5.1 Colour.
5.2 Quantum Chromo Dynamics (QCD)
5.3 Heavy Quark Bound States
5.4 The Strong Coupling Constant and Asymptotic Freedom.
5.5 Quark-Gluon Plasma.
5.6 Jets and Gluons.
5.7 Colour Counting.
5.8 Deep Inelastic Scattering and Nucleon Structure.
Scaling
Quark-Parton model
Scaling violation and structure functions
27/04/14
F. Ould-Saada
1
à
How do quarks interact? What is the role of gluons in strong interactions (SI)? ¡
Why are observed hadrons of the form qqq
, q q q , ?q q
¡
Difficulties in Quark Parton Model (QPM) à introduction of colour quantum number §  Non-­‐observation of free quarks §  Disagreement between theory and experiment §  Wave function of baryons with identical quarks Δ+ + ≡ u ↑ u ↑ u ↑
Ω ≡s↑s↑s↑
−
27/04/14
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2
Symmetric WF
Pauli principle!
Way out: 3 colors
à Anti-symmetric WF
27/04/14
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3
Experimentally:
¡
Coulour counting in e+e-­‐ à section 5.7 27/04/14
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4
¡
¡
¡
Quarks come in 3 colours – r,g,b 1
[r1g2b3 + g1b2r3 + b1r2 g3 − r1b2 g3 − b1g2r3 − g1r2b3 ]
ψ
=
C colour
“Colour isospin” -­‐ I3
6
“Colour hypercharge” -­‐ YC §  Only states with zero “colour charges” are observable as free particles – colour singlets §  qqq states with r,g,b have both I3C and YC equal to zero §  q-­‐qbar states allowed §  qq, qqqq, … forbidden – not observed §  Exotics: qqq qqbar, qqbarg, gg, ggg, … allowed but not observed … 27/04/14
Quarks
I3C
YC
Antiquarks
I3C
YC
r
1/2
1/3
rbar
-1/2
-1/3
g
-1/2
1/3
gbar
1/2
-1/3
b
0
-2/3
bbar
0
2/3
F. Ould-Saada
5
¡
At fundamental level, SI §  takes place between quarks and gluons §  Controls collisions between two quarks, ▪  interaction between three quarks to form a baryon, ▪  or between a quark and an antiquark to form a meson ¡
Nuclear force is “residual SI” §  Analogous to residual EM force between 2 atoms or 2 molecules ¡
EM §  QED: Abelian U(1) §  Source: one type of electric charge (Q=±) §  Mediator: 1 photon – no charge ¡
SI §  QCD: non-­‐Abelian SU(3)c §  Mediator: 8 gluons – with color charge §  Source: color charge with 3 degrees of freedom (r,g,b) for quarks 3 for antiquarks necessary to describe the dynamics ¡
Gauge symmetry of strong interactions: SU(3)c §  Exact symmetry: color charges absolutely conserved, gluon is massless §  Color charge structure more complex than that of the electric charge ¡
Charges are in fundamental representations of the group §  For U(1): singlet
§  SU(3) has 2 representations: 3 and 3 §  3 charges (R,G,B) : Quarks have color and antiquarks antichlor -­‐ ¡
Strong force depends on color, NOT flavour nor electric charge 27/04/14
7
¡
¡
Gluons belong to the octet obtained by combining color and ant color 3 ⊗ 3 = 1⊕ 8
Similar situation as in quark model with 3 flavors, hence SU(3) §  Singlet completely symmetric. Does not interact with quarks à no single gluon g0 =
1
(RR + BB + GG )
3
§  By analogy with meson octet: g1 = RG ; g 2 = RB ; g 3 = GR ; g 4 = GB ; g 5 = BR ; g 6 = BG
27/04/14
1
1
(RR − GG ) ; g8 = (RR − 2BB + GG )
g7 =
2
6
8
g1 = RG
g2 = RB
3 ⊗ 3 = 8a ⊕1s
g3 = GR
g4 = GB
g5 = BR
1
g0 =
RR + BB + GG )
(
3
27/04/14
g6 = BG
1
g7 =
RR − GG )
(
2
1
g8 =
(RR + GG − 2BB )
6
9
¡
Gluons are colored §  can interact coupled by continuous color lines (a) §  Gauge symmetry of strong interactions: SU(3)c ¡
+ self-­‐gluon interactions! Gluon-­‐gluon scattering g + g → g + g can happen by exchanging another gluon (b) §  further contribution is the 4-­‐gluon vertex (c) 27/04/14
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10
! QED
Interaction amplitude
proportional to z1z2α
! QCD
¡
Incoming and outgoing fermions may be different §  gluon has color of one quark and the opposite of the color of the other quark ¡
Vertex contains coupling constant AND color factor §  ci and cj : colors of 2 quarks -­‐ λ : gluon type (color lines continuous) 27/04/14
11
g1 = RG
g2 = RB
g3 = GR
g4 = GB
κ 1RG = 1
κ 1RG = −1
κ 2RB = 1
κ 2RB = −1
κ 3GR = 1
κ 3GR = −1
κ 4GB = 1
..................
