Presentation on theme: "C4 Lectures Dynamic Quarks and (introduction to) QCD."— Presentation transcript:
1C4 Lectures Dynamic Quarks and (introduction to) QCD. 5 lectures HT 2013Tony WeidbergFeedback very much welcomed!Please ask questions in lecturesCome and find me in DWB 629Corrections for draft chapter (prize)See handbook for suggested textbooksSlides and draft chapter on C4 website
2Outline Review of evidence for quarks from static quark model. How to study particle structureReview of Rutherford scattering and how to measure size of nucleus (form factors).Scattering off nucleons in nucleiSimple Quark Parton Modelve scatteringnq scatteringnp scatteringep and mp scatteringPredictions of the QPM direct evidence for spin ½ fractionally charged partons quarks are real!Gluons and extension to QCDScaling violations gluon distributionHadron – hadron interactions.222222
3Static Quark Model SU(3) and quark multiplets Allowed understanding of bizarre number of “elementary particles”.SU(3) badly broken so predictions approximate.Charmonimum and bbar StatesSpectra strikingly similar to positroniumCan we get more direct evidence for quarks?Yes we can with DIS These lectures!333333
4Rutherford Scattering (1) Alpha –gold scattering; prototype for all scattering experiments.Use Fermi Golden RuleH’ is Coulomb interactionPlane waves for initial and final states444444
5Rutherford Scattering (2) Matrix elementSub for V(r) and define momentum transferUse spherical polar r axis // q q.r=qr cos(θ)555555
6Rutherford Scattering (3) Integral is divergent (cf classical result)Modify potential by factor exp(-r/a) and let a∞Let a∞ gives matrix element666666
7Rutherford Scattering (4) Density of statesCross section and transition rates: T=s F and flux F=v.Ignore nuclear recoil:Gives Rutherford scattering cross section777777
8Form Factor (1)Rutherford scattering cross section doesn’t depend on any fixed scale in problem.We had assumed point like nucleus.What happens for finite sized nuclei?Coulomb potential modified:Matrix element modified888888
9Form Factor (2)Matrix element modified compared to pure Coulomb by form factorForm factor is Fourier transform of charge distribution in 3D space to momentum space.In principle: measure deviations from pure Coulomb potential F(q2) inverse FT gives ρ(r).Why doesn’t this work in practice?999999
10Form Factor (3)Requires probe with λ<<R ie qR>>1 (high energy!).Assume toy model for charge distribution ieGives form factor (work it out yourself!)q << m F(q2) ~ constant because resolution too poor.q >> m F(q2) ~ 1/q4 and σ suppressed by 1/q8101010101010
11Form Factor (4) Real example! Data and fit to charge distribution. 153 MeV e- scattering on AuLarge difference between data and point nucleus.Fits to two models for charge distribution.111111111111
12Approximate fit to uniform charge distribution (sharp edge) Good fit to realistic charge distribution121212121212
13Summary so far … Low energy scattering shows Scaling cross section evidence for point like nucleus (within resolution l= h/p)Higher energy scattering showsFixed scale in cross section evidence for finite size nucleusScattering data form factors charge distribution in nucleus131313131313
14Conservation of 4 momentum and energy Nucleon Scattering (1)Next steps into the nucleus … lets look for evidence of nucleons inside the nucleus from scattering experiments …Conservation of 4 momentum and energy1414141414
15Nucleon Scattering (2) Recoil invariant mass W and 4-momtum transfer Elastic scattering off entire nucleus W=M151515151515
16Nucleon Scattering (3) e- He scattering fixed angle 450,Ee=400 MeVSharp peak: elastic scattering off nucleus smeared peak: elastic scattering off p (Fermi motion) What do we expect with partons inside proton?1616161616
17Quark-Parton ModelAssume QPM is correct predictions for DIS compare with data.(0) Kinematics for DIS(1)(2)(3) Generalise to n quarks(4) Generalise to n nucleon(5) Extend to e/m nucleon(6) Compare with data!1717171717
18Kinematics (1) Assume quarks have mass m=xM Elastic scattering off quark:Define Q2=-q2nb only need scattered electron measurement1818181818
19Kinematics (2)x can also represent fractional proton momentum carried by quark.DIS large Q2 neglect quark mass compared to energy evaluate quark mass after scattering.x can represent either mass fraction or momentum fraction (infinite momentum frame)p.q LI use lab frame1919191919
20Kinematics (3) Relation between CMS angle and lab energies. LT from CMS to lab (b=1) for scattered and initial lepton. Define y=n/E (0<y<1)CMS scattering angleq* = 0 for incident leptonElastic E=E’2020202020
21Anti-neutrino e scattering (1) Consider W contributionW couples to LH e- and RH anti nhigh energy chirality = helicity.Very simple spin structureAmplitude from projecting m=1/2 to m’=1/22121212121
22(Anti) Neutrino e scattering (2) Cross section proportional to |A|2Phase space proportional to p2=s/4.GF Fermi coupling constantNumerical factors requires Feynman rules2222222222
