QCD (for LHC)Lecture 2: Parton Distribution Functions
Gavin Salam
LPTHE, CNRS and UPMC (Univ. Paris 6)
At the 2009 European School of High-Energy PhysicsJune 2009, Bautzen, Germany
QCD
Lecture 2(The contents of the proton: Parton Distribution
Functions)
QCD lecture 2 (p. 3)
PDF introduction Factorization & parton distributions
Cross section for some hardprocess in hadron-hadroncollisions
x2 p
2
p1 p2
x 1p 1
σ
Z H
σ =
∫
dx1fq/p(x1, µ2)
∫
dx2fq/p(x2, µ2) σ(x1p1, x2p2, µ
2) , s = x1x2s
◮ Total X-section is factorized into a ‘hard part’ σ(x1p1, x2p2, µ2) and
‘normalization’ from parton distribution functions (PDF).
◮ Measure total cross section ↔ need to know PDFs to be able to testhard part (e.g. Higgs electroweak couplings).
◮ Picture seems intuitive, but◮ how can we determine the PDFs? NB: non-perturbative◮ does picture really stand up to QCD corrections?
QCD lecture 2 (p. 3)
PDF introduction Factorization & parton distributions
Cross section for some hardprocess in hadron-hadroncollisions
x2 p
2
p1 p2
x 1p 1
σ
Z H
σ =
∫
dx1fq/p(x1, µ2)
∫
dx2fq/p(x2, µ2) σ(x1p1, x2p2, µ
2) , s = x1x2s
◮ Total X-section is factorized into a ‘hard part’ σ(x1p1, x2p2, µ2) and
‘normalization’ from parton distribution functions (PDF).
◮ Measure total cross section ↔ need to know PDFs to be able to testhard part (e.g. Higgs electroweak couplings).
◮ Picture seems intuitive, but◮ how can we determine the PDFs? NB: non-perturbative◮ does picture really stand up to QCD corrections?
QCD lecture 2 (p. 3)
PDF introduction Factorization & parton distributions
Cross section for some hardprocess in hadron-hadroncollisions
x2 p
2
p1 p2
x 1p 1
σ
Z H
σ =
∫
dx1fq/p(x1, µ2)
∫
dx2fq/p(x2, µ2) σ(x1p1, x2p2, µ
2) , s = x1x2s
◮ Total X-section is factorized into a ‘hard part’ σ(x1p1, x2p2, µ2) and
‘normalization’ from parton distribution functions (PDF).
◮ Measure total cross section ↔ need to know PDFs to be able to testhard part (e.g. Higgs electroweak couplings).
◮ Picture seems intuitive, but◮ how can we determine the PDFs? NB: non-perturbative◮ does picture really stand up to QCD corrections?
QCD lecture 2 (p. 3)
PDF introduction Factorization & parton distributions
Cross section for some hardprocess in hadron-hadroncollisions
x2 p
2
p1 p2
x 1p 1
σ
Z H
σ =
∫
dx1fq/p(x1, µ2)
∫
dx2fq/p(x2, µ2) σ(x1p1, x2p2, µ
2) , s = x1x2s
◮ Total X-section is factorized into a ‘hard part’ σ(x1p1, x2p2, µ2) and
‘normalization’ from parton distribution functions (PDF).
◮ Measure total cross section ↔ need to know PDFs to be able to testhard part (e.g. Higgs electroweak couplings).
◮ Picture seems intuitive, but◮ how can we determine the PDFs? NB: non-perturbative◮ does picture really stand up to QCD corrections?
QCD lecture 2 (p. 3)
PDF introduction Factorization & parton distributions
Cross section for some hardprocess in hadron-hadroncollisions
x2 p
2
p1 p2
x 1p 1
σ
Z H
σ =
∫
dx1fq/p(x1, µ2)
∫
dx2fq/p(x2, µ2) σ(x1p1, x2p2, µ
2) , s = x1x2s
◮ Total X-section is factorized into a ‘hard part’ σ(x1p1, x2p2, µ2) and
‘normalization’ from parton distribution functions (PDF).
◮ Measure total cross section ↔ need to know PDFs to be able to testhard part (e.g. Higgs electroweak couplings).
◮ Picture seems intuitive, but◮ how can we determine the PDFs? NB: non-perturbative◮ does picture really stand up to QCD corrections?
QCD lecture 2 (p. 4)
PDF introduction
DIS kinematicsDeep Inelastic Scattering: kinematics
Hadron-hadron is complex because of two incoming partons — so startwith simpler Deep Inelastic Scattering (DIS).
e+
xp
k
p
"(1−y)k"
q (Q 2 = −q2)
proton
Kinematic relations:
x =Q2
2p.q; y =
p.q
p.k; Q2 = xys
√s = c.o.m. energy
◮ Q2 = photon virtuality ↔ transverseresolution at which it probes protonstructure
◮ x = longitudinal momentum fraction ofstruck parton in proton
◮ y = momentum fraction lost by electron(in proton rest frame)
QCD lecture 2 (p. 4)
PDF introduction
DIS kinematicsDeep Inelastic Scattering: kinematics
Hadron-hadron is complex because of two incoming partons — so startwith simpler Deep Inelastic Scattering (DIS).
e+
xp
k
p
"(1−y)k"
q (Q 2 = −q2)
proton
Kinematic relations:
x =Q2
2p.q; y =
p.q
p.k; Q2 = xys
√s = c.o.m. energy
◮ Q2 = photon virtuality ↔ transverseresolution at which it probes protonstructure
◮ x = longitudinal momentum fraction ofstruck parton in proton
◮ y = momentum fraction lost by electron(in proton rest frame)
QCD lecture 2 (p. 5)
PDF introduction
DIS kinematicsDeep Inelastic scattering (DIS): example
Q2 = 25030 GeV 2; y = 0:56;
e+
x=0.50
e+
Q2
x
proton
e+
jet
proton
jet
QCD lecture 2 (p. 6)
PDF introduction
DIS X-sectionsE.g.: extracting u & d distributions
Write DIS X-section to zeroth order in αs (‘quark parton model’):
d2σem
dxdQ2≃ 4πα2
xQ4
(1 + (1− y)2
2F em
2 +O (αs)
)
∝ F em2 [structure function]
F2 = x(e2uu(x) + e2
dd(x)) = x
(4
9u(x) +
1
9d(x)
)
[u(x), d(x): parton distribution functions (PDF)]
NB:
◮ use perturbative language for interactions of up and down quarks
◮ but distributions themselves have a non-perturbative origin.
QCD lecture 2 (p. 6)
PDF introduction
DIS X-sectionsE.g.: extracting u & d distributions
Write DIS X-section to zeroth order in αs (‘quark parton model’):
d2σem
dxdQ2≃ 4πα2
xQ4
(1 + (1− y)2
2F em
2 +O (αs)
)
∝ F em2 [structure function]
F2 = x(e2uu(x) + e2
dd(x)) = x
(4
9u(x) +
1
9d(x)
)
[u(x), d(x): parton distribution functions (PDF)]
NB:
◮ use perturbative language for interactions of up and down quarks
◮ but distributions themselves have a non-perturbative origin.
F2 gives us combination of u and d .How can we extract them separately?
QCD lecture 2 (p. 7)
Quark distributions
u & d
Using neutrons
Assumption (SU(2) isospin): neutron is just proton with u ⇔ d :
proton = uud; neutron = ddu
Isospin: un(x) = dp(x) , dn(x) = up(x)
F p2 =
4
9up(x) +
1
9dp(x)
F n2 =
4
9un(x) +
1
9dn(x) =
4
9dp(x) +
1
9up(x)
Linear combinations of F p2 and F n
2 give separately up(x) and dp(x).
Experimentally, get F n2 from deuterons: F d
2 =1
2(F p
2 + F n2 )
QCD lecture 2 (p. 7)
Quark distributions
u & d
Using neutrons
Assumption (SU(2) isospin): neutron is just proton with u ⇔ d :
proton = uud; neutron = ddu
Isospin: un(x) = dp(x) , dn(x) = up(x)
F p2 =
4
9up(x) +
1
9dp(x)
F n2 =
4
9un(x) +
1
9dn(x) =
4
9dp(x) +
1
9up(x)
Linear combinations of F p2 and F n
2 give separately up(x) and dp(x).
Experimentally, get F n2 from deuterons: F d
2 =1
2(F p
2 + F n2 )
QCD lecture 2 (p. 7)
Quark distributions
u & d
Using neutrons
Assumption (SU(2) isospin): neutron is just proton with u ⇔ d :
proton = uud; neutron = ddu
Isospin: un(x) = dp(x) , dn(x) = up(x)
F p2 =
4
9up(x) +
1
9dp(x)
F n2 =
4
9un(x) +
1
9dn(x) =
4
9dp(x) +
1
9up(x)
Linear combinations of F p2 and F n
2 give separately up(x) and dp(x).
Experimentally, get F n2 from deuterons: F d
2 =1
2(F p
2 + F n2 )
QCD lecture 2 (p. 8)
Quark distributions
u & d
NMC proton & deuteron data
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
x
Q2 = 27 GeV2
NMC data
xf(x
)
3F p2 − 6
5F d2 →“xu(x)”
−3F p2 + 24
5 F d2 →“xd(x)”
Combine F p2 & F d
2 data,deduce u(x), d(x):
◮ Definitely more up thandown (✓)
How much u and d?
◮ Total U =∫
dx u(x)
◮ F2 = x(49u + 1
9d)
◮ u(x) ∼ d(x) ∼ x−1.25
non-integrabledivergence
So why do we sayproton = uud?
QCD lecture 2 (p. 8)
Quark distributions
u & d
NMC proton & deuteron data
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
x
Q2 = 27 GeV2
NMC data
CTEQ6D fit
xf(x
)
3F p2 − 6
5F d2 →“xu(x)”
−3F p2 + 24
5 F d2 →“xd(x)”
Combine F p2 & F d
2 data,deduce u(x), d(x):
◮ Definitely more up thandown (✓)
How much u and d?
◮ Total U =∫
dx u(x)
◮ F2 = x(49u + 1
9d)
◮ u(x) ∼ d(x) ∼ x−1.25
non-integrabledivergence
So why do we sayproton = uud?
QCD lecture 2 (p. 8)
Quark distributions
u & d
NMC proton & deuteron data
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
x
Q2 = 27 GeV2
NMC data
CTEQ6D fit
xf(x
)
3F p2 − 6
5F d2 →“xu(x)”
−3F p2 + 24
5 F d2 →“xd(x)”
Combine F p2 & F d
2 data,deduce u(x), d(x):
◮ Definitely more up thandown (✓)
How much u and d?
◮ Total U =∫
dx u(x)
◮ F2 = x(49u + 1
9d)
◮ u(x) ∼ d(x) ∼ x−1.25
non-integrabledivergence
So why do we sayproton = uud?
QCD lecture 2 (p. 8)
Quark distributions
u & d
NMC proton & deuteron data
0
0.2
0.4
0.6
0.8
1
0 0.2 0.4 0.6 0.8 1
x
Q2 = 27 GeV2
NMC data
CTEQ6D fit
xf(x
)
3F p2 − 6
5F d2 →“xu(x)”
−3F p2 + 24
5 F d2 →“xd(x)”
Combine F p2 & F d
2 data,deduce u(x), d(x):
◮ Definitely more up thandown (✓)
How much u and d?
◮ Total U =∫
dx u(x)
◮ F2 = x(49u + 1
9d)
◮ u(x) ∼ d(x) ∼ x−1.25
non-integrabledivergence
So why do we sayproton = uud?
