8 Probing the Proton: Electron - proton scattering

Transcription

1 8 Probing the Proton: Electron - proton scattering Scattering of charged leptons by protons is an electromagnetic interaction. Electron beams have been used to probe the structure of the proton (and neutron) since the 1960s, with the most recent results coming from a high energy electron-proton collider called HERA at DESY in Hamburg. These experiments provide direct evidence for the composite nature of protons and neutrons, and measure the distributions of the quarks and gluons inside the nucleon. The results of e p e p scattering depend strongly on the wavelength λ = E/c. At very low electron energies λ>>r p, where r p is the radius of the proton, the scattering is equivalent to that from a point-like spin-less object At low electron energies λ r p the scattering is equivalent to that from a extended charged object At high electron energies λ<r p : the wavelength is sufficiently short to resolve sub-structure. Scattering from constituent quarks At very high electron energies λ << r p : the proton appears to be a sea of quarks and gluons. 8.1 Form Factors Extended object - like the proton - have a matter density ρ(r), normalised to unit volume: d 3 r ρ (r ) = 1. The Fourier Transform of ρ(r) is the form factor, F (q): F (q )= d 3 r exp{iq r } ρ(r ) F (0) = 1 (8.1) Cross section from extended objects are modified by the form factor: dσ dσ F (q ) 2 (8.2) dω extended dω point like For e p e p scattering we need form factors are required: F 1 to describe the distribution of the electric charge F 2 to describing the recoil of the proton 8.2 Elastic Scattering The elastic scattering of a pointlike spin-1/2 electron by a pointlike spin 1/2 target is described in the relativistic limit p e = E e by the Mott formula: dσ α 2 2 θ2 q2 dω = cos sin 2 θ (8.3) point 4p 2 e sin 4 θ 2m 2 2 p 2 51

2 In this formula θ and p e are in the Lab frame. The no recoil limit corresponds to a very massive target m 2 p q 2. In the non-relativistic limit p e m e this reduces to Rutherford scattering: dσ α 2 dω = NR 4m 2 eve 4 sin 4 θ 2 (8.4) e (p 1 ) ieγ µ q e (p 3 ) p(p 2 ) iek µ p(p 4 ) 2m p ν = Q 2 = q 2 (8.5) Note that ν>0 by energy conservation, so Q 2 > 0 and the mass squared of the virtual photon is negative, q 2 < 0! 8.3 Form Factors Deviations from the point-like Mott scattering are described in the no recoil limit by a form factor F (q 2 ), related to the finite size of a charge distribution inside the proton: dσ dω = dσ dω F (q 2 ) 2 (8.6) point At low q 2 the distances probed are large compared to the size of the proton, so the scattering still appears point-like with F (0) = 1. As q 2 gets larger the electron probes deeper into the proton, and F (q 2 ) is found to decrease. Mathematically, the form factor is the Fourier transform of the charge distribution inside the proton: F (q 2 )= ρ(x ) exp(iq x )d 3 x (8.7) The charge distribution ρ(x ) is assumed to be spherically symmetric, and normalised to one. It has a mean square radius <r 2 >: ρ(r)d 3 r =1 <r 2 >= r 2 ρ(r)d 3 r (8.8) The matrix element for elastic scattering is written in terms of electron and proton currents: M(e p e e 2 p)= (p 1 p 3 ) (ū 3γ µ u 2 1 )(ū 4 K µ u 2 ) (8.9) 52

3 where the proton is treated as an extended structure, with a current operator K µ that is more complex than the point-like γ µ : K µ = γ µ F 1 (q 2 )+ iκ p 2m p F 2 (q 2 )σ µν q ν + q ν F 3 (q 2 ) (8.10) In this general form there are three form factors F 1, F 2 and F 3 which are functions of q 2. However, electromagnetic current conservation δ µ (ū 4 K µ u 2 ) = 0 implies that F 3 = 0. F 1 is the electrostatic form factor, while F 2 is associated with the recoil of the proton. The anomalous magnetic moment of the proton is defined by: µ p = e(1 + κ p) 2m p κ p =1.79 (8.11) The differential cross section for elastic electron-proton scattering is dσ α 2 E 3 dω = F 21 κ2 q 2 F 2 2 θ2 q2 lab 4E1 2 sin 4 θ E 2 1 4m 2 2 cos (F p 2m κf 2 ) 2 sin 2 θ p 2 (8.12) where the two form factors F 1 and F 2 are functions of q 2 which parameterise the structure of the proton. They have to be determined by experiment. For a point-like spin 1/2 particle, F 1 = 1, κ = 0, and the above equation reduces to the Mott scattering result. It is common to use linear combinations of the form factors: G E = F 1 + κq2 F 4m 2 2 G M = F 1 + κf 2 (8.13) p which are referred to as the electric and magnetic form factors, respectively. The differential cross section can be rewritten as: dσ α 2 E 3 G 2 dω = E + τg 2 M lab 4E1 2 sin 4 θ E τ cos 2 θ 2 +2τG2 M sin 2 θ 2 (8.14) where we have used the abbreviation τ = Q 2 /4m 2 p. The experimental data on the form factors as a function of q 2 are in good agreement with a dipole fit: G E = G M β 2 2 = β =0.84GeV (8.15) µ p β 2 + Q 2 This corresponds to an exponential charge distribution: ρ(r) =ρ 0 exp( r/r 0 ) 1/r 2 0 =0.71GeV 2 <r 2 >=0.81fm 2 (8.16) The whole of the above discussion can be repeated for electron-neutron scattering, with similar form factors for the neutron. 53

4 Note that even though the neutron has zero total charge, it has an anomalous magnetic moment: µ n = eκ n 2m n κ n = 1.91 (8.17) The anomalous magnetic moments are themselves evidence that the proton and neutron are not pointlike Dirac fermions. 54

