8 Probing the Proton: Electron - Proton Scattering

Home , Elastic scattering, Electron, Proton, Scattering

Scattering of electrons and 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 the high energy HERA electron-proton collider 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.

8.1 Electron - Proton Scattering

The results of e−p e−p scattering depends strongly on the wavelength λ = hc/E. →

At very low electron energies λ rp, where rp is the radius of the proton, the • scattering is equivalent to that from a point-like spin-less object.

At low electron energies λ rp the scattering is equivalent to that from a extended • charged object. ∼

At high electron energies λ

At very high electron energies λ rp: the proton appears to be a sea of quarks • and gluons.

8.2 Form Factors

Extended object - like the proton - have a matter density ρ(r), normalised to unit volume: d3rρ (r ) = 1. The Fourier Transform of ρ(r) is the form factor, F (q):

F (q )= d3r 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 two form factors are required: F to describe the distribution → 1 of the electric charge, F2 to describe the recoil of the proton.

8.3 Elastic Electron-Proton Scattering

Elastic electron-proton scattering is illustrated in ﬁgure 8.3. As the proton is a composite object the vertex factor is modiﬁed by Kµ (compared to γµ): 2 e µ (e−p e−p)= (¯u γ u )(¯u K u ) (8.3) M → (p p )2 3 1 4 µ 2 1 − 3 51 e−(p1) e−(p3) ieγµ q

ieKµ − p(p2) p(p4)

Figure 8.1: Elastic electron proton scattering. As the proton is a composite object the vertex factor is Kµ.

µ µ 2 iκp 2 µν K = γ F1(q )+ F2(q )σ qν (8.4) 2mp

F1 is the electrostatic form factor, while F2 is associated with the recoil of the proton (as described above). F1 and F2 parameterise the structure of the proton, and are functions of the momentum transferred by the photon (q2).

8.4 Elastic Scattering

The elastic scattering in the relativistic limit pe = Ee by the Mott formula:

dσ α2 θ q2 θ = cos2 sin2 (8.5) dΩ 4p2 sin4 θ 2 − 2m2 2 point e 2 p For θ and pe are in the Lab frame. In the non-relativistic limit p m this reduces to Rutherford scattering: e e dσ α2 = (8.6) dΩ 4m2v4 sin4 θ NR e e 2

8.4.1 Q2 and ν

We deﬁne a two new quantities: Q2 and ν, which are useful in describing scattering.

2 2 2 p2 q Q q =(p1 p3) > 0 ν · (8.7) ≡− − ≡ Mp

Note that ν>0 so Q2 > 0 and the mass squared of the virtual photon is negative, q2 < 0!

52 8.4.2 Higher Energy Scattering

At higher energy, we have to account for the recoil of the proton. The diﬀerential cross section for electron-proton scattering becomes:

2 2 2 2 dσ α E3 2 κ q 2 2 θ q 2 2 θ = F F cos (F1 + κF2) sin (8.8) dΩ 4E2 sin4 θ E 1 − 4m2 2 2 − 2m2 2 lab 1 2 1 p p 1 For a point-like spin- 2 particle, F1 = 1, κ = 0, and the above equation reduces to the Mott scattering result, equation (8.5). It is common to use linear combinations of the form factors: κq2 GE = F1 + 2 F2 GM = F1 + κF2 (8.9) 4mp

which are referred to as the electric and magnetic form factors, respectively. The diﬀerential cross section can be rewritten as: dσ α2 E G2 + τG2 θ θ = 3 E M cos2 +2τG2 sin2 (8.10) dΩ 4E2 sin4 θ E 1+τ 2 M 2 lab 1 2 1 2 2 where we have used the abbreviation τ = Q /4mp. The experimental data on the form factors as a function of q2 are correspond to an exponential charge distribution:

ρ(r)=ρ exp( r/r )1/r2 =0.71 GeV2 r2 =0.81 fm2 (8.11) 0 − 0 0 The proton is an extended object with an rms radius of 0.81 fm. The whole of the above discussion can be repeated for electron-neutron scattering, with similar form factors for the neutron.

8.5 Deep Inelastic Scattering

During inelastic scattering the proton can break up into its constituent quarks which then form a hadronic jet. At high q2 this is known as deep inelastic scattering (DIS).

e−(p1) e−(p3) ieγµ

q X

p(p2)

53 Figure 8.2: Diﬀerential cross section for E1 =10 GeV and θ =6◦, as a function of the hadronic jet mass mX (labeled W in plot). The peaks represent elastic scattering, and then the excitation of baryonic resonances at higher mass. Above 2 GeV there is a continuum of non-resonant scattering. This data was taken by Friedman, Kendall and Taylor at SLAC in the 1960 and 1970’s and was accepted as the ﬁrst experimental proof of quarks. Friedman, Kendall and Taylor won the Nobel prize in 1990 for this work.

We introduce a third variable, known as Bjorken x:

Q2 x (8.12) ≡ 2p q 2 · where again q2 is the four-momentum transferred by the photon.

