p The colour average matrix element contains the average colour factor For rr rr,.. rb rb,.. rb br,..

41 Quark-Quark Scattering Previously derived the Lorentz Invariant cross section for e e elastic scattering in the ultra-relativistic limit. QED For ud ud in QCD replace and multiply by QCD Never see colour, but enters through colour factors. Can tell QCD is SU(3) Here is the centre-of-mass energy of the uark-uark collision The calculation of hadron-hadron scattering is very involved, need to include parton structure functions and include all possible interactions e.g. two jet production in proton-antiproton collisions

42 QED Running Coupling Constants Running Coupling Constants bare charge of electron screened by virtual e + e pairs behaves like a polarizable dielectric In terms of Feynman diagrams: -Q +Q Same final state so add matrix element amplitudes: Giving an infinite series which can be summed and is euivalent to a single diagram with running coupling constant Note sign

43 Running Coupling Constants Might worry that coupling becomes infinite at i.e. at OPAL Collaboration, Eur. Phys. J. C33 (2004) But uantum gravity effects would come in way below this energy and it is highly unlikely that QED as is would be valid in this regime In QED, running coupling increases very slowly Atomic physics: High energy physics:

44 Running of α S Running of s QCD Similar to QED but also have gluon loops Fermion Loop Boson Loops Remembering adding amplitudes, so can get negative interference and the sum can be smaller than the original diagram alone Bosonic loops interfere negatively with = no. of colours = no. of uark flavours S decreases with Q2 Nobel Prize for Physics, 2004 (Gross, Politzer, Wilczek)

45 Colour Charge Strength s for strong α S ~ 0.2 at E = 30 GeV Compare with electromagnetic interaction α EM ~ 1/137 ~ Low energy Long distance High energy Short distance

46 Running of α S Measure S in many ways: jet rates DIS tau decays bottomonium decays α (µ) 0.2 QCD Prediction As predicted by QCD, S decreases with Q µ (GeV) At low : S is large, e.g. at find S ~ 1 Can t use perturbation theory! This is the reason why QCD calculations at low energies are so difficult, e.g. properties hadrons, hadronisation of uarks to jets, At high : S is rather small, e.g. at find S ~ 0.12 Asymptotic Freedom Can use perturbation theory and this is the reason that in DIS at high uarks behave as if they are uasi-free (i.e. only weakly bound within hadrons)

47 QCD in e + e - Annihilation e - e + Cleanest laboratory for studying perturbative QCD: No hadrons in initial state No remnants of initial proton(s) Quarks and gluons produced at short distances and emerge as jets

48 Two Jet Event e - e + Distribution follows dσ d(cosϑ) cm (1 + cos 2 ϑ) Just like e + e - -> µ + µ - -> uarks have spin 1/2

49 Proof that Gluons Exist e- e- e- e- Gluons fragment into jets similarly to uarks so expect 3 jet events

50 Proof that Gluons Exist 3 jet events observed at rate consistent with expectations

51 The Ratio R [ some threshold kinematics factors] Sum over all contributing uarks Charge of μ 2 As uarks come in 3 colours: The fact that the factor of 3 is necessary is very strong evidence in support of the colour hypothesis

52 The Ratio R Flavour : u d s c b t Charge 2 : (⅔) 2 (-⅓) 2 (-⅓) 2 (⅔) 2 (-⅓) 2 (⅔) 2 For ECM ~ 2 GeV have uu, dd, ss R = 3 (4/9 + 1/9 + 1/9) = 2 For ECM ~ 5 GeV have uu, dd, ss, cc R = 3 (4/9 + 1/9 + 1/9 + 4/9) = 3⅓ Need ~3 GeV to make cc Need ~10 GeV to make bb For ECM ~ 11 GeV have uu, dd, ss, cc, bb R = 3 (4/9 + 1/9 + 1/9 + 4/9 + 1/9) = 3⅔