5 8.4 Deep Inelastic Scattering During inelastic scattering the proton can break up into its constituent quarks which then form a hadronic jet. At high q 2 this is known as deep inelastic scattering (DIS). e (p 1 ) ieγ µ e (p 3 ) q X p(p 2 ) The invariant mass, W, of the final state hadronic jet is: W 2 = m 2 p +2m p ν + q 2 (8.18) Since W = m p the four-momentum and energy transfer, q 2 and ν, are two independent variables in DIS, and it is necessary to measure E 1, E 3 and θ in the Lab frame to determine the full kinematics. It is useful to introduce two dimensionless variables, a parton energy x, and a rapidity y, which replace ν and q 2 x = Q2 2m p ν = q2 2m p ν y = p 2 q p 2 p 1 = ν E 1 (8.19) It is an exercise to show that the allowed kinematic ranges of these variables are 0 x 1 and 0 y Structure Functions The matrix element squared for DIS can be factorised into lepton and hadron currents: M 2 = e4 q 2 Lµν e (W hadron ) µν (8.20) where the hadronic part W µν hadron describes the inelastic breakup of the proton. As in elastic scattering, there are two independent form factors in W hadron, which are known as structure functions W 1 (ν, Q 2 ) and W 2 (ν, Q 2 ). In the Lab frame, the doubly differential cross section for deep inelastic scattering is: dσ α 2 de 3 dω = W lab 4E1 2 sin 4 θ 2 (ν, Q 2 )cos 2 θ 2 +2W 1(ν, Q 2 ) sin 2 θ (8.21)

6 Bjorken predicted that at high energy the structure functions should exhibit a property called scaling: m p W 1 (ν, Q 2 ) F 1 (x) νw 2 (ν, Q 2 ) F 2 (x) (8.22) where F 1 (x) and F 2 (x) are now functions of x alone. Note that the structure functions F 1 (x) and F 2 (x) in DIS are different from the elastic form factors F 1 (q 2 ) and F 2 (q 2 )! The experimental data for F 2 are shown in figure 8.1. For intermediate regions of x and Q 2 scaling holds, but at high Q 2 and low x there is a significant amount of scaling violation which we will discuss later. Figure 8.1: Structure function F 2 for large Q 2 and small x, as measured at HERA using collisions between 30 GeV electrons and 830 GeV protons. 56

7 8.6 The Parton Model The parton model was proposed by Feynman in 1969, to describe deep inelastic scattering in terms of point-like constituents inside the nucleon known as partons with an effective mass m<m p. Nowadays partons are identified as being quarks or gluons. e (p 1 ) e (p 3 ) m q m The parton model restores the elastic scattering relationship between q 2 and ν, with m replacing m p : ν + q2 2m = 0 (8.23) and the cross section for elastic electron-parton scattering is: dσ dω = Z2 α 2 lab 4E1 2 sin 4 θ 2 E 3 E 1 2 θ2 q2 cos θ 2m 2 sin2 2 where Z is the charge of the parton (+2/3 or 1/3 for quarks). (8.24) Effectively the DIS structure functions for scattering off a single parton, have become delta functions: 2W 1 = Q2 Q2 δ(ν 2m2 2m ) W 2 = δ(ν Q2 2m ) (8.25) It can be seen that the parton energy variable: x = is the fraction of the proton rest mass carried by the parton. Q2 2m p ν = m m p (8.26) 8.7 Parton Distribution Functions We introduce the parton distribution functions, f i (x), defined as the probability that a parton of type i carries a fraction x of the proton mass. The structure functions can then be written: W 1 (x) = F 1(x) F 1 (x) = 1 Z 2 i f i (x) (8.27) m p 2 W 2 (x) = F 2(x) ν F 2 (x) = i 57 i xz 2 i f i (x) (8.28)

8 0.8 CT10.00 PDFs x* f(x, µ=2 GeV) g/5 u d ubar dbar s c X 3*x 5/3 * f(x, µ=85 GeV) CT10.00 PDFs (area proportional to momemtum fraction) g u d ubar dbar sc X The structure functions F 1 and F 2 satisfy the Callan-Gross relation: 2xF 1 (x) =F 2 (x) = i xz 2 i f i (x) (8.29) The valence quark distributions are written u(x) and d(x). In a proton they have normalisations: 1 1 u(x)dx =2 d(x)dx = 1 (8.30) and the contribution of the valence quarks to F 2 are: For a neutron u and d are interchanged: 0 0 F p 2 (x) = 4 9 xu(x)+1 xd(x) (8.31) 9 F2 n (x) = 4 9 xd(x)+1 xu(x) (8.32) 9 58

9 There is an additional sea of quarks and antiquarks at low x, which also contribute to F 2, shown as ū(x) = d(x) = s(x). 8.8 Quark, Antiquark & Gluon Fractions The total number of valence quarks can be obtained from: F CC 3 (νn)= u(x) ū(x)+d(x) d(x) (8.33) 1 The fraction of sea quarks is obtained from: q(x)dx q(x)dx = 3R 1 3 R 0 F CC 3 (νn)dx = 3 (8.34) 0.1 (8.35) where R is the ratio of the antineutrino to neutrino total cross-sections. For a pointlike spin 1/2 fermion, f: R = σ( νf) σ(νf) = 1 (8.36) 3 whereas for an antifermion R = 3. If we integrate the area under the electron DIS structure function F 2 (x) we measure the total fraction carried by the valence and sea quarks. 1 0 F p 2 (x)dx = 4 9 f u f d We obtain the surprising results: 1 0 F p 2 (x)dx = F n 2 (x)dx = 4 9 f d f u (8.37) F n 2 (x)dx =0.12 (8.38) f u =0.36 f d =0.18 (8.39) The quarks constitute only 54% of the proton rest mass! The remainder is carried by gluons which must also be considered as partons. Understanding the gluon component of the proton presents a problem, since the lowest order scattering processes of electrons and neutrinos only couple to the quarks. There are additional scattering processes that involve gluons either in the initial state ( hard scattering ) or in the final state ( gluon emission ), where there is an additional hadronic jet from the gluon. These have different kinematics from the lowest order scattering. 59

10 Figure 8.2: Gluon emission from a final state quark γ q qg Figure 8.3: Hard scattering off an initial state gluon γ g q q 9 Quantum Chromodynamics Quantum Chromodynamics or QCD is the theory of strong interactions between quarks and gluons. It is a quantum field theory similar to QED but with some crucial differences. 9.1 Feynman rules for QCD amplitudes The calculation of Feynman diagrams containing quarks and gluons has the following changes compared to QED: The coupling constant α becomes α s (where α s = g s ). α s (q 2 ) decreases rapidly as a function of q 2. At small q 2 it is large, and QCD is a non-perturbative theory. At large q 2 QCD becomes perturbative like QED. A quark has one of three colour states (replacing electric charge). Antiquarks have anticolour states. A gluon propagator has one of eight colour-anticolour states. A quark-gluon vertex has a factor ig s λ a γ ν. As a consequence of the gluon colour states, gluons can self-interact in three or four gluon vertices. 60

