Lecture II. QCD and its basic symmetries. Renormalisation and the running coupling constant

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1 Lecture II QCD and its basic symmetries Renormalisation and the running coupling constant Experimental evidence for QCD based on comparison with perturbative calculations

2 The road to QCD: SU(3) quark model

3 Symmetry considerations: group theory Hadron multipletts are product representations of the fundamental triplett representation Analogous to spin, SU(2): =1, = 3 2 Analogous to spin, SU(3): quarks in fundamental rep., triplet and anti-triplet = = ψ = ψ u ψ d ψ s

4 The need for colour Contradiction with Pauli principle: Total wave function must be anti-symmetric! Colour Cure: introduce new, unobservable, quantum number: colour Observable states must be colourless! Colour

5 Quark spinors now in colour triplet: ψ = ψ r ψ g ψ b Also explains absence of qq or qqqq states: = contains singlet 3 3 = contains singlet 3 3= 3+6 does not contain singlet! etc.

6 QCD, theory of strong interactions L QCD = 1 4 F a µν(x)f aµν (x)+ N c c=1 N f f=1 ψ c,f (x)(iγ µ D µ m f ) ψ c,f (x) Quark fields: spinor ψ α,c,f (x) colour flavour α = 1,... 4 (spin up/down, particle/anti-particle) c = 1, 2, 3 (red,blue,green) f = u, d, s, c, b, t Covariant derivative: D µ = µ igt a A a µ(x) Field strength tensor: F a µν = µ A a ν ν A a µ gf abc A b µa c ν Parameters: α s (q 2 ) = g2 (q 2 ) 4π m u 3MeV,m d 6MeV m s 120MeV,m c 1.5GeV m b 4.5GeV,m t 175GeV

7 QCD, theory of strong interactions gluon self-interaction!

8 Interaction dictated by gauge symmetry ψiγ µ D µ ψ = ψiγ µ ( µ iqa µ )ψ ψiγ µ D µ ψ = ψiγ µ ( µ ig s T a G aµ )ψ Parameters: couplings and masses need to be determined by experiment

9 Feynman rules for perturbative QCD L QCD = 1 4 F a µν(x)f aµν (x)+ N c c=1 F a µν = µ A a ν ν A a µ gf abc A b µa c ν N f f=1 ψ c,f (x)(iγ µ D µ m f ) ψ c,f (x) D µ = µ igt a A a µ(x)

10 Global symmetries: example isospin ψ(x) = ( ψu ψ d ) ψ = Uψ, U SU(2), U U =1 ψ = ψ γ 0 = ψ U γ 0 = ψ γ 0 U = ψu If m u = m d we can write ψ (iγ µ D µ m)ψ = ψu (iγ µ D µ m)uψ = ψ(iγ µ D µ m)u Uψ For non-degenerate masses this does not work! ψ(x)iγ µ D µ ψ(x) m u ψu (x)ψ u (x) m d ψd (x)ψ d (x)

11 Symmetries of the QCD Lagrangian Local SU(3) c transformations ψ c(x) = ( e iθa (x)t a) cc ψ c (x) For degenerate quarks, m f1 =... = m fnf global SU(n f ) transformations ψ f (x) = a =1,... N 2 c 1 ( e iθa T a) ff ψ f (x) a =1,... n 2 f 1 Global U(1) transformations: ψ (x) = e iθ ψ(x) For massless quarks, m f1 =... = m fnf =0: Global axial SU(n f ) transformations ψ f (x) = (e iθa T a γ 5 ) ff a =1,... n 2 f 1 ψ f (x) Global axial U(1) transformations, ψ (x) = e iθγ 5 ψ(x) anomalous, broken by quantum effects

12 Symmetries for parameter values realised by nature SU(3) c U(1) B gauge symmetry, exact, only colour singlets observable baryon number, exact SU(2) isospin approximate, O(few %), m u m d SU(3) flavour approximate, O(few 10 %), m u m d m s (quark model!) SU(2) axial approximate m u m d 0 SU(2) L SU(2) R approximate chiral symmetry m u m d 0 = isospin+axial flavour symmetry combined

13 Elementary perturbative processes in QCD

14 Higher order processes as in QED do not exist in QED! Gluon self coupling: besides quark loops also gluon loops can occur!

