Basics of QCD Lecture 1: the Core Ingredients

Basics of QCD Lecture 1: the core ingredients

Gavin Salam CERN Theory Unit

ICTP–SAIFR school on QCD and LHC physics July 2015, S˜aoPaulo, Brazil [What is QCD] QCD

QUANTUM CHROMODYNAMICS

The theory of quarks, gluons and their interactions

It’s central to all modern colliders. (And QCD is what we’re made of)

Gavin Salam (CERN) QCD Basics 1 2 / 24 [What is QCD] QCD predictions v. data for many processes

Status: March 2015 dt Standard Model Production Cross Section Measurements L 1 Reference R[fb− ] pp 8 8 10− Nucl. Phys. B, 486-548 (2014) total ATLAS Preliminary × Jets R=0.4 0.1 < pT < 2 TeV arXiv:1410.8857 [hep-ex] y <3.0 4.5 | | Dijets R=0.4 0.3 < mjj < 5 TeV JHEP 05, 059 (2014) y <3.0, y <3.0 Run 1 √s = 7, 8 TeV 4.5 | | ∗ W PRD 85, 072004 (2012) total 0.035 Z 0.035 PRD 85, 072004 (2012) total t¯t 4.6 Eur. Phys. J. C 74: 3109 (2014) fiducial 20.3 Eur. Phys. J. C 74: 3109 (2014) PRD 90, 112006 (2014) tt chan 4.6 −total 20.3 ATLAS-CONF-2014-007 WW 4.6 PRD 87, 112001 (2013) total 20.3 ATLAS-CONF-2014-033 γγ JHEP 01, 086 (2013) fiducial 4.9 Wt 2.0 PLB 716, 142-159 (2012) total 20.3 ATLAS-CONF-2013-100 WZ 4.6 EPJC 72, 2173 (2012) total 13.0 ATLAS-CONF-2013-021 ZZ 4.6 JHEP 03, 128 (2013) total LHC pp √s = 7 TeV 20.3 ATLAS-CONF-2013-020 Wγ 4.6 PRD 87, 112003 (2013) fiducial Theory arXiv:1407.1618 [hep-ph] WW+WZ JHEP 01, 049 (2015) fiducial Observed 4.6 stat Zγ stat+syst 4.6 PRD 87, 112003 (2013) fiducial arXiv:1407.1618 [hep-ph] t¯tW 20.3 ATLAS-CONF-2014-038 total 95% CL upper limit LHC pp √s = 8 TeV t¯tZ 4.7 ATLAS-CONF-2012-126 total 20.3 ATLAS-CONF-2014-038 ¯ Theory ttγ 4.6 arXiv:1502.00586 [hep-ex] fiducial Observed Zjj EWK stat 20.3 JHEP 04, 031 (2014) fiducial stat+syst H γγ 20.3 Preliminary fiducial→

Wγγ arXiv:1503.03243 [hep-ex] ﬁducial, njet=0 20.3 W±W±jj EWK 20.3 PRL 113, 141803 (2014) ﬁducial 95% CL upper limit ts chan 0.7 ATLAS-CONF-2011-118 −total 95% CL upper limit 20.3 arXiv:1410.0647 [hep-ex] 3 2 1 1 2 3 4 5 6 11 10− 10− 10− 1 10 10 10 10 10 10 10 0.5 1 1.5 2 σ [pb] observed/theory

Gavin Salam (CERN) QCD Basics 1 3 / 24 [What is QCD] QCD for Higgs physics

QCD is especially relevant in order to deduce information about the Higgs boson. [cf. lectures by Andre Sznajder & Claude Duhr]

Gavin Salam (CERN) QCD Basics 1 4 / 24 [What is QCD] QCD for Higgs physics

ATLAS Input measurements Individual analysis ± 1σ on µ mH (GeV)

+0.27 Overall: µ = 1.17 125.4 → γγ -0.27 H ggF: µ = 1.32+0.38 -0.38 125.4 VBF: µ = 0.8+0.7 -0.7 125.4 WH: µ = 1.0+1.6 -1.6 125.4 ZH: µ = 0.1+3.7 -0.1 125.4

