Introduction to Quantum Chromodynamics...Introduction to Quantum Chromodynamics Michal Sumbera Nuclear Physics Institute ASCR, Prague December 15, 2009 Michal Sumbera (NPI ASCR, Prague)

Introduction toQuantum Chromodynamics

Michal Šumbera

Nuclear Physics Institute ASCR, Prague

December 15, 2009

Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, 2009 1 / 58

QCD improved quark-parton mode

1 Introduction

2 General framework

3 Partons within partonsBranching functions in QCDMultigluon emission and Sudakov formfactors of partons

4 Partons within hadronsEvolution equations at the leading orderMoments of structure functions and sum rulesExtraction of gluon PDF from scaling violationsEvolution equations at the next-to-leading orderHard scattering and the factorization theorem

5 Brief survey of methods of solving the evolution equations

6 Exercises


Literature

Our discussion is based on

Quarks, partons and Quantum Chromodynamicsby Jǐŕı Chýla

Available at http://www-hep.fzu.cz/ chyla/lectures/text.pdf


Introduction

In this lecture:

How the naive QPM is modified within the framework of pQCD.

Why pQCD modification to QPM is such that the basic concepts of theQPM maintain their meaning and are useful even in a theory whichsimultaneously aspires to describe the confinement of colored partons.

We’ll try to distinguish the effects which can be calculated in perturbationtheory (in which partons are treated essentially as massless observableparticles), from those where the color confinement plays a crucial role andperturbation theory is inapplicable.


General framework

Hard collision A + B → F + . . . , where A,B are h, ` or gauge bosons.“hard” ≡ process dominated by short distance interaction.pQCD description retains basic QPM strategy of dividing space–timeevolution of collision into three distinct stages.

Figure 1: The general scheme of QCD improved QPM


General framework

I: Initial evolution is represented by PDF Da/A(x1,M,FS1), Db/B(x2,M,FS2)of partons a, b and by factorization scheme. Incalculable in pQCD. Onlytheir dependence on the factorization scale M is calculable.

II: Hard scattering of partons : a + b → c + d is described by parton levelcross–section

σab→cd(s, x1, x2, pc , pd , µ,M1,FS1,M2,FS2), (1)where s = (pa + pb)

2, µ is the hard scattering scale, in general differentfrom the factorization scales Mi , and FS1, FS2 are factorization schemesdefining the distribution functions Da/A,Db/B . (1) is calculable in pQCDLeftrightarrow some measure of “hardness” is large compared to the mass ofproton. QPM is zeroth order approximation to (1) which is thensystematically improved in pQCD.

III: Hadronization of partonic state produced in hard parton scattering is theleast understood stage of collision.pQCD is not applicable here and various sorts of models must be employed.In the independent fragmentation model, Dh/p(z ,M,FS) acquire,similarly to PDFs inside hadrons, dependence on the M and FS.


General framework

In this lecture only stages I and II will be discussed.

General formula in pQCD is the same as in QPM:

σ(A + B → F + anything) = (2)∑abcd

∫ ∫dx1dx2Da/A(x1,M1)Db/B(x2,M2)σab→cd(s, x1, x2, pc , pd ,M1,M2)

⊗Dhadr (pc , pd ,PF )︸︷︷︸model dependent

Sum runs over all parton combinations leading to required final hadronicstate F and σab→cd contains all appropriate δ–functions expressing theoverall momentum conservation on the partonic level.

Convolution ⊗ stands for further integrations over the momenta of finalstate partons pc , pd , which lead to the final hadronic state F .

Initial PDFs depend on factorization scales M1,M2 which are, similarly tothe hard scale µ, unphysical and the cross–sections of physical processesmust, if evaluated to all orders, be independent of them.


Partons within partons

Basic idea of partons within partons picture is contained already inWeizsäcker-Williams approximation (WWA) in QED, avoiding at the sametime conceptual problem of parton confinement in hadrons.

Consider e−p in the region of small Q2. This interaction cannot be describedby an incoherent sum of cross-sections of interactions on individual partons.

Our interest is now structure of the beam electron not of the proton.

⇒⇒⇒ We stay within pQED and concentrate on the upper vertex in Fig.2.

Figure 2: The relation between cross–sections of ep (a) and γp (b) interactions inlow Q2 region. Kinematics of the branching e− →e− + γ (c) and the vertexdescribing the branching γ →e+e− (d).


Equivalent photon approximation

Basic idea of WWA:Interpret e + p → e . . . as emission from the beam electron of a nearly realphoton followed by its interaction with the target proton.

Correspondingly express σtot(ep) in terms of σtot(γp) of the realphoton–proton collision γ + p → . . . .

Basic assumption: for small photon virtuality (mass) γ∗ + p ≈ γ + p.Measure of “smallness” of the photon virtuality is not directly Q2, butrather its ratio Q2/W 2, where W 2 ≡ sγp.

Small virtuality: ⇒ in the overall ep CMS the photon in Fig. 2 behaves asnearly real and parallel to the beam electron, which thus looks like beingaccompanied by the “beam” of photons with certain momentum distributionfunction.



Determination of this distribution function proceeds in several steps:

1 In the proton rest frame (with proton mass M),

dσ =(2π)4

| ~v |e4

2E 2MLµνWµν(q, p)

d3k ′

(2π)32Ek′

1

Q4=

=(2π)4e4

2E 2MLµνWµν(q, p)

d4q(2π)3Q4

δ(k ′2), (3)

where Lµν is the lepton tensor and Wµν(q, p) is the hadronic tensorassociated with the lower vertex in Fig. 2a. Wµν is related to DIS structurefunctions Fi , but for low Q

2 it cannot be expressed in terms of PDFs.Neglecting me we set in the second part of (3) | ~v |= 1.

2 Similarly we express

σtot(γp) =(2π)4e2

2Eγ2M

(−1

2gµν

)Wµν(Q

2 = 0, pq), (4)

where Wµν(Q2 = 0) is the same hadronic tensor as in (3) and 12 results

from the averaging over the spin states of the incoming photon.



