Q 2 - JLab Computer Center
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Transcript Q 2 - JLab Computer Center
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
1018 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 (11cos )
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 ) 2GM
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
espin 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 Ga 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 gsaG
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 Ga Ga g s f abc GaGa
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