qcd (for lhc) lecture 2: parton distribution functions · qcd lecture 2 (p. 3) pdf introduction...

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?


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)


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


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.


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?


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 )


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?


Quark distributions

u & d


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



How much u and d?

◮ Total U =∫

dx u(x)

◮ F2 = x(49u + 1

9d)

◮ u(x) ∼ d(x) ∼ x−1.25




Quark distributions

u & d


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



How much u and d?

◮ Total U =∫

dx u(x)

◮ F2 = x(49u + 1

9d)

◮ u(x) ∼ d(x) ∼ x−1.25




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)


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−


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


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. . .


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



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



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?



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.



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.



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)



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



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



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



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



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




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








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








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








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








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








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








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







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.




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








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








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








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








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








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








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







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


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!


Determining gluon


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





g(x ,Q20 ) = 0


Complete failure!


Determining gluon


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





g(x ,Q20 ) = 0


Complete failure!


Determining gluon


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





g(x ,Q20 ) = 0


Complete failure!


Determining gluon


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





g(x ,Q20 ) = 0


Complete failure!


Determining gluon


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





g(x ,Q20 ) = 0


Complete failure!


Determining gluon


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 ),




g(x ,Q20 ) = 0


Complete failure!


Determining gluon


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 ),




g(x ,Q20 ) = 0


Complete failure!


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!


Determining gluon


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






Success!


Determining gluon


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






Success!


Determining gluon


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






Success!


Determining gluon


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






Success!


Determining gluon


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






Success!


Determining gluon


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.





Success!


Determining gluon


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.





Success!


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



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



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



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 + · · ·



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



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



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



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.



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.



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



PDF uncertainties

How well do we know the 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



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



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



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.



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



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



PDF uncertaintiesEvolution uncertainty

unce

rtai

nty

on g

(x, Q

= 1

00 G

eV)

x

Uncert. on gluon ev. from 2 to 100 GeV


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




unce

rtai

nty

on g

(x, Q

= 1

00 G

eV)

x



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







data uncertainties




unce

rtai

nty

on g

(x, Q

= 1

00 G

eV)

x



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







data uncertainties


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.

qcd (for lhc) lecture 2: parton distribution functions · qcd lecture 2 (p. 3) pdf introduction...

Documents