3 Most general form of the symmetric Wµν(Q2) for unpolarized e + p

scattering consisten with Lorentz invariance and parity conservation is:

Wµν = C1(Q2, pq)gµν+C2(Q

2, pq)pµpν+C3(Q2, pq)qµqν+C4(Q

2, pq) (pµqν + pνqµ) ,(5)

where Ci (Q2, pq) contain all information about the structure of the

unpolarized proton. In γp rest frame as W 2 = 2pq − Q2 + M2. In thefollowing the dependence on pq will not be explicitly written out.

4 Requirement of gauge invariance imposes further restriction:

qµWµν(Q2) = 0⇒ qν

[C1 + C3q

2 + C4(pq)]

+ pν[C2(pq) + C4q

2]

= 0, (6)

which implies that only two of the four functions Ci are independent

C4(Q2) = −C1(Q2)

1

(pq)− C3(Q2)

q2

(pq), (7)

C2(Q2) = C1(Q

2)q2

(pq)2+ C3(Q

2)q4

(pq)2. (8)



5 For the real photon only C1(Q2 = 0) contributes to the contraction:

− 12 gµνWµν(Q

2 = 0)) = −C1(Q2 = 0). (9)

6 For the virtual photon only the terms proportional to C1,C2 contribute aftercontraction with the leptonic tensor Lµν :

LµνWµν = 2[−2C1(Q2)(kk ′ − 2m2) + C2(Q2)

(2(kp)(k ′p)−M2(kk ′)

)]. (10)

7 Recalling basic kinematical relations from lecture on QPM, taking intoaccount (8) and keeping in (10) the first two leading terms in Q2, we get:

LµνWµν = −2C1(Q2)Q2[

1 + 2

(kp

qp

)2− 2

(kp

qp

)− 2m

2

Q2

]

= −2C1(Q2)Q2[

1 + (1− y)2

y 2− 2m

2

Q2

], (11)

where y ≡ qp/kp is the fraction of incoming electron energy, carried awayby the exchanged virtual, nearly parallel, photon.


Weizsäcker-Williams approximation

8 Express d3k

′

2Ek′in terms of y and the transverse (w.r.t. to the incoming e)

momentum qT of the emitted γ∗. In the collinear kinematics, i.e. for

qT � q‖ (see Fig. 2c) we have:∫d3k

′

2Ek′=

∫δ(k ′2)d4q =

∫1

4Ek′dφdq‖dq2T

.=

∫π

2(1− y)dydq2T

.=π

2

∫dydQ2

(12)where in the last equality approximation y ≈ q‖/Ek was used. In thecollinear kinematics we also have

Q2 = 4EkEk′ sin2(ϑ/2)

.=

q2T1− y

, (13)

i.e. photon virtuality for fixed y is proportional to q2T .

9 [ (13) + (12) + (11) ] 7→ (3), leading to the following total cross–section ofe + p collision at low Q2:

σep(S ,Q2max) =

∫ ∫dy

dq2Tq2T

[α

2π

(1 + (1− y)2

y− 2m

2y

Q2

)]σγp(yS),

(14)where integral is taken over values of y , qT satisfying the conditionQ2 < Q2max . This is the mentioned Weizsäcker-Williams approximation.



Several aspects of this formula are worth to comment.

y 2 → y in the denominator of (11) was used to convert Ek → Eγ .(14) naturally suggests to interpret the function

α

2π

1

Q2

[1 + (1− y)2

y− 2m

2y

Q2

](15)

as the probability to find a photon with virtuality Q2 and momentumyP inside an electron of momentum P.

Although formally of the order O(m2) the second term in (15) gives actuallya finite contribution after integration over Q2 (or q2T ) in some interval(Q2min,Q

2max) : σep(S ,Q

2max ,Q

2min)

.=∫

dyα

2π

([1 + (1− y)2

y

]ln

Q2maxQ2min

− 2m2y[

1

Q2min− 1

Q2max

])︸︷︷︸

fγ/e(y ,Q2max ,Q

2min)

σγp(yS), (16)

where the lower limit follows from kinematics: Q2min =m2y 2

1− y. (17)



Considering Q2max (relabeled now as Q2), as a free parameter, we get:

fγ/e(y ,Q2) =

α

2π

[1 + (1− y)2

yln

Q2(1− y)m2y 2

− 2(1− y)y

+ O(m2/Q2)

]=

α

2π

[1 + (1− y)2

yln

Q2

m2+

1 + (1− y)2

yln

1− yy 2− 2(1− y)

y

](18)

where m2 from matrix element got canceled by 1/m2 coming from phasespace integration, giving finite contribution to fγ/e(y ,Q

2).

This function is then interpreted as the distribution function of photons,carrying fraction y of electron energy and having virtuality up to Q2.

Crucial feature of WWA is the presence of the logarithmic term ln(Q2/m2),which is due to the fact that the matrix element (11) is proportional to Q2.



One can also introduce complementary distribution function –probability of finding electrons inside an electron:

fe/e(y ,Q2) = δ(1− y) + α

2π

[1 + y 2

1− y

]ln

Q2

m2, (19)

where the δ–function corresponds to the case of no photon radiation.

To include contribution of virtual diagrams in (19) one needs to replace[. . . ] 7→ [. . . ]+.

This modification is crucial for the probability interpretation of fe/e(x ,Q2):∫ 1

0

fe/e(x ,Q2)dx = 1 (20)

total number of electrons minus the number of positrons inside an electronis conserved (and equal to 1 for a physical electron).


Weizsäcker-Williams approximation: Real photon

Same ideas can be applied also to the case that the beam particle is a realphoton which can split into a e+e− pair ( see Fig. 2d). The distributionfunction of electrons (or positrons) inside a photon, carrying fraction y of itsenergy and having virtuality up to Q2, is given as

fe/γ(y ,Q2) =

α

2π

[y 2 + (1− y)2

]ln

Q2

m2(21)

with the same provision about the neglected terms as for previous case.Notice that there is no IR singularity in (21) and consequently no “+”distribution in this expression.

