QCD (for colliders) Lecture 1: Introduction

Transcription

1 QCD (for colliders) Lecture 1: Introduction Gavin Salam CERN At the Fourth Asia-Europe-Pacific School of High-Energy Physics September 2018, Quy Nhon, Vietnam

2 QCD lecture 1 (p. 2) 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)

3 QCD lecture 1 (p. 3) What is QCD The ingredients of QCD Quarks (and anti-quarks): they come in 3 colours Gluons: a bit like photons in QED But there are 8 of them, and they re colour charged And a coupling, α s, that s not so small and runs fast At LHC, in the range 0.08(@ 5 TeV) to O (1)(@ 0.5 GeV)

4 QCD lecture 1 (p. 4) What is QCD Aims of this course I ll try to give you a feel for: How QCD works How theorists handle QCD at high-energy colliders How you can work with QCD at high-energy colliders

5 A proton-proton collision: INITIAL STATE proton proton

6 A proton-proton collision: FINAL STATE µ B + π K... µ + B (actual final-state multiplicity ~ several hundred hadrons)

7 ATLAS H WW* ANALYSIS [ ] 3 Signal and background models The ggf and VBF production modes for H WW are modelled at next-to-leading order (NLO) in the strong coupling α S with the Powheg MC generator [22 25], interfaced with Pythia8[26] (version 8.165) for the parton shower, hadronisation, and underlying event. The CT10 [27] PDF set is used and the parameters of the Pythia8 generator controlling the modelling of the parton shower and the underlying event are those corresponding to the AU2 set [28]. The Higgs boson mass set in the generation is GeV, which is close to the measured value. The Powheg ggf model takes into account finite quark masses and a running-width Breit Wigner distribution that includes electroweak corrections at NLO [29]. To improve the modelling of the Higgs boson p T distribution, a reweighting scheme is applied to reproduce the prediction of the next-to-next-to-leading-order (NNLO) and next-to-next-to-leading-logarithm (NNLL) dynamic-scale calculation given by the HRes 2.1 program [30]. Events with 2 jets are further reweighted to reproduce the pt H spectrum predicted by the NLO Powheg simulation of Higgs boson production in association with two jets (H + 2 jets) [31]. Interference with continuum WW production [32, 33] has a negligible impact on this analysis due to the transverse-mass selection criteria described in Section 4 and is not included in the signal model. The inclusive cross sections at s 8 TeV for a Higgs boson mass of GeV, calculated at NNLO NNL Jets are reconstructed from topological clusters of calorimeter cells [50 52] using the anti-k t algorithm with a radius parameter of R = 0.4 [53]. Jet energies are corrected for the effects of calorimeter noncompensation, signal losses due to noise threshold e ects, energy lost in non-instrumented regions, con-

8 ATLAS H WW* ANALYSIS [ ] Events / 5 GeV Data - Bkg ATLAS -1 s = 8 TeV, 20.3 fb H WW* eνµν, 0 jets Data SM bkg (sys stat) H WW Other VV W+jet Top Z/γ* Multijet 60 Data-Bkg SM bkg (sys stat) H [GeV] m T That whole paragraph was just for the red part of this distribution (the Higgs signal). Complexity of modelling each of the backgrounds is comparable (a) N jet = 0

9 QCD lecture 1 (p. 9) What is QCD Quarks 3 colours: ψ a = Quark part of Lagrangian: ψ 1 ψ 2 ψ 3 Lagrangian + colour Let sl q write = ψ a (iγ down µ µ δ ab QCD g s γ µ tab C in AC µ full m)ψ detail b SU(3) (There s local gauge a lot symmetry to absorb here 8 (= 3 2 but 1) it should generators become tab 1. more.. t8 ab corresponding palatable to 8 gluons as we Areturn 1 µ... Ato 8 µ. individual elements later) A representation is: t A = 1 2 λa, λ 1 = 1 0 0, λ 2 = i λ 5 = i 0 0, λ 6 = 0 i 0 i 0 0, λ 3 = , λ 7 = , λ 4 = i, λ 8 = 0 i , ,

10 QCD lecture 1 (p. 9) What is QCD Quarks 3 colours: ψ a = Quark part of Lagrangian: ψ 1 ψ 2 ψ 3 Lagrangian + colour L q = ψ a (iγ µ µ δ ab g s γ µ t C ab AC µ m)ψ b SU(3) local gauge symmetry 8 (= 3 2 1) generators t 1 ab... t8 ab corresponding to 8 gluons A 1 µ... A 8 µ. A representation is: t A = 1 2 λa, λ 1 = 1 0 0, λ 2 = i λ 5 = i 0 0, λ 6 = 0 i 0 i 0 0, λ 3 = , λ 4 = , λ 7 = 0 0 i, λ 8 = i , ,

11 QCD lecture 1 (p. 9) What is QCD Quarks 3 colours: ψ a = Quark part of Lagrangian: ψ 1 ψ 2 ψ 3 Lagrangian + colour L q = ψ a (iγ µ µ δ ab g s γ µ t C ab AC µ m)ψ b SU(3) local gauge symmetry 8 (= 3 2 1) generators t 1 ab... t8 ab corresponding to 8 gluons A 1 µ... A 8 µ. A representation is: t A = 1 2 λa, λ 1 = 1 0 0, λ 2 = i λ 5 = i 0 0, λ 6 = 0 i 0 i 0 0, λ 3 = , λ 4 = , λ 7 = 0 0 i, λ 8 = i , ,

