Basics of QCD Lecture 3: PDFs and DGLAP

Basics of QCDLecture 3: PDFs and DGLAP

Gavin Salam

CERN Theory Unit

ICTP–SAIFR school on QCD and LHC physicsJuly 2015, Sao Paulo, Brazil

Factorization & parton distributions[PDFs]

Cross section for some hardprocess in hadron-hadroncollisions

x2 p

2

p1 p2

x 1p 1

σ

Z H

σ =∑i ,j

∫dx1fi/p(x1, µ

2F )

∫dx2fj/p(x2, µ

2F ) σij(s, µ2

R , µ2F ) , s = x1x2s

I Total X-section is factorized into a ‘hard part’ σ(x1p1, x2p2, µ2) and

‘normalization’ from parton distribution functions (PDF).

I Measure total cross section ↔ need to know PDFs to be able to testhard part (e.g. Higgs electroweak couplings).

I Picture seems intuitive, butI how can we determine the PDFs? NB: non-perturbativeI does picture really stand up to QCD corrections?

Gavin Salam (CERN) QCD basics 3 2 / 32

Deep Inelastic Scattering: kinematics[PDFs]

[DIS 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

I Q2 = photon virtuality ↔ transverseresolution at which it probes protonstructure

I x = longitudinal momentum fraction ofstruck parton in proton

I y = momentum fraction lost by electron(in proton rest frame)


Deep Inelastic scattering (DIS): example[PDFs]

[DIS kinematics]

Q2 = 25030 GeV 2; y = 0:56;

e+

x=0.50

e+

Q2

x

proton

e+

jet

proton

jet


E.g.: extracting u & d distributions[PDFs]

[DIS X-sections]

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 em

2 [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:

I use perturbative language for interactions of up and down quarks

I but distributions themselves have a non-perturbative origin.

F2 gives us combination of u and d .How can we extract them separately?


Extracting full flavour structure?[Quark distributions]

[u & d ]

I Using neutrons and isospin

F n2 =

4

9un(x) +

1

9dn(x)

' 4

9dp(x) +

1

9up(x)

I Using charged-current (W±) scattering[neutrinos instead of electrons in initial or final-state]

I W + interacts only with d , uI angular structure of interaction differs between d and u


Extracting full flavour structure?[Quark distributions]

[u & d ]

I Using neutrons and isospin

F n2 =

4

9un(x) +

1

9dn(x) ' 4

9dp(x) +

1

9up(x)

I Using charged-current (W±) scattering[neutrinos instead of electrons in initial or final-state]

I W + interacts only with d , uI angular structure of interaction differs between d and u


All quarks[Quark distributions]

[u & d ]

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 GeV

2

uVdV

dS

uSsS

cS

CTEQ6D fit

These & other methods→ whole setof quarks & antiquarks

NB: also strange and charm quarks

I valence quarks (uV = u − u) arehardx → 1 : xqV (x) ∼ (1− x)3

quark counting rules

x → 0 : xqV (x) ∼ x0.5

Regge theory

I sea quarks (uS = 2u, . . . ) fairlysoft (low-momentum)x → 1 : xqS(x) ∼ (1− x)7

x → 0 : xqS(x) ∼ x−0.2


Momentum sum rule[Quark distributions]

[u & d ]

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: it’s crucial to know it for gg → H

To discuss gluons we must go beyond ‘naive’leading order picture, and bring in QCD split-ting. . .


Recall final-state splitting[Initial-state splitting]

[1st order analysis]

Yesterday: calculated q → qg (θ � 1, E � p) for final state of arbitraryhard process (σh):

σh+g ' σhαsCF

π

dE

E

dθ2

θ2

ω =

pzp

(1−z)p

θσh

Rewrite with different kinematic variables

σh+g ' σhαsCF

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 ' −σhαsCF

π

dz

1− z

dk2t

k2t

p pσ

h


Initial-state splitting[Initial-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 pσ

h

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 σhGavin Salam (CERN) QCD basics 3 10 / 32

Initial-state collinear divergence[Initial-state splitting]


σg+h + σV+h 'αsCF

π

∫ Q2

0

dk2t

k2t︸︷︷︸

infinite

∫dz

1− z[σh(zp)− σh(p)]︸︷︷︸

finite

I In soft limit (z → 1), σh(zp)− σh(p)→ 0: soft divergence cancels.

I 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?


Collinear cutoff[Initial-state splitting]


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 µ2F .


Collinear cutoff[Initial-state splitting]


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

F )

σ1 'αsCF

π

∫ Q2

µ2F

dk2t

k2t︸︷︷︸

finite (large)

∫dx dz

1− z[σh(zxp)− σh(xp)] q(x , µ2

F )︸︷︷︸finite

In general: replace 1 GeV2 cutoff with arbitrary factorization scale µ2F .


Summary so far[Initial-state splitting]


I Collinear divergence for incoming partons not cancelled by virtuals.Real and virtual have different longitudinal momenta

I Situation analogous to renormalization: need to regularize (but in IRinstead of UV).

Technically, often done with dimensional regularization

I 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

I 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)


Summary so far[Initial-state splitting]


I Collinear divergence for incoming partons not cancelled by virtuals.Real and virtual have different longitudinal momenta

I Situation analogous to renormalization: need to regularize (but in IRinstead of UV).

Technically, often done with dimensional regularization

I 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

I 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

u

u

u

g

gg

du

ud d

ug

gu

u


DGLAP equation (q ← q)[Initial-state splitting]

[DGLAP]

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

xdz 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) = CF1 + z2

1− z

Until now we approximated it in soft (z → 1) limit, pqq ' 2CF

1−z


DGLAP rewritten[Initial-state splitting]

[DGLAP]

Awkward to write real and virtual parts separately. Use more compactnotation:

dq(x , µ2)

d lnµ2=αs

2π

∫ 1

xdz 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


DGLAP flavour structure[Initial-state splitting]

[DGLAP]

Proton contains both quarks and gluons — so DGLAP is a matrix in flavourspace:

d

d ln Q2

(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.