κ 5BR = 1
g5 = BR
g6 = BG
1
g7 =
RR − GG )
(
2
1
g8 =
RR + GG − 2BB )
(
6
κ 6BG = 1
Colour factors of antiquark are opposite to those of quarks κ 6RG = 1
1
1
; κ 7GG = −
2
2
1
1
2
=
; κ 8GG =
; κ 8BB = −
6
6
6
κ 7RR =
κ 8RR
g1 à g6 have colour and anti-­‐colour g7 has 2 colours and 2 anti-­‐coulours 27/04/14
g8 has 3 colours and 3 anti-­‐coulours 12
BB→BB
B
B
B
B
q+ q→ q+ q
κ 8BB = −
2
6
1 BB 1 BB 1 ⎛ −2 ⎞ ⎛ −2 ⎞ 1
κ8
κ 8 = ⎜
⎟ ⎜
⎟ =
2 ⎝ 6 ⎠ ⎝ 6 ⎠ 3
2
2
RR→RR
R
¡
q + R q →R q + R q
1
2
1
=
6
§
κ 7RR =
κ 8RR
1 RR 1 RR 1 RR 1 RR 1 ⎛ 1 ⎞ ⎛ 1 ⎞ 1 ⎛ 1 ⎞ ⎛ 1 ⎞ 1
κ7
κ7 +
κ8
κ 8 = ⎜
⎟ ⎜
⎟ + ⎜
⎟ ⎜
⎟ =
2 ⎝ 2 ⎠ ⎝ 2 ⎠ 2 ⎝ 6 ⎠ ⎝ 6 ⎠ 3
2
2
2
2
27/04/14
As expected from symmetry, force between R and R is the same as between B and B. ¡
verify intensity of force between R and G + sign à repulsive force: as in QED same-­‐sign color charges repel each other 13
¡
Hadrons have no color charge, but are made of colored quarks §  ⇒ Color charges of quarks form “neutral” combination §  In QED: analogue to atom (neutral because it has as many + charges as – charges) §  QCD ⇒ neutrality is color singlet state ¡
3 ⊗ 3=8A ⊕1S
Mesons: quark-­‐antiquark bound states §  Binding because their product contains singlet (qq )singoletto
singlet
=
¡
¡
1
3
( q q+
R
R
B
q B q + G qG q
)
Mesons: quark-­‐antiquark “interaction” à factor -­‐4/3 αs à attractive force è Baryons: qqq “interaction” à factor -­‐2/3 αs à attractive force 27/04/14
14
B
B
q q
κ 2RB = 1;
κ 2RB = −1
κ 4GB = 1;
κ 4GB = −1
κ 8BB = −
2
;
6
κ 8BB = +
Symmetry ⇒ interaction between 2 pairs equal ⇒ calculate for one and multiply by 3
2
1 ⎛ 4
4 α ⎛ 1 ⎞ 1 BB BB
⎞
RB RB
GB GB
⎡
⎤
3⎜
κ
κ
+
κ
κ
+
κ
κ
=
−
−1−1
=
−
s
⎜⎝
⎟⎠
5
5
6
6 ⎦
⎝ 3 ⎟⎠ 2 ⎣ 8 8
2
6
3
αs
Attractive force!