12 QCD lecture 1 (p. 10) What is QCD Lagrangian: gluonic part Field tensor: F A µν = µ A A ν ν A A ν g s f ABC A B µ A C ν [t A, t B ] = if ABC t C f ABC are structure constants of SU(3) (antisymmetric in all indices SU(2) equivalent was ɛ ABC ). Needed for gauge invariance of gluon part of Lagrangian: L G = 1 4 F µν A F A µν

13 QCD lecture 1 (p. 10) What is QCD Lagrangian: gluonic part Field tensor: F A µν = µ A A ν ν A A ν g s f ABC A B µ A C ν [t A, t B ] = if ABC t C f ABC are structure constants of SU(3) (antisymmetric in all indices SU(2) equivalent was ɛ ABC ). Needed for gauge invariance of gluon part of Lagrangian: L G = 1 4 F µν A F A µν

14 QCD lecture 1 (p. 11) Basic methods Solving QCD Two main approaches to solving it Numerical solution with discretized space time (lattice) Perturbation theory: assumption that coupling is small Also: effective theories

15 QCD lecture 1 (p. 12) Basic methods Lattice Lattice QCD Put all the quark and gluon fields of QCD on a 4D-lattice NB: with imaginary time Figure out which field configurations are most likely (by Monte Carlo sampling). You ve solved QCD image credits: fdecomite [Flickr]

16 QCD lecture 1 (p. 13) Basic methods Lattice Lattice hadron masses Lattice QCD is great at calculation static properties of a single hadron. E.g. the hadron mass spectrum Durr et al 08

17 QCD lecture 1 (p. 14) Basic methods Lattice Lattice for LHC? How big a lattice do you need for an LHC 14 TeV? Lattice spacing: Lattice extent: 1 14 TeV 10 5 fm non-perturbative dynamics for quark/hadron near rest takes place on 1 timescale t 0.4 fm/c 0.5 GeV But quarks at LHC have effective boost factor 10 4 So lattice extent should be 4000 fm Total: need lattice units in each direction, or nodes total. Plus clever tricks to deal with high particle multiplicity, imaginary v. real time, etc.

18 QCD lecture 1 (p. 15) Basic methods Perturbation theory Perturbation theory Relies on idea of order-by-order expansion small coupling, α s 1 α s + α 2 s }{{} small + α 3 s }{{} smaller Interaction vertices of Feynman rules: +... }{{} negligible? A, µ A, µ D, σ A, µ p a ig s t A ba γµ b C, ρ r q B, ν g s f ABC [(p q) ρ g µν +(q r) µ g νρ +(r p) ν g ρµ ] C, ρ B, ν ig 2 s f XAC f XBD [g µν g ρσ g µσ g νγ ] + (C, γ) (D, ρ) + (B, ν) (C, γ) These expressions are fairly complex, so you really don t want to have to deal with too many orders of them! i.e. α s had better be small...

19 QCD lecture 1 (p. 16) Basic methods Perturbation theory What do Feynman rules mean physically? A, µ A, µ b a ψ b ( ig s tba A γµ )ψ a b ( ) } {{ } ψ b a } {{ } tab }{{} ψ a A gluon emission repaints the quark colour. A gluon itself carries colour and anti-colour.

20 QCD lecture 1 (p. 17) Basic methods Perturbation theory What does ggg Feynman rule mean? A, µ A, µ p p C, ρ r q B, ν g s f ABC [(p q) ρ g µν +(q r) µ g νρ +(r p) ν g ρµ ] C, ρ r q B, ν 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)

21 QCD lecture 1 (p. 18) Basic methods Perturbation theory Quick guide to colour algebra Tr(t A t B ) = T R δ AB, T R = 1 2 A B A ta ab ta bc = C F δ ac, C F = N2 c 1 2N c = 4 3 C,D f ACD f BCD = C A δ AB, C A = N c = 3 a A c B t A ab ta cd = 1 2 δ bcδ ad 1 2N c δ ab δ cd (Fierz) b c a = d N N c number of colours = 3 for QCD

22 QCD lecture 1 (p. 18) Basic methods Perturbation theory Quick guide to colour algebra Tr(t A t B ) = T R δ AB, T R = 1 2 A B A ta ab ta bc = C F δ ac, C F = N2 c 1 2N c = 4 3 C,D f ACD f BCD = C A δ AB, C A = N c = 3 a A c B t A ab ta cd = 1 2 δ bcδ ad 1 2N c δ ab δ cd (Fierz) b c a = d N N c number of colours = 3 for QCD

23 QCD lecture 1 (p. 18) Basic methods Perturbation theory Quick guide to colour algebra Tr(t A t B ) = T R δ AB, T R = 1 2 A B A ta ab ta bc = C F δ ac, C F = N2 c 1 2N c = 4 3 C,D f ACD f BCD = C A δ AB, C A = N c = 3 a A c B t A ab ta cd = 1 2 δ bcδ ad 1 2N c δ ab δ cd (Fierz) b c a = d N N c number of colours = 3 for QCD

24 QCD lecture 1 (p. 18) Basic methods Perturbation theory Quick guide to colour algebra Tr(t A t B ) = T R δ AB, T R = 1 2 A B A ta ab ta bc = C F δ ac, C F = N2 c 1 2N c = 4 3 C,D f ACD f BCD = C A δ AB, C A = N c = 3 a A c B t A ab ta cd = 1 2 δ bcδ ad 1 2N c δ ab δ cd (Fierz) b c a = d N N c number of colours = 3 for QCD

25 QCD lecture 1 (p. 19) Basic methods Perturbation theory How big is the coupling? All couplings run (QED, QCD, EW), i.e. they depend on the momentum scale (Q 2 ) of your process. The QCD coupling, α s (Q 2 ), runs fast: Q 2 α s Q 2 = β(α s), β(α s ) = α 2 s (b 0 + b 1 α s + b 2 α 2 s +...), b 0 = 11C A 2n f, b 1 = 17C A 2 5C An f 3C F n f 12π 24π 2 = n f 24π 2 Note sign: Asymptotic Freedom, due to gluon to self-interaction 2004 Novel prize: Gross, Politzer & Wilczek At high scales Q, coupling becomes small quarks and gluons are almost free, interactions are weak At low scales, coupling becomes strong quarks and gluons interact strongly confined into hadrons Perturbation theory fails.