The invariant mass, MX , of the ﬁnal state hadronic jet is:

2 2 2 2 2 MX = p4 =(q + p2) = mp +2p2q + q (8.13)

The system X will be a baryon. The proton is the lightest baryon, therefore MX >mp. Since M = m q2 and ν, are two independent variables in DIS, and it is necessary to X p measure E1, E3 and θ in the Lab frame to determine the full kinematics. Futhermore Q2 =2p q + M 2 M 2 Q2 < 2p q (8.14) 2 · p − X ⇒ 2 · Implying the allowed range of x is 0 to 1. 0

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

54 Figure 8.3: The structure function F2 as measured at HERA using collisions between 30 GeV electrons and 830 GeV protons.

55 e−(p1) e−(p3)

q m

The parton model restores the elastic scattering relationship between q2 and ν, with m (parton mass) replacing mp: q2 ν + = 0 (8.15) 2m and the cross section for elastic electron-parton scattering is:

2 2 d σ α 1 2 2 θ 2 2 2 θ = F2(x, Q )cos + F1(x, Q ) sin (8.16) dE dΩ 4E2 sin4 θ ν 2 M 2 3 1 2

The structure functions are sums over the charged partons in the proton:

2 2xF1(x)=F2(x)= xeqq(x) (8.17) q

where eq is the charge of the parton (+2/3 or 1/3 for quarks). Essentially the structure functions are just describing the parton content− of the proton!

8.7 Parton Distribution Functions

We introduce the parton distribution functions, fi(x), defined as the probability that to find a parton in the proton the carries energy between x and x + dx. The structure functions can then be written: 1 F (x)= e2f (x) (8.18) 1 2 q i i 2 F2(x)= xeqfi(x) (8.19) i Where the sum is over all the the different flavour of partons in the proton, and eq = 1/3, +2/3 is the charge of the parton. The partons in the protons are: −

The valance quarks, uud. • The sea quarks, which come in quark anti-quark pairs: u¯u, dd¯, s¯s, c¯c, bb.¯ • Gluons, g. • 56 The parton distribution functions must describe a proton with total fractional momentum x = 1:

1 dx x[u(x) + ¯u(x) + d(x)+d(¯ x) + s(x) + ¯s(x)+...] = 1 (8.20) 0

1 ¯ 1 dx[d(x)+d(x)] = 1 0 dx[u(x) + ¯u(x)] = 2 (8.21) 0 1 1 dx[s(x) + ¯s(x)] = 0 0 dx[c(x) + ¯c(x)] = 0 (8.22) 0 8.8 Structure Functions and Quarks in the Nucleon

Scattering experiments can be used to measure the structure functions q(x), these are shown in ﬁgure 8.6.

The structure function F2(x) (equation 8.19) for a proton, consisting of just uud valance quarks is: 4 1 F p(x)= xe2q(x)= xu(x)+ xd(x) (8.23) 2 q 9 9 q=u,d

If we integrate the area the structure function F2(x) we measure the total fraction carried by the valence and sea quarks:

1 4 1 1 4 1 F p(x)dx = f + f F n(x)dx = f + f (8.24) 2 9 u 9 d 2 9 d 9 u 0 0

Where fu (fd) is the fraction of momentum carried by the up (down) quarks. Using isospin symmetry (see chapter 9) the neutron can be described by d u: ↔ 1 4 1 F n(x)dx = f + f (8.25) 2 9 d 9 u 0

Experimental measurement give:

1 1 p n F2 (x)dx =0.18 F2 (x)dx =0.12 (8.26) 0 0 Implying fu =0.36 fd =0.18 (8.27) 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.

57 CT10.00 PDFs 0.8 g/5 0.7 u d 0.6 ubar dbar 0.5 s c =2 GeV) 0.4 µ

0.3 x* f(x, 0.2

0.1

0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 X

CT10.00 PDFs (area proportional to momemtum fraction) 1.2 g u 1 d ubar 0.8 dbar s

=85 GeV) c µ 0.6 * f(x,

5/3 0.4 3*x 0.2

0 10-5 10-3 10-2 .05 .1 .2 .3 .4 .5 .6 .7 .8 .9 X

Figure 8.4: Measured parton distribution functions xq(x) ﬁtted from Hera data for Q = 2 GeV (top) and 85 GeV (bottom) (taken from arXiv1007.2241). Note that the main contribution is from the up and down quarks. However contributions from strange and charm quarks are also observed, and there is a huge contribution from low energy gluons at small values of x.

58 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 diﬀer- ences.

9.1 Colour Charge

Quarks carry one of three colour states, red r, green g, and blue b. It is an intrinsic charge, a quark always carries a colour charge (just as an electron always carries an electric charge). Antiquarks carry the anticolour states,r ¯,¯g, and ¯b. We deﬁne three quark eigenstates r, g, and b: 1 0 0 r = 0 g = 1 b = 0 (9.1)       0 0 1 The quark wavefunction is describes by a spinor (see section 5) plus the above colour wavefunction. The colour states are related by an SU(3) symmetry group which deﬁnes rotations in colour. Strong interactions are invariant under SU(3) colour rotations. The coupling is independent of the colour states of the quarks and gluons.