27/04/14
15
2
6
¡
Let’s illustrate some QCD discussions e+e− → J / ψ + X → e+e− / µ +µ − / hadrons + X
§  Static potential between Heavy quark and antiquark §  Bound states: Charmonium and Bottomonium (analogy with positronium) ¡
Charmonium J/ψ discovery at BNL and SLAC p + N → J/ψ + X → e + e − + X
27/04/14
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16
J PC (J / ψ ) = J PC (γ ) = 1−−
σ e+e− →J /ψ →hadrons = π ! 2
¡
Hadronic decay à §
Add EM contribution at 3 GeV 2J +1
Γ ee Γ h
(2s1 +1) (2s2 +1) ( E − Mc 2 )2 +Γ 2 / 4
$ Γ ee Γ h &
=
≅ 1.197) 2 * = 0.07mb
2
2
$
&
% Γ '
4 %( E − M ) +Γ / 4'
3π ! 2 Γ ee Γ h
M = 3097MeV, J = 1 ; ! = "c / pc ≅ 197MeV fm / 2 × 3097MeV ≅ 0.127 fm
¡
Why is J/ψ resonance so much narrower than other hadronic resonances (~10’s MeV)? J / ψ (3097) ≡ cc
with QNs
: n = 1, 2S+1LJ = 3S1
J / ψ → hadrons (86%) but Γ J /ψ ~ 90keV !!
/ DD : M J /ψ < 2M D
J /ψ →
¡
¡
Lowest order decay to hadrons diagram is not allowed 27/04/14
F. Ould-Saada
Higher orders allowed but contribute much less! 18
J PC ( J /ψ ) = 1− −
C − parity = −
3
⇒ ggg ⇒ (α s )
§  g-­‐exchange à no suppression but m<2mD à not possible §  3-­‐gluon exchange à suppression à narrow width J / ψ (3097) → ηc (2980) + γ
Charmonium spectroscopy 27/04/14
F. Ould-Saada
ψ (3686) → ηc (2980) + γ
ψ (3686) → χ ci + γ
i = 1, 3
m > 2mD ⇒ decay to open charm
⇒ decay width ~MeV
19
27/04/14
F. Ould-Saada
20
¡
Energy levels for (a) positronium and (b) charmonium. §  Scale on ordinate is [eV] in (a) and [MeV] in (b) & a(!c)
( −
(
r
V (r) = '
( + b⋅r
() !c
•  Similarities à flavor independence! 27/04/14
F. Ould-Saada
r ≤ 0.1 fm ⇒ Asymptotic Freedom
r ≥ 1 fm ⇒ Confinement
a(!c) b ⋅ r
+
r
!c
cc + bb ⇒ a ≈ 0.48 , b ≈ 0.18 GeV 2
V (r) = −
22
¡
¡
Strong force – among other things – binds quarks into hadrons Asymptotic freedom and running coupling constant §  Interaction gets weaker at short distancesà large momentum transfers ""%
!