26 QCD lecture 1 (p. 19) Basic methods Perturbation theory How big is the coupling? All couplings run (QED, QCD, EW), i.e. they depend on the momentum scale (Q 2 ) of your process. The QCD coupling, α s (Q 2 ), runs fast: Q 2 α s Q 2 = β(α s), β(α s ) = α 2 s (b 0 + b 1 α s + b 2 α 2 s +...), b 0 = 11C A 2n f, b 1 = 17C A 2 5C An f 3C F n f 12π 24π 2 = n f 24π 2 Note sign: Asymptotic Freedom, due to gluon to self-interaction 2004 Novel prize: Gross, Politzer & Wilczek At high scales Q, coupling becomes small quarks and gluons are almost free, interactions are weak At low scales, coupling becomes strong quarks and gluons interact strongly confined into hadrons Perturbation theory fails.

27 QCD lecture 1 (p. 20) Basic methods Perturbation theory Running coupling (cont.) Solve Q 2 α s Q 2 = b 0α 2 s α s (Q 2 ) = α s (Q 2 0 ) 1 + b 0 α s (Q0 2 Q2 ) ln Q0 2 = 1 b 0 ln Q2 Λ 2 Λ 0.2 GeV (aka Λ QCD ) is the fundamental scale of QCD, at which coupling blows up. Λ sets the scale for hadron masses (NB: Λ not unambiguously defined wrt higher orders) Perturbative calculations valid for scales Q Λ.

28 QCD lecture 1 (p. 20) Basic methods Perturbation theory Running coupling (cont.) Solve Q 2 α s Q 2 = b 0α 2 s α s (Q 2 ) = α s (Q 2 0 ) 1 + b 0 α s (Q0 2 Q2 ) ln Q0 2 = 1 b 0 ln Q2 Λ 2 Λ 0.2 GeV (aka Λ QCD ) is the fundamental scale of QCD, at which coupling blows up. Λ sets the scale for hadron masses (NB: Λ not unambiguously defined wrt higher orders) α s (Q 2 ) τ decays (N 3 LO) DIS jets (NLO) Heavy Quarkonia (NLO) e + e jets & shapes (res. NNLO) e.w. precision fits (N 3 LO) ( ) pp > jets (NLO) pp > tt (NNLO) April 2016 Perturbative calculations valid for scales Q Λ. 0.1 QCD α s (M z ) = ± Q [GeV] 1000

29 Baikov Davier Pich Boito SM review HPQCD (Wilson loops) HPQCD (c-c correlators) Maltmann (Wilson loops) JLQCD (Adler functions) PACS-CS (vac. pol. fctns.) ETM (ghost-gluon vertex) BBGPSV (static energy) ABM BBG JR NNPDF MMHT ALEPH (jets&shapes) OPAL(j&s) JADE(j&s) Dissertori (3j) JADE (3j) DW (T) Abbate (T) Gehrm. (T) Hoang (C) GFitter CMS (tt cross section) τ-decays lattice structure e+e- annihilation functions electroweak precision fits hadron collider STRONG-COUPLING DETERMINATIONS Most consistent set of independent determinations is from lattice Bethke, Dissertori & GPS in PDG 16 Three best determinations are from lattice QCD (HPQCD, , , ALPHA ) α s(m Z) = ± (0.6%) [heavy-quark correlators] α s(m Z) = ± (0.6%) [Wilson loops] α s(m Z) = ± (0.7%) [Schrodinger Functional] Many determinations quote small uncertainties ( 1%). All are disputed! Some determinations quote anomalously small central values (~0.113 v. world avg. of ±0.0011). Also disputed 6

30 QCD lecture 1 (p. 22) Basic methods Perturbation theory QCD perturbation theory (PT) & LHC? α s (Q 2 ) e.w. precision fits (N 3 LO) ( ) pp > jets (NLO) pp > tt (NNLO) QCD αs(mz) = ± Q [GeV] τ decays (N 3 LO) DIS jets (NLO) Heavy Quarkonia (NLO) e + e jets & shapes (res. NNLO) April Higgs, SM and searches at colliders probe scales Q p t 50 GeV 5 TeV The coupling certainly is small there! But we re colliding protons, m p 0.94 GeV The coupling is large! When we look at QCD events (this one is interpreted as e + e Z q q), we see: hadrons (PT doesn t hold for them) lots of them so we can t say 1 quark/gluon 1 hadron, and we limit ourselves to 1 or 2 orders of PT.

31 QCD lecture 1 (p. 22) Basic methods Perturbation theory QCD perturbation theory (PT) & LHC? α s (Q 2 ) e.w. precision fits (N 3 LO) ( ) pp > jets (NLO) pp > tt (NNLO) QCD αs(mz) = ± Q [GeV] τ decays (N 3 LO) DIS jets (NLO) Heavy Quarkonia (NLO) e + e jets & shapes (res. NNLO) April Higgs, SM and searches at colliders probe scales Q p t 50 GeV 5 TeV The coupling certainly is small there! But we re colliding protons, m p 0.94 GeV The coupling is large! When we look at QCD events (this one is interpreted as e + e Z q q), we see: hadrons (PT doesn t hold for them) lots of them so we can t say 1 quark/gluon 1 hadron, and we limit ourselves to 1 or 2 orders of PT.