QCD lecture 2 (p. 8)
Quark distributions
u & d
NMC proton & deuteron data
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
x
Q2 = 27 GeV2
NMC data
CTEQ6D fit
f(x
)
F p,d2 /x →“u(x)”
F p,d2 /x →“d(x)”
Combine F p2 & F d
2 data,deduce u(x), d(x):
◮ Definitely more up thandown (✓)
How much u and d?
◮ Total U =∫
dx u(x)
◮ F2 = x(49u + 1
9d)
◮ u(x) ∼ d(x) ∼ x−1.25
non-integrabledivergence
So why do we sayproton = uud?
QCD lecture 2 (p. 8)
Quark distributions
u & d
NMC proton & deuteron data
0
2
4
6
8
10
0 0.2 0.4 0.6 0.8 1
x
Q2 = 27 GeV2
NMC data
CTEQ6D fit
f(x
)
F p,d2 /x →“u(x)”
F p,d2 /x →“d(x)”
Combine F p2 & F d
2 data,deduce u(x), d(x):
◮ Definitely more up thandown (✓)
How much u and d?
◮ Total U =∫
dx u(x)
◮ F2 = x(49u + 1
9d)
◮ u(x) ∼ d(x) ∼ x−1.25
non-integrabledivergence
So why do we sayproton = uud?
QCD lecture 2 (p. 9)
Quark distributions
Sea & valenceAnti-quarks in proton
u
d
u
u
d
u
uu
u
d
u
uu
d
d
How can there be infinite number ofquarks in proton?
Proton wavefunction fluctuates — extrauu, dd pairs (sea quarks) can appear:
Antiquarks also have distributions, u(x), d(x)
F2 =4
9(xu(x) + xu(x)) +
1
9(xd(x) + xd(x))
NB: photon interaction ∼ square of charge → +ve
◮ Previous transparency: we were actually looking at ∼ u + u, d + d
◮ Number of extra quark-antiquark pairs can be infinite, so∫
dx (u(x) + u(x)) =∞
as long as they carry little momentum (mostly at low x)
QCD lecture 2 (p. 9)
Quark distributions
Sea & valenceAnti-quarks in proton
u
d
u
u
d
u
uu
u
d
u
uu
d
d
How can there be infinite number ofquarks in proton?
Proton wavefunction fluctuates — extrauu, dd pairs (sea quarks) can appear:
Antiquarks also have distributions, u(x), d(x)
F2 =4
9(xu(x) + xu(x)) +
1
9(xd(x) + xd(x))
NB: photon interaction ∼ square of charge → +ve
◮ Previous transparency: we were actually looking at ∼ u + u, d + d
◮ Number of extra quark-antiquark pairs can be infinite, so∫
dx (u(x) + u(x)) =∞
as long as they carry little momentum (mostly at low x)
QCD lecture 2 (p. 9)
Quark distributions
Sea & valenceAnti-quarks in proton
u
d
u
u
d
u
uu
u
d
u
uu
d
d
How can there be infinite number ofquarks in proton?
Proton wavefunction fluctuates — extrauu, dd pairs (sea quarks) can appear:
Antiquarks also have distributions, u(x), d(x)
F2 =4
9(xu(x) + xu(x)) +
1
9(xd(x) + xd(x))
NB: photon interaction ∼ square of charge → +ve
◮ Previous transparency: we were actually looking at ∼ u + u, d + d
◮ Number of extra quark-antiquark pairs can be infinite, so∫
dx (u(x) + u(x)) =∞
as long as they carry little momentum (mostly at low x)
QCD lecture 2 (p. 10)
Quark distributions
Sea & valence“Valence” quarks
When we say proton has 2 up quarks & 1 down quark we mean
∫
dx (u(x)− u(x)) = 2 ,
∫
dx (d(x)− d(x)) = 1
u − u = uV is known as a valence distribution.
How do we measure difference between u and u? Photon interactsidentically with both → no good. . .
Question: what interacts differently with particle & antiparticle?
Answer: W + or W−
QCD lecture 2 (p. 10)
Quark distributions
Sea & valence“Valence” quarks
When we say proton has 2 up quarks & 1 down quark we mean
∫
dx (u(x)− u(x)) = 2 ,
∫
dx (d(x)− d(x)) = 1
u − u = uV is known as a valence distribution.
How do we measure difference between u and u? Photon interactsidentically with both → no good. . .
Question: what interacts differently with particle & antiparticle?
Answer: W + or W−
QCD lecture 2 (p. 10)
Quark distributions
Sea & valence“Valence” quarks
When we say proton has 2 up quarks & 1 down quark we mean
∫
dx (u(x)− u(x)) = 2 ,
∫
dx (d(x)− d(x)) = 1
u − u = uV is known as a valence distribution.
How do we measure difference between u and u? Photon interactsidentically with both → no good. . .
Question: what interacts differently with particle & antiparticle?
Answer: W + or W−
QCD lecture 2 (p. 11)
Quark distributions
SummaryAll quarks
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
x
quarks: xq(x)
Q2 = 10 GeV2
uVdV
dS
uSsScS
CTEQ6D fit
These & other methods→ whole setof quarks & antiquarks
NB: also strange and charm quarks
◮ valence quarks (uV = u − u) arehardx → 1 : xqV (x) ∼ (1− x)3
quark counting rules
x → 0 : xqV (x) ∼ x0.5
Regge theory
◮ sea quarks (uS = 2u, . . . ) fairlysoft (low-momentum)x → 1 : xqS(x) ∼ (1− x)7
x → 0 : xqS(x) ∼ x−0.2
QCD lecture 2 (p. 11)
Quark distributions
SummaryAll quarks
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
x
quarks: xq(x)
Q2 = 10 GeV2
uVdV
dS
uSsScS
CTEQ6D fit
These & other methods→ whole setof quarks & antiquarks
NB: also strange and charm quarks
◮ valence quarks (uV = u − u) arehardx → 1 : xqV (x) ∼ (1− x)3
quark counting rules
x → 0 : xqV (x) ∼ x0.5
Regge theory
◮ sea quarks (uS = 2u, . . . ) fairlysoft (low-momentum)x → 1 : xqS(x) ∼ (1− x)7
x → 0 : xqS(x) ∼ x−0.2
QCD lecture 2 (p. 11)
Quark distributions
SummaryAll quarks
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1
x
quarks: xq(x)
Q2 = 10 GeV2
uVdV
dS
uSsScS
CTEQ6D fit
These & other methods→ whole setof quarks & antiquarks
NB: also strange and charm quarks
◮ valence quarks (uV = u − u) arehardx → 1 : xqV (x) ∼ (1− x)3
quark counting rules
x → 0 : xqV (x) ∼ x0.5
Regge theory
◮ sea quarks (uS = 2u, . . . ) fairlysoft (low-momentum)x → 1 : xqS(x) ∼ (1− x)7
x → 0 : xqS(x) ∼ x−0.2
QCD lecture 2 (p. 12)
Quark distributions
SummaryMomentum sum rule
Check momentum sum-rule (sum over all species carries all momentum):
∑
i
∫
dx xqi (x) = 1
qi momentum
dV 0.111uV 0.267dS 0.066uS 0.053sS 0.033cS 0.016
total 0.546
Where is missing momentum?Only parton type we’ve neglected so far is the
gluon
Not directly probed by photon or W±.NB: need to know it for gg → H
To discuss gluons we must go beyond ‘naive’leading order picture, and bring in QCD split-ting. . .
QCD lecture 2 (p. 12)
Quark distributions
SummaryMomentum sum rule
Check momentum sum-rule (sum over all species carries all momentum):
∑
i
∫
dx xqi (x) = 1
qi momentum
dV 0.111uV 0.267dS 0.066uS 0.053sS 0.033cS 0.016
total 0.546
Where is missing momentum?Only parton type we’ve neglected so far is the
gluon
Not directly probed by photon or W±.NB: need to know it for gg → H
To discuss gluons we must go beyond ‘naive’leading order picture, and bring in QCD split-ting. . .
QCD lecture 2 (p. 12)
Quark distributions
SummaryMomentum sum rule
Check momentum sum-rule (sum over all species carries all momentum):
∑
i
∫
dx xqi (x) = 1
qi momentum
dV 0.111uV 0.267dS 0.066uS 0.053sS 0.033cS 0.016
total 0.546
Where is missing momentum?Only parton type we’ve neglected so far is the
gluon
Not directly probed by photon or W±.NB: need to know it for gg → H
To discuss gluons we must go beyond ‘naive’leading order picture, and bring in QCD split-ting. . .
QCD lecture 2 (p. 12)
Quark distributions
SummaryMomentum sum rule
Check momentum sum-rule (sum over all species carries all momentum):
∑
i
∫
dx xqi (x) = 1
qi momentum
dV 0.111uV 0.267dS 0.066uS 0.053sS 0.033cS 0.016
total 0.546
Where is missing momentum?Only parton type we’ve neglected so far is the
gluon
Not directly probed by photon or W±.NB: need to know it for gg → H
To discuss gluons we must go beyond ‘naive’leading order picture, and bring in QCD split-ting. . .
QCD lecture 2 (p. 12)
Quark distributions
SummaryMomentum sum rule
Check momentum sum-rule (sum over all species carries all momentum):
∑
i
∫
dx xqi (x) = 1
qi momentum
dV 0.111uV 0.267dS 0.066uS 0.053sS 0.033cS 0.016
total 0.546
Where is missing momentum?Only parton type we’ve neglected so far is the
gluon
Not directly probed by photon or W±.NB: need to know it for gg → H
To discuss gluons we must go beyond ‘naive’leading order picture, and bring in QCD split-ting. . .
QCD lecture 2 (p. 13)
Initial-state splitting
1st order analysisRecall final-state splitting
Previous lecture: calculated q → qg (θ ≪ 1, E ≪ p) for final state ofarbitrary hard process (σh):
σh+g ≃ σh
αsCF
π
dE
E
dθ2
θ2
ω =
pzp
(1−z)p
θσh
Rewrite with different kinematic variables
σh+g ≃ σh
αsCF
π
dz
1− z
dk2t
k2t
E = (1− z)pkt = E sin θ ≃ Eθ
If we avoid distinguishing q + g final state from q (infrared-collinear safety),then divergent real and virtual corrections cancel
σh+V ≃ −σh
αsCF
π
dz
1− z
dk2t
k2t
p pσh
QCD lecture 2 (p. 14)
Initial-state splitting
1st order analysisInitial-state splitting
For initial state splitting, hard process occurs after splitting, andmomentum entering hard process is modified: p → zp.
σg+h(p) ≃ σh(zp)αsCF
π
dz
1− z
dk2t
k2t
zpp
(1−z)p
σh
For virtual terms, momentum entering hard process is unchanged
σV+h(p) ≃ −σh(p)αsCF
π
dz
1− z
dk2t
k2t
p pσh
Total cross section gets contribution with two different hard X-sections
σg+h + σV+h ≃αsCF
π
∫dk2
t
k2t
dz
1− z[σh(zp)− σh(p)]
NB: We assume σh involves momentum transfers ∼ Q ≫ kt , so ignore extra
transverse momentum in σh
QCD lecture 2 (p. 15)
Initial-state splitting
1st order analysisInitial-state collinear divergence
σg+h + σV+h ≃αsCF
π
∫ Q2
0
dk2t
k2t
︸ ︷︷ ︸
infinite
∫dz
1− z[σh(zp)− σh(p)]
︸ ︷︷ ︸
finite
◮ In soft limit (z → 1), σh(zp)− σh(p)→ 0: soft divergence cancels.
◮ For 1− z 6= 0, σh(zp)− σh(p) 6= 0, so z integral is non-zero but finite.
