Gavin Salam, CERN PSI Summer School Exothiggs, Zuoz, August 2016

INGREDIENTS FOR ACCURATE COLLIDER PHYSICS Gavin Salam, CERN PSI Summer School Exothiggs, Zuoz, August 2016

1 LHC – TWO ROLES – A DISCOVERY MACHINE AND A PRECISION MACHINE

Today ➤ 20 fb-1 at 8 TeV ➤ 13 fb-1 at 13 TeV

Future ➤ 2018: 100 fb-1 @ 13 TeV ATLAS: general purpose CMS: general purpose ➤ 2023: 300 fb-1 @ 1? TeV ➤ 2035: 3000 fb-1 @ 14 TeV

Interconnection between 1 fb-1 = 1014 collisions two “dipoles” (bending ALICE: heavy-ion physicsmagnets) in the LHCb: LHC B-physics tunnel.

Increase in luminosity brings

+TOTEM,LHCf discovery reach and precision LHC – TWO ROLES – A DISCOVERY MACHINE AND A PRECISION MACHINE

Post/pre-dictionsZ’ exclusion for sequential reach Z' v.exclusion lumi reach 8 GPSreference & Weiler (ATLAS) 14 TeV 7 cern.ch/collider-reachextrapolations 2035 [preliminary plot] ATLAS / CDF 6 CMS / D0 2023 13 TeV 5 2018

4 mid 2016 end 2012

Z' reach [TeV] 3 8 TeV ATLAS: general purpose CMS: general purpose 7 TeV 2 − 1.96 TeV, pp 1

Interconnection between 0 two “dipoles” (bending 0.01 0.1 1 10 100 1000 10000 ALICE: heavy-ion physicsmagnets) in the LHCb: LHC B-physics -1 tunnel. integrated lumi [fb ]

Increase in luminosity brings

+TOTEM,LHCf discovery reach and precision LHC – TWO ROLES – A DISCOVERY MACHINE AND A PRECISION MACHINE

Higgs couplings

ATLAS and CMS H → γγ 3 LHC Run 1 f VBF+VH H → ZZ µ H → WW H → ττ 2 H → bb

ATLAS: general purpose CMS: general purpose 0

−1 68% CL Best fit SM expected Interconnection between two “dipoles” (bending 0 1 2 3 ALICE: heavy-ion physics LHCb: B-physics µf magnets) in the LHC ggF+ttH tunnel.

f f Figure 14: Negative log-likelihood contours at 68% CL in the (µggF+ttH, µVBF+VH) plane for the combination of ATLAS and CMS, as obtained from the ten-parameter fit described in the text for each of the five decay channels H ZZ, H WW, H , H ⌧⌧, and H bb. The best fit values obtained for each of the five decay ! ! ! ! ! channels are also shown, together with the SM expectation.

mass measurementsIncrease in the di↵erent in channels. luminosity Several BSM models predict, brings for example, a superposition of states with indistinguishable mass values [121–124], possibly with di↵erent coupling structures to the +TOTEM,LHCf SM particles.discovery With such an assumption, reach it may be possible and to distinguish precision between single and multiple states by measuring the cross sections of individual production processes independently for each decay mode, as described in Section 4.1.1. Several methods have been proposed to assess the compatibility of the data with a single state [125, 126]. A test for the possible presence of overlapping Higgs boson states is performed, based on a proﬁle likelihood ratio suggested in Ref. [127]. This test accounts both for missing measurements, such as the H bb decay mode in the ggF and VBF production processes, ! and for uncertainties in the measurements, including their correlations.

35 LONG-TERM HIGGS PRECISION?

�� NAIVELY EXTRAPOLATE�� 7+8 TEV RESULTS (based on lumi and σ) official HL-LHC �� forecasts �� today’s �� TH syst ��

�� CMS naively extrapolate 7/8 TeV results�� (based on lumi and 50% TH syst. ATLAS �� no TH syst.

�� Extrapolation suggests that �� we get value from full lumi only if we aim for O(1%) �� µ or better precision

�� 3 σ �� µ ) ��

Naive extrapolation suggests LHC has long-term potential to do Higgs physics at 1% accuracy 5 Phenomenology: lecture 1 (9/101) Recall of SM (EW part) Higgs mechanism THE HIGGS SECTOR Higgs ﬁelds: complex scalar doublet The theory is old (1960s-70s). V( ) + φ φ µ φ = , H =(D φ)†(Dµφ) V (φ) But the particle and it’s theory are φ0 L − unlike anything we’ve seen in nature. ! " Potential has form ➤ A fundamental scalar φ, i.e. spin 0 2 2 (all other particles are spin 1 or 1/2) V (φ)= µ φ†φ + λ(φ†φ) − 2 † † 2 ➤ + Α potential V(φ) ~ -μ (φφ ) + λ(φφ ) , 0 |φ | which leads to a Vacuum Ex- which until now was limited to being |φ | pectation Value (VEV): φ = 4 2 | | theorists’ “toy model” (φ ) µ /2λ = v/√2. SU(2) symmetry of conﬁgurations with#φ = v/√2. Choose gauge ➤ ”Yukawa” interactions responsible for | | transformation (unitary gauge) to map fermion masses, y i ¯ , with couplings (y ) spanning 5 orders of magnitude 0 i φ → (v + H)/√2 ! "

6 Phenomenology: lecture 1 (9/101) Recall of SM (EW part) Higgs mechanism THE HIGGS SECTOR Higgs ﬁelds: complex scalar doublet The theory is old (1960s-70s). V( ) + φ φ µ φ = , H =(D φ)†(Dµφ) V (φ) But the particle and it’s theory are φ0 L − unlike anything we’ve seen in nature. ! " Potential has form ➤ A fundamental scalar φ, i.e. spin 0 2 2 (all other particles are spin 1 or 1/2) V (φ)= µ φ†φ + λ(φ†φ) − 2 † † 2 ➤ + Α potential V(φ) ~ -μ (φφ ) + λ(φφ ) , 0 |φ | which leads to a Vacuum Ex- which until now was limited to being |φ | pectation Value (VEV): φ = 4 2 | | theorists’ “toy model” (φ ) µ /2λ = v/√2. SU(2)Higgs symmetry sector of conﬁgurationsneeds with#φ = v/√2. Choose gauge ➤ ”Yukawa” interactions responsible for | | transformationstress-testing (unitary gauge) to map fermion masses, y i ¯ , with couplings (y ) spanning 5 orders of magnitude 0 i φ Is Higgs fundamental or composite?→ (v + H)/√2 If fundamental, is it “minimal”?! " 4 Is it really φ ? Are Yukawa couplings responsible for all fermion masses?

6 up) interactions are modelled by overlaying minimum-bias interactions generated using Pythia8. Further detail of all MC generators and cross sections used is given in Ref. [19].

4 Event selection

This section describes the reconstruction-level definition of the signal region. The definition of physics objects reconstructed in the detector follows that of Ref. [19] exactly and is summarised here. All objects are defined with respect to a primary interaction vertex, which is required to have at least three associated tracks with pT 400 MeV. If more than one such vertex is present, the one with the largest value of 2 (pT), where the sum is over all tracks associated with that vertex, is selected as the primary vertex. P 4.1 Object reconstruction and identification

Electron candidates are built from clusters of energy depositions in the EM calorimeter with an associated well-reconstructed track. They are required to have ET > 10 GeV, where the transverse energy ET is defined as E sin(✓). Electrons reconstructed with ⌘ < 2.47 are used, excluding 1.37 < ⌘ < 1.52, which A three-level trigger system reduces the event rate| to| about 400 Hz [21]. The Level-1| trigger| is imple- corresponds to the transition region between the barrel and the endcap calorimeters. Additional identi- mented in hardware and uses a subset of detector information to reduce the event rate to a design value fication criteria are applied to reject background, using the calorimeter shower shape, the quality of the of at most 75 kHz. The two subsequent trigger levels, collectively referred to as the High-Level Trigger match between the track and the cluster, and the amount of transition radiation emitted in the ID [46–48]. (HLT), are implemented in software. ForATLAS electrons H → withWW* 10 GeV ANALYSIS< ET < 25[1604.02997] GeV, a likelihood-based electron selection at the “very tight” oper- ating point is used for its improved background rejection. For ET > 25 GeV, a more ecient “medium” selection is used because background is less of a concern. The eciency of these requirements varies 3strongly Signal as a and function background of ET, starting models from 65–70% for ET < 25 GeV, jumping to about 80% with the change in identification criteria at ET = 25 GeV, and then steadily increasing as a function of ET [47]. The ggF and VBF production modes for H WW are modelled at next-to-leading order (NLO) in the Muon candidates are selected from tracks! reconstructed⇤ in the ID matched to tracks reconstructed in strong coupling ↵ with the Powheg MC generator [22–25], interfaced with Pythia8[26] (version 8.165) the muon spectrometer.S Tracks in both detectors are required to have a minimum number of hits to for the parton shower, hadronisation, and underlying event. The CT10 [27] PDF set is used and the para- ensure robust reconstruction. Muons are required to have ⌘ < 2.5 and p > 10 GeV. The reconstruction meters of the Pythia8 generator controlling the modelling| of| the parton showerT and the underlying event eciency is between 96% and 98%, and stable as a function of p [49]. are those corresponding to the AU2 set [28]. The Higgs boson massT set in the generation is 125.0 GeV, whichAdditional is close criteria to the are measured applied to value. electrons The and Powheg muonsggF to reducemodel takes backgrounds into account from finitenon-prompt quark massesleptons andand a electromagnetic running-width Breit–Wigner signatures produced distribution by hadronic that includes activity. electroweak Lepton isolation corrections is atdefined NLO [ using29]. To track- im- provebased the and modelling calorimeter-based of the Higgs quantities. boson p AllT distribution, isolation variables a reweighting used are scheme normalised is applied relative to reproduce to the trans- the predictionverse momentum of the next-to-next-to-leading-order of the lepton, and are optimised (NNLO) for the andH next-to-next-to-leading-logarithmWW e⌫µ⌫ analysis, resulting in (NNLL) stricter ! ⇤! dynamic-scalecriteria for better calculation background given rejection by the HR ates lower2.1 programp and looser [30]. Events criteria with for better2 jets e areciency further at reweighted higher p . T T H owheg / toSimilarly, reproduce requirements the pT spectrum on the predicted transverse by impact-parameter the NLO P significancesimulation ofd0 Higgsd0 and boson the production longitudinal in im- as- sociationpact parameter with twoz0 are jets made. (H + The2 jets) eciency [31]. Interference of the isolation with and continuum impact-parameterWW production requirements [32, 33 for] has elec- a negligibletrons satisfying impact all on of this the identification analysis due to criteria the transverse-mass requirements ranges selection from criteria 68% for described 10 GeV in< SectionET < 154 GeVand isto not greater included than in 90% the forsignal electrons model. with ET > 25 GeV. For muons, the equivalent eciencies are 60– 96%. The inclusive cross sections at ps = 8 TeV for a Higgs boson mass of 125.0 GeV, calculated at NNLO+NNLL inJets QCD are reconstructed and NLO in fromthe electroweak topological couplings, clusters of are calorimeter 19.3 pb cells and 1.58 [50–52 pb] for using ggF the and anti- VBFkt algorithm respect- ivelywith a [34 radius]. The parameter uncertainty of onR = the0. ggF4[53 cross]. Jet section energies has are approximately corrected for equal the e↵ contributionsects of calorimeter from QCD non- scalecompensation, variations signal (7.5%) losses and PDFs due to (7.2%). noise threshold For the VBF e↵ects, production, energy lost the in uncertainty non-instrumented on the crossregions, section con- istributions 2.7%, mainly from in-time from PDF and variations. out-of-time The pile-up,WH and andZH theprocesses position areof the modelled primary with interaction Pythia8 vertex and norm- [50, alised54]. Subsequently, to cross sections the jets of 0.70 are pb calibrated and 0.42 to pb the respectively, hadronic energy calculated scale at [50 NNLO, 55]. inTo QCD reduce and the NLO chance in the of7 electroweak couplings [34]. The uncertainty is 2.5% on the WH cross section and 4.0% on the ZH cross section.