Smallness of Q2 needed for validity of WWA depends on process in the lowervertex of Fig. 2ab, in which γ∗ interacts with the target particle, as well ason the kinematical region considered.

WWA works when photon virtuality is small compared to values of basickinematical variables describing its interaction with the target particle. Forthe case of the total cross–section in (3) required Q2/sγp = Q

2/yS � 1, forthe photoproduction of jets in e + p (HERA) Q2 should be much smallerthan the transverse momentum squared of the produced jets.


Weizsäcker-Williams approximation: Real photon

The essence of the WWA is contained in the branching functions

Pγe(x) =1 + (1− x)2

x, (22)

Pee(x) =

[1 + x2

1− x

]+

, (23)

Peγ(x) = x2 + (1− x)2, (24)

all of which reflect the basic QED vertex eγe.

For processes dominated by low Q2 the incoming electron behaves as abeam of nearly real electrons and photons, described by fγ/e and fe/e .

Similarly, a single photon can be viewed as a photon accompanied by abeam of nearly real electrons and positrons, distributed in the photonaccording to the function fe/γ(x ,Q

2).

Finally since all QED branching functions appear always in the product withQED couplant α, the “admixture” of photons inside an electron andelectrons in a photon are small effects. whch are entirely due to theinteraction of electrons and photons with their own electromagnetic field.


Branching functions in QCD

Straightforward generalization of WWA: e → q and γ → G and inclusion ofthe 3–gluon vertex, which leads to PGG (x) branching function having thesame physical interpretation as the other branchings.

Due to color QCD branchings acquire additional nf dependent factors:

P(0)qq (x) = P(0)qq (x) =

4

3

[1 + x2

1− x

]+

, (25)

P(0)Gq (x) = P

(0)Gq (x) =

4

3

[1 + (1− x)2

x

], (26)

P(0)qG (x) = P

(0)qG (x) =

[x2 + (1− x)2

2

], (27)

P(0)GG (x) = 6

{[x

1− x

]+

+1− x

x+ x(1− x) +

(33− 2nf

36− 1)δ(1− x)

}(28)

These branching functions are the same for all quark flavors qi , qi .Parton branchings are now due to interaction of q and G with their ownchromodynamic field. The nf –dependent term in (28) comes from virtual qqloop correction to gluon propagator.


Multigluon emission

When discussing KLN theorem we saw how the integration over thetransverse momentum of the gluon emitted from the target quark, togetherwith the virtual corrections yields for large Q2 the process independentcontribution to the cross–section of the process e− + q → e− + q + g :

σ ∝ αsP(0)qq (x) lnQ2

m2g(29)

This equation is correct provided 0 ≤ ε < x < 1− ε ≤ 1.

The logarithmic terms come from the integration over the transversemomentum around the singularity at qT = 0. In the case of two gluonemission the leading term proportional to α2s ln

2(Q2/m2g ) comes fromintegration over the region of strongly ordered virtualities t1, t2 in Fig. 3a

|t2| ≤ � |t1| , |t1| ≤ �Q2, (30)

where � is some small parameter introduced to quantify the meaning of“strongly ordered”. As we shall see the coefficients in front of the leadinglogarithms do not depend on this parameter.


Multigluon emission

Figure 3: Multigluon emission from incoming parton leg (a) in eq collision (a),and from outgoing quarks in e+e− →qq annihilation (b).


Multigluon emission

Omitting the branching functions and concentrating on the transversemomenta of radiated gluons we find that the contribution of the diagramwith two radiated gluons is proportional to (we define τ = |t| for allvirtualities)

α2s (µ)

∫ �Q2m2g

dτ1τ1

∫ �τ1m2g

dτ2τ2

= α2s (µ)

∫ �Q2m2g

dτ1τ1

ln�τ1m2g

=

α2s (µ)

[1

2ln2

Q2

m2g+ f1(ln �) ln

Q2

m2g+ f2(ln �)

], (31)

where fi (ln �) are simple polynomials in ln �.

Calculation can be generalized to n emitted gluons: coefficient An in front ofthe term with the same power of αs and ln Q

2 (the so called leading log(LL) )

σ(n) ∝ αns(

An lnn

(Q2

m2g

)︸︷︷︸

LL

+ Bn lnn−1

(Q2

m2g

)︸︷︷︸

NLL

+ · · ·+ fn(x))

(32)

comes entirely from the region (30) of strongly ordered virtualities, is�–independent and equals 1/n!.


Multigluon emission

In (32) the scale µ must to be used in αs(µ). The “best” µ in vertices ofthe ladder of Fig. 3a, where quark of virtuality τi+1 emits a quark withvirtuality τi and a real gluon, is µ

2 = τi , i.e., µ2 should be identified with the

highest virtuality of all partons interacting in this vertex.⇒ Scales of αs are different at different vertices along the ladder andincrease when moving up from the proton to the qγq vertex.

Consider the correction to the emission of the first gluon in Fig. 3a, broughtabout by the emission and subsequent reabsorption of another gluon. UVrenormalization of this loop leads to the appearance of ln(µ/τi ) where µ isthe argument of αs(µ) at the loop vertices.

Large UV logarithms can be avoided by setting µ ∼ τi . For strongly orderedvirtualities the smaller one in each ladder link can be neglected and thus thevertex effectively describes the interaction of one off mass–shell and twomassless partons. As in the mentioned logarithm µ enters scaled bysomething describing the kinematics of the vertex, the maximal virtuality atthat vertex is essentially the scale available.


Multigluon emission and Sudakov formfactors of partons

Consider again e+e− → qq as discussed in connection with the KLN.Probability of emitting a single real gluon is ∝ ln2 β, while the radiativecorrections (virtual emissions) give negative contributions, which cancel bothsingle and double logs of β. By the KLN theorem this mechanism operatesat each order of αs .