15 Higher order corrections and renormalisation Vertex correction: p 1 k p 2 p 1 k p 2 + k d 4 k (2π) 4 (p 2µ + k µ )γ µ + m p p 2k + k 2 m 2 (p 1µ k µ )γ µ + m 1 p 2 1 2p 1k + k 2 m 2 k 2 k d 4 k k2 ln k k6 Regularisation by momentum cut-off: Λ d 4 k k2 k 6 ln Λ finite Altogether there are quadratic, linear and logarithmic divergencies from vacuum corrections to different n-point functions

16 Similar: correction to the mass Remember Feynman propagator, free field Pole mass: parameter from Lagrangian, mass of the non-interacting particle (One) correction in the interacting theory: Full propagator: d 4 p (2π) 4 e ipx p 2 m Σ(p2 ) Pole of the propagator gets shifted! Correction to the mass: p 2 m Σ(p 2 )=0, m 2 = m δm 2

17 Renormalisation Observation: bare parameters from L do not correspond to physical couplings, masses not measurable! Physical parameters: calculated from sum of all Feynman diagrams e = e 0 ( 1+a 1 e 2 0 ln Λ µ +... ) m = m 0 ( 1+b 1 e 2 0 ln Λ µ +... ) e 0 = e 0 (e, m, Λ),m 0 = m 0 (e, m, Λ) Observable O(e 0,m 0 )=O(e, m, Λ) Renormalisation: absorb cut-off dependence in Z-factors, order by order m 0 = Zm 1/2 e 0 = Ze 1/2 m e O R (e, m) = lim Z(e, m, Λ)O(e, m, Λ) Λ

18 Running coupling constants: the vacuum as a medium Free electron : idealisation, exists only in perturbation theory Vacuum permits quantum fluctuations, Pair production of virtual electron-positron pairs Physical electron in continuous interaction with vacuum, surrounded by cloud of electrons, positrons, photons Polarisation of virtual dipoles, screening!

19 Effect of vacuum polarisation on the QED charge Measure charge by scattering with 4-momentum transfer Q 2 Small = large distance small coupling Q 2 α(q 2 ) Large Q 2 = small distance large coupling α(q 2 )

20 Effect of vacuum polarisation on the QCD charge Vacuum polarisation: quark anti-quark pairs Screening as in QED? Yes, but... Gluon loops in addition! In total: colour charge mostly surrounded by same charge gluonic cloud, anti-screening Q 2 Small = large distance large coupling Q 2 Large = small distance small coupling

21 Comparing calculation and experiment QED QCD Running much faster than in QED!

22 Confronting theory with experiment: 1. Perturbative high energy regime

23 Fragmentation (non-perturbative) and jets Example e-p scattering remainder of proton scattered quark

24 Test for Flavours and Colours via a QED process Fermion anti-fermion production f Q f = e f

25 Evidence for 3 colours per quark flavour

26 Evidence for 3 colours in pion and tau decay c Γ M 2 N 2 c Same principle: both processes sensitive to number of contributing quark states!

27 Evidence for gluons: 3-jet events Discovery: PETRA storage ring (DESY Hamburg), 1979 Three-quark final state not possible with leading order QCD Feynman rules One jet comes from a gluon!

28 The gluon spin from 3-jet events Theta: angle between axis of highest energy jet and the direction of the other two jets in their CMS

29 Strong coupling from 2-jet and 3-jet events

30 Evidence for quarks: Deep Inelastic Scattering

31 The difference to the quark model:

32 Confronting theory with experiment: I1. Non-perturbative low energy regime

33 Non-relativistic QCD (NRQCD) Consider Dirac eqn for free particle + plane wave solution: (iγ µ µ m)ψ(x) =0 (E m)ϕ σ pχ = 0 (E + m)χ σ pϕ = 0 E =( p 2 + m 2 ) 1/2 = m ( ) p 2 1/2 ) m 2 +1 = m (1+ p2 2m expansion in small v/c χ σ p 2m ϕ negligible, only ϕ left, two components (spin) Wave equation: i t ψ = 2 ψ Schrödinger! Generalises to interactions 2m

34 Bound states of heavy quarks Short distance part perturbative V (r) = C α s r Singlet: C = 4 3, Octet: C = 1 3

35 Confinement, qualitative Fields between charge and anti-charge: dipole field colour electric fluc tube Field energy in the QCD flux tube grows linearly with separation

36 At some point pair creation of light quarks is possible: String breaking Formation of heavy light mesons, saturation of the potential

37 Discovery of J/psi (1974)

38 Predictions for spectra of quarkonia c,b,t quarks are non-relativistic solve Schrödinger eqn with potential V (r) = 4 3 α s r + kr cc charmonia bb bottomonia