BUT: kt integral is just a factor, and is infiniteThis is a collinear (kt → 0) divergence.
Cross section with incoming parton is not collinear safe!
This always happens with coloured initial-state particlesSo how do we do QCD calculations in such cases?
QCD lecture 2 (p. 15)
Initial-state splitting
1st order analysisInitial-state collinear divergence
σg+h + σV+h ≃αsCF
π
∫ Q2
0
dk2t
k2t
︸ ︷︷ ︸
infinite
∫dz
1− z[σh(zp)− σh(p)]
︸ ︷︷ ︸
finite
◮ In soft limit (z → 1), σh(zp)− σh(p)→ 0: soft divergence cancels.
◮ For 1− z 6= 0, σh(zp)− σh(p) 6= 0, so z integral is non-zero but finite.
BUT: kt integral is just a factor, and is infiniteThis is a collinear (kt → 0) divergence.
Cross section with incoming parton is not collinear safe!
This always happens with coloured initial-state particlesSo how do we do QCD calculations in such cases?
QCD lecture 2 (p. 15)
Initial-state splitting
1st order analysisInitial-state collinear divergence
σg+h + σV+h ≃αsCF
π
∫ Q2
0
dk2t
k2t
︸ ︷︷ ︸
infinite
∫dz
1− z[σh(zp)− σh(p)]
︸ ︷︷ ︸
finite
◮ In soft limit (z → 1), σh(zp)− σh(p)→ 0: soft divergence cancels.
◮ For 1− z 6= 0, σh(zp)− σh(p) 6= 0, so z integral is non-zero but finite.
BUT: kt integral is just a factor, and is infiniteThis is a collinear (kt → 0) divergence.
Cross section with incoming parton is not collinear safe!
This always happens with coloured initial-state particlesSo how do we do QCD calculations in such cases?
QCD lecture 2 (p. 15)
Initial-state splitting
1st order analysisInitial-state collinear divergence
σg+h + σV+h ≃αsCF
π
∫ Q2
0
dk2t
k2t
︸ ︷︷ ︸
infinite
∫dz
1− z[σh(zp)− σh(p)]
︸ ︷︷ ︸
finite
◮ In soft limit (z → 1), σh(zp)− σh(p)→ 0: soft divergence cancels.
◮ For 1− z 6= 0, σh(zp)− σh(p) 6= 0, so z integral is non-zero but finite.
BUT: kt integral is just a factor, and is infiniteThis is a collinear (kt → 0) divergence.
Cross section with incoming parton is not collinear safe!
This always happens with coloured initial-state particlesSo how do we do QCD calculations in such cases?
QCD lecture 2 (p. 16)
Initial-state splitting
1st order analysisCollinear cutoff
Q 2
pxp
zxp
(1−z)xp
σh
1 GeV 2
By what right did we go to kt = 0?
We assumed pert. QCD to be valid forall scales, but below 1 GeV it becomesnon-perturbative.
Cut out this divergent region, & insteadput non-perturbative quark distributionin proton.
σ0 =
∫
dx σh(xp) q(x , 1 GeV2)
σ1 ≃αsCF
π
∫ Q2
1 GeV2
dk2t
k2t
︸ ︷︷ ︸
finite (large)
∫dx dz
1− z[σh(zxp)− σh(xp)] q(x , 1 GeV2)
︸ ︷︷ ︸
finite
In general: replace 1 GeV2 cutoff with arbitrary factorization scale µ2.
QCD lecture 2 (p. 16)
Initial-state splitting
1st order analysisCollinear cutoff
Q 2
pxp
zxp
(1−z)xp
σh
µ 2
By what right did we go to kt = 0?
We assumed pert. QCD to be valid forall scales, but below 1 GeV it becomesnon-perturbative.
Cut out this divergent region, & insteadput non-perturbative quark distributionin proton.
σ0 =
∫
dx σh(xp) q(x , µ2)
σ1 ≃αsCF
π
∫ Q2
µ2
dk2t
k2t
︸ ︷︷ ︸
finite (large)
∫dx dz
1− z[σh(zxp)− σh(xp)] q(x , µ2)
︸ ︷︷ ︸
finite
In general: replace 1 GeV2 cutoff with arbitrary factorization scale µ2.
QCD lecture 2 (p. 17)
Initial-state splitting
1st order analysisSummary so far
◮ Collinear divergence for incoming partons not cancelled by virtuals.Real and virtual have different longitudinal momenta
◮ Situation analogous to renormalization: need to regularize (but in IRinstead of UV).
Technically, often done with dimensional regularization
◮ Physical sense of regularization is to separate (factorize) protonnon-perturbative dynamics from perturbative hard cross section.
Choice of factorization scale, µ2, is arbitrary between 1 GeV2 and Q2
◮ In analogy with running coupling, we can vary factorization scale and geta renormalization group equation for parton distribution functions.
Dokshizer Gribov Lipatov Altarelli Parisi equations (DGLAP)
QCD lecture 2 (p. 17)
Initial-state splitting
1st order analysisSummary so far
◮ Collinear divergence for incoming partons not cancelled by virtuals.Real and virtual have different longitudinal momenta
◮ Situation analogous to renormalization: need to regularize (but in IRinstead of UV).
Technically, often done with dimensional regularization
◮ Physical sense of regularization is to separate (factorize) protonnon-perturbative dynamics from perturbative hard cross section.
Choice of factorization scale, µ2, is arbitrary between 1 GeV2 and Q2
◮ In analogy with running coupling, we can vary factorization scale and geta renormalization group equation for parton distribution functions.
Dokshizer Gribov Lipatov Altarelli Parisi equations (DGLAP)
Q2
increase
Q2
increase
uuu
gg
gdu
ud d
ug
gu u
QCD lecture 2 (p. 18)
Initial-state splitting
DGLAPDGLAP equation (q ← q)
Change convention: (a) now fix outgoing longitudinal momentum x ; (b)take derivative wrt factorization scale µ2
p
x
xp
x
x/z x(1−z)/z
(1+δ)µ2(1+δ)µ2
µ 2µ 2
+
dq(x , µ2)
d lnµ2=
αs
2π
∫ 1
x
dz pqq(z)q(x/z , µ2)
z− αs
2π
∫ 1
0dz pqq(z) q(x , µ2)
pqq is real q ← q splitting kernel: pqq(z) = CF
1 + z2
1− z
Until now we approximated it in soft (z → 1) limit, pqq ≃ 2CF
1−z
QCD lecture 2 (p. 19)
Initial-state splitting
DGLAPDGLAP rewritten
Awkward to write real and virtual parts separately. Use more compactnotation:
dq(x , µ2)
d lnµ2=
αs
2π
∫ 1
x
dz Pqq(z)q(x/z , µ2)
z︸ ︷︷ ︸
Pqq⊗q
, Pqq = CF
(1 + z2
1− z
)
+
This involves the plus prescription:
∫ 1
0dz [g(z)]+ f (z) =
∫ 1
0dz g(z) f (z)−
∫ 1
0dz g(z) f (1)
z = 1 divergences of g(z) cancelled if f (z) sufficiently smooth at z = 1
QCD lecture 2 (p. 20)
Initial-state splitting
DGLAPDGLAP flavour structure
Proton contains both quarks and gluons — so DGLAP is a matrix in flavourspace:
d
d lnQ2
(qg
)
=
(Pq←q Pq←g
Pg←q Pg←g
)
⊗(
qg
)
[In general, matrix spanning all flavors, anti-flavors, Pqq′ = 0 (LO), Pqg = Pqg ]
Splitting functions are:
Pqg (z) = TR
[z2 + (1− z)2
], Pgq(z) = CF
[1 + (1− z)2
z
]
,
Pgg (z) = 2CA
[z
(1− z)++
1− z
z+ z(1 − z)
]
+ δ(1− z)(11CA − 4nf TR)
6.
Have various symmetries / significant properties, e.g.
◮ Pqg , Pgg : symmetric z ↔ 1− z (except virtuals)
◮ Pqq, Pgg : diverge for z → 1 soft gluon emission
◮ Pgg , Pgq: diverge for z → 0 Implies PDFs grow for x → 0
QCD lecture 2 (p. 21)
Initial-state splitting
Example evolutionEffect of DGLAP (initial quarks)
0
0.5
1
1.5
2
2.5
3
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 12.0 GeV2
xg(x,Q2)xq + xqbar
Take example evolution starting withjust quarks:
∂lnQ2q = Pq←q ⊗ q∂lnQ2g = Pg←q ⊗ q
◮ quark is depleted at large x
◮ gluon grows at small x
QCD lecture 2 (p. 21)
Initial-state splitting
Example evolutionEffect of DGLAP (initial quarks)
0
0.5
1
1.5
2
2.5
3
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 15.0 GeV2
xg(x,Q2)xq + xqbar
Take example evolution starting withjust quarks:
∂lnQ2q = Pq←q ⊗ q∂lnQ2g = Pg←q ⊗ q
◮ quark is depleted at large x
◮ gluon grows at small x
QCD lecture 2 (p. 21)
Initial-state splitting
Example evolutionEffect of DGLAP (initial quarks)
0
0.5
1
1.5
2
2.5
3
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 27.0 GeV2
xg(x,Q2)xq + xqbar
Take example evolution starting withjust quarks:
∂lnQ2q = Pq←q ⊗ q∂lnQ2g = Pg←q ⊗ q
◮ quark is depleted at large x
◮ gluon grows at small x
QCD lecture 2 (p. 21)
Initial-state splitting
Example evolutionEffect of DGLAP (initial quarks)
0
0.5
1
1.5
2
2.5
3
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 35.0 GeV2
xg(x,Q2)xq + xqbar
Take example evolution starting withjust quarks:
∂lnQ2q = Pq←q ⊗ q∂lnQ2g = Pg←q ⊗ q
◮ quark is depleted at large x
◮ gluon grows at small x
QCD lecture 2 (p. 21)
Initial-state splitting
Example evolutionEffect of DGLAP (initial quarks)
0
0.5
1
1.5
2
2.5
3
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 46.0 GeV2
xg(x,Q2)xq + xqbar
Take example evolution starting withjust quarks:
∂lnQ2q = Pq←q ⊗ q∂lnQ2g = Pg←q ⊗ q
◮ quark is depleted at large x
◮ gluon grows at small x
QCD lecture 2 (p. 21)
Initial-state splitting
Example evolutionEffect of DGLAP (initial quarks)
0
0.5
1
1.5
2
2.5
3
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 60.0 GeV2
xg(x,Q2)xq + xqbar
Take example evolution starting withjust quarks:
∂lnQ2q = Pq←q ⊗ q∂lnQ2g = Pg←q ⊗ q
◮ quark is depleted at large x
◮ gluon grows at small x
QCD lecture 2 (p. 21)
Initial-state splitting
Example evolutionEffect of DGLAP (initial quarks)
0
0.5
1
1.5
2
2.5
3
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 90.0 GeV2
xg(x,Q2)xq + xqbar
Take example evolution starting withjust quarks:
∂lnQ2q = Pq←q ⊗ q∂lnQ2g = Pg←q ⊗ q
◮ quark is depleted at large x
◮ gluon grows at small x
QCD lecture 2 (p. 21)
Initial-state splitting
Example evolutionEffect of DGLAP (initial quarks)
0
0.5
1
1.5
2
2.5
3
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 150.0 GeV2
xg(x,Q2)xq + xqbar
Take example evolution starting withjust quarks:
∂lnQ2q = Pq←q ⊗ q∂lnQ2g = Pg←q ⊗ q
◮ quark is depleted at large x
◮ gluon grows at small x
QCD lecture 2 (p. 22)
Initial-state splitting
Example evolutionEffect of DGLAP (initial gluons)
0
1
2
3
4
5
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 12.0 GeV2
xg(x,Q2)xq + xqbar
2nd example: start with just gluons.