Probability of multiple gluon emission grows with the order of perturbationexpansion as there are more powers of ln2 β. As a result probability of nogluon emission must accordingly be suppressed. This important phenomenonis expressed quantitatively in the so called Sudakov formfactors of quarksand gluons.

Imagine that the quark (or antiquark) produced in e+e− annihilations has apositive virtuality τ and assume for simplicity that τ is small with respect tothe CMS energy squared s = 4E 2. Kinematical bounds on the fraction z ofthe parent quark energy, carried by the daughter quark after the gluonemission in Fig. 3b are given as

zmin =1

2

[1−

√1− τ/E 2

].

=τ

s; zmax =

1

2

[1 +

√1− τ/E 2

].

= 1− τs. (33)


Sudakov formfactors of partons

The time–like Sudakov formfactor gives the probability that there will be nogluon radiation with the virtuality of the outgoing off mass–shell quarkbetween τ0 and τ and the fraction z between zmin and zmax

S(τ0, τ,E ) ≡ exp

{−∫ ττ0

dτ ′

τ ′

∫ zmaxzmin

αs(√τ ′)

2πP(0)qq (z)dz

}, (34)

where {. . . } gives probability of gluon radiation from the quark withτ ∈ (τ0, τ) and will carry energy fraction 1− z ∈ (1− zmax , 1− zmin).

Note the analogy to usual non-decay probability of an unstable system.

Neglecting dependence αs(√τ) integration in (34) can be performed:

S(τ0, τ,E ) ≈ exp[−αs

2π

4

3ln

(τ

τ0

)ln

(4E 2

τ0

)+O(ln τ0)

]. (35)

Notice that [. . . ] contains a double log of τ0 and that the Sudakovformfactor vanishes as τ0 → 0.


Partons within hadrons

Up to now quarks and gluons were treated as free particles before and afterthe collision, despite the experimental evidence that they exist merely insidehadrons and behave like free particles only if probed at short distances.

Basic idea to get around this (Fig. 4) is reminiscent of the UVrenormalization of electric or color charges.

Figure 4: Graphical representation of the definition of “dressed” partondistribution functions inside a given hadron for the nonsinglet quark distributionfunction. τi denotes absolute value of the virtuality of a given intermediate state.


Evolution equations at the leading order

Define primordial (sometimes also called “bare”) distribution functions ofpartons inside hadrons, q0(x),G0(x), which depend on x only and areinterpreted in the sense of the QPM.

Introduce nonsinglet (NS) quark distribution functions, which at the LOcoincide with the valence distribution functions of QPM. We need toconsider only the effects of multiple gluon emissions off the primordialquarks, described by qNS,0(x), as sketched in Fig. 4.

Summing up contributions from ladders in Fig. 4 we define therenormalized, or “dressed”, NS quark distribution function

qNS(x ,M) ≡ qNS,0(x) +∫ 1

x

dyy

[P(0)qq

(x

y

)∫ M2m2

dτ1τ1

αs(τ1)

2π

]qNS,0(y) + (36)

∫ 1x

dyy

∫ 1y

dww

∫ M2m2

dτ1τ1

∫ τ1m2

dτ2τ2

αs(τ1)

2π

αs(τ2)

2π

[P(0)qq

(x

y

)P(0)qq

( yw

)]qNS,0(w)+· · · ,

where newly introduced scale M has the meaning of maximal virtuality of thequark interacting with γ in the upper, QED, vertex of Fig. 4a and coincidingwith the factorization scale introduced at the beginning of this lecture.


Factorization scale M vs. renormalization scale µ

In both cases scales emerge when “bare” quantities are replaced with theirdressed counterparts. Important difference between them are that:

µ emerged in the process of ultraviolet renormalization, i.e. concernsshort distance properties of the theoryM has been introduced to deal with parallel singularities, i.e. concernslarge distances.

In evolution equations M is interpreted as the upper bound on partonvirtualities included in the definition of dressed PDFs, without specificationof its relation to kinematic variables of any physical process.

In DIS it is common to set M2 = Q2 and thus include in the dressed PDFsgluon emission even if it is actually far from parallel. This is legal to do asthere is no sharp dividing line between “small” and “large” virtualities, butfor large virtualities the pole term 1/t, which, after integration, leads toln(Q2/m2), no longer dominates the exact matrix element for the gluonradiation.

In the LL approximation there are no good arguments in flavour of this, orany other, choice of M and all values of M are in principle equally good.


Evolution equations at the leading order: DGLAP

Similarly to step from bare to the renormalized electric charge in QED (orcolor charge in QCD) we take derivative of both sides in (36) with respect tothe scale M: dqNS(x ,M)

d ln M2=αs(M)

2π

∫ 1x

dyy

P(0)qq

(x

y

)× (37)[

qNS,0(y) +

∫ 1y

dww

∫ M2m2

dτ2τ2

αs(τ2)

2πP(0)qq

( yw

)qNS,0(w) + · · ·

]︸︷︷︸

≡ qNS(y ,M)

,

i .e.dqNS(x ,M)

d ln M2=αs(M)

2π

∫ 1x

dyy

P(0)qq

(x

y

)qNS(y ,M) =

αs2π

∫dz∫

dyP(0)qq (z)qNS(y)δ(x − yz) ≡αs2π

P(0)qq ⊗ qNS, (38)

which is the so called Dokshitzer-Gribov-Lipatov-Altarelli-Parisi(DGLAP) evolution equation for the dressed NS quark distribution functionqNS(x ,M).



Derivation of DGLAP is based on two essential ingredients:

1 Strong ordering of virtualities in the ladder of Fig. 3a.2 The fact that the argument of αs(µ) at each vertex of this ladder is

given by the upper (i.e. the largest) virtuality. As a consequence theonly dependence on M appears in the upper bound on the integrationover the largest virtuality τ1 in each ladder of Fig. 4.

Note that parallel logarithms have completely disappeared in the process oftaking derivative with respect to ln Q2! Moreover, this equation effectivelyresums the LL series (36).