9.2 Gluons

Gluons are the propagator of the strong force. They exchange colour charge between quarks and are therefore themselves coloured. The allowed gluon states are described by the eight Gell-Mann matrices which describe some properties (are the generators of) the SU(3) group:

010 0 i 0 100 − λ1 = 100 λ2 = i 00 λ3 = 0 10 (9.2)       000 000 000−       001 00 i 000 − λ4 = 000 λ5 = 00 0 λ6 = 001 (9.3)       100 i 00 010       00 0 10 0 1 λ7 = 00 i λ8 = 01 0 (9.4)   √   0 i −0 3 00 2 −     The λ matrices can be identiﬁed with the eight gluon states. The operators λ1,2,4,5,6,7 are the ones that change quark colour states. The diagonal operators λ3,8 are the ones that do not change the colour states.

59 The gluon states are:

g = 1 (r¯b + br¯) g = 1 (r¯b br¯) g = 1 (rr¯ b¯b) 1 √2 2 √2 3 √2 1 1 − 1 − ¯ g4 = (rg¯ + gr¯) g5 = (rg¯ gr¯) g3 = (bg¯ gb) (9.5) √2 √2 − √2 − g = 1 (bg¯ + g¯b) g = 1 (rr¯ + b¯b 2gg¯) 7 √2 8 √6 − You do not have to learn these matrices or the allowed gluon colour states, just how to use them if they are given to you.

9.3 Feynman rules for QCD amplitudes

The calculation of Feynman diagrams containing quarks and gluons has the following changes compared to QED:

The coupling constant is α (instead of α): α g /4π. • S S ≡ S 2 2 2 αS(q ) decreases rapidly as a function of q . At small q it is large, and QCD is • a non-perturbative theory. At large q2 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 1 g λaγµ. • 2 s As a consequence of the gluon colour states, gluons can self-interact in three or • four gluon vertices.

60 a, µ b, ν b, ν c, λ

q1 q2 q3

c, λ a, µ d, ρ Figure 9.1: Gluon self-interaction. Left: three gluon vertex; Right: four gluon vertex. No further gluon self-interaction vertices are allowed.

Figure 9.2: A color ﬂux tube.

9.4 Gluon Self-Interactions

The SU(3) symmetry of the strong forces implies interactions between the gluons themselves: gluons may undergo three gluon and four gluon interactions as shown in ﬁg- ure 9.1. If we try to separate quarks (or quark and an anti-quark), the gluons being exchange between the gluons will be attracted to each other, forming a colour ﬂux tube. The potential to separate the quarks grows as the distance between the quarks grows. At long distances the potential grows as the distance between the quarks:

Vq¯q(r)=kr (9.6)

The constant k is measured to be 1 GeV/fm, making it impossible to separate quarks! Gluon-gluon interactions are therefore∼ responsible for holding quarks inside mesons and baryons. This is known as confinement (see section 10.2). A pictorial way of thinking of this is as a colour flux tube (see figure 9.2) 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, due to the interactions of the gluons with other gluons. As a consequence of the positive term in the strong interaction potential, a qq¯ pair cannot be separated since infinite energy would be required. Instead the flux tube can break creating an additional qq¯ pair in the middle which combines with the original qq¯ pair to form two separate hadrons.

61 p2,k p4,l gS

p1,i p3,j Figure 9.3: Quark–anti-quark scattering. The i, j, k, l represent the colour labels of the quarks.

9.5 Quark–Anti-quark Scattering

Quark–anti-quark scattering is shown in ﬁgure 9.3. Recall, to write the matrix element we follow the fermion arrows backwards. For the quark line j i we get a term λ at → ji the vertex. For the antiquark line we get a term λkl at the vertex. The matrix element for quark-antiquark scattering is written: g gµν g = u¯ S λa γµu δab v¯ S λb γνv (9.7) M j 2 ji i q2 k 2 kl l where uj,ui,vk,vl are the spinors for the quarks, with the subscripts representing the colour part of the wavefunction. g2 λa λa = S ji kl [¯u γµu ][¯v γµv ] (9.8) M q2 4 j i k l

This looks very similar to the matrix element for electron-positron scattering, except that e (EM coupling strength) is replaced by gs (strong coupling strength) and there is a colour factor f: 1 1 f = λa λa = λa λa (9.9) 4 ji kl 4 ji kl a where sum is over all eight λ matrices. At the lowest order we can describe this t-channel scattering similar to electromagnetic scattering in term of a Coulomb-like potential: fα V = S (9.10) q¯q − r where f is deﬁned in equation (9.9).

9.5.1 Colour factor for q¯q q¯q → For the calculation of the colour factors we can choose particular colours for q and ¯q. QCD is invariant under rotations in colour space, therefore any choice of colours will give us the same answer. There are three colour options:

62 ¯ ¯ 1 a a 1. i =1,k = 1 j =1,l = 1, e.g. rr¯ rr¯: f1 = 4 a λ11λ11 The only matrices that contribute→ to this sum (elements→ non-zero) are λ3 and λ8: 1 1 1 1 1 1 f = λa λa = [λ3 λ3 + λ8 λ8 ]= [(1)(1) + ( )( )] = (9.11) 1 4 11 1 4 11 11 11 11 4 √ √ 3 a 3 3 2. i =1,k = 2¯ j =1,l = 2,¯ e.g. r¯b r¯b: f = 1 λa λa : → → 2 4 a 11 22 1 1 1 1 1 1 f = λa λa = [λ3 λ3 + λ8 λ8 ]= [(1)( 1) + ( )( )] = (9.12) 2 4 11 22 4 11 22 11 22 4 − √ √ −6 a 3 3 3. i =1,k = 1¯ j =2,l = 2,¯ e.g. rr¯ b¯b: f = 1 λa λa : → → 3 4 a 21 12 1 1 1 1 f = λa λa = (λ1 λ1 + λ2 λ2 )= [( i)(i) + (1)(1)] = (9.13) 3 4 21 12 4 21 12 12 21 4 − 2 a