q = Ο$ '
#r&
2
! E
µ ≡ q 2 − 2q
c
2
§  Strength of interaction depends on §  QCD predicts dependence of αs on scale µ §  Λ : Scale parameter extracted from experiments αs =
6π
#µ&
33−
2N
ln
% (
(
f)
$Λ'
µ 2 >> 1GeV 2
¡
¡
αs measurements at various µ à Λ~(0.2±0.1) GeV f = u, d, s, c, b,... : ( 2mq2 < µ 2 )
QCD vs QED … 27/04/14
F. Ould-Saada
23
¡
Single electrons emit and reabsorb photons continuously (a) – quantum fluctuations – §  or photon may be absorbed by another electron nearby à scattering (b) ¡
Higher orders §  Initial electron emits photons and (indirectly) e+e-­‐ pairs – sea or virtual e+e-­‐ à vacuum polarisation effects §  Production of virtual e+e-­‐ pairs leads to a shielding effect such that αQED is altered Coulomb potential :φeff =
α eff
27/04/14
α eff (r)!c
r
! %
'
= α ≈ 1 /137 for r >> rC ≡
α (µ 0 )
me c & α em (µ ) =
0
* µ -3
2
'
r ≤ rC ⇒ α decreases
21− α (µ 0 )ln , /5
(
+ µ 0 .4
1 3π
F. Ould-Saada
24
¡  Gluon self-­‐coupling è anti-­‐screening α s (µ 0 )
α s (µ ) =
1+
"µ%
α s (µ 0 )
33−
2N
ln
$ '
(
f)
6π
# µ0 &
µ 2 >> 1GeV 2
N f < 17 ⇒ Asymtotic Freedom
27/04/14
F. Ould-Saada
25
¡
αs measurements 27/04/14
F. Ould-Saada
α s (M Z ) = 0.118 ± 0.002
26
¡
Confinement §  In QCD at normal energy-­‐densities à q and g confined within hadrons (nature of ¡
At extremely high E-­‐densities QCD predicts de-­‐
confinement of q an g across volume larger compared to Tc
that of a hadron à new state of matter: quark-­‐
gluon plasma §  first µs after Temperature
¡
Big-­‐Bang? Neutron stars? §  Lattice QCD à transition energy~160-­‐190 GeV à 1012K Early universe
confinement not fully understood) quark-gluon plasma
hadron gas
nucleon gas
nuclei
ρ0
net baryon density
27
¡
Stages in formation of q-­‐g plasma and subsequent hadron emission (a)  2 heavy nuclei collide at HE (b)  interact via color field (c)  de-­‐confinement and plasma formation and possible radiation of photons and lepton-­‐pairs (d)  As plasma cools, hadrons condensate and are emitted 27/04/14
F. Ould-Saada
28
¡
¡
RHIC gold-­‐gold ion collisions at 200 GeV per nucleon Head-­‐on collisions à several 1000 final state particles produced §  View of a 200 GeV gold-­‐gold interaction in STAR detector at RHIC §  ALICE, ATLAS, CMS data at LHC, CERN at some TeV/nucleon à ¡
q-­‐g-­‐plasma signatures: §  Copious production of strange particles in excess of N-­‐N collisions gg → ss
§  gg
→
cc
→
J / ψ à suppression – open charm (D-­‐mesons) favored ¡
¡
Under which conditions can a q-­‐g plasma be made? What are the rules governing evolution and transition to and from this kind of matter? 27/04/14role in understanding F. Ould-Saada
Crucial basic nature of confinement! 29
Pb
Pb
Heavy ion collisions p
π-
π+
27/04/14
¡
Z0à e+e ¡
J/ψà µ+µ-­‐ F. Ould-Saada
31
§  Sign of quark-­‐gluon plasma? §  Jet Quenching Experimental Particle Physics @ UiO
32
Observation of a centrality-­‐dependent dijet asymmetry in lead-­‐lead collisions at √sNN = 2.76 TeV with the ATLAS detector at the LHC http://prl.aps.org/abstract/PRL/v105/i25/e252303 ¡
More on High Energy F.HOuld-Saada
eavy Ions collisions in Chapter 9 and as extra lecture 27/04/14
33
e +e − →qq → jet − jet
+ −
e
€ e → qq → hadrons
2m q ≤
¡
¡
s , pq =
s
2
Quarks not seen as free particles Quark and antiquark hadronise and appear as a flux in a narrow solid angle with the shape of a jet §  Typical momenta of hadrons ~0.5-­‐1 GeV §  Opening angle of “jet” of hadrons à pT
0.5
1
≈
=
p
s /2
s
s = 30GeV ⇒ φ ~ few °
§ 27/04/14
At low energies, hadrons distributed over all solid angle: no jet structure 34
JADE detector at PETRA e+e- collider 27/04/14
35
e + e − → qq → jet + jet
dσ z 2α 2
2
=
1+
cos
θ)
(
dΩ
s
! jet !
