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QCD Lagrangian Gauge Invariance Colour Algebra Feynman Rules 2.Tree Level Amplitudes Application of Feynman Rules Polarisation Sums 3

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QCD Lagrangian Gauge Invariance Colour Algebra Feynman Rules 2.Tree Level Amplitudes Application of Feynman Rules Polarisation Sums 3 QCD and precision calculations Lecture 1: basics Gudrun Heinrich Max Planck Institute for Physics, Munich [email protected] PREFIT School, March 2, 2020 1 Outline 1. Basics of QCD QCD Lagrangian Gauge invariance Colour algebra Feynman rules 2.Tree level amplitudes Application of Feynman rules Polarisation sums 3. NLO calculations (details see lectures 2-4) 4. Beyond NLO 2 Some literature 3 QCD QCD is a very rich field! asymptotic freedom confinement strong CP-problem QCD at finite temperature quark-gluon-plasma flavour puzzles spectroscopy lattice gauge theory disclaimer: I am not doing justice to it by just considering perturbative QCD, and even that only at fixed order However, as the focus will be on precision calculations, I will make a (highly biased) choice 4 Motivation • at hadron colliders, QCD is everywhere • without QCD corrections, (most of) the data are not well described , Grazzini, Kallweit, Rathlev ‘17 GH, Jahn, Jones, Kerner, Pires ‘17 5 QCD corrections perturbation theory in the strong coupling next-to-next-to-leading order leading order next-to-leading order NLO corrections typically NNLO corrections typically a few % but there are prominent exceptions (e.g. Higgs production, NLO corr. ~100%) Why can we do such an expansion at all? important concepts: • asymptotic freedom • factorisation 6 Factorisation parton distribution functions partonic cross power-suppressed non-perturbative (PDFs) section corrections factorisation scale (separate short- and long-distance dynamics) renormalisation scale (UV divergences from loops, see later) PDFs : from fits to data, only evolution with scale is calculable perturbatively contains hard scattering matrix element, calculable in perturbation theory sum over parton types i, j 7 Asymptotic freedom CMS +0.0060 0.24 CMS Incl.Jet, s = 8TeV, α (M ) = 0.1164 (Q) S z -0.0043 S CMS Incl.Jet, s = 8TeV α 0.22 CMS R32 , s = 7TeV CMS Incl.Jet , s = 7TeV 0.2 CMS tt , s = 7TeV CMS 3-Jet Mass , s = 7TeV 0.18 D0 Incl.Jet D0 Angular Correlation 0.16 H1 ZEUS World Avg α (M ) = 0.1185 ± 0.0006 0.14 S z 0.12 0.1 0.08 5 6 7 8 10 20 30 40 100 200 300 1000 2000 Q (GeV) strong coupling becomes weaker as energy scale increases will be discussed in more detail later 8 LHC event 9 non-perturbative stuff visualisation of pile-up in the ATLAS tracker (Run I) 10 QCD or New Physics? a typical distribution, decreasing with energy 11 QCD or New Physics? NLO LO small discrepancy to data at max. available energy due to lack of NNLO corrections? or onset of New Physics? 12 New Physics! (Z-boson 1983) 13 matrix element image: P.Uwer Z-boson resonance note also: (see Effective Field Theory lecture) 14 Today no sign of a resonance up to di-jet invariant masses of ~7 TeV if New Physics scale is very large, only indirect effects visible need precision 15 QCD Lagrangian strong interactions: gauge theory “colours” of quarks non-Abelian structure of SU(3) related to gluon-self-interactions purely gluonic part: Yang-Mills Lagrangian convention: doubly occurring indices are summed over : structure constants of SU(3) (totally antisymmetric) generators of SU(3) 8 gluons, in the adjoint representation of SU(3) 16 QCD Lagrangian fermionic part: quark fields for flavour f: : colour index quarks are in the fundamental representation of SU(3) , define generators of SU(3) in fundamental representation Gell-Mann matrices 17 Evidence for Colour hadronic R-ratio: 2 2 R(s)=Nc e ✓(s 4m ) N =3 f − f c f=u,d,s,c,...X 2 1 1 threshold energy R =3 ( )2 +( )2 +( )2 + ... 3 −3 −3 to produce ✓ ◆ quark-antiquark-pair u d s 18 of mass Gauge invariance local gauge transformation with , , remember covariant derivative for to be invariant 19 gauge invariance note: mass term ~ would break gauge invariance 20 Colour Algebra generators of SU(3): fundamental representation: Gell-Mann matrices (traceless, hermitian) 21 Colour Algebra adjoint representation: dimension of the vector space on which it acts is equal to the dimension of the group itself ( dimension of the group = number of generators, ) matrices (8x8) - matrices group invariants: : eigenvalues of Casimir operators in fundamental/adjoint representation usually (convention) 22 Colour Algebra pictorially 23 Colour Algebra pictorially more: see exercises 23 colour algebra exercise calculate the colour factors for the following diagrams 24 Colour Algebra explicit example for a