∂lnQ2q = Pq←g ⊗ g∂lnQ2g = Pg←g ⊗ g
◮ gluon is depleted at large x .
◮ high-x gluon feeds growth ofsmall x gluon & quark.
QCD lecture 2 (p. 22)
Initial-state splitting
Example evolutionEffect of DGLAP (initial gluons)
0
1
2
3
4
5
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 15.0 GeV2
xg(x,Q2)xq + xqbar
2nd example: start with just gluons.
∂lnQ2q = Pq←g ⊗ g∂lnQ2g = Pg←g ⊗ g
◮ gluon is depleted at large x .
◮ high-x gluon feeds growth ofsmall x gluon & quark.
QCD lecture 2 (p. 22)
Initial-state splitting
Example evolutionEffect of DGLAP (initial gluons)
0
1
2
3
4
5
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 27.0 GeV2
xg(x,Q2)xq + xqbar
2nd example: start with just gluons.
∂lnQ2q = Pq←g ⊗ g∂lnQ2g = Pg←g ⊗ g
◮ gluon is depleted at large x .
◮ high-x gluon feeds growth ofsmall x gluon & quark.
QCD lecture 2 (p. 22)
Initial-state splitting
Example evolutionEffect of DGLAP (initial gluons)
0
1
2
3
4
5
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 35.0 GeV2
xg(x,Q2)xq + xqbar
2nd example: start with just gluons.
∂lnQ2q = Pq←g ⊗ g∂lnQ2g = Pg←g ⊗ g
◮ gluon is depleted at large x .
◮ high-x gluon feeds growth ofsmall x gluon & quark.
QCD lecture 2 (p. 22)
Initial-state splitting
Example evolutionEffect of DGLAP (initial gluons)
0
1
2
3
4
5
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 46.0 GeV2
xg(x,Q2)xq + xqbar
2nd example: start with just gluons.
∂lnQ2q = Pq←g ⊗ g∂lnQ2g = Pg←g ⊗ g
◮ gluon is depleted at large x .
◮ high-x gluon feeds growth ofsmall x gluon & quark.
QCD lecture 2 (p. 22)
Initial-state splitting
Example evolutionEffect of DGLAP (initial gluons)
0
1
2
3
4
5
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 60.0 GeV2
xg(x,Q2)xq + xqbar
2nd example: start with just gluons.
∂lnQ2q = Pq←g ⊗ g∂lnQ2g = Pg←g ⊗ g
◮ gluon is depleted at large x .
◮ high-x gluon feeds growth ofsmall x gluon & quark.
QCD lecture 2 (p. 22)
Initial-state splitting
Example evolutionEffect of DGLAP (initial gluons)
0
1
2
3
4
5
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 90.0 GeV2
xg(x,Q2)xq + xqbar
2nd example: start with just gluons.
∂lnQ2q = Pq←g ⊗ g∂lnQ2g = Pg←g ⊗ g
◮ gluon is depleted at large x .
◮ high-x gluon feeds growth ofsmall x gluon & quark.
QCD lecture 2 (p. 22)
Initial-state splitting
Example evolutionEffect of DGLAP (initial gluons)
0
1
2
3
4
5
0.01 0.1 1
x
xq(x,Q2), xg(x,Q2)
Q2 = 150.0 GeV2
xg(x,Q2)xq + xqbar
2nd example: start with just gluons.
∂lnQ2q = Pq←g ⊗ g∂lnQ2g = Pg←g ⊗ g
◮ gluon is depleted at large x .
◮ high-x gluon feeds growth ofsmall x gluon & quark.
QCD lecture 2 (p. 23)
Initial-state splitting
Example evolutionDGLAP evolution
◮ As Q2 increases, partons lose longitudinal momentum; distributions allshift to lower x .
◮ gluons can be seen because they help drive the quark evolution.
Now consider data
QCD lecture 2 (p. 24)
Determining gluon
Evolution versus dataDGLAP with initial gluon = 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 12.0 GeV2
DGLAP: g(x,Q02) = 0
ZEUS
NMC
Fit quark distributions to F2(x ,Q20 ),
at initial scale Q20 = 12 GeV2.
NB: Q0 often chosen lower
Assume there is no gluon at Q20 :
g(x ,Q20 ) = 0
Use DGLAP equations to evolve tohigher Q2; compare with data.
Complete failure!
QCD lecture 2 (p. 24)
Determining gluon
Evolution versus dataDGLAP with initial gluon = 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 15.0 GeV2
DGLAP: g(x,Q02) = 0
ZEUS
NMC
Fit quark distributions to F2(x ,Q20 ),
at initial scale Q20 = 12 GeV2.
NB: Q0 often chosen lower
Assume there is no gluon at Q20 :
g(x ,Q20 ) = 0
Use DGLAP equations to evolve tohigher Q2; compare with data.
Complete failure!
QCD lecture 2 (p. 24)
Determining gluon
Evolution versus dataDGLAP with initial gluon = 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 27.0 GeV2
DGLAP: g(x,Q02) = 0
ZEUS
NMC
Fit quark distributions to F2(x ,Q20 ),
at initial scale Q20 = 12 GeV2.
NB: Q0 often chosen lower
Assume there is no gluon at Q20 :
g(x ,Q20 ) = 0
Use DGLAP equations to evolve tohigher Q2; compare with data.
Complete failure!
QCD lecture 2 (p. 24)
Determining gluon
Evolution versus dataDGLAP with initial gluon = 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 35.0 GeV2
DGLAP: g(x,Q02) = 0
ZEUS
NMC
Fit quark distributions to F2(x ,Q20 ),
at initial scale Q20 = 12 GeV2.
NB: Q0 often chosen lower
Assume there is no gluon at Q20 :
g(x ,Q20 ) = 0
Use DGLAP equations to evolve tohigher Q2; compare with data.
Complete failure!
QCD lecture 2 (p. 24)
Determining gluon
Evolution versus dataDGLAP with initial gluon = 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 46.0 GeV2
DGLAP: g(x,Q02) = 0
ZEUS
NMC
Fit quark distributions to F2(x ,Q20 ),
at initial scale Q20 = 12 GeV2.
NB: Q0 often chosen lower
Assume there is no gluon at Q20 :
g(x ,Q20 ) = 0
Use DGLAP equations to evolve tohigher Q2; compare with data.
Complete failure!
QCD lecture 2 (p. 24)
Determining gluon
Evolution versus dataDGLAP with initial gluon = 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 60.0 GeV2
DGLAP: g(x,Q02) = 0
ZEUS
NMC
Fit quark distributions to F2(x ,Q20 ),
at initial scale Q20 = 12 GeV2.
NB: Q0 often chosen lower
Assume there is no gluon at Q20 :
g(x ,Q20 ) = 0
Use DGLAP equations to evolve tohigher Q2; compare with data.
Complete failure!
QCD lecture 2 (p. 24)
Determining gluon
Evolution versus dataDGLAP with initial gluon = 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 90.0 GeV2
DGLAP: g(x,Q02) = 0
ZEUS Fit quark distributions to F2(x ,Q20 ),
at initial scale Q20 = 12 GeV2.
NB: Q0 often chosen lower
Assume there is no gluon at Q20 :
g(x ,Q20 ) = 0
Use DGLAP equations to evolve tohigher Q2; compare with data.
Complete failure!
QCD lecture 2 (p. 24)
Determining gluon
Evolution versus dataDGLAP with initial gluon = 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 150.0 GeV2
DGLAP: g(x,Q02) = 0
ZEUS Fit quark distributions to F2(x ,Q20 ),
at initial scale Q20 = 12 GeV2.
NB: Q0 often chosen lower
Assume there is no gluon at Q20 :
g(x ,Q20 ) = 0
Use DGLAP equations to evolve tohigher Q2; compare with data.
Complete failure!
QCD lecture 2 (p. 25)
Determining gluon
Evolution versus dataDGLAP with initial gluon 6= 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 12.0 GeV2
DGLAP (CTEQ6D)
ZEUS
NMC If gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.
➥ faster rise of F2
Find a gluon distribution that leadsto correct evolution in Q2.
Done for us by CTEQ, MRST, . . .
PDF fitting collaborations.
Success!
QCD lecture 2 (p. 25)
Determining gluon
Evolution versus dataDGLAP with initial gluon 6= 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 15.0 GeV2
DGLAP (CTEQ6D)
ZEUS
NMC If gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.
➥ faster rise of F2
Find a gluon distribution that leadsto correct evolution in Q2.
Done for us by CTEQ, MRST, . . .
PDF fitting collaborations.
Success!
QCD lecture 2 (p. 25)
Determining gluon
Evolution versus dataDGLAP with initial gluon 6= 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 27.0 GeV2
DGLAP (CTEQ6D)
ZEUS
NMC If gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.
➥ faster rise of F2
Find a gluon distribution that leadsto correct evolution in Q2.
Done for us by CTEQ, MRST, . . .
PDF fitting collaborations.
Success!
QCD lecture 2 (p. 25)
Determining gluon
Evolution versus dataDGLAP with initial gluon 6= 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 35.0 GeV2
DGLAP (CTEQ6D)
ZEUS
NMC If gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.
➥ faster rise of F2
Find a gluon distribution that leadsto correct evolution in Q2.
Done for us by CTEQ, MRST, . . .
PDF fitting collaborations.
Success!
QCD lecture 2 (p. 25)
Determining gluon
Evolution versus dataDGLAP with initial gluon 6= 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 46.0 GeV2
DGLAP (CTEQ6D)
ZEUS
NMC If gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.
➥ faster rise of F2
Find a gluon distribution that leadsto correct evolution in Q2.
Done for us by CTEQ, MRST, . . .
PDF fitting collaborations.
Success!
QCD lecture 2 (p. 25)
Determining gluon
Evolution versus dataDGLAP with initial gluon 6= 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 60.0 GeV2
DGLAP (CTEQ6D)
ZEUS
NMC If gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.
➥ faster rise of F2
Find a gluon distribution that leadsto correct evolution in Q2.
Done for us by CTEQ, MRST, . . .
PDF fitting collaborations.
Success!
QCD lecture 2 (p. 25)
Determining gluon
Evolution versus dataDGLAP with initial gluon 6= 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 90.0 GeV2
DGLAP (CTEQ6D)
ZEUSIf gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.
➥ faster rise of F2
Find a gluon distribution that leadsto correct evolution in Q2.
Done for us by CTEQ, MRST, . . .
PDF fitting collaborations.
Success!
QCD lecture 2 (p. 25)
Determining gluon
Evolution versus dataDGLAP with initial gluon 6= 0
0
0.4
0.8
1.2
1.6
0.001 0.01 0.1 1
x
F2p (x,Q2)
Q2 = 150.0 GeV2
DGLAP (CTEQ6D)
ZEUSIf gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.
➥ faster rise of F2
Find a gluon distribution that leadsto correct evolution in Q2.
Done for us by CTEQ, MRST, . . .
PDF fitting collaborations.
Success!
QCD lecture 2 (p. 26)
Determining full PDFs
Global fitsGluon distribution
0
1
2
3
4
5
6
0.01 0.1 1
x
xq(x), xg(x)
Q2 = 10 GeV2
uV
dS, uS
gluon
CTEQ6D fitGluon distribution is HUGE!
Can we really trust it?
◮ Consistency: momentum sum-ruleis now satisfied.
NB: gluon mostly at small x
◮ Agrees with vast range of data
QCD lecture 2 (p. 27)
Determining full PDFs
Global fitsDIS data and global fits
x
Q2 /
GeV
2
y =
1
y = 0.