There is a simple relation between particle densities in the intervals of z andx , where x is the daughter particle momentum fraction (see Fig. 4) wheny ≡ x/z is held fixed: d ln x = d ln z . The contribution to the density D(x)of particles having after the branching x in the interval (x , x + dx) andcoming from the interval (y , y + dy) of the primordial momentum y is equalto

x = yz ⇒ dx = ydz ⇒ D(x) ≡ dNdx

=1

y

dNdz

=1

yD(z). (39)



Branching function P(0)qG (z), describing the probability to find a quark (of

any flavor) inside the gluon, also contributes to the LL evolution equationsfor dressed∗ quark/antiquark distribution functions

dqi (x ,M)d ln M

=αs(M)

π

[∫ 1x

dyy

P(0)qq

(x

y

)qi (y ,M) +

∫ 1x

dyy

P(0)qG

(x

y

)G (y ,M)

](40)

dqi (x ,M)d ln M

=αs(M)

π

[∫ 1x

dyy

P(0)qq

(x

y

)qi (y ,M) +

∫ 1x

dyy

P(0)qG

(x

y

)G (y ,M)

](41)

where the gluon distribution function G (x ,M) satisfies similar evolutionequation

dG (x ,M)d ln M

= (42)

αs(M)

π

[nf∑i=1

∫ 1x

dyy

P(0)Gq

(x

y

)(qi (y ,M) + qi (y ,M)) +

∫ 1x

dyy

P(0)GG

(x

y

)G (y ,M)

]

* From now on we drop the adjective “dressed” and write the derivative with respect to

ln M instead of ln M2.Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, 2009 31 / 58

Evolution equations at the leading order: Kernels

Figure 5: Graphical representation of the kernels of the evolution equations forthe quark (a) and gluon(b) distribution functions inside the proton.

Solution requires boundary conditions containing information on thex–dependence of PDFs at some arbitrary initial scale M0. Later cannot becalculated from pQCD and must be taken from experimental data.

N.B. results truncated to any finite order do depend on the choice of M0.

The evolution equations therefore determine merely the M–dependence ofparton distribution functions, not their absolute magnitude.


PDF evolution in x − Q2 plane: DGLAP, BFKL etc.

Y =

ln 1

/x

non-

pert

urba

tive

regi

on

ln Q2

ln Q2s(Y) saturation region

BK/JIMWLK

DGLAP

BFKL

ln Λ2QCDαs

PDF at Q2 = 10GeV 2: at small x the sea is BIG!

010-4 10-3 10-2 10-1 1

HERA-I PDF (prel.)

experimental uncertainty

model uncertainty

x

xf

Q2 = 10 GeV2

HERA Structure Functions Working GroupNucl. Phys. B 181-182 (2008) 57–61

vxu

vxd

xg (×1/20)

xS (×1/20)

0.2

0.4

0.6

0.8

1

Figure 7: Comparison between valencequark and sea PDF: sea distributions arescaled by a factor of 1/20.

010-4 10-3 10-2 10-1 1

HERA-I PDF (prel.)

experimental uncertainty

model uncertainty

x

xf

Q2 = 10 GeV2

HERA Structure Functions Working GroupNucl. Phys. B 181-182 (2008) 57–61

20

4

8

12

16

xg

xS

vxu

vxd

Figure 8: Comparison between valencequark and sea PDF: sea distributions arenow unscaled.


F2(x ,Q2) up to HERA

ZEUS NLO QCD fit

H1 PDF 2000 fit

0

1

2

3

4

5

1 10 102

103

104

105

F 2 em-lo

g 10(x)

Q2(GeV2)

x=6.32 10-5x=0.000102x=0.000161

x=0.000253x=0.0004

x=0.0005x=0.000632

x=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

BCDMS

E665

NMC

H1 94-00

H1 (prel.) 99/00

ZEUS 96/97

HERA F2


Solving DGLAPMost straightforward method solves the system of coupled evolution equations bymeans of sophisticated numerical algorithms on powerful computers. Thestandard analysis of experimental data on DIS proceeds in the following steps:

The initial M0 is chosen.

Some parameterization of the boundary condition for parton distributionfunction Di (x ,M0), i = q, q,G , for instance,

Di (x ,M0) = Aixαi (1− x)βi (1 + γix) (43)

where Ai , αi , βi , γi are free parameters, is chosen.

These parameters, together with the value of the basic QCD parameter Λ,entering αs(M/Λ), are then varied within some reasonable ranges and foreach such set the evolution equations (40)-(43) are solved. This yieldsDi (x ,M) at all x and M.

In this way obtained theoretical predictions are fitted to experimental data,allowing thereby the determination of the parameters Λ,Ai , αi , βi , γi for eachflavor of the quarks as well as for the gluon.


Moments of structure functions and sum rules

DGLAP looks particularly simple in terms of moments:

f (n) ≡∫ 1

0

xnf (x)dx . (44)

Instead of convolutions in (40)-(43) we get simple multiplications for themoments

dqi (n,M)d ln M

=αs(M)

π

(P(0)qq (n)qi (n,M) + P

(0)qG (n)G (n,M)

), (45)

dqi (n,M)d ln M

=αs(M)

π

(P(0)qq (n)qi (n,M) + P

(0)qG (n)G (n,M)

), (46)

dG (n,M)d ln M

=αs(M)

π

(P

(0)Gq (n)

nf∑i=1

(qi (n,M) + qi (n,M)) + P(0)GG (n)G (n,M)

)(47)



A particularly simple equation holds for moments of the nonsinglet quarkdistribution function (a ≡ αs/π)

dqNS(n,M)d ln M

=αs(M)

πP(0)qq (n)qNS(n,M) ⇒ qNS(n,M) = An

(ca(M)

1 + ca(M)

)−P(0)qq (n)/b(48)

An are unknown constants, which play the same role as the boundarycondition (43) on the distribution functions and which must also bedetermined from experimental data.

Two features of the above solution are worth noting.

Due to the fact that, trivially, P(0)qq (0) = 0 the integral over qNS is

M–independent.