I Pqg , Pgg : symmetric z ↔ 1− z (except virtuals)

I Pqq, Pgg : diverge for z → 1 soft gluon emission

I Pgg , Pgq: diverge for z → 0 Implies PDFs grow for x → 0

2015 EPS HEP prize to Bjorken, Altarelli, Dokshitzer, Lipatov & Parisi


Higher-order calculations[Initial-state splitting]

[DGLAP]

P (1)ps (x) = 4 CF nf

(20

9

1

x− 2 + 6x − 4H0 + x2

[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 + x2

[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 CF

2(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− x2

)−

2

3(1 + x)H0 −

2

3δ(1 − x)

)+ 4 CA

2(

27

+(1 + x)

[11

3H0 + 8H0,0 −

27

2

]+ 2pgg(−x)

[H0,0 − 2H−1,0 − ζ2

]−

67

9

(1

x− x2

)− 12H0

−44

3x2

H0 + 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


NNLO splitting functions[Initial-state splitting]

[DGLAP]

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

✞4

3

☎ 1

x

✁x2✆✟ 13

3H✠ 1✡ 0

☛ 14

9H0

✁ 1

2H✠ 1ζ2

☛ H✠ 1✡✠ 1✡ 0☛ 2H✠ 1✡ 0✡ 0

☛ H✠ 1✡ 2

☞✁ 2

3

☎ 1

x☛ x2✆✟ 16

3ζ2

✁H2✡ 1

✁9ζ3

✁ 9

4H1✡ 0

☛ 6761

216

✁ 571

72H1

✁ 10

3H2

✁H1ζ2

☛ 1

6H1✡ 1

☛ 3H1✡ 0✡ 0✁

2H1✡ 1✡ 0✁

2H1✡ 1✡ 1

☞✁☎

1☛ x✆✟ 182

9H1

✁ 158

3

✁ 397

36H0✡ 0

☛ 13

2H✠ 2✡ 0

✁3H0✡ 0✡ 0✡ 0

✁ 13

6H1✡ 0

✁3xH1✡ 0

✁H✠ 3✡ 0

✁H✠ 2ζ2

✁2H✠ 2✡✠ 1✡ 0

✁3H✠ 2✡ 0✡ 0

✁ 1

2H0✡ 0ζ2

✁ 1

2H1ζ2

☛ 9

4H1✡ 0✡ 0

☛ 3

4H1✡ 1

✁H1✡ 1✡ 0

✁H1✡ 1✡ 1

☞✁☎

1✁

x✆✟

7

12H0ζ2

✁ 31

6ζ3

✁ 91

18H2

✁ 71

12H3

✁ 113

18ζ2

☛ 826

27H0

✁ 5

2H2✡ 0

✁ 16

3H✠ 1✡ 0

✁6xH✠ 1✡ 0

✁ 31

6H0✡ 0✡ 0

☛ 17

6H2✡ 1

✁ 117

20ζ2

2✁9H0ζ3

✁ 5

2H✠ 1ζ2

✁2H2✡ 1✡ 0

✁ 1

2H✠ 1✡ 0✡ 0

☛ 2H✠ 1✡ 2✁

H2ζ2☛ 7

2H2✡ 0✡ 0

✁H✠ 1✡✠ 1✡ 0

✁2H2✡ 1✡ 1

✁H3✡ 1

☛ 1

2H4

☞✁

5H✠ 2✡ 0✁

H2✡ 1

✁H0✡ 0✡ 0✡ 0

☛ 1

2ζ2

2✁4H✠ 3✡ 0

✁4H0ζ3

☛ 32

9H0✡ 0

☛ 29

12H0

☛ 235

12ζ2

☛ 511

12☛ 97

12H1

✁ 33

4H2

☛ H3

☛ 11

2H0ζ2

☛ 11

2ζ3

☛ 3

2H2✡ 0

☛ 10H0✡ 0✡ 0✁ 2

3x2

✟83

4H0✡ 0

☛ 243

4H0

✁10ζ2

✁ 511

8

✁ 97

8H1

☛ 4

3H2

☛ 4ζ3☛ H0ζ2✁

H3✁

H2✡ 0☛ 6H✠ 2✡ 0

☞✌✁

16CFnf2

✞2

27H0

☛ 2☛ H2✁

ζ2✁ 2

3x2

✟

H2☛ ζ2✁

3

☛ 19

6H0

☞✁ 2

9

☎ 1

x☛ x2✆✟

H1✡ 1✁ 5

3H1

✁ 2

3

☞✁☎

1☛ x✆✟ 1

6H1✡ 1

☛ 7

6H1

✁xH1

✁ 35

27H0

✁ 185

54

☞

✁ 1

3

☎1

✁x

✆✟ 4

3H2

☛ 4

3ζ2

✁ζ3

✁H2✡ 1

☛ 2H3✁

2H0ζ2✁ 29

6H0✡ 0

✁H0✡ 0✡ 0

☞✌✁

16CF2nf

✞85

12H1

☛ 25

4H0✡ 0

☛ H0✡ 0✡ 0✁ 583

12H0

☛ 101

54

✁ 73

4ζ2

☛ 73

4H2

✁H3

☛ 5H2✡ 0☛ H2✡ 1☛ H0ζ2✁

x2✟

55

12

☛ 85

12H1

☛ 22

3H0✡ 0

☛ 109

6☛ 13

54H0

✁ 28

9ζ2

☛ 28

9H2

☛ 16

3H0ζ2

✁ 16

3H3

✁4H2✡ 0

✁ 4

3H2✡ 1

☛ 26

3ζ3

✁ 22

3H0✡ 0✡ 0

☞✁ 4

3

☎ 1

x☛ x2✆✟ 23

12H1✡ 0

☛ 523

144H1

☛ 3ζ3✁ 55

16

✁ 1

2H1✡ 0✡ 0

✁H1✡ 1

☛ H1✡ 1✡ 0☛ H1✡ 1✡ 1

☞

✁☎1☛ x

✆✟

1

2H1✡ 0✡ 0

✁ 7

12H1✡ 1

☛ 2743

72H0

☛ 53

12H0✡ 0

☛ 251

12H1

☛ 5

4ζ2

✁ 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

☛ 37

10ζ2

2

☛ 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

✁ 9

4H1✡ 1✡ 0

✁3H2✡ 1✡ 0

✁H0ζ3

☛ 2H2✡ 1✡ 1✁

4H2ζ2☛ 173

12H0ζ2

☛ 551

72H0✡ 0

✁ 64

3ζ3

☛ ζ22☛ 49

4H2

☛ 3

2H1✡ 0✡ 0✡ 0

☛ 1

3H1✡ 0✡ 0

16

385

72H1 0

31

2H1 1

113

12H1

49

4H2 0

5

2H1ζ2

79

6H0 0 0

173

12H3

1259

32

2833

216H0

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 349

4ζ2 pqg x

17

2H 1ζ3

5

2H 1 1 0

5

2H 1 2

9

2H 1 0

5

2H 2 0

3

2H 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 2 011

2H 1 0 0 0

6H 1 31

xx2 55

124ζ3

23

9H1 0

4

3H1 1 0

1

xx2 2

3H1 0 0

371

108H1

23

9H1 1

2

3H1 1 1 1 x 6H2 1 0 3H2 1 1

5

6H1 1 1 7H2 0 0 2H1 2 39H0ζ3 4H2ζ2

16

3ζ3

H1 1 0154

3H0ζ2

899

24H0 0

121

10ζ2

