Basics of QCD Lecture 1: the core ingredients - ICTP … · Basics of QCD Lecture 1: the core...

Basics of QCDLecture 1: the core ingredients

Gavin Salam

CERN Theory Unit

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

QCD[What is QCD]

QUANTUM CHROMODYNAMICS

The theory of quarks, gluons and their interactions

It’s central to allmodern colliders.

(And QCD is whatwe’re made of)

Gavin Salam (CERN) QCD Basics 1 2 / 24

QCD predictions v. data for many processes[What is QCD]

∫L dt

[fb−1] Reference

ts−chantotal

95% CL upper limit 0.7 ATLAS-CONF-2011-11895% CL upper limit 20.3 arXiv:1410.0647 [hep-ex]

W±W±jj EWKfiducial

20.3 PRL 113, 141803 (2014)

Wγγfiducial, njet=0 20.3 arXiv:1503.03243 [hep-ex]

H→γγfiducial

20.3 Preliminary

Zjj EWKfiducial

20.3 JHEP 04, 031 (2014)

ttγfiducial

4.6 arXiv:1502.00586 [hep-ex]

ttZtotal

95% CL upper limit 4.7 ATLAS-CONF-2012-126

20.3 ATLAS-CONF-2014-038

ttWtotal

20.3 ATLAS-CONF-2014-038

Zγfiducial

4.6 PRD 87, 112003 (2013)arXiv:1407.1618 [hep-ph]

WW+WZfiducial

4.6 JHEP 01, 049 (2015)

Wγfiducial

4.6 PRD 87, 112003 (2013)arXiv:1407.1618 [hep-ph]

ZZtotal

4.6 JHEP 03, 128 (2013)

20.3 ATLAS-CONF-2013-020

WZtotal

4.6 EPJC 72, 2173 (2012)

13.0 ATLAS-CONF-2013-021

Wttotal

2.0 PLB 716, 142-159 (2012)

20.3 ATLAS-CONF-2013-100

γγfiducial

4.9 JHEP 01, 086 (2013)

WWtotal

4.6 PRD 87, 112001 (2013)

20.3 ATLAS-CONF-2014-033

tt−chantotal

4.6 PRD 90, 112006 (2014)

20.3 ATLAS-CONF-2014-007

ttfiducial

4.6 Eur. Phys. J. C 74: 3109 (2014)

20.3 Eur. Phys. J. C 74: 3109 (2014)

Ztotal

0.035 PRD 85, 072004 (2012)

Wtotal

0.035 PRD 85, 072004 (2012)

Dijets R=0.4|y |<3.0, y∗<3.0 4.5 JHEP 05, 059 (2014)0.3 < mjj < 5 TeV

Jets R=0.4|y |<3.0 4.5 arXiv:1410.8857 [hep-ex]0.1 < pT < 2 TeV

pptotal 8×10−8 Nucl. Phys. B, 486-548 (2014)

σ [pb]10−3 10−2 10−1 1 101 102 103 104 105 106 1011

observed/theory0.5 1 1.5 2

LHC pp√s = 7 TeV

TheoryObservedstatstat+syst

LHC pp√s = 8 TeV

Theory

Observedstatstat+syst

Standard Model Production Cross Section Measurements Status: March 2015

ATLAS Preliminary

Run 1√s = 7, 8 TeV


QCD for Higgs physics[What is QCD]

QCD is especially relevant inorder to deduce information

about the Higgs boson.

[cf. lectures by Andre Sznajder& Claude Duhr]


QCD for Higgs physics[What is QCD]

QCD is especially relevant inorder to deduce information

about the Higgs boson.