For n > 0 the moments of P(0)qq read:

P(0)qq (n) =4

3

[−2S1(n) +

1

n + 1+

1

n + 2− 3

2

], S1(n) ≡

n∑k=1

1

k= ψ(n+1)+γE ,

(49)which implies negative value of P

(0)qq (n > 0). This provides the

basis for one of the methods for solving the evolution equations.



Combining (45)-(47) we obtain evolution equation for the sum:

S(n,M) ≡nf∑i=1

(qi (n,M) + qi (n,M))+G (n,M) = q(n,M)+q(n,M)+G (n,M) (50)

which reads:dS(n,M)d ln M

= (51)

αs(M)

π

[(P(0)qq (n) + P

(0)Gq (n)

)(q(n,M) + q(n,M)) +

(2nf P

(0)qG (n) + P

(0)GG (n)

)G (n,M)

]Since S(1,M) represents the fractional sum of the momenta carried by allpartons inside the proton (or other hadrons) (51) for n = 1 should be equalto unity at any scale M and therefore the derivative (51) should vanish. Thishappens provided

P(0)qq (1) + P(0)Gq (1) =

∫ 10

dzz(

P(0)qq (z) + P(0)Gq (z)

)= 0, (52)

2nf P(0)qG (1) + P

(0)GG (1) =

∫ 10

dzz(

2nf P(0)qG (z) + P

(0)GG (z)

)= 0. (53)

Straightforward evaluation of the above integrals using (25)-(28) shows thatin QCD these conditions are, indeed, satisfied.


Extraction of gluon PDF from scaling violations

Contrary to quark distribution functions measured directly in `+ p DIS thegluon distribution function enters only indirectly as one of the sources (theother being quark distributions themselves) of scaling violations, describedby the evolution equations (40) and (41).

The relative importance of the two branchings q → q + G and G → q + qdepends on x :

while q → q + G is dominant at large xG → q + q is imporant at small (x ≤ 10−2) region.

Dropping in small x region the first term in brackets of (40) and (41), theevolution equations for quark and antiquark distribution functions qi (x ,M),qi (x ,M) become identical and read

dqi (x ,M)d ln M

=dqi (x ,M)

d ln M=αs(M)

π

[∫ 1x

dyy

P(0)qG

(x

y

)G (y ,M)

]. (54)


Extraction of gluon PDF from scaling violations

Number of effectively massless quarks to be taken into account in (54)depends on the scale M. For nf = 4, these equations imply the followingexpression for the derivative of F ep2 (x ,M)

dF ep2 (x ,M)d ln M

=αs(M)

π

(2

nf =4∑i=1

e2i

)︸︷︷︸

20/9

∫ 1x

dz ((x/z)G (x/z)) P(0)qG (z). (55)

Eq. (55) is nonlocal, in the sense that the value of its left hand side at somex depends on gluon distribution function G (x) in the whole interval (x , 1).In order to get a local relation between dF ep2 (x ,M)/dx and G (x).

Prytz has suggested to expand H(x/z ,M) ≡ (x/z)G (x/z ,M) around z = 12 ,the symmetry point of P

(0)qG (z) = (z

2 + (1− z)2)/2:

H(x/z ,M) = H(2x ,M)+H ′(2x ,M)(z−1/2)+H ′′(2x ,M)(z−1/2)2/2!+· · ·(56)

and keeping only the first two terms in (56).


Extraction of gluon PDF: Prytz approximation

At small x the lower integration bound in (55) can safely be set to zero.⇒⇒⇒ Second and higher order terms in (56) will vanish:

dF ep2 (x ,M)d ln M

=20αs(M)

9π2xG (2x ,M)

∫ 10

dzP(0)qG (z)︸︷︷︸1/3

=20αs(M)

27π2xG (2x ,M) (57)

The price for simplicity of (57): in practice data do not allow us todetermine dF2(x ,M)/d ln M locally at each M, but only as some average forthe measured range of M2 = Q2.

⇒⇒⇒ Extracted gluon PDF cannot be attached to any well–defined scale M.

This simple approximate formula was used at HERA to determine averagegluon distribution function from scaling violations, see Fig. 9a.

Results of complete analysis at all x and Q2 using the full set of coupledDGLAP evolution equations (40-43) is shown in Fig. 9b. Note the sizablevariation of G (x ,Q) as Q2 increases from 5 to 20 GeV!


Extraction of gluon PDF: Prytz approximation

Figure 9: a) Gluon distribution functions measured at HERA an in NMCexperiment, using several different methods, including that based on (57) anddenoted “Prytz”; b) Gluon density xG (x ,M) at two values of the scale M = Q.


Higher orders contribution to gluon PDF: NNLO

Figure 10: Valence quark dissociatinginto qG → qGG → qGGqq̄ Figure 11: G-PDF as measured in higher

orders of DIS


Evolution equations at the next-to-leading order

In the LL approximation (32) at each order of αs only the term with thesame power of the parallel logarithm ln(M2/m2), coming from theintegration over the region of strongly ordered parton virtualities was kept.

In the next-to-leading logarithm (NLL) approximation we extend theintegration range by taking into account also the configuration where one ofthe emissions is not strongly ordered, thereby contributing one power of theparallel log less, i.e. at each order of αs we include also the terms

αks lnk−1

(M2

m2g

). (58)

Inclusion of these terms entails several novel features.

The relation between PDFs in the NLL approximation and observablestructure functions Fi (x ,Q

2) becomes more complicated.Branching functions Pij become perturbative expansions in powers ofthe couplant and thus functions of both z and αs(M)

Pij(z , αs(M)) = P(0)ij (z) +

αs(M)

πP

(1)ij (z) + · · · . (59)


NLL Evolution equations

Contrary to LL branching functions P(0)ij (z) which correspond directly to

basic QCD interaction vertices qG q and 3g , evolution functions P(1)ij (z)

come from the whole set of diagrams.

