1

Short Introduction to QCD• Renormalisation• Quark Parton Model and its improvement by QCD• Factorisation and the Altarelli-Parisi splitting functions• Evolution equations: DGLAP and BFKL• Determination of the proton parton density• Parton shower• NLO calculation principles…

Deep-inelastic scattering Proton-proton collisions:

2

Principle of Renormalisation - QED

Loop momentum can be anything ultraviolet divergence

QM

log 3

lim~))f(q(

lim ~ q)-k( k

1 kd ~ )I(q

2

2e

2e

M

m2

20

M222

22

M22m

42

π

α

κ

κdκ

In fact, thisintegration is rathercomplicated(see Aitchison&Hey problem 6.13)

momentum loop on off-cut M2

2

20

2

20

2

202

QM

log3

1

......QM

log32

1QM

log3

1 )Q(

π

αα

π

α

π

ααα

0

2

0

Geometrical series

Result depend on unphysical cut-off Mbut physically measured coupling contains all ordersand must be independent of M

k

q-k

q q

bare charge bare charge screenedphysical charge

2e

2

mM2

22

22

log~)I(q

:GeV 0Q For

GeV 1 Q

for valid only

mass electron m2e

2k)k (-i 2

0κ

α 0α

q of function

dcomplicate

loop each

for (-1) factor

:Notel

3

2

2

2

2

2

2

22

2

2

2

2

2

2

2

20

2

Qlog

31-

M

log MQ

log 31-

)(1

)(Q1

Mlog

31

)(

1

QM

log 31

)(Q

1

QM

log3

1 )(Q

πμπμαα

μπα

1

μα

πα

1

α

π

αα

0

0

0

Q

log 3

)1

) )(Q

2

22

πα(μ

α(2

2

Infinities removed at

the prise of renormalisation scale,but results depends onarbitrary parameter

Note: physical observabledo not depend on

charge experimentalistmeasures depends on scale“running coupling constant”

1371

1281

2ZMem

Relation betweenbare and physical chargehas to specified at aparticular value of thephoton momentum

Renormalisation - QED

1

_

! finite and Q

scales between Difference22 μ

)(Q2α

2Q

4

Loops in QED and QCD

QED: QCD:

320β

Low resolution:charge is screenedby ee-pairsHigh resolution :charge is big

factors colour NC ,21

T

colours of number N

fermions of number N

CAF

C

F

3

N 11N 2

C611

N T 32

CF

AFF

0β

QED) in (as

nsfluctuatio qq

by generated ninteractio self gluonic

small very

is charge resolution

infinite at i.e. gluons, by

out-spread is charge

screening-anti

16N for F

00β

Q

log )b 1

) )(Q

2

2

0

2

μ(μα

(μαα

2s

2s

s

10b since term, positive

π

β

4-

b where 00

Q

log ) 2

1

)

2

20

μα(μ

α(μ

2

22

)(Q

5

The QCD Scale Parameter

π 12

N 2 - 33b with

Qlog )b 1

) )(Q

:loop-1 At

F0

2

2

0

2

μα(μ

α(μα

2

2

states bound form gluons and quark : ΛQ

applicable QCD veperturbati small )(Qα :ΛQ

Λlog b

1Q

log blog b

1 )Q(

:Therefore

Λlog b

1 )( or ))( exp(1/b

large becomes coupling effective where point : Choose

ation)renormalis from over-(left parameter arbitrary is

22

2s

22

0

2

0

2

0

2

0

022

2

2

22

2

2

22

Q

μΛ

μα

μαμαμΛ

Λμ

μ

s

ss

mass hadron of order of be to expected

experiment by determined be to has

theory, of parameter free isΛ

“Confinement region”:coupling gets very large

Asymptotic freedom:unique to non-abelian theories

“soft” “hard”

QCD explains confinement of colour andallows calculations of hard hadronic processes via perturbative expansion of coupling !