9.5.2 Colour factor for Mesons

Mesons are colourless q¯qstates in a ”colour singlet”: rr¯ + b¯b + gg¯. When the quarks inside a meson are heavy (e.g. charm or bottom quarks) the main interactions will take place due to t-channel scattering as we have just calculated. There are two possible contributions to the colour factor, the quarks stay the same colour (f1 =1/3 above), or the quark and antiquark change colour (f3 =1/2 above). There are two ways for this second contribution e.g. the an initial red–anti-red pair, they can turn into green–anti-green or blue–anti-blue. This have the same colour factor, this is guaranteed by the SU(3) symmetry. Note that f2 does not contribute as the quark and anti-quark must have equal and opposite colour charge inside a meson.

The colour factor for a heavy meson is therefore f = f1 +2f3 =4/3, and the potential is therefore: 4 α V = S (9.14) q¯q −3 r This is the short-distance contribution to the potential (before we have to consider gluon self-interactions).

9.5.3 QCD Potential for Mesons

The overall QCD potential is shown in ﬁgure 9.4 for mesons containing heavy quarks is the sum of terms in equation (9.14) and equation (9.6): 4 α V = S + kr (9.15) q¯q −3 r

This potential describes the allowed states of charmonimum (c¯c)and bottomonium (bb)¯ states very well.

63 Figure 9.4: QCD potential. The dotted line shows the short distance potential; the solid line includes the long distance potential.

9.6 Hadronisation and Jets

Consider a quark and anti-quark produced in a high energy collision, as shown in ﬁgure 9.5. The processes illustrated are:

1. Initially the quark and anti-quark separate at high velocity.

2. Colour ﬂux tube forms between the quarks.

3. Energy stored in the ﬂux tube suﬃcient to produce q¯qpairs

4. Process continues until quarks pair up into jets of colourless hadrons

This process is called hadronisation. It is not (yet) calculable. At collider experiments quarks and gluons observed as jets of particles. Jet events from the ATLAS experiment are illustrated in ﬁgure 9.6.

64 Gluon self-Interactions and Confinement , Gluon self-interactions are believed to give e+ q rise to colour confinement , Qualitative picture: •Compare QED with QCD •In QCD “gluon self-interactions squeeze lines of force into a flux tube” e- q , What happens when try to separate two coloured objects e.g. qq q q

•Form a flux tube of interacting gluons of approximately constant energy density

•Require infinite energy to separate coloured objects to infinity •Coloured quarks and gluons are always confined within colourless states •In this way QCD provides a plausible explanation of confinement – but not yet proven (although there has been recent progress with Lattice QCD)

Prof. M.A. Thomson Michaelmas 2011 257

Hadronisation and Jets ,Consider a quark and anti-quark produced in electron positron annihilation i) Initially Quarks separate at q q high velocity q q ii) Colour flux tube forms between quarks iii) Energy stored in the q q q q flux tube sufficient to produce qq pairs iv) Process continues until quarks pair up into jets of colourless hadrons , This process is called hadronisation. It is not (yet) calculable. Figure 9.5: Hadronisation of a quark and anti-quark , The main consequence is that at collider experiments quarks and gluons observed as jets of particles + q e !

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10.1 Strong Coupling Constant α = no. of colours with S = no. of quark flavours

The coupling strength αS is large, which means that higher order diagrams are impor- 2 2 tant. At low q (where q is the momentum transferred2 Nobel by Prize the gluon) for Physics, higher 2004 order !S decreases with Q amplitudes are larger than the lowest order diagrams, so the(Gross, sum Politzer, of all diagrams Wilczek) does notProf. converge, M.A. Thomson and QCD is non-perturbativeMichaelmas 2011. At high q2 the sum does converge, 278 and QCD becomes perturbative.

10.1.1 Renormalisation of αS

The coupling constant αS can be renormalised at a scale µ in a similar way to α: α(µ2) α(q2)= (QED) (10.1) 2 2 1 α(µ ) log q − 3π µ2 In QCD the renormalization is attributed both to the colour screening effect of virtual qq¯ pairs and to anti-screening effects from gluons since they have colour-anticolour states, see figure 10.1. This leads to a running of the strong coupling constant: α (µ2) α (q2)= S α (q2) (10.2) S 2 2 S 1+ αS (µ ) (11n 2n ) ln q 12π C − f µ2 where nC = 3 is the number of colours, and nf 6 is the number of active quark flavours which is a function of q2. ≤ The anti-screening effect of the gluons dominates, and the strong coupling constant decreases rapidly as a function of q2. The running of the coupling constant is usually written: 12π α (q2)= (10.3) S 2 (11n 2n ) ln q C − f Λ2 where ΛQCD 217 25 MeV is a reference scale which defines the onset of a strongly coupled theory∼ where± α 1. S ≈ 67 Figure 10.2: The strong coupling constant as a function of the energy scale at which it is measured. Note that at low energy scales the coupling is large!