For all of the background processes, with the exception6 of W + jets and multijet events, MC simulation is used to model event kinematics and as an input to the background normalisation. The W + jets and multijet background models are derived from data as described in Section 5. For the dominant WW and top-quark backgrounds, the MC generator is Powheg +Pythia6[35] (version 6.426), also with CT10 for the input PDFs. The Perugia 2011 parameter set is used for Pythia6[36]. For the WW background with N 2, to better model the additional partons, the Sherpa [37] program (version 1.4.3) with the CT10 jet PDF set is used. The Drell–Yan background, including Z/ ⌧⌧, is simulated with the Alpgen [38] ⇤ ! program (version 2.14). It is interfaced with Herwig [39] (version 6.520) with parameters set to those of the ATLAS Underlying Event Tune 2 [40] and uses the CTEQ6L1 [41] PDF set. The same conﬁguration is applied for W events. Events in the Z/⇤ sample are reweighted to the MRSTmcal PDF set [42]. For the W⇤ and Z/ backgrounds, the Sherpa program is used, with the same version number and PDF set as the WW background with 2 jets. Additional diboson backgrounds, from WZ and ZZ, are modelled using Powheg +Pythia8. For all MC samples, the ATLAS detector response is simulated [43] using either Geant4[44] or Geant4 combined with a parameterised Geant4-based calorimeter simulation [45]. Multiple proton–proton (pile-

5 ATLAS H →WW* ANALYSIS [1604.02997]

400 300 ATLAS Data SM bkg (sys ⊕ stat) That ATLASwhole Data SM bkg (sys ⊕ stat) H WW H WW 350 -1 -1 s = 8 TeV, 20.3 fb Other VV W+jet paragraph250 s = 8 TeV, was 20.3 fb just Top Other VV Top Z/γ* W+jet Z/γ* 300 H→WW*→eνµν, 0 jets H→WW*→eνµν, 1 jet Multijet for the red part of Multijet 200 Events / 5 GeV 250 Events / 10 GeV this distribution 200 (the150 Higgs signal). 150 Complexity100 of 100 modelling50 each of the 50 backgrounds is

60 Data-Bkg comparable60 Data-Bkg SM bkg (sys ⊕ stat) SM bkg (sys ⊕ stat) 40 H 40 H 20 20 Data - Bkg 0 Data - Bkg 0 -200 50 100 150 200 250 0 50 100 150 200 250

mT [GeV] mT [GeV]

(a) Njet = 0 (b) Njet =81

140 ATLAS Data SM bkg (sys ⊕ stat) H Top -1 120 s = 8 TeV, 20.3 fb WW Z/γ* H→WW*→eνµν, ≥ 2j Other VV W+jet Multijet 100 Events / 10 GeV 80

40 Data-Bkg SM bkg (sys ⊕ stat) 20 H 0 Data - Bkg -20 0 50 100 150 200 250

mT [GeV] (c) N 2 jet

Figure 1: Observed distributions of mT with signal and background expectations after all other selection criteria have been applied for the N = 0 (top left), N = 1 (top right) and N 2 (bottom) signal regions. The background jet jet jet contributions are normalised as described in Section 5. The SM Higgs boson signal prediction shown is summed over all production processes. The hatched band shows the sum in quadrature of statistical and systematic uncertainties of the sum of the backgrounds. The vertical dashed lines indicate the lower and upper selection boundaries on mT at 85 and 125 GeV.

half the number of events that ggF does, and constitute about 3% of the total background. The Njet distribution and other shapes are taken from simulation.

For the Njet = 0 and Njet = 1 categories, the WW background is normalised using control regions distin- guished from the SR primarily by m``, and the shape is taken from simulated events generated using Powheg +Pythia6 as described in Section 3. For the N 2 category, WW is normalised using the NLO jet

10 AIMS OF THESE LECTURES

➤ Give you basic understanding of the “jargon” of theoretical collider prediction methods and inputs ➤ Give you insight into the power & limitations of diﬀerent techniques for making collider predictions

9 A proton-proton collision: INITIAL STATE

proton proton

10 A proton-proton collision: FINAL STATE

+ − B µ π K ...

+ µ B −

(actual final-state multiplicity ~ several hundred hadrons) 11 QCD lecture 1 (p. 12) Basic methods What do Feynman rules mean physically? Perturbation theory

A, A, IT’SQCD MOSTLY lecture 1 (p. QUANTUM 5) CHROMODYNAMICS (QCD) µ µ What is QCD Lagrangian + colour

ψ1 Quarks — 3 colours: ψa = ψ ⎛ 2 ⎞ ψ3 ba ba ¯ A µ Quark part of Lagrangian: ⎝ ⎠ ψb(−igs tbaγ )ψa ¯ µ µ C C ( 0 1 0 ) 010 1 Lq = ψa(iγ ∂µδab − gs γ tabAµ − m)ψb 100 0 2 1 8 ⎛ ⎞ ⎛ ⎞ SU(3) local gauge symmetry ↔ 8(=3 − 1) generators tab ...tab 000 0 corresponding to 8 gluons A1 ...A8 . ⎝ ⎠ ⎝ ⎠ µ µ ¯ 1 ψa ψb tab A 1 A A representation is: t = 2 λ , ! "# $ ! "# $ ! "# $ 010 0 −i 0 100 Agluonemission001 repaints the quark colour. λ1 = ⎛ 100⎞ , λ2 = ⎛ i 00⎞ , λ3 = ⎛ 0 −10⎞ , λ4 = ⎛ 000A gluon⎞ itself, carries colour and anti-colour. ⎝ 000⎠ ⎝ 000⎠ ⎝ 000⎠ ⎝ 100⎠

−i 1 00 00 000 00 0 ⎛√3 ⎞ λ5 ⎛ ⎞ , λ6 ⎛ ⎞ , λ7 ⎛ −i ⎞ , λ8 0 1 0 , = 00 0 = 001 = 00 = √3 2 i 00 010 0 i 0 ⎜ 00− ⎟ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ √3 ⎠ 12 QCD lecture 1 (p. 11) QCD lecture 1 (p. 11) Basic methods QCD lecture 1 (p. 6) Basic methods What is QCD Lagrangian:Perturbation gluonic part theory Perturbation theory Perturbation theory Perturbation theory

Relies on idea of order-by-orderIT’S MOSTLY QUANTUM expansion CHROMODYNAMICS small (QCD) coupling, αs ≪ 1 Relies on idea of order-by-order expansion small coupling, αs ≪ 1 2 3 Field tensor:α2 F A = ∂αA3A − ∂ AA − g...f AB AC [tA, tB ]=if tC αs + α s +µν αµ sν +ν ν ...s ABC µ ν ABC αs + s + s + fABC are structure constants of SU(3) (antisymmetric in all indices — small smallerABC negligible? SU(2)small equivalentsmaller was ϵ ). Needednegligible? for gauge invariance of gluon part of Lagrangian: 1 L = − F µν F A µν !"#$ !"#$ G !"#$4 A InteractionInteraction vertices vertices of of Feynman Feynman!"#$ rules: rules:!"#$ !"#$ A, µ A, µ D, σ A, µ A, µ A, µ D, σ A, µ p p r r C, ρ C, ρ q C, ρ B, ν q C, ρ B, ν a b B, ν 2 XAC XBD µν ρσ a b B, ν XAC XBD A µ ABC ρ µν −ig2s f f [gµν gρσ − −ig tA γµ −g fABC [(p − q)ρ gµν −igs fµσ νγf [g g − s ba s µσ νγ −igs tbaγ −gs f [(p −µq)νρg g g ]+(C, γ) ↔ +(q − r)µ gνρ g g ]+(C, γ) ↔ +(q − r) g (D, ρ)+(B, ν) ↔ (C, γ) 13 ν ρµ (D, ρ)+(B, ν) ↔ (C, γ) +(+(rr−−pp))νggρµ]]