23Neutrino e scattering Jz=0 so J=0 or J=1 not as simple … Crossing symmetry: anti-particles & amplitudesMatrix element for crossed diagrams same with p_incoming - p_outgoing.Compare ne and ne p1<->p3Isotropic, phase space + GF + numerical factor3neq*12e-e-42323232323
24Neutrino Quark Scattering (1) Universality of CC weak interactions.In nq scattering:Now write down cross sections for (anti) neutrino on (anti) quark scattering by analogy with (anti) neutrino scattering on e-24242424
25Neutrino quark scattering q and q’ not same (CKM). Also ignore threshold effects25252525
26Neutrino Nucleon Scattering QPM: scattering of leptons from nucleon is incoherent sum of scattering off quarksAssumption only valid for large q2>>1 GeV2Justification from QCD and experimentLet q(x) be the pdf for finding quark with momentum fraction x in a proton.What does q(x) mean?QPM can’t predict q(x) need fits to data.Many clear predictions can be checked with data26262626
27Neutrino Nucleon Scattering (1) Combine all ingredients cross section for measurable quantities!27272727
28Neutrino Detector High energy beam nm. Detector shape and size ? What final state particles do we want to detect?Detector shape and size ?Passive material?Active material?Readout?
29Neutrino Nucleon Scattering (2) Compare y distributions with dataIf anti-quarks negligible:nN : flat y anti-n N: (1-y)2Data fits combination of flat and (1-y)2 quarks are spin ½ and interact by a parity violating interactions.29292929
30Neutrino Nucleon Scattering (3) Integrate in x and y to get total cross sections.Assume quarks dominanty integrals trivialx integral ~ unityNeutrino larger than anti-neutrinoSee next slide for dataCross sections scale like s quarks are point likeIf quarks had finite size form factor suppression.30303030
31Neutrino Nucleon Scattering (4) S=2mEs/E constant vs EQuarks are point like!313131
32Parton Distribution Functions (1) How to determine the PDFs qi(x) from data?Rewrite QPM prediction to allow fits to data.Compare with formula allowed by Lorentz Invariance323232
33Parton Distribution Functions (2) Fit F2 and F3 to data q and anti-qWhich quarks are we considering?Assume nucleons only contain u,d,s and anti-quarks333333
34Parton Distribution Functions (3) Which quarks do n see ?343434
35Parton Distribution Functions (4) What about neutrons?Assume SU(2) symmetry:353535
36Nuclear Targets What about nuclei? Assume isoscalar target I=0 Just average n and p and assume s=sbar363636
37Sum Rules (1) Important consistency checks of QPM Divide q(x) into valence and sea2 valence u + 1 valence d in proton Gross-Llewellyn-Smith sum ruleCheck with data 373737
38GLS Sum Rule Agrees with data at large Q2 Disagreement at low Q2 can be explained by higher order QCD corrections.383838
39Addler Sum Rule Addler Sum Rule: Two valence u and one valence d SA=12SA=2.02 ± 0.4 (D. Allasia, et al., Z. Phys. C 28 (1985) 321.)Agrees with data!393939
40Momentum Sum Integrate F2 momentum fraction carried by quarks Naïve expectiation: I=1 but I=0.44 at Q2 =10 GeV2.~ ½ momentum is carried by particles that don’t experience weak forceFirst evidence for gluons!More direct evidence when we look at QCD.404040
41Charged Lepton Probes Advantages of e/m Disadvantages? Higher intensity beamsep (HERRA) access to larger range in x and q2.Disadvantages?How to calculate the lepton – nucleon s?Calculate e m scatteringGeneralise to e q scatteringGeneralise to eN scattering (QPM)414141
42em Scattering (1) Work by analogy with elastic ne scattering EM interactions, photon massless and strength given by a.Need to evaluate spin factor F424242
43em Scattering (2) Spin factor F Calculate amplitude for each spin configurationSum over final spin statesAverage over initial spin statesFor (1) and (2): JZ=0 same angular distribution as ne isotropic.For (3) : JZ=1 same rotation matrices as for434343
44em Scattering (3) For (4): JZ=-1 rotation matrices gives Averaging over initial spins and summing over final spin statesEM parity conserving equal amplitudes for LH and RH (high energy –ive and +ive H)444444
45Elastic e Quark Scattering Fractional charge of quark flavour i, is qiMomentum fraction x cms energy squared of e quark:4545
46e/m Nucleon DIS Same assumptions as for n DIS: Incoherent scattering off individual quarksMomentum distribution functions fi(x)cf general phenomenological formula4646
47Quark Spin (1)Equating powers y QPM and phenomenological formula (Callan-Gross)Data agrees with spin ½ and excludes spin 0.Simple explanation?Quark spin also measured in e+e- 2 jets./24747
48Quark Spin (2) e+e- qq 2 jets Assume jet direction = quark Helicity conservation for EM interactionCoupling is LR and LRJz=+1 or Jz=-1Rotation matrix for J=1, m=1 ,m’=1 and m=-1,m’=-1 amplitude
49Quark Spin (3) Data from e+e- annihilation at Petra Agrees with 1+cos2qConfirms quark spin=1/2.