11 Figure 8.4: Evidence for gluon emission from transverse momentum squared of jets in deep inelastic scattering of muons. The difference between the solid and dashed lines is due to gluon emission. a, µ b, ν q 1 q 3 q 2 b, ν c, λ c, λ a, µ d, ρ There is a complicated factor for a three-gluon vertex: g S f abc [g µν (q 1 q 2 ) λ + g νλ (q 2 q 3 ) µ + g λµ (q 3 q 1 ) ν ] (9.1) where f abc are known as colour structure constants. 9.2 SU(3) Colour Quarks carry one of three colour states, red r, green g, and blue b. Antiquarks carry the anticolour states, r, ḡ, and b. We define three quark eigenstates r, g, and b: 1 c r = 0 0 c g = 1 0 c b = 0 (9.2) The colour states are related by an SU(3) symmetry group which defines rotations in colour. Strong interactions are invariant under SU(3) colour. The coupling is independent of the colour states of the quarks and gluons Gluon States At a quark-gluon vertex a quark can either change colour or remain the same. A change of colour requires that gluons also have colour states. 61

12 This differs from QED where the photon has no charge. A gluon carries a colour and an anti-colour state c i c j. Naively you would think that there are nine possible gluon states: r b b r rḡ g r bḡ g b r r b b gḡ of which the last three are apparently colour neutral. The symmetry properties of the SU(3) group actually classify the states as 3 3 = 8 1. There is a colour-octet of allowed gluon states which are colour antisymmetric: G 1 = 1 2 r b + b r G 2 = 1 2 r b b r G 4 = 1 2 (rḡ + g r) G 5 = 1 2 (rḡ g r) G 6 = 1 2 bḡ + g b G 7 = 1 2 bḡ g b G 3 = 1 2 r r b b G 8 = 1 6 r r + b b 2gḡ and a colour-singlet, which is the only colour symmetric state: (9.3) G 0 = 1 3 r r + gḡ + b b (9.4) The singlet state G 0 is forbidden for gluons, because it would give rise to long-range strong interactions The λ matrices There are eight λ matrices which are the generators of SU(3): i λ 1 = λ 2 = i 0 0 λ 3 = λ 4 = λ 7 = λ 5 = i 0 i i i 0 0 λ 8 = 1 3 λ 6 = The λ matrices can be identified with the eight gluon states. The operators λ 1,2,4,5,6,7 are the ones that change quark colour states. The diagonal operators λ 3,8 are the ones that do not change the colour states. The SU(3) structure constants describe the commutators of the λ matrices: λ a,λ b =2i c f abc λ c (9.5) 62

13 Non-zero values are: f 123 =1 f 458 = f 678 = 3 2 (9.6) f 147 = f 165 = f 246 = f 257 = f 345 = f 376 = 1 2 (9.7) The f abc are related to each other by the property that thay are antisymmetric under the interchange of any pair of indices, e.g. f 132 = f 213 = f 321 = 1 and f 231 = f 312 =1 from the value f 123 = 1. The f abc not covered by permutations of the values given above are zero. More familiar are the three Pauli matrices σ which are the generators of SU(2). They can be seen as subsets of λ 1,4,6 (σ 1 ), λ 2,5,7 (σ 2 ), and λ 3 (σ 3 ). Note that the Pauli matrices have only one structure constant ijk = f Non-Abelian Gauge Symmetry A rotation in colour space is written as: U = e iαa λa (9.8) where α a are the equivalent of angles in colour space. QCD amplitudes can be shown to be invariant under this non-abelian gauge transformation. The transformations of the quark and gluon states are: q (1 + iα a λ a )q G a µ G a µ 1 g s µ α a f abc α b G c µ (9.9) 9.3 Quark-antiquark scattering p 2,c 2 g S p 4,c 4 g p 1,c 1 g S p 3,c 3 The matrix element for quark-antiquark scattering is written: M = ū 3 c ig s 3 2 λa γ µ [u 1 c 1 ] gµν δ ab q 2 v 2 c ig s 2 2 λb γ ν [v 4 c 4 ] (9.10) M = α s 4q 2 [ū 3γ µ u 1 ][ v 2 γ µ v 4 ](c 3λ a c 1 )(c 2λ a c 4 ) (9.11) 63

14 This looks very similar to the matrix element for electron-positron scattering, except that α is replaced by α s and there is a colour factor c f : c f = 1 2 (c 3λ a c 1 )(c 2λ a c 4 ) (9.12) The calculation of the c f can be found in Halzen & Martin P : quark states gluon states c f rr rr G 7,G 8 +2/3 r r r r G 7,G 8 2/3 rb rb G 8 1/3 rb br G 1 +1 r r b b G 1 1 r b r b G 8 +1/3 9.4 Strong Coupling Constant α s The coupling strength α s is large, which means that higher order diagrams are important. At low q 2 higher order amplitudes are larger than the lowest order diagrams, so the sum of all diagrams does not converge, and QCD is non-perturbative. At high q 2 the sum does converge, and QCD becomes perturbative. The coupling constant α s can be renormalised at a scale µ in a similar way to α: α(q 2 )= 1 α(µ2 ) 3π α(µ 2 ) log q 2 µ 2 (QED) In QCD the renormalization is attributed both to the colour screening effect of virtual q q pairs and to anti-screening effects from gluons since they have colour-anticolour states. This leads to a running of the strong coupling constant: α s (q 2 )= α s (µ 2 ) 1+βα s (µ 2 ) ln q 2 µ 2 β = where N c = 3 is the number of colours, and N f flavours which is a function of q 2. (11N c 2N f ) 12π (QCD) 6 is the number of active quark From the positive sign of β it can be seen that the anti-screening effect of the gluons dominates, and the strong coupling constant decreases rapidly as a function of q 2. The running of the coupling constant is usually written: α s (q 2 )= 12π (33 2N f )ln q 2 Λ 2 QCD (9.13) where Λ QCD = 217 ± 25 MeV is a reference scale which defines the onset of a strongly coupled theory where α s 1. There are many independent determinations of α s at different scales µ, which are summarized in the figures on the next page. The results are compared with each other by adjusting them to a common scale µ = M Z where α s =0.1184(7). 64