Overall: µ = 1.44+0.40 125.36 H → ZZ* -0.33 ggF+ttH: µ = 1.7+0.5 -0.4 125.36 VBF+VH: µ = 0.3+1.6 -0.9 125.36 +0.24 Overall: µ = 1.16 125.36 H → WW* -0.21 QCD is especially relevant in +0.29 ggF: µ = 0.98 -0.26 125.36 VBF: µ = 1.28+0.55 -0.47 125.36 order to deduce information VH: µ = 3.0+1.6 -1.3 125.36 Overall: µ = 1.43+0.43 125.36 H → ττ -0.37 +1.5 about the . ggF: µ = 2.0 125.36 Higgs boson -1.2 VBF+VH: µ = 1.24+0.59 -0.54 125.36 Overall: µ = 0.52+0.40 125.36 → -0.40 VH Vbb +0.65 WH: µ = 1.11 125 [cf. lectures by Andre Sznajder -0.61 ZH: µ = 0.05+0.52 -0.49 125

& Claude Duhr] H → µµ µ +3.7 Overall: = -0.7 -3.7 125.5

→ γ H Z Overall: µ = 2.7+4.5 -4.3 125.5

bb: µ = 1.5+1.1 125 ttH -1.1 Multilepton: µ = 2.1+1.4 -1.2 125 γγ: µ = 1.3+2.62 -1.75 125.4

-1 − s = 7 TeV, 4.5-4.7 fb 2 0 2 4

s = 8 TeV, 20.3 fb-1 Signal strength (µ)

Gavin Salam (CERN) QCD Basics 1 4 / 24 [What is QCD] QCD and searches for new physics

Search for stop squarks

19.4 fb-1 (8 TeV) 140 CMS Data tt/W→ τ +X 120 hadr tt/W→ e,µ+X Z→νν Identiﬁcation and/or exclusion of 100 ~ ∼0 t→ tχ (500, 100) ~ ∼0 t→ tχ (650, 50) potential new physics relies in 80 diverse ways on a QCD-based Events / 50 GeV understanding of signals and 60 backgrounds. 40

0 2 1 miss 0 p [GeV] -1 T -2 Data 200 250 300 350 400 450 500

Data - MC pmiss [GeV] T

Gavin Salam (CERN) QCD Basics 1 5 / 24 [What is QCD] QCD and searches for new physics

Summary of CMS SUSY Results* in SMS framework ICHEP 2014 m(mother)-m(LSP)=200 GeV m(LSP)=0 GeV

~ ∼ 0 g → qq χ SUS 13-019 L=19.5 /fb ~ ∼ 0 g → bb χ SUS-14-011 SUS-13-019 L=19.3 19.5 /fb ~ ∼ 0 g → tt χ SUS-13-007 SUS-13-013 L=19.4 19.5 /fb ~ ~ ∼ 0 g → t(t → t χ ) SUS-13-008 SUS-13-013 L=19.5 /fb ~ ∼ ± ∼ 0 x = 0.20 g → qq(χ → Wχ ) SUS-13-013 L=19.5 /fb x = 0.50 0 gluino production ~ ~ ∼ ± ∼ g → b(b → t(χ → Wχ )) SUS-13-008 SUS-13-013 L=19.5 /fb

~ ∼ 0 q → q χ SUS-13-019 L=19.5 /fb squark ~ ∼ 0 t → t χ SUS-14-011 L=19.5 /fb ~ ∼ + ∼ 0 x = 0.25 x = 0.50 t → b(χ → Wχ ) SUS-13-011 L=19.5 /fb x = 0.75 ~ ∼ 0 ∼ 0 t → t b χ ( χ → H G) SUS-13-014 L=19.5 /fb ∼ 0 stop ~ ~ t → ( t → t χ ) Z SUS-13-024 SUS-13-004 L=19.5 /fb ~2 ~1 ∼ 01 t → ( t → t χ ) H SUS-13-024 SUS-13-004 L=19.5 /fb 2 1 1

