Introduction to QCD and perturbative QCD ELFT Summer School, May 24, 2005

Three lectures:1. Symmetries

– exact– approximate

2. Asymptotic freedom– renormalisation group– β function

3. Basics of pQCD– fixed order– Resummation

Should be understood by everybody, so will be trivial

When you measure what you are speaking about and express it in numbers, you know something about it, but when you cannot express it in numbers your knowledge about is of a meagre and unsatisfactory kind.

Lord Kelvin

Field content

• Quark fields: qf f=1,…,6 B=1/3

f 1 2 3 4 5 6

qfu d s c b t

mf~5Me

V~7Me

V~100MeV 1.2GeV 4.2GeV 174GeV

mf running masses (see later) at =2GeV, approximate

values; Nf is the number of light quarks (e.g. 3)

•Gluon fields: Aɑ ɑ=1,…,8

The QCD Lagrangian

• QCD is a QFT, part of the SM• The SM is a gauge theory with underlying

SUc(3) SUL(2) UY(1)

sources0QCDQCDQCD LLL

GhostGFCQCD LLLL 0

The Classical Lagrangian

fNf

fff

N

ffC qmDqqDq

f

f

6

11

ii4

1

aa FFL

aaTAgD si

g, 2 05 ,

aT

cabcba Tf,TT i

are the SU(Nc) generators with algebra:

abba RTTT Tr 2/1RT

fundamental representation:

adjoint representation:

aaa tT 2

1

abc

abcbc

a FT i

The Classical Lagrangian

cba

bcaaaF AAFgAA si

c

cRFF N

NTCCtt

12 bcbc

aa

Colour factors (like eigenvalues of J2):

•in the adjoint representation:

source of non-Abelian nature: gluon self coupling

cRAA NTCCFF 2 bcbcaa

•in the fundamental representation:

Note: in lattice QCDaa AAg s

aa FF s

1

g

Exact symmetries of the classical Lagrangian

• Quantum effects may violate (e.g. scale invariance, axial anomaly)

• Continuous: local gauge invariance(suppress flavour and Dirac indices)

,)()()()(: xqxUxqxqU jijii cNSUTxxU

8

1a

aa )(iexp)(

• The covariant derivative transforms as the field itself

aTAxxgx a A,)q()U(Ai)(qD s

jijii

xqDxUxqDxqDU )()()()(:

)q()U(Ai)q()U()q()U( s xxgxxxx

)q(A)U(i)q()U( s xxgxx 11

s

UAUUUi

A g1UUFF

Exact symmetries of the classical Lagrangian

• Almost supersymmetric:– massless QCD for one flavour

– SUSY Yang-Mills :

qDqQCDC

i

4

1 aa FFL

DSYM

C i4

1 aa FFL

• q transforms under the fundamental, under the adjoint representation of SU(Nc)

• Advantageous in deriving matrix elements

Exact symmetries of the classical Lagrangian

• The quark mass term is, the gluon mass term is not gauge invariant

• Discrete: C, P and T in agreement with observed properties of strong interactions (C, P and T violating strong decays are not observed)

• Note: additional gauge invariant dimension-four operator, the -term:

F2

1F~

,F~

F32 2

s

g

L

Conventional normalisation

•violates P and T (corresponds to EB in electrodyn.) is small (<10-9 experimentally), set =0 in pQCD

Approximate symmetries of the classical Lagrangian

Related to the quark mass matrix

Introduce:

3

2

1

q

q

q

s

d

u

512

1 P

1,, 2 PPPPPPPP

…and from Dirac algebra: PP

Let: P

525 1Eigenvector of 5:

P

0

Approximate symmetries of the classical Lagrangian

The quark sector of the Lagrangian can be written:

PPDPPDChir iiL

PDPPDP ii

RL LLLL

• would not work if gluons were not vectors (in D)

• the left- and right-handed fields are not coupled LChir is invariant under UL(Nf) UR(Nf)

• the group elements can be parametrised in terms of 2 Nf

2 real numbers: aa

aa

aa

aa TTTT

RL gg -ii-iiiiii eeee,eeee,

fRfLAV NSUNSUUU 11

DD ii

Approximate symmetries of the classical Lagrangian

• This symmetry acts separately on left- and right-handed fields: chiral symmetry

• Has vector subgroups SUV(Nf) UV(1)

• The axial transformations do not form a subgroup

cbaaa

aa

aa Tf,TT,hh abcTTT ieee, 55

i-ii 5

fTTT Ngg Viii SUee,e, a

aa

aa

a

• Chiral symmetry is not observed in the QCD spectrum, it is spontaneously broken to SUV(Nf) UV(1)

Chiral perturbation theory

• In QCD it is believed that the vacuum has a non-zero VEV of the light-quark operator

• This quark condensate breaks chiral symmetry because it connects left- and right-handed fields