[cf. lectures by Andre Sznajder& Claude Duhr]

) µSignal strength (

2− 0 2 4

ATLASIndividual analysis

-1 = 7 TeV, 4.5-4.7 fbs

-1 = 8 TeV, 20.3 fbs

0.27-

0.27+ = 1.17µOverall:

0.38-

0.38+ = 1.32µggF:

0.7-

0.7+ = 0.8µVBF:

1.6-

1.6+ = 1.0µWH:

0.1-

3.7+ = 0.1µZH:

γγ →H 125.4

125.4

125.4

125.4

125.4

0.33-

0.40+ = 1.44µOverall:

0.4-

0.5+ = 1.7µggF+ttH:

0.9-

1.6+ = 0.3µVBF+VH:

ZZ*→H 125.36

125.36

125.36

0.21-

0.24+ = 1.16µOverall:

0.26-

0.29+ = 0.98µggF:

0.47-

0.55+ = 1.28µVBF:

1.3-

1.6+ = 3.0µVH:

WW*→H 125.36

125.36

125.36

125.36

0.37-

0.43+ = 1.43µOverall:

1.2-

1.5+ = 2.0µggF:

0.54-

0.59+ = 1.24µVBF+VH:

ττ →H 125.36

125.36

125.36

0.40-

0.40+ = 0.52µOverall:

0.61-

0.65+ = 1.11µWH:

0.49-

0.52+ = 0.05µZH:

b Vb→VH 125.36

125

125

3.7-

3.7+ = -0.7µOverall: µµ →H 125.5

4.3-

4.5+ = 2.7µOverall: γ Z→H 125.5

1.1-

1.1+ = 1.5µ: bb

1.2-

1.4+ = 2.1µMultilepton:

1.75-

2.62+ = 1.3µ: γγ

ttH125

125

125.4

(GeV)Hm

Input measurements

µ on σ 1±


QCD and searches for new physics[What is QCD]

Identification and/or exclusion ofpotential new physics relies indiverse ways on a QCD-basedunderstanding of signals and

backgrounds.

Search for stop squarks

[GeV]missT

pE

vent

s / 5

0 G

eV0

20

40

60

80

100

120

140Data

+Xhadrτ →/Wtt+Xµ e,→/Wtt

νν→Z (500, 100)0χ∼ t→t~

(650, 50)0χ∼ t→t~

CMS

(8 TeV)-119.4 fb

[GeV]missT

p200 250 300 350 400 450 500D

ata

Dat

a -

MC

-2-1012


QCD and searches for new physics[What is QCD]