53 The R Ratio e - e + PDG

54 The Ratio R u + d + s 2 u + d + s + c 3.3 u + d + s + c + b 3.7 colour factor of 3

55 u, d, s: R 0 = 3 ( u2 + d2 + s2 ) = 2 u, d, s, c: R 0 = 3 ( u2 + d2 + 2 s + c2 ) = 10/3 = 3.3 u, d, s, c, b: R 0 = 3 ( u2 + d2 + 2 s + 2 c + b2 ) = 11/3 = 3.7 u, d, s, c, b, t: R 0 = 3 ( u2 + d2 + 2 s + 2 c + 2 b + t2 ) = 5

56 Any more Quarks? We have 3 full generations. We only need one (u, d, e, ѵe) for everyday life So if 3 why not more? Since it is not understood why there are 3 generations there is no theoretical reason why there shouldn't be more (Periodic Table again?) However, experimental evidence now suggests only 3 In early 1990s LEP measured the width of the Z 0 to very high precision. The Z 0 can decay into any fermion pairs with a combined mass less than the Z 0 mass (~90 GeV/c 2 ) f = u, d, s, c, b, e, μ, τ, ѵe, ѵμ, ѵτ The more ways the Z 0 can decay, the shorter its lifetime τ and the larger its width Γ (Γ τ ~ ħ)

57 Any more Quarks? Generation u c t h... Z 0 tt since M Z 0 < 2M t d s b l... e μ τ χ... ѵe ѵμ ѵτ ν χ... The Z 0 can decay into all these The Z 0 would decay into any new neutrinos assuming they were massless (or light compared with M Z 0)

58 The Width of the Z 0 We know how much each uark/lepton pair contributes to Γ Z 0 Γ Z 0 = Γ uu + Γ dd + Γ ss + Γ cc + Γ bb + Γ e + e + Γ μ + μ + Γ τ + τ + N ѵ Γ ѵѵ Each of the Γ uu etc are known as partial widths A measurement of Γ Z 0 is euivalent to measuring the number of neutrino species N ѵ If there is one neutrino species per generation as now total number of generations Result from LEP: N ѵ = ± i.e. only 3 generations This is only true if the new neutrinos are light but they cannot be too heavy or they would contribute too much mass to the Universe Astrophysical constraint on neutrino masses

59 The Width of the Z 0 Data points

60 Deep Inelastic Scattering Shoot high energy electrons at protons Scattering angle θ related to momentum transfer = (k k') x is the fraction of the proton's momentum carried by the struck uark 0 x 1 The total energy of the hadrons from the struck uark is + xp Measure hadron energy + scattering angle θ deduce x

61 Scaling Violations x is a 'scaled' parameter i.e. x = p uark / p proton - so naively expect uark and antiuark distributions to depend on x and not energy But see 'scaling violations' at higher energies probe deeper into the proton and see more 'structure i.e. more of the proton's momentum appears to be carried by gluons and pairs and less by valence uarks valence uarks high energy valence uarks low energy Distribution depends on x and energy

62 Structure of the Proton Protons are very complicated and have a lot more in them than just two up uarks and a down uark. There are zillions of gluons, antiuarks, and uarks in a proton. The standard shorthand, the proton is made from two up uarks and one down uark, is really a statement that the proton has two more up uarks than up antiuarks, and one more down uark than down antiuarks. The proton consists of partons: three valence uarks (uud), an infinite sea of light uarkantiuark pairs (sea uarks), and gluons. The protons viewed at ever higher resolution appears more and more as field energy (soft glue).