004
HERA Data:
H1 1994-2000
ZEUS 1994-1997
ZEUS BPT 1997
ZEUS SVX 1995
H1 SVX 1995, 2000
H1 QEDC 1997
Fixed Target Experiments:
NMC
BCDMS
E665
SLAC
10-1
1
10
10 2
10 3
10 4
10-6
10-5
10-4
10-3
10-2
10-1
1
HERA F2
0
1
2
3
4
5
1 10 102
103
104
105
F2 em
-log
10(x
)
Q2(GeV2)
ZEUS NLO QCD fit
H1 PDF 2000 fit
H1 94-00
H1 (prel.) 99/00
ZEUS 96/97
BCDMS
E665
NMC
x=6.32E-5 x=0.000102x=0.000161
x=0.000253
x=0.0004x=0.0005
x=0.000632x=0.0008
x=0.0013
x=0.0021
x=0.0032
x=0.005
x=0.008
x=0.013
x=0.021
x=0.032
x=0.05
x=0.08
x=0.13
x=0.18
x=0.25
x=0.4
x=0.65
QCD lecture 2 (p. 27)
Determining full PDFs
Global fitsDIS data and global fits
y=1 (
HERA √s=32
0 GeV
)
x
Q2 (
GeV
2 )
E665, SLAC
CCFR, NMC, BCDMS,
Fixed Target Experiments:
D0 Inclusive jets η<3
CDF/D0 Inclusive jets η<0.7
ZEUS
H1
10-1
1
10
10 2
10 3
10 4
10 5
10-6
10-5
10-4
10-3
10-2
10-1
1
HERA F2
0
1
2
3
4
5
1 10 102
103
104
105
F2 em
-log
10(x
)
Q2(GeV2)
ZEUS NLO QCD fit
H1 PDF 2000 fit
H1 94-00
H1 (prel.) 99/00
ZEUS 96/97
BCDMS
E665
NMC
x=6.32E-5 x=0.000102x=0.000161
x=0.000253
x=0.0004x=0.0005
x=0.000632x=0.0008
x=0.0013
x=0.0021
x=0.0032
x=0.005
x=0.008
x=0.013
x=0.021
x=0.032
x=0.05
x=0.08
x=0.13
x=0.18
x=0.25
x=0.4
x=0.65
QCD lecture 2 (p. 28)
Determining full PDFs
Back to factorizationCrucial check: other processes
Factorization of QCD cross-sections into convolution of:
◮ hard (perturbative) process-dependent partonic subprocess
◮ non-perturbative, process-independent parton distribution functions
e+
Q2
x1 x2
q(x, Q 2) q1(x1, Q2) g2(x2, Q2)
proton proton 1 proton 2
qg −> 2 jetse+q −> e+ + jet
x
σep = σeq ⊗ q σpp→2 jets = σqg→2 jets ⊗ q1 ⊗ g2 + · · ·
QCD lecture 2 (p. 28)
Determining full PDFs
Back to factorizationCrucial check: other processes
Factorization of QCD cross-sections into convolution of:
◮ hard (perturbative) process-dependent partonic subprocess
◮ non-perturbative, process-independent parton distribution functions
e+
Q2
x1 x2
q(x, Q 2) q1(x1, Q2) g2(x2, Q2)
proton proton 1 proton 2
qg −> 2 jetse+q −> e+ + jet
x
σep = σeq ⊗ q σpp→2 jets = σqg→2 jets ⊗ q1 ⊗ g2 + · · ·
QCD lecture 2 (p. 29)
Determining full PDFs
Jet productionJet production in pp
[GeV/c]Tp100 200 300 400 500 600
[p
b /
(G
eV
/c)]
⟩ T
/ d
pσ
d⟨
10-3
10-2
10-1
1
10
102
103
|y| < 0.5
1.5 < |y| < 2.0
2.0 < |y| < 2.4
NLO (JETRAD) CTEQ6MmaxT = 0.5 pRµ = Fµ=1.3, sep R
Run II preliminaryOD
-1 = 143 pbintL
=1.96TeVs
data , Cone R=0.7OD
Jet production in proton-antiprotoncollisions is good test of large gluondistribution, since there are large di-rect contributions from
gg → gg , qg → qg
NB: more complicated to interpretthan DIS, since many channels, andx1, x2 dependence.
pT ∼√
x1x2s jet transverse mom.∼ Q
y ∼ 1
2log
x1
x2y = log tan θ
2
jet angle wrt pp beams
Good agreement confirms factorization
QCD lecture 2 (p. 29)
Determining full PDFs
Jet productionJet production in pp
Jet production in proton-antiprotoncollisions is good test of large gluondistribution, since there are large di-rect contributions from
gg → gg , qg → qg
NB: more complicated to interpretthan DIS, since many channels, andx1, x2 dependence.
pT ∼√
x1x2s jet transverse mom.∼ Q
y ∼ 1
2log
x1
x2y = log tan θ
2
jet angle wrt pp beams
Good agreement confirms factorization
QCD lecture 2 (p. 29)
Determining full PDFs
Jet productionJet production in pp
Jet production in proton-antiprotoncollisions is good test of large gluondistribution, since there are large di-rect contributions from
gg → gg , qg → qg
NB: more complicated to interpretthan DIS, since many channels, andx1, x2 dependence.
pT ∼√
x1x2s jet transverse mom.∼ Q
y ∼ 1
2log
x1
x2y = log tan θ
2
jet angle wrt pp beams
Good agreement confirms factorization
QCD lecture 2 (p. 30)
Determining full PDFs
Jet productionWhich PDF channels contribute?
Inclusive jet cross sections with MSTW 2008 NLO PDFs
(GeV)T
p100
Fra
ctio
nal c
ontr
ibut
ions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 = 1.96 TeVsTevatron,
jets→gg
jets→gq
jets→qq
0.1 < y < 0.7
(GeV)T
p100
Fra
ctio
nal c
ontr
ibut
ions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(GeV)T
p100 1000
Fra
ctio
nal c
ontr
ibut
ions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 = 14 TeVsLHC,
algorithm with D = 0.7TkT
= pF
µ = R
µfastNLO with
jets→gg
jets→gq
jets→qq 0.0 < y < 0.8
(GeV)T
p100 1000
Fra
ctio
nal c
ontr
ibut
ions
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
A large fraction of jets are gluon-induced
QCD lecture 2 (p. 31)
Determining full PDFs
Jet productionTaking PDFs from HERA to LHC
Suppose we produce a system ofmass M at LHC from partons withmomentum fractions x1, x2:
◮ M =√
x1x2s
◮ rapidity y =1
2ln
x1
x2pseudorapidity ≡ η ≡ ln tan θ
2
= rapidity for massless objects
. 5 at LHC
Are PDFs being used in region wheremeasured?
Only partial kinematic overlap
◮ DGLAP evolution is essential forthe prediction of PDFs in theLHC domain.
QCD lecture 2 (p. 31)
Determining full PDFs
Jet productionTaking PDFs from HERA to LHC
⇐=
DGLAP
⇐=
DGLAP
Suppose we produce a system ofmass M at LHC from partons withmomentum fractions x1, x2:
◮ M =√
x1x2s
◮ rapidity y =1
2ln
x1
x2pseudorapidity ≡ η ≡ ln tan θ
2
= rapidity for massless objects
. 5 at LHC
Are PDFs being used in region wheremeasured?
Only partial kinematic overlap
◮ DGLAP evolution is essential forthe prediction of PDFs in theLHC domain.
QCD lecture 2 (p. 32)
Determining full PDFs
Jet productionBy how much do PDFs evolve?
g(x,
Q =
100
GeV
) / g
(x, Q
= 2
GeV
)
x
Gluon evolution from 2 to 100 GeV
Input: CTEQ61 at Q = 2 GeVEvolution: HOPPET 1.1.1
LO evolution
0.01
0.1
1
10
100
0.0001 0.001 0.01 0.1 1
Illustrate for the gluon distributionHere using fixed Q scales
But for HERA → LHC
relevant Q range is x-dependent
◮ See factors ∼ 0.1− 10
◮ Remember: LHC involves productof two parton densities
It’s crucial to get this right!
Without DGLAP evolution, you
couldn’t predict anything at LHC
QCD lecture 2 (p. 33)
Determining full PDFs
PDF uncertainties
How well do we know the PDFs?
QCD lecture 2 (p. 34)
Determining full PDFs
PDF uncertaintiesPDF uncertainties
Major activity is translationof experimental errors (andtheory uncertainties) into un-certainty bands on extractedPDFs.
PDFs with uncertainties allowone to estimate degree of reli-ability of future predictions
QCD lecture 2 (p. 35)
Determining full PDFs
PDF uncertaintiesUncertainties on predictions
x-410 -310 -210 -110
Rat
io t
o M
ST
W 2
008
NN
LO
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
2 = (120 GeV)2H = M 2Gluon at Q
at Tevatron y = 0
at LHCy = 0
MSTW 2008 NNLO (68% C.L.)
at +68% C.L. limitS
αFix
at - 68% C.L. limitS
αFix
x-410 -310 -210 -110
Rat
io t
o M
ST
W 2
008
NN
LO
0.95
0.96
0.97
0.98
0.99
1
1.01
1.02
1.03
1.04
1.05
General message
Data-related errors on PDFs aresuch that uncertainties are just afew % for many key Tevatron andLHC observables
QCD lecture 2 (p. 36)
Determining full PDFs
PDF uncertainties
It’s not enough for data-related errors to be small.
DGLAP evolution must also be well constrained.
So evolution must be done with more than justleading-order DGLAP splitting functions
QCD lecture 2 (p. 37)
Determining full PDFs
PDF uncertaintiesHigher-order calculations
Earlier, we saw leading order (LO) DGLAP splitting functions, Pab = αs2π P
(0)ab :
P(0)qq (x) = CF
[1 + x2
(1− x)++
3
2δ(1 − x)
]
,
P(0)qg (x) = TR
[x2 + (1− x)2
],
P(0)gq (x) = CF
[1 + (1− x)2
x
]
,
P(0)gg (x) = 2CA
[x
(1− x)++
1− x
x+ x(1− x)
]
+ δ(1 − x)(11CA − 4nf TR)
6.
QCD lecture 2 (p. 37)
Determining full PDFs
PDF uncertaintiesHigher-order calculations
P(1)ps
(x) = 4 CF nf
„
20
9
1
x− 2 + 6x − 4H0 + x
2»
8
3H0 −
56
9
–
+ (1 + x)
»
5H0 − 2H0,0
–«
P(1)qg
(x) = 4 CAnf
„
20
9
1
x− 2 + 25x − 2pqg(−x)H−1,0 − 2pqg(x)H1,1 + x
2»
44
3H0 −
218
9
–
+4(1 − x)
»
H0,0 − 2H0 + xH1
–
− 4ζ2x − 6H0,0 + 9H0
«
+ 4 CF nf
„
2pqg(x)
»
H1,0 + H1,1 + H2
−ζ2
–
+ 4x2
»
H0 + H0,0 +5
2
–
+ 2(1 − x)
»
H0 + H0,0 − 2xH1 +29
4
–
−
15
2− H0,0 −
1
2H0
«
P(1)gq
(x) = 4 CACF
„
1
x+ 2pgq(x)
»
H1,0 + H1,1 + H2 −
11
6H1
–
− x2
»
8
3H0 −
44
9
–
+ 4ζ2 − 2
−7H0 + 2H0,0 − 2H1x + (1 + x)
»
2H0,0 − 5H0 +37
9
–
− 2pgq(−x)H−1,0
«
− 4 CF nf
„
2
3x
−pgq(x)
»
2
3H1 −
10
9
–«
+ 4 CF2
„
pgq(x)
»
3H1 − 2H1,1
–
+ (1 + x)
»
H0,0 −
7
2+
7
2H0
–
− 3H0,0
+1 −
3
2H0 + 2H1x
«
P(1)gg
(x) = 4 CAnf
„
1 − x −
10
9pgg(x) −
13
9
„
1
x− x
2«
−
2
3(1 + x)H0 −
2
3δ(1 − x)
«
+ 4 CA2
„
27
+(1 + x)
»
11
3H0 + 8H0,0 −
27
2
–
+ 2pgg(−x)
»
H0,0 − 2H−1,0 − ζ2
–
−
67
9
„
1
x− x
2«
− 12H0
−
44
3x2H0 + 2pgg (x)
»
67
18− ζ2 + H0,0 + 2H1,0 + 2H2
–
+ δ(1 − x)
»
8
3+ 3ζ3
–«
+ 4 CF nf
„
2H0
+2
3
1
x+
10
3x2− 12 + (1 + x)
»
4 − 5H0 − 2H0,0
–
−
1
2δ(1 − x)
«
.