3MeV2500||00||0 dduuqq

• The SSB of chiral symmetry implies the existence of Nf

2-1massless Goldstone bosons

• The light quarks are not exactly massless the chiral symmetry is not exact, the Goldstone bosons are not massless: pseudoscalar meson octet

• The mf are treated as perturbation PT

0||00||0 LRRL qqqqqq

Topics of QCD (T=0)

• Low-energy properties (<GeV)

• High energy collisions (>GeV)

• Perturbative

• Non-perturbative

PT (light quark masses)

• Jet physics

• Sum rules, lattice QCD

Approximate symmetries of the classical Lagrangian

Choose Weyl representation:

10

01

0

0

01

1050

i

ii

~0

0

~

pxpx ~iexp~

two-componentWeyl spinors

~

Define

0~0~i ppx

helicityeigenstatesif m=0, g=0

0~~ pppE ii

pp ~~p̂σ

Asymptotic freedom

• At the heart of QCD Nobel prize 2004• Consider a dimensionless physical observable

R=R(Q), with Q being a large energy scale, Q any other dimensionful parameter (e.g. mf)

set mf =0 (check later if R (mf =0) is OK)

• Classically dimR = 1 0

2lim0d

dRR

Q

RQ

• In a renormalized QFT we need an additional scale: renormalization scale R = R (Q2/2) is not a constant: scaling violation

the „small” parameter in the perturbative expansion of R, s() also depends on the scale choice

Asymptotic freedom

• But is an arbitrary, non-physical parameter (LCl does not depend on it) physical quantites cannot depend on

0d

d

s2s2

222

s2

2

22

R,

QR

• Let t = ln (Q 2/2), (s) =

0e ss

s

,Rt

t

fixed2s2

0s

s

Asymptotic freedom

• To solve this renormalization-group equtaion, we introduce the running coupling s(Q 2):

2s

2s

dQ

x

xt

• If 2 =Q 2 et = 1, the scale-dependence in R enters through s(Q 2)

• All this was non-perturbative yet

t

Q

Q

2

s2

s

11

The function in perturbation theory

• We solve

• known coefficients:

fRA NTC3

4

3

110

2

2s22

s Q

QQQ

1

0

sss 4

n

nn

in PT (we analyse the validity of PT a little later)

fRAfRFA NTCNTCC3

204

3

34 21

22 54

325

18

5033

2

2857ff NN

323 5.19.4053.694629243 fff NNN

The function in perturbation theory

• if s(Q 2) can be treated as small parameter, we can truncate the series, keep the first two terms:

• LO:

4

00 b

22s

2s1

22s02

2s2 O1 QQbQbQ

QQ

with0

11 4

b

0s

2s

1b

t

tbQ 02

s2

s

11

tb

Q0

2s

1~if t:

• Relation between s(Q 2) and s( 2) if both small

• Q 2 0 s(Q 2) 0: asymptotic freedom (sign!)

1

2s0

2s

2s 1

1

tbQ

The function in perturbation theory

• The running coupling resums logs: if R = R1 s+O(s

2)

• R2 s2 gives one less log in each term

• NLO (b10):

1

02

s2

s12

s1j

jtbRQ,R

0s

s12s 1

1b

tb

tb

b

Qbb

Qb

Q 02s1

2s1

12s

2s

12s

2s 1

1lnln

11

QCD

• A more traditional approach to solving the renormalization-group equation: introduce

indicates the scale at which s(Q 2) gets strong

• LO (b00, bi=0): 2

2

0

2s

ln

1

Qb

Q

2

s

dln

2

2

Qx

xQ

2

2

02s1

2s1

12s

ln1

ln1

Qb

Qb

Qbb

Q

• NLO (b10):

• The two solutions differ by subleading terms that are important in present day precision measurements

The running coupling

The quark masses

• Assume one flavour with renormalized mass m: yet another mass scale

0s22

ss

s22

Qm,,QRm

mm

m is the mass anomalous dimension, in PT:

2s1

22s0 1 QcQcm

72

10303110

fNcc

• R is dimensionless

0s22

22

22

22

Qm,,QRm

mQ

Q

02

1s

22s

ss2

2

Qm,,QRm

mQ

Q m

The running quark mass

• To solve this renormalization-group equtaion, we introduce the running quark mass m(Q 2):

• the derivative terms (if finite) are suppressed by at least an inverse power of Q at high Q 2

dropping the quark masses is justified only IR-safe observables can be computed

QQm,Q,RQm,,QR 22ss

22 1

N

n

n

n

,Q,RQ

Qm

n,Q,R

1

2s

22

s 01!