For instance P(1)qjqi is associated with both diagrams in Fig. 12 and its flavor

structure can be decomposed as follows:

P(1)qjqi = P(1)Vqq δji + P

(1)Sqq . (60)

Figure 12: Diagrams contributing to the NLO branching functions P(1)qiqj and Pqiqj .


NLL Evolution equations

Starting at NLL approximation we encounter branching functions Pqiqjdescribing probability of finding antiquarks inside quarks and vice versa.They are associated with the diagram in Fig. 12a and can be decomposedsimilarly to (60):

P(1)qjqi

= P(1)Vqq δji + P

(1)Sqq , (61)

where the singlet parts in qq and qq channels, coming from the gluonbranching in Fig. 12a, are equal, while the valence parts, originating fromboth diagrams, differ:

P(1)Sqq = P(1)Sqq ; P

(1)Vqq 6= P

(1)Vqq . (62)

The nonequality in (62) comes from Fig. 12a when i = j , two identicalfermions in the final state.

Contrary to the LL branching functions P(0)ij , which are universal,

P(1)ij (z) are similarly as the coefficients ci of the β–function:

da(µ, ci )d lnµ

= −ba2(µ, ci )[1 + ca(µ, ci ) + c2a

2(µ, ci ) + · · ·]

(63)

ambiguous.


NLL Evolution equations: Factorization scheme

The set FS={P(k)ij ; k ≥ 1} of nonuniversal higher order branching functionsdefines the factorization scheme. Corresponding evolution equation:

dqNS(x ,M)d ln M

≡ αs(M)π

∫ 1x

dyy

qNS(y ,M)

[P

(0)NS

(x

y

)+αs(M)

πP

(1)NS

(x

y

)+ · · ·

](64)

is thus a definition equation of qNS(x ,M), similarly as (63) is a definitionequation of the QCD couplant αs(M)!

There are, however, two types of quark nonsiglet distribution functions

1 the valence type q(−)NS,i ≡ qi − qi ; i = u, d , s, · · · , (65)

2 the nonvalence type q(+)NS ≡ qi + qi −

1

nf

nf∑k=1

(qk + qk) , (66)

which for nf = 2 reduces to

u(+) =1

2

(u + u − d − d

)=

1

2

[(u − d)− (d − u)

]. (67)


NLL Evolution equations: Factorization scheme

In the LL approximation q PDFs are generated exclusively by gluon splitting⇒ we expect u = d . Consequently (67) coincides with the difference12 (u

(−) − d (−)) of the valence-like NS distributions.

In the NLL approximation the q(+) and q(−) NS PDFs evolve according to

different kernels, expressed as combinations of P(1)Vqq and P

(1)Vqq :

P(1)(−) ≡ P

(1)Vqq − P

(1)Vqq , P

(1)(+) ≡ P

(1)Vqq + P

(1)Vqq . (68)

In FS used in phenomenological analyses P(1)Vqq is very small ⇒ difference

between these two types of NS distributions negligible.

Converted into moments the NLL NS evolution equation (64) implies:

dqNS(n,M)d ln M

=αsπ

qNS(n,M)

[P

(0)NS(n) +

αs(M)

πP

(1)NS(n)

], (69)

⇒ qNS(n,M) = An[

ca(M)

1 + ca(M)

]−P(0)NS(n)/b(1 + ca(M))−P

(1)NS(n)/bc , (70)

where a(M) ≡ αs(M)/π.Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, 2009 49 / 58

Hard scattering and the factorization theorem

Basic idea of factorization is to split range of virtualities τ ≡| t | corresponding toFeynman diagram in Fig. 13a into two parts:

τ ≤ τ0: The integral over nearly parallel configurations is absorbed in thedressed PDFs qNS(x ,

√τ0) and similarly for PDFs of q and G .

τ > τ0: Integration yields finite result CNS(z ,Q/M,FS) called hardscattering cross–section in the NS channel. Beside Q2,M2 = τ0 and z it

also depends on the FS of PDFs, specified at NLL by P(1)ij (z), and hard

scattering scale µ, which in general is different from factorization scale M.

Figure 13: KLN in DIS. t in a) denotes virtuality of intermediate quark.



Hard scattering cross–section is itself given as expansion in αs(µ)

CNS

(z ,

Q

M,FS

)= δ

(k)NS

[δ(1− z) + αs(µ)

πC

(1)NS

(z ,

Q

M,FS

)+ · · ·

], (71)

where the superscript “(k)” distinguishes between different NS channels and

the quantities δ(k)NS include electromagnetic (or weak) couplings of quarks.

According to the factorization theorem a generic NS structure functionFNS(x ,Q

2) can be written as the convolution

FNS(x ,Q2) =

∫ 1x

dyy

qNS(y ,M,FS)CNS

(x

y,

Q

M,FS

), (72)

where all dependence Q resides in the hard-scattering cross–section CNS.Note that both qNS(x ,M,FS) and CNS(z ,Q/M,FS) depend on thefactorization scale M, as well as on the factorization convention FS.

N.B. physical quantity FNS(x ,Q2) in (72) depends neither on the

factorization scale M, nor on the renormalization scale µ.



Factorization theorem guarantees that all parallel singularities can beabsorbed into qNS and also that dependence on unphysical quantities M and

P(k)ij , which define the FS, cancel in the convolution (72), provided

perturbative expansions in (71) as well as (64) are summed to all orders.

Cancelation of the dependences of qNS and CNS on the FS is guaranteed by:

C(1)NS

(z ,

Q

M,FS

)= δ

(k)NS

(P

(0)NS(z) ln

Q

M+

P(1)NS(z)

b+ k(z)

), (73)

which expresses C(1)NS explicitly as a function of P

(1) and where k(z) is a

function of z , independent of M and P(1)NS.