There are many independent determinations of αS at diﬀerent scales µ, which are summarised in the ﬁgure 10.2. The results are compared with each other by adjusting them to a common scale µ = MZ where αS =0.1184(7).

10.2 Conﬁnement and Asymptotic Freedom

In deep inelastic scattering experiments at large q2, the quarks and gluons inside the proton can be observed as independent partons. This property is known as asymptotic 2 freedom. It is associated with the small value of αS(q ), 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 q2 strong interactions has been well established over the past 20 years by experiments at high energy colliders. In contrast, at low q2 the quarks and gluons are tightly bound into hadrons. This is known as colour confinement. For large αS 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. The colour flux tube discussed in section 9.1, is an example of confinement. There are no free quarks or gluons!

68 + 10.3 e e− Hadrons →

+ Figure 10.3: e e− hadrons events. Top: 2 jets produced from an initial q¯qpair. Bottom: 3 jets produced→ from an initial q¯qg state.

+ In the electromagnetic process e e− qq¯ (figure 10.4) the flavour and colour charge of the quark and anti-quark q andq ¯ must→ be the same. The final state quark and antiquark each form a jets as described in section 9.6. The two jets follow the directions of the quarks, and are back-to-back in the center-of-mass system. An illustration of an event + from the ALEPH detector e e− 2 jet is shown in figure 10.3. → + The cross section for e e− qq¯ 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 = 3: → C 2 + + e + µ + µ (e e− µ µ−)= [¯v(e )γ u(e−)][v(µ )γ u¯(µ−)] (10.4) M → q2 + eeq + µ µ (e e− q¯q) = [¯v(e )γ u(e−)][v(¯q)γ u¯(q)] (10.5) M → q2

Where eq is the charge of the quarks either eq = +2/3, 1/3. Summing over all the quark ﬂavours that can be produced at a given energy: −

+ σ(e e− hadrons) R → =3 e2 (10.6) ≡ σ(e+e µ+µ ) q − − q → 69 e+(p2) q(p4)

e−(p1) q¯(p3)

+ Figure 10.4: e e− hadrons → Where the sum is over the number of quarks that can be produced at the centre of mass energy √s = q2 and the factor of three comes about because of the three available colours of each of the quark ﬂavours. At centre of mass energies 1 GeV < √s<3 GeV: u, d and s quark pairs can be • produced, giving R 2. ≈ At 4 GeV < √s<9 GeV: u, d, s, c quark pairs can be produced, giving R 10/9. • ≈ At √s>10 GeV: u, d, s, c, b quark pairs can be produced, giving R 11/9. • ≈ + Measurements of the cross section e e− hadrons and R are shown in ﬁgure 10.3. → Note the importance of the colour factor nC = 3. The measurement of the ratio R shows that there are three colour states of quarks!

10.4 Gluon Jets

Sometimes one of the quarks radiates a hard gluon which carries a large fraction of the + quark energy, e e− qqg¯ . In this process there are three jets, with one coming from the gluon. This provided→ the first direct evidence for the gluon in 1979. The jets are no longer back-to-back, and the gluon jet has a different pT distribution from the quark jets, as illustrated in figure 10.3. The production rate of three-jet versus two-jet events is proportional to αS. Including higher order processes, the ratio R is: α (q2) R =3 e2 1+ S (10.7) q π q This is used to measure the strong coupling constant as a function of q2.

10.5 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 1995. In 2010 the Large 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.

70 6 41. Plots of cross sections and related quantities

+ σ and R in e e− Collisions

-2 10 ω φ J/ψ -3 10 ψ(2S) ρ Υ -4 ρ ! 10 Z

-5 10 [mb]

σ -6 10

-7 10

-8 10 2 1 10 10

Υ 3 10 J/ψ ψ(2S) Z

10 2 φ R ω 10 ρ!

1 ρ -1 10 2 1 10 10 √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− + → 2 → µ µ−,s)=4πα (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 Figure 10.5: Top: measurements of the cross section for e e− hadrons. Bottom: 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 the ratioparameterizationsR as deﬁned of J/ψ, ψ(2S), in and equationΥ(nS),n =1, 2, 10.6.3, 4arealsoshown.Thefulllistofreferen Note the jumps atces√ tos the original4 and data and 10 the GeV. details of At the R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable≈ data ﬁles are available at √s http://pdg.lbl.gov/current/xsect/4 GeV charm quark pairs. (Courtesy can of the be COMPAS made (Protvino) in addition and HEPDATA to u, (Durham) d and Groups, s. MayAt 2010.)√s See full-color10 GeV version on color pages at end of book. bottom≈ quark pairs can be made in addition to u, d, s and c. The peaks labelled≈ J/ψ, ψ,Υare resonances (see section 12.1) and Z is due to Z jets (chapter 17). →

71 to the Fermilab accelerator complex. In addition, both the CDF [11] and DØ detectors [12] were upgraded. The results reported here utilize an order of magnitude higher integrated luminosity than reported previously [5].