6

P x

e

e

q

P

2Q

)Qx,(F y)-(1 1 Q x

2

dQ dxd 2

22

42

2

2πασDIS

scattering-eq

of dependence spin contains Y

α

proton in density parton of sum :)Qx,(F 22

0 1 x1/3

0 1 x1/3

Gluons

Naive picture:

f(x)

f(x)

1/R q - Q

deep""2p

22

M q) (p W inelastic"" 2p

22

Q

2

λ

momentum its of fraction with

proton in quark find to yprobabilit )(f with q

ξ

ξ

Quark-Parton Model and Deep-Inelastic Scattering

q

q2q

q

1

q2q2 (x)f x e )-x( )(f x e d (x)F

0

ζδξξ

7

The Proton Structure Function

q

q2q

q

1

q2q2 (x)f x e )-x( )(f x e d (x)F

0

ζδξξ

q

)-(x )(f x e d )y)-(1 (121

Q x 4

dxdQ

dq

2q

24

2

2 ξδξξπασ

2F

But number of parton is stable

20Q

1 ~ r

visible become structure proton

of details more and more

:power resolving Q 2

) (scaling Q on not x, on depend does function structure Proton

:QPM In2

8

QCD improved Parton Model

(scaling) Q on not x, on depend does

QPM in function structure Proton :F

(x)f e )-x( )(f e d x(x)F

2

2

qq

2q

q

1

q2q

2 0

ζδξξ

QPM:

QCD:

zz1

34

2

k1

~dkd

up)-back (see

: gives ncalculatio

2s

2T

2T

1

later) (see

onscontributi virtual with cancel

limit) red-(infra 1z at pole

)z(P

function splitting

Parisi-Altarelli

qq

20

2

qq

2s

Q

2T

2T

qqs

qgq

Qlog (z)P

2 ~

kdk

(z)P 2

~2

20

*

μπ

α

π

ασ

μγ

Integral diverges ! Need to introduce artificial regulator.non-perturbative scale where pQCD breaks down

z)-(1 momentum with gluon a

emitted having quark parent a of z fraction

momentum with quark find to yProbabilit

:)z(P of tioninterpreta Physical qq

p ξ

ξ

xTk

k

ξ

x z

ξ x

Q2

Q2

ysingularit collinear

kz-1

k-p

1 ~M 2

T2

2

9

Factorisation

ξ

2

2

qqqq

1

x

2q

2 Q log

xP

2

x )(f

d e

xF

0

1μξπ

α

ξδξ

ξ

ξ s

1

x

2qqq

s2

22 )x/z,(f )z(P

zdz

2

d

)df(x,μ

π

α

μ

μμ

DGLAP*-equation:

Scaling violation of F2 caused by gluon emission !*DGLAP: Dokshitzer, Gribov, Lipatov, Altarelli, Parisi

ξ

ξ

ξ

d

zdz

x

z

From iteration

ξ

xTk

k

ξ

x z

ξ x

Q2

1

xqqq

s2

22

2

2

2

1

xqqq2

2s22

1

xqqq2

0

2s2

2

1

xqqq2

0

2s2

2

2renq

bareq

)x/z(f )z(P zdz

2

d

)df(x,

log d )Qdf(x,

0

:it on depend not should )Qf(x, therfore

arbitrary, is of choice

(2)-(1) )x/z(f )z(P zdz

Q

log 2

)f(x, )Qf(x,

(2) )x/z(f )z(P zdz

log 2

f(x) )f(x,

:scale) (large scale at density Parton

(1) )x/z(f )z(P zdz

Q

log 2

f(x) )Qf(x,

:Q scale at density Parton

)Q,(f )(f: densities bare into iessingularit Factorise

π

α

μ

μμ

μ

μ

μπ

αμ

μ

μ

π

αμ

μ

μπ

α

ξξ

10

z-1

z134

P2

qgq

2z-1z2

n P 2

2f

qqg z)-(1 zz))-(1 z-(1

3P2

ggg

z

z-11

34

P qq

2 g

DGLAP equation are the basis for parton shower model in MC

1

x

2gg

q

2gq

2s

2

2

1

x

2qg

2qq

2s

2

2

)Qg( )x/(P)Qq( )x/(Pd

2 )Q(

dlogQ

)Qdg(x,

)Qg( )x/(P)Qq( )x/(P d

2 )Q(

dlogQ

)Qdq(x,

ξ,ξξ,ξξ

ξ

π

α

ξ,ξξ,ξξ

ξ

π

α

Summary DGLAP-EquationsDGLAP-equation

Altarelli-Parisi splitting functions:

Given a parton density f at Q0 the DGLAP evolution predicts f at any Q2 !