50ScalingMost critcial prediction of QPM is scaling SF depend on x but not Q2 at high Q2 and n.Data show approximate scaling.If quarks had finite size very large form factor suppression.Slow (log) variations with Q2 can be explained by QCD.Quarks are point like within resolution of current experiments5050
51Quark Charges (1) Compare eN and nN <q2> Assume isospin symmetry up=dn ubar=dbar (?)Isoscalar target5151
52Quark Charges (2)cf nN & eN SFs and assuming s negligible (ok at x>~0.2)?Agreement confirms quark charge assignments.Quark charges also confirmed by(see appendix for plot)x 18/5 F2(ep)● F2(np)5252
53Gottfried Sum Rule Compare ep and en DIS Split quark = valence + sea How can we have a n target ???Split quark = valence + seaDefine Gottfried sum53
55NMC Data Naïve QPM IG=1/3 Lets look at data from mp DIS for H and D targetsEm=90 & 280 GeVNo data at very low xWhy?Extrapolate x=0IG=0.235 ± 0.026Why ???NMC, Phys. Rev. D 50, R1–R3 (1994)55
56DIS Summary Scaling cross section point like partons Quark spin = ½ confirmed.Comparison of n and charged lepton data consistent with quark charges.Quarks are real !Momentum sum: more to proton than quarks: gluons important.Why does the QPM work at all?
57Gluons & QCD Direct evidence for gluons Running coupling constant Asymptotic freedomConfinementScaling violations and gluon distributionHadron-hadron collisionsDrell-Yann: m+m-, W & Z.Jet productionLook to the future: LHC57
58Direct Evidence for Gluons Indirect evidence from momentum sum.Look at e+e- annihilationClear 3 jets: gluon bremsstrhalungAngular distribution for 3rd jet wrt 2nd jet.
593 Jet EventsJade experiment at PETRACMS E ~ 30 GeV
60Gluon Spin Measure angle of 3rd jet sensitive to gluon spin Boost 2 lowest pt jets to CMS and plot angle jets make with thrust axis Consistent with 1.2132q13
61QCD We start from experiment with: Spin 1 gluonsNumber of quark colours = 3.Assume theory is given by SU(3) colour.The group theory for SU(3) colour identical to SU(3) flavour but colour is conserved exactly (cf electric charge) but SU(3) flavour is approximate.Quarks come in 3 colours, red, blue, green (nothing to do with colour of light! Just convenient labels).
62SU(3) Unitary matrices U†U=UU†=I. Diagonal terms must be real 3 constraints.6 off-diagonal terms but if U†ij =0 U†ji=0.Leaves 3 complex elements 3*2=6 constraints.Det(U†U)=Det((UT)*) Det(U)=1Det(U)* Det(U)=1 Det(U)= ±1
63SU(3) – continued SU(3): select Det(U)=+1. 3*3 complex matrix: Quarks: Start with 2*9=18 numbersConstraints 3(diagonal) + 6 (off-diagonal) + 1 (det U=+1)SU(3) Needs 8 parameters 8 generators for SU(3) 8 gluons.Quarks:fundamental representation of SU(3), triplet of states labeled |r>, |b>, |g>
64SU(3) OctetGluon wave functions (same maths as for SU(3) flavour)
65Consequences of SU(3)8 gluon states (transform into each other under SU(3) rotations).Mathematically we could have a colour singlet gluonHow do we know that such a gluon state doesn’t exist?Why is it ok to have colour singlet states for mesons?How do we know that this choice is correct?