NLO:
Pab =αs
2πP(0)+
α2s
16π2P(1)
Curci, Furmanski
& Petronzio ’80
QCD lecture 2 (p. 38)
Determining full PDFs
PDF uncertaintiesNNLO splitting functions
Divergences for x� 1 are understood in the sense of�
-distributions.
The third-order pure-singlet contribution to the quark-quark splitting function (2.4), corre-sponding to the anomalous dimension (3.10), is given by
P�
2�
ps�
x�� 16CACFnf
� 43
� 1x
�x2�� 13
3H 1 0
� 149
H0� 1
2H 1ζ2
� H 1 1 0� 2H 1 0 0
� H 1 2
�� 2
3
� 1x
� x2�� 163
ζ2�
H2 1�
9ζ3� 9
4H1 0
� 6761216
� 57172
H1� 10
3H2
�H1ζ2
� 16
H1 1
� 3H1 0 0�
2H1 1 0�
2H1 1 1
���
1� x�� 182
9H1
� 1583
� 39736
H0 0� 13
2H 2 0
�3H0 0 0 0
� 136
H1 0�
3xH1 0�
H 3 0�
H 2ζ2�
2H 2 1 0�
3H 2 0 0� 1
2H0 0ζ2
� 12
H1ζ2� 9
4H1 0 0
� 34
H1 1�
H1 1 0�
H1 1 1
���
1�
x�� 7
12H0ζ2
� 316
ζ3� 91
18H2
� 7112
H3� 113
18ζ2
� 82627
H0
� 52
H2 0� 16
3H 1 0
�6xH 1 0
� 316
H0 0 0� 17
6H2 1
� 11720
ζ22�
9H0ζ3� 5
2H 1ζ2
�2H2 1 0
� 12
H 1 0 0� 2H 1 2�
H2ζ2� 7
2H2 0 0
�H 1 1 0
�2H2 1 1
�H3 1
� 12
H4
��
5H 2 0�
H2 1
�H0 0 0 0
� 12
ζ22�
4H 3 0�
4H0ζ3� 32
9H0 0
� 2912
H0� 235
12ζ2
� 51112
� 9712
H1� 33
4H2
� H3
� 112
H0ζ2� 11
2ζ3
� 32
H2 0� 10H0 0 0� 2
3x2
� 834
H0 0� 243
4H0
�10ζ2
� 5118
� 978
H1� 4
3H2
� 4ζ3� H0ζ2�
H3�
H2 0� 6H 2 0
� �
16CFnf2
� 227
H0� 2� H2� ζ2� 2
3x2
�
H2� ζ2�
3
� 196
H0
�� 2
9
� 1x
� x2��
H1 1� 5
3H1
� 23
���
1� x�� 1
6H1 1
� 76
H1�
xH1� 35
27H0
� 18554
�
� 13
�1
�x
�� 43
H2� 4
3ζ2
� ζ3�
H2 1� 2H3�
2H0ζ2� 29
6H0 0
�H0 0 0
� �
16CF2nf
� 8512
H1
� 254
H0 0� H0 0 0� 583
12H0
� 10154
� 734
ζ2� 73
4H2
�H3
� 5H2 0� H2 1� H0ζ2�
x2� 55
12
� 8512
H1� 22
3H0 0
� 1096
� 1354
H0� 28
9ζ2
� 289
H2� 16
3H0ζ2
� 163
H3�
4H2 0� 4
3H2 1
� 263
ζ3
� 223
H0 0 0
�� 4
3
� 1x
� x2�� 2312
H1 0� 523
144H1
� 3ζ3� 55
16� 1
2H1 0 0
�H1 1
� H1 1 0� H1 1 1
�
��1� x
�� 1
2H1 0 0
� 712
H1 1� 274372
H0� 5312
H0 0� 25112
H1� 54
ζ2� 5
4H2� 8
3H1 0
�3xH1 0
�3H0ζ2
� 3H3� H1 1 0� H1 1 1
���
1�
x�� 1669
216� 5
2H0 0 0
�4H2 1
�7H2 0
�10xζ3
� 3710
ζ22
� 7H0ζ3�
6H0 0ζ2� 4H0 0 0 0�
H2 0 0� 2H2 1 0� 2H2 1 1� 4H3 0� H3 1� 6H4
� � (4.12)
Due to Eqs. (3.11) and (3.12) the three-loop gluon-quark and quark-gluon splitting functions read
P�
2�
qg�
x�� 16CACFnf
�
pqg�
x�� 39
2H1ζ3
� 4H1 1 1�
3H2 0 0� 15
4H1 2
� 94
H1 1 0�
3H2 1 0
�H0ζ3
� 2H2 1 1�
4H2ζ2� 173
12H0ζ2
� 55172
H0 0� 64
3ζ3
� ζ22� 49
4H2
� 32
H1 0 0 0� 1
3H1 0 0
16
38572
H1 0312
H1 111312
H1494
H2 052
H1ζ2796
H0 0 017312
H31259
322833216
H0
6H2 1 3H1 2 0 9H1 0ζ2 6H1 1ζ2 H1 1 0 0 3H1 1 1 0 4H1 1 1 1 3H1 1 2 6H1 2 1
6H1 3494
ζ2 pqg x172
H 1ζ352
H 1 1 052
H 1 292
H 1 052
H 2 032
H 1 0 0
2H3 1 2H4 6H 2 2 6H 2 1 0 6H 2 0 0 2H0 0ζ2 9H 2ζ2 3H 1 2 0 2H 1 2 1
6H 1 1 1 0 6H 1 1 0 0 6H 1 1 2 9H 1 0ζ2 9H 1 1ζ2 2H 1 2 0112
H 1 0 0 0
6H 1 31x
x2 5512
4ζ3239
H1 043
H1 1 01x
x2 23
H1 0 0371108
H1239
H1 1
23
H1 1 1 1 x 6H2 1 0 3H2 1 156
H1 1 1 7H2 0 0 2H1 2 39H0ζ3 4H2ζ2163
ζ3
H1 1 0154
3H0ζ2
89924
H0 012110
ζ22 607
36H2
52
H1ζ2656
H1 0 02912
H1 01318
H1 1
1189108
H1673
H2 1 29H2 094936
ζ2672
H0 0 0142
3H3
21532
398948
H0 2H 3 0
1 x H 1 0 0 10H 2ζ2 6H 2 0 0 2H0 0ζ2 9H 1 1 0 7H 1 2 9H 2 0 2H3 1
4H 2 1 0 4H4 4H3 0 4H0 0 0 0372
H 1 052
1 x H 1ζ2 4H 2 0 0 2H0 0ζ2
H2ζ2 3H1 1 0 2H0 0 0 0 H 3 0 9H2 1 092
H2 1 1113
H1 1 1192
H2 0 092
H1 2
912
H0ζ3 8H 2ζ252
H 1 1 052
H 1 292
H 1 0392
H 2 047312
H0ζ21853
48H0 0
21712
ζ3594
ζ22 169
18H2
134
H1ζ223
H1 0 016724
H1 019118
H1 11283108
H118512
H2 1
754
H2 0170
9ζ2
854
H0 0 042512
H37693192
365948
H0 2x xH2 2 4H3 0 4H 2 2
16CAnf2 1
6pqg x H1 2 H1ζ2 H1 0 0 H1 1 0 H1 1 1
22918
H043
H0 0112
x16
H2
5318
H0176
H0 0 ζ31118
ζ2139108
13
pqg x H 1 0 053
1621x
x2 29
1 x 6H0 0 0
76
xH1 H0 072
xH1 179
x 1 x H 1 074
H01954
H1 H0 0 059
H1 159
H 1 0
85216
16CA2nf pqg x 3H1 3
316
H1 0 0172
H2 175
ζ22 55
12H1 1 0
3112
H3312
H1ζ3
512
H2 0638
H1 02312
H1 2155
6ζ3
2524
H22537
27H0
8678
232
H 1 0 0 3H4 H1 1 1
38372
H1 1252
H 2 038
ζ274
H1ζ2 3H0 0ζ23112
H0ζ2103216
H152
H1 0 0 02561
72H0 0
H1 1 1 2H2 0 0 3H1 2 0 5H1 0ζ2 3H0 0 0 H1 1ζ2 H1 1 0 0 4H1 1 1 0 2H1 1 1 1
2H1 1 2 2H1 2 0 pqg x H 1 1ζ2 2H 1 2 6H 1 1 0 H1 1 1 2H 2ζ2 H 2 0 0
72736
H 1 0 H 1ζ2 2H 2 252
H 1ζ3 H 1 2 0 2H 1 1 0 0 2H 1 1 232
H 1 0 0 0
17
6H 1 1 1 0 2H 1 3 2H 1 2 11x
x2 23
H2 1329
ζ2 2H1 0 043
H1 1 0109
H1 1
83
H 1 0 032
H1 0 6ζ316136
H12351108
23
1x
x2 263
H 1 0289
H0 2H 1 1 0
2H 1 2 H1ζ2 H 1ζ2103
H2 H1 1 1 1 x 15H0 0 0 0 5H2ζ2656
ζ3116
H1 1 1
32
H452
H0 0ζ2 H1 1 0316
H2 01712
H1 055120
ζ22 29
4H1 0 0
1134
H218691
72H0
2243108
2656
H 1 0 0332
H2 0 0 19H2 13112
H1 1232
H 2 049736
ζ2296
H1ζ214312
H3
116
H1 1 11912
H0ζ21223
72H1
436
H0 0 03011
36H0 0 1 x 8H2 1 0 4H 1 2
7H 1 1 0356
H1 1 1 5H 2ζ2 11H 2 0 013
H 1 0152
H 1ζ2 8H3 1 10H 2 1 0
5H2ζ2 4H2 1 1 H 3 0 36H0ζ3 5H2ζ2 2H 1 2 6H 1 1 0 6H2 1 0 3H2 1 1
11H0 0 0 0 5H3 1254
H1 1 1132
H 2ζ2272
H 2 0 0112
H 3 0132
H2ζ2174
H1 0 0
13H 2 1 01712
H1 1 134
H414
H0 0ζ2 H1 2112
H1 1 07912
H2 0678
H1 0263
8ζ2
2
1193
ζ396724
H230512
H 1 0 24H0ζ3 H 1ζ213375
72H0
188918
38H 1 0 0212
H2 1
794
H2 0 021724
H1 172
H 2 07972
ζ243
H1ζ21712
H1 1 11712
H0ζ23118
H1 3H0 0 0
14512
H31553
24H0 0 16CFnf
2 76
H0 0 01136
H173996
16324
H0724
H0 0 2H0 0 0 0
59
H1 159
H25
18H1 0
59
ζ216
pqg x H2 1912
353
H0223
H0 0 H1 1 1 6H0 0 0
ζ3 2H1 0 079
H17781
1x
x2 1 x112
H16463432
4H0 0 0 0163
H0 0 079
xH1 1
79
xH289
xH1 079
xζ2 1 x3475216
H010312
H0 0 16CF2nf pqg x 7H1 3 7H4
2H 3 0 7H1ζ3 5H2 2 6H3 0 6H3 1 H2 1 0 4H2 0 0 3H2 1 2H2 1 152
H2 0
618
H2618
ζ2878
H1112
H1 2618
H1 1172
H1 0 7H0 0ζ252
H1 0 052
H1 1 0192
ζ3
8132
112
H3112
H0ζ272
H1ζ2152
H0 0 0878
H0115
ζ22 3H1 1 1 5H2ζ2 7H0ζ3
11H0 0 2H1 2 0 7H1 0ζ2 3H1 0 0 0 5H1 1ζ2 4H1 1 0 0 H1 1 1 0 2H1 1 1 1 5H1 1 2