15 Figure 9.1: Top: Running of the strong coupling constant α s (q 2 ). Bottom: Measurements of α s (M Z ), the equivalent strong coupling at q 2 = M 2 Z. 9.5 Confinement and Asymptotic Freedom In deep inelastic scattering experiments at large q 2, the quarks and gluons inside the proton can be observed as partons. This property is known as asymptotic freedom. It is associated with the small value of α s (q 2 ), which allows the strong interaction corrections from gluon emission and hard scattering to be calculated using a perturbative expansion of QCD. The perturbative QCD treatment of high q 2 strong interactions has been well established over the past 20 years by experiments at high energy colliders. In contrast, at low q 2 the quarks and gluons are tightly bound into hadrons. This is known as confinement. For large α s QCD is not a perturbative theory and different mathematical methods have to be used to calculate the properties of hadronic systems. A rigorous numerical approach is provided by Lattice gauge theories. The breakdown of the perturbation series is due to the colour states of the gluons and the contributions from three-gluon and four-gluon vertices which do not have an analogue in QED. A pictorial way of thinking of this is as a colour flux tube, connecting the quarks in a hadron. Starting from the familiar dipole field between two charges, imagine the colour field lines as being squeezed down into a tight line between the two quarks: The additional energy density stored in the flux tube as the q q pair are pulled apart can be parametrized by a string tension k, and the QCD potential can be written as the sum of a Coulomb-like potential and the string potential energy: V q q (r) = 4 α s + kr (9.14) 3 r As a consequence of the positive term in the strong interaction potential, a q q pair cannot be separated since infinite energy would be required. Instead the flux tube can 65

16 break creating an additional q q pair in the middle which combines with the original q q pair to form two separate hadrons. There are no free quarks or gluons! 66

17 10 QCD at Colliders 10.1 e + e Hadrons In the electromagnetic process e + e q q, the flavour of the q and q must be the same. This is known as associated production. The final state quark and antiquark each form a jet by a process known as fragmentation. The two jets follow the directions of the quarks, and are back-to-back in the center-of-mass system. e + (p2) q(p4) γ e (p1) q(p3) There is an initial colour flux between the quark and antiquark, but this is broken by the energy of the collision. The quarks then radiate gluons, which couple to quarkantiquark pairs, eventually forming a large number of bound states known as hadrons. The fragmentation process is characterised by the transverse momentum, p T, of the hadrons relative to the jet axis. The cross section for e + e q q can be calculated using QED. The matrix element is similar to e + e µ + µ apart from the final state charges and a colour factor, N c =3 σ(e + e q q) =N c Z 2 q σ(e + e µ + µ ) (10.1) Summing over all the active quark flavours gives the ratio: R σ(e+ e hadrons) σ(e + e µ + µ ) 67 =3 q Z 2 q (10.2)

18 Note the importance of the factor 3. The measurement of the ratio R shows that there are three colour states of quarks! σ and R in e + e Collisions ρ ω φ ρ J/ψ ψ(2s) Υ Z σ[mb] J/ψ ψ(2s) Υ Z R ω φ ρ ρ s [GeV] Figure 41.6: World data on the total cross section of e + e hadrons and the ratio R(s) =σ(e + e hadrons, s)/σ(e + e µ + µ,s). σ(e + e hadrons, s) is the experimental cross section corrected for initial state radiation and electron-positron vertex loops, σ(e + e µ + µ,s)=4πα 2 (s)/3s. Data errors are total below 2 GeV and statistical above 2 GeV. The curves are an educative guide: the broken one (green) is a naive quark-parton model prediction, and the solid one (red) is 3-loop pqcd prediction (see Quantum Chromodynamics section of this Review, Eq.(9.7) or, for more details, K. G. Chetyrkin et al., Nucl. Phys. B586, 56 (2000) (Erratum ibid. B634, 413 (2002)). Breit-Wigner parameterizations of J/ψ, ψ(2s), and Υ(nS),n = 1, 2, 3, 4 are also shown. The full list of references to the original data and the details of the R ratio extraction from them can be found in [arxiv:hep-ph/ ]. Corresponding computer-readable data files are available at (Courtesy of the COMPAS (Protvino) and HEPDATA (Durham) Groups, May 2010.) See full-color version on color pages at end of book Gluon Jets Sometimes one of the quarks radiates a hard gluon which carries a large fraction of the quark energy, e + e q qg. In this process there are three jets, with one coming from the gluon. This provided the first direct evidence for the gluon in The jets are no longer back-to-back, and the gluon jet has a different p T distribution from the quark jets. The production rate of three-jet versus two-jet events is proportional to α s. Including higher order processes, the ratio R is: R =3 Zq 2 1+ α s(q 2 ) π q (10.3) This is used to measure the strong coupling constant as a function of q Hadron Colliders Proton-antiproton colliders were used to discover the W and Z bosons at CERN in the 1980s, and the top quark at the Tevatron (Fermilab) in In 2010 the Large 68

19 69

20 Hadron Collider (LHC) began operation at CERN. This collides two proton beams with an initial CM energy of 7 TeV. This is half the eventual design energy, but is already the world s highest energy collider Parton Level Scattering Figure 10.1: Stylized representation of a hadron-hadron collision from Jet Physics at the Tevatron by A.Bhatti & D.Lincoln, arxiv: Each initial hadron provides one parton, which can be either a quark, antiquark or gluon. At lower energy colliders the valence quarks dominate, whereas at the LHC most scattering is between low x gluons. The remnants of the two protons form backward and forward jets similar to the jet in electron-proton DIS. These remnants are known as the underlying event, and are usually regarded as background. The partons undergo hard scattering, a strong interaction with a large q 2 transfer. This leads to two or more quarks or gluons in the final state. The amplitude can be calculated using perturbative QCD. The final state quarks and gluons form hadronic jets in a similar way to the quarks (and gluons) in e + e q q(g). Additional gluons can be emitted from either the initial or final state quarks (and gluons). A distinction is made between hard gluons, and soft collinear gluons. Most collisions have no missing energy and no high mass final state particles W, Z, H, b, t. These events are said to be minimum biased. A large transverse momentum (high p t ) jet or isolated charged lepton is a signature for the production of a high mass final state particle. 70