ξ

xTk

k

ξ

x z

ξ x

11

The Parton Shower Approximation

FSRISR 22 :in n2 factorize

0Q :shower likespace

:Radiation State Inital2i

0Q :shower liketime

:Radiation State Final2i

Hard 22 process calculation has all (external leg) partons on mass shellHowever, partons can be off-shell for short times (uncertainty principle)close to the hard interaction

Incoming partons radiate harder and harder partons

Outgoing partons radiate softer and softer partons

probablity unit with showers parton afterwards add

ME 22 by n2 :represent

neglected be can Q all :QQ all for if 2i

22i

For more complexreaction often notclear which subdiagramShould be treated ashardest double counting

12S

caling violations

Scaling Violations

α

sα

2Q

α

2Q

x

x

20Q

1 ~ r

Gluon indirectly determinedby scaling violationsSensitivity logarithmic

2Q

)Qx,(F y)-(1 1 Q x

2dQ dx

d 22

242

2

2πασDIS

13)C )Qx,((f Q dlog

)Qx,(df

:equations DGLAP via Q with evolution predicts QCD -

parameters free 20 about

g)c,...,s,d,(u, flavour each for

P(x) x)-(1 x f e.g.

GeV, 2-1Q scale starting at edparametris -

)Qx,(f functions density parton

i2

is2

2i

2

20

2i

C fx)x/(C )Q,(f d

x)Q,x(Fi

iiiig,qq,i

1

x2

ξξξ

ξ 22

Determination of the Parton Densities Functions

0.03 xGeV 100E For

S

2Eee

S

Ex

T

TyyT 21

central

Par

ton

dens

ity

ξ

x

2QFactorisation

scale

Most events at LHCIn low-x region

n

nns C as calculable

function tcoefficien

α

PDF

correct is DGLAP

:Assume

x 1 P(x)

:ioncollaborat CTEQ

x x 1 P(x)

:(MRS) Stirling Roberts, Martin,

i

i

εγ

γε

i

i

LHC is gluon-gluon collider !

14

A part of Wilczek’s comments upon the Nobel Prize announcement…

Scaling violations

Scaling

gluo

n de

nsity

LHC is gluon-gluon collider

)Qg(x, x ~dlogQ

)Q(x,dF

:L.O. in

2s2

22 α

F2

F2

15

Calculation of Hadron-Hadron Cross-Section

ij

ij2

2j2

1i21 t/dσd )μ,(xf )μ,(xf tddx dxσ

2T

2T2

22

2ij /pdpt

us

st/dσd :LO in scattering-qq :e.g.

2

9

4s

p 1p

3p

2p4p

p

241

231

211

)p (p u

)p (p t

)p (p s

:variables mMandelstam

Diverges for low PT0

Calculation of exact matrix elements: LO, NLO done, NNLO close-by (many processes)(loops, divergences, cancellations between large positive/negative numbers)

Factorisation Theorem:PDF is universalOnce extracted cancalculate anycross-section withinsame theoreticalscheme

16

Global NLO QCD AnalysisParton densities are from „global“ fits, i.e. from all available data:

Recently (2002): PDF with uncertainties using phenomenological analysisQuantifiable uncertainties on PDF and physical predictionsProblem: complexity of global analysis as results from many experiments from variety of physical processes with diverse characteristics and errors often mutally not compatible and theoretical uncertainty can not be rigoursly quantified

like-point not photon:Q low

νFe in scorrection mass target e.g. terms(1/x)log α:xlow

:iesuncertaint model termsx)(1log α :xlarge

tsmeasuremen among nscorrelatio :QED

ncorrelatio /PDF

errors systematic of )O( :NNLO :order higher

estimation over-under (x)s s(x) :violation isospin

iesuncertaint alExperiment iesuncertaint lTheoretica

2

1-2nns

1-2nns

em3s

3s

s

Tung (2004):„PDF-users mustbe well informedabout nature ofuncertainties !“

17

Determination of Parton Density Function

Experimental data and errors e.g. DIS structure functions

Theoretical frameworke.g. NLO DGLAP fit, MS scheme etc.

