1. C4 Lectures Dynamic Quarks and (introduction to) QCD.
5 lectures HT 2013
Tony Weidberg
Feedback very much welcomed!
Please ask questions in lectures
Come and find me in DWB 629
Corrections for draft chapter (prize)
See handbook for suggested textbooks
Slides and draft chapter on C4 website

2. Outline Review of evidence for quarks from static quark model.
How to study particle structure
Review of Rutherford scattering and how to measure size of nucleus (form factors).
Scattering off nucleons in nuclei
Simple Quark Parton Model
ve scattering
nq scattering
np scattering
ep and mp scattering
Predictions of the QPM  direct evidence for spin ½ fractionally charged partons  quarks are real!
Gluons and extension to QCD
Scaling violations  gluon distribution
Hadron – hadron interactions. 2 2 2 2 2 2

3. Static 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 States
Spectra strikingly similar to positronium
Can we get more direct evidence for quarks?
Yes we can with DIS  These lectures! 3 3 3 3 3 3

4. Rutherford Scattering (1)
Alpha –gold scattering; prototype for all scattering experiments.
Use Fermi Golden Rule
H’ is Coulomb interaction
Plane waves for initial and final states 4 4 4 4 4 4

5. Rutherford Scattering (2)
Matrix element
Sub for V(r) and define momentum transfer
Use spherical polar r axis // q  q.r=qr cos(θ) 5 5 5 5 5 5

6. Rutherford Scattering (3)
Integral is divergent (cf classical result)
Modify potential by factor exp(-r/a) and let a∞
Let a∞ gives matrix element 6 6 6 6 6 6

7. Rutherford Scattering (4)
Density of states
Cross section and transition rates: T=s F and flux F=v.
Ignore nuclear recoil:
Gives Rutherford scattering cross section 7 7 7 7 7 7

8. Form 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 modified 8 8 8 8 8 8

9. Form Factor (2)
Matrix element modified compared to pure Coulomb by form factor
Form 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? 9 9 9 9 9 9

10. Form Factor (3)
Requires probe with λ<<R ie qR>>1 (high energy!).
Assume toy model for charge distribution ie
Gives 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/q8 10 10 10 10 10 10

11. Form Factor (4) Real example! Data and fit to charge distribution.
153 MeV e- scattering on Au
Large difference between data and point nucleus.
Fits to two models for charge distribution. 11 11 11 11 11 11

12. Approximate fit to uniform charge distribution (sharp edge)
Good fit to realistic charge distribution 12 12 12 12 12 12

13. Summary so far … Low energy scattering shows
Scaling cross section  evidence for point like nucleus (within resolution l= h/p)
Higher energy scattering shows
Fixed scale in cross section  evidence for finite size nucleus
Scattering data  form factors  charge distribution in nucleus 13 13 13 13 13 13

14. Conservation 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 energy 14 14 14 14 14

15. Nucleon Scattering (2) Recoil invariant mass W and 4-momtum transfer
Elastic scattering off entire nucleus W=M 15 15 15 15 15 15

16. Nucleon Scattering (3) e- He scattering
fixed angle 450,
Ee=400 MeV
Sharp peak: elastic scattering off nucleus smeared peak: elastic scattering off p (Fermi motion) What do we expect with partons inside proton? 16 16 16 16 16

17. Quark-Parton Model
Assume 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! 17 17 17 17 17

18. Kinematics (1) Assume quarks have mass m=xM
Elastic scattering off quark:
Define Q2=-q2
nb only need scattered electron measurement 18 18 18 18 18

19. Kinematics (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 frame 19 19 19 19 19

20. Kinematics (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 angle
q* = 0 for incident lepton
Elastic E=E’ 20 20 20 20 20

21. Anti-neutrino e scattering (1)
Consider W contribution
W couples to LH e- and RH anti n
high energy  chirality = helicity.
Very simple spin structure
Amplitude from projecting m=1/2 to m’=1/2 21 21 21 21 21

22. (Anti) Neutrino e scattering (2)
Cross section proportional to |A|2
Phase space proportional to p2=s/4.
GF Fermi coupling constant
Numerical factors requires Feynman rules 22 22 22 22 22