II. PERTURBATIVE QCD

The theory of QCD describes the behavior of those particles (quarks q and gluons g) that experience the strong force. It is broadly modeled on the theory of Quantum Electrodynam- ics (QED), which describes the interactions between electrically-charged particles. However, unlike the electrically-neutral photon of QED, the gluons, the force-mediating bosons of the strong interaction, carry the strong charge. This fact greatly increases the complexity in calculating the behavior of matter undergoing interactions via the strong force. The mathematical techniques required to make these calculations can be found in text- books (e.g. [13]). Instead of giving an exhaustive description of those techniques here, we 10.5.1focus Hadron on those Collider aspects of Dictionary* the calculations employed most frequently in the experimental analysis, thereby clarifying the phenomena experimentalists investigate.

FIG. 1: Stylized hadron-hadron collision, with relevant features labeled. Note that a LO calculation Figure 10.6: Production of Jets at a Hadron Collider of the hard scatter (dashed line) will assign a jet to ﬁnal state radiation that would be included in the hard scatter calculation by a NLO calculation (dotted line). Jet production at hadron colliders (e.g. LHC) is illustrated in ﬁgure 10.6.

The hard scatter is an initial scattering at high q2 between partons (gluons, quarks, 3 • antiquarks). The underlying event is the interactions of what is left of the protons after parton • scattering. Initial and final state radiation (ISR and FSR) are high energy gluon emissions • from the scattering partons. Fragmentation is the process of producing final state particles from the parton • produced in the hard scatter. A hadronic jet is a collimated cone of particles associated with a final state parton, • produced through fragmentation. Transverse quantities are measured transverse to the beam direction. An event • with high transverse momentum (pT ) jets or isolated leptons, is a signature for the production of high mass particles (W, Z, H,t). An event with missing transverse energyE ( ) is a signature for neutrinos, or other missing neutral particles. T

10.5.2 Description of Jets*

The hadrons that constitute a jet are summed to provide the kinematic information about the jet:

72 Jet energy E = i Ei and momentum p = i p i The direction of p deﬁnes the • jet axis. Jet invariant mass W 2 = E2 p 2 • −| | Transverse energy ﬂow within the jet p 2 = p p 2 • | T | i | i − | Transverse jet momentum p = (p2 + p2), relative to beam axis. • t x y “Pseudorapidity” y = ln [(E + p )/(E p )]/2], related to cos θ. This goes to • z − z ∞ along the beam axis, and is zero at 90◦.

1 Azimuthal angle φ = tan− (p /p ) • y x

+ In the case of e e− qq¯(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:

R2 =(y y )2 +(φ φ )2 (10.8) i i − jet i − jet A typical value used to form jets is R =0.7.

73 11 Mesons and Baryons

Notation: In this section J represents the total angular momentum of a state. J = L+S where L is the orbital angular momentum and S is the spin angular momentum.

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: 1 χc = (rr¯ + gg¯ + b¯b) (11.1) √3 Baryons are formed from three quarks, all with different colour states, with an antisymmetric colour wavefunction: 1 χc = (rgb rbg + gbr grb + brg bgr) (11.2) √6 − − −

11.2 Quark and Hadron Masses

There are several deﬁnitions of the quark masses, depending on how renormalisation is deﬁned. The following discussion uses a renormalisation scheme called MS.

The masses of the u and d quarks, mu and md are only a few MeV, and the mass of the s quark, m 100 MeV. s ≈ Most of the mass of hadrons is due to the QCD interactions between the quarks. There- fore in discussing hadron masses and magnetic moments we use constituent quark masses m = m 300 MeV and m 500 MeV. u d ≈ s ≈

11.3 Strong Isospin

Strong interactions are the same for u and d quarks. This observation motivates SU(2) flavour symmetry known as as isospin or strong isospin (to differentiate it from the weak isospin we will meet later). Isospin, like spin, has two quantum numbers: total isospin (I) and the third component of isospin (I3). I can take on positive integer and half-integer values, I can take value I = I, I +1,...,I. 3 3 − − The u and d quarks are assigned to an ”isospin doublet”, as they both have the same value of I = +1/2, they are differentiated by their value of I3 (isospin up or isospin down): 1 1 1 1 u: I = ,I=+ d: I = ,I= (11.3) 2 3 2 2 3 −2

74 11.3.1 Pions

The lowest lying meson states are the pions, with spin J =0( ). These states are known as ”pseudoscalars”, as they appear to be scalars with J ↑↓= 0, but are built of two objects with J = 0. Pions form an triplet of states with I = 1:

+ 0 1 π [I =1,I3 = +1] = ud¯ π [I =1,I3 = 0] = (u¯u dd)¯ π− [I+1,I3 = 1] = d¯u √2 − − (11.4) There is also a singlet state with I = 0, the eta meson: 1 η [I =0,I3 = 0] = (u¯u+ dd)¯ (11.5) √2

Isospin symmetry implies that the three pions have the same strong interactions.

11.3.2 Nucleons

For baryons the lowest lying states are the J =1/2( ) proton and neutron: ↑↑↓ 1 1 1 1 p [I = ,I =+ ] = uud n [I = ,I = ] = ddu (11.6) 2 3 2 2 3 −2

Isospin symmetry means that protons and neutrons have the same strong interactions.