21 Large missing transverse energy (missing E t ) is a signature for neutrinos or other weakly interacting neutral particles Description of Jets The hadrons that constitute a jet are summed to provide the kinematic information about the jet: Jet energy E = i E i and momentum p = i p i The direction of p defines the jet axis. Jet invariant mass W 2 = E 2 p 2 Transverse energy flow within the jet p T 2 = i p i p 2 Transverse jet momentum p t = (p 2 x + p 2 y), relative to beam axis. Pseudorapidity y = ln [(E + p z )/(E p z )]/2], related to cos θ. This goes to along the beam axis, and is zero at 90. Azimuthal angle φ = tan 1 (p y /p x ) In the case of e + e q q(g) it is straightforward to decide which hadrons belong to which jets. At a hadron collider a minimum bias event has at least two jets from the hard scatter, as well as two jets from the underlying event. With large numbers of jets it is hard to decide which hadrons belong to which jets. What is done is to define a radial distance from the jet axis for each hadron: R 2 i =(y i y jet ) 2 +(φ i φ jet ) 2 (10.4) An initial estimate is made of y jet and φ jet using a subset of the highest momentum hadrons. Then a cone radius R is defined containing all the i hadrons that make up the jet. These are then used to recalculate y jet and φ jet. The process can be iterated until it converges. The choice of radius for the cone is critical. A typical value is R =0.7. The cross-section for jet production as a function of p t and y can be measured and compared to pertubative QCD calculations. These are usually carried out to next-toleading order (NLO), with one additional hard gluon, and sometimes to next-to-nextto-leading order (NNLO). The measurements are used to constrain gluon and quark parton density functions, and to determine the strong coupling constant α s (q 2 ) Jet Fragmentation The initial stages of jet fragmentation involve the emission of soft collinear gluons from the scattered partons. These can still be understood using perturbative QCD. Eventually the energy scales of the fragmentation become too low, and hadronisation into bound states begins. This part is modelled in several different ways: 71

22 PYTHIA breaks colour flux tubes between quarks and gluons. HERWIG clusters the quarks and gluons according to colour matching. SHERPA uses a cutoff in the transverse energy of a quark or gluon within a jet to stop the production of lower energy jet fragments. The fragmentation models are compared to data and tuned appropriately. The most relevant jet parameter for this is the tranverse energy flow within the jet p T 2, although the hadron multiplicity and individual momenta can also be looked at Production of Heavy Quarks Discovery of the J/ψ In November 1974, two groups simultaneously announced the discovery of a narrow J/ψ resonance with a mass of 3100 MeV. At SLAC the resonance was observed in e + e q q, while at Brookhaven the inverse process was studied, e + e pair production by a proton beam on a Beryllium target. Figure 10.2: Discovery of J/ψ in 1974 at Brookhaven (left) and SLAC (right). The J/ψ is identified as the lightest of the c c states, known as charmonium. Its width Γ= MeV, is much smaller than the experimental resolution, which is a surprise b b Production The narrow b b bound state Υ(1S) with mass 9.5 GeV, was first identified in hadron collisions at Fermilab in It is also produced in e + e collisions in a similar way to 72

23 the J/ψ. The production of b quarks has a significant role at hadron colliders, because it is possible to tag jets that come from b quarks. This is done by measuring detached decay vertices due to the finite lifetime of the b quark, τ b =1.5ps. Measurements of the production cross-section for b jets can then be compared with perturbative QCD calculations. These are particularly useful for constraining the antiquark and gluon parton density functions, since b b pairs are produced either by quark-antiquark annihilation or by gluon fusion. An important use of b tagging is to identify jets that have been produced by the decay of a heavy particle into a b quark. Examples of these are t bw, Z b b, and eventually for a light Higgs boson, H b b The Top Quark The top quark was not discovered until 1995, because of its unexpectedly large mass m t = ± 1.6 GeV. It was observed at the Tevatron(Fermilab) in proton-antiproton collisions at a centre-of-mass energy of 1.8 TeV. The main production mechanism is quark-antiquark annihilation giving t t, followed by the decays t bw : t W + b t W b W + + ν W q q (10.5) There are no narrow t t resonances because of the short lifetime of the top quark. The total cross-section for t t production at the Tevatron is 7.5 ± 0.5 pb. There is also single top production via a W coupling to the valence quarks, W t b, with a cross-section 2.3 ± 0.6 pb. Recently the CDF experiment at the Tevatron has measured a large forward-backward asymmetry in t t production from proton-antiproton collisions. This is not expected in the Standard Model! (see preprint arxiv: ) 73

24 11 Mesons and Baryons 11.1 Formation of Hadrons As a result of the colour confinement mechanism of QCD only bound states of quarks are observed as free particles, known as hadrons. They are colour singlets, with gluon exchange between the quarks inside the hadrons, but no colour field outside. Mesons are formed from a quark and an antiquark with colour and anticolour states with a symmetric colour wavefunction: χ c = 1 3 (r r + gḡ + b b) (11.1) Baryons are formed from three quarks, all with different colour states, with an antisymmetric colour wavefunction: χ c = 1 6 (rgb rbg + gbr grb + brg bgr) (11.2) There may be other types of hadrons which are colour singlets: four quark states (q qq q), pentaquarks (qqqq q), hybrid mesons (q qg), or glueballs (gg, ggg). There is some experimental evidence for these, but it is not yet convincing Isospin Strong interactions are the same for u and d quarks. This is an SU(2) flavour symmetry, known rather confusingly as isospin. The u and d quarks are assigned to an isospin doublet: u : I = 1 2, I 3 =+ 1 2 d : I = 1 2, I 3 = 1 2 (11.3) The lowest lying meson states are the pions, which are pseudoscalars with spin J =0 ( ). They form an I= 1 triplet: π + [1, 1] = u d π 0 [1, 0] = 1 2 (uū d d) π [1, 1] = dū (11.4) There is also an I= 0 singlet, the eta meson: η [0, 0] = 1 2 (uū + d d) (11.5) For baryons the lowest lying states are the J =1/2 proton and neutron: p [1/2, +1/2] = uud n [1/2, 1/2] = ddu (11.6) Isospin symmetry means that protons, neutrons and pions all have the same strong interactions. 74