In terms of moments the relations (72) and (73) read

C(1)NS

(n,

Q

M,FS

)= δ

(k)NS

(P

(0)NS(n) ln

Q

M+

P(1)NS(n)

b+ k(n)

)(74)

andFNS(n,Q

2) = qNS(n,M,FS)CNS

(n,

Q

M,FS

). (75)


Solving the evolution equations: Buras–Gaemers method

Historically first method used to solve evolution equations in the LO. Itstarts by assuming that the initial conditions on NS, i.e. valence uv and dvquark PDFs are given by so–called beta–distribution:

qv(x ,M0) = Aqxη

q1 (1− x)η

q2 , (76)

which implies for the moments

qv(n,M0) = AqB(ηq1 + n + 1, η

q2 + 1), B(x , y) ≡

Γ(x)Γ(y)

Γ(x + y). (77)

Using these initial conditions in DGLAP evolution equations (38) forqv(n,M) yields explicit analytic expressions for their scale dependence

∗.

* Note, however, that the simplicity of these explicit solutions is lost when we attempt

to translate them back into the x–space.


Solving the evolution equations: Buras–Gaemers method

BG assumed that these solutions can be reasonably approximated at all Mby the form (77) in which the coefficients Aq, ηq1 , η

q2 are allowed to depend

on M as:

ηqi (s) = ηqi +η

′qi s, i = 1, 2; A

q =δq

B(ηq1 (s), 1 + ηq2 (s))

, s ≡ ln[

ln(M/Λ)

ln(M0/Λ)

](78)

where δu = 2 and δd = 1.

These later conditions guarantee the validity of fundamental QPM sumrules, like the Gross-Llewellyn-Smith sum rule:∫ 1

0

F νN3 (x ,Q2)dx =

∫ 10

(uv (x) + dv (x))dx , (79)

The variable s defined in (78) is usually called the “evolution variable”.

ηqi , η′qi were determined by fitting, for a chosen initial M0, the approximation

based on (78) to the exact LO solutions for the first 10 moments.

Λ appearing in (78) can not be directly associated with any renormalizationscheme and is usually denoted ΛLO. A number of experiments used thismethod to determine ΛLO from fits to experimental data.


Numerical and Jacobi polynomials methods

Sophisticated numerical methods developed by Duke, Owens andcollaborators in 1980th. Due to the immense increase of CPU and memoryof modern computers the original limits of this method have disappearedand now most of the groups use these methods. The numerical routines areusually made available by their authors, so that it is relatively easy andstraightforward to use them even for nonexperts and experimentalists.

Another technique is based on expansion of the convolution (72) into(orthogonal) Jacoby polynomials:

Θαβk (x) ≡k∑

j=0

cαβkj xj ,

∫ 10

dxxα(1− x)βΘαβk (x)Θαβl (x) = δkl , (80)

In terms of the Jacobi moments aαβk (Q2),FNS(x ,Q

2) can be expanded as:

FNS(x ,Q2) = xα(1− x)β

∞∑k=0

Θαβk (x)aαβk (Q

2) (81)

former being given as linear combinations of conventional momentsFNS(j ,Q

2):aαβk (Q

2) ≡∫ 1

0

dxFNS(x ,Q2)Θαβk (x) =

k∑j=0

cαβkj FNS(j ,Q2).

(82)Michal Šumbera (NPI ASCR, Prague) Introduction to QCD December 15, 2009 55 / 58

Jacobi polynomials methods, PDFLIB

Substituting (70) and (74) into (75) and this subsequently into (82) and(81), we get an explicit expression for FNS(x ,Q

2) as a function of theunknown constants An and parameter the ΛMS.

Instead of An we can also start with some initial distribution, like (76),specifying qNS(x ,M0) at some M0, and use it to evaluate the constants Anby means of the expression (70). Having determined An we can thenproceed as outlined before.

Over the last decade a large number of phenomenological analyses ofexperimental data have been carried out and PDFs of nucleons, pions andrecently also photons, determined. Several groups of theorists andphenomenologists have been systematically improving these analyses byincorporating new and more precise data as well as by employing moresophisticated theoretical methods. All these distribution functions are nowavailable, as functions of both the momentum fraction x and thefactorization scale M, in the computerized form in the CERN libraryPDFLIB.


F2(x ,Q2) up to HERA

ZEUS NLO QCD fit

H1 PDF 2000 fit

0

1

2

3

4

5

1 10 102

103

104

105

F 2 em-lo

g 10(x)

Q2(GeV2)

x=6.32 10-5x=0.000102x=0.000161

x=0.000253x=0.0004

x=0.0005x=0.000632

x=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

BCDMS

E665

NMC

H1 94-00

H1 (prel.) 99/00

ZEUS 96/97

HERA F2


Exercises

1 Argue on physical terms why P(0)GG (x) must contain the “+” distribution.

2 Show explicitly that the integration over the region of unordered virtualitiesdoesn’t produce the leading logs.

3 Derive (33) and evaluate the integrals in (34), neglecting the dependence ofαs on τ .

4 Show what results if the argument µ of αs(µ) were the same and equal toM2 for all the vertices in the ladder of Fig. 4a.

5 Show that the NS combinations in (66) and (65) do not mix underbranching.

6 Working in the LL approximation and assuming the parametrization (43),express the constants A(n) as functions of the parameters A, α, β, γ, settingfor simplicity γ = 0.

7 Determine the value of the coupling δNS in (71) for the NS structurefunction defined as 12 (F

ep2 − F en2 ).

8 Derive the WW approximation for the scalar charged particle. Show that thecorresponding branching function Pγe(x) ∝ 1−xx .


IntroductionGeneral frameworkPartons within partonsBranching functions in QCDMultigluon emission and Sudakov formfactors of partons

Partons within hadronsEvolution equations at the leading orderMoments of structure functions and sum rulesExtraction of gluon PDF from scaling violationsEvolution equations at the next-to-leading orderHard scattering and the factorization theorem

Brief survey of methods of solving the evolution equationsExercises

Documents

Introduction to Quantum Chromodynamics...Introduction to Quantum Chromodynamics Michal Sumbera Nuclear Physics Institute ASCR, Prague December 15, 2009 Michal Sumbera (NPI ASCR, Prague)