1. How to probe the quarks?
• Scatter high-energy
electron off a proton:
Deep-Inelastic
Scattering (DIS)
• Highest energy e-p
collider: HERA at DESY
in Hamburg: ~ 300 GeV
• Relevant scales:
d probed   

 1018 m
p
Deep-Inelastic Electron Scattering
• DIS kinematics:
• Four-momentum transfer:
 
 
q  ( E  E ' )  (k  k ' )  (k  k ' ) 
 
2
2
 me  me '  2( EE' | k | | k ' | cos ) 
2
2
  4 EE' sin 2   Q 2
• Mott Cross Section (c=1):
d
d Mott
( )

L : lepton ten
sor
W : hadron tensor


4 2 E '2
Q4
4 2 E '2
16 E 2 E '2 sin 4 2
 2 cos2 2
4 E 2 sin 4 2
2 

cos
2 

cos
 EE'
 1 E (11cos )
M
 1 E ( 21sin 2  )
M
2
Electron scattering of a spinless point particle
Electron-proton scattering
• Effect of proton spin:
• Nucleon form factors:
– Mott cross section:
 Mott 
4 2 E '2
Q4
2
cos 

E'
E

2 
cos
Ruth

– Effect proton spin 
 ep   Mott [ A(Q 2 )  B(Q 2 ) tan2  ]
2
with:
GE2  GM2
2
2
A(Q ) 
and B(Q )  2GM
1
2
• helicity conservation
• 0 deg.: ep(magnetic)  0
• The size of the nucleon:
• 180 deg.: spin-flip!
magn ~ Ruth
sin2(/2)
GEp (0)  1 and GMp (0) 
gp
2
 N  2.79 N
~ Mott
tan2(/2)
GEn (0)  0 and GMn (0) 
gn
2
 N  -1.91 N
 espin    Mott  [1  2 tan2  ]
1
2
• with
2

Q2
– proton: rp = 0.86 fm
– neutron: rn = 0.10 fm
4 M 2c 2
Mass of target = proton
Hofstadter, R., et al., Phys. Rev. 92, 978 (1953)
Excited states of the nucleon
D(1232)
• Scatter 4.9 GeV electrons
from a hydrogen target:
• Why use invariant energy?
• Evaluate invariant energy of
virtual-photon proton system:
W 2  (Pp  q)2  P2  2Pq  q2
• In the lab-frame: P = (mp,0) 
W 2  m2p  2m p  Q 2
• Observation excited states:
Nucleons are composite
→ What do we see in the
data for W > 2 GeV ?
Looking deep inside the proton
• First SLAC experiment (‘69):
– expected from proton form factor:
2

d / dE ' d 
1
  Q 8
 
2
2 
(d / d) Mott  (1  Q / 0.71) 
• First data show big surprise:
– very weak Q2-dependence:
– scattering off point-like objects?
• How to proceed:
– Find more suitable variable
– What is the meaning of
 ep /  Mott  "structurefunction"
or the ‘effective form factor’?
As often at such a moment….
…. introduce a clever model!
Stanford Linear Accelerator Center
Look familiar…?.....
The Quark-Parton Model
• Assumptions:
– Neglect masses and pT’’s
e’
e
– Proton constituent = Parton
– Impulse Approximation:
P
Quasi-elastic scattering off partons
parton
• Lets assume: pquark = xPproton
2
( xP  q)2  p'2quark  mquark
0
– Since xP2  M2 <<Q2 it follows:
Q2
Q2
2 xP  q  q  0  x 

2Pq 2M
2
• Check limiting case:
1
W 2  M p2  2M p  Q2 x

M p2
• Therefore:
x = 1: elastic scattering
and 0 < x < 1
Definition Bjorken scaling variable
Cross section for deep-inelastic scattering
• Cross section:
d
  Mott  W2 ( )  2W1 ( ) tan2  / 2 


dE ' d
electron
quark
• with
– Mott cross section  Mott :
scattering off point charge
– Structure functions W1, W2
with dimension [GeV]-1
Inelastic Electron Scattering
– Key issue: if quark is not a
fermion we will find W1=0
Structure Functions F1, F2
homework
• Introduce dimensionless structure functions:
F1  MW1 and F2  W2 
d
2
 d  1 