32 QCD lecture 1 (p. 23) Basic methods Perturbation theory Neither lattice QCD nor perturbative QCD can offer a full solution to using QCD at colliders What the community has settled on is perturbative QCD inputs + non-perturbative modelling/factorisation Rest of this lecture: take a simple environment (e + e hadrons) and see how PT allows us to understand why QCD events look the way they do. Next lectures: dealing with incoming protons, jets, modern predictive tools

33 QCD lecture 1 (p. 24) Soft-collinear emission Start with γ q q: M q q = ū(p 1 )ie q γ µ v(p 2 ) Soft gluon amplitude ie γ µ p 1 Emit a gluon: p 2 M q qg = ū(p 1 )ig s ɛ/t A i p/ 1 + /k ie qγ µ v(p 2 ) i ū(p 1 )ie q γ µ p/ 2 + /k ig s ɛ/t A v(p 2 ) Make gluon soft k p 1,2 ; ignore terms suppressed by powers of k: ( M q qg ū(p 1 )ie q γ µ t A p1.ɛ v(p 2 ) g s p 1.k p ) 2.ɛ p 2.k p/v(p) = 0, p//k + /kp/ = 2p.k

34 QCD lecture 1 (p. 24) Soft-collinear emission Start with γ q q: M q q = ū(p 1 )ie q γ µ v(p 2 ) Soft gluon amplitude ie γ µ p 1 Emit a gluon: p 2 M q qg = ū(p 1 )ig s ɛ/t A i p/ 1 + /k ie qγ µ v(p 2 ) i ū(p 1 )ie q γ µ p/ 2 + /k ig s ɛ/t A v(p 2 ) ie γ µ p 1 k,ε ie γ µ p 1 p 2 k,ε Make gluon soft k p 1,2 ; ignore terms suppressed by powers of k: ( M q qg ū(p 1 )ie q γ µ t A p1.ɛ v(p 2 ) g s p 1.k p ) 2.ɛ p 2.k p 2 p/v(p) = 0, p//k + /kp/ = 2p.k

35 QCD lecture 1 (p. 24) Soft-collinear emission Soft gluon amplitude Start with γ q q: p 1 ie γ ū(p M q q = ū(p 1 )ie q γ µ v(p 2 ) µ 1 )ig s ɛ/t A i p/ 1 + /k ie qγ µ v(p 2 ) = ig s ū(p 1 )ɛ/ p/ 1 + /k (p 1 + k) 2 e qγ µ t A v(p 2 ) Use /A/B = 2A.B /B /A: p 2 Emit a gluon: 1 = ig s ū(p 1 )[2ɛ.(p 1 + k) (p/ 1 + /k)ɛ/] M q qg = ū(p 1 )ig s ɛ/t A i p/ + /k ie (p 1 + k) 2 e qγ µ t A v(p 2 ) p 1 p 1 qγ µ v(p 2 ) ie γ µ ie γ k,ε µ Use ū(p 1 )p/ 1 = 0 and k p 1 (p 1, k massless) i ū(p 1 )ie q γ µ p/ 2 + /k ig s ɛ/t A p v(p 2 ) 2 1 ig s ū(p 1 )[2ɛ.p 1 ] (p Make gluon soft k p 1,2 ; ignore terms 1 + k) suppressed 2 e qγ µ t A v(p 2 ) p by powers of k: 1.ɛ = ig s ū(p 1 )e ( q γ µ t A v(p 2 ) p 1.k } M q qg ū(p 1 )ie q γ µ t A p1 {{.ɛ v(p 2 ) g s p 1.k p } ) pure QED spinor structure 2.ɛ p/v(p) = 0, p 2.k p//k + /kp/ = 2p.k p 2 k,ε

36 QCD lecture 1 (p. 24) Soft-collinear emission Start with γ q q: M q q = ū(p 1 )ie q γ µ v(p 2 ) Soft gluon amplitude ie γ µ p 1 Emit a gluon: p 2 M q qg = ū(p 1 )ig s ɛ/t A i p/ 1 + /k ie qγ µ v(p 2 ) i ū(p 1 )ie q γ µ p/ 2 + /k ig s ɛ/t A v(p 2 ) ie γ µ p 1 k,ε ie γ µ p 1 p 2 k,ε Make gluon soft k p 1,2 ; ignore terms suppressed by powers of k: ( M q qg ū(p 1 )ie q γ µ t A p1.ɛ v(p 2 ) g s p 1.k p ) 2.ɛ p 2.k p 2 p/v(p) = 0, p//k + /kp/ = 2p.k

37 QCD lecture 1 (p. 25) Soft-collinear emission Squared amplitude Mq 2 qg ( ū(p 1)ie q γ µ t A p1.ɛ v(p 2 ) g s p 1.k p ) 2.ɛ 2 p 2.k A,pol = M 2 q q C F g 2 s ( p1 p 1.k p ) 2 2 = Mq 2 p 2.k q C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Include phase space: dφ q qg M 2 q qg (dφ q q M 2 q q ) d 3 k 2E(2π) 3 C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Note property of factorisation into hard q q piece and soft-gluon emission piece, ds. ds = EdE dcos θ dφ 2π 2α sc F 2p 1.p 2 π (2p 1.k)(2p 2.k) θ θ p1 k φ = azimuth