6H1 2 0 6H1 2 1 4pqg x H0 0 0 0 H 2 0 H 1 1 0 H 2 0 012
H 1 2 058
H 1 0
54
H 1 0 012
H 3 012
H 1ζ2 H 1 1 0 014
H 1 0 0 0 2 1 x H2 1 0 H2 0 0 H2 2
H3 1 2H3 0 2H 1ζ2 H1 2 H1 0 0 H1 1 0 H2ζ2 ζ22 43
8H2
498
ζ2138
H1 1
18
3316
H152
H1 072
H0 0ζ2214
ζ347964
12
H1 1 112
H314
H2 112
H2 1 132
H0ζ2
12
H0ζ372
H4 H1ζ2192
H0 0 023916
H0 040532
H0 8 1 x H 1 1 0 H 1 0 0
H0 0 0 034
H 2 094
H 1 0 4H 1 1 0 5H0 0 0 0 5H 1 0 0 13H 2 012
H 1 0
4xH 2 0 0113
8H2
718
ζ2354
H1112
H1 2338
H1 172
H1 072
H0 0ζ252
H1 0 0
52
H1 1 052
ζ315764
94
H1 1 194
H354
H2 112
H2 1 114
H0ζ2 H2ζ252
H0ζ3
95
ζ22 7
2H4
72
H1ζ2494
H0 0 039116
H0 040116
H0 H2 0 0 H2 1 0 H2 2 H3 1
2H3 0 6H 1ζ212
H2 0 2H 2ζ2 4H 2 1 0 (4.13)
and
P 2gq x 16CACFnf
29
x2 256
H1131
43ζ2 H 1 0 3H2 H1 1
1256
H0 H0 0
56
pgq x H1 2 H2 1967120
25190
H13910
H1 1 3ζ325
H0ζ215
H1ζ243
H1 0 H1 1 0
25
H1 0 0 H1 1 125
H2 023
pgq x 2H 1ζ274
ζ24112
H 1 015172
H012
H 2 0
53
H2 2H 1 1 0 H 1 0 0 H 1 223
1 x H 2 0 2ζ3 H3 1 x179108
H1
59
ζ2259
H 1 05
36H1 1
16736
H0 013
H2 143
H0ζ219372
14
H119
H 1 0 4H2
14
H1 122718
H03512
H0 0 H2 123
H0ζ2103
H 2 0 3ζ3 2H323
H0 0 0 x114
ζ2
523144
1936
H2271108
H056
H1 0 16CACF2 x2 7
217354
H1 2ζ323
H1 1 1269
H1 1
6H2 2H2 1 6ζ233554
H0289
H0 083
H0 0 0 pgq x32
H1ζ316332
5ζ2274
ζ3
6503432
H129
H1 1353
H1 1 1 4H292
H2 1 4H1 0 0 2H2 0 0 H2ζ24112
H1 2 H2 2
19124
H1 0 3H2 0 2H2 1 132
H 1ζ25912
H1ζ2 5H1 2 0 H1 0ζ252
H1 0 0 0 2H1 1ζ2
112
H1 1 0 5H1 1 0 0 3H1 1 1 0 4H1 1 1 1 H1 1 2 2H1 2 1 H2 1 0 pgq x H 1 0
H 1 0ζ232
H 1 0 02710
ζ22 3H 1 1 0
112
H 1ζ3 3H 1 2 032
H 1 0 0 0 3H 1 2
5H 1 1ζ2 4H 1 1 0 0 2H 1 1 2 6H 1 1 1 0 2H 1 2 1 1 x H2ζ2 H2 2
2312
H1 07061432
H04631144
H0 0383
H0 0 0 H 3 0 2H3 04433432
H1 2H2 0 0212
H1 0 0
25
ζ22 7
2H1 2
232
H1ζ2 4H0ζ3 1 x496
H3 H 2 0556
H0ζ212
H3 1115936
ζ2
19
� 655576
� 1516
ζ3� 185
18H1 1
� 16
H1 1 1� 95
9H2
� 296
H2 1� 171
4H 1 0
� 12H 1 0 0�
7H 1ζ2
�16H 1 1 0
� 53
H2 0� 3
2H2 1 1
�4H0 0 0 0
�� 35H 2 0� 179
27H0
� 2041144
H0 0� 19
6H0 0 0
� 2H3 0� 13
2H0ζ2
� 13H 3 0� 13
2H3 1
� 152
H3� 2005
64� 157
4ζ2
�8ζ3
� 1291432
H1� 55
12H1 1
� 32
H2� 1
2H2 1
� 274
H 1 0� 11
2H1 0 0
� 8H2 0 0� 4ζ2
2� 32
H1 2� H2 2� 5
2H1ζ2
�8H 1 1 0
�4H2 0
� 32
H2 1 1� H 1ζ2�
7H2ζ2�
6H 2ζ2�
12H 2 1 0� 6H 2 0 0�
x�
3H1 1 1� H0 0ζ2
� 92
H 1 0 0� 35
8H1 0
�2H4
�3H1 1 0
�H 1 2
� �
16CA2CF
�
x2� 2
3H1ζ2
� 210581
� 7718
H0 0
� 6H3� 16
3ζ3
� 10H 1 0� 14
3H2 0
� 23
H 1ζ2� 14
3H0 0 0
� 1049
H2� 4
3H1 0 0
� 379
H1 1
� 43
H 1 1 0� 104
9ζ2
� 83
H2 1� 145
18H1 0
� 43
H 1 2� 2
3H1 1 1
� 10927
H1� 8
3H 1 0 0
�6H0ζ2
�4H 2 0
� 58427
H0
��
pgq�
x�� 7
2H1ζ3
� 1383052592
� 13
H2 0� 13
4H 1ζ2
�2H2 1 1
� 112
H1 0 0
�4H3 1
� 436
H1 1 1� 109
12ζ2
� 173
H2 1� 71
24H1 0
� 116
H 2 0� 21
2ζ3
� 32
H1 0 0 0� H1 2 0
� 39554
H0� 2H1 0ζ2� H1 1ζ2� 55
12H1 1 0
�2H1 1 0 0
�4H1 1 1 0
�2H1 1 1 1
�4H1 1 2
� 5512
H1 2
�6H1 2 0
�4H1 2 1
�4H1 3
�3H2 1 0
�3H2 2
��
pgq�� x�� 23
2H 1ζ3
�5H 2ζ2
�2H 2 1 0
� 10912
H 1 0�
H0ζ3� 17
5ζ2
2� 16
H1ζ2�
2H2ζ2� 65
24H1 1
� 192
H 1 1 0� 4H3 0� 3H2 0 0
� 7H 2 0 0� 3
2H 1 2
� 3379216
H1� 4H 2 2� 49
6H 1 0 0
� 112
H 1 0 0 0� 13H 1 1ζ2� 8H 1 3
� 6H 1 1 1 0�
12H 1 1 0 0�
10H 1 1 2�
10H 1 0ζ2�
5H 1 2 0� 2H 1 2 0� 2H 1 2 1
� 116
H0ζ2
���
1� x�� 41699
2592� 3H 2 1 0� 3
2H 2ζ2
� 1289
ζ2� 4H3 0� 26
3ζ3
� 52
H 2 0 0
� 7H1ζ2� 97
12H1 0 0
� 103
H 1 0 0� 245
12H3
� 8H0 0 0 0
���
1�
x��
4H3 1� H2 1 1� 29
6H 1 2
� 176
H 2 0� 12H2 0� 31
12H2 1
� 12
H2 0 0� H2ζ2� 61
36H1 0
� 4H0ζ3� 13
3H 1ζ2
� 463
H 1 1 0
� 254
H4� 93
4H0ζ2
� 559
H1 1� 71
18H2
� 4918
H0 0� 13
2H0 0ζ2
� 4740
ζ22
�� 6131
2592� 31
2H 2ζ2
� 15H 2 1 0� 9
2H 1 0 0
� 3H2 1 1� 9
4H2 1
� 533
H 2 0� 1
2H 2 0 0
� 5H2 0� 7
6H1 1 1
� 8H0ζ3
� 6740
ζ22� 29
6H 1 2
� H 1 0�
8H 2 2�
25H0ζ2� 412
9H1
� 9289
H0� 1
4H4
� 65H3� 38H0 0
� 9H 3 0� 17
3H0 0 0
�x
� 272
H 1 0� 1
2H0 0 0 0
� 34
H0 0ζ2� 1
2H 3 0
� 14H0 0 0� 1
12H1 1 1
� 4336
ζ2� 1
2H2ζ2
� 772
H0� 749
54H1
� 1354
ζ3� 97
24H1 0
� 4312
H1ζ2� 85
12H 1ζ2
� 133
H1 0 0
20
5312
H2394
H1 1 2H3 1136
H 1 1 074
H2 0 0 4H1 1 0 4H1 2 16CFnf2 1
919
1x
29
x16
xH116
pgq x H1 153
H1 16CF2nf
49
x2 H0 0116
H072
H 1 0
13
pgq x H1 2 H1 0 H1ζ2 9ζ38312
H1 1 2H 2 07
36H1 2H0ζ2
162548
32
H1 0 0
2H1 1 052
H1 1 13118
pgq x9593
H0 ζ2 H 1 013
2 x 6H0 0 0 0 H313051288
132
ζ3 4H 2 0 H2 012
H1 012
H2 1 2H0 0 065324
H0 0 1 x H0ζ21187216
H0
89
H28518
H 1 010118
ζ28027
H02318
ζ213
H1 154
xH1 119
H13712
xH12318
H 1 0
150154
H0ζ2 H0 0 0101
3H0 0
13
H1 0 16CF3 pgq x 3H1 1ζ2 3H1ζ2
72
ζ2
238
H1 1 8H1ζ3 6H1 2 0 2H1 0ζ2 3H1 1 0 3H1 1 0 0 H1 1 1 0 2H1 1 1 1 3H1 1 2
2H1 2 0 2H1 2 192
H1 1 132
H1 0 04716
4716
H1152
ζ3 pgq x 2H 1 2 0
6H 1 1 0 3H 1ζ274
H1 0165
ζ22 6H 1 0 0
72
H 1 0 4H 1 1 0 0 2H 1 0ζ2
H 1 0 0 0 1 x 9H1 0 0 H1 1 1 10H1ζ2 3H0ζ3 H2 2 H2ζ2 H0 0 0 5H2 0 0
4H3 H2 1 1 3H0 0ζ2 3H3 1 3H421116
H14920
ζ22 1 x 11ζ3
14
H1 114
H1 0
9116
H0 36H 1 0 8H 1 0 0 14H 1 1 0 7H 1ζ2 2H1 2 4H0ζ2 H2 1 2H 2 0 0
5H 2 0112
H2 2H0 0 0 0 2H 1 1 0 H 1ζ2134
ζ294
H1 09
20ζ2
2 28732
1116
H1
4H 1 0 0 16H 3 0 4H 2ζ2 8H 2 1 0 5H2ζ2194
H2 H2 2358
H0 0 9H0ζ3
25H 2 0 6H 2 0 032
x583
ζ273
H1ζ2 4H1 132
H1 1 152
H1 0 017596
H3 1193
ζ3
2H2 0 14H0 H0 0ζ2 H 1 0 H432
H2 113
H2 1 1 3H2 0 056
H3 H1 276
H0ζ2
23
H1 1 0296
H0 0 0185
8H0 0 (4.14)
Finally the Mellin inversion of Eq. (3.13) yields the NNLO gluon-gluon splitting function
P 2gg x 16CACFnf x2 4
9H2 3H1 0
9712
H183
H 2 023
H0ζ210327
H0163
ζ2 2H3
6H 1 0 2H2 012718
H0 051112
pgg x 2ζ35524
43
1x
x2 1724
H1 04318
H0
521144
H16923432
12
H2 1 2H1ζ2 H0ζ2 2H1 0 0112
H1 1 H1 1 0 H1 1 117512
H2
6H 1 0 8H0ζ3 6H 2 0536
H0ζ2492
H0185
4ζ2
51112
12
H2 0 3H1 0 4H0 0 0 0
21
6712
H0 0432
ζ3 H2 19712
H1 4ζ22 9
2H3 8H 3 0
332
H0 0 043
1x
x2 12
H2 H2 0
113
H 1 0 H 2 0196