66Colour Factors (1)We can use gluon wave functions to calculate relative amplitudes from different qq scattering processes (colour factors).qq qq: same colour, chose r (b or g would give same answer): rrrrqqg vertex requires term g3 and g8Amplitudes from gluon wave functions (previous page):
67Colour Factors (2)Next consider two different colours, arbitrary choice, take rbAdd amplitudes for (1) rbrb and (2) rbbr(1) needs and terms g3 and g8(2) needs and terms g1 and g2
68Colour Factors (3)Antiquarks -ive sign at vertex (cf –ive electric charge in QED).Compare withSame gluons exchanged, flip signCompare with
69Colour Factors Summary Now we’ve done the maths we can do some physics …Learn something about quark binding in hadronsCalculate scattering amplitudes
70Quark Binding in Mesons (1) Now consider colour singlet state ofCalculate colour factor for single gluon exchange for singlet using our previous results3 terms (colours) likeGives total = -3*2/9 = -2/3Same term forMultiply *3 for 3 colours = 3*(-2/3)=-2Final result for colour singlet a=-2/3-2 =-8/3
71Quark Binding in Mesons (2) Amplitude also containscoupling “constant” from each vertexPropagator gives 1/q2 (4-momentum transfer)In non-relativistic (NR) limit Fourier Transform of potential scattering amplitude.Inverse FT (amplitude) potential V(r) ~ 1/rColour factor =-8/3 negative suggests that colour singlet should be bound state.Similar calculation for colour octet gives +ive colour factor.Suggestive of confinement for colour singlets but beware this is NR limit in which coupling constant large so can’t use perturbation theory.
72Quark scatteringDon’t have free quarks so can’t measure the qq scattering cross sections directly but …We can do hadron-hadron (mainly pp or pp) scattering and if we know quark PDFs we can combine this with these scattering amplitudes to calculate measurable cross sections (see later).
73QCD Assume SU(3) colour gauge symmetry. Consider transformation: g s coupling constant and Ta are the generators of SU(3)Infinitesimal transformationAdd 8 gluon fields to keep gauge invariance fabc are structure constants of SU(3).
74QCD (2) Replace derivative with covariant derivative (cf QED) KE term for gluon fieldsThis givesKE for gluons and qqg vertices (cf QED) and3g and 4g vertices (unlike QED).
75Running Coupling Constant - QED QED: example of running coupling constantShielding by virtual e+e- pairs a increases at shorter distance, higher Q2.Bare charge & renormalised chargeThis term depends on number of “active” flavours m<Q)Burcham & Jobes p
76Running Coupling Constant - QCD Similar shielding from gq qbarAnti-shielding from gggNet result:as(Q2) decreases as Q2 increases asymptotic freedom.Explains why naïve QPM works at large Q2: as(Q2) small so lowest order calculation gives good approximation.Infrared slavery quark confinement.
77Running Coupling Constant – QCD (2) Experiments measure as(Q) at different values of Q2.tau decay BR: how?Quarkonia: how?e+e-3 jets: how?Good fit to QCD prediction withas (MZ)= ±10100Q (GeV)
78Experimental Tests SU(3) Running of as with Q2 is implicit test of SU(3)Can we do explicit test of SU(3)?SU(3) predicts colour factors for qqg and ggg vertices: CF=4/3 and CA=3.Several variables sensitive to this in e+e- hadrons. Some examples:Charged particle multiplicity in q and g jetsAngular distributions in 4 jet eventsEvent shapes, e.g. Thrust
79Experimental FitFit for CF and CA consistent with SU(3) and inconsistent with other choices for the symmetry.CA = 2.89 ± 0.21CF = 1.30 ± 0.09S. Kluth, Tests of QCD at e+e- collidersdoi: / /69/6/R04
80Scaling Violations Expect log scaling violations in QCD qq ggq qbargg gAs Q2 increases, resolve more splitting.Valence q decreasesSea q and gluons increase at low x
81Scaling Violations Need to look over large range of x and Q2. HERA data gives good coverage (high energy!)At low x, F2 increases as Q2 increasesAt high x, F2 decreases as Q2 increasesUse these scaling violations to fit the gluon distribution figures
93Dijet Angular Distributions Dijet LHC: mainly gg gg.Angular distribution dominated by g exchange (cf Rutherford scattering) propagator 1/Q4Gives angular distribution ~ 1/sin4(q/2)Change variables to get flatter distribution:
94Dijet Angular Distribution Sensitive to any new contact interactioncf 4 Fermi theory more isotropic distribution excess of events at low c.L > % cl d< fm
95DIS & QCD Summary Quarks are real! QCD Scaling point like constituents of protonSpin ½ and charges from quark modelMore to p than uud: sea quarks + gluons.QCDRunning coupling constant, explains success of naïve QPM.Measurements of as(Q2).Extension to hadron-hadron collisions
98Crossing Symmetry From the solutions to the Dirac equation we formally represent the states which apparently have negativeenergy as positive energy states travelling backwards in time.crossing symmetry relates the amplitudes of reactionsparticle anti-particle.Fortwo crossed diagrams structure of the matrix elementsare the same replace the momentum of theincoming (outgoing) particles minus that of the outgoing(incoming) anti-particles.