101

The running quark mass

• All non-trivial scale dependence of R can be included in the running of mass and coupling:

2

s2

222

2

2

dexp Q

Q

QmQm m

Q

• c/b > 0 the running mass vanishes with the running coupling at high Q 2

2s2

2 QmQ

mQ m

Solution:

1dexp 2

ss

ss

2

2s

2s

bcQ

m Qmm

2 hard photons in CMS

4 muons in CMS

4 muons in CMS

Basics of Perturbative QCD

• Vast subject – only give the flavour• Will use a specific example:

ee

qqee

ee

eeR qhadrons

ffee

2222

22222222

2

cos12 cos d

d

ZZZ

ffeefMMs

sVAVAQ

s

• 2→2 scattering has one free kinematical parameter, the θ scattering angle

• The differential cross section for

The total cross section • below the Z pole • on the Z pole

22

0 3

4fQ

s

22222

22

0 3

4ffee

Z

VAVAs

The total hadronic cross section

• LO: the hadronic cross section is obtained by counting the possible final states:

023 RQR

qq

• With q = u,d,s,c,b R =11/3=3.67 and RZ =20.09 • The measured value at LEP is RZ =20.79±0.04• The 3.5% difference is mainly due to QCD

effects:– Real

– virtual

gluon emission

22

22

3 VA

VA

R qqq

Z

NLO: real gluon emission

• Three-body phase space has 5 independent variables: 2 energies and 3 angles

• Integrate over the angles and use yij = 2pi·pj/s scaled two-particle subenergies, y12+y13+y23=1

• The real contribution to the total cross section:

23132313

12

23

13

13

23s1

0

23

1

0

1300R 1

2

2dd yy

yy

y

y

y

y

yCyyR F

• Divergent along the boundaries at yi3 = 0:

• Unphyisical singularities - quarks and gluons are never on (zero) mass shell: Breakdown of PT 333 cos12 iii EEsy

• Divergent when E3→0 (soft gluon), or θi3 →0 (collinear gluon)

NLO: the real and virtual contribution in d 4

• To make sense of the real contribution, we use dimensional regularization:

2

21

2

dd

2313

12

23

13

13

23s1

0 23

231

0 13

1300

R

yy

y

y

y

y

yC

y

y

y

yHR F

• Has to be combined with the virtual contribution

O1O2

1932 2200

HHR

O8

32 2200

V HR

• The sum of the real and virtual contributions is finite in d = 4:(same for RZ)

2

ss

0 O1

RR

The total hadronic cross section at O(αs3)

• The total cross section can be computed more easily using the optical theorem fIm

4s

32s2

2

2

220111023

22s2

2

0122

s10

Olnln2

ln1

QQbcbcbcc

QbcccRR

33221

85.12409.11

ccc

• Satisfies the renormalization-group equation to order αs

4

The total hadronic cross section at O(αs3)

The total hadronic cross section at O(αs4)

(non-singlet contribution)

Jet cross sections

Typical final states in high energy electron-positron collisions

2 jets 3 jets

Modelling of events with jets

Production probability pattern:2jets : 3jets : 4jets ~ O(αs

0) : O(α s1) : O(α

s2) ⇒ jets reflect the partonic structure

Jet cross sections

• We average over event orientation ⇒ |M2|2 has no dependence on the parton momenta

21212

1

0

12

2

2LO 1d ,ppJyy M

32132313

12

23

13

13

23s1

0

23

1

0

13

2

2R 2

2dd ,p,ppJ

yy

y

y

y

y

yCyy F

M

21212

1

0

12

22

2s2

2V

1d

O8324

1

1

2

,ppJyy

sCF

M

• Cannot combine the integrands (like for σtot)

• NLO corrections:

The subtraction scheme

• Process and observable independent solutionNLO2

NLO3

NLO

2122

231313

12

13

23

2313

12

23

13

13

23

yyy

y

y

y

yy

y

y

y

y

y

2AVNLO

22A

3RNLO

3 dddddd JJJ

• Made possible by the process-independent factorisation properties of QCD matrix elements

211221

13122313

231312

13

yyyy

yyy

y

211

111

21

13

231

13113

y

yz

yzy

The subtraction scheme

• The approximate cross section:

O3

10324

1

1

2

2

2

2s2

2 sCFM

21

11dd4

1

1d 2,131113

2113

1

0

1

1

0

13

2A

D-ε-εε- zzyyzys

11

13113

s2

23212,13 1111

21

2zz

yzyC,p,pp F

MD

= I(ε) |M2|2

• with universal factorisation:

The subtraction scheme

• The integrated approximate cross section:

21212

1

0

12

22

2s2

2V

1d

O8324

1

1

2

,ppJyy

sCF

M

21~~1d 21322,132,13A p,pJyy

21212

1

0

12

2s2

2NLO2 1dO

31 ,ppJyyCF

M

Parton showers and resummation

• The universal factorisation

can be used to describe parton showers (neglecting colour correlation of soft emissions and azimuthal correlations of gluon splitting - not transparent in the simple example considered here)

ii

ijiijFmkjikij zz

yzyC,p,pp 11

11

21

2s2

,

MD

General picture of high-energy collisions

R at low energies