23. Neutrino e scattering Jz=0 so J=0 or J=1  not as simple …
Crossing symmetry: anti-particles & amplitudes
Matrix element for crossed diagrams same with p_incoming  - p_outgoing.
Compare ne and ne  p1<->p3
Isotropic, phase space + GF + numerical factor 3 ne q* 1 2 e- e- 4 23 23 23 23 23

24. Neutrino 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- 24 24 24 24

25. Neutrino quark scattering
q and q’ not same (CKM). Also ignore threshold effects 25 25 25 25

26. Neutrino Nucleon Scattering
QPM: scattering of leptons from nucleon is incoherent sum of scattering off quarks
Assumption only valid for large q2>>1 GeV2
Justification from QCD and experiment
Let 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 data 26 26 26 26

27. Neutrino Nucleon Scattering (1)
Combine all ingredients  cross section for measurable quantities! 27 27 27 27

28. Neutrino 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?

29. Neutrino Nucleon Scattering (2)
Compare y distributions with data
If anti-quarks negligible:
nN : flat y anti-n N: (1-y)2
Data fits combination of flat and (1-y)2  quarks are spin ½ and interact by a parity violating interactions. 29 29 29 29

30. Neutrino Nucleon Scattering (3)
Integrate in x and y to get total cross sections.
Assume quarks dominant
y integrals trivial
x integral ~ unity
Neutrino larger than anti-neutrino
See next slide for data
Cross sections scale like s  quarks are point like
If quarks had finite size  form factor suppression. 30 30 30 30

31. Neutrino Nucleon Scattering (4)
S=2mE
s/E constant vs E
Quarks are point like! 31 31 31

32. Parton 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 Invariance 32 32 32

33. Parton Distribution Functions (2)
Fit F2 and F3 to data  q and anti-q
Which quarks are we considering?
Assume nucleons only contain u,d,s and anti-quarks 33 33 33

34. Parton Distribution Functions (3)
Which quarks do n see ? 34 34 34

35. Parton Distribution Functions (4)
What about neutrons?
Assume SU(2) symmetry: 35 35 35

36. Nuclear Targets What about nuclei? Assume isoscalar target I=0
Just average n and p and assume s=sbar 36 36 36

37. Sum Rules (1) Important consistency checks of QPM
Divide q(x) into valence and sea
2 valence u + 1 valence d in proton Gross-Llewellyn-Smith sum rule
Check with data  37 37 37

38. GLS Sum Rule Agrees with data at large Q2
Disagreement at low Q2 can be explained by higher order QCD corrections. 38 38 38

39. Addler Sum Rule Addler Sum Rule:
Two valence u and one valence d  SA=1
2SA=2.02 ± 0.4 (D. Allasia, et al., Z. Phys. C 28 (1985) 321.)
Agrees with data! 39 39 39

40. Momentum 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 force
First evidence for gluons!
More direct evidence when we look at QCD. 40 40 40

41. Charged Lepton Probes Advantages of e/m Disadvantages?
Higher intensity beams
ep (HERRA)  access to larger range in x and q2.
Disadvantages?
How to calculate the lepton – nucleon s?
Calculate e m scattering
Generalise to e q scattering
Generalise to eN scattering (QPM) 41 41 41

42. em Scattering (1) Work by analogy with elastic ne scattering
EM interactions, photon massless and strength given by a.
Need to evaluate spin factor F 42 42 42

43. em Scattering (2) Spin factor F
Calculate amplitude for each spin configuration
Sum over final spin states
Average over initial spin states
For (1) and (2): JZ=0  same angular distribution as ne  isotropic.
For (3) : JZ=1 same rotation matrices as for 43 43 43

44. em Scattering (3) For (4): JZ=-1 rotation matrices gives
Averaging over initial spins and summing over final spin states
EM parity conserving  equal amplitudes for LH and RH (high energy –ive and +ive H) 44 44 44

45. Elastic e Quark Scattering
Fractional charge of quark flavour i, is qi
Momentum fraction x  cms energy squared of e quark: 45 45