~ ∼ 0 b → b χ SUS-13-018 L=19.4 /fb ~ ∼ 0 b → tW χ SUS-13-008 SUS-13-013 L=19.5 /fb ~ ∼ 0

sbottom b → bZ χ SUS-13-008 L=19.5 /fb

∼ 0 ∼ ± ∼ 0 ∼ 0 x = 0.05 χ χ → lll ν χ χ SUS-13-006 L=19.5 /fb x = 0.50 2 x = 0.95 ∼ +∼ - + ∼ 0∼ 0 χ χ → l l-ν ν χ χ SUS-13-006 L=19.5 /fb ∼ 0 ∼ 0 ∼ 0 ∼ 0 χ χ → Z Z χ χ SUS-14-002 L=19.5 /fb ∼ ±2∼ 02 ∼ 0 ∼ 0 χ χ → W Z χ χ 2 SUS-13-006 L=19.5 /fb CMS Preliminary ∼ 0 ∼ 0 ∼ 0 ∼ 0 χ χ → H Z χ χ SUS-14-002 L=19.5 /fb 2 2 ∼ ± ∼ 0 ∼ 0 ∼ 0 For decays with intermediate mass, χ χ → H W χ χ SUS-14-002 L=19.5 /fb ∼ 0 ∼ ±2 ∼ 0 ∼ 0 x = 0.05 χ χ → llτ ν χ χ x = 0.50 ⋅ ⋅ EWK gauginos 2 SUS-13-006 L=19.5 /fb x = 0.95 m = x m +(1-x) m ∼ 0 ∼ ± ∼ 0 ∼ 0 intermediate mother lsp χ χ → τττ ν χ χ SUS-13-006 L=19.5 /fb 2

~ ∼ 0 l → l χ SUS-13-006 L=19.5 /fb

slepton ~ g → qllν λ 122 SUS-12-027 L=9.2 /fb ~ g → qllν λ SUS-12-027 L=9.2 /fb ~ 123 g → qllν λ 233 SUS-12-027 L=9.2 /fb ~ g → qbtµ λ ' 231 SUS-12-027 L=9.2 /fb ~ g → qbtµ λ ' SUS-12-027 L=9.2 /fb ~ 233 g → qqb λ '' 113/223 EXO-12-049 L=19.5 /fb ~ g → qqq λ '' EXO-12-049 L=19.5 /fb ~ 112 g → tbs λ '' 323 SUS-13-013 L=19.5 /fb ~ g → qqqq λ '' SUS-12-027 L=9.2 /fb ~ 112 q → qllν λ 122 SUS-12-027 L=9.2 /fb ~ q → qllν λ 123 SUS-12-027 L=9.2 /fb ~ q → qllν λ

RPV SUS-12-027 L=9.2 /fb ~ 233 q → qbtµ λ ' 231 SUS-12-027 L=9.2 /fb ~ q → qbtµ λ ' SUS-12-027 L=9.2 /fb ~ 233 q → qqqq λ '' R 112 SUS-12-027 L=9.2 /fb ~ t → µ e ν t λ R 122 SUS-13-003 L=19.5 9.2 /fb ~ t → µ τν t λ SUS-12-027 L=9.2 /fb ~R 123 t → µ τν t λ R 233 SUS-13-003 L=19.5 9.2 /fb ~ t → tbtµ λ ' R 233 SUS-13-003 L=19.5 /fb 0 200 400 600 800 1000 1200 1400 1600 1800 *Observed limits, theory uncertainties not included Mass scales [GeV] Only a selection of available mass limits Probe *up to* the quoted mass limit

Gavin Salam (CERN) QCD Basics 1 6 / 24 [What is QCD] Aims of the course

The school will cover many aspects of QCD, from the advanced calculation techniques to the experimental consequences. This course: A reminder of what QCD is

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

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

Gavin Salam (CERN) QCD Basics 1 7 / 24 [What is QCD] The ingredients of QCD

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

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

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

Gavin Salam (CERN) QCD Basics 1 8 / 24 [What is QCD] Lagrangian + colour   ψ1 Quarks — 3 colours: ψa =  ψ2  ψ3 Quark part of Lagrangian:

µ µ C C L = ψ¯ (iγ ∂µδ − g γ t A − m)ψ Let’sq writea downab QCDs ab inµ full detailb SU(3) local gauge symmetry ↔ 8 (= 32 − 1) generators t1 ... t8 (There’s a lot to absorb here — but it should becomeab moreab 1 8 correspondingpalatable to 8 gluons as weA returnµ ... A toµ. individual elements later) A 1 A A representation is: t = 2 λ ,  0 1 0   0 −i 0   1 0 0   0 0 1  1 2 3 4 λ =  1 0 0  , λ =  i 0 0  , λ =  0 −1 0  , λ =  0 0 0  , 0 0 0 0 0 0 0 0 0 1 0 0         √1 0 0 0 0 −i 0 0 0 0 0 0 3 λ5 = 0 0 0 , λ6 = 0 0 1 , λ7 = 0 0 −i , λ8 =  0 √1 0  ,        3  i 0 0 0 1 0 0 i 0 0 0 √−2 3