TheseThese expressions expressions are are fairly fairly complex, complex, soso you you really really don’t don’t want want to to have have to to deal deal withwith too too many many orders orders of of them! them! i.e.i.e. ααss hadhad better better be be small. small. . . . . IT’S MOSTLY QUANTUM CHROMODYNAMICS (QCD)

The only complete solution uses lattice QCD ➤ put all quark & gluon fields on a 4d lattice (NB: imaginary time) ➤ Figure out most likely configurations (Monte Carlo sampling) image credit fdecomite [flickr]

14 IT’S MOSTLY QUANTUM CHROMODYNAMICS (QCD)

The only complete solution hadron spectrum from lattice QCD uses lattice QCD ➤ put all quark & gluon ﬁelds on a 4d lattice (NB: imaginary time) ➤ Figure out most likely conﬁgurations

(Monte Carlo sampling) Durr et al, arXiv:0906.3599

15 IT’S MOSTLY QUANTUM CHROMODYNAMICS (QCD)

The only complete solution hadron spectrum from lattice QCD uses lattice QCD ➤ put all quark & gluon ﬁelds on a 4d lattice (NB: imaginary time) ➤ Figure out most likely conﬁgurations

(Monte Carlo sampling) Durr et al, arXiv:0906.3599

For LHC reactions, lattice would have to ➤ Resolve smallest length scales (2 TeV ~ 10-4 fm) ➤ Contain whole reaction (pion formed on timescale of 1fm, with boost of 10000 — i.e. 104 fm) That implies 108 nodes in each dimension, i.e. 1032 nodes — unrealistic 15 A proton-proton collision: FILLING IN THE PICTURE

+ − B µ π K ...

+ µ B −

proton proton

16 A proton-proton collision: FILLING IN THE PICTURE

+ − B µ π K ... _ + µ b B − b Z H

σ _ u u

proton proton

17 A proton-proton collision: SIMPLIFYING IN THE PICTURE

_ µ− b + µ b Z H

σ _ u u

proton proton

18 WHY IS SIMPLIFICATION “ALLOWED”? KEY IDEA #1 FACTORISATION

➤ Proton’s dynamics occurs on timescale O(1 fm) Final-state hadron dynamics occurs on timescale O(1fm) ➤ Production of Higgs, Z (and other “hard processes”) occurs on timescale + 1/MH ~ 1/125 GeV ~ 0.002 fm − B µ π K ... _ + µ b B − b Z H

σ _ u u

proton proton

That means we can separate — “factorise” — the hard process, i.e. treat it as independent from all the hadronic dynamics 19 WHY IS SIMPLIFICATION “ALLOWED”? KEY IDEA #2 SHORT-DISTANCE QCD CORRECTIONS ARE PERTURBATIVE

➤ On timescales 1/MH ~ 1/125 GeV ~ 0.002 fm you can take advantage of asymptotic freedom ➤ i.e. you can write results in terms of an expansion in the (not

so) strong coupling constant αs(125 GeV) ~ 0.11

ˆ =ˆ (1 + c ↵ + c ↵2 + ) 0 1 s 2 s ··· _ µ− b + LO µ b Z H (Leading Order) σ _ u u

20 proton proton WHY IS SIMPLIFICATION “ALLOWED”? KEY IDEA #2 SHORT-DISTANCE QCD CORRECTIONS ARE PERTURBATIVE

➤ On timescales 1/MH ~ 1/125 GeV ~ 0.002 fm you can take advantage of asymptotic freedom ➤ i.e. you can write results in terms of an expansion in the (not

so) strong coupling constant αs(125 GeV) ~ 0.11

ˆ =ˆ (1 + c ↵ + c ↵2 + ) 0 1 s 2 s ··· _ µ− b + NLO µ b Z H (Next-to-Leading Order) σ _ u u

21 proton proton WHY IS SIMPLIFICATION “ALLOWED”? KEY IDEA #2 SHORT-DISTANCE QCD CORRECTIONS ARE PERTURBATIVE

➤ On timescales 1/MH ~ 1/125 GeV ~ 0.002 fm you can take advantage of asymptotic freedom ➤ i.e. you can write results in terms of an expansion in the (not

so) strong coupling constant αs(125 GeV) ~ 0.11

ˆ =ˆ (1 + c ↵ + c ↵2 + ) 0 1 s 2 s ··· _ µ− b + NNLO µ b Z H (Next-to-next-to-Leading Order) σ _ u u

22 proton proton 8 1. Quantum chromodynamics

The PDFs’ resulting dependence on µF is described by the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) equations [43], which to leading order (LO) read∗ 2 2 1 ∂fi/p x, µ αs µ dz x µ2 F = F P (1) (z) f ,µ2 , (1.14) F 2 2π z i←j j/p z F ∂!µF " ! " x #j $ % & (1) 2 2 with, for example, Pq←g(z)=TR(z +(1 z) ). The other LO splitting functions are listed − 2 3 in Sec. 16 of this Review,whileresultsuptoNLO,αs,andNNLO,αs,aregiveninRefs. 44 and 45 respectively. Beyond LO, the coefficient functions are also µF dependent, for µ2 example C(1)(x, Q2,µ2 ,µ2 )=C(1)(x, Q2,µ2 ,Q2) ln F 1 dz C(0)( x )P (1) (z). 2,i R F 2,i R − Q2 j x z 2,j z j←i As with the renormalization scale, the choice of factorizat! " 'ion( scale is arbitrary, but if one has an infinite number of terms in the perturbative series, the µF -dependences of the coefficient functions and PDFs will compensate each other fully. Given only N terms of the series, a residual (αN+1)uncertaintyisassociatedwiththeambiguityin O s the choice of µF .AswithµR,varyingµF provides an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choice for the scales is usually µR = µF = Q. As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered to be part of the proton’s structure, while above it is considered to be part of the proton’s structure and evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with massless coefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near the threshold, it is instead necessaryTHE MASTER to appropriately EQUATION adapt the standard massive coefficient functions to account for the heavy-quark contribution already included in the PDFs [46,47,48]. Hadron-hadron collisions. The extension to processes with two initial-state hadrons can be illustrated with the example of the total (inclusive) cross section for W boson production in collisions of hadrons h1 and h2,whichcanbewrittenas ∞ n 2 2 2 σ (h1h2 ZHW + X)= α µ dx1dx2 f x1,µ f x2,µ → s R i/h1 F j/h2 F n#=0 % & #i,j $ % & % & 2 (n) 2 2 Λ σîj→WZH+X+X x1x2s, µR,µF + 4 , (1.15) × O )M * % & W ∗ LO is generally taken to mean the_ lowest order at which a quantity is non-zero. This µ− definition is nearly always unambiguous,b the one major exception being for the case of the hadronic branching ratio+ of virtual photons, Z, τ, etc.,forwhichtwoconventionsexist: µ b LO can either mean the lowest order that contributes to the hadronic branching fraction, Z H i.e. the term “1” in Eq. (1.7); or it can mean the lowest order at which the hadronic branching ratio becomes sensitive to the coupling, n =1inEq.(1.8), as is relevant when σ _ extracting the value of theu coupling from a measurementu of thebranchingratio.Because of this ambiguity, we avoid use of the term “LO” in that context.

May 5, 2016 21:57 proton proton 23 8 1. Quantum chromodynamics

The PDFs’ resulting dependence on µF is described by the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) equations [43], which to leading order (LO) read∗ 2 2 1 ∂fi/p x, µ αs µ dz x µ2 F = F P (1) (z) f ,µ2 , (1.14) F 2 2π z i←j j/p z F ∂!µF " ! " x #j $ % & (1) 2 2 with, for example, Pq←g(z)=TR(z +(1 z) ). The other LO splitting functions are listed − 2 3 in Sec. 16 of this Review,whileresultsuptoNLO,αs,andNNLO,αs,aregiveninRefs. 44 and 45 respectively. Beyond LO, the coefficient functions are also µF dependent, for µ2 example C(1)(x, Q2,µ2 ,µ2 )=C(1)(x, Q2,µ2 ,Q2) ln F 1 dz C(0)( x )P (1) (z). 2,i R F 2,i R − Q2 j x z 2,j z j←i As with the renormalization scale, the choice of factorizat! " 'ion( scale is arbitrary, but if one has an infinite number of terms in the perturbative series, the µF -dependences of the coefficient functions and PDFs will compensate each other fully. Given only N terms of the series, a residual (αN+1)uncertaintyisassociatedwiththeambiguityin O s the choice of µF .AswithµR,varyingµF provides an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choice for the scales is usually µR = µF = Q. As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered to be part of the proton’s structure, while above it is considered to be part of the proton’s structure and evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with massless coefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near the threshold, it is instead necessaryTHE MASTER to appropriately EQUATION adapt the standard massive coefficient functions to account for the heavy-quark contribution already included in the PDFs [46,47,48]. Hadron-hadron collisions. The extension to processes with two initial-state hadrons can be illustrated with the example of the total (inclusive) cross section for W boson production in collisions of hadrons h1 and h2,whichcanbewrittenas ∞ n 2 2 2 σ (h1h2 ZHW + X)= α µ dx1dx2 f x1,µ f x2,µ → s R i/h1 F j/h2 F n#=0 % & #i,j $ % & % & 2 (n) 2 2 Λ σîj→WZH+X+X x1x2s, µR,µF + 4 , (1.15) × O )M * % & W ∗ LO is generally taken to mean the_ lowest order at which a quantity is non-zero. This µ− definition is nearly always unambiguous,b the one major exception being for the case of the hadronic branching ratio+ of virtual photons, Z, τ, etc.,forwhichtwoconventionsexist: µ b LO can either mean the lowest order that contributes toParton the ha dronicdistribution branching function fraction, Z H i.e. the term “1” in Eq. (1.7); or it can mean the lowest(PDF order): e.g. at number which of the up hadronicanti- branching ratio becomes sensitive to the coupling, n =1inEq.(1quarks carrying.8), as fraction is relevant x2 of when σ _ extracting the value of theu coupling from a measurementu of thproton’sebranchingratio.Because momentum of this ambiguity, we avoid use of the term “LO” in that context.