Theoretical assumptions and prejudicese.g. omit certain data, correct for non-perturbative effects (nucleon shadowing etc)

gd,...,u, i

Q Q for )Qx,(f

densities parton20

22i

Nets Neural on based approaches New

H1Zeus

Alekhin

(MRS) al. et Martin

(CTEQ) al. et Tung

(EHLQ) al. et Eichten

(GHR) al. et Glück

(DO) Owens & Duke

:sets-PDF

1984

2009

LO

NLO

NNLO

18

Parton-Parton Luminosities at LHC

ijij

ij

1

021

ij2

2j2

1iij

1

021

σ s dysd

dLdy

ssd

dxdxσ

σ s )Q,(xf )Q,(xfdxdxσ

)Q,(xf )Q,(xfs1

dysd

dL2

2j2

1iij

section-cross hard σ

energy cms parton-parton s

ijx1

x2

Example:

s

i

iii gq q gq g

gluon-gluon

i

ii qq q q

0η

2η4η

6η

19

Example: Single Inclusive Cross-section

Main systematic errors ?

At the LHC the statistical uncertainties on the jet cross-section will be small.

Large momentum transfers and small-x !

Theory uncertainty ?

2

)t,sA( SMjetjet

2s/Ms)f(-e )t,sA(

where

Rather general string theory toy-model (hep-ph/0111298)

At very small distances, particles disappearinto curled extra-dimensions

LHC reach in thefirst year

- test of pQCD in an energy regime never probed!- validate our understanding of pQCD at high momentum transfers

from ren+fac scale

from PDF

TeV 20

TeV 40

TeV 100

s

s

s

M

M

M

TeVatron reachends here

20

ξ

nT,n k,x

nf

1-nf

BFKL- Evolution Equation

)k,x(f )k,K(k dk xdx

)k,x(f 21-nT,1-n1-n

21-nT,

2nT,

21-nT,

1

x 1-n

1-n2nT,nn

n

)kf(x, kdk

)Qg(x, x

:)kf(x, ondistributi gluon edUnintegrat

2T

Q

2T

2T2

2T

2

2μ

)k,f(x )k,kK( dk

x1

dlog

)k,df(x 21-nt,1-n

21-nt,

2nt,

2nt,

2nt,n

2Q ln

x1

log

BFKL

space phase k full over nintegratio needs

terms n!Qlog

giving k of ordering strong relax Must

dependence Q full keeping by

orders all to x1

log terms of nresummatio Needs

? x1

log logQ :when happens what But

T

2n2T

2

s

s2

s

Recursive BFKL equation:

21

Typical Evolution in an Event

ix1

lnη

BFKLDGLAP

diffusion pattern along the ladder leads to strong ordering of transverse momenta

(x))-(t-

exp (x)x

~ )kf(x, :form the of

xlog 2

kk

log-exp

xlog 2

xx

~ )kf(x,

-2t

0

2t

2t

0

02t

σ

μ

σ

xλ

xλπ

λ

λ

:BFKL of Solution

xx

log (x) 0σ

2Tklog

0xx

log

2tklog

0

region

veperturbati-non

1.202(3) with

(3) 28 3

0.5 log2 4 3

with

s

s

ς

ςπ

αλ

π

αλ

22

Physical Interpretation of Evolution EquationsAt low-x probability that parton radiates becomes largeStruck parton originates most likely from a cascade initiated by a parton with large longitudinal momentum

• describes change of parton densities with varying spatial resolution of the probe• leads to strong ordering of transverse momenta from photon to proton end

•describes how high momentum parton in the proton is dressed by a cloud of gluons localised in fixed transverse spatial region of the proton•diffusion pattern along the ladder

BFKL

DGLAP

Non-perturbative region

x

BFKL

2Q

DGLAPCCFM

23

1) BFKL evolution equation - resummation of log (1/x) terms -unintegrated gluon distribution

-ordering in rapidity, unordered in virtuality

2) CFFM evolution: resummation of log Q2-log (1/x) terms unintegrated parton distribution angular ordering

3) Skewed parton distribution

• Parton collinear in proton

• DGLAP evolution, i.e. resummation of log Q2-terms

ordering in parton virtualities

)Q,(xf )Q,(xf )Q,(σ dxdx)Q(σ 22j

21i

2ij

ij

1

021

2 xs 2

Standard picture

Alternative picture

)Q,x,(xf 221i

Pythia, HerwigSherpa, AlpgenNLOJET++,MCFM

)kf(ε( (x/ εxσ dk εdε

σ

:)kf(x, ondistributi gluon edUnintegrat

2T

Q

μ

2T

2T

2

2

Cascade

Colour dipole showers (?)