2



F
(
x
)

F
(
x
)
tan

/
2

2
1

dE' d  d  M  
M
2
• Rewrite this in terms of :   Q2 / 4mquark
(elastic e-q scatt.: 2mq = Q2 )
2

d
1
Q 2 4mq 
 d 
2
F1 ( x) tan  / 2 

   F2 ( x)  2 2 2
dE' d  d  M  
4mq Q M

Solution
1
  F2 ( x)  2  2 xF1 ( x) tan2  / 2

 
if F2 ( x)  2 xF1 ( x)
1
 F2 ( x) 1  2 tan2  / 2



/
• Experimental data for 2xF1(x) / F2(x)
→ quarks have spin 1/2
(if bosons: no spin-flip  F1(x) = 0)
Interpretation of F1(x) and F2(x)
• In the quark-parton model:
F1 ( x)   f 12 z 2f [q f ( x)  q f ( x)]
[and F2 = 2xF1 analogously]
Quark momentum distribution
• Heisenberg requires:
– Gluon emission: presence
of virtual
qq-pairs
• Distinguish
– Valence quarks (N-prop.)
– Sea quarks

– derived from: F3  (q f ( x)  q f ( x))
 DIS,  (1  y) F2  y 2 xF1  y(1 2y ) xF3
The quark structure of nucleons
• Quark quantum numbers:
• Structure functions:
– Spin: ½  Sp,n = () = ½
F2p  x[ 19 (d vp  d sp  d sp )  94 (uvp  usp  u sp )  19 ( ss  ss )]
– Isopin: ½  Ip,n = () = ½
F2n  x[ 19 (d vn  d sn  d sn )  94 (uvn  u sn  usn )  19 ( ss  ss )]
• Why fractional charges?
– Extreme baryons: Z = (1,2)
1  3zq  2  - 13  zq   23
– Assign: zup =+ 2/3, zdown = - 1/3
• Three families:
– Isospin symmetry:
uvn  dvp , dvn  uvp , usn  dsn  usp  dsp
– ‘Average’ nucleon F2(x)
with q(x) = qv(x) + qs(x) etc.
F2N  12 ( F2p  F2n )
 185 x   (q( x)  q ( x))  19 x  [ ss ( x)  ss ( x)]
u ,d
 u  c  t 
   
 d  s  b 
 z   23 ; mu  mc (  1.5 GeV)  mt
 z   ; md  ms (  0.3 GeV)  mb
1
3
– mc,b,t >> mu,d,s : no role in p,n
• Neutrinos:
F2  x[(d v  d s  d s )  (uv  us  us )  ( ss  ss )]
 x[(d  u  s)  (d s  us  ss )]  x  (q( x)  q ( x))
u ,d , s
Fractional quark charges
• Neglect strange quarks 
F2e, N
5

F2 , N 18
– Data confirm factor 5/18:
Evidence for fractional charges
• Fraction of proton momentum
carried by quarks:
1
F
 ,N
2
0
1
( x)dx  185  F2e, N ( x)dx  0.5
0
– 50% of momentum due to nonelectro-weak particles:
Evidence for gluons
Quarks in protons & neutrons
• If qsp(x) = qsn(x) and x  0:
F2n x[ 19 (d sn  d sn )  94 (usn  usn )  19 (ss  ss )]
 1 p
1
p
p
p
p
4
1
F2
x[ 9 (d s  d s )  9 (us  us )  9 (ss  ss )]
• In the limit x  1:
– assume same high-x tail:
d  u
p
v
1
2
p
v
and u  d
n
v
1
2
n
v
3d vn
F2n
2
 p  1 p 
F2
4 2 uv
3
– assume instead isospin symmetry:
x[ 19 d  94 u ] isospin
F
symmetry
 1