2 607

36H2

5

2H1ζ2

65

6H1 0 0

29

12H1 0

13

18H1 1

1189

108H1

67

3H2 1 29H2 0

949

36ζ2

67

2H0 0 0

142

3H3

215

32

3989

48H0 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 0 037

2H 1 0

5

21 x H 1ζ2 4H 2 0 0 2H0 0ζ2

H2ζ2 3H1 1 0 2H0 0 0 0 H 3 0 9H2 1 09

2H2 1 1

11

3H1 1 1

19

2H2 0 0

9

2H1 2

91

2H0ζ3 8H 2ζ2

5

2H 1 1 0

5

2H 1 2

9

2H 1 0

39

2H 2 0

473

12H0ζ2

1853

48H0 0

217

12ζ3

59

4ζ2

2 169

18H2

13

4H1ζ2

2

3H1 0 0

167

24H1 0

191

18H1 1

1283

108H1

185

12H2 1

75

4H2 0

170

9ζ2

85

4H0 0 0

425

12H3

7693

192

3659

48H0 2x xH2 2 4H3 0 4H 2 2

16CAnf2 1

6pqg x H1 2 H1ζ2 H1 0 0 H1 1 0 H1 1 1

229

18H0

4

3H0 0

11

2x

1

6H2

53

18H0

17

6H0 0 ζ3

11

18ζ2

139

108

1

3pqg x H 1 0 0

53

162

1

xx2 2

91 x 6H0 0 0

7

6xH1 H0 0

7

2xH1 1

7

9x 1 x H 1 0

7

4H0

19

54H1 H0 0 0

5

9H1 1

5

9H 1 0

85

21616CA

2nf pqg x 3H1 331

6H1 0 0

17

2H2 1

7

5ζ2

2 55

12H1 1 0

31

12H3

31

2H1ζ3

5

12H2 0

63

8H1 0

23

12H1 2

155

6ζ3

25

24H2

2537

27H0

867

8

23

2H 1 0 0 3H4 H1 1 1

383

72H1 1

25

2H 2 0

3

8ζ2

7

4H1ζ2 3H0 0ζ2

31

12H0ζ2

103

216H1

5

2H1 0 0 0

2561

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

727

36H 1 0 H 1ζ2 2H 2 2

5

2H 1ζ3 H 1 2 0 2H 1 1 0 0 2H 1 1 2

3

2H 1 0 0 0

17

6H 1 1 1 0 2H 1 3 2H 1 2 11

xx2 2

3H2 1

32

9ζ2 2H1 0 0

4

3H1 1 0

10

9H1 1

8

3H 1 0 0

3

2H1 0 6ζ3

161

36H1

2351

108

2

3

1

xx2 26

3H 1 0

28

9H0 2H 1 1 0

2H 1 2 H1ζ2 H 1ζ210

3H2 H1 1 1 1 x 15H0 0 0 0 5H2ζ2

65

6ζ3

11

6H1 1 1

3

2H4

5

2H0 0ζ2 H1 1 0

31

6H2 0

17

12H1 0

551

20ζ2

2 29

4H1 0 0

113

4H2

18691

72H0

2243

108

265

6H 1 0 0

33

2H2 0 0 19H2 1

31

12H1 1

23

2H 2 0

497

36ζ2

29

6H1ζ2

143

12H3

11

6H1 1 1

19

12H0ζ2

1223

72H1

43

6H0 0 0

3011

36H0 0 1 x 8H2 1 0 4H 1 2

7H 1 1 035

6H1 1 1 5H 2ζ2 11H 2 0 0

1

3H 1 0

15

2H 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 125

4H1 1 1

13

2H 2ζ2

27

2H 2 0 0

11

2H 3 0

13

2H2ζ2

17

4H1 0 0

13H 2 1 017

12H1 1 1

3

4H4

1

4H0 0ζ2 H1 2

11

2H1 1 0

79

12H2 0

67

8H1 0

263

8ζ2

2

119

3ζ3

967

24H2

305

12H 1 0 24H0ζ3 H 1ζ2

13375

72H0

1889

1838H 1 0 0

21

2H2 1

79

4H2 0 0

217

24H1 1

7

2H 2 0

79

72ζ2

4

3H1ζ2

17

12H1 1 1

17

12H0ζ2

31

18H1 3H0 0 0

145

12H3

1553

24H0 0 16CFnf

2 7

6H0 0 0

11

36H1

739

96

163

24H0

7

24H0 0 2H0 0 0 0

5

9H1 1

5

9H2

5

18H1 0

5

9ζ2

1

6pqg x H2 1

91

2

35

3H0

22

3H0 0 H1 1 1 6H0 0 0

ζ3 2H1 0 07

9H1

77

81

1

xx2 1 x

1

12H1

6463

4324H0 0 0 0

16

3H0 0 0

7

9xH1 1

7

9xH2

8

9xH1 0

7

9xζ2 1 x

3475

216H0

103

12H0 0 16CF

2nf 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 15

2H2 0

61

8H2

61

8ζ2

87

8H1

11

2H1 2

61

8H1 1

17

2H1 0 7H0 0ζ2

5

2H1 0 0

5

2H1 1 0

19

2ζ3

81

32

11

2H3

11

2H0ζ2

7

2H1ζ2

15

2H0 0 0

87

8H0

11

5ζ2

23H1 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 01

2H 1 2 0

5

8H 1 0

5

4H 1 0 0

1

2H 3 0

1

2H 1ζ2 H 1 1 0 0

1

4H 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

49

8ζ2

13

8H1 1

18

and

✁ 655

576☛ 151

6ζ3

☛ 185

18H1✡ 1

✁ 1

6H1✡ 1✡ 1

☛ 95

9H2

✁ 29

6H2✡ 1

☛ 171

4H✠ 1✡ 0

☛ 12H✠ 1✡ 0✡ 0✁

7H✠ 1ζ2

✁16H✠ 1✡✠ 1✡ 0

✁ 5

3H2✡ 0

✁ 3

2H2✡ 1✡ 1

✁4H0✡ 0✡ 0✡ 0

☞☛ 35H✠ 2✡ 0☛ 179

27H0

✁ 2041

144H0✡ 0

☛ 19

6H0✡ 0✡ 0

☛ 2H3✡ 0☛ 13

2H0ζ2

☛ 13H✠ 3✡ 0☛ 13

2H3✡ 1

✁ 15

2H3

☛ 2005

64

✁ 157

4ζ2

✁8ζ3

✁ 1291

432H1

✁ 55

12H1✡ 1

✁ 3

2H2

✁ 1

2H2✡ 1

