QCD Daniel de Florian Dpto. de Física- FCEyN- UBA 1
DISCLAIMER(S) Purpose(s) of these lectures: Introduction to QCD Refresh your knowledge on QCD (another view) Understand the vocabulary! New developments in the field (Lectures 3 and 4) pQCD 2
‣ In the LHC era, QCD is everywhere! a H, γ , Z, W b jet ‣ In these lectures : pQCD as precision QCD for Colliders 3
‣ LHC was incredibly successful at 7 & 8 TeV ‣ Everything SM like (including Higgs) LHC cross section measurements Standard Model Total Production Cross Section Measurements Status: July 2014 σ [pb] 10 11 80 µ b − 1 Run 1 √ s = 7, 8 TeV ATLAS Preliminary 10 6 LHC pp √ s = 7 TeV LHC pp √ s = 8 TeV 10 5 Theory Theory 35 pb − 1 Data Data 35 pb − 1 10 4 10 3 20.3 fb − 1 4.6 fb − 1 20.3 fb − 1 10 2 20.3 fb − 1 4.6 fb − 1 20.3 fb − 1 20.3 fb − 1 4.7 fb − 1 4.6 fb − 1 13.0 fb − 1 2 4.6 fb − 1 10 1 20.3 fb − 1 4.8 fb − 1 2.0 fb − 1 4.6 fb − 1 20.3 fb − 1 1 20.3 fb − 1 2 20.3 fb − 1 ! 10 − 1 ! pp t t − chan W Z t¯ WW H ggF Wt WZ ZZ H VBF t¯ t¯ t WW + WZ tW tZ total total total total total total total total total total total total total total No deviation from Standard Model observed so far..... 4
‣ Next run at 13 TeV ... will find evidence of new physics or not? 5
discovery ... as for Higgs at LHC ‣ Observe new particles: Need good understanding of background • Involve High multiplicities at LHC 6
‣ Very likely: New physics might show up in the detail • Flavor Physics • Contribution from new particles at loop level • Need to be precise on cross-sections and SM parameters EW vacuum stability m H , m t , α s , ... • Explore Higgs sector with precision • Multiple Gauge boson and HQ production (gauge/couplings to new physics) Precision is the name of the game These Lectures Toolkit for precise TH predictions at the LHC 7
Outline of the lecture 1 ✤ Basics of QCD : Lagrangian and Feynman rules ✤ QCD at work: beta function and running coupling ✤ QCD at work in e + e − ✤ Infrared Safety in QCD ✤ Jets in QCD 8
Outline of the lecture 2 ✤ Deep Inelastic Scattering ✤ Parton Model ✤ Scaling Violations and Evolution ✤ Factorization ✤ Parton Distribution Functions 9
Outline of the lecture 3 ✤ QCD at Colliders ✤ LO calculations : tools and recursions for amplitudes ✤ Why higher orders? ✤ How to do NLO ✤ Automated tools at NLO 10
Outline of the lecture 4 ✤ NNLO ✤ Higgs at NNLO and beyond ✤ Resummation : when fixed order fails ✤ Parton Showers ✤ Matching Parton showers and NLO 11
Some bibliography (and much material on the web) • QCD and Collider Physics, R.K.Ellis, W.J.Stirling and B.R.Webber , Cambridge University Press Sons (1999) • Foundations of Quantum Chromodynamics, T. Muta, World Scientific (1998) • Gauge Theory and Elementary Particle Physics, T. Cheng and L. Li, Oxford Science Publications (1984) • The theory of quark and gluon interactions, F.J. Ynduráin, Springer-Verlag (1999) • Collider Physics, V. Barger and R. Phillips, Addison-Wesley (1996) • Quantum Chromodynamics: High Energy Experiments and Theory, G. Dissertori, I. Knowles and M. Schmelling, International Series of Monograph on Physics (2009) 12
The eightfold way (1961) Gell-Mann and Ne’eman Everything starts by organizing hadron spectrum to show some pattern of symmetry (such as Mendeleev did for atoms in periodic table) One still missing by that time, but predicted following pattern Then one asks ... what is the reason for this pattern? 13
Quarks (1964) Gell-Mann and Zweig propose the existence of elementary (spin 1/2) particles named quarks : with 3 of them (plus antiquarks) can explain the composition of all known hadrons u m u ≈ 3 − 9 MeV d m d ≈ 1 − 5 M s m s ≈ 75 − 170 Baryon Bound states are only made by 3 quarks (baryon) qqq or by a quark+antiquark (meson). No other Meson q ¯ q structure observed. 14
c J/ Ψ = ( c ¯ c ) m c ≈ 1 . 1 − 1 . 3 GeV (1974) Discovered at SLAC and Brookhaven. Expected due to strong theoretical arguments Computer reconstruction of a ψ′ decay in the Mark I (GIM mechanism) detector at SLAC, making a near-perfect image of the Greek letter ψ The “bump” at 9.5 GeV • 1975: tau discovered at SLAC that lead to the discovery of the bottom quark at b FNAL in 1977 Υ =( b ¯ 1977: disco • 1977: discovered at Fermilab (E288) b ) Υ =( b ¯ b ) • • m b ≈ 4 . 0 − 4 . 4 GeV (1977) Discovered at Fermilab (E288) 3rd family of quarks needed to account for CP violation 15