25 11.3 SU(3) Flavour Symmetry In 1961, before the discovery of quarks, Gell-Mann proposed a structure to classify the hadrons, which was known as the eightfold way. This classification is now understood as an approximate SU(3) flavour symmetry between the three lightest quarks u, d, s. This symmetry is broken by the mass of the strange quark. The s quark is assigned a strangeness, S= 1 and I= 0, and the u and d quarks have S= 0 and I= 1/2. The mathematics of SU(3) flavour is identical to that of SU(3) colour which was discussed in lecture 8. The eight generators of the group are the λ a matrices. The matrices λ 1, λ 2, and λ 3 form the SU(2) isospin part of SU(3) flavour which deals with the u and d quarks. The two diagonal matrices, λ 3 and λ 8 are related to eigenstates of isospin, I 3, and hypercharge, Y, respectively: I 3 = 1 2 λ3 Y=S+B= 1 3 λ 8 (11.7) where B= 1/3 is the baryon number of the quarks. The quarks charges can be written as: Q=I 3 + Y (11.8) 2 The quark and antiquark flavours can be represented as 2-dimensional SU(3) multiplets of isospin and hypercharge: 3 d!!!!! i Y 2/3 1/3!!!!! 1+- i 2-1/2 1/2-1/3-2/3 s!!!!!! + - i 4 5 u I 3-3 u i!!!!! 7 Y 2/3-1/3-2/3 s!!!!!! - + i 4 5 1/3-1/2 1/2!!!!! 1+- i 2 d I Hadron Multiplets The SU(3) multiplet structure for the q q meson states is 3 3 =8 1, i.e. it contains a flavour octet and a flavour singlet. The lowest lying J =0( ) pseudoscalar and J =1 ( ) vector mesons are shown on the next pages. Three of the nine J = 0 mesons are neutral with I 3 = Y = 0. These states are π 0, η 1 and η 8, where η 1 is the SU(3) singlet. The physically observed states, η and η, are mixtures of η 1 and η 8. For the J = 1 mesons, the three neutral states that are observed are ρ 0, ω and φ, where the φ is purely an s s state. According to SU(3) symmetry, the baryon states are classified as = , i.e. the three light quarks form a decuplet, two octets and a singlet. The lowest lying 75

26 baryon octet with J=1/2 ( ) contains p, n, Λ, Σ +,Σ 0,Σ,Ξ 0 and Ξ states. The decuplet has J =3/2 ( ). It consists of, Σ,Ξ states and the Ω which is an sss state and Proton Wavefunctions The flavour wavefunctions for decuplet baryons are symmetric, e.g.: 1 6 (dus + uds + sud + sdu + dsu + usd) (11.9) The ++ (uuu) has symmetric (S) flavour, spin and orbital wavefunctions, and an antisymmetric (A) colour wavefunction. It is overall antisymmetric, as it must be for a system of identical fermions. The wavefunctions for the lowest baryon octet have a combined symmetry of the flavour and spin parts: uud( + 2 )+udu( + 2 )+duu( + 2 ) (11.10) Hence the proton also has an overall antisymmetric wavefunction, ψ: Hadron χ c χ f χ S χ L ψ ++ A S S S A p A S S A Note that there are no J=1/2 states uuu, ddd, sss because the flavour symmetric part would have to be combined with an antisymmetric spin part Hadron Masses* In the MS renormalization scheme of QCD the masses of the u and d quarks, m u and m d are only a few MeV, and the mass of the s quark, m s 100 MeV. In discussing hadron masses and magnetic moments we use constituent quark masses m u = m d 300 MeV and m s 500 MeV. Gell-mann and Okubo introduced a semi-empirical mass relation for baryons: M = M 0 +YM 1 + M 2 I(I + 1) Y 2 /4 (11.11) For the J =3/2 baryon decuplet Y = B + S = 2 (I 1), and the formula reduces to: For the J =1/2 baryon octet the masses are related by: M = M +(m d m s )S (11.12) 3M Λ + M Σ =2M N +2M Ξ (11.13) 76

27 Y,S +1 K 0 (d s) K + (u s) Pseudoscalar mesons J PC =0 + 0 π (dū) π 0 η 8 η 1 π + (u d) π 0 =(d d uū)/ 2 η 8 =(d d + uū 2s s)/ 6 η 1 =(d d + uū + s s)/ 3 1 K (sū) K 0 (s d) I3 1 1/2 0 +1/2 +1 Y,S +1 K 0 (d s) K + (u s) Vector mesons J PC =1 0 ρ (dū) ρ 0 ω φ ρ + (u d) ρ 0 =(d d uū)/ 2 ω =(d d + uū)/ 2 φ = s s 1 K (sū) K 0 (s d) I3 1 1/2 0 +1/

28 M MeV/c 2 S Y 0 +1 ddd 0 ddu + duu ++ uuu Σ dds Σ 0 dus Σ + uus Ξ dss Ξ 0 uss Ω sss 1672 I 3 3/2 1 1/2 0 +1/ /2 S 0 Y +1 n ddu p uud M MeV/c Σ dds Σ 0 Λ uds Σ + uus Ξ dss Ξ 0 uss 1318 I 3 1 1/2 0 +1/

29 These formulae are accurate to 1%. The semi-empirical mass relation fails for mesons, but the mass differences between J=0 and J=1 mesons and between J=1/2 and J=3/2 baryons can be understood as hyperfine splitting due to spin-spin coupling between quarks Resonances The J=1 vector mesons decay to the J=0 pseudoscalar mesons through strong interactions in which a second quark-antiquark pair is produced, e.g. ρ 0 π + π. Similarily the J=3/2 baryons (except for the Ω ), decay to the J=1/2 baryons through strong interactions, e.g. ++ π + p. The decay widths for these processes are 100 MeV, corresponding to lifetimes of s. Hadronic states that can decay through strong interactions are known as resonances. They are observed as broad mass peaks in the combinations of their daughter particles. Figure 11.1: The ++ resonance observed in π + p scattering. Figure 11.2: Quark line diagram of a ρ + meson coupling to π + π 0. The Feynman diagram for resonance production can be drawn in reduced form showing just the quark lines, which are continuous and do not change flavour. Since the process is a low energy strong interaction, there are lots of gluons coupling to the quark lines which form a colour flux between them. 79

30 11.6 Heavy Quark States The c and b quarks can replace the lighter quarks to form heavy hadrons. However, the t quark is too short-lived to form observable hadrons. The equivalent of the K and K mesons are known as charm, D ( ), and beauty, B ( ) mesons. There are three possible D states (and antipartners): and four possible B states: D + (c d) D 0 (cū) D + s (c s) (11.14) B + ( bu) B 0 ( bd) B 0 s ( bs) B + c ( bc) (11.15) Similarly there are heavy baryons such as Λ c and Λ b, where a heavy quark replaces one of the light quarks. The masses of heavy quark states are dominated by the heavy quark constituent masses m c 1.5 GeV and m b 4.6 GeV Charmonium and Bottomonium The J/ψ meson is identified as the lowest 3 S 1 bound state of c c. The width of this state is narrow because M J/ψ < 2M D, so it cannot decay into a D D meson pair. It decays to light hadrons by quark-antiquark annihilation into gluons. The quark line diagram is drawn as being disconnected: c c Light hadrons There is a complete spectroscopy of c c charmonium states with angular momentum states equivalent to atomic spectroscopy. Only the states with M(c c) < 2M D have narrow widths. A similar spectroscopy is observed for b b bottomonium states. 80