✁ 27

4H✠ 1✡ 0

☛ 11

2H1✡ 0✡ 0

☛ 8H2✡ 0✡ 0☛ 4ζ2

2✁ 3

2H1✡ 2

☛ H2✡ 2✁ 5

2H1ζ2

✁8H✠ 1✡✠ 1✡ 0

✁4H2✡ 0

✁ 3

2H2✡ 1✡ 1

☛ H✠ 1ζ2✁

7H2ζ2✁

6H✠ 2ζ2✁

12H✠ 2✡✠ 1✡ 0☛ 6H✠ 2✡ 0✡ 0✁

x

✟

3H1✡ 1✡ 1☛ H0✡ 0ζ2

✁ 9

2H✠ 1✡ 0✡ 0

☛ 35

8H1✡ 0

✁2H4

✁3H1✡ 1✡ 0

✁H✠ 1✡ 2

☞✌✁

16CA2CF

✞

x2✟

2

3H1ζ2

☛ 2105

81☛ 77

18H0✡ 0

☛ 6H3✁ 16

3ζ3

☛ 10H✠ 1✡ 0☛ 14

3H2✡ 0

☛ 2

3H✠ 1ζ2

☛ 14

3H0✡ 0✡ 0

✁ 104

9H2

☛ 4

3H1✡ 0✡ 0

✁ 37

9H1✡ 1

✁ 4

3H✠ 1✡✠ 1✡ 0

☛ 104

9ζ2

☛ 8

3H2✡ 1

✁ 145

18H1✡ 0

✁ 4

3H✠ 1✡ 2

✁ 2

3H1✡ 1✡ 1

☛ 109

27H1

✁ 8

3H✠ 1✡ 0✡ 0

✁6H0ζ2

✁4H✠ 2✡ 0

✁ 584

27H0

☞✁

pgq

☎x

✆✟ 7

2H1ζ3

✁ 138305

2592☛ 1

3H2✡ 0

✁ 13

4H✠ 1ζ2

✁2H2✡ 1✡ 1

✁ 11

2H1✡ 0✡ 0

✁4H3✡ 1

☛ 43

6H1✡ 1✡ 1

☛ 109

12ζ2

☛ 17

3H2✡ 1

☛ 71

24H1✡ 0

☛ 11

6H✠ 2✡ 0

☛ 21

2ζ3

✁ 3

2H1✡ 0✡ 0✡ 0

☛ H1✡✠ 2✡ 0

✁ 395

54H0

☛ 2H1✡ 0ζ2☛ H1✡ 1ζ2☛ 55

12H1✡ 1✡ 0

✁2H1✡ 1✡ 0✡ 0

✁4H1✡ 1✡ 1✡ 0

✁2H1✡ 1✡ 1✡ 1

✁4H1✡ 1✡ 2

☛ 55

12H1✡ 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

✁ 109

12H✠ 1✡ 0

✁H0ζ3

✁ 17

5ζ2

2✁ 1

6H1ζ2

✁2H2ζ2

☛ 65

24H1✡ 1

☛ 19

2H✠ 1✡✠ 1✡ 0

☛ 4H3✡ 0☛ 3H2✡ 0✡ 0

☛ 7H✠ 2✡ 0✡ 0☛ 3

2H✠ 1✡ 2

✁ 3379

216H1

☛ 4H✠ 2✡ 2☛ 49

6H✠ 1✡ 0✡ 0

☛ 11

2H✠ 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

✁ 11

6H0ζ2

☞✁☎

1☛ x✆✟ 41699

2592☛ 3H✠ 2✡✠ 1✡ 0☛ 3

2H✠ 2ζ2

☛ 128

9ζ2

☛ 4H3✡ 0✁ 26

3ζ3

☛ 5

2H✠ 2✡ 0✡ 0

☛ 7H1ζ2✁ 97

12H1✡ 0✡ 0

✁ 10

3H✠ 1✡ 0✡ 0

✁ 245

12H3

☛ 8H0✡ 0✡ 0✡ 0

☞✁☎

1✁

x✆✟

4H3✡ 1☛ H2✡ 1✡ 1✁ 29

6H✠ 1✡ 2

✁ 17

6H✠ 2✡ 0

☛ 12H2✡ 0☛ 31

12H2✡ 1

✁ 1

2H2✡ 0✡ 0

☛ H2ζ2✁ 61

36H1✡ 0

☛ 4H0ζ3☛ 13

3H✠ 1ζ2

☛ 46

3H✠ 1✡✠ 1✡ 0

✁ 25

4H4

✁ 93

4H0ζ2

☛ 55

9H1✡ 1

☛ 71

18H2

✁ 49

18H0✡ 0

☛ 13

2H0✡ 0ζ2

☛ 47

40ζ2

2☞✁ 6131

2592☛ 31

2H✠ 2ζ2

☛ 15H✠ 2✡✠ 1✡ 0✁ 9

2H✠ 1✡ 0✡ 0

☛ 3H2✡ 1✡ 1☛ 9

4H2✡ 1

✁ 53

3H✠ 2✡ 0

☛ 1

2H✠ 2✡ 0✡ 0

☛ 5H2✡ 0☛ 7

6H1✡ 1✡ 1

☛ 8H0ζ3

☛ 67

40ζ2

2✁ 29

6H✠ 1✡ 2

☛ H✠ 1✡ 0✁

8H✠ 2✡ 2✁

25H0ζ2✁ 412

9H1

✁ 928

9H0

✁ 1

4H4

☛ 65H3☛ 38H0✡ 0

☛ 9H✠ 3✡ 0☛ 17

3H0✡ 0✡ 0

✁x

✟27

2H✠ 1✡ 0

☛ 1

2H0✡ 0✡ 0✡ 0

✁ 3

4H0✡ 0ζ2

✁ 1

2H✠ 3✡ 0

☛ 14H0✡ 0✡ 0✁ 1

12H1✡ 1✡ 1

☛ 43

36ζ2

☛ 1

2H2ζ2

✁ 7

72H0

✁ 749

54H1

✁ 135

4ζ3

✁ 97

24H1✡ 0

✁ 43

12H1ζ2

☛ 85

12H✠ 1ζ2

☛ 13

3H1✡ 0✡ 0

20

53

12H2

39

4H1 1 2H3 1

13

6H 1 1 0

7

4H2 0 0 4H1 1 0 4H1 2 16CFnf

2 1

9

1

9

1

x2

9x

1

6xH1

1

6pgq x H1 1

5

3H1 16CF

2nf

4

9x2 H0 0

11

6H0

7

2H 1 0

1

3pgq x H1 2 H1 0 H1ζ2 9ζ3

83

12H1 1 2H 2 0

7

36H1 2H0ζ2

1625

48

3

2H1 0 0

2H1 1 05

2H1 1 1

31

18pgq x

95

93H0 ζ2 H 1 0

1

32 x 6H0 0 0 0 H3

13051

28813

2ζ3 4H 2 0 H2 0

1

2H1 0

1

2H2 1 2H0 0 0

653

24H0 0 1 x H0ζ2

1187

216H0

8

9H2

85

18H 1 0

101

18ζ2

80

27H0

23

18ζ2

1

3H1 1

5

4xH1 1

1

9H1

37

12xH1

23

18H 1 0

1501

54H0ζ2 H0 0 0

101

3H0 0

1

3H1 0 16CF

3 pgq x 3H1 1ζ2 3H1ζ27

2ζ2

23

8H1 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 19

2H1 1 1

3

2H1 0 0

47

16

47

16H1

15

2ζ3 pgq x 2H 1 2 0

6H 1 1 0 3H 1ζ27

4H1 0

16

5ζ2

26H 1 0 0

7

2H 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 3H4211

16H1

49

20ζ2

21 x 11ζ3

1

4H1 1

1

4H1 0

91

16H0 36H 1 0 8H 1 0 0 14H 1 1 0 7H 1ζ2 2H1 2 4H0ζ2 H2 1 2H 2 0 0

5H 2 011

2H2 2H0 0 0 0 2H 1 1 0 H 1ζ2

13

4ζ2

9

4H1 0

9

20ζ2

2 287

32

11

16H1

4H 1 0 0 16H 3 0 4H 2ζ2 8H 2 1 0 5H2ζ219

4H2 H2 2

35

8H0 0 9H0ζ3

25H 2 0 6H 2 0 03

2x

58

3ζ2

7

3H1ζ2 4H1 1

3

2H1 1 1

5

2H1 0 0

175

96H3 1

19

3ζ3

2H2 0 14H0 H0 0ζ2 H 1 0 H43

2H2 1

1

3H2 1 1 3H2 0 0

5

6H3 H1 2

7

6H0ζ2

2

3H1 1 0

29