Mass scales [GeV]0 200 400 600 800 1000 1200 1400 1600 1800

233'λ µ tbt→

Rt~

233λt ντµ →

Rt~ 123

λt ντµ → R

t~

122λt νeµ →

Rt~

112''λ qqqq →

Rq~ 233

'λ µ qbt→ q~

231'λ µ qbt→ q

~ 233λ ν qll→ q

~123

λ ν qll→ q~

122λ ν qll→ q

~ 112''λ qqqq → g

~323

''λ tbs → g~ 112

''λ qqq → g~

113/223''λ qqb → g

~ 233'λ µ qbt→ g

~231'λ µ qbt→ g

~233

λ ν qll→ g~ 123

λ ν qll→ g~

122λ ν qll→ g

~

0χ∼ l → l~

0

χ∼ 0

χ∼ν τττ → ±χ∼ 2

0χ∼

0

χ∼ 0

χ∼ν τ ll→ ±χ∼ 2

0χ∼

0χ∼

0χ∼ H W →

2

0χ∼ ±χ∼

0χ∼

0χ∼ H Z →

2

0χ∼

2

0χ∼

0χ∼

0χ∼ W Z →

2

0χ∼ ±χ∼

0χ∼

0χ∼ Z Z →

2

0χ∼

2

0χ∼

0χ∼

0χ∼νν-l

+ l→

-χ∼

+χ∼

0

χ∼ 0

χ∼ν lll → ±χ∼ 2

0χ∼

0χ∼ bZ → b

~

0χ∼ tW → b

~

0χ∼ b → b

~

) H 1

0χ∼ t →

1t~

(→ 2

t~

) Z 1

0χ∼ t →

1t~

(→ 2

t~

H G)→ 0

χ∼(0

χ∼ t b → t~

)0

χ∼ W→ +

χ∼ b(→ t~

0χ∼ t → t

~

0χ∼ q → q

~

))0

χ∼ W→ ±

χ∼ t(→ b~

b(→ g~

)0

χ∼ W→±

χ∼ qq(→ g~

)0

χ∼ t→ t~

t(→ g~

0χ∼ tt → g

~

0χ∼ bb → g

~

0χ∼ qq → g

~

SUS-13-006 L=19.5 /fb

SUS-13-008 SUS-13-013 L=19.5 /fb

SUS-13-011 L=19.5 /fbx = 0.25 x = 0.50

x = 0.75

SUS-14-002 L=19.5 /fb

SUS-13-006 L=19.5 /fbx = 0.05

x = 0.50x = 0.95

SUS-13-006 L=19.5 /fb

SUS-12-027 L=9.2 /fb

SUS-13-007 SUS-13-013 L=19.4 19.5 /fb

SUS-12-027 L=9.2 /fb

SUS 13-019 L=19.5 /fb

SUS-14-002 L=19.5 /fb

SUS-12-027 L=9.2 /fbSUS-13-003 L=19.5 9.2 /fb

SUS-13-006 L=19.5 /fb

SUS-12-027 L=9.2 /fb

EXO-12-049 L=19.5 /fb

SUS-14-011 L=19.5 /fb

SUS-12-027 L=9.2 /fb

SUS-13-008 L=19.5 /fb

SUS-12-027 L=9.2 /fb

EXO-12-049 L=19.5 /fb

SUS-12-027 L=9.2 /fb

SUS-12-027 L=9.2 /fb

SUS-13-024 SUS-13-004 L=19.5 /fb

SUS-13-003 L=19.5 /fb

SUS-12-027 L=9.2 /fb

SUS-13-019 L=19.5 /fb

SUS-13-018 L=19.4 /fb

SUS-13-014 L=19.5 /fb

SUS-14-011 SUS-13-019 L=19.3 19.5 /fb

SUS-13-008 SUS-13-013 L=19.5 /fb

SUS-13-024 SUS-13-004 L=19.5 /fb

SUS-13-013 L=19.5 /fb x = 0.20x = 0.50

SUS-12-027 L=9.2 /fb

SUS-13-003 L=19.5 9.2 /fb

SUS-12-027 L=9.2 /fb

SUS-13-008 SUS-13-013 L=19.5 /fb

SUS-12-027 L=9.2 /fb

SUS-14-002 L=19.5 /fb

SUS-12-027 L=9.2 /fb

SUS-13-013 L=19.5 /fb

SUS-13-006 L=19.5 /fb x = 0.05x = 0.50x = 0.95

SUS-13-006 L=19.5 /fb

RP

Vgl

uino

pro

duct

ion

squa

rkst

opsb

otto

mE

WK

gau

gino

ssl

epto

n

Summary of CMS SUSY Results* in SMS framework

CMS Preliminary

m(mother)-m(LSP)=200 GeV m(LSP)=0 GeV

ICHEP 2014

lspm⋅+(1-x)

motherm⋅ = xintermediatem

For decays with intermediate mass,

Only a selection of available mass limits*Observed limits, theory uncertainties not included

Probe *up to* the quoted mass limit


Aims of the course[What is QCD]

The school will cover many aspects of QCD, from theadvanced calculation techniques to the experimental

consequences.

This course:

A reminder of what QCD is

A few core equations & concepts(running coupling, collider cross sectionsinfrared divergences and infrared safety)

More depth on topics not covered in other lectures(parton distribution functions, jets)


The ingredients of QCD[What is QCD]

I Quarks (and anti-quarks): they come in 3 colours

I Gluons: a bit like photons in QEDBut there are 8 of them, and they’re colour charged

I And a coupling, αs, that’s not so small and runs fastAt LHC, in the range 0.08(@ 5 TeV) to O (1)(@ 0.5 GeV)


Lagrangian + colour[What is QCD]

Quarks — 3 colours: ψa =

ψ1

ψ2

ψ3

Quark part of Lagrangian:

Lq = ψa(iγµ∂µδab − gsγµtCabAC

µ −m)ψb

SU(3) local gauge symmetry ↔ 8 (= 32 − 1) generators t1ab . . . t

8ab

corresponding to 8 gluons A1µ . . .A8

µ.