• 1980: naked beauty Λ b = ( udb ) t • 1995: top quark identified at m t ≈ 171 GeV (1995) Discovered at Tevatron EW precision measurements predicted mass with accuracy Several orders of magnitude in masses 10 0 up-type quarks t quark charge mass (approx.) down-type quarks 10 − 1 b Yukawa coupling u 2/3 ~4 MeV 10 − 2 c d -1/3 ~ 7 MeV 10 − 3 s c 2/3 ~ 1.3 GeV s -1/3 ~150 MeV 10 − 4 d u t 2/3 ~171 GeV 10 − 5 proton b -1/3 ~4.4 GeV 10 − 6 16
Spin-statistics issue ∆ ++ = u ↑ u ↑ u ↑ Wave function (flavor+spin) completely symmetric : forbidden by Pauli exclusion principle u u u Introduce new additional quantum number : color ∆ ++ = ✏ ijk u i ↑ u j ↑ u k ↑ wave function becomes antisymmetric 17
Will see that experiment directly confirms 3 colors σ ( e + e − → hadrons ) σ ( e + e − → µ + µ − ) Upgrade color to “charge of the strong interactions” So strong that only hadrons observed in nature are those combinations of quarks that result in color singlets! Baryon qqq Only states results in color singlets Meson q ¯ q 3 colors explain observed spectrum of hadrons! ψ i color SU (3) color is an exact symmetry of nature f flavor 18
QCD: non-abelian gauge theory under SU(3) Simple recipe: take free Lagrangian for fermions Force it to be invariant under non-abelian local transformation with 8 generators obeying 2 λ A 3x3 Gell-Mann matrices (1 representation) t A = 1 2 0 1 0 1 0 1 0 1 0 1 0 0 − i 0 1 0 0 0 0 1 λ 1 = A , λ 2 = A , λ 3 = A , λ 4 = 1 0 0 i 0 0 0 − 1 0 0 0 0 @ @ @ @ A 0 0 0 0 0 0 0 0 0 1 0 0 0 1 1 0 0 0 1 0 1 0 1 0 0 − i 0 0 0 0 0 0 √ 3 λ 5 = A , λ 6 = A , λ 7 = A , λ 8 = 1 0 0 0 0 0 0 0 1 0 0 − i B C √ @ @ @ 3 @ A − 2 i 0 0 0 1 0 0 i 0 0 0 √ 3 @ 19
Original Lagrangian not invariant due to derivative of α a ( x ) To correct for that change derivative to covariant derivative adding extra spin-1 fields (one per generator) D transforms as the quark field Add all gauge invariants! ( F is not invariant in non-abelian theories, but..) QCD Lagrangian (+ gauge fixing terms and eventually ghosts) α S ≡ g 2 s one single coupling constant 4 π . no mass term for gluon (gauge invariance) m 2 A µ A µ 20
g Feynman rules − gf a 1 a 2 a 3 � µ 2 a 2 g µ 1 µ 2 ( p 1 − p 2 ) µ 3 µ + g µ 2 µ 3 ( p 2 − p 3 ) µ 1 p 2 g + g µ 3 µ 1 ( p 3 − p 1 ) µ 2 � α β p 1 p 3 i j − ig ( t a ) ij ( γ µ ) αβ µ 1 a 1 µ 3 a 3 µ 2 a 2 µ 3 a 3 p 3 p 2 − ig 2 � f ba 1 a 2 f ba 3 a 4 ( g µ 1 µ 3 g µ 2 µ 4 − g µ 1 µ 4 g µ 2 µ 3 ) p 1 p 4 � +(2 ↔ 3) + (2 ↔ 4) µ 1 a 1 µ 4 a 4 21
Propagators α β p i ( / p + m ) αβ Quark p 2 − m 2 + i ϵ δ ij i j µ p ν i Gluon p 2 + i ϵ d µ ν ( p ) δ ab a b ∗ spin polarization tensor � d µ ν ( p ) = ( λ ) ( p ) ε ν ε µ ( λ ) ( p ) λ Explicit expression depends on gauge propagation of physical and unphysical polarizations − g µ ν + (1 − α ) p µ p ν covariant gauges p 2 + i ϵ d µ ν ( p ) = − g µ ν + p µ n ν + p ν n µ − n 2 p µ p ν axial gauges ( p · n ) 2 p · n propagation of physical (transverse) polarizations only 22
In covariant gauges Lorentz invariance is manifest but ghosts must be included to cancel effect of unphysical polarizations in propagator 2 2 2 ! = =+1, ! 1 ! =+1, ! 1,0 ! µ b a p b gf abc p µ i p 2 + i ϵδ ab a c Similar trick can be used to simplify calculations when gluon (initial of final state) polarization enters in any amplitude 2 ✏ µ ( λ ) ∗ � ( λ ) ( p ) ε ν ε µ ( p ) = ( λ ) ( p ) λ 23
gg ! qq Example do it! In QED it is OK to use But in QCD one needs to use physical polarizations - k is a light-like vector, Alternatively on could add ghosts in the initial state and use again + 24
Color algebra Conventional normalization Tr ( t a t b ) = T R δ ab T R = 1 / 2 Fundamental representation 3 i,j,.. quark C F = N 2 c − 1 ( t a t a ) il = C F δ il a,b,.. gluon 2 N c Adjoint representation 8 � f adc f bdc = C A δ ab C A = N c j = 1 1 Very useful Fierz identity ( t a ) i k ( t a ) l 2 δ i j δ l δ i k δ l k − j 2 N c 1 1 ! = 2 N c 2 25
Most relevant color structures Compute those! quark gluon gluon quark gluon gluon 26
QCD at work QCD can not be solved exactly: use perturbation theory Coupling constant “large” : many orders needed for precision Several problems appear in the calculation of perturbative corrections Ultraviolet (UV) and InfraRed (IR) divergences 27
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