6H0 0 0

185

8H0 0 (4.14)

Finally the Mellin inversion of Eq. (3.13) yields the NNLO gluon-gluon splitting function

P2

gg x 16CACFnf x2 4

9H2 3H1 0

97

12H1

8

3H 2 0

2

3H0ζ2

103

27H0

16

3ζ2 2H3

6H 1 0 2H2 0127

18H0 0

511

12pgg x 2ζ3

55

24

4

3

1

xx2 17

24H1 0

43

18H0

521

144H1

6923

432

1

2H2 1 2H1ζ2 H0ζ2 2H1 0 0

1

12H1 1 H1 1 0 H1 1 1

175

12H2

6H 1 0 8H0ζ3 6H 2 053

6H0ζ2

49

2H0

185

4ζ2

511

12

1

2H2 0 3H1 0 4H0 0 0 0

21

67

12H0 0

43

2ζ3 H2 1

97

12H1 4ζ2

2 9

2H3 8H 3 0

33

2H0 0 0

4

3

1

xx2 1

2H2 H2 0

11

3H 1 0 H 2 0

19

6ζ2 2ζ3 H 1ζ2 4H 1 1 0

1

2H 1 0 0 H 1 2 1 x 9H1ζ2

12H0 0 0 0293

108

61

6H0ζ2

7

3H1 0

857

36H1 9H0ζ3 16H 2 1 0 4H 2 0 0 8H 2ζ2

13

2H1 0 0

3

4H1 1 H1 1 0 H1 1 1 1 x

1

6H2 0

95

3H 1 0

149

36H2

3451

108H0

7H 2 0302

9H0 0

19

6H3

991

36ζ2

163

6ζ3

35

3H0 0 0

17

6H2 1

43

10ζ2

213H 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 0241

288δ 1 x 16CAnf

2 19

54H0

1

24xH0

1

27pgg x

13

54

1

xx2 5

3H1

1 x11

72H1

71

216

2

91 x ζ2

13

12xH0

1

2H0 0 H2

29

288δ 1 x

16CA2nf x2 ζ3

11

9ζ2

11

9H0 0

2

3H3

2

3H0ζ2

1639

108H0 2H 2 0

1

3pgg x

10

3ζ2

209

368ζ3 2H 2 0

1

2H0

10

3H0 0

20

3H1 0 H1 0 0

20

3H2 H3

10

9pgg x ζ2

2H 1 03

10H0ζ2 H0 0

1

3

1

xx2 H3 H0ζ2

13

3H2

5443

1083H1ζ2

205

36H1

13

3H1 0 H1 0 0

1

xx2 151

54H0

8

3ζ2

1

3H 1ζ2 ζ3 2H 1 1 0

2

3H 1 0 0

37

9H 1 0

2

3H 1 2 1 x

5

6H 2 0 H 3 0 2H0 0 0

269

36ζ2

4097

2163H 2ζ2

6H 2 1 0 3H 2 0 07

2H1ζ2

677

72H1 H1 0

7

4H1 0 0 1 x

193

36H2

11

2H 1ζ2

39

20ζ2

2 7

12H3

53

9H0 0

7

12H0ζ2

5

2H0 0ζ2 5ζ3 7H 1 1 0

77

6H 1 0

9

2H 1 0 0

2H 1 2 3H2ζ22

3H2 0

3

2H2 0 0

3

2H4

1

9ζ2 7H 2 0 2H2

458

27H0 H0 0ζ2

3

2ζ2

24H 3 0 x

131

12H0 0

8

3H0ζ2

7

2H3 H0 0 0 0

7

6H0 0 0

1943

216H0 6H0ζ3

δ 1 x233

288

1

6ζ2

1

12ζ2

2 5

3ζ3 16CA

3 x2 33H 2 0 33H0ζ21249

18H0 0

44H0 0 0110

3H3

44

3H2 0

85

6ζ2

6409

108H0 pgg x

245

24

67

9ζ2

3

10ζ2

2 11

3ζ3

4H 3 0 6H 2ζ2 4H 2 1 011

3H 2 0 4H 2 0 0 4H 2 2

1

6H0 7H0ζ3

67

9H0 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

134

9H1 0

11

6H1 0 0 8H1 2 0 8H1 3

134

9H2 4H2ζ2 8H3 1 8H2 2

11

6H3 10H3 0

8H2 1 0 pgg x11

2ζ2

2 11

6H0ζ2 4H 3 0 16H 2ζ2 12H 2 2

134

9H 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

The large-

NNLO, P(2)ab : Moch, Vermaseren & Vogt ’04


Effect of (LO) DGLAP: initial quarks[Initial-state splitting]

[Example evolution]

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 12.0 GeV

2

xg(x,Q2)

xq + xqbar

Take example evolution starting withjust quarks:

∂lnQ2q = Pq←q ⊗ q∂lnQ2g = Pg←q ⊗ q

I quark is depleted at large x

I gluon grows at small x



[Example evolution]

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 15.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 27.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 35.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 46.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 60.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 90.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

0.5

1

1.5

2

2.5

3

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 150.0 GeV

2

xg(x,Q2)

xq + xqbar






Effect of (LO) DGLAP: initial gluons[Initial-state splitting]

[Example evolution]

0

1

2

3

4

5

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 12.0 GeV

2

xg(x,Q2)

xq + xqbar

2nd example: start with just gluons.