A representation is: tA = 12λ

A,

λ1 =

0 1 01 0 00 0 0

, λ2 =

0 −i 0i 0 00 0 0

, λ3 =

1 0 00 −1 00 0 0

, λ4 =

0 0 10 0 01 0 0

,

λ5 =

0 0 −i0 0 0i 0 0

, λ6 =

0 0 00 0 10 1 0

, λ7 =

0 0 00 0 −i0 i 0

, λ8 =

1√3

0 0

0 1√3

0

0 0 −2√3

,


Let’s write down QCD in full detail

(There’s a lot to absorb here — but it should become morepalatable as we return to individual elements later)



ψ1

ψ2

ψ3



µ −m)ψb


8ab


µ.


A,

λ1 =

0 1 01 0 00 0 0

, λ2 =

0 −i 0i 0 00 0 0

, λ3 =

1 0 00 −1 00 0 0

, λ4 =

0 0 10 0 01 0 0

,

λ5 =

0 0 −i0 0 0i 0 0

, λ6 =

0 0 00 0 10 1 0

, λ7 =

0 0 00 0 −i0 i 0

, λ8 =

1√3

0 0

0 1√3

0

0 0 −2√3

,




ψ1

ψ2

ψ3



µ −m)ψb


8ab


µ.


A,

λ1 =

0 1 01 0 00 0 0

, λ2 =

0 −i 0i 0 00 0 0

, λ3 =

1 0 00 −1 00 0 0

, λ4 =

0 0 10 0 01 0 0

,

λ5 =

0 0 −i0 0 0i 0 0

, λ6 =

0 0 00 0 10 1 0

, λ7 =

0 0 00 0 −i0 i 0

, λ8 =

1√3

0 0

0 1√3

0

0 0 −2√3

,


Lagrangian: gluonic part[What is QCD]

Field tensor: FAµν = ∂µAA

ν − ∂νAAν − gs fABCAB

µACν [tA, tB ] = ifABC t

C

fABC are structure constants of SU(3) (antisymmetric in all indices —SU(2) equivalent was εABC ). Needed for gauge invariance of gluon part ofLagrangian:

LG = −1

4FµνA FAµν

Exercise

Consider gauge transformations

ψ → U(x)ψ , U(x) = e iuA(x)tA

How should the gluon field ACµ transform in order for Lq to be gague

invariant. Show that with the same transformations, LG is gague invariant.






C


LG = −1

4FµνA FAµν

Exercise










C


LG = −1

4FµνA FAµν

Exercise






Perturbation theory[What is QCD]

[Perturbation theory]

Relies on idea of order-by-order expansion small coupling, αs � 1

αs + α2s︸︷︷︸

small

+ α3s︸︷︷︸

smaller

+ . . .︸︷︷︸negligible?

Interaction vertices of Feynman rules:A, µ

ba

−igstAbaγµ

A, µ

B, ν

C, ρ

p

q

r

−gs f ABC [(p − q)ρgµν

+(q − r)µgνρ

+(r − p)νgρµ]

B, ν

D, σ

C, ρ

A, µ

−ig2s f

XAC f XBD [gµνgρσ −gµσgνγ ] + (C , γ)↔

(D, ρ) + (B, ν)↔ (C , γ)

These expressions are fairly complex, so you reallydon’t want to have to deal with too many orders ofthem! i.e. αs had better be small. . .

[Zvi Bern will show you how to do things more easily!]


What do Feynman rules mean physically?[What is QCD]


A, µ

b a

ψb(−igstAbaγµ)ψa

A, µ

b a

( 0 1 0 )

︸︷︷︸ψb

0 1 01 0 00 0 0

︸︷︷︸

t1ab

100

︸︷︷︸

ψa

A gluon emission repaints the quark colour.A gluon itself carries colour and anti-colour.