q = ∑ hi
i
Quark is spin ½ pointlike particle Absolute value because
quark and antiquark jets
cannot be differentiated
27/04/14
36
e+ e − → qq g → 3 − jets
¡
At typical e+e– energies of 30-­‐100 GeV, a third jet appears in the detector in ~ αs≈ 10% of the time ¡
“Hard” gluon hadronises to a jet in a similar way as the quark and antiquark Guon and quark jets are in general similar
¡
JADE detector at PETRA e+e- collider §  Classify jet energies: E1< E2<E3 §  Gluon=jet 3 in ~70% of the cases 27/04/14
37
e+ e − → qq g → 3 − jets
¡
Define and plot angle φ defined as §  Transform to jet1-jet2 c.o.m system and
compute angle φ between direction of
pair and jet3 §  distribution depends on spin of gluon
TASSO at PETRA
Gluon is spin 1 JP=1– 27/04/14
38
Braibant et al. σ (e e
+ −
R = R0 ≡
u,d,s
u,d,s,c
u,d,s,c,b
€
σ ee
→ hadrons ) ∑ (
=
σ ( e+ e− → µ +µ − )
+ −
n
→ qn qn )
σ ( e+ e− → µ +µ − )
Nf
= N c ∑Qn2
n=1
*# 2 & 2 # 1 & 2 # 1 & 2 ⇒ R = N c ,% + ( + % − ( + % − ( / = 2
+$ 3 ' $ 3 ' $ 3 ' .
*# 2 & 2 # 1 & 2 # 1 & 2 # 2 & 2 - 10
⇒ R = N c ,% + ( + % − ( + % − ( + % + ( / =
+$ 3 ' $ 3 ' $ 3 ' $ 3 ' . 3
*# 2 & 2 # 1 & 2 # 1 & 2 # 2 & 2 # 1 & 2 - 11
⇒ R = N c ,% + ( + % − ( + % − ( + % + ( + % − ( / =
+$ 3 ' $ 3 ' $ 3 ' $ 3 ' $ 3 ' . 3
& αs )
2 + 3 − jets ⇒ R = R0 (1+ +
'
π*
σ (e e
R=
σ (e e
+ −
σ (e e
∑
)
→ hadrons
=
+ −
+ −
f
→ µ +µ − )
→ qf qf )
σ (e+ e− → µ + µ − )
Nq
= N c ∑ e2f
f =1
⎡⎛ 2 ⎞ 2 ⎛ 1 ⎞ 2 ⎛ 1 ⎞ 2 ⎤
⇒ R = 3⎢⎜ + ⎟ + ⎜ − ⎟ + ⎜ − ⎟ ⎥ = 2
⎣⎢⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ ⎦⎥
u, d , s
u , d , s, c
u , d , s , c, b
⎡⎛ 2 ⎞ 2 ⎛ 1 ⎞ 2 ⎛ 1 ⎞ 2 ⎛ 2 ⎞ 2 ⎤ 10
⇒ R = 3⎢⎜ + ⎟ + ⎜ − ⎟ + ⎜ − ⎟ + ⎜ + ⎟ ⎥ =
⎢⎣⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ ⎥⎦ 3
⎡⎛ 2 ⎞ 2 ⎛ 1 ⎞ 2 ⎛ 1 ⎞ 2 ⎛ 2 ⎞ 2 ⎛ 1 ⎞ 2 ⎤ 11
⇒ R = 3⎢⎜ + ⎟ + ⎜ − ⎟ + ⎜ − ⎟ + ⎜ + ⎟ + ⎜ − ⎟ ⎥ =
⎢⎣⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ ⎝ 3 ⎠ ⎥⎦ 3
27/04/14
Question: Evaluate αs at 40 GeV (from figure)
41
¡
Chapter 2 §  Scattering of electrons from nuclei §  à determination of radial charge distributions ▪  Assuming parameterised form of charge distribution ▪  Calculating resulting form factor (Fourrier transform of charge distribution) ▪  Determining unknown parameters by fitting experimental cross-­‐sections ¡
DIS: high energy inelastic scattering §  à determine charge distribution within nucleons §  à first definite evidence for the existence of quarks in 1960s §  à nucleons have sub-­‐structure of point-­‐like charged constituents 27/04/14
F. Ould-Saada
42
2
¡
Scattering of spin-­‐0 point-­‐like projectile of unit charge from a fixed point-­‐like target with electric charge Ze ¡
Electron spin à Mott ¡
¡
¡
Recoil of target à E/E’ term " dσ %
$
'
# dΩ &Mott
Z 2α 2 ( !c )
" dσ %
=
$
'
"θ %
# dΩ &Rutherford
4E 2 sin 4 $ '
#2&
)
,
" dσ %
2
2 "θ %
=$
'
+1− β sin $ '.