∂lnQ2q = Pq←g ⊗ g∂lnQ2g = Pg←g ⊗ g

I gluon is depleted at large x .

I high-x gluon feeds growth ofsmall x gluon & quark.



[Example evolution]

0

1

2

3

4

5

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 15.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

1

2

3

4

5

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 27.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

1

2

3

4

5

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 35.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

1

2

3

4

5

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 46.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

1

2

3

4

5

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 60.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

1

2

3

4

5

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 90.0 GeV

2

xg(x,Q2)

xq + xqbar







[Example evolution]

0

1

2

3

4

5

0.01 0.1 1

x

xq(x,Q2), xg(x,Q

2)

Q2 = 150.0 GeV

2

xg(x,Q2)

xq + xqbar






DGLAP evolution[Initial-state splitting]

[Example evolution]

I As Q2 increases, partons lose longitudinal momentum; distributions allshift to lower x .

I gluons can be seen because they help drive the quark evolution.

Now consider data


DGLAP with initial gluon = 0[Determining gluon]

[Evolution versus data]

0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 12.0 GeV

2

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!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 15.0 GeV

2

DGLAP: g(x,Q02) = 0

ZEUS

NMC





g(x ,Q20 ) = 0


Complete failure!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 27.0 GeV

2

DGLAP: g(x,Q02) = 0

ZEUS

NMC





g(x ,Q20 ) = 0


Complete failure!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 35.0 GeV

2

DGLAP: g(x,Q02) = 0

ZEUS

NMC





g(x ,Q20 ) = 0


Complete failure!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 46.0 GeV

2

DGLAP: g(x,Q02) = 0

ZEUS

NMC





g(x ,Q20 ) = 0


Complete failure!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 60.0 GeV

2

DGLAP: g(x,Q02) = 0

ZEUS

NMC





g(x ,Q20 ) = 0


Complete failure!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 90.0 GeV

2

DGLAP: g(x,Q02) = 0

ZEUS Fit quark distributions to F2(x ,Q20 ),




g(x ,Q20 ) = 0


Complete failure!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 150.0 GeV

2

DGLAP: g(x,Q02) = 0

ZEUS Fit quark distributions to F2(x ,Q20 ),




g(x ,Q20 ) = 0


Complete failure!


DGLAP with initial gluon 6= 0[Determining gluon]


0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 12.0 GeV

2

DGLAP (CTEQ6D)

ZEUS

NMC If gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.

å faster rise of F2

Find a gluon distribution that leadsto correct evolution in Q2.

Done for us by CTEQ, MRST, . . .

PDF fitting collaborations.

Success!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 15.0 GeV

2

DGLAP (CTEQ6D)

ZEUS






Success!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 27.0 GeV

2

DGLAP (CTEQ6D)

ZEUS






Success!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 35.0 GeV

2

DGLAP (CTEQ6D)

ZEUS






Success!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 46.0 GeV

2

DGLAP (CTEQ6D)

ZEUS






Success!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 60.0 GeV

2

DGLAP (CTEQ6D)

ZEUS






Success!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 90.0 GeV

2

DGLAP (CTEQ6D)

ZEUSIf gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.





Success!




0

0.4

0.8

1.2

1.6

0.001 0.01 0.1 1

x

F2p (x,Q

2)

Q2 = 150.0 GeV

2

DGLAP (CTEQ6D)

ZEUSIf gluon 6= 0, splitting g → qq gen-erates extra quarks at large Q2.





Success!


Gluon distribution[Determining full PDFs]

[Global fits]

0

1

2

3

4

5

6

0.01 0.1 1

x

xq(x), xg(x)

Q2 = 10 GeV

2

uV

dS, uS

gluon

CTEQ6D fitGluon distribution is HUGE!

Can we really trust it?

I Consistency: momentum sum-ruleis now satisfied.

NB: gluon mostly at small x

I Agrees with vast range of data


DIS data and global fits[Determining full PDFs]

[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

102

103

104

10-6

10-5

10-4

10-3

10-2

10-1

1


DIS data and global fits[Determining full PDFs]

[Global fits]

x6

105

10 4103

10 210 110 1

]2

[ G

eV

2 T /

p2

/ M

2Q

1

10

210

310

410

510

610

710FT DIS

HERA1

FT DY

TEV EW

TEV JET

ATLAS EW

LHCB EW

LHC JETS

HERA2

ATLAS JETS 2.76TEV

ATLAS HIGH MASS

ATLAS WpT

CMS W ASY

CMS JETS

CMS WC TOT

CMS WC RAT

LHCB Z

TTBAR

NNPDF3.0 NLO dataset


Other processes[Determining full PDFs]

[Back to factorization]

Factorization of QCD cross-sections into convolution of:

I hard (perturbative) process-dependent partonic subprocess

I non-perturbative, process-independent parton distribution functions

e+

Q2

x1 x2

q(x, Q2) 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 + · · ·Gavin Salam (CERN) QCD basics 3 26 / 32

Taking PDFs from HERA to LHC[Determining full PDFs]


Suppose we produce a system ofmass M at LHC from partons withmomentum fractions x1, x2:

I M =√

x1x2s

I 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

I DGLAP evolution is essential forthe prediction of PDFs in theLHC domain.


Taking PDFs from HERA to LHC[Determining full PDFs]


⇐=DGLAP

⇐=DGLAP

Suppose we produce a system ofmass M at LHC from partons withmomentum fractions x1, x2:

I M =√

x1x2s

I 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

I DGLAP evolution is essential forthe prediction of PDFs in theLHC domain.


Comparisons to hadron-collider data[Determining full PDFs]

[Back to factorization]M

uon

char

ge a

sym

met

ry

|ηµ|

CMS, √s=7 TeV, L=4.7 [fb]-1

PT > 25 GeV

CT14, 68% C.L.

