Quantum Chromodynamics Lecture 1: All about color Hadron Collider Physics Summer School 2010 John Campbell, Fermilab
References and thanks • Useful references for this short course are: • QCD and Collider Physics R. K. Ellis, W. J. Stirling and B. R. Webber Cambridge Monographs on Particle Physics, Nuclear Physics and Cosmology • Hard Interactions of Quarks and Gluons: a Primer for LHC Physics J. C., J. W. Huston and W. J. Stirling Rept. Prog. Phys. 70, 89 (2007) [hep-ph/0611148] • Resource Letter: Quantum Chromodynamics A. S. Kronfeld and C. Quigg arXiv:1002.5032 [hep-ph] (for the American Journal of Physics) • Thanks to R. K. Ellis and G. Zanderighi, for lecture notes from previous schools - upon which much of these lectures will be based. Quantum Chromodynamics - John Campbell - 2
QCD: why we care proton - (anti)proton cross sections 9 9 10 10 • It is no surprise that hadron colliders 8 8 require an understanding of QCD. 10 10 ! tot 7 7 10 10 Tevatron LHC • This plot demonstrates the extent 6 6 10 10 to which we must have a good 5 5 10 10 understanding, ! b 4 4 10 10 -1 -2 s 3 3 • cross sections for inclusive 10 10 33 cm jet > " s/20) ! jet (E T bottom production and final 2 2 events/sec for L = 10 10 10 ! (nb) states with jets of hadrons 1 ! W 1 10 10 ! Z are near the top. 0 0 10 10 jet > 100 GeV) ! jet (E T -1 -1 10 10 • Higgs boson cross sections -2 -2 10 10 are at the bottom. -3 -3 10 10 ! t • Discovering such New Physics jet > " s/4) -4 ! jet (E T -4 10 10 requires a sophisticated, ! Higgs (M H = 150 GeV) -5 -5 10 10 quantitative understanding of QCD. -6 -6 10 10 ! Higgs (M H = 500 GeV) -7 -7 10 10 • In these lectures, we will develop the 0.1 1 10 tools necessary for such a task. " s (TeV) Quantum Chromodynamics - John Campbell - 3
QCD: why we care even more • If a Higgs-like signal is observed, to confirm its interpretation as the Higgs boson requires measurement of its couplings and quantum numbers. • need an accurate understanding of the production/decay mechanisms. • Hopefully, we will see more than just a Higgs boson. supersymmetry? extra dimensions? technicolor? • All of these models of New Physics introduce new particles that will (most likely) decay as they traverse the detectors, into “old” colored particles → QCD interactions. Quantum Chromodynamics - John Campbell - 4
The challenge of QCD L QCD = − 1 µ ν F µ ν � 4 F A A + q i ( iD ¯ / − m ) ij q j flavors Quantum Chromodynamics - John Campbell - 5
Tasks for today • Understand why the Lagrangian looks like this: • why color and why SU(3)? • Understand some features of this Lagrangian: • in practical terms, how does QCD differ from QED? • Understand how to use this Lagrangian: • how can we use it to make predictions? Quantum Chromodynamics - John Campbell - 6
Quarks and color up charm top • The quark model is a useful Q=+2/3 m u ~4 MeV m c ~1.5 GeV m t ~172 GeV way of categorizing mesons (baryons) in terms of two down strange bottom (three) constituent quarks. Q=-1/3 m d ~7 MeV m s ~135 MeV m b ~5 GeV • Simple picture must be amended due to, for example, Δ ++ =(u,u,u) in a symmetric spin state. • The baryons should obey the Pauli principle: the overall wavefunction should be antisymmetric. • In order to accommodate this, the antisymmetry should be carried by another quantum number: color. Baryon decuplet (S=3/2) • Observed particles are colorless. Quantum Chromodynamics - John Campbell - 7
Probing color • Subsequent realization that color could be probed directly in e + e - collisions. e + f • production of fermion pairs through a virtual photon sensitive to electric charge of fermion and the number - of degrees of freedom allowed. cross section f e - ~ Q f2 • Hence investigate quarks through “R-ratio”: R = σ ( e + e − → hadrons) � Q 2 = N c quark f σ ( e + e − → µ + µ − ) charge f sum over active quarks assume N c colors of quark (this is at least the most basic expectation - corrections later) • Each active quark is produced in N c colors: must be above the kinematic threshold for each quark in the sum, i.e. √ s > 2m q . Quantum Chromodynamics - John Campbell - 8