# dΩ &Rutherford *
# 2 &-
; β =v/c
,
" dσ %
" dσ % )
q2
2 "θ %
=
1+
tan
$
'
$
' +
$ '.
# dΩ &spin−1/2,recoil # dΩ &Mott * 2M 2
# 2 &"θ %
! !
4EE '
2
−Q 2 = q 2 = ( p − p') = 2me2 c 2 − p p' cosθ ≈ − 2 sin 2 $ '
#2&
c
0
At HE à magnetic moment " d σ
% " d σ % - q 2 2 2 " θ %
2
)
+
=
1+
2(1+
κ
)
tan
+
κ
.
1
$
' $
'
$ '
,
of target in addition to electric 2*
# dΩ & # dΩ &spin−1/2,recoil / 4M
#2&
2
charge µ N = (1+ κ )µ 0
µ 0 = e! / 2M (Dirac point-like)
Spatial extension of nucleus §
à form factor F §  ! Experimental cross section 27/04/14
F. Ould-Saada
" dσ %
" dσ % ! 2 2
=
$
'
$
' F(q)
# dΩ &expt # dΩ &
43
¡
Form Factors, Electric charge distribution and Magnetisation 0
" dσ % " dσ %
q2 )
2
2
2 2+
2 "θ %
2 2
2
=
F
(q
)
+
2(F
(q
)
+
κ
F
(q
))
tan
+
κ
F
(q
)
.
1
$
' $
'
$ '
1
1
2
2
,
2*
# dΩ & # dΩ &spin−1/2,recoil /
#2&
4M
2
F1p (0) = F2p (0) = F2n (0) = 1 ; F1n (0) = 0
¡
Electric and Magnetic Form Factors (normalised) q2
p,n
2
G (q ) = F1 (q ) −
κ
F
(q
)
2
4M 2
GMp,n (q 2 ) = F1p,n (q 2 ) + κ F2p,n (q 2 )
p,n
E
2
p,n
2
GEp (0) = 1; GMp(0) = +2.79; GEn (0) = 0; GMn(0) = −1.91
¡
Rosenbluth formula for ep scattering §
Interference between electric and magnetic form factors 27/04/14
) 2
q2
2
G
+
G
2
+ E
+
M
2
" dσ %
" dσ %
q
2
2 "θ %
4M
+
GM tan $ '.
*
$
' =$
'
2
q2
# dΩ &Ros # dΩ &spin−1/2,recoil +
# 2 &+
2M
1+
,
/
4M 2
F. Ould-Saada
44
¡
Summary §  ep scattering §  From Rutherford to Rosenbluth Parameterisation " dσ %
" dσ %
"θ %
= A(q 2 ) + B(q 2 )tan 2 $ '
$
' /$
'
# dΩ &Ros # dΩ &spin−1/2,recoil
#2&
¡
Scaling law & Dipole formula ¡
GMp(q 2 ) GMn(q 2 )
G(q ) = G (q ) =
=
µp
µn
2
p
E
¡
¡
Scattering mediated by single photon exchange Electric and magnetic form factors of the proton and magnetic form factor of the neutron. 2
GEn (q 2 ) = 0
%
(
'
*
1
*
G(q 2 ) = '
2
2
' Q !"(GeV / c) #$ *
' 1+
*
&
)
0.71
¡
¡
2
f(r)-­‐ Fourier transform of G HE: elastic FF very small, inelastic scattering much more lF.ikely! 27/04/14
Ould-Saada
f (r)dipole = 3.06 e−4.25r
#
1
G (q ) ≈ f (0)%1− 2 2
%$ 6q r
p
E
2
&
( ; r 2 = 0.81 fm
('
46
¡
¡
¡
¡
¡
¡
Spectator Quark Model Variables ν =E-­‐E’ and scaling variable x M: proton mass W: invariant mass final-­‐
state hadrons Q2: squared energy-­‐
momentum transfer 2 independent variables x, ν 27/04/14
F. Ould-Saada
!M $
!E$
! E '$
Proton P = # ! & ; electron p = # ! & ; p' = # ! &
"p%
" p' %
"0 %
!W $
!ν = E − E '$
; Hadronic
system
P
'
=
#! &
! ! &
" q = p − p' %
" P '%
Photon q = # !