0.10

0.15

0.20

0.25

0.30

0 0.5 1 1.5 2 2.5

Muo

n ch

arge

asy

mm

etry

|ηµ|

CMS, √s=7 TeV, L=4.7 [fb]-1

PT > 35 GeV

CT14, 68% C.L.

0.10

0.15

0.20

0.25

0.30

0 0.5 1 1.5 2 2.5

Ele

ctro

n ch

arge

asy

mm

etry

|ηe|

CMS, √s=7 TeV, L=840 [pb]-1

PT > 35 GeV

CT14, 68% C.L.0.08

0.12

0.16

0.20

0.24

0 0.5 1 1.5 2 2.5

FIG. 20: Charge asymmetry of decay muons and electrons from W ± production measured by the

CMS experiment. The data values have pT ℓ > 25 or 35 GeV for the muon data and pT ℓ > 35

GeV for the electron data. The vertical error bars on the data points include both statistical and

systematic uncertainties. The curve shows the CT14 theoretical calculation; the shaded region is

the PDF uncertainty at 68% C.L.

the missing transverse energy to be greater than 25 GeV, and the lepton-neutrino transverse

mass to be greater than 40 GeV.

The curve in Fig. 19 shows the NNLO theoretical calculation based on CT14 NNLO

PDFs. The shaded region is the PDF uncertainty at 68% C.L. Again the points with error

bars represent the unshifted data with total experimental errors added in quadrature. The

data fluctuate around the CT14 predictions and are described well by the CT14 error band.

Figure 20 presents a similar comparison of the unshifted data and CT14 NNLO theory for

the charge asymmetry of decay muons [46] and electrons [47] from inclusive W ± production

from the CMS experiment at the LHC 7 TeV. The asymmetry for muons is measured with

4.7 pb−1 of integrated luminosity, with pT ℓ > 25 and 35 GeV; the asymmetry for electrons is

36

Ele

ctro

n ch

arge

asy

mm

etry

|ηe|

D0, L=9.7 fb-1

CT14 NNLOCT14 NNLOCT10 NNLOCTEQ6.6 NLOE

eT > 25 GeV, ET,miss > 25 GeV

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0 0.5 1 1.5 2 2.5 3

Ele

ctro

n ch

arge

asy

mm

etry

|ηe|

D0, L=0.75 fb-1

CT10 NNLOCT10 NNLOCT14 NNLOCTEQ6.6 NLOE

eT > 25 GeV, ET,miss > 25 GeV

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0 0.5 1 1.5 2 2.5 3

FIG. 22: Charge asymmetry of decay electrons from W ± production measured by the DØ exper-

iment in Run-2 at the Tevatron. The data values have EeT > 25 GeV and ET,miss > 25 GeV. The

curve shows the CT14 theoretical calculation; the shaded region is the PDF uncertainty at 68%

C.L.

The

ory

- S

hift

ed D

ata

|ηe|

D0, L=9.7 fb-1

CT14 NNLOCT14 NNLOCT10 NNLOCTEQ6.6 NLO

-0.10

-0.05

0.00

0.05

0 0.5 1 1.5 2 2.5 3

The

ory

- S

hift

ed D

ata

|ηe|

D0, L=0.75 fb-1

CT10 NNLOCT10 NNLOCT14 NNLOCTEQ6.6 NLO

-0.2

-0.1

0.0

0.1

0.2

0 0.5 1 1.5 2 2.5 3

FIG. 23: Same as Fig. 22, plotted as the Charge asymmetry as the difference between theory and

shifted data for Ach from DØ Run-2 (9.7 fb−1).

E. Constraints on strangeness PDF from CCFR, NuTeV, and LHC experiments

Let us now turn to the strangeness PDF s(x, Q), which has become smaller at x > 0.05 in

CT14 compared to our previous analyses, CT10 and CTEQ6.6. Although the CT14 central

s(x, Q) lies within the error bands of either earlier PDF set, it is important to verify that the

new result is consistent with the data that are sensitive to the strange-quark PDF. With our

selection of experiments, four related fixed-target measurements are known to be sensitive

to s(x, Q): these are the measurements of dimuon production in neutrino and antineutrino

39

NLO

NNLO1�

d�d|yZ |

CMS Z, 20 < M < 30 GeV

.

0.05

0.04

0.03

0.02

0.01

.

|yZ|

Data

/T

heo

ry

21.510.50

1.2

1.1

1

0.9

NLO

NNLO1�

d�d|yZ |

CMS Z, 30 < M < 45 GeV

.

0.06

0.05

0.04

0.03

0.02

0.01

.

|yZ|

Data

/T

heo

ry

21.510.50

1.1

1

0.9

NLO

NNLO1�

d�d|yZ |

CMS Z, 45 < M < 60 GeV

.

0.03

0.02

0.01

.

|yZ|

Data

/T

heo

ry

21.510.50

1.2

1.1

1

0.9

NLO

NNLO1�

d�d|yZ |

CMS Z, 60 < M < 120 GeV

.

0.7

0.6

0.5

0.4

0.3

0.2

0.1

.

|yZ|

Data

/T

heo

ry

21.510.50

1.1

1

0.9

NLO

NNLO1�

d�d|yZ |

CMS Z, 120 < M < 200 GeV

.

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

.

|yZ|

Data

/T

heo

ry

21.510.50

1.2

1.1

1

0.9

0.8

NLO

NNLO1�

d�d|yZ |

CMS Z, 200 < M < 1500 GeV

.

0.002

0.0015

0.001

0.0005

.

|yZ |

Data

/T

heo

ry

21.510.50

1.4

1.2

1

0.8

0.6

Figure 11: The fit quality for the CMS double di↵erential Drell-Yan data for (1/�Z) · d�/d|yZ |versus |yZ |, in [78], for the lowest two mass bins (20 < M < 30 GeV and 30 < M < 45 GeV)

(top), the mass bins (45 < M < 60 GeV and 60 < M < 120 GeV) (middle) and the mass bins

(120 < M < 200 GeV and 200 < M < 1500 GeV) (bottom), at NLO and NNLO. Note that

correlated uncertainties are made available in the form of a correlation matrix, so the shift of data

relative to theory cannot be shown.

23

MMHT v. Z rapidity @

CMS

Lepton charge asym. v. CT14 @ D0 & CMS

NLO, ATLAS jets (7 TeV), 0.0 < |y| < 0.3

.

pT [GeV]

Dat

a/T

heo

ry

1000100

1


.

pT [GeV]

Dat

a/T

heo

ry

1000100

1


.

pT [GeV]

Dat

a/T

heo

ry

1000100

1


.

pT [GeV]

Dat

a/T

heo

ry

1000100

1


.

pT [GeV]

Dat

a/T

heo

ry

1000100

1


.

pT [GeV]

Dat

a/T

heo

ry

1000100

1


.

pT [GeV]

Dat

a/T

heo

ry

1000100

1

Figure 13: The fit quality for the ATLAS 7 TeV jet data in various rapidity intervals [99] at

NLO. The red points represent unshifted data and theory, and the black points (clustering around

Data/Theory=1) correspond to data and theory shifted using correlated systematics.