4
F
x[ 9 d  9 u ]
n
2
p
2
1
9
1
9
n
v
p
v
n
v
p
v
uvp  94 d vp uvp  d vp 19 uvp 1
 
 4 p 
p
4 p
d v  9 uv
4
9 uv
• Extract F2n/ F2p from data:
F  (F  F ) 
d
2
1
2
n
2
p
2
F2n
F2d
 2 p 1
F2p
F2
→ u-quark dominance
Modern data
• First data (1980):
• “Scaling violations”:
– weak Q2 dependence
– rise at low x
– what physics??
….. QCD
PDG 2002
Quantum Chromodynamics (QCD)
• Field theory for strong interaction:
q
– quarks interact by gluon exchange
– quarks carry a ‘colour’ charge
– exchange bosons (gluons) carry
colour  self-interactions (cf. QED!)
• Hadrons are colour neutral:
– RR, BB, GG or RGB
– leads to confinement:
| q, | qq or | qqq  forbidden
• Effective strength ~ #gluons exch.
– low Q2: more g’s: large eff. coupling
– high Q2: few g’s: small eff. coupling
q
gg
s
s
q
q
The QCD Lagrangian
Lqcd  i qj  (D ) jk qk   mq qj qk  14 Ga Ga
q
q
(j,k = 1,2,3 refer to colour; q = u,d,s refers to flavour; a = 1,..,8 to gluon fields)
Covariant derivative:
D     i gsaG
1
2
qg-interactions
SU(3) generators:
([a , b ]  i 12 f abc c )
a

Free
quarks
 0 1 0
0  i



1   1 0 0  2   i 0
 0 0 0
0 0



0 0  i
0 0



5   0 0 0  6   0 0
i 0 0 
0 1



Gμνa    Ga   Ga  g s f abc GaGa

 
 
Gluon kinetic
energy term
Gluon selfinteraction
0
 1 0 0
0 0




0  3   0  1 0  4   0 0
 0 0 0
1 0
0 



0
0 0 0 
1



1 
1  7   0 0  i  8   0
3
0 i 0 
0 


0
1

0
0 
0 0

1 0
0  2 
QCD predictions: the running of s
• pQCD valid if s << 1:

Q2
> 1.0
(GeV/c)2
CERN 2004
PDG 2002
• pQCD calculation:
 s (Q 2 ) 
12
33  2n f )  ln(Q 2 / L2 )
– with Lexp = 250 MeV/c:
Q2     s  0
 asymptotic freedom
Q2  0   s  
 confinement
Running coupling constant is
best quantitative test of QCD.
QCD predictions: scaling violations
• Originally: F2 = F2(x)
– but also Q2-dependence
• Why scaling violations?
– if Q2 increases:
 more resolution (~1/ Q2)
 more sea quarks +gluons
• QCD improved QPM:
F2 ( x, Q 2 )
x

2
2
+
+
• Officially known as: Altarelli-Parisi Equations (“DGLAP”)
2
(x,Q )
QCD fits of F2
data
• Free parameters:
– coupling constant:
s 
Quarks
12
 0.16
 n f ) ln(Q 2 / L )
– quark distribution q(x,Q2)
– gluon distribution g(x,Q2)
• Successful fit:
Corner stone of QCD
• Nucleon structure:
Unique self-replicating structure
Gluons
Summary of key QCD successes
• The data on the structure
function F2(x,Q2):
• The ‘converted’ distance
dependence of s:
The problem of QCD
• Extrapolate s to the size
of the proton, 10-15 m:
• If s >1 perturbative
expansions fail…
 Non-perturbative QCD:
– Proton structure & spin
– Confinement
– Nucleon-Nucleon forces
– Higher twist…..
l  rproton   s  1
Lattice QCD
simulations…
Summary
• Quarks are the constituents of the proton
• Quark carry only 50% of the proton momentum
• QCD describes quark-gluon interactions:
– Successful description scaling violations
– Running coupling constant
– But non-pQCD is insufficient at r ~ rproton
• What JLab (and others) are looking into:
–
–
–
–
–
Non-pQCD effects
The origin of the spin of the proton
The role of gluons and orbital angular momentum
Generalized parton distributions
Transversity