May 5, 2016 21:57 proton proton 24 8 1. Quantum chromodynamics

The PDFs’ resulting dependence on µF is described by the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) equations [43], which to leading order (LO) read∗ 2 2 1 ∂fi/p x, µ αs µ dz x µ2 F = F P (1) (z) f ,µ2 , (1.14) F 2 2π z i←j j/p z F ∂!µF " ! " x #j $ % & (1) 2 2 with, for example, Pq←g(z)=TR(z +(1 z) ). The other LO splitting functions are listed − 2 3 in Sec. 16 of this Review,whileresultsuptoNLO,αs,andNNLO,αs,aregiveninRefs. 44 and 45 respectively. Beyond LO, the coefficient functions are also µF dependent, for µ2 example C(1)(x, Q2,µ2 ,µ2 )=C(1)(x, Q2,µ2 ,Q2) ln F 1 dz C(0)( x )P (1) (z). 2,i R F 2,i R − Q2 j x z 2,j z j←i As with the renormalization scale, the choice of factorizat! " 'ion( scale is arbitrary, but if one has an infinite number of terms in the perturbative series, the µF -dependences of the coefficient functions and PDFs will compensate each other fully. Given only N terms of the series, a residual (αN+1)uncertaintyisassociatedwiththeambiguityin O s the choice of µF .AswithµR,varyingµF provides an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choice for the scales is usually µR = µF = Q. As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered to be part of the proton’s structure, while above it is considered to be part of the proton’s structure and evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with massless coefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near the threshold, it is instead necessaryTHE MASTER to appropriately EQUATION adapt the standard massive coefficient functions to account for the heavy-quark contribution already included in the PDFs [46,47,48]. Hadron-hadron collisions. The extension to processes with two initial-state hadrons can be illustrated with the example of the total (inclusive) cross section for W boson production in collisions of hadrons h1 and h2,whichcanbewrittenas ∞ n 2 2 2 σ (h1h2 ZHW + X)= α µ dx1dx2 f x1,µ f x2,µ → s R i/h1 F j/h2 F n#=0 % & #i,j $ % & % & 2 (n) 2 2 Λ σîj→WZH+X+X x1x2s, µR,µF + 4 , (1.15) × O )M * % & W ∗ LO is generally taken to mean the_ lowest order at which a quantity is non-zero. This µ− definition is nearly always unambiguous,b the one major exception being for the case of the hadronic branching ratio+ of virtual photons, Z, τ, etc.,forwhichtwoconventionsexist: µ b LO can either mean the lowest order that contributes toParton the ha dronicdistribution branching function fraction, Z H i.e. the term “1” in Eq. (1.7); or it can mean the lowest(PDF order): e.g. at number which of the up hadronicanti- branching ratio becomes sensitive to the coupling, n =1inEq.(1quarks carrying.8), as fraction is relevant x1 of when σ _ extracting the value of theu coupling from a measurementu of thproton’sebranchingratio.Because momentum of this ambiguity, we avoid use of the term “LO” in that context.

May 5, 2016 21:57 proton proton 25 8 1. Quantum chromodynamics

The PDFs’ resulting dependence on µF is described by the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) equations [43], which to leading order (LO) read∗ 2 2 1 ∂fi/p x, µ αs µ dz x µ2 F = F P (1) (z) f ,µ2 , (1.14) F 2 2π z i←j j/p z F ∂!µF " ! " x #j $ % & (1) 2 2 with, for example, Pq←g(z)=TR(z +(1 z) ). The other LO splitting functions are listed − 2 3 in Sec. 16 of this Review,whileresultsuptoNLO,αs,andNNLO,αs,aregiveninRefs. 44 and 45 respectively. Beyond LO, the coefficient functions are also µF dependent, for µ2 example C(1)(x, Q2,µ2 ,µ2 )=C(1)(x, Q2,µ2 ,Q2) ln F 1 dz C(0)( x )P (1) (z). 2,i R F 2,i R − Q2 j x z 2,j z j←i As with the renormalization scale, the choice of factorizat! " 'ion( scale is arbitrary, but if one has an infinite number of terms in the perturbative series, the µF -dependences of the coefficient functions and PDFs will compensate each other fully. Given only N terms of the series, a residual (αN+1)uncertaintyisassociatedwiththeambiguityin O s the choice of µF .AswithµR,varyingµF provides an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choice for the scales is usually µR = µF = Q. As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered to be part of the proton’s structure, while above it is considered to be part of the proton’s structure and evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with massless coefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near the threshold, it is instead necessaryTHE MASTER to appropriately EQUATION adapt the standard massive coefficient functions to account for the heavy-quark contribution already included in the PDFs [46,47,48]. Hadron-hadron collisions. The extensionPerturbative to processes sum with over two powers initial-state of the hadrons can be illustrated with the example of thestrong total coupling: (inclusive) typicallycross section we use forfirstW 2-3boson production in collisions of hadrons h1 and h2,whichcanbewrittenasorders ∞ n 2 2 2 σ (h1h2 ZHW + X)= α µ dx1dx2 f x1,µ f x2,µ → s R i/h1 F j/h2 F n#=0 % & #i,j $ % & % & 2 (n) 2 2 Λ σîj→WZH+X+X x1x2s, µR,µF + 4 , (1.15) × O )M * % & W ∗ LO is generally taken to mean the_ lowest order at which a quantity is non-zero. This µ− definition is nearly always unambiguous,b the one major exception being for the case of the hadronic branching ratio+ of virtual photons, Z, τ, etc.,forwhichtwoconventionsexist: µ b LO can either mean the lowest order that contributes to the hadronic branching fraction, Z H i.e. the term “1” in Eq. (1.7); or it can mean the lowest order at which the hadronic branching ratio becomes sensitive to the coupling, n =1inEq.(1.8), as is relevant when σ _ extracting the value of theu coupling from a measurementu of thebranchingratio.Because of this ambiguity, we avoid use of the term “LO” in that context.

May 5, 2016 21:57 proton proton 26 8 1. Quantum chromodynamics

The PDFs’ resulting dependence on µF is described by the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) equations [43], which to leading order (LO) read∗ 2 2 1 ∂fi/p x, µ αs µ dz x µ2 F = F P (1) (z) f ,µ2 , (1.14) F 2 2π z i←j j/p z F ∂!µF " ! " x #j $ % & (1) 2 2 with, for example, Pq←g(z)=TR(z +(1 z) ). The other LO splitting functions are listed − 2 3 in Sec. 16 of this Review,whileresultsuptoNLO,αs,andNNLO,αs,aregiveninRefs. 44 and 45 respectively. Beyond LO, the coefficient functions are also µF dependent, for µ2 example C(1)(x, Q2,µ2 ,µ2 )=C(1)(x, Q2,µ2 ,Q2) ln F 1 dz C(0)( x )P (1) (z). 2,i R F 2,i R − Q2 j x z 2,j z j←i As with the renormalization scale, the choice of factorizat! " 'ion( scale is arbitrary, but if one has an infinite number of terms in the perturbative series, the µF -dependences of the coefficient functions and PDFs will compensate each other fully. Given only N terms of the series, a residual (αN+1)uncertaintyisassociatedwiththeambiguityin O s the choice of µF .AswithµR,varyingµF provides an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choice for the scales is usually µR = µF = Q. As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered to be part of the proton’s structure, while above it is considered to be part of the proton’s structure and evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with massless coefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near the threshold, it is instead necessaryTHE MASTER to appropriately EQUATION adapt the standard massive coefficient functions to account for the heavy-quark contribution already included in the PDFs [46,47,48]. Hadron-hadron collisions. The extension to processes with two initial-state hadrons can be illustrated with the example of the total (inclusive) cross section for W boson production in collisions of hadrons h1 and h2,whichcanbewrittenas ∞ n 2 2 2 σ (h1h2 ZHW + X)= α µ dx1dx2 f x1,µ f x2,µ → s R i/h1 F j/h2 F n#=0 % & #i,j $ % & % & 2 (n) 2 2 Λ σîj→WZH+X+X x1x2s, µR,µF + 4 , (1.15) × O )M * % & W ∗ LO is generally taken to mean the_ lowest order at which a quantity is non-zero. This µ− At each perturbative order n definition is nearly always unambiguous,b the one major exception being for the case of the hadronic branching ratio+ of virtual photons, Z, τ, etc.,forwhichtwoconventionsexist:we have a specific “hard µ b LO can either mean the lowest order that contributes tomatrix the ha element”dronic branching (sometimes fraction, Z H i.e. the term “1” in Eq. (1.7); or it can mean the lowestseveral orderfor different at which subprocesses) the hadronic branching ratio becomes sensitive to the coupling, n =1inEq.(1.8), as is relevant when σ _ extracting the value of theu coupling from a measurementu of thebranchingratio.Because of this ambiguity, we avoid use of the term “LO” in that context.