28

ATLAS inclusive jetsratio to MMHT


Precision of today’s PDFs (from PDF4LHC)[Determining full PDFs]


x 5−10 4−10 3−10 2−10 1−10

) [r

ef]

2)

/ u (

x, Q

2u

( x,

Q

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

NNPDF3.0

CT14

MMHT14

2 = 100 GeV2NNLO, Q

up quark

x 5−10 4−10 3−10 2−10 1−10

) [r

ef]

2)

/ d (

x, Q

2d

( x,

Q

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

NNPDF3.0

CT14

MMHT14

2 = 100 GeV2NNLO, Q

down quark

x 5−10 4−10 3−10 2−10 1−10

) [r

ef]

2)

/ g (

x, Q

2g

( x,

Q

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

NNPDF3.0

CT14

MMHT14

2 = 100 GeV2NNLO, Q

gluon

x 5−10 4−10 3−10 2−10 1−10

) [r

ef]

2 (

x, Q

+)

/ s2

( x

, Q+ s

0.8

0.9

1

1.1

1.2

1.3

1.4

NNPDF3.0

CT14

MMHT14

2 = 100 GeV2NNLO, Q

s + s


Precision of today’s PDFs (from PDF4LHC)[Determining full PDFs]


x 5−10 4−10 3−10 2−10 1−10

) [r

ef]

2)

/ u (

x, Q

2u

( x,

Q

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

NNPDF3.0

CT14

MMHT14

2 = 100 GeV2NNLO, Q

up quark

x 5−10 4−10 3−10 2−10 1−10

) [r

ef]

2)

/ d (

x, Q

2d

( x,

Q

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

NNPDF3.0

CT14

MMHT14

2 = 100 GeV2NNLO, Q

down quark

x 5−10 4−10 3−10 2−10 1−10

) [r

ef]

2)

/ g (

x, Q

2g

( x,

Q

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

1.3

NNPDF3.0

CT14

MMHT14

2 = 100 GeV2NNLO, Q

gluon

x 5−10 4−10 3−10 2−10 1−10

) [r

ef]

2 (

x, Q

+)

/ s2

( x

, Q+ s

0.8

0.9

1

1.1

1.2

1.3

1.4

NNPDF3.0

CT14

MMHT14

2 = 100 GeV2NNLO, Q

s + s


In the range 10−3 < x < 0.1 manyparton distributions are known with

precisions of a couple of percent.

Conclusions on PDFs[Conclusions]

I Factorization is key to our ability carry out calculations for hadroncolliders

I Beyond leading order, factorization implies shuffling of divergencesbetween the PDFs and the perturbative part of the calculation

I Equation for (factorisation) scale dependence of PDFs is DGLAP

I Our knowledge of PDFs comes both from the direct DIS measurements(for quarks) and from the scale-dependence of those measurements (forgluons), as well as Tevatron & LHC data.

I Today’s precision on (some) PDFs approaches the percent level, which iscrucial for Higgs precision prospects.


PDF exercise 1[Conclusions]

Check that LO DGLAP evolution preserves the momentum sum ruleunchanged.

Hints:

I One way of doing this is to use Mellin moments: f(N)i/p (µ2) =

∫ 1

0dx xN fi/p(x , µ2)

I Which Mellin moment tells you about the momentum carried by parton flavouri?

I Show that the N th Mellin moment of the DGLAP convolution, i.e.∫ 1

0dx xN

∫ 1

xdzz P(z)f (x/z) is given by the product of the N th Mellin moments of

P and f , i.e. P(N)f (N) ≡∫ 1

0dz zNP(z)×

∫ 1

0dx xN f (x).

I Then work out the appropriate Mellin moments and you should be able to provemomentum conservation.


PDF exercise 2[Conclusions]

Show that at asymptotically large Q2, the fraction of a hadron’smomentum that is contained in the gluons tends to the value

2CF

2CF + nf TR

(and compare this to the result shown earlier in the slides)

Hints:

I write the evolution equation for the momentum in the quarks and gluons interms of Mellin moments as a differential matrix equation:

∂lnµ2

(Σ(N)(µ2)g (N)(µ2)

)=αs(µ

2)

2π

(P

(N)ΣΣ P

(N)Σg

P(N)gΣ P

(N)gg

)(Σ(N)(µ2)g (N)(µ2)

)where Σ corresponds to a sum over all quark flavours. Watch out for the factors

of 2 and nf in P(N)Σg .

I Examine the asymptotic solutions of this equation


EXTRAS


Anti-quarks in proton[Extras]

[Sea & valence]

u

d

u

u

d

u

u

u

u

d

u

u

u

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

I Previous transparency: we were actually looking at ∼ u + u, d + d

I 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)


“Valence” quarks[Extras]

[Sea & valence]

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−


By how much do PDFs evolve?[Extras]

[Size of PDF evolution]g(x

, Q

= 1

00 G

eV

) / g(x

, Q

= 2

GeV

)

x

Gluon evolution from 2 to 100 GeV

Input: CTEQ61 at Q = 2 GeV

Evolution: 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

I See factors ∼ 0.1− 10

I Remember: LHC involves productof two parton densities

It’s crucial to get this right!

Without DGLAP evolution, you

couldn’t predict anything at LHC


Which PDF channels contribute?[Extras]

[Channels for dijet production]

Inclusive jet cross sections with MSTW 2008 NLO PDFs

(GeV)T

p100

Fra

cti

on

al co

ntr

ibu

tio

ns

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

cti

on

al co

ntr

ibu

tio

ns

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

cti

on

al co

ntr

ibu

tio

ns

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 = 0.7T

kT

= pF

µ = R

µfastNLO with

jets→gg

jets→gq

jets→qq

0.0 < y < 0.8

(GeV)T

p100 1000

Fra

cti

on

al co

ntr

ibu

tio

ns

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


Evolution uncertainty[Extras]

[Higher order uncertainties]uncert

ain

ty o

n 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

I with LO evolution, uncertainty is∼ 30%

I NLO brings it down to ∼ 5%

I NNLO → 2% Commensurate with

data uncertainties




ain

ty o

n 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




ain

ty o

n 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


Basics of QCD Lecture 3: PDFs and DGLAP

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Transcript of Basics of QCD Lecture 3: PDFs and DGLAP