quark-gluon vertex figure: Peter Skands gluon represented as double line of two colours (second term because therefore relation of traceless generators) 25 QCD Lagrangian • the gluon propagator is constructed from the inverse of the bilinear term in the gluon fields in the action • in momentum space it should fulfil (1) however as Eq. (1) has zero modes, so [.] is not invertible reason: contains physically equivalent configurations need gauge fixing: add constraint on gluon fields with a Lagrange multiplier to the Lagrangian 26 QCD Lagrangian covariant gauges: add condition : Feynman gauge : Landau gauge 27 QCD Lagrangian • however in covariant gauges unphysical (non-transverse) degrees of freedom also can propagate • their effect is cancelled by ghost fields coloured complex scalars obeying Fermi statistics Faddeev-Popov matrix in Feynman gauge: 28 QCD Lagrangian • unphysical degrees of freedom and the ghost fields can be avoided by choosing axial (physical) gauges in axial gauges: vector with gluon propagator: light-cone gauge: 29 Feynman rules Propagators: gluon quark ghost covariant gauge light-cone gauge 30 Feynman rules Vertices: quark-gluon ghost-gluon 31 Feynman rules 3-gluon 4-gluon 32 Feynman rules spinors: incoming fermion incoming anti-fermion outgoing fermion outgoing anti-fermion polarisation vectors: incoming vector boson outgoing vector boson 33 Feynman rules further rules: • momentum conservation at each vertex • factor (-1) for each closed fermion loop • factor (-1) for switching identical external fermions • integrate over loop momenta with 34 spinor relations useful identities: (Dirac equation) (orthogonality) (completeness) 35 colour decomposition using the “double line notation”, we can express gluon amplitudes entirely in termsColor of generators in pictures Insertbased on and where is color factor for qqg vertex into typical string of fabc structure constants for a Feynman diagram: •Always single traces (at tree level) + all non-cyclic permutations •comes only from those planar(withdiagrams corresponding signs) with cyclic ordering of external legs fixed to 1,2,…,n L. Dixon, similarly7/20/06 Higher Order QCD: Lect. 1 + permutations 20 A, µ B, ν A, µ p B, ν 1 p a b A B TR CA a b A B 2Nc T C 1 colour R A 2Nc = colour = 2 Nc − 1 2 CF = Nc − 1 2N CF = c 2Nc A T r(t ) = 0 T r(tA) = 0 A B AB A 1B AB 1 T r(t t ) = T δ , TT r(=t t ) = TRδ , TR = R R 2 2 A A A A tactcb = CF δab tactcb = CF δab A A ! ! CDA CDB AB CDA CDB ABf f = CA δ , CA = Nc f f = CA δC,D , CA = Nc C,D ! ! A A 1 1 (t )ab(t )cd = δadδbc − δabδcd A A 1 1 2 2Nc (t )ab(t )cd = δadδbc − δabδcd 2 2Nc f ABEf CDE + f BCEf ADE + f CAEf BDE = 0 f ABEf CDE + f BCEf ADE + f CAEf BDE = 0 A (t )ab ABC A i f (t )ab generators ofABSCU(Nc): 2 i f A Nc − 1 hermitean traceless matrices (t )ab generators of SU(Nc): ABC A B C N 2 − 1 hermitean traceless matrices (tA) f = −2 i Tr([t , t ] t ) c colourab decomposition A1 A2 A f ABC = −2TirT(r([t ttA, ·tB· ·]ttCn) + all non-cyclic permutations • therefore the colour part of the amplitude can be qq¯gggg . ⇒ T r(tA1 tA2 · · · tAn ) + permutations T r(tA1 tA2 · ·completely· tAn ) + all non-cyclicseparatedp ermfromutations the kinematicab part tree n−2 A1 A2 A tree h1 h2 h M ({pA,1aA,2h }) =Ang T r(t t · · · t n ) M (1 , 2 . n n ) + all non-cyclic permutations qq¯gggg . ⇒nT r(t i ti ·i · · t )ab + permutations n momenta, colour, helicities colour ordered partial amplitude, colour factors stripped off tree n−2 A1 A2 An tree h1 h2 hn Mn ({pi, ai, hi}) = g usefulT r( tproperty:t · · · t as) Mn (1 , 2 . n ) comes+ all non-cyclic from diagramsperm withutations cyclic ordering of external legs, it only can have infrared singularities in adjacent invariants 2 si,i+1 = (pi + pi+1) • colour decomposition at loop level is straightforward, but then single and double trace structures are generated figure: Lance Dixon 37 A A tactcb = CF δab !A CDA CDB AB f f = CA δ , CA = Nc C,D ! A A 1 1 (t )ab(t )cd = δadδbc δabδcd 2 − 2Nc f ABEf CDE + f BCEf ADE + f CAEf BDE = 0 A (At A)ab tactcb = CF δab ABC !A i f CDA CDB AB f f = CA δ , CA = Nc generators of SU(Nc): 2 C,D A Nc 1 hermitean traceless! matrices (t )ab − A A 1 1 (t )ab(t )cd = δadδbc δabδcd f ABC = 2 i2Tr([tA−, t2BN] ctC ) − f ABEf CDE + f BCEf ADE + f CAEf BDE = 0 T r(tA1 tA2 tAn ) + all non-cyclic permutations Color· · · sumsA Color sums(t )ab qq¯gggg . T r(tA1 tA2i f ABtCAn ) + permutations ⇒ · · · ab In theIn the end, end, we we want generatorswant to tosum/average ofsum/averageSU(Nc): over over final/initial final/initial colors colors N 2 1 hermitean traceless matrices (tA) (as(as well well as ashelicities): helicities):c − ab tree( p , a , h ) = gn−2T r(tA1 tA2 tAn ) M tree(1h1 , 2h2 .
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