May 5, 2016 21:57 proton proton 27 8 1. Quantum chromodynamics

The PDFs’ resulting dependence on µF is described by the Dokshitzer-Gribov-Lipatov- Altarelli-Parisi (DGLAP) equations [43], which to leading order (LO) read∗ 2 2 1 ∂fi/p x, µ αs µ dz x µ2 F = F P (1) (z) f ,µ2 , (1.14) F 2 2π z i←j j/p z F ∂!µF " ! " x #j $ % & (1) 2 2 with, for example, Pq←g(z)=TR(z +(1 z) ). The other LO splitting functions are listed − 2 3 in Sec. 16 of this Review,whileresultsuptoNLO,αs,andNNLO,αs,aregiveninRefs. 44 and 45 respectively. Beyond LO, the coefficient functions are also µF dependent, for µ2 example C(1)(x, Q2,µ2 ,µ2 )=C(1)(x, Q2,µ2 ,Q2) ln F 1 dz C(0)( x )P (1) (z). 2,i R F 2,i R − Q2 j x z 2,j z j←i As with the renormalization scale, the choice of factorizat! " 'ion( scale is arbitrary, but if one has an infinite number of terms in the perturbative series, the µF -dependences of the coefficient functions and PDFs will compensate each other fully. Given only N terms of the series, a residual (αN+1)uncertaintyisassociatedwiththeambiguityin O s the choice of µF .AswithµR,varyingµF provides an input in estimating uncertainties on predictions. In inclusive DIS predictions, the default choice for the scales is usually µR = µF = Q. As is the case for the running coupling, in DGLAP evolution one can introduce flavor thresholds near the heavy quark masses: below a given heavy quark’s mass, that quark is not considered to be part of the proton’s structure, while above it is considered to be part of the proton’s structure and evolves with massless DGLAP splitting kernels. With appropriate parton distribution matching terms at threshold, such a variable flavor number scheme (VFNS), when used with massless coefficient functions, gives the full heavy-quark contributions at high Q2 scales. For scales near the threshold, it is instead necessaryTHE MASTER to appropriately EQUATION adapt the standard massive coefficient functions to account for the heavy-quark contribution already included in the PDFs [46,47,48]. Hadron-hadron collisions. The extension to processes with two initial-state hadrons can be illustrated with the example of the total (inclusive) cross section for W boson production in collisions of hadrons h1 and h2,whichcanbewrittenas ∞ n 2 2 2 σ (h1h2 ZHW + X)= α µ dx1dx2 f x1,µ f x2,µ → s R i/h1 F j/h2 F n#=0 % & #i,j $ % & % & 2 (n) 2 2 Λ σîj→WZH+X+X x1x2s, µR,µF + 4 , (1.15) × O )M * % & W ∗ LO is generally taken to mean the_ lowest order at which a quantity is non-zero. This µ− Additional corrections from definition is nearly always unambiguous,b the one major exception being for the case of the hadronic branching ratio+ of virtual photons, Z, τ, etc.non-perturbative,forwhichtwoconventionsexist: effects (higher µ b LO can either mean the lowest order that contributes to“twist”, the ha suppresseddronic branching by powers fraction, of Z H i.e. the term “1” in Eq. (1.7); or it can mean the lowestQCD order scale at(Λ which) / hard the scale) hadronic branching ratio becomes sensitive to the coupling, n =1inEq.(1.8), as is relevant when σ _ extracting the value of theu coupling from a measurementu of thebranchingratio.Because of this ambiguity, we avoid use of the term “LO” in that context.

May 5, 2016 21:57 proton proton 28 THE STRONG COUPLING

29 QCD lecture 1 (p. 15) Basic methods How big is the coupling? RUNNINGPerturbation COUPLING theory All couplings run (QED, QCD, EW), i.e. they depend on the momentum scale (Q2) of your process.

2 The QCD coupling, αs(Q ), runs fast: ∂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 2004 Novel prize: Gross, Politzer & Wilczek ! At high scales Q,couplingbecomessmall ➥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.

CA = 3, nf = number of light quark flavours; Q (→ μR) is the “renormalisation scale” 30 QCD lecture 1 (p. 16) THEBasic STRONG methods COUPLING V. SCALE Running coupling (cont.) Perturbation theory ∂α α (Q2) 1 Solve Q2 s = −b α2 ⇒ α (Q2)= s 0 = ∂Q2 0 s s 2 Q2 Q2 1+b0αs(Q0 )ln 2 b0 ln 2 Q0 Λ

40 1. Quantum chromodynamics Λ ≃ 0.2GeV(akaΛQCD )isthe fundamental scale of QCD, at which April 2016 α 2 τ decays (N3LO) coupling blows up. s(Q ) DIS jets (NLO) Heavy Quarkonia (NLO) – 0.3 e+e jets & shapes (res. NNLO) ! e.w. precision fits (NNLO) Λ sets the scale for hadron masses (–) pp –> jets (NLO) (NB: Λ not unambiguously pp –> tt (NNLO) deﬁned wrt higher orders) 0.2

! Perturbative calculations valid for 0.1 scales Q ≫ Λ. QCD αs(Mz) = 0.1181 ± 0.0011 1 10 100 1000 Q [GeV]

QCD αs(Mz) = 0.1181 ± 0.0013

Figure 1.3: Summary of measurements of αs as a function of the energy scale Q. PDG World Average: αs(MTheZ) respective = 0.1181 degree of QCD perturbation ± 0.0011 theory used in (0.9%) the extraction of αs is indicated in brackets (NLO: next-to-leading order; NNLO: next-to-next-to leading order; res. NNLO: NNLO matched with resummed next-to-leading logs; N313LO: next-to-NNLO).

5. R. Brock et al.,[CTEQCollab.],Rev.Mod.Phys.67,157(1995),seealso http://www.phys.psu.edu/~cteq/handbook/v1.1/handbook.pdf. 6. A.S. Kronfeld and C. Quigg, Am. J. Phys. 78,1081(2010). 7. T. Plehn, Lect. Notes Phys. 844,1(2012). 8. R. Stock (Ed.), Relativistic Heavy Ion Physics, Springer-Verlag Berlin, Heidelberg, 2010. 9. Proceedings of the XXIV International Conference on Ultrarelativistic Nucleus– Nucleus Collisions, Quark Matter 2014,Nucl.Phys.A,volume931. 10. R. Hwa and X. N. Wang (Ed.), Quark Gluon Plasma 4, World Scientiﬁc Publishing Comparny, 2010. 11. T. van Ritbergen, J.A.M. Vermaseren, and S.A. Larin, Phys. Lett. B400,379(1997). 12. M. Czakon, Nucl. Phys. B710,485(2005). 13. W.A. Bardeen et al.,Phys.Rev.D18,3998(1978).

May 5, 2016 21:57 τ

Baikov -decays Davier Pich STRONG-COUPLING DETERMINATIONS Boito SM review Bethke, Dissertori & GPS in PDG ‘16

HPQCD (Wilson loops) ➤ HPQCD (c-c correlators) Most consistent set of independent lattice Maltmann (Wilson loops) determinations is from lattice JLQCD (Adler functions) PACS-CS (vac. pol. fctns.) ETM (ghost-gluon vertex) ➤ BBGPSV (static energy) Two best determinations are from same

functions structure ABM group (HPQCD, 1004.4285, 1408.4169) BBG αs(MZ) = 0.1183 ± 0.0007 (0.6%) JR NNPDF [heavy-quark correlators] MMHT

e+e- annihilation e+e- αs(MZ) = 0.1183 ± 0.0007 (0.6%) ALEPH (jets&shapes) OPAL(j&s) [Wilson loops] JADE(j&s) Dissertori (3j) JADE (3j) ➤ Many determinations quote small DW (T) Abbate (T) uncertainties (≲1%). All are disputed! Gehrm. (T) Hoang (C) ➤ electroweak Some determinations quote GFitter precision fts CMS hadron anomalously small central values (tt cross section) collider (~0.113 v. world avg. of 0.1181±0.0011). Also disputed 32 PARTON DISTRIBUTION FUNCTIONS (PDFs)

33 [PDFs] DEEP[DIS kinematics] INELASTIC SCATTERING Deep Inelastic Scattering: kinematics

Hadron-hadron is complex because of two incoming partons — so start with simpler Deep Inelastic Scattering (DIS).

Kinematic relations:

"(1−y)k" Q2 p.q x = ; y = ; Q2 = xys e+ 2p.q p.k k q (Q 2 = −q2) ps = c.o.m. energy 2 I Q = photon virtuality transverse $ resolution at which it probes proton xp structure p I x = longitudinal momentum fraction of struck parton in proton proton I y = momentum fraction lost by electron (in proton rest frame)

Gavin Salam (CERN) QCD basics 3 3/32 34 [PDFs] DEEP[DIS kinematics] INELASTIC SCATTERINGDeep Inelastic scattering (DIS): example

2 2 Q = 25030 GeV ; y = 0:56; x=0.50

e+ proton

e+ jet Q2 x

proton jet

Gavin Salam (CERN) QCD basics 3 4/32 35 F2 gives us combination of u and d. How can we extract them separately?

DEEP[PDFs] INELASTIC SCATTERING [DIS X-sections] E.g.: extracting u & d distributions

Write DIS X-section to zeroth order in ↵s (‘quark parton model’):

d 2em 4⇡↵2 1+(1 y)2 F em + (↵ ) dxdQ2 ' xQ4 2 2 O s ✓ ◆ F em [structure function] / 2

4 1 F = x(e2u(x)+e2d(x)) = x u(x)+ d(x) 2 u d 9 9 ✓ ◆ [u(x), d(x): parton distribution functions (PDF)] NB:

I use perturbative language for interactions of up and down quarks I but distributions themselves have a non-perturbative origin.