38 QCD lecture 1 (p. 25) Soft-collinear emission Squared amplitude Mq 2 qg ( ū(p 1)ie q γ µ t A p1.ɛ v(p 2 ) g s p 1.k p ) 2.ɛ 2 p 2.k A,pol = M 2 q q C F g 2 s ( p1 p 1.k p ) 2 2 = Mq 2 p 2.k q C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Include phase space: dφ q qg M 2 q qg (dφ q q M 2 q q ) d 3 k 2E(2π) 3 C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Note property of factorisation into hard q q piece and soft-gluon emission piece, ds. ds = EdE dcos θ dφ 2π 2α sc F 2p 1.p 2 π (2p 1.k)(2p 2.k) θ θ p1 k φ = azimuth

39 QCD lecture 1 (p. 25) Soft-collinear emission Squared amplitude Mq 2 qg ( ū(p 1)ie q γ µ t A p1.ɛ v(p 2 ) g s p 1.k p ) 2.ɛ 2 p 2.k A,pol = M 2 q q C F g 2 s ( p1 p 1.k p ) 2 2 = Mq 2 p 2.k q C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Include phase space: dφ q qg M 2 q qg (dφ q q M 2 q q ) d 3 k 2E(2π) 3 C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Note property of factorisation into hard q q piece and soft-gluon emission piece, ds. ds = EdE dcos θ dφ 2π 2α sc F 2p 1.p 2 π (2p 1.k)(2p 2.k) θ θ p1 k φ = azimuth

40 QCD lecture 1 (p. 25) Soft-collinear emission Squared amplitude Mq 2 qg ( ū(p 1)ie q γ µ t A p1.ɛ v(p 2 ) g s p 1.k p ) 2.ɛ 2 p 2.k A,pol = M 2 q q C F g 2 s ( p1 p 1.k p ) 2 2 = Mq 2 p 2.k q C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Include phase space: dφ q qg M 2 q qg (dφ q q M 2 q q ) d 3 k 2E(2π) 3 C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Note property of factorisation into hard q q piece and soft-gluon emission piece, ds. ds = EdE dcos θ dφ 2π 2α sc F 2p 1.p 2 π (2p 1.k)(2p 2.k) θ θ p1 k φ = azimuth

41 QCD lecture 1 (p. 25) Soft-collinear emission Squared amplitude Mq 2 qg ( ū(p 1)ie q γ µ t A p1.ɛ v(p 2 ) g s p 1.k p ) 2.ɛ 2 p 2.k A,pol = M 2 q q C F g 2 s ( p1 p 1.k p ) 2 2 = Mq 2 p 2.k q C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Include phase space: dφ q qg M 2 q qg (dφ q q M 2 q q ) d 3 k 2E(2π) 3 C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Note property of factorisation into hard q q piece and soft-gluon emission piece, ds. ds = EdE dcos θ dφ 2π 2α sc F 2p 1.p 2 π (2p 1.k)(2p 2.k) θ θ p1 k φ = azimuth

42 QCD lecture 1 (p. 25) Soft-collinear emission Squared amplitude Mq 2 qg ( ū(p 1)ie q γ µ t A p1.ɛ v(p 2 ) g s p 1.k p ) 2.ɛ 2 p 2.k A,pol = M 2 q q C F g 2 s ( p1 p 1.k p ) 2 2 = Mq 2 p 2.k q C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Include phase space: dφ q qg M 2 q qg (dφ q q M 2 q q ) d 3 k 2E(2π) 3 C F gs 2 2p 1.p 2 (p 1.k)(p 2.k) Note property of factorisation into hard q q piece and soft-gluon emission piece, ds. ds = EdE dcos θ dφ 2π 2α sc F 2p 1.p 2 π (2p 1.k)(2p 2.k) θ θ p1 k φ = azimuth

43 QCD lecture 1 (p. 26) Soft-collinear emission Soft & collinear gluon emission Take squared matrix element and rewrite in terms of E, θ, 2p 1.p 2 (2p 1.k)(2p 2.k) = 1 E 2 (1 cos 2 θ) So final expression for soft gluon emission is NB: ds = 2α sc F π de E dθ sin θ dφ 2π It diverges for E 0 infrared (or soft) divergence It diverges for θ 0 and θ π collinear divergence Soft, collinear divergences derived here in specific context of But they are a very general property of QCD

44 QCD lecture 1 (p. 26) Soft-collinear emission Soft & collinear gluon emission Take squared matrix element and rewrite in terms of E, θ, 2p 1.p 2 (2p 1.k)(2p 2.k) = 1 E 2 (1 cos 2 θ) So final expression for soft gluon emission is NB: ds = 2α sc F π de E dθ sin θ dφ 2π It diverges for E 0 infrared (or soft) divergence It diverges for θ 0 and θ π collinear divergence Soft, collinear divergences derived here in specific context of But they are a very general property of QCD

45 QCD lecture 1 (p. 26) Soft-collinear emission Soft & collinear gluon emission Take squared matrix element and rewrite in terms of E, θ, 2p 1.p 2 (2p 1.k)(2p 2.k) = 1 E 2 (1 cos 2 θ) So final expression for soft gluon emission is NB: ds = 2α sc F π de E dθ sin θ dφ 2π It diverges for E 0 infrared (or soft) divergence It diverges for θ 0 and θ π collinear divergence Soft, collinear divergences derived here in specific context of But they are a very general property of QCD

46 QCD lecture 1 (p. 26) Soft-collinear emission Soft & collinear gluon emission Take squared matrix element and rewrite in terms of E, θ, 2p 1.p 2 (2p 1.k)(2p 2.k) = 1 E 2 (1 cos 2 θ) So final expression for soft gluon emission is NB: ds = 2α sc F π de E dθ sin θ dφ 2π It diverges for E 0 infrared (or soft) divergence It diverges for θ 0 and θ π collinear divergence Soft, collinear divergences derived here in specific context of But they are a very general property of QCD

47 QCD lecture 1 (p. 27) Total X-sct Real-virtual cancellations: total X-sctn Total cross section: sum of all real and virtual diagrams p 1 2 ie γ µ ie γ µ ie γ k,ε µ + x p 2 Total cross section must be finite. If real part has divergent integration, so must virtual part. (Unitarity, conservation of probability) ( σ tot = σ q q 1 + 2α sc F de dθ R(E/Q, θ) π E sin θ 2α ) sc F de dθ V (E/Q, θ) π E sin θ R(E/Q, θ) parametrises real matrix element for hard emissions, E Q. V (E/Q, θ) parametrises virtual corrections for all momenta.

