QCD Theory — some essentials Lecture 1 M. Diehl Deutsches Elektronen-Synchroton DESY DESY Summer Student Programme 2014, Hamburg DESY Introduction Renormalization Summary Plan of lectures M. Diehl I Brief introduction I The running coupling renormalization and scale dependence I e+ e− → hadrons some basics of applied perturbation theory I Factorization parton densities and scale evolution QCD Theory — some essentials 2 Introduction Renormalization Summary Quantum chromodynamics I gauge theory describing the interactions between quarks and gluons embedded in the SU(3)×SU(2)×U(1) group of the Standard Model I phenomenology very different from weak and e.m. interactions because coupling is strong at small momentums scales • quarks and gluons are confined inside bound states (hadrons) • perturbative expansion in αs only for high momentum scales I M. Diehl focus of these lectures: theory concepts that allow us to use perturbation theory, although all measurements are made with hadrons, not quarks and gluons QCD Theory — some essentials 3 Introduction Renormalization Summary Quantum chromodynamics I gauge theory describing the interactions between quarks and gluons embedded in the SU(3)×SU(2)×U(1) group of the Standard Model I phenomenology very different from weak and e.m. interactions because coupling is strong at small momentums scales • quarks and gluons are confined inside bound states (hadrons) • perturbative expansion in αs only for high momentum scales I not here: theory tools to deal with the strongly interacting sector of QCD I I I M. Diehl lattice gauge theory (Feynman path integral for discretized space-time, after going from physical to imaginary time) chiral perturbation theory (effective theory for hadron dynamics at scales 1 GeV, based on the chiral symmetry of QCD and its breaking) models for hadrons and for the transition quarks + gluons ↔ hadrons QCD Theory — some essentials 4 Introduction Renormalization Summary Basics of perturbation theory I split QCD Lagrangian into free and interacting parts: LQCD = Lfree + Lint I I I I from Lfree : free quark and gluon propagators I I I Lint : interaction terms ∝ g or g 2 expand process amplitudes, cross sections, etc. in g Feynman graphs visualize individual terms in expansion in position space: propagation from xµ to y µ in momentum space: propagation with four-momentum k µ from Lint : elementary vertices ∝g M. Diehl ∝g QCD Theory — some essentials ∝ g2 5 Introduction Renormalization Summary Basics of perturbation theory I split QCD Lagrangian into free and interacting parts: LQCD = Lfree + Lint I I I I M. Diehl Lint : interaction terms ∝ g or g 2 expand process amplitudes, cross sections, etc. in g Feynman graphs visualize individual terms in expansion gauge fixing Lfree + Lint + Lgauge ghost fields (propagate and couple to gluons) QCD Theory — some essentials 6 Introduction Renormalization Summary Loop corrections I in loop corrections find ultraviolet divergences I only appear in corrections to propagators elementary vertices nF nF Exercise: Draw the remaining one-loop graphs for all propagators and elementary vertices M. Diehl QCD Theory — some essentials 7 Introduction Renormalization Summary I origin of UV divergences: region of ∞ ly large loop momenta ↔ quantum fluctuations at ∞ ly small space-time distances I idea: remove this when describe physics at some given scale µ renormalization technically: I 1. regularize: artificial change of theory making div. terms finite • physically intuitive: momentum cutoff • in practice: dimensional regularization 2. renormalize: absorb would-be infinities into • quark masses mq (µ) • coupling constant αs (µ) • quark and gluon fields (wave function renormalization) 3. remove regulator: quantities are finite when expressed in terms of renormalized parameters and fields I renormalization scheme: choice of which terms to absorb “∞” is as good as “∞ + log(4π)” M. Diehl QCD Theory — some essentials 8 Introduction Renormalization Summary Renormalization group equations (RGE) I scale dependence of renormalized quantities described by differential equations ` ´ d αs (µ) = β αs (µ) 2 d log µ ` ´ d mq (µ) = mq (µ)γm αs (µ) 2 d log µ β, γm = perturbatively calculable functions in region where αs (µ) is small enough ˆ ˜ β = −b0 αs2 1 + b1 αs + b2 αs2 + b3 αs3 + . . . ˆ ˜ γm = −c0 αs 1 + c1 αs + c2 αs2 + c3 αs3 + . . . coefficients known including b3 , c3 ` ´ 1 b0 = 4π 11 − 23 nF M. Diehl QCD Theory — some essentials c0 = 1 π 9 Introduction Renormalization Summary The running of αs I βQCD < 0 ⇒ αs (µ) decreases with µ 2004 for Nobel prize Gross, Politzer and Wilczek • asymptotic freedom at large µ plot: Review of Particle Properties 2012 • perturbative expansion becomes invalid at low µ quarks and gluons are strongly bound inside hadrons: confinement momenta below 1 GeV ↔ distances above 0.2 fm M. Diehl QCD Theory — some essentials 10 Introduction Renormalization Summary The running of αs I truncating β = −b0 αs2 (1 + b1 αs ) get αs (µ) = 1 b1 log L 1 − + O b0 L (b0 L)2 L3 with L = log I more detail µ2 Λ2QCD blackboard plot: Review of Particle Properties 2012 M. Diehl QCD Theory — some essentials 11 Introduction Renormalization Summary Scale dependence of observables I observables computed in perturbation theory depend on renormalization scale µ • implicitly through αs (µ) • explicitly through terms ∝ log(µ2 /Q2 ) where Q = typical scale of process e.g. Q = pT for production of particles with high pT Q = MH for decay Higgs → hadrons Q = c.m. energy for e+ e− → hadrons I µ dependence of observables must cancel at accuracy of the computation see how this works M. Diehl blackboard QCD Theory — some essentials 12 Introduction Renormalization Summary Scale dependence of observables I for generic observable C have expansion µ2 C(Q) = αsn (µ) C0 + αs (µ) C1 + nb0 C0 log 2 + O αs2 Q I Exercise: check that this satisfies d C = O αsn+2 2 d log µ ⇒ residual scale dependence when truncate perturbative series I term with αsk in brackets comes with up to k powers of log(µ2 /Q2 ) • calculation at lower orders + RGE → logarithmically enhanced terms at higher orders • choose µ ∼ Q so that αs log(µ/Q) 1 otherwise higher-order terms spoil series expansion M. Diehl QCD Theory — some essentials 13 Introduction Renormalization Summary Example I inclusive hadronic decay of Higgs boson via top quark loop (i.e. without direct coupling to b¯b) I in perturbation theory: H → 2g, H → 3g, . . . Baikov, Chetyrkin 2006 known to N3 LO 500 LO NLO NNLO NNNLO Γ [keV] 400 I scale dependence decreases at higher orders I scale variation by factor 2 up- and downwards often taken as estimate of higher-order corrections I choice µ < MH more appropriate 300 200 100 0 0.1 0.5 1 µ /MH 5 plot for mH = 125 GeV M. Diehl QCD Theory — some essentials 14 Introduction Renormalization Summary Summary of lecture 1: renormalization I I beyond all technicalities reflects physical idea: eliminate details of physics at scales scale µ of problem running of αs characteristic features of QCD: • asymptotic freedom at high scales use perturbation theory • strong interactions at low scales need other methods I M. Diehl dependence of observable on µ governed by RGE reflects (and estimates) particular higher-order corrections . . . but not all QCD Theory — some essentials 15
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