36 Gavin Salam (CERN) QCD basics 3 5/32 [Initial-state splitting] [1st order analysis] Summary so far

I Collinear divergence for incoming partons not cancelled by virtuals. PARTON DISTRIBUTION AND DGLAPReal and virtual have di↵erent longitudinal momenta

I Situation analogous to renormalization: need to regularize (but in IR ➤ Write up-quark distribution in proton as instead of UV). Technically, often2 done with dimensional regularization u(x, µF ) I Physical sense of regularization is to separate (factorize) proton

➤non-perturbativeμF is the factorisation dynamics scale from — perturbativea bit like the hardrenormalisation cross section. scale 2 2 2 (μR) for theChoice running of factorization coupling. scale, µ ,isarbitrarybetween1GeV and Q I ➤InAs analogy you vary with the running factorisation coupling, scale, we the can partonvary factorizationdistributions scale evolveand get a renormalizationwith a renormalisation-group group equation typefor equation parton distribution functions. Dokshizer Gribov Lipatov Altarelli Parisi equations (DGLAP)

u u g u increase increase d g d g g d 2 u 2 Q Q u u g u u

Gavin Salam (CERN) QCD basics 3 13 / 32

Dokshitzer-Gribov-Lipatov-Altarelli-Parisi (DGLAP) equations 37 [Initial-state splitting] [DGLAP] DGLAP equation (q q) DGLAP EQUATION Change convention: (a) now ﬁx outgoing longitudinal momentum x;(b) take derivative wrt factorization scale µ2

(1+δ)µ2 (1+δ)µ2 x x µ 2 µ 2 +

x/z x(1−z)/z x p p

dq(x,µ2) ↵ 1 q(x/z,µ2) ↵ 1 = s dz p (z) s dz p (z) q(x,µ2) d ln µ2 2⇡ qq z 2⇡ qq Zx Z0

1+z2 p is real q q splitting kernel: p (z)=C qq qq F 1 z 2CF Until now we approximated it in soft (z 1) limit, pqq 1 z ! ' Gavin Salam (CERN) QCD basics 3 14 / 32 38 [Initial-state splitting] [DGLAP] DGLAP rewritten DGLAP EQUATION

Awkward to write real and virtual parts separately. Use more compact notation:

dq(x,µ2) ↵ 1 q(x/z,µ2) 1+z2 = s dz P (z) , P = C d ln µ2 2⇡ qq z qq F 1 z Zx ✓ ◆+ Pqq q ⌦ This involves the plus| prescription:{z }

1 1 1 dz [g(z)] f (z)= dz g(z) f (z) dz g(z) f (1) + Z0 Z0 Z0 z = 1 divergences of g(z)cancellediff (z)suciently smooth at z =1

Gavin Salam (CERN) QCD basics 3 15 / 32 39 DGLAP[Initial-state EQUATION splitting] [DGLAP] DGLAP flavour structure Proton contains both quarks and gluons — so DGLAP is a matrix in flavour space: d q Pq q Pq g q 2 = d ln Q g Pg q Pg g ⌦ g ✓ ◆ ✓ ◆ ✓ ◆ [In general, matrix spanning all flavors, anti-flavors, Pqq0 =0(LO),Pqg¯ = Pqg ] Splitting functions are: 1+(1 z)2 P (z)=T z2 +(1 z)2 , P (z)=C , qg R gq F z  ⇥ z ⇤1 z (11C 4n T ) P (z)=2C + + z(1 z) + (1 z) A f R . gg A (1 z) z 6  + Have various symmetries / significant properties, e.g.

I P , P : symmetric z 1 z (except virtuals) qg gg $ I P , P : diverge for z 1 soft gluon emission qq gg ! I P , P : diverge for z 0 Implies PDFs grow for x 0 gg gq ! !

2015 EPS HEP prize to Bjorken, Altarelli, Dokshitzer, Lipatov & Parisi

Gavin Salam (CERN) QCD basics 3 16 / 32 40 [Initial-state splitting] [DGLAP]NLO DGLAP Higher-order calculations

NLO:

20 1 8 56 P (1)(x)=4C n 2+6x 4H + x2 H +(1+x) 5H 2H ps F f 0 0 0 0,0 ✓ 9 x  3 9  ◆ ↵s (0) Pab = P + 20 1 44 218 P (1)(x)=4C n 2+25x 2p ( x)H 2p (x)H + x2 H 2⇡ qg A f qg 1,0 qg 1,1 0 ✓ 9 x  3 9 ↵2 +4(1 x) H 2H + xH 4⇣ x 6H +9H +4C n 2p (x) H + H + H s (1) 0,0 0 1 2 0,0 0 F f qg 1,0 1,1 2 P  ◆ ✓  2 5 29 15 1 16⇡ ⇣ +4x2 H + H + +2(1 x) H + H 2xH + H H 2 0 0,0 0 0,0 1 0,0 0  2  4 2 2 ◆ Curci, Furmanski 1 11 8 44 P (1)(x)=4C C +2p (x) H + H + H H x2 H +4⇣ 2 gq A F gq 1,0 1,1 2 1 0 2 ✓ x  6  3 9 & Petronzio ’80 37 2 7H +2H 2H x +(1+x) 2H 5H + 2p ( x)H 4 C n x 0 0,0 1 0,0 0 gq 1,0 F f  9 ◆ ✓ 3 2 10 7 7 p (x) H +4C 2 p (x) 3H 2H +(1+x) H + H 3H gq 1 F gq 1 1,1 0,0 0 0,0  3 9 ◆ ✓   2 2 3 +1 H +2H x 0 1 2 ◆

10 13 1 2 2 P (1)(x)=4C n 1 x p (x) x2 (1 + x)H (1 x) +4C 2 27 gg A f gg 0 A ✓ 9 9 ✓ x ◆ 3 3 ◆ ✓ 11 27 67 1 +(1 + x) H +8H +2p ( x) H 2H ⇣ x2 12H 0 0,0 gg 0,0 1,0 2 0  3 2  9 ✓ x ◆ 44 67 8 x2H +2p (x) ⇣ + H +2H +2H + (1 x) +3⇣ +4C n 2H 0 gg 2 0,0 1,0 2 3 F f 0 3  18  3 ◆ ✓ 2 1 10 1 + + x2 12 + (1 + x) 4 5H 2H (1 x) . 0 0,0 3 x 3  2 ◆

Gavin Salam (CERN) QCD basics 3 17 / 32 41 [Initial-state splitting] [DGLAP] NNLO splitting functions NNLO DGLAP