Gavin Salam (CERN) QCD Basics 1 9 / 24 [What is QCD] Lagrangian + colour   ψ1 Quarks — 3 colours: ψa =  ψ2  ψ3 Quark part of Lagrangian:

¯ µ µ C C Lq = ψa(iγ ∂µδab − gs γ tabAµ − m)ψb

2 1 8 SU(3) local gauge symmetry ↔ 8 (= 3 − 1) generators tab ... tab 1 8 corresponding to 8 gluons Aµ ... Aµ. A 1 A A representation is: t = 2 λ ,  0 1 0   0 −i 0   1 0 0   0 0 1  1 2 3 4 λ =  1 0 0  , λ =  i 0 0  , λ =  0 −1 0  , λ =  0 0 0  , 0 0 0 0 0 0 0 0 0 1 0 0         √1 0 0 0 0 −i 0 0 0 0 0 0 3 λ5 = 0 0 0 , λ6 = 0 0 1 , λ7 = 0 0 −i , λ8 =  0 √1 0  ,        3  i 0 0 0 1 0 0 i 0 0 0 √−2 3

Gavin Salam (CERN) QCD Basics 1 9 / 24 [What is QCD] Lagrangian + colour   ψ1 Quarks — 3 colours: ψa =  ψ2  ψ3 Quark part of Lagrangian:

¯ µ µ C C Lq = ψa(iγ ∂µδab − gs γ tabAµ − m)ψb

Gavin Salam (CERN) QCD Basics 1 9 / 24 Exercise

Consider gauge transformations

A A ψ → U(x)ψ , U(x) = eiu (x)t

C How should the gluon ﬁeld Aµ transform in order for Lq to be gauge invariant? Show that with the same transformations, LG is gauge invariant.

[What is QCD] Lagrangian: gluonic part

A A A B C A B C Field tensor: Fµν = ∂µAν − ∂νAν − gs fABC Aµ Aν [t , t ] = ifABC t fABC are structure constants of SU(3) (antisymmetric in all indices — SU(2) equivalent was ABC ). Needed for gauge invariance of gluon part of Lagrangian: 1 L = − F µνF A µν G 4 A

Gavin Salam (CERN) QCD Basics 1 10 / 24 Exercise

Consider gauge transformations

A A ψ → U(x)ψ , U(x) = eiu (x)t

C How should the gluon ﬁeld Aµ transform in order for Lq to be gauge invariant? Show that with the same transformations, LG is gauge invariant.

[What is QCD] Lagrangian: gluonic part

Gavin Salam (CERN) QCD Basics 1 10 / 24 [What is QCD] Lagrangian: gluonic part

Exercise

Consider gauge transformations

A A ψ → U(x)ψ , U(x) = eiu (x)t

C How should the gluon ﬁeld Aµ transform in order for Lq to be gauge invariant? Show that with the same transformations, LG is gauge invariant.

Gavin Salam (CERN) QCD Basics 1 10 / 24 [What is QCD] [Perturbation theory] Perturbation theory

Relies on idea of order-by-order expansion small coupling, αs 1 2 3 αs + αs + αs + ... |{z} |{z} |{z} small smaller negligible?

Interaction vertices of Feynman rules:

A, µ A, µ D, σ A, µ p

r C, ρ C, q ρ B, ν B, ν a b −ig 2f XAC f XBD [g µνg ρσ − A µ ABC ρ µν s −igs tbaγ −gs f [(p − q) g g µσg νγ] + (C, γ) ↔ µ νρ +(q − r) g (D, ρ) + (B, ν) ↔ (C, γ) +(r − p)νg ρµ] These expressions are fairly complex, so you really don’t want to have to deal with too many orders of them! i.e. αs had better be small. . . [Zvi Bern will show you how to do things more easily!]

Gavin Salam (CERN) QCD Basics 1 11 / 24 [What is QCD] [Perturbation theory] What do Feynman rules mean physically?

A, µ A, µ

b a b a ¯ A µ ψb(−igs tbaγ )ψa (010)  0 1 0   1   1 0 0   0  0 0 0 0 | {z } | {z } | {z } ¯ 1 ψ ψb tab a

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

Gavin Salam (CERN) QCD Basics 1 12 / 24 [What is QCD] [Perturbation theory] What does “ggg” Feynman rule mean?