11.4 SU(3) Flavour Symmetry

The isospin symmetry can be enlarged to a three-fold SU(3) symmetry between u, d and s quarks. This symmetry is broken by the strange quark mass, which is much larger than the up and down quark masses. However the strong interactions are almost invariant under this SU(3) symmetry. A strangeness quantum number is assigned to the strange quark. A strange quark has S = 1 and I = 0. The u and d quarks have S = 0 and I =1/2. (This SU(3) symmetry was first proposed by Gell-Mann in 1961 as a way to classify hadrons even before the discovery of quarks!) The SU(3) symmetry simply implies that any rotations in the flavour space of the u, d and s quarks leaves the strong interactions unchanged. This is illustrated in figure 11.1. This triangle structure of the symmetry with the appropriate quantum numbers will be used to ”build” plots of the allowed hadron states below.

11.5 Hadron Multiplets

The lowest lying meson states have total spin J = 0, they are pseudoscalars, the spin of the individual quarks is . They are illustrated in ﬁgure 11.2. Three of the nine ↑↓ 75 Y - Y 3 3 2/3 s 2/3 +i h h!!!!!12- h 7 4 !!!!!! h - + du i i +- h !!!!!6 1/3 h 1/3 5 -1/2 1/2 I3 I3 -1/2 1/2 h -1/3 5 -1/3 !!!!! h 6- + i i + h - !!!!!!4 u +i d 7 h h!!!!!12- h -2/3 -2/3 s

Figure 11.1: Illustration of SU(3) ﬂavour symmetry.

0 J = 0 mesons are neutral with I3 = 0. These states are π (I = 1), η1 (I = 0) and η8 (I = 1). The physically observed states, η and η, are mixtures of η1 and η8. The lowest lying states have total spin J = 1, they are vector mesons with . They are illustrated in ﬁgures 11.3. Three neutral states that are observed are ρ0,↑↑ω and φ, where the φ is purely an s¯sstate. The lowest lying baryon octet (8 states) with J =1/2( ) contains p, n,Λ,Σ+,Σ0, 0 ↑↑↓ Σ−,Ξ and Ξ− states, as shown in ﬁgure 11.5.

The decuplet (10 states) has J =3/2( ). It consists of ∆,Σ∗,Ξ∗ states and the Ω− which is an sss state. These are illustated↑↑↑ in ﬁgure 11.4.

11.5.1 ∆++ and Proton Wavefunctions

The flavour wavefunctions for decuplet baryons are symmetric, e.g.: 1 (dus + uds + sud + sdu + dsu + usd) (11.7) √6 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 ) (11.8) ↑↓↑ ↓↑↑ − ↑↑↓ ↓↑↑ ↑↑↓ − ↑↓↑ ↑↑↓ ↑↓↑ − ↓↑↑ Hence the proton also has an overall antisymmetric wavefunction, ψ:

Hadron χc χf χS χL 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.

76 ✻ S K0(d¯s) K+(u¯s) Pseudoscalar mesons +1 J =0

π0 = (dd¯ u¯u)/√2 π π0 π+ ¯ − 0 − η8 = (dd+u¯u 2s¯s)/√6 (d¯u) η8 η1 (ud)¯ − η1 = (dd¯ + u¯u+ s¯s)/√3

1 0 − K−(s¯u) K (sd)¯ ✲ I3 1 1/2 0 +1/2 +1 − −

Figure 11.2: The lightest pseudoscalar mesons with J = 0. The y-axis represents strangeness, S and the x-axis represents I3.

✻ S 0 + K∗ (d¯s) K∗ (u¯s) Vector mesons +1 J =1

ρ0 = (dd¯ u¯u)/√2 0 + − ρ− ρ ρ ω = (dd¯ + u¯u)/√2 0 (d¯u) ωφ (ud)¯ φ =s¯s

1 0 − K∗−(s¯u) K∗ (sd)¯ ✲ I3 1 1/2 0 +1/2 +1 − −

Figure 11.3: The lightest vector mesons, with J = 1. The y-axis represents strangeness, S and the x-axis represents I3.

77 M MeV/c2

S ✻ 0 + ++ 0 ∆− ∆ ∆ ∆ 1232 ddd ddu duu uuu

0 + 1 Σ − Σ Σ 1385 − dds dus uus

0 2 Ξ − Ξ 1533 − dss uss

3 Ω− 1672 − sss ✲ I3 3/2 1 1/2 0 +1/2 +1 +3/2 − − −

Figure 11.4: The lightest J =3/2 baryons states. The y-axis represents strangeness, S and the x-axis represents I3.

78 M MeV/c2 S ✻ 0 n p 938.9 ddu uud

0 Σ + 1193 Σ− Σ 1 Λ 1116 − dds uds uus

0 2 Ξ− Ξ 1318 − dss uss

✲ I3 1 1/2 0 +1/2 +1 − −

Figure 11.5: The lightest baryons with J =1/2.

79 12 Decays of Hadrons

12.1 Resonances

24 Hadrons which decay due to the strong force have a very short lifetime, e.g. τ 10− s. Evidence for the existence of these hadrons can be found through resonances∼produced in scattering. Resonances are broad mass peaks in the combinations of their daughter particles, see figure 12.1. The J = 1 vector mesons decay to J = 0 pseudoscalar mesons through strong interac- 0 + tions in which a second quark-antiquark pair is produced, e.g. ρ π π−. Similarly → the J =3/2 baryons (except for the Ω−), decay to the J =1/2 baryons through strong interactions, e.g. ∆++ π+p (figure 12.1). → 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, see figure 12.2. Since the process is a low energy strong interaction, there are lots of gluons coupling to the quark lines which form a colour flux tube between them.