385 31 113 49 5 79 173 1259 2833 1 2 32 4 10 Divergences for x 1areunderstoodinthesenseof -distributions. H H H H H ζ H H H 6H 2H 2H x2 H ζ 2H H H 72 10 2 11 12 1 4 20 2 1 2 6 000 12 3 32 216 0 1110 13 121 x 3 21 9 2 100 3 110 9 11 The third-order pure-singlet contribution to the quark-quark splitting function (2.4), corre- 6H 3H 9H ζ 6H ζ H 3H 4H 3H 6H 8 3 161 2351 2 1 26 28 21 120 10 2 11 2 1100 1110 1111 112 121 H H 6ζ H x2 H H 2H sponding to the anomalous dimension (3.10), is given by 49 17 5 5 9 5 3 3 100 2 10 3 36 1 108 3 x 3 10 9 0 110 6H ζ p x H ζ H H H H H 13 4 2 qg 2 1 3 2 110 2 12 2 10 2 20 2 100 10 65 11 2 4 1 2 13 14 1 2H 12 H1ζ2 H 1ζ2 H2 H111 1 x 15H0000 5H2ζ2 ζ3 H111 Pps x 16C C n x H 1 0 H0 H 1ζ2 H 1 1 0 2H 1 0 0 2H 2H 6H 6H 6H 2H ζ 9H ζ 3H 2H 3 6 6 A F f 3 x 3 9 2 31 4 22 210 200 00 2 2 2 120 121 11 3 5 31 17 551 2 29 113 18691 2 1 2 16 9 6761 571 10 1 6H 6H 6H 9H ζ 9H ζ 2H H H4 H00ζ2 H110 H20 H10 ζ2 H100 H2 H0 H 1 2 x ζ2 H2 1 9ζ3 H1 0 H1 H2 H1ζ2 H1 1 1110 1100 112 10 2 112 120 2 1000 2 2 6 12 20 4 4 72 3 x 3 4 216 72 3 6 2243 265 33 31 23 497 29 143 182 158 397 13 1 2 55 23 4 1 2 2 371 23 H H 19H H H ζ H ζ H 3H 2H 2H 1 x H H H 3H 6H 13 x 4ζ3 H10 H110 x H100 H1 H11 108 6 100 2 200 21 12 11 2 20 36 2 6 1 2 12 3 1 0 0 1 1 0 1 1 1 9 1 3 36 0 0 2 2 0 0 0 0 0 x 12 9 3 x 3 108 9 2 5 16 11 19 1223 43 3011 13 1 1 9 H 1 x 6H 3H H 7H 2H 39H ζ 4H ζ ζ H111 H0ζ2 H1 H000 H00 1 x 8H210 4H 12 H1 0 3xH1 0 H 3 0 H 2ζ2 2H 2 1 0 3H 2 0 0 H0 0ζ2 H1ζ2 H1 0 0 3 111 210 211 6 111 200 12 0 3 2 2 3 3 6 12 72 6 36 6 2 2 4 35 1 15 3 7 31 91 71 113 826 154 899 121 2 607 5 65 29 13 7H H 5H ζ 11H H H ζ 8H 10H H H H 1 x H ζ ζ H H ζ H H1 10 H0ζ2 H0 0 ζ2 H2 H1ζ2 H1 00 H1 0 H1 1 110 6 1 11 2 2 200 3 10 2 1 2 3 1 210 4 1 1 1 1 0 1 1 1 12 0 2 6 3 18 2 12 3 18 2 27 0 3 24 10 36 2 6 12 18 1189 67 949 67 142 215 3989 5 16 31 17 117 2 5 5H2ζ2 4H211 H 30 36H0ζ3 5H2ζ2 2H 12 6H 110 6H210 3H211 and H2 0 H 1 0 6xH 1 0 H0 0 0 H2 1 ζ2 9H0ζ3 H 1ζ2 2H2 1 0 H1 H2 1 29H2 0 ζ2 H0 00 H3 H0 2H 30 2 3 6 6 20 2 108 3 36 2 3 32 48 25 13 27 11 13 17 1 7 1 11H0000 5H31 H111 H 2ζ2 H 200 H 30 H2ζ2 H100 H 2H H ζ H H 2H H H 5H H 1 x H 100 10H 2ζ2 6H 200 2H00ζ2 9H 110 7H 12 9H 20 2H31 4 2 2 2 2 4 2 1 0 0 1 2 2 2 2 2 0 0 1 1 0 2 1 1 3 1 2 4 2 0 2 1 37 5 17 3 1 11 79 67 263 2 1 32 29 235 511 97 33 13H 210 H111 H4 H00ζ2 H12 H110 H20 H10 ζ2 H ζ 2 4H 4H ζ H H ζ H H H 4H 210 4H4 4H30 4H0000 H 10 1 x H 1ζ2 4H 200 2H00ζ2 12 4 4 2 12 8 8 0 0 0 0 2 2 3 0 0 3 9 0 0 12 0 12 2 12 12 1 4 2 3 2 2 9 11 19 9 119 967 305 13375 1889 21 11 11 3 2 83 243 511 97 4 ζ3 H2 H 10 24H0ζ3 H 1ζ2 H0 38H 100 H21 H ζ ζ H 10H x2 H H 10ζ H H H2ζ2 3H1 10 2H0 000 H 30 9H2 10 H2 11 H1 11 H2 00 H1 2 3 24 12 72 18 2 2 0 2 2 3 2 2 0 0 0 0 3 4 0 0 4 0 2 8 8 1 3 2 2 3 2 2 91 5 5 9 39 473 1853 79 217 7 79 4 17 17 31 2 2 H200 H11 H 20 ζ2 H1ζ2 H111 H0ζ2 H1 3H000 4ζ H ζ H H 6H 16C n 2 H 2 H ζ x2 H ζ 3 H0ζ3 8H 2ζ2 H 110 H 12 H 10 H 20 H0ζ2 H00 4 24 2 72 3 12 12 18 3 0 2 3 2 0 2 0 F f 27 0 2 2 3 2 2 2 2 2 2 2 12 48 217 59 169 13 2 167 191 1283 185 145 1553 2 7 11 739 163 7 19 2 1 5 2 1 7 35 185 2 H3 H00 16C n H000 H1 H0 H00 2H0000 H x2 H H 1 x H H xH H ζ3 ζ2 H2 H1ζ2 H100 H10 H11 H1 H21 12 24 F f 6 36 96 24 24 6 0 9 x 1 1 3 1 3 6 1 1 6 1 1 27 0 54 12 4 18 4 3 24 18 108 12 75 170 85 425 7693 3659 5 5 5 5 1 91 35 22 1 4 4 29 85 H1 1 H2 H1 0 ζ2 pqg x H2 1 H0 H0 0 H1 11 6H0 00 1 x H ζ ζ H 2H 2H ζ H H 16C 2n H H20 ζ2 H000 H3 H0 2xxH22 4H30 4H 22 9 9 18 9 6 2 3 3 3 3 2 3 2 3 2 1 3 0 2 6 0 0 0 0 0 F f 12 1 4 9 4 12 192 48 1 229 4 11 1 7 77 1 2 1 6463 16 7 25 583 101 73 73 55 2 ζ3 2H100 H1 x 1 x H1 4H0000 H000 xH11 H H H ζ H H 5H H H ζ x2 16CAnf pqg x H12 H1ζ2 H100 H110 H111 H0 H00 x H2 9 81 x 12 432 3 9 4 0 0 0 0 0 12 0 54 4 2 4 2 3 2 0 2 1 0 2 12 6 18 3 2 6 53 17 11 139 1 53 1 2 7 8 7 3475 103 2 85 22 109 13 28 28 16 16 4 26 2 xH2 xH1 0 xζ2 1 x H0 H0 0 16CF nf pqg x 7H13 7H4 H H H ζ H H ζ H 4H H ζ H0 H00 ζ3 ζ2 pqg x H 100 x 1 x 6H000 9 9 9 216 12 12 1 3 0 0 6 54 0 9 2 9 2 3 0 2 3 3 2 0 3 2 1 3 3 18 6 18 108 3 162 x 9 7 7 7 7 19 5 5 5 22 4 1 23 523 55 1 2H 30 7H1ζ3 5H22 6H30 6H31 H210 4H200 3H21 2H211 H20 H x2 H H 3ζ H H H H xH1 H0 0 xH1 1 x 1 x H 10 H0 H1 H0 00 H1 1 H 10 2 3 0 0 0 3 x 12 1 0 144 1 3 16 2 1 0 0 1 1 1 1 0 1 1 1 6 2 9 4 54 9 9 85 31 17 7 55 31 31 61 61 87 11 61 17 5 5 19 1 7 2743 53 251 5 5 8 2 2 H2 ζ2 H1 H1 2 H1 1 H1 0 7H0 0ζ2 H1 00 H1 10 ζ3 1 x H H H H H ζ H H 3xH 16CA nf pqg x 3H13 H100 H21 ζ2 H110 H3 H1ζ3 8 8 8 2 8 2 2 2 2 2 1 0 0 12 1 1 72 0 12 0 0 12 1 4 2 4 2 3 1 0 1 0 216 6 2 5 12 12 2 5 63 23 155 25 2537 867 23 81 11 11 7 15 87 11 2 1669 5 37 H3 H0ζ2 H1ζ2 H0 00 H0 ζ2 3H1 11 5H2ζ2 7H0ζ3 3H ζ 3H H H 1 x H 4H 7H 10xζ ζ 2 H20 H10 H12 ζ3 H2 H0 H 100 3H4 H111 32 2 2 2 2 8 5 0 2 3 1 1 0 1 1 1 216 2 0 0 0 2 1 2 0 3 10 2 12 8 12 6 24 27 8 2 383 25 3 7 31 103 5 2561 11H00 2H120 7H10ζ2 3H1000 5H11ζ2 4H1100 H1110 2H1111 5H112 H H ζ H ζ 3H ζ H ζ H H H 7H0ζ3 6H0 0ζ2 4H0 0 0 0 H2 0 0 2H2 1 0 2H2 1 1 4H3 0 H3 1 6H4 (4.12) 72 11 2 20 8 2 4 1 2 00 2 12 0 2 216 1 2 1000 72 00 1 5 6H1 20 6H1 21 4pqg x H0 000 H 20 H 110 H 200 H 120 H 10 H1 11 2H2 00 3H1 20 5H1 0ζ2 3H0 00 H1 1ζ2 H1 100 4H1 110 2H1 111 2 8 Due to Eqs. (3.11) and (3.12) the three-loop gluon-quark and quark-gluon splitting functions read 5 1 1 1 2H1 12 2H1 20 pqg x H 11ζ2 2H 12 6H 110 H1 11 2H 2ζ2 H 200 H 100 H 30 H 1ζ2 H 1100 H 1000 21 x H2 10 H2 00 H2 2 2 39 15 9 4 2 2 4 Pqg x 16CACFnf pqg x H1ζ3 4H1 1 1 3H2 0 0 H1 2 H1 1 0 3H2 1 0 727 5 3 2 43 49 13 2 4 4 H H ζ 2H H ζ H 2H 2H H H31 2H30 2H 1ζ2 H12 H100 H110 H2ζ2 ζ2 H2 ζ2 H11 173 551 64 49 3 1 36 10 1 2 22 2 1 3 120 1100 112 2 1000 8 8 8 H ζ 2H 4H ζ H ζ H ζ ζ 2 H H H 0 3 2 1 1 2 2 12 0 2 72 0 0 3 3 2 4 2 2 1 0 0 0 3 1 0 0