48 QCD lecture 1 (p. 27) Total X-sct Real-virtual cancellations: total X-sctn Total cross section: sum of all real and virtual diagrams p 1 2 ie γ µ ie γ µ ie γ k,ε µ + x p 2 Total cross section must be finite. If real part has divergent integration, so must virtual part. (Unitarity, conservation of probability) ( σ tot = σ q q 1 + 2α sc F de dθ R(E/Q, θ) π E sin θ 2α ) sc F de dθ V (E/Q, θ) π E sin θ R(E/Q, θ) parametrises real matrix element for hard emissions, E Q. V (E/Q, θ) parametrises virtual corrections for all momenta.

49 QCD lecture 1 (p. 27) Total X-sct Real-virtual cancellations: total X-sctn Total cross section: sum of all real and virtual diagrams p 1 2 ie γ µ ie γ µ ie γ k,ε µ + x p 2 Total cross section must be finite. If real part has divergent integration, so must virtual part. (Unitarity, conservation of probability) ( σ tot = σ q q 1 + 2α sc F de dθ R(E/Q, θ) π E sin θ 2α ) sc F de dθ V (E/Q, θ) π E sin θ R(E/Q, θ) parametrises real matrix element for hard emissions, E Q. V (E/Q, θ) parametrises virtual corrections for all momenta.

50 QCD lecture 1 (p. 28) Total X-sct σ tot = σ q q ( 1 + 2α sc F π de E dθ sin θ Total X-section (cont.) ) (R(E/Q, θ) V (E/Q, θ)) From calculation: lim E 0 R(E/Q, θ) = 1. For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel: Result: lim (R V ) = 0, E 0 lim (R V ) = 0 θ 0,π corrections to σ tot come from hard (E Q), large-angle gluons Soft gluons don t matter: Physics reason: soft gluons emitted on long timescale 1/(Eθ 2 ) relative to collision (1/Q) cannot influence cross section. Transition to hadrons also occurs on long time scale ( 1/Λ) and can also be ignored. Correct renorm. scale for α s : µ Q perturbation theory valid.

51 QCD lecture 1 (p. 28) Total X-sct σ tot = σ q q ( 1 + 2α sc F π de E dθ sin θ Total X-section (cont.) ) (R(E/Q, θ) V (E/Q, θ)) From calculation: lim E 0 R(E/Q, θ) = 1. For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel: Result: lim (R V ) = 0, E 0 lim (R V ) = 0 θ 0,π corrections to σ tot come from hard (E Q), large-angle gluons Soft gluons don t matter: Physics reason: soft gluons emitted on long timescale 1/(Eθ 2 ) relative to collision (1/Q) cannot influence cross section. Transition to hadrons also occurs on long time scale ( 1/Λ) and can also be ignored. Correct renorm. scale for α s : µ Q perturbation theory valid.

52 QCD lecture 1 (p. 28) Total X-sct σ tot = σ q q ( 1 + 2α sc F π de E dθ sin θ Total X-section (cont.) ) (R(E/Q, θ) V (E/Q, θ)) From calculation: lim E 0 R(E/Q, θ) = 1. For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel: Result: lim (R V ) = 0, E 0 lim (R V ) = 0 θ 0,π corrections to σ tot come from hard (E Q), large-angle gluons Soft gluons don t matter: Physics reason: soft gluons emitted on long timescale 1/(Eθ 2 ) relative to collision (1/Q) cannot influence cross section. Transition to hadrons also occurs on long time scale ( 1/Λ) and can also be ignored. Correct renorm. scale for α s : µ Q perturbation theory valid.

53 QCD lecture 1 (p. 28) Total X-sct σ tot = σ q q ( 1 + 2α sc F π de E dθ sin θ Total X-section (cont.) ) (R(E/Q, θ) V (E/Q, θ)) From calculation: lim E 0 R(E/Q, θ) = 1. For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel: Result: lim (R V ) = 0, E 0 lim (R V ) = 0 θ 0,π corrections to σ tot come from hard (E Q), large-angle gluons Soft gluons don t matter: Physics reason: soft gluons emitted on long timescale 1/(Eθ 2 ) relative to collision (1/Q) cannot influence cross section. Transition to hadrons also occurs on long time scale ( 1/Λ) and can also be ignored. Correct renorm. scale for α s : µ Q perturbation theory valid.

54 QCD lecture 1 (p. 28) Total X-sct σ tot = σ q q ( 1 + 2α sc F π de E dθ sin θ Total X-section (cont.) ) (R(E/Q, θ) V (E/Q, θ)) From calculation: lim E 0 R(E/Q, θ) = 1. For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel: Result: lim (R V ) = 0, E 0 lim (R V ) = 0 θ 0,π corrections to σ tot come from hard (E Q), large-angle gluons Soft gluons don t matter: Physics reason: soft gluons emitted on long timescale 1/(Eθ 2 ) relative to collision (1/Q) cannot influence cross section. Transition to hadrons also occurs on long time scale ( 1/Λ) and can also be ignored. Correct renorm. scale for α s : µ Q perturbation theory valid.