46. e/m Nucleon DIS Same assumptions as for n DIS:
Incoherent scattering off individual quarks
Momentum distribution functions fi(x)
cf general phenomenological formula 46 46

47. Quark 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. /2 47 47

48. Quark Spin (2) e+e-  qq  2 jets Assume jet direction = quark
Helicity conservation for EM interaction
Coupling is LR and LR
Jz=+1 or Jz=-1
Rotation matrix for J=1, m=1 ,m’=1 and m=-1,m’=-1 amplitude

49. Quark Spin (3) Data from e+e- annihilation at Petra
Agrees with 1+cos2q
Confirms quark spin=1/2.

50. Scaling
Most 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 experiments 50 50

51. Quark Charges (1) Compare eN and nN <q2>
Assume isospin symmetry up=dn ubar=dbar (?)
Isoscalar target 51 51

52. Quark 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) 52 52

53. Gottfried Sum Rule Compare ep and en DIS Split quark = valence + sea
How can we have a n target ???
Split quark = valence + sea
Define Gottfried sum 53

55. NMC Data Naïve QPM  IG=1/3
Lets look at data from mp DIS for H and D targets
Em=90 & 280 GeV
No data at very low x
Why?
Extrapolate x=0
IG=0.235 ± 0.026
Why ???
NMC, Phys. Rev. D 50, R1–R3 (1994) 55

56. DIS 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?

57. Gluons & QCD Direct evidence for gluons Running coupling constant
Asymptotic freedom
Confinement
Scaling violations and gluon distribution
Hadron-hadron collisions
Drell-Yann: m+m-, W & Z.
Jet production
Look to the future: LHC 57

58. Direct Evidence for Gluons
Indirect evidence from momentum sum.
Look at e+e- annihilation
Clear 3 jets: gluon bremsstrhalung
Angular distribution for 3rd jet wrt 2nd jet.

59. 3 Jet Events
Jade experiment at PETRA
CMS E ~ 30 GeV

60. Gluon 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. 2 1 3 2 q 1 3

61. QCD We start from experiment with:
Spin 1 gluons
Number 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).

62. SU(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)=1
Det(U)* Det(U)=1  Det(U)= ±1

63. SU(3) – continued SU(3): select Det(U)=+1. 3*3 complex matrix: Quarks:
Start with 2*9=18 numbers
Constraints 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>

64. SU(3) Octet
Gluon wave functions (same maths as for SU(3) flavour)

65. Consequences of SU(3)
8 gluon states (transform into each other under SU(3) rotations).
Mathematically we could have a colour singlet gluon
How 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?

66. Colour 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): rrrr
qqg vertex requires term  g3 and g8
Amplitudes from gluon wave functions (previous page):

67. Colour Factors (2)
Next consider two different colours, arbitrary choice, take rb
Add amplitudes for (1) rbrb and (2) rbbr
(1) needs and terms  g3 and g8
(2) needs and terms  g1 and g2

68. Colour Factors (3)
Antiquarks -ive sign at vertex (cf –ive electric charge in QED).
Compare with
Same gluons exchanged, flip sign
Compare with

69. Colour Factors Summary
Now we’ve done the maths we can do some physics …
Learn something about quark binding in hadrons
Calculate scattering amplitudes

70. Quark Binding in Mesons (1)
Now consider colour singlet state of
Calculate colour factor for single gluon exchange for singlet using our previous results
3 terms (colours) like
Gives total = -3*2/9 = -2/3
Same term for
Multiply *3 for 3 colours = 3*(-2/3)=-2
Final result for colour singlet a=-2/3-2 =-8/3

71. Quark Binding in Mesons (2)
Amplitude also contains
coupling “constant” from each vertex
Propagator gives 1/q2 (4-momentum transfer)
In non-relativistic (NR) limit Fourier Transform of potential  scattering amplitude.
Inverse FT (amplitude)  potential V(r) ~ 1/r
Colour 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.

72. Quark scattering
Don’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).

73. QCD Assume SU(3) colour gauge symmetry. Consider transformation:
g s coupling constant and Ta are the generators of SU(3)
Infinitesimal transformation
Add 8 gluon fields to keep gauge invariance fabc are structure constants of SU(3).