16 17 18

655 151 185 1 95 29 171 53 39 13 7 1 1 1 67 43 97 9 33 4 1 1 ζ H H H H H 12H 7H ζ H H 2H H H 4H 4H 16C n 2 H ζ H H 4ζ 2 H 8H H x2 H H 576 6 3 18 1 1 6 1 1 1 9 2 6 2 1 4 1 0 1 0 0 1 2 12 2 4 11 31 6 110 4 200 110 12 F f 9 9 x 12 00 2 3 21 12 1 2 2 3 30 2 000 3 x 2 2 20 5 3 179 2041 19 2 1 1 5 4 11 7 11 19 1 16H H H 4H 35H H H H x xH p x H H 16C 2n x2 H H H H H ζ 2ζ H ζ 4H H H 1 x 9H ζ 1 1 0 3 2 0 2 2 1 1 0 0 0 0 2 0 27 0 144 0 0 6 0 0 0 9 6 1 6 gq 1 1 3 1 F f 9 0 0 6 0 2 10 3 10 20 6 2 3 1 2 110 2 100 12 1 2 13 13 15 2005 157 1291 55 1 83 7 1625 3 293 61 7 857 2H H ζ 13H H H ζ 8ζ H H p x H H H ζ 9ζ H 2H H 2H ζ H 12H H ζ H H 9H ζ 16H 4H 8H ζ 3 0 2 0 2 3 0 2 3 1 2 3 64 4 2 3 432 1 12 1 1 3 gq 12 10 1 2 3 12 11 20 36 1 0 2 48 2 100 0000 108 6 0 2 3 1 0 36 1 0 3 210 200 2 2 3 1 27 11 3 5 5 31 95 1 13051 13 3 1 95 149 3451 H H H H 8H 4ζ 2 H H H ζ 8H 2H H p x H ζ H 2 x 6H H H H H H 1 x H H H H 2 2 2 2 1 4 1 0 2 1 0 0 2 0 0 2 2 1 2 2 2 2 1 2 1 1 0 110 2 111 18 gq 93 0 2 10 3 0000 3 288 2 100 4 11 110 111 6 20 3 10 36 2 108 0 3 13 1 1 653 1187 302 19 991 163 35 17 43 2 4H2 0 H2 1 1 H 1ζ2 7H2ζ2 6H 2ζ2 12H 2 1 0 6H 2 0 0 x 3H1 1 1 H0 0ζ2 ζ 4H H H H 2H H 1 x H ζ H 7H H H ζ ζ H H ζ 13H ζ 2 2 3 20 2 0 2 1 0 2 2 1 0 00 24 0 0 0 2 216 0 20 9 0 0 6 3 36 2 6 3 3 0 00 6 2 1 10 2 1 2 9 35 2 2 2 2105 77 8 85 101 80 23 1 5 1 37 23 18H 110 H3 1 6H4 4H 12 6H0 0ζ2 8H2ζ2 7H2 00 2H2 10 2H2 11 4H3 0 H 1 0 0 H1 0 2H4 3H1 1 0 H 1 2 16CA CF x H1ζ2 H0 0 H2 H 10 ζ2 H0 ζ2 H1 1 xH1 1 H1 xH1 H 10 2 8 3 81 18 9 18 18 27 18 3 4 9 12 18 241 2 19 1 1 13 1 2 5 9H 100 δ 1 x 16CAnf H0 xH0 pgg x x H1 16 14 2 14 104 4 37 1501 101 1 3 7 288 54 24 27 54 x 3 6H3 ζ3 10H 1 0 H2 0 H 1ζ2 H0 0 0 H2 H1 0 0 H1 1 H ζ H H H 16C p x 3H ζ 3H ζ ζ 3 3 3 3 9 3 9 54 0 2 000 3 00 3 10 F gq 11 2 1 2 2 2 11 71 2 13 1 29 1 x H 1 x ζ xH H H δ 1 x 4 104 8 145 4 2 109 8 23 72 1 216 9 2 12 0 2 0 0 2 288 H 1 1 0 ζ2 H2 1 H1 0 H 1 2 H1 1 1 H1 H 1 0 0 6H0ζ2 H 8H ζ 6H 2H ζ 3H 3H H 2H 3H 3 9 3 18 3 3 27 3 8 11 1 3 120 10 2 110 1100 1110 1111 112 11 11 2 2 1639 1 10 16C 2n x2 ζ ζ H H H ζ H 2H p x ζ 584 7 138305 1 13 11 9 3 47 47 15 A f 3 9 2 9 00 3 3 3 0 2 108 0 20 3 gg 3 2 4H 2 0 H0 pgq x H1ζ3 H2 0 H 1ζ2 2H2 1 1 H1 0 0 2H 2H H H H ζ p x 2H 27 2 2592 3 4 2 120 121 2 111 2 100 16 16 1 2 3 gq 120 209 1 10 20 20 10 8ζ3 2H 20 H0 H00 H10 H100 H2 H3 pgg x ζ2 43 109 17 71 11 21 3 7 16 2 7 36 2 3 3 3 9 4H3 1 H1 1 1 ζ2 H2 1 H1 0 H 2 0 ζ3 H1 0 0 0 H1 2 0 6H 3H ζ H ζ 6H H 4H 2H ζ 6 12 3 24 6 2 2 110 1 2 4 10 5 2 100 2 10 1100 10 2 3 1 1 13 5443 205 2H H ζ H x2 H H ζ H 3H ζ H 395 55 55 10 10 0 2 0 0 3 x 3 0 2 3 2 108 1 2 36 1 H0 2H1 0ζ2 H1 1ζ2 H1 1 0 2H1 1 0 0 4H1 1 1 0 2H1 1 1 1 4H1 1 2 H1 2 H 1000 1 x 9H100 H111 10H1ζ2 3H0ζ3 H22 H2ζ2 H000 5H200 54 12 12 13 1 2 151 8 1 2 211 49 1 1 H10 H100 x H0 ζ2 H 1ζ2 ζ3 2H 110 H 100 23 2 3 x 54 3 3 3 6H1 2 0 4H1 2 1 4H1 3 3H2 1 0 3H2 2 pgq x H 1ζ3 5H 2ζ2 2H 2 1 0 4H3 H211 3H00ζ2 3H31 3H4 H1 ζ2 1 x 11ζ3 H11 H10 2 16 20 4 4 37 2 5 269 4097 H H 1 x H H 2H ζ 3H ζ 109 17 2 1 65 19 91 10 12 20 30 000 2 2 2 H 1 0 H0ζ3 ζ2 H1ζ2 2H2ζ2 H1 1 H 1 1 0 4H3 0 3H2 0 0 H0 36H 10 8H 100 14H 110 7H 1ζ2 2H12 4H0ζ2 H21 2H 200 9 3 6 36 216 12 5 6 24 2 16 7 677 7 193 11 3 3379 49 11 11 13 9 9 2 287 11 6H 210 3H 200 H1ζ2 H1 H1 0 H100 1 x H2 H 1ζ2 7H 2 0 0 H 1 2 H1 4H 2 2 H 1 0 0 H 1 0 0 0 13H 1 1ζ2 8H 1 3 5H 20 H2 2H0000 2H 110 H 1ζ2 ζ2 H10 ζ2 H1 2 72 4 36 2 2 216 6 2 2 4 4 20 32 16 39 7 53 7 5 77 9 ζ 2 H H H ζ H ζ 5ζ 7H H H 6H 1 1 1 0 12H 1 1 0 0 10H 1 1 2 10H 1 0ζ2 5H 1 2 0 2H 1 2 0 2H 1 2 1 19 35 2 3 00 0 2 00 2 3 110 10 100 4H 100 16H 30 4H 2ζ2 8H 210 5H2ζ2 H2 H22 H00 9H0ζ3 20 12 9 12 2 6 2 11 41699 3 128 26 5 4 8 2 3 3 1 458 H0ζ2 1 x 3H 2 1 0 H 2ζ2 ζ2 4H3 0 ζ3 H 2 0 0 3 58 7 3 5 175 19 2H 12 3H2ζ2 H2 0 H2 00 H4 ζ2 7H 20 2H2 H0 H00ζ2 6 2592 2 9 3 2 25H 6H x ζ H ζ 4H H H H ζ 3 2 2 9 27 20 200 2 3 2 3 1 2 11 2 111 2 100 96 31 3 3 97 10 245 29 3 2 131 8 7 7 1943 7H1ζ2 H1 0 0 H 1 0 0 H3 8H0 0 0 0 1 x 4H3 1 H2 1 1 H 1 2 3 1 5 7 ζ2 4H 30 x H00 H0ζ2 H3 H0000 H000 H0 6H0ζ3 12 3 12 6 2H 14H H ζ H H H H 3H H H H ζ 2 12 3 2 6 216 2 0 0 0 0 2 10 4 2 2 1 3 2 11 2 00 6 3 1 2 6 0 2 17 31 1 61 13 46 233 1 1 2 5 3 2 1249 H 2 0 12H2 0 H2 1 H2 0 0 H2ζ2 H1 0 4H0ζ3 H 1ζ2 H 1 1 0 2 29 185 δ 1 x ζ2 ζ2 ζ3 16C x 33H 20 33H0ζ2 H00 6 12 2 36 3 3 H H H (4.14) 288 6 12 3 A 18 3 110 6 000 8 00 25 93 55 71 49 13 47 2 6131 31 110 44 85 6409 245 67 3 2 11 H4 H0ζ2 H1 1 H2 H0 0 H0 0ζ2 ζ2 H 2ζ2 44H0 00 H3 H2 0 ζ2 H0 pgg x ζ2 ζ2 ζ3 4 4 9 18 18 2 40 2592 2 Finally the Mellin inversion of Eq. (3.13) yields the NNLO gluon-gluon splitting function 3 3 6 108 24 9 10 3 9 9 53 1 7 11 1 67 15H H 3H H H H 5H H 8H 2 1 0 1 0 0 2 1 1 2 1 2 0 2 0 0 2 0 1 1 1 0ζ3 4 97 8 2 103 16 4H 30 6H 2ζ2 4H 210 H 20 4H 200 4H 22 H0 7H0ζ3 H00 2 4 3 2 6 P 2 x 16C C n x2 H 3H H H H ζ H ζ 2H 3 6 9 67 29 412 928 1 gg A F f 9 2 10 12 1 3 20 3 0 2 27 0 3 2 3 8H ζ 4H 6H ζ 4H 10H 6H ζ 8H 8H 8H ζ 2 H H 8H 25H ζ H H H 65H 38H 0 0 2 0 000 1 3 1 20 2 00 1 0 2 1 000 1 100 4 40 2 6 1 2 1 0 2 2 0 2 9 1 9 0 4 4 3 0 0 127 511 55 4 1 17 43 6H 2H H p x 2ζ x2 H H 134 11 134 11 17 27 1 3 1 1 10 20 18 00 12 gg 3 24 3 x 24 10 18 0 H10 H100 8H120 8H13 H2 4H2ζ2 8H31 8H22 H3 10H30 9H H x H H H ζ H 14H H 9 6 9 6 3 0 3 0 0 0 2 1 0 2 0 0 0 0 4 0 0 2 2 3 0 0 0 0 12 1 1 1 521 6923 1 1 175 The large- H H 2H ζ H ζ 2H H H H H 11 2 11 134 43 1 7 749 135 97 43 85 13 144 1 432 2 21 1 2 0 2 100 12 11 110 111 12 2 8H210 pgg x ζ2 H0ζ2 4H 30 16H 2ζ2 12H 22 H 10 2H2ζ2 ζ H ζ H H ζ H H ζ H ζ H 2 6 9 36 2 2 2 2 72 0 54 1 4 3 24 1 0 12 1 2 12 1 2 3 1 0 0 53 49 185 511 1 6H 8H ζ 6H H ζ H ζ H 3H 4H 8H 210 12H 1ζ3 18H 200 8H 120 16H 11ζ2 24H 1100 16H 112 10 0 3 20 6 0 2 2 0 4 2 12 2 2 0 1 0 0 000

20 21 22 (2) NNLO, Pab : Moch, Vermaseren & Vogt ’04

Gavin Salam (CERN) QCD basics 3 18 / 3242 DGLAP evolution (initial quarks only) [Initial-state splitting] [Example evolution] E↵ect of (LO) DGLAP: initial quarks 2 2 xq(x,Q ), xg(x,Q2 ) 2 3 xq(x,Q ), xg(x,Q ) 3 2 xg(x,Qxg(x,Q) 2) xq + xqbar 2.5 2.5 xq + xqbar Take example evolution starting with 2 2 just quarks: 2 2 Q Q= 212.0 = 12.0 GeV GeV2