Mainly for soft and diffractive processes )Q,x,(xf 221i

xx 21

x1 x2

0 x2 xx 21

)Q,x,(xf 221i

Quark is reabsorbedSecond quark is emitted

proton

24

Back-up

25

Renormalisation Group Equation (RGE)

μ dependence scale ationrenormalis

the on depend not does R observable physical A

0)(,/QR ddR 2

222

22

22

2 μμαμ

αμ

μμ

μ ss

s αμ

22

22

μ

αμαβ

μμμμ

ss

)( and

t-

log ),/log(Q t with

222

0)(,/QR )( t

2

ss

22 μαμα

αβ s

RGE

RGE ensures that entire Q2 dependence of Rcomes from running of the strong coupling constant(not shown here)

26

Example to second order

)O( )( )CQ

log b C()( C 1 ))(,/R(Q

(2) )(Q

log b)(Q

log b )(1

)()Q(

(1) )Q( C)Q( C 1 ))Q(R(1,

22

2

012

s1

(1) in (2)2

s22

22

02

s2

02

s

2s2

s

22

2s1

2s

3s

2s2

2s2

2

2s

αμαμ

αμαμ

μαμ

μα

μμα

μαα

ααα

)( by dcompensate is change their but

, on depends )( of tcoefficien2s

2s

2

2

μα

μμα

)O()( b C ))( (-b C

)O()( b C dlog

)(d C

dlogdR

3s

2s01

2s01

3s

2s012

s12

αμαμα

αμαμ

μα

μ22

22

)O( dlog

dR 3

s2 αμ

02s

2

02s

2s

blog

)()(

1- 0

Qlog b

)(1

)Q(

1 since

2

2

2

2α

s

! oncontributi order higher of estimate variation scale ationrenormalis

)O( at changed be will )O( to calculated observable an truncated is series When 1ns

ns

Consider a quantity R:

27

Reminder: Next-To-Leading-Order calculations

Born: First-Order: RealFirst Order: Virtual

qqee

:)O( gqqge ee :..

:) O( s

qqees

:)O(

:0x3cancel each other (KNL-theorem), ifinfra-red singularities

One can show that for any observable where the NLO prediction is:

(x) dxd

BB

(x)V 2

dxd

B

sV x

xRs

R

)(dxd

Loop diagram

BR(x) lim 0x

Real and virtual contributions can be regularised by introducing integral in d=4-2dim.

2

)( )log(

1

1

10

1

1

0

21

0

xxdx

xdx

In this case: LO

sVR )( lim

0

BR(x) lim 0x

1

0

2

dxd

dxd

dxd

O(x))-(O limdOd

RVB

xdx

0

where:

(infra-red safeness)

28

Subtraction Method

1

02

1

02

010

11

)))(())(()())((

dOd

xOOBxOOxR

dxx

dxOOB ssR

))(( ))(()( 0

s 0OOxB

xOOxxR sx

Add and subtract locally a counter-term with same point-wise singular behaviour as R(x):

BR(x) lim 0x

Ellis, Ross, Terrano (1981)

Since

1

021

1

0

2ε )( )(

)(dOd

xxR

dxxx

xRdxxxR

dx sssR

1

0

regularised

Let us look at the real contribution:

By construction this integral is finite

Add and subtract counter-term

The only divergent term has B&V kinematicsand gets cancels against s B/2term of virtual contribution cancellation independent of Observable

1

0

00

xOOBxOOxR

dxOOB

ss

R

)))(())(()())((

2-dOd

22-2))((

)())((dxlim

dOd 1

0 xBB

VB

BOOxxR

xOO ss

0

0

29

Infinite momentum frame:proton is moving with infinite momentum(all masses can be neglected)

flat, flat, frozen, frozen, unexpectedunexpected

time dilation: partons frozen (no interaction) they can be treated as ‘free’ during the short time they interact with photon

Photon-Proton interaction can be expressedas sum of incoherent scattering from point-like quark !