55 QCD lecture 1 (p. 28) Total X-sct σ tot = σ q q ( 1 + 2α sc F π de E dθ sin θ Total X-section (cont.) ) (R(E/Q, θ) V (E/Q, θ)) From calculation: lim E 0 R(E/Q, θ) = 1. For every divergence R(E/Q, θ) and V (E/Q, θ) should cancel: Result: lim (R V ) = 0, E 0 lim (R V ) = 0 θ 0,π corrections to σ tot come from hard (E Q), large-angle gluons Soft gluons don t matter: Physics reason: soft gluons emitted on long timescale 1/(Eθ 2 ) relative to collision (1/Q) cannot influence cross section. Transition to hadrons also occurs on long time scale ( 1/Λ) and can also be ignored. Correct renorm. scale for α s : µ Q perturbation theory valid.

56 QCD lecture 1 (p. 29) Total X-sct total X-section (cont.) Dependence of total cross section on only hard gluons is reflected in good behaviour of perturbation series: ( σ tot = σ q q α ( ) s(q) αs (Q) 2 ( ) ) αs (Q) π π π (Coefficients given for Q = M Z )

57 QCD lecture 1 (p. 30) How many gluons are emitted? Let s look at more exclusive quantities structure of final state

58 QCD lecture 1 (p. 31) How many gluons are emitted? Naive gluon multiplicity Let s try and integrate emission probability to get the mean number of gluons emitted off a a quark with energy Q: N g 2α sc F π Q de E π/2 dθ θ This diverges unless we cut the integral off for transverse momenta (k t Eθ) below some non-perturbative threshold, Q 0 Λ QCD. On the grounds that perturbation no longer applies for k t Λ QCD Language of quarks and gluons becomes meaningless With this cutoff, result is: N g α sc F π ln2 Q Q 0 + O (α s ln Q)

59 QCD lecture 1 (p. 31) How many gluons are emitted? Naive gluon multiplicity Let s try and integrate emission probability to get the mean number of gluons emitted off a a quark with energy Q: N g 2α sc F π Q de E π/2 dθ θ Θ(Eθ > Q 0) This diverges unless we cut the integral off for transverse momenta (k t Eθ) below some non-perturbative threshold, Q 0 Λ QCD. On the grounds that perturbation no longer applies for k t Λ QCD Language of quarks and gluons becomes meaningless With this cutoff, result is: N g α sc F π ln2 Q Q 0 + O (α s ln Q)

60 QCD lecture 1 (p. 32) How many gluons are emitted? Naive gluon multiplicity (cont.) Suppose we take Q 0 = Λ QCD, how big is the result? Let s use α s = α s (Q) = 1/(2b ln Q/Λ) [Actually, over most of integration range this is optimistically small] N g α sc F π ln2 Q Λ QCD C F 2bπ ln Q Λ QCD NB: given form for α s, this is actually 1/α s Put in some numbers: Q = 100 GeV, Λ QCD 0.2 GeV, C F = 4/3, b 0.6, N g 2.2 Perturbation theory assumes that first-order term, α s should be 1. But the final result is 1/α s > 1... Is perturbation theory completely useless?

61 QCD lecture 1 (p. 32) How many gluons are emitted? Naive gluon multiplicity (cont.) Suppose we take Q 0 = Λ QCD, how big is the result? Let s use α s = α s (Q) = 1/(2b ln Q/Λ) [Actually, over most of integration range this is optimistically small] N g α sc F π ln2 Q Λ QCD C F 2bπ ln Q Λ QCD NB: given form for α s, this is actually 1/α s Put in some numbers: Q = 100 GeV, Λ QCD 0.2 GeV, C F = 4/3, b 0.6, N g 2.2 Perturbation theory assumes that first-order term, α s should be 1. But the final result is 1/α s > 1... Is perturbation theory completely useless?

62 QCD lecture 1 (p. 33) How many gluons are emitted? Given this failure of first-order perturbation theory, two possible avenues. 1. Continue calculating the next order(s) and see what happens 2. Try to see if there exist other observables for which perturbation theory is better behaved

63 QCD lecture 1 (p. 34) How many gluons are emitted? k p θ k p Gluon emission from quark: 2α sc F π Gluon emission from gluon: 2α sc A π de E de E dθ θ dθ θ Both expressions valid only if θ 1 and energy soft relative to parent Same divergence structures, regardless of where gluon is emitted from All that changes is the colour factor (C F = 4/3 v. C A = 3) Expect low-order structure (α s ln 2 Q) to be replicated at each new order

64 QCD lecture 1 (p. 35) How many gluons are emitted? Picturing a QCD event q q Start of with q q

65 QCD lecture 1 (p. 35) How many gluons are emitted? Picturing a QCD event q q A gluon gets emitted at small angles

66 QCD lecture 1 (p. 35) How many gluons are emitted? Picturing a QCD event q q It radiates a further gluon

67 QCD lecture 1 (p. 35) How many gluons are emitted? Picturing a QCD event q q And so forth

68 QCD lecture 1 (p. 35) How many gluons are emitted? Picturing a QCD event q q Meanwhile the same happened on other side of event

69 QCD lecture 1 (p. 35) How many gluons are emitted? Picturing a QCD event q q And then a non-perturbative transition occurs

70 QCD lecture 1 (p. 35) How many gluons are emitted? Picturing a QCD event q π, K, p,... q Giving a pattern of hadrons that remembers the gluon branching Hadrons mostly produced at small angle wrt q q directions or with low energy

71 QCD lecture 1 (p. 36) How many gluons are emitted? Gluon v. hadron multiplicity It turns out you can calculate the gluon multiplicity analytically, by summing all orders (n) of perturbation theory: N g n 1 (n!) 2 ( CA πb ln Q ) n Λ exp 4CA πb ln Q Λ Compare to data for hadron multiplicity (Q s) Including some other higher-order terms and fitting overall normalisation Agreement is amazing! charged hadron multiplicity in e + e events adapted from ESW

72 QCD lecture 1 (p. 37) Infrared and Collinear safety It s great that putting together all orders of gluon emission works so well! This, together with a hadronisation model, is part of what s contained in Monte Carlo event generators like Pythia, Herwig & Sherpa. But are there things that we can calculate about the final state using just one or two orders perturbation theory?

