Introduction to perturbative QCD and factorization

Transcription

1 Introduction to perturbative QCD and factorization Part 1 M. Diehl Deutsches Elektronen-Synchroton DESY Ecole Joliot Curie 2018 DESY

2 Plan of lectures 0. Brief introduction 1. Renormalisation, running coupling, running masses scale dependence of observables 2. e + e hadrons some basics of applied perturbation theory 3. Factorisation and parton densities using perturbation theory in ep and pp collisions 4. GPDs and exclusive processes a more detailed look into proton structure 5. TMDs probing the transverse momentum of partons 6. Double parton scattering a new case for factorisation M. Diehl Introduction to perturbative QCD and factorization 2

3 Some references for all lectures: more on short-distance factorization J Collins, hep-ph/ and hep-ph/ J Collins, Foundations of Perturbative QCD, CUP 2011 short overview of GPDs and TMDS MD, arxiv: full bibliography for GPDs e.g. in reviews S Boffi and B Pasquini, arxiv: A Belitsky and A Radyushkin, hep-ph/ MD, hep-ph/ K Goeke et al., hep-ph/ overviews of TMDs A Bacchetta et al., hep-ph/ S Mert Aybat and T Rogers, arxiv: T Rogers, arxiv: M. Diehl Introduction to perturbative QCD and factorization 3

4 Some references for all lectures: more on short-distance factorization J Collins, hep-ph/ and hep-ph/ J Collins, Foundations of Perturbative QCD, CUP 2011 multiparton interactions Multiple Parton Interactions at LHC, eds. P Bartalini and J Gaunt, (individual chapters on arxiv) MD, summer school lectures (2014) M. Diehl Introduction to perturbative QCD and factorization 4

5 Quantum chromodynamics (QCD) theory of interactions between quarks and gluons very different from weak and electromagnetic interactions because coupling α s is large at small momentum scales quarks and gluons are confined inside bound states: hadrons (proton, neutron, pion,... ) perturbative expansion in α s only at high momentum scales symmetries gauge invariance: group SU(3) colour charge electromagnetism: U(1) electric charge Lorentz invariance and discrete symmetries: P (parity = space inversion) T (time reversal) C (charge conjugation) chiral symmetry for zero masses of u, d and s embedded in Standard Model: quarks couple to γ, W, Z and H M. Diehl Introduction to perturbative QCD and factorization 5

6 Why care about QCD? without quantitative understanding of QCD would have very few physics results from LHC, Belle,... α s and quark masses are fundamental parameters of nature need e.g. m t for precision fits in electroweak sector Higgs physics α s to discuss possible unification of forces QCD is the one strongly interacting quantum field theory we can study in experiment many interesting phenomena: structure of proton confinement breaking of chiral symmetry convergence of perturbative series M. Diehl Introduction to perturbative QCD and factorization 6

7 Basics of perturbation theory split Lagrangian into free and interacting parts: L QCD = L free + L int L int : interaction terms g or g 2 expand process amplitudes, cross sections, etc. in g Feynman graphs visualise individual terms in expansion from L free : free quark and gluon propagators in position space: propagation from x µ to y µ in momentum space: propagation with four-momentum k µ from L int : elementary vertices g g g 2 M. Diehl Introduction to perturbative QCD and factorization 7

8 Loop corrections in loop corrections find ultraviolet (UV) divergences only appear in corrections to propagators elementary vertices n F n F Exercise: Draw the remaining one-loop graphs for all propagators and elementary vertices M. Diehl Introduction to perturbative QCD and factorization 8

9 origin of UV divergences: region of ly large loop momenta quantum fluctuations at ly small space-time distances idea: encapsulate UV effects in (a few) parameters when describe physics at a given scale µ renormalisation n F n F M. Diehl Introduction to perturbative QCD and factorization 9

10 origin of UV divergences: region of ly large loop momenta quantum fluctuations at ly small space-time distances idea: encapsulate UV effects in (a few) parameters when describe physics at a given scale µ renormalisation technically: 1. regulate: artificial change of theory making div. terms finite physically intuitive: momentum cutoff in practice: dimensional regularisation 2. renormalise: absorb UV effects into coupling constant α s(µ) quark masses m q(µ) quark and gluon fields (wave function renormalisation) 3. remove regulator: quantities are finite when expressed in terms of renormalised parameters and fields renormalisation scheme: choice of which terms to absorb is as good as + log(4π) M. Diehl Introduction to perturbative QCD and factorization 10

11 Dimensional regularisation in a nutshell choice of regulator choice between evils dim. reg.: little (any?) physics intuition, but keeps intact essential symmetries (gauge and Lorentz invariance) idea: integrals for Feynman graphs become UV finite in lower space-time dimension, e.g. d D k 1 1 log. div. for D = 4 (2π) D k 2 m 2 (k p) 2 m 2 converg. for D = 3, 2, 1 procedure: 1. formulate theory in D dimensions (with D small enough) 2. analytically continue results from integer to complex D original divergences appear as poles in 1/ɛ (D = 4 2ɛ) 3. renormalise 4. take ɛ 0 M. Diehl Introduction to perturbative QCD and factorization 11