Quark Parton ModelInteraction of hadrons due to interaction of partonsStructure of hadron describable by distribution of partons at any timechanges in number and momenta of partons should be small during time they are probed

1x 0

1 x

i

ii

γ/R 2 p

P

P xi

pR 2

31

Quarks interact-> redistribution of momenta

! x small at bigxdx

~ dxdE

:lungBremsstrah

QPM

QCD

32

33

)(xt )u(k )v(k kdxdx

)u(k )x(t )u(k

)x(t )u(k )k,(xf and

)x(t )u(k )k,(xf

(1) )k,(xf )v(k kdxdx

)u(k )k,(xf

)k,(xf )k,K(k kdxdx

)k,(xf

:equation recursion BFKL

1-n1-n2

1-nT,2

1-nT,2

1-nT,x 1-n

1-n2nT,

)(

nn2

nT,

1-n1-n2

1-nT,2

1-nT,1-n1-n

nn2

nT,2

nT,nn

21-nT,1-n1-n

21-nT,

21-nT,

x 1-n

1-n2nT,

2nT,nn

21-nT,1-n1-n

21-nT,

2nT,

21-nT,

x 1-n

1-n2nT,nn

n

n

n

11

1

1

BFKL Toy Modelfactorises Kernel )v(k )u(k )k,K(k :Assume 2

1-nT,2

nT,2

1-nT,2

nT,

)x(t nn)x(t )u(k 1-n1-n1-nT,

)x(tx

dx )x(t 1-n1-n

1-n

1-nnn

1

λnx

0

21-nT,

21-nT,

21-nT, )u(k )v(k dk

with

λ

x1

log2

xlog2

x log

x

dx - t

x1

log x

dx t 1, t :e.g.

2 2

21-n

x 1-n

1-n2

x 1-n

1-n10

λ

21

2

1

λλ

λλ

x1

logn!

)x(t :general In nn

n

λ

x1

logexp )u(k

x1

logn!

)u(k

)x(t )u(k )k,f(x

2nT,

nn

2nT,

nn2

nT,2

nT,n

λ

λ

0

0

n

n

λ-2nT,

2nT,n x )u(k )k,f(x

0.5 log2 4 3

:gives ncalculatio real""

s π

αλ

! expected gluon

of rise Steep

n

nf f

34

Relation PDF and cross-section/F2

Structure Function/cross-sectionParton densityfunction

)Qx,(F 22 )g(x, ),q(x, 22 μμ

Physical observable Theoretical construct

Model independent Model dependent

well definedDefinition depends on:1) order of alpha_s2) factorisation scheme3) factorisation scale

35

LHC gives access:• to high momentum transfers at relatively low-x• to high-x

DGLAP evolution

DIS:o) theoretically well definedo) experimentally clean

HERA:o) measure strong coupling and parton densitieso) verifiy/falsify DGLAP evolutiono) develop techniques to constrain theory uncertainties from LHC data

P x

ee

q

P

GeV 27.5

GeV 820/920

2Q

36

W-Boson Production at LHC

…after effort of 10 years a differential NNLO calculation is available !

Scale dependence at y=0:LO: 30% NLO: 6% NNLO: 0.6%No change in shape from NLO->NNLO

W and Z production~105 events containing W (pT

W > 400 GeV) ~104 events containing Z (pT

Z > 400 GeV)

“Standard candles” at LHC:- Luminosity - detector calibration -- constrain quark and anti-quark densities in the proton.

Precision measurements MW etc.

Huge statistical samples & clean experimental

channel.

rapidity

Precision from theorychallenge for experiment !

37

PDF Impact on W-Boson Cross-section at LHC

PDF uncertainties:2 NNLO sets by MRSTMode=4 gives better description ofTevatron High Et jet data

At NLO: PDF uncertainties are absorbedin scale dependenceAt NNLO: PDF uncertainties are larger !1-2% difference visible/measurable at LHC ?

NNLO NLO

rapidity