73 QCD lecture 1 (p. 38) Infrared and Collinear safety Infrared and Collinear Safety (definition) For an observable s distribution to be calculable in [fixed-order] perturbation theory, the observable should be infra-red safe, i.e. insensitive to the emission of soft or collinear gluons. In particular if p i is any momentum occurring in its definition, it must be invariant under the branching p i p j + p k whenever p j and p k are parallel [collinear] or one of them is small [infrared]. [QCD and Collider Physics (Ellis, Stirling & Webber)] Examples Multiplicity of gluons is not IRC safe Energy of hardest particle is not IRC safe [modified by soft/collinear splitting] [modified by collinear splitting] Energy flow into a cone is IRC safe [soft emissions don t change energy flow collinear emissions don t change its direction]

74 QCD lecture 1 (p. 38) Infrared and Collinear safety Infrared and Collinear Safety (definition) For an observable s distribution to be calculable in [fixed-order] perturbation theory, the observable should be infra-red safe, i.e. insensitive to the emission of soft or collinear gluons. In particular if p i is any momentum occurring in its definition, it must be invariant under the branching p i p j + p k whenever p j and p k are parallel [collinear] or one of them is small [infrared]. [QCD and Collider Physics (Ellis, Stirling & Webber)] Examples Multiplicity of gluons is not IRC safe Energy of hardest particle is not IRC safe [modified by soft/collinear splitting] [modified by collinear splitting] Energy flow into a cone is IRC safe [soft emissions don t change energy flow collinear emissions don t change its direction]

75 QCD lecture 1 (p. 39) Infrared and Collinear safety Sterman-Weinberg jets The original (finite) jet definition An event has 2 jets if at least a fraction (1 ɛ) of event energy is contained in two cones of half-angle δ. ( σ 2 jet = σ q q 1 + 2α sc F π ( ( ) de dθ E R E sin θ Q, θ ( ( ) ) E 1 Θ Q ɛ Θ(θ δ) V For small E or small θ this is just like total cross section full cancellation of divergences between real and virtual terms. δ ( ))) E Q, θ For large E and large θ a finite piece of real emission cross section is cut out. Overall final contribution dominated by scales Q cross section is perturbatively calculation.

76 QCD lecture 1 (p. 39) Infrared and Collinear safety Sterman-Weinberg jets The original (finite) jet definition An event has 2 jets if at least a fraction (1 ɛ) of event energy is contained in two cones of half-angle δ. ( σ 2 jet = σ q q 1 + 2α sc F π ( ( ) de dθ E R E sin θ Q, θ ( ( ) ) E 1 Θ Q ɛ Θ(θ δ) V For small E or small θ this is just like total cross section full cancellation of divergences between real and virtual terms. δ ( ))) E Q, θ For large E and large θ a finite piece of real emission cross section is cut out. Overall final contribution dominated by scales Q cross section is perturbatively calculation.

77 QCD lecture 1 (p. 39) Infrared and Collinear safety Sterman-Weinberg jets The original (finite) jet definition An event has 2 jets if at least a fraction (1 ɛ) of event energy is contained in two cones of half-angle δ. ( σ 2 jet = σ q q 1 + 2α sc F π ( ( ) de dθ E R E sin θ Q, θ ( ( ) ) E 1 Θ Q ɛ Θ(θ δ) V For small E or small θ this is just like total cross section full cancellation of divergences between real and virtual terms. δ ( ))) E Q, θ For large E and large θ a finite piece of real emission cross section is cut out. Overall final contribution dominated by scales Q cross section is perturbatively calculation.

78 QCD lecture 1 (p. 40) Infrared and Collinear safety Real 2-jet event Near perfect 2-jet event 2 well-collimated jets of particles. Nearly all energy contained in two cones. Cross section for this to occur is σ 2 jet = σ q q (1 c 1 α s + c 2 α 2 s +...) where c 1, c 2 all 1.

79 QCD lecture 1 (p. 41) Infrared and Collinear safety 3-jet event How many jets? Most of energy contained in 3 (fairly) collimated cones Cross section for this to happen is σ 3 jet = σ q q (c 1α s + c 2α 2 s +...) where the coefficients are all O (1) Cross section for extra gluon diverges Cross section for extra jet is small, O (α s ) NB: Sterman-Weinberg procedure gets complex for multi-jet events. 4th lecture will discuss modern approaches for defining jets.

80 QCD lecture 1 (p. 42) Summary 1st lecture summary QCD at colliders mixes weak and strong coupling No calculation technique is rigorous over that whole domain Gluon emission repaints a quark s colour That implies that gluons carry colour too Quarks emit gluons, which emit other gluons: this gives characteristic shower structure of QCD events, and is the basis of Monte Carlo simulations To use perturbation theory one must measure quantities that are insensitive to the (divergent) soft & collinear splittings, like jets.