What does “ggg” Feynman rule mean?[What is QCD]


A, µ

B, ν

C, ρ

p

q

r

−gs f ABC [(p − q)ρgµν

+(q − r)µgνρ

+(r − p)νgρµ]

A, µ

B, ν

C, ρ

p

q

r

A gluon emission also repaints the gluon colours.

Because a gluon carries colour + anti-colour, it emits ∼twice as strongly as a quark (just has colour)


Quick guide to colour algebra[What is QCD]


Tr(tAtB) = TRδAB , TR = 1

2

A B

∑A tAabt

Abc = CF δac , CF =

N2c − 1

2Nc=

4

3a c

∑C ,D f ACD f BCD = CAδ

AB , CA = Nc = 3A B

tAabtAcd =

1

2δbcδad −

1

2Ncδabδcd (Fierz)

1

2 2N

−1

b a

c d

=

Nc ≡ number of colours = 3 for QCD





2

A B

∑A tAabt


N2c − 1

2Nc=

4

3a c


AB , CA = Nc = 3A B

tAabtAcd =

1

2δbcδad −

1

2Ncδabδcd (Fierz)

1

2 2N

−1

b a

c d

=






2

A B

∑A tAabt


N2c − 1

2Nc=

4

3a c


AB , CA = Nc = 3A B

tAabtAcd =

1

2δbcδad −

1

2Ncδabδcd (Fierz)

1

2 2N

−1

b a

c d

=






2

A B

∑A tAabt


N2c − 1

2Nc=

4

3a c


AB , CA = Nc = 3A B

tAabtAcd =

1

2δbcδad −

1

2Ncδabδcd (Fierz)

1

2 2N

−1

b a

c d

=



Exercise[What is QCD]


Use the Fierz identity (fourth line of previous slide) to derive the secondline.


The running coupling[What is QCD]

[Running coupling]

All couplings run (QED, QCD, EW), i.e. they depend on the momentumscale (Q2) of your process.

The evolution equation for the QCD coupling, αs(Q2), is:

Q2 ∂αs

∂Q2= β(αs) , β(αs) = −α2

s (b0 + b1αs + b2α2s + . . .) ,

b0 =11CA − 2nf

12π, b1 =

17C 2A − 5CAnf − 3CFnf

24π2=

153− 19nf24π2

Note sign: Asymptotic Freedom, due to gluon to self-interaction

I At high scales Q, coupling becomes smallåquarks and gluons are almost free, interactions are weak

I At low scales, coupling becomes strongåquarks and gluons interact strongly — confined into hadrons

Perturbation theory fails.


The running coupling[What is QCD]

[Running coupling]

All couplings run (QED, QCD, EW), i.e. they depend on the momentumscale (Q2) of your process.

The evolution equation for the QCD coupling, αs(Q2), is:

Q2 ∂αs

∂Q2= β(αs) , β(αs) = −α2

s (b0 + b1αs + b2α2s + . . .) ,

b0 =11CA − 2nf

12π, b1 =

17C 2A − 5CAnf − 3CFnf

24π2=

153− 19nf24π2

Note sign: Asymptotic Freedom, due to gluon to self-interaction

I At high scales Q, coupling becomes smallåquarks and gluons are almost free, interactions are weak

I At low scales, coupling becomes strongåquarks and gluons interact strongly — confined into hadrons

Perturbation theory fails.


Running coupling (cont.)[What is QCD]

[Running coupling]

Solve Q2 ∂αs

∂Q2= −b0α

2s ⇒ αs(Q

2) =αs(Q

20 )

1 + b0αs(Q20 ) ln Q2

Q20

=1

b0 ln Q2

Λ2

Λ ' 0.2 GeV (aka ΛQCD) isthe fundamental scale of QCD,at which coupling blows up.