4 − momentum transfer
2
− Q 2 = q 2 = ( p − p') = −4EE 'sin 2 (θ / 2)
2
W 2 = ( P + q ) = M 2 + 2P ⋅ q + q 2 = M 2 + 2M ⋅ ν − Q 2 > M 2
Q2
2M ν ≡ W c + Q − M c ;
x≡
2M ν
# ( E − E ') 2
! ! 2&
2
2
Q = −q = −%
− ( p − p') (
2
%$ c
('
ν = E − E ' in proton restframe
2 2
2
2 2
47
¡
Electric and magnetic (not yet taken into account) form factors (Slides 44, 46) à *
d 2σ
4πα 2 E '
2
2
2 !θ $
=
)W2 (Q , ν ) + 2W1 (Q , ν )tan # &,
2
4
" 2 %+
dQ dν
Q E '(
θ : scattering angle
ν :E − E'
W1,W2 : structure functions
F1 (x,Q 2 ) ≡ MW1 (Q 2 , ν )
¡
¡
F2 (x,Q 2 ) ≡ ν W2 (Q 2 , ν )
At fixed value of x , dimensionless structure functions F1,2 have only very weak dependence on Q2 à Scaling behavior 27/04/14
F. Ould-Saada
48
¡
At fixed x , structure functions F1,2 have very weak dependence on Q2 as shown by data for Q2 from 2 to 18 GeV2 §  à Scaling behaviour ¡
As Fourier transform of a spherically symmetric point-­‐like distribution is a constant §  à proton has a sub-­‐
structure of point-­‐like charge constituents 27/04/14
F. Ould-Saada
49
27/04/14
F. Ould-Saada
–
(a) λγ >> dN: the photon "sees" a point-­‐like nucleon –
(b) λγ ~ dN: the cross section depends on q2 through a form factor, F(q2/M2N), corresponding to the charge density of the nucleon. To keep the form factor dimensionless, a mass scale is necessary, taken to be the mass of the nucleon, MN. –
(c) λγ << dN: the photon interacts directly with a parton, independently on the rest of the nucleon. The cross section becomes simpler. 50
¡
Interpretation of scaling simplest in a reference frame where target is moving with very high velocity (Infinite RF) §  pT and rest masses of constituents (partons) may be neglected ¡
Parton Model (partons=quarks+gluons) §  Target nucleon= stream of partons with 4-­‐momentum xP §  x= fraction of nucleon 3-­‐momentum carried by parton in infinite RF §  If one parton (mass: m) scattered by photon (4-­‐momentum: q) !
!
P = ( p, p) ; p = p (M ≈ 0)
( xP + q)
2
= ( x 2 P 2 + 2xP⋅ q + q 2 ) = m 2c 2 ≈ 0
q2
Q2
If x P = x M c << Q ⇒ x = −
=
2P⋅ q 2Mν
2
27/04/14
2
F. Ould-Saada
2
2 2
2
Invariant
P.q
evaluated
in lab
51
¡
x= fraction of nucleon 3-­‐momentum carried by parton in infinite RF ¡
Equivalent to parton of mass m stationary in lab System, with elastic relation €
Q2
x=
2Mν
2
Q
m
Q 2 = 2mν ; if Q 2 >> M 2c 2 ⇒ x =
=
2 Mν M
¡
x= fraction of nucleon mass carried by struck parton ¡
To identify the constituent partons with quarks ¡  need to know spin and electric charge ¡
Parton Spin: Callan-­‐Gross relation $ dσ '
$ dσ ' $ '0
q2 4
2 θ
=&
&
)
) /1 − 2τ tan & )2 ; τ = −
6
% dΩ ( spin −1/ 2 % dΩ ( Mott .