A, µ A, µ p p

r r C, ρ C, ρ q q B, ν B, ν ABC ρ µν −gs f [(p − q) g +(q − r)µg νρ +(r − p)νg ρµ]

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)

Gavin Salam (CERN) QCD Basics 1 13 / 24 [What is QCD] [Perturbation theory] Quick guide to colour algebra

AB A B AB 1 Tr(t t ) = TR δ , TR = 2

N2 − 1 4 P A A c a c A tabtbc = CF δac , CF = = 2Nc 3

AB P ACD BCD AB C,D f f = CAδ , CA = Nc = 3

b a 1 −1 A A 1 1 = tabtcd = δbc δad − δabδcd (Fierz) 2 2N 2 2Nc c d

Nc ≡ number of colours = 3 for QCD

Gavin Salam (CERN) QCD Basics 1 14 / 24 [What is QCD] [Perturbation theory] Quick guide to colour algebra

AB A B AB 1 Tr(t t ) = TR δ , TR = 2

N2 − 1 4 P A A c a c A tabtbc = CF δac , CF = = 2Nc 3

AB P ACD BCD AB C,D f f = CAδ , CA = Nc = 3

b a 1 −1 A A 1 1 = tabtcd = δbc δad − δabδcd (Fierz) 2 2N 2 2Nc c d

Nc ≡ number of colours = 3 for QCD

Gavin Salam (CERN) QCD Basics 1 14 / 24 [What is QCD] [Perturbation theory] Quick guide to colour algebra

AB A B AB 1 Tr(t t ) = TR δ , TR = 2

N2 − 1 4 P A A c a c A tabtbc = CF δac , CF = = 2Nc 3

AB P ACD BCD AB C,D f f = CAδ , CA = Nc = 3

b a 1 −1 A A 1 1 = tabtcd = δbc δad − δabδcd (Fierz) 2 2N 2 2Nc c d

Nc ≡ number of colours = 3 for QCD

Gavin Salam (CERN) QCD Basics 1 14 / 24 [What is QCD] [Perturbation theory] Quick guide to colour algebra

AB A B AB 1 Tr(t t ) = TR δ , TR = 2

N2 − 1 4 P A A c a c A tabtbc = CF δac , CF = = 2Nc 3

AB P ACD BCD AB C,D f f = CAδ , CA = Nc = 3

b a 1 −1 A A 1 1 = tabtcd = δbc δad − δabδcd (Fierz) 2 2N 2 2Nc c d

Nc ≡ number of colours = 3 for QCD

Gavin Salam (CERN) QCD Basics 1 14 / 24 [What is QCD] [Perturbation theory] Exercise

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

Gavin Salam (CERN) QCD Basics 1 15 / 24 [What is QCD] [Running coupling] The running coupling

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

2 The evolution equation for the QCD coupling, αs(Q ), is: ∂α Q2 s = β(α ) , β(α ) = −α2(b + b α + b α2 + ...) , ∂Q2 s s s 0 1 s 2 s

11C − 2n 17C 2 − 5C n − 3C n 153 − 19n b = A f , b = A A f F f = f 0 12π 1 24π2 24π2 Note sign: Asymptotic Freedom, due to gluon to self-interaction

I At high scales Q, coupling becomes small ¯quarks and gluons are almost free, interactions are weak

I At low scales, coupling becomes strong ¯quarks and gluons interact strongly — conﬁned into hadrons Perturbation theory fails.

Gavin Salam (CERN) QCD Basics 1 16 / 24 [What is QCD] [Running coupling] The running coupling

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

2 The evolution equation for the QCD coupling, αs(Q ), is: ∂α Q2 s = β(α ) , β(α ) = −α2(b + b α + b α2 + ...) , ∂Q2 s s s 0 1 s 2 s

11C − 2n 17C 2 − 5C n − 3C n 153 − 19n b = A f , b = A A f F f = f 0 12π 1 24π2 24π2 Note sign: Asymptotic Freedom, due to gluon to self-interaction

I At high scales Q, coupling becomes small ¯quarks and gluons are almost free, interactions are weak

I At low scales, coupling becomes strong ¯quarks and gluons interact strongly — conﬁned into hadrons Perturbation theory fails.