74. QCD (2) Replace derivative with covariant derivative (cf QED)
KE term for gluon fields
This gives
KE for gluons and qqg vertices (cf QED) and
3g and 4g vertices (unlike QED).

75. Running Coupling Constant - QED
QED: example of running coupling constant
Shielding by virtual e+e- pairs  a increases at shorter distance, higher Q2.
Bare charge & renormalised charge
This term depends on number of “active” flavours m<Q)
Burcham & Jobes p

76. Running Coupling Constant - QCD
Similar shielding from gq qbar
Anti-shielding from ggg
Net 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.

77. Running 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 with
as (MZ)= ± 10
100
Q (GeV)

78. Experimental 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 jets
Angular distributions in 4 jet events
Event shapes, e.g. Thrust

79. Experimental Fit
Fit for CF and CA consistent with SU(3) and inconsistent with other choices for the symmetry.
CA = 2.89 ± 0.21
CF = 1.30 ± 0.09
S. Kluth, Tests of QCD at e+e- colliders
doi: / /69/6/R04

80. Scaling Violations Expect log scaling violations in QCD
qq g
gq qbar
gg g
As Q2 increases, resolve more splitting.
Valence q decreases
Sea q and gluons increase at low x

81. Scaling 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 increases
At high x, F2 decreases as Q2 increases
Use these scaling violations to fit the gluon distribution  figures

82. Kinematic Plane

83. Measurements from HERA + Fixed Target
Low x: F2 increases as Q2 increases
Approximate scaling at intermediate x
High x: F2 decreases as Q2 increases

84. PDF Fits
Q2=10,000 GeV2
Q2=10 GeV2
LHC : mainly gluons !

85. Hadron-Hadron Collisions (1)
Convolution of PDFs with parton-parton cross section
Master Picture p a
PDFs from DIS fits + QCD evolution to appropriate scale c
ab  cd d b p
Need to calculate parton-parton cross section abcd

86. Hadron-Hadron Collisions (2)
Master equation
Convolution integral over parton distribution functions and parton-parton cross section
Parton kinematics
Q2: scale of parton –parton reaction, e.g. M2

87. Drell-Yann (1) m+m- via virtual photon: parton-parton cross section.
Master formula:
Change variables (appendix)
Scaling cross section:

88. Drell-Yann (2)
Scaling

89. Drell-Yann (3)
Use DY (+QCD) predict Z cross sections for pbar p and pp

90. W Charge Asymmetry (1)
Define asymmetry in terms of rapidity of charged lepton
Assume
W+ from u valence dbar sea
W- from d valence ubar sea

91. W Charge Asymmetry (2) Assume sea symmetric ubar=dbar
u(x)> d(x)  expect A +ive.
u(x)/d(x) increasing with x. Rapidity lepton correlated with rapidity W  expect A to increase with h.

92. W Charge Asymmetry (3)

93. Dijet Angular Distributions
Dijet LHC: mainly gg  gg.
Angular distribution dominated by g exchange (cf Rutherford scattering) propagator  1/Q4
Gives angular distribution ~ 1/sin4(q/2)
Change variables to get flatter distribution:

94. Dijet Angular Distribution
Sensitive to any new contact interaction
cf 4 Fermi theory  more isotropic distribution  excess of events at low c.
L > % cl  d< fm

95. DIS & QCD Summary Quarks are real! QCD
Scaling  point like constituents of proton
Spin ½ and charges from quark model
More to p than uud: sea quarks + gluons.
QCD
Running coupling constant, explains success of naïve QPM.
Measurements of as(Q2).
Extension to hadron-hadron collisions

96. Appendix

97. e+e- R

98. Crossing Symmetry From the solutions to the Dirac equation we
formally represent the states which apparently have negative
energy as positive energy states travelling backwards in time.
crossing symmetry relates the amplitudes of reactions
particle anti-particle.
Fortwo crossed diagrams structure of the matrix elements
are the same  replace the momentum of the
incoming (outgoing) particles  minus that of the outgoing
(incoming) anti-particles.