12 Dimensional regularisation in a nutshell choice of regulator choice between evils dim. reg.: little (any?) physics intuition, but keeps intact essential symmetries (gauge and Lorentz invariance) idea: integrals for Feynman graphs become UV finite in lower space-time dimension, e.g. d D k 1 1 log. div. for D = 4 (2π) D k 2 m 2 (k p) 2 m 2 converg. for D = 3, 2, 1 enter: a mass scale µ coupling in 4 2ɛ dimensions is µ ɛ g with g dimensionless needed to get dimensionless action d D x L any other regularisation introduces a mass parameter as well renormalised quantities depend on µ M. Diehl Introduction to perturbative QCD and factorization 12

13 Renormalisation group equations (RGE) scale dependence of renormalised quantities described by differential equations: d d log µ αs(µ) = β( α ) s(µ) 2 d d log µ mq(µ) = ( ) 2 mq(µ)γm αs(µ) β, γ m = perturbatively calculable functions in region where α s (µ) is small enough β = b 0 αs 2 [ 1 + b1 α s + b 2 αs 2 + b 3 αs ] [ γ m = c 0 α s 1 + c1 α s + c 2 αs 2 + c 3 αs ] coefficients known including b 4, c 4 (b 4 since 2016) ( b 0 = 1 4π n ) F c 0 = 1 π M. Diehl Introduction to perturbative QCD and factorization 13

14 The running of α s β QCD < 0 α s (µ) decreases with µ 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 Nobel prize 2004 for Gross, Politzer and Wilczek asymptotic freedom at large µ 0.1 QCD α s (M z ) = ± Q [GeV] 1000 plot: Review of Particle Properties 2018 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 Introduction to perturbative QCD and factorization 14

15 The running of α s truncating β = b 0 α 2 s(1 + b 1 α s ) get α s (µ) = 1 b 0 L b 1 log L (b 0 L) 2 + O ( 1 L 3 ) 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 with L = log µ2 Λ 2 QCD 0.1 QCD α s (M z ) = ± Q [GeV] 1000 plot: Review of Particle Properties 2018 dimensional transmutation: mass scale Λ QCD not in Lagrangian, reflects quantum effects more detail blackboard M. Diehl Introduction to perturbative QCD and factorization 15

16 Scale dependence of observables observables computed in perturbation theory depend on renormalisation scale µ implicitly through α s (µ) explicitly through terms log(µ 2 /Q 2 ) where Q = typical scale of process e.g. Q = p T for production of particles with high p T Q = M H for decay Higgs hadrons Q = c.m. energy for e + e hadrons µ dependence of observables must cancel at accuracy of the computation see how this works blackboard M. Diehl Introduction to perturbative QCD and factorization 16

17 Scale dependence of observables for generic observable C have expansion [ } C(Q) = αs n (µ) C 0 + α s (µ) {C 1 + nb 0 C 0 log µ2 Q 2 + O ( αs 2 ) ] Exercise: check that this satisfies d d log µ 2 C = O( αs n+2 ) residual scale dependence when truncate perturbative series at higher orders: α n+k s (µ) comes with up to k powers of log(µ 2 /Q 2 ) choose µ Q so that α s log(µ/q) 1 otherwise higher-order terms spoil series expansion M. Diehl Introduction to perturbative QCD and factorization 17

18 Example inclusive hadronic decay of Higgs boson via top quark loop (i.e. without direct coupling to b quark) in perturbation theory: H 2g, H 3g,... known to N 3 LO Baikov, Chetyrkin 2006 H t Γ [kev] LO NLO NNLO NNNLO µ /M H scale dependence decreases at higher orders scale variation by factor 2 up- and downwards often taken as estimate of higher-order corrections choice µ < M H more appropriate M. Diehl Introduction to perturbative QCD and factorization 18

19 Quark masses recall: α s and m q depend on renormalisation scheme standard in QCD: MS scheme running α s (µ) and m q (µ) for heavy quarks c, b, t can also use pole mass def. by condition: quark propagator has pole at p 2 = m 2 pole possible in perturbation theory, but in nature quarks confined scheme transformation: [ m pole = m(µ) 1 + αs(µ) π MS masses from Review of Particle Properties 2018 ( ) ] 4 3 log m2 (µ) + O(αs) 2 µ 2 m u = MeV m d = MeV m s = MeV m c = GeV m b = GeV m t = GeV at µ = 2 GeV with m q(µ = m q) = m q M. Diehl Introduction to perturbative QCD and factorization 19

20 Summary of part 1: renormalisation beyond all technicalities reflects physical idea: eliminate details of physics at scales scale Q of problem running of α s characteristic features of QCD: asymptotic freedom at high scales use perturbation theory strong interactions at low scales need other methods introduces mass scale Λ QCD into theory dependence of observable on µ governed by RGE reflects (and estimates) particular higher-order corrections... but not all M. Diehl Introduction to perturbative QCD and factorization 20