Gavin Salam (CERN) QCD Basics 1 16 / 24 Sept. 2013 α τ decays (N 3LO) s(Q ) Lattice QCD (NNLO) DIS jets (NLO) 0.3 Heavy Quarkonia (NLO) – e+e jets & shapes (res. NNLO) Z pole fit (N 3LO) (–) pp –> jets (NLO) 0.2

0.1 QCD αs(Mz) = 0.1185 ± 0.0006 1 10 100 1000 Q [GeV]

[What is QCD] [Running coupling] Running coupling (cont.)

2 2 ∂αs 2 2 αs(Q0 ) 1 Solve Q = −b0α ⇒ αs(Q ) = = ∂Q2 s 2 Q2 Q2 1 + b0αs(Q0 ) ln 2 b0 ln 2 Q0 Λ

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

I Λ sets the scale for hadron masses (NB: Λ not unambiguously deﬁned wrt higher orders)

I Perturbative calculations valid for scales Q Λ.

Gavin Salam (CERN) QCD Basics 1 17 / 24 [What is QCD] [Running coupling] Running coupling (cont.)

2 2 ∂αs 2 2 αs(Q0 ) 1 Solve Q = −b0α ⇒ αs(Q ) = = ∂Q2 s 2 Q2 Q2 1 + b0αs(Q0 ) ln 2 b0 ln 2 Q0 Λ

Λ ' 0.2 GeV (aka ΛQCD ) is Sept. 2013 α τ decays (N 3LO) s(Q ) the fundamental scale of QCD, Lattice QCD (NNLO) DIS jets (NLO) at which coupling blows up. 0.3 Heavy Quarkonia (NLO) – e+e jets & shapes (res. NNLO) Z pole fit (N 3LO) (–) pp –> jets (NLO) I Λ sets the scale for hadron masses 0.2 (NB: Λ not unambiguously deﬁned wrt higher orders) 0.1 QCD αs(Mz) = 0.1185 ± 0.0006 1 10 100 1000 I Perturbative calculations valid Q [GeV] for scales Q Λ.

Gavin Salam (CERN) QCD Basics 1 17 / 24 [What is QCD] [Running coupling] Nobel prize citation (annotated by Skands)

Gavin Salam (CERN) QCD Basics 1 18 / 24 Exercise

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

scheme-B scheme-A scheme-A 2 αs = αs + cB (αs )

where cB is some scheme-conversion coeﬃcient.

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

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

Gavin Salam (CERN) QCD Basics 1 19 / 24 [Basic methods] A dilemna

Sept. 2013 I We want to investigate collisions at α τ decays (N 3LO) s(Q ) Lattice QCD (NNLO) high energies (∼ 100 GeV to few DIS jets (NLO) 0.3 Heavy Quarkonia (NLO) – TeV), where the coupling is small e+e jets & shapes (res. NNLO) Z pole fit (N 3LO) (–) pp –> jets (NLO) → perturbative methods are the 0.2 natural choice

0.1 I But the LHC collides protons, QCD αs(Mz) = 0.1185 ± 0.0006 m ' 0.94 GeV, which deﬁnitely 1 10 100 1000 Q [GeV] involve strong coupling physics

There is no escape from non-perturbative physics

Gavin Salam (CERN) QCD Basics 1 20 / 24 [Basic methods] [Lattice] Lattice QCD

I Put all the quark and gluon ﬁelds of QCD on a 4D-lattice NB: with imaginary time

I Figure out which ﬁeld conﬁgurations are most likely (by Monte Carlo sampling).

I You’ve solved QCD

image credits: fdecomite [Flickr]

Gavin Salam (CERN) QCD Basics 1 21 / 24 [Basic methods] [Lattice] Lattice hadron masses

Lattice QCD is great at calculation static properties of a single hadron.

E.g. the hadron mass spec- trum

Durr et al ’08

Gavin Salam (CERN) QCD Basics 1 22 / 24 [Basic methods] [Lattice] Lattice for LHC?

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

1 Lattice spacing: ∼ 10−5 fm 14 TeV

Lattice extent:

I non-perturbative dynamics for quark/hadron near rest takes place on 1 timescale t ∼ ∼ 0.4 fm/c 0.5 GeV 4 I But quarks at LHC have eﬀective boost factor ∼ 10

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.

Gavin Salam (CERN) QCD Basics 1 23 / 24 Neither lattice QCD nor perturbative QCD can oﬀer a 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”).

Gavin Salam (CERN) QCD Basics 1 24 / 24