63 Parton Distribution Functions Introduced by Feynman (1969) in the parton model, to explain Bjorken scaling in deep inelastic scattering data The number densities of the partons in the proton depend on the probing energy scale, Q 2, and parton momentum fraction, x, with respect to the momentum of the proton, expressed by a parton distribution function (PDF), f i (x,q 2 ), for each parton i. Interpretation as probability distributions: P uv `x1, x 2, Q 2 Z x2 According to the QCD factorization theorem for inclusive hard scattering processes, universal distributions containing long-distance structure of hadrons; related to parton model distributions at leading order, but with logarithmic scaling violations. PDFs are extracted from fits to the deep inelastic lepton-hadron scattering experimental data using the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) evolution euations. x 1 dxu V (x, Q 2 )

64 Proton Structure The proton consists of u u d + pairs At energies ~ 10 GeV Valence uarks Sea uarks valence uarks antiuarks

65 Parton Distribution Functions The curves all fall off very uickly as x increases; you are very unlikelyindeed to hit anything with more than 50% of the proton s energy. Most particles in the proton carry much less than 10% (that is, x< 0.1) of the proton s energy, and you are more likely to hit one of those, with lower energies than with higher ones. If you do hit something that carries less than 10% of the proton s energy, then it is most likely a gluon; and uarks and antiuarks are about eually likely once you get below about 2% of the proton s energy. If you hit something with much above 20% (that is, x>0.2) of the proton s energy which is very rare it is most likely a uark, and about twice as likely to be an up uark as a down uark

66 Structure of the Proton U U d d d s s Proton is composed of u,d valence uarks which carry uantum numbers and " sea" of u, u, d, d, s, s, g with _ no _ net _ uantum _ number

67 Constraints on Parton Density Functions The charm, beauty and top compositions are negligible so proton is composed of u, d, s uarks and anti uarks and glue. The observed uantum number constraints No net strangeness ( s( x) s( x)) dx = 0 Charge =1 Isospin I 3 =1/2 1 1 [ ( u( x) u( x)) ( d( x) d ( x))] dx = ( u( x) u( x)) ( d( x) d ( x)) ( s( x) s( x)) dx = Leads to: ( u( x) u( x)) dx = 2 ( d( x) d ( x)) dx = 1 Two u uarks and 1 d uark as expected. These are called sum rules and there is one for every uantum number. Momentum sum rule: x[ u( x) + u( x) + d( x) + d ( x) + s( x) + s( x) g( x)] dx = 1

68 Early Data on Proton Structure SLAC-MIT ~20 GeV electrons, protons at rest, Q 2 = few GeV 2 BCDMS GeV muons, protons at rest, Q 2 = tens of GeV 2

69 PDFs u(x) and d(x) parton distributions for proton µ = Q x(x) u(x) d(x) x

70 PDFs u(x) and u(x) parton distributions for proton x(x) u(x) valence + sea u ( x) sea only x

71 Gluon Distribution Quark distributions can be measured by e or µ scattering but not the gluon distribution as gluons are not electrically charged. A nonzero gluon distribution can be inferred from the momentum sum rule: so: x( u( x) + u( x) + d( x) + d ( x) + s( x) + s( x) + g( x) ) dx = 1. 0 xg( x) dx 1.0 ( u( x) + u( x) + d( x) + d ( x) + s( x) + = s( x)) dx More generally, Q 2 evolution of (x,q 2 ), and hence F i (x,q 2 ), involves g(x,q 2 ), so gluon distribution can be extracted that way. Nevertheless, it is not as well known as the uark distributions.

72 PDFs Gluons carry 50% of proton momentum xg ( x) dx = 0. 5 x(x) g(x) u(x) +d(x) x

73 PDFs At high Q 2 = µ 2 see more of sea x(x) u(x) 500 GeV u(x) 5 GeV x

74 PDFs At high Q 2 see less valence uarks more sea antiuarks x(x) u( x) 500 GeV u( x) 5 GeV u(x) 5 GeV x

75 PDFs More low x gluons less high x valence uarks at high Q 2 x(x) g( x) 500 g( x) 5 GeV GeV x

76 Quark uantum numbers Flavor I I 3 S C B* T Q/e u 1/2 1/ /3 d 1/2-1/ /3 s /3 c /3 b /3 t /3 Q/e = I (B + C + S + B *+T) 2 Baryon number :1/3 for all uarks