31 (2S) c (2S) c1 (1P) h c (1P) c2 (1P) hadrons, 0 hadrons c0 (1P) hadrons hadrons 0 hadrons c (1S) J/ (1S) hadrons hadrons radiative J PC! 0 " 1 0 "" 1 "" 1 " 2 "" Figure 11.3: Spectroscopy of lightest charmonium states with M(c c) < 2M D. Note that out of these states only the J/ψ and ψ(2s) are produced in e + e collisions. The singlet η c states, and the P-wave χ c and h c states, are observed via electromagnetic transitions. 81

32 12 Weak Decays of Hadrons 12.1 Selection Rules for Decays There are selection rules to decide if a hadronic decay is weak, electromagnetic or strong, based on which quantities are conserved: For strong interactions total isospin I, and its projection I 3 are conserved. Strong interactions always lead to hadronic final states. For electromagnetic decays I is not conserved, but I 3 is. There are often photons or charged lepton-antilepton pairs in the final state. For weak decays I and I 3 are not conserved. They are the only ones that can change quark flavour, e.g. S = 1 when s u. Weak decays can lead to hadronic, semileptonic, or leptonic final states. A neutrino in the final state is a clear signature of a weak interaction. The lightest hadrons cannot decay to other hadrons via strong interactions. The π 0, η and Σ 0 decay electromagnetically. The π ±, K ±, K 0, n, Λ,Σ ±, Ξ and Ω decay via flavour-changing weak interactions, with long lifetimes Pion and Kaon Decays Neutral Pion Decay π 0 γγ The π 0 meson is a (uū d d) state which decays through the annihilation of its uū or d d quarks into a pair of photons. This is an electromagnetic decays that conserves I 3 and charge conjugation symmetry C. The quantum numbers are [I, I 3 ] π 0 = [1, 0], C π 0 = +1 and I γ = 0, C γ = 1, so the allowed decay is to two photons. The decay to three photons has not been observed, B(π 0 3γ) < The lifetime τ π 0 = (8.4 ± 0.6) s, is the shortest lifetime ever measured experimentally. The decay width is: Γ(π 0 2γ) =α 2 N 2 c g 2 π 0m3 π 0 (12.1) where N c = 3 is the colour factor, and g π 0 = 92 MeV is the π 0 decay constant, which represents the probability that the quark-antiquark pair will meet inside the meson and annihilate. 82

33 Charged Pion Decay π + µ + ν µ u µ + (p 3 ) f π W Charged pions decay mainly to a muon and a neutrino: d ν µ (p 4 ) π + µ + ν µ π µ ν µ (12.2) This occurs through the annihilation of the u d quark-antiquark into a charged W boson, which is described in terms of a pion decay constant, f π. The matrix element for the decay is: M = V udg F f π q µ ū µ γ µ (1 γ5 )u νµ (12.3) The matrix element squared in the rest frame of the pion is: and the total decay rate is: M 2 =4 V ud 2 G 2 F f 2 πm 2 µ[p 3.p 4 ] (12.4) Γ= 1 τ π = V ud 2 G 2 F 8π f 2 πm π m 2 µ 2 1 m2 µ (12.5) m 2 π From the charged pion lifetime, τ π + = 26ns, the pion decay constant can be deduced, f π = 131 MeV. Note that this is very similar to the charged pion mass, m π + = MeV. The decay of a charged pion to an electron and a neutrino π + e + ν e, is helicity suppressed. This can be seen from the factor m 2 µ/m 2 π in the decay rate. Replacing m µ with m e reduces the decay rate by a factor The suppression is associated with the helicity states of neutrinos, which force the spin-zero π + to decay to a left-handed neutrino and a left-handed µ +, or the π to decay to a right-handed antineutrino and a righthanded µ. The precise experimental measurement B(π + e + ν e )=1.230(4) 10 4, is a test of the lepton universality of weak couplings Charged Kaon Decays The charged Kaon has a mass of 494MeV and a lifetime τ K + = 12ns. Its main decay modes are: Purely leptonic decays, where the su annihilate, similar to π + decay: B(K + µ + ν µ ) = 63.4% (12.6) The Kaon decay constant f K flavour symmetry breaking. = 160 MeV is 20% larger than f π due to SU(3) 83

34 Semileptonic decays, where s ū and the W + boson couples to either e + ν e or µ + ν µ : B(K + π 0 + ν )=8.1% (12.7) Semileptonic decays satisfy the Q = S rule, i.e. the change in strangeness is equal to the lepton charge, K + + and K. Hadronic decays, where s ū and the W + boson couples to u d: B(K + π + π 0 ) = 21.1% (12.8) B(K + π + π + π )=5.6% (12.9) Hadronic weak decays prefer an isospin change I= 1/2 to I= 3/2. The origins of this rule are not well understood The Cabibbo Angle The relationship between the couplings of the W boson to u d and u s quarks is described by the Cabibbo angle, θ C = 12.7 : d cos θc sin θ s = C d (12.10) sin θ C cos θ C s and the vertex couplings which modify the usual weak coupling G F are: V ud =cosθ C =0.974(1) V us = sin θ C =0.220(3) (12.11) A factor V us 2 appears in all the decay rates for charged Kaons Decays of Heavy Quarks Hadrons with charm decay mainly by weak c s transitions, while beauty hadrons decay via weak b c transitions. There are also suppressed c d and b u couplings to the W. The lifetimes are quite short but measurable: τ D 0 = 410(2)fs and τ B 0 =1.53(1)ps Semileptonic decays Semileptonic decays of heavy quarks are used to determine the W couplings. D Kν and D πν give V cs = cos θ C and V cd = sin θ C. More interestingly B D ( ) ν is used to measure V cb 10 2, and B πν is used to measure V ub The extraction of these couplings requires a knowledge of hadronic form factors, which describe the B D ( ) or π transitions in terms of the overlap of initial and final state hadronic wavefunctions. These are usually calculated non-perturbatively using Lattice QCD. 84