ζ2 2ζ3 H 1ζ2 4H 1 1 012
H 1 0 0 H 1 2 1 x 9H1ζ2
12H0 0 0 0293108
616
H0ζ273
H1 085736
H1 9H0ζ3 16H 2 1 0 4H 2 0 0 8H 2ζ2
132
H1 0 034
H1 1 H1 1 0 H1 1 1 1 x16
H2 0953
H 1 014936
H23451108
H0
7H 2 0302
9H0 0
196
H399136
ζ2163
6ζ3
353
H0 0 0176
H2 14310
ζ22 13H 1ζ2
18H 1 1 0 H3 1 6H4 4H 1 2 6H0 0ζ2 8H2ζ2 7H2 0 0 2H2 1 0 2H2 1 1 4H3 0
9H 1 0 0241288
δ 1 x 16CAnf2 19
54H0
124
xH01
27pgg x
1354
1x
x2 53
H1
1 x1172
H171216
29
1 x ζ21312
xH012
H0 0 H229
288δ 1 x
16CA2nf x2 ζ3
119
ζ2119
H0 023
H323
H0ζ21639108
H0 2H 2 013
pgg x103
ζ2
20936
8ζ3 2H 2 012
H0103
H0 0203
H1 0 H1 0 0203
H2 H3109
pgg x ζ2
2H 1 03
10H0ζ2 H0 0
13
1x
x2 H3 H0ζ2133
H25443108
3H1ζ220536
H1
133
H1 0 H1 0 01x
x2 15154
H083
ζ213
H 1ζ2 ζ3 2H 1 1 023
H 1 0 0
379
H 1 023
H 1 2 1 x56
H 2 0 H 3 0 2H0 0 026936
ζ24097216
3H 2ζ2
6H 2 1 0 3H 2 0 072
H1ζ267772
H1 H1 074
H1 0 0 1 x19336
H2112
H 1ζ2
3920
ζ22 7
12H3
539
H0 07
12H0ζ2
52
H0 0ζ2 5ζ3 7H 1 1 0776
H 1 092
H 1 0 0
2H 1 2 3H2ζ223
H2 032
H2 0 032
H419
ζ2 7H 2 0 2H245827
H0 H0 0ζ2
32
ζ22 4H 3 0 x
13112
H0 083
H0ζ272
H3 H0 0 0 076
H0 0 01943216
H0 6H0ζ3
δ 1 x233288
16
ζ21
12ζ2
2 53
ζ3 16CA3 x2 33H 2 0 33H0ζ2
124918
H0 0
44H0 0 0110
3H3
443
H2 0856
ζ26409108
H0 pgg x24524
679
ζ23
10ζ2
2 113
ζ3
4H 3 0 6H 2ζ2 4H 2 1 0113
H 2 0 4H 2 0 0 4H 2 216
H0 7H0ζ3679
H0 0
8H0 0ζ2 4H0 0 0 0 6H1ζ3 4H1 2 0 10H2 0 0 6H1 0ζ2 8H1 0 0 0 8H1 1 0 0 8H4
1349
H1 0116
H1 0 0 8H1 2 0 8H1 3134
9H2 4H2ζ2 8H3 1 8H2 2
116
H3 10H3 0
8H2 1 0 pgg x112
ζ22 11
6H0ζ2 4H 3 0 16H 2ζ2 12H 2 2
1349
H 1 0 2H2ζ2
8H 2 1 0 12H 1ζ3 18H 2 0 0 8H 1 2 0 16H 1 1ζ2 24H 1 1 0 0 16H 1 1 2
22
18H 1 0ζ2 16H 1 0 0 0 4H 1 2 0 16H 1 3 5H0ζ3 8H0 0ζ2 4H0 0 0 0 2H3 0
679
ζ2679
H0 0 8H41x
x2 16619162
223
H2 0552
ζ3112
H0ζ2679
H2679
H1 0
413108
H1112
H1ζ2332
H1 0 0 111x
x2 7154
H016
H3389198
ζ223
H 2 012
H 1ζ2
H 1 1 0523198
H 1 083
H 1 0 0 H 1 2 1 x3136
H1272
H1 0252
H1 0 0 4H 3 0
26312
H0 0293
H0 0 0193
H 2 011317
1084H 2ζ2 8H 2 1 0 12H 2 0 0
32
H1ζ2
1 x272
H0ζ2436
H3293
H2 04651216
H032918
ζ2112
1 x ζ3435
ζ22 215
6H 1 0
22H0 0ζ2 8H0ζ3 3H 1 1 0 38H 1 0 0 25H 1 2 10H2 0 0 4H2ζ2 16H3 0 26H4
1589
H2532
H 1ζ2 29H0 0403
H0 0 0 27H 2 0413
H0ζ2 20H3 24H2 0536
ζ2
60112
H0 24ζ3 2ζ22 27H2 4H0 0ζ2 16H0ζ3 16H 3 0 28xH0 0 0 0 δ 1 x
7932
ζ2ζ316
ζ21124
ζ22 67
6ζ3 5ζ5 16CFnf
2 29
x2 116
H0 H2 ζ2 2H0 0 913
H2
13
ζ2103
H013
H0 0 229
1x
x2 83
H1 2H1 0 H1 17718
1 x13
H1 016
H1 1
49
136
H1 xH113
1 x689
H043
H243
ζ2296
H0 0 ζ3 2H0ζ2 H0 0 0 2H3
H2 1 2H2 011144
δ 1 x 16CF2nf
43
x2 16316
278
H072
H0 0 H2 0 ζ294
H1 0
H2 112
H0 0 08516
H1 H2 2H 2 032
ζ343
1x
x2 3116
H11116
54
H1 012
H1 0 0
H1ζ2 H1 1 H1 1 0 H1 1 1 ζ343
1x
x2 H 1ζ2 2H 1 1 0 H 1 0 021512
H0 0
203
H0131
63H2 0
20512
xζ2 3H1 0 H2 18512
H1114
H2 8H 2 0 2ζ22 H0ζ2
H3 6H0ζ3 8H 3 0 4xH0 0 0 1 x10712
H156
H1 0 4ζ2 H0ζ3 8H 2 1 0
4H 2ζ2 4H 2 0 0 4H1ζ272
H1 0 07
12H1 1 H1 1 0 H1 1 1 1 x
54
H2338
994
H0 0 8H2 054124
H0 4H2 132
H0 0 0 2xζ392
ζ22 5H0ζ2 5H3 4H 1ζ2
8H 1 1 0673
H 1 0 4H 1 0 0 2H0 0ζ2 2H0 0 0 0 4H2ζ2 3H2 0 0 2H2 1 0
2H2 1 1 H3 1 2H41
16δ 1 x (4.15)
The large-x behaviour of the gluon-gluon splitting function P 2gg x is given by
P 2gg x 1 x
Ag3
1 xBg
3 δ 1 x Cg3 ln 1 x O 1 (4.16)
23
NNLO, P(2)ab : Moch, Vermaseren & Vogt ’04
QCD lecture 2 (p. 39)
Determining full PDFs
PDF uncertaintiesEvolution uncertainty
unce
rtai
nty
on g
(x, Q
= 1
00 G
eV)
x
Uncert. on gluon ev. from 2 to 100 GeV
Input: CTEQ61 at Q = 2 GeVEvolution: HOPPET 1.1.1
LO evolution
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.0001 0.001 0.01 0.1 1
Estimate uncertainties on evolutionby changing the scale used for αs in-side the splitting functions
Talk more about such
tricks in next lecture
◮ with LO evolution, uncertainty is∼ 30%
◮ NLO brings it down to ∼ 5%
◮ NNLO → 2% Commensurate with
data uncertainties
QCD lecture 2 (p. 39)
Determining full PDFs
PDF uncertaintiesEvolution uncertainty
unce
rtai
nty
on g
(x, Q
= 1
00 G
eV)
x
Uncert. on gluon ev. from 2 to 100 GeV
Input: CTEQ61 at Q = 2 GeVEvolution: HOPPET 1.1.1
LO evolution
NLO evolution
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.0001 0.001 0.01 0.1 1
Estimate uncertainties on evolutionby changing the scale used for αs in-side the splitting functions
Talk more about such
tricks in next lecture
◮ with LO evolution, uncertainty is∼ 30%
◮ NLO brings it down to ∼ 5%
◮ NNLO → 2% Commensurate with
data uncertainties
QCD lecture 2 (p. 39)
Determining full PDFs
PDF uncertaintiesEvolution uncertainty
unce
rtai
nty
on g
(x, Q
= 1
00 G
eV)
x
Uncert. on gluon ev. from 2 to 100 GeV
Input: CTEQ61 at Q = 2 GeVEvolution: HOPPET 1.1.1
LO evolution
NLO evolution
NNLO evolution
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
0.0001 0.001 0.01 0.1 1
Estimate uncertainties on evolutionby changing the scale used for αs in-side the splitting functions
Talk more about such
tricks in next lecture
◮ with LO evolution, uncertainty is∼ 30%
◮ NLO brings it down to ∼ 5%
◮ NNLO → 2% Commensurate with
data uncertainties
QCD lecture 2 (p. 40)
Closing remarks Conclusions on PDFs
◮ Experiments tell us that proton really is what we expected (uud)
◮ Plus lots more: large number of ‘sea quarks’ (qq), gluons (50% ofmomentum)
◮ Factorization is key to usefulness of PDFs◮ Non-trivial beyond lowest order
◮ PDFs depend on factorization scale, evolve with DGLAP equation
◮ Pattern of evolution gives us info on gluon (otherwise hard to measure)
◮ PDFs really are universal!
◮ Precision of data & QCD calculations is striking.
◮ Crucial for understanding future signals of new particles, e.g. HiggsBoson production at LHC.
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