Broad support Experimental measurements for N c =3 �� 2 � 2 � � 2 � 2 � � − 1 − 1 R u,d,s = 3 × + + 3 3 3 = 2 � 2 � 2 R u,d,s,c = R u,d,s + 3 × 3 10 = 3 � 2 � − 1 R u,d,s,c,b = R u,d,s,c + 3 × 3 11 = 3 Quantum Chromodynamics - John Campbell - 9
QCD interactions • In QCD, the color quantum number is mediated by the gluon, analogous to the photon in QED. • it will be responsible for changing quarks from one color to another; as such it must also carry a color charge (not neutral, as in QED). • 1st try: mediating quark and anti-quark of 3 different colors → 3 x 3 = 9 gluons. R red (R) - or as gluon R - (RB) “color flow” B - - blue (B) B • In fact we should take six such combinations, plus three mutually orthogonal combinations of same-color states. - - - - RB RG (RR - BB)/ √ 2 - - - - - (RR + BB - 2 GG)/ √ 6 GB GR - - - - - (RR + BB + GG)/ √ 3 BR BG Quantum Chromodynamics - John Campbell - 10
QCD interactions • Since color is an internal degree of freedom, we expect invariance of the theory under rotations in this color space. • this requires that eight of our color combinations share the same coupling: - - - - RB RG (RR - BB)/ √ 2 - - - - - (RR + BB - 2 GG)/ √ 6 GB GR - - BR BG • the remaining combination only transforms into itself - it is a color singlet: - - - (RR + BB + GG)/ √ 3 • Such a combination is not present in QCD: we are left with 8 gluons. • The color charge of each gluon is represented by a matrix in color space. • the eight combinations result in eight matrices, T A , with A=1,..8. • a conventional choice is to write these in terms of the Gell-Mann matrices, which are just an extension of Pauli Matrices: T A = 1 2 λ A Quantum Chromodynamics - John Campbell - 11
Gell-Mann matrices 0 − i 0 0 1 0 1 0 0 λ 2 = λ 1 = λ 3 = i 0 0 1 0 0 0 − 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 − i 0 0 0 λ 4 = λ 5 = λ 6 = 0 0 0 0 0 0 0 0 1 1 0 0 i 0 0 0 1 0 1 0 0 0 0 0 1 λ 8 = λ 7 = 0 1 0 0 0 − i √ 3 0 0 − 2 0 i 0 • These matrices are Hermitian, ( λ A ) † = λ A ,and traceless. • only two diagonal matrices: the color singlet would not have been traceless. completely antisymmetric • They obey the two relations: set of real constants, f ABC λ A λ B � = 2 δ AB , λ A , λ B � = 2 if ABC λ C � � Tr Quantum Chromodynamics - John Campbell - 12
Color matrices • Translating back to color matrices, we have: = if ABC T C , T A , T B � T A T B � = T R δ AB � � Tr ( with T R = 1 / 2) • The first of these relations reflects that fact that: • the matrices T A are the generators of the SU(3) group, A=1,...,8; • the antisymmetric set, f ABC , contains the SU(3) structure constants. • The second relation is just a normalization convention. • The group structure is also characterized by two other relations: f ACD f BCD = C A δ AB � with C A = N c = 3 C,D with C F = N 2 c − 1 = 4 T A T A = C F 1 � 2 N c 3 A “Casimir” 3x3 identity matrix Quantum Chromodynamics - John Campbell - 13
Further support for SU(3) • These color sums are exactly the quantities which will appear when we compute cross sections involving QCD. • In particular, the cross section for 4-jet production in e + e - annihilation at LEP is sensitive to both C A and C F . • At this point, no one expected that SU(3) was not the correct description. • However, demonstrates that the group structure is an important phenomenological aspect - not just math! Quantum Chromodynamics - John Campbell - 14
The QCD Lagrangian • The quantum field theory of QCD is then based on the Lagrangian: L QCD = − 1 µ ν F µ ν � 4 F A q i ( iD µ γ µ − m ) ij q j A + ¯ flavors field strength tensor, gluon in the non-interacting case, the degrees of freedom Dirac term for quark d.o.f. • Color plays a crucial role in the Lagrangian: F A µ ν = ∂ µ A A ν − ∂ ν A A µ − g s f ABC A B µ A C ν µ : field for the spin-1 gluon (just like A A self-interaction term for the photon in QED, but with an gluon fields: called “non- extra color label) Abelian” since it arises from the SU(3) structure Quantum Chromodynamics - John Campbell - 15
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