35 B τν and D s µν The purely leptonic decays of heavy mesons have small branching fractions, B(D s µν) = and B(B + τν)= The decay B τν was first observed by the Belle experiment in It measures the combination f 2 B V ub 2, where f B 190MeV is the B decay constant, also obtained from Lattice QCD. The measured branching fraction is larger than expected from other determinations of V ub b sγ The flavour-changing neutral current (FCNC) transitions b s, b d and c u are not allowed at first order in the Standard Model. They do occur as a second order weak interaction through a penguin diagram containing a loop with a W boson and a t quark (or b quark in the case of c u). The decay b sγ was first observed by the CLEO experiment in It has a branching fraction B(b sγ) = This is a strong constraint on new physics beyond the Standard Model, because new heavy particles could replace the W and t in the loop, giving significant changes to the decay rate. The decay b dγ has recently been observed by the BaBar and Belle experiments, but c u transitions have not yet been seen The Cabibbo-Kobayashi-Maskawa (CKM) Matrix The 3 3 CKM matrix is an extension of the 2 2 Cabibbo matrix to include the heavy quarks: d V ud V us V ub d s = V cd V cs V cb s (12.12) b V td V ts V tb b where the mass eigenstates of the Q = 1/3 quarks are d, s, b and the weak eigenstates which couple to the W are d,s,b. The matrix is unitary, and its elements satisfy: Vij 2 =1 Vij 2 = 1 (12.13) i V ij V ik =0 i j V ij V kj = 0 (12.14) By considering the above unitarity constraints it can be deduced that the CKM matrix can be written in terms of just four parameters. Starting from the definition of the Cabibbo angle, the four CKM parameters can be writen as three angles s i = sin θ i, c i =cosθ i, and a complex phase δ: c 1 s 1 c 3 s 1 s 3 s 1 c 3 c 1 c 2 c 3 s 2 s 3 e iδ c 1 c 2 s 3 + s 2 c 3 e iδ s 1 s 2 c 1 s 2 c 3 c 2 s 3 e iδ c 1 s 2 s 3 + c 2 c 3 e iδ j (12.15) It is more common to see the CKM matrix written as an expansion in powers of the Cabibbo angle λ = sin θ C. This is known as the Wolfenstein parametrization. It satisfies 85

36 unitarity to O(λ 3 ): 1 λ 2 /2 λ Aλ 3 (ρ iη) λ 1 λ 2 /2 Aλ 2 Aλ 3 (1 ρ iη) Aλ 2 1 (12.16) The relative sizes of the CKM elements in powers of λ vary from 1 on the diagonal, to λ 3 in the off-diagonal corners. Note that the complex phase η is associated with the smallest elements of the matrix V ub and V td. Figure 12.1: Constraints on the ρ and η parameters of the CKM matrix. Plot comes from the CKM fitter group at More details on the measurements can be found at the Heavy Flavour Averaging Group The measured values of the CKM elements are: V cs =0.97(12) from D Kν. V cd =0.224(12) from D πν and neutrino production of charm νd c. V cb =0.042(1) from b cν. This determines the parameter A. V ub =0.0044(4) from b uν. V td and V ts 0.04 from m d and m s in B d and B s mixing (see Lecture 13). V tb 1 from top decays at the Tevatron. The constraints on the ρ and η parameters of the CKM matrix are shown in Figure 12. We will discuss K and the angles α, β and γ of the unitarity triangle in Lecture 13. From a full fit of all the constraints on the CKM matrix the Wolfenstein parameters are: λ =0.225(1) A =0.81(2) ρ =0.14(3) η =0.34(2) These are fundamental parameters of the Standard Model. 86

37 13 Symmetries in Particle Physics Symmetries play in important role in particle physics. The mathematical description of symmetries uses group theory, examples of which are SU(2) and SU(3): A serious student of elementary particle physics should plan eventually to study this subject in far greater detail. (Griffiths P.115) There is a relation between symmetries and conservation laws which is known as Noether s theorem. Examples of this in classical physics are: invariance under change of time conservation of energy invariance under translation in space conservation of momentum invariance under rotation conservation of angular momentum In particle physics there are many examples of symmetries and their associated conservation laws. There are also cases where a symmetry is broken, and the mechanism has to be understood. The breaking of electroweak symmetry and the associated Higgs field will be discussed in lectures 15 and Gauge Symmetries The Lagrangian is L = T V, where T and V are the kinetic and potential energies of a system. It can be used to obtain the equations of motion. The Dirac equation follows from a Lagrangian of the form: L = i ψγ µ δ µ ψ m ψψ (13.1) It can be seen that this Lagrangian is invariant under a phase transformation: This is an example of a gauge invariance. ψ e iα ψ ψ e iα ψ (13.2) U(1) Symmetry of QED The Lagrangian for QED is written: L = ψ(iγ µ δ µ m)ψ + e ψγ µ A µ ψ 1 4 F µνf µν (13.3) where A µ represents the photon, and the second term can be thought of as J µ A µ where J µ = e ψγ µ ψ is an electromagnetic current. F µν is the electromagnetic field tensor: F µν = δ µ A ν δ ν A µ (13.4) 87

38 The gauge transformation is: ψ e iα ψ A µ A µ + 1 e δ µα (13.5) This has the property that it conserves the current, and hence conserves electric charge. A mass term ma µ A µ is forbidden in the Lagrangian by gauge invariance. This explains why the photon must be massless. The gauge invariance of QED is described mathematically by a U(1) group SU(3) Symmetry of QCD This gauge symmetry was already introduced in Lecture 8. A rotation in colour space is written as: U = e iαa λa (13.6) where α a are the equivalent of angles, and λ a are the generators of SU(3). QCD amplitudes can be shown to be invariant under this non-abelian gauge transformation. The transformations of the quark and gluon states are: q (1 + iα a λ a )q G a µ G a µ 1 g s µ α a f abc α b G c µ (13.7) The Lagrangian for QCD is written: L = q(iγ µ δ µ m)q + g s qγ µ λ a G µ aq 1 4 Ga µνg µν a (13.8) where the gluon states G a µ replace the photon, and g s replaces e. The gluon field energy contains a term for the self-interactions of the gluons: G a µν = δ µ G a ν δ ν G a µ g s f abc G b µg c ν (13.9) The absence of a mass term mg a µg µ a makes the gluon massless Flavour Symmetries In Lecture 10 we met isospin symmetry, which is a flavour symmetry of the strong interactions between u and d quarks. It is described by SU(2), and can be extended to SU(3) with the addition of the s quark. The SU(3) symmetry is partially broken by the s quark mass. In principle this could be extended further to an SU(6) symmetry between all the quark flavours, but at this point the level of symmetry breaking becomes rather large. The interesting question, to which we do not yet have an answer, is what causes the breaking of quark flavour symmetry. Similar questions can be asked about lepton flavour symmetry. There are precise tests of lepton flavour conservation, and of the universality of lepton couplings in electromagnetic and weak interactions. However, flavour symmetry is broken by the lepton 88