I Λ sets the scale for hadronmasses(NB: Λ not unambiguouslydefined wrt higher orders)

I Perturbative calculations validfor scales Q � Λ.

QCD αs(Mz) = 0.1185 ± 0.0006

Z pole fit

0.1

0.2

0.3

αs (Q)

1 10 100Q [GeV]

Heavy Quarkonia (NLO)

e+e– jets & shapes (res. NNLO)

DIS jets (NLO)

Sept. 2013

Lattice QCD (NNLO)

(N3LO)

τ decays (N3LO)

1000

pp –> jets (NLO)(–)


Running coupling (cont.)[What is QCD]

[Running coupling]

Solve Q2 ∂αs

∂Q2= −b0α

2s ⇒ αs(Q

2) =αs(Q

20 )

1 + b0αs(Q20 ) ln Q2

Q20

=1

b0 ln Q2

Λ2

Λ ' 0.2 GeV (aka ΛQCD) isthe fundamental scale of QCD,at which coupling blows up.

I Λ sets the scale for hadronmasses(NB: Λ not unambiguouslydefined wrt higher orders)

I Perturbative calculations validfor scales Q � Λ.

QCD αs(Mz) = 0.1185 ± 0.0006

Z pole fit

0.1

0.2

0.3

αs (Q)

1 10 100Q [GeV]



DIS jets (NLO)

Sept. 2013

Lattice QCD (NNLO)

(N3LO)

τ decays (N3LO)

1000



Nobel prize citation (annotated by Skands)[What is QCD]

[Running coupling]


Exercise

There is a freedom in defining the “scheme” for αs. E.g.

αscheme-Bs = αscheme-A

s + cB(αscheme-As )2

where cB is some scheme-conversion coefficient.

The β-function coefficients, usually given in the so-calledMS renormalisation scheme, are known up to b3.

Which of the b0, b1, b2, etc. coefficients is modified if thescheme is changed?


A dilemna[Basic methods]

QCD αs(Mz) = 0.1185 ± 0.0006

Z pole fit

0.1

0.2

0.3

αs (Q)

1 10 100Q [GeV]



DIS jets (NLO)

Sept. 2013

Lattice QCD (NNLO)

(N3LO)

τ decays (N3LO)

1000


I We want to investigate collisions athigh energies (∼ 100 GeV to fewTeV), where the coupling is small→ perturbative methods are thenatural choice

I But the LHC collides protons,m ' 0.94 GeV, which definitelyinvolve strong coupling physics

There is no escape from non-perturbative physics


Lattice QCD[Basic methods]

[Lattice]

I Put all the quark and gluon fieldsof QCD on a 4D-lattice

NB: with imaginary time

I Figure out which fieldconfigurations are most likely (byMonte Carlo sampling).

I You’ve solved QCD

image credits: fdecomite [Flickr]


http://www.flickr.com/photos/fdecomite/2615572026/

Lattice hadron masses[Basic methods]

[Lattice]

Lattice QCD is great at cal-culation static properties of asingle hadron.

E.g. the hadron mass spec-trum

Durr et al ’08


Lattice for LHC?[Basic methods]

[Lattice]

How big a lattice do you need for an LHC collision @ 14 TeV?

Lattice spacing:1

14 TeV∼ 10−5 fm

Lattice extent:

I non-perturbative dynamics for quark/hadron near rest takes place on

timescale t ∼ 1

0.5 GeV∼ 0.4 fm/c

I But quarks at LHC have effective boost factor ∼ 104

I So lattice extent should be ∼ 4000 fm

Total: need ∼ 4× 108 lattice units in each direction, or 3× 1034 nodes total.Plus clever tricks to deal with high particle multiplicity,

imaginary v. real time, etc.


Neither lattice QCD nor perturbative QCD can offera full solution to using QCD at colliders

What the community has settled on is

1) factorisation of initial state non-perturbative problem

from

2) the “hard process,” calculated perturbatively

supplemented with

3) non-perturbative modelling of final-state hadronic-scale processes(“hadronisation”).


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