% 2 (1
4m 2c 2 6 2W1
5⇒
= 2τ
2
W
0
$ dσ '
$θ '
dσ
6
2
=&
) /W 2(Q2,ν) + 2W1(Q2,ν)tan 2 & ) 2 6
% 2 (1 7
dΩdE' % dΩ ( Mott .
- F1
F2 0
ν F1
Q2 4
/.W1 → Mc 2 ; W 2 → ν 21 ⇒ Mc 2 F = 4m 2c 2 6
6
2
5
2
Q
6
Q2 = 2mν ⇒ m =
= xM
6
7
2ν
⇒ 2xF1 = F2
F1 (x,Q2 ) =0 ⇒spin = 0
2xF1 (x,Q2 ) = F2 (x,Q2 ) ⇒spin =
27/04/14
F. Ould-Saada
1
2
53
¡
¡
¡
qf(x) : momentum distribution of quark of flavour f qf(x) dx: probability of finding in a nucleon a quark of flavour f with momentum fraction in interval x to x+dx Nucleon= valence quarks (carry observed quantum numbers)+ sea quarks (q-­‐qbar pairs from radiated gluons) [
]
F2 (x) = x ∑ e 2f q f (x) + q f (x)
f
#1 p
&
4 p
1 p
lp
p
p
p
F
(x)
=
x
d
+
d
+
u
+
u
+
s
+
s
) 9(
) 9(
)('
2
%$ 9 (
#1 n
&
4 n
1 n
ln
n
n
n
F2 (x) = x % ( d + d ) + ( u + u ) + ( s + s ) (
$9
'
9
9
Isospin symmetry :u ↔ d ⇒ n ↔ p
⇒ u p (x) = d n (x) ≡ u(x) ; d p (x) = u n (x) ≡ d(x) ; s p (x) = sn (x) ≡ s(x)
€ target : N p = N n
Isoscalar
1 lp
5
1
ln
F
(x)
+
F
(x)
=
x
q(x)
+
q
(x)
+
[
] x [ s(x) + s (x)]
∑
[
]
2
2
2
18 q =u,d
9
27/04/14
F. Ould-Saada
F2lN (x) =
54
[
]
F2 (x) = x ∑ e 2f q f (x) + q f (x)
f
F2lN (x) =
5
x ∑ [q(x) + q (x)]
18 q =u,d
1
+ x [ s(x) + s (x)]
9
F2νN ( x ) = x ∑ [q( x ) + q ( x )]
q = u ,d
⇒ F2νN ( x ) ≤
18 lN
F2 ( x )
5
2
1
⎡18
⎤
Data ⇒ ⎢ F2lN ( x ) ≅ F2νN ( x )⎥ ⇒ Parton Charges : + and −
3
3
⎣ 5
⎦
27/04/14
F. Ould-Saada
55
¡
Scaling is approximately correct but not exact ¡
Deviations from scaling due to QCD corrections to QPM §  Quark can radiate gluon §  Gluon can split into qqbar or gg ¡
Analysis of data with QCD corrections à αs and Λ 27/04/14
F. Ould-Saada
56
à Extraction of parton distributions from combination of cross sections Q 2 = 10(GeV / c)2
Q(x) = d(x) + u(x)
Q (x) = d (x) + u (x)
Qv (x) ≡ Q(x) − Q (x)
Qv at x ≈ 0.2
•  Qv is concentrated at x~0.2 and €
dominates except at small x €
where anti-­‐quarks in sea Q
are important •  Integral over all x à ~0.5 •  50% of momentum carried by Gluons! 27/04/14
F. Ould-Saada
57
27/04/14
F. Ould-Saada
58
27/04/14
F. Ould-Saada
59
¡  Feynman diagrams in QCD §  Interaction of quarks, anti-­‐quarks and gluons ¡  Page 174 §  5.3, 5.4, 5.5 ¡  Page 175 §  5.6, 5.11 27/04/14
F. Ould-Saada
61
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62