12.2 Heavy Quark Mesons and Baryons

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):

+ ¯ 0 + D (cd) D (c¯u) Ds (c¯s) (12.1) and four possible B states:

+ ¯ 0 ¯ 0 ¯ + ¯ B (bu) B (bd) Bs (bs) Bc (bc) (12.2)

Similarly there are heavy baryons such as Λc and Λb, 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 1.5 GeV and m 4.6 GeV. c ≈ b ≈

12.2.1 Charmonium and Bottomonium

Charmonium and bottomonium are bound states of c¯cand bb¯ states respectively. The J/ψ meson is identiﬁed as the lowest J = 1 bound state of c¯c.The width of this state is narrow because MJ/ψ < 2MD, so it cannot decay into a DD¯ meson pair. It decays to light hadrons by quark-antiquark annihilation into gluons. The quark line diagram is drawn as being disconnected as shown in ﬁgure 12.3.

80 14 41. Plots of cross sections and related quantities

10 2

π+ p ⇓ total !"#$$%$&'()#*%+,-.

+ π p elastic

Plab GeV/c

-1 2 10 1 10 10 1.2 2 3 4 5 6 7 8 9 10 20 30 40 2010 — Subatomic:πp Particle Physics 2 √s GeV πd 2.2 3 4 5 6 7 8 9 10 20 30 40 50 60

10 2 2. Draw a Feynman diagram for+ the process pπ+ ∆++ pπ+. Figure 12.1: Cross section for pπ scattering. The resonance→ at √→s 1.2 GeV is due to the production and decay of the ∆++ baryon.⇓ ∼ π∓ d total

⇓ π− p total

!"#$$%$&'()#*%+,-. 10

π− p elastic

Plab GeV/c

-1 2 10 1 10 10

Figure 41.13: Total and elastic cross sections for π±p and π±d (total only) collisions as a function of laboratory beam momentum and total center-of-mass energy. Corresponding computer-readable data ﬁles may be found at http://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS Group, IHEP, Protvino, August 2005) You have to think of the constituents quarks. We have uud+du¯ uuu uud+du.¯ Figure2007/08 12.2: — Physics Feynman 3: Particle diagram Physics for pπ+ ∆++ pπ+ →scattering.5 → The down and anti-down quarks annihilate by→ producing→ a gluon. The gluon can be absorbed by any of the quarks. Remember as long as the ﬂavour of the quark doesn’t change, a gluon can be emitted or absorbed by any quark. After about 24 10− s (any) one of the quarks emits a gluon which turns into a dd¯ pair. It decays so quickly as the strong coupling constant is large, so the quarks are very likely to emit an energetic enough gluon to make a dd,¯ and because the minimum energy state is favoured. The mass of the pion and the proton is smaller than the mass of the ∆++. Figure 12.3: IllustrationThe cross section of charmonium is therefore proportional decay. to α Note6 . This thegive a charm longer lifetime and than anti-charm if quarks This looks like a pretty silly Feynman diagram,S but the uuu state actually lives for annihilate with eachonly one other, gluon was either exchanged. via a photon or an odd number, n 3 gluons. slightly longer than just three quarks would+ if they0 didn’t form a bound state. This makes the matrix element (J/ψ π π−π ) small compared to most strong≥ decays. Therefore there is relativelyM small→ probability per unit time that this decay 3. The φ mesonhappens, decays giving via the J/ theψ a relatively strong long force lifetime. as follows: + In fact it’s a long enough lifetime to give the J/ψ µ µ− decay a chance to contribute as well. 81 →

7. What is meant by colour+ conﬁnement? φ K K− 49% Colour conﬁnement→ describes the phenomena where quarks are only found in hadrons. The gluons exchangedK0 betweenK¯ 0 the quarks self interact. The potential34% between two quarks, due to the gluon interactions is V (r)=kr. The energy stored between the → + 0 quarks increases asπ theyπ− areπ pulled apart! 17% → What is the amount of kinetic energy produced in the decay, for these decays (also know as the Q-value of the decay). Draw Feynman diagrams of the above decays and explain why the decays to kaons is favoured despite the low Q-value? Hint: You have to consider the colour charge of the gluons.

4. What is meant by colour conﬁnement? Colour conﬁnement describes the phenomena where quarks are only found in hadrons. The gluons exchanged between the quarks self interact. The potential between two quarks, due to the gluon interactions is V (r)=kr. The energy stored between the quarks increases as they are pulled apart! There is a complete spectroscopy of cc¯ charmonium states with angular momentum states equivalent to atomic spectroscopy. Only the states with M(cc¯) < 2MD have narrow widths. A similar spectroscopy is observed for b¯b bottomonium states.

s (2S) a dc(2S) rc2(1P) a a h (1P) rc1(1P) c hadrons hadrons rc0(1P) hadrons /0 hadrons a hadrons a 0 a d,/ a //

J/s (1S) a dc(1S)

hadrons hadrons a radiative

JPC ! 0 <" 1 << 0 "" 1 "" 1 "< 2 ""

Figure 12.4: Spectroscopy of lightest charmonium states with M(cc¯) < 2MD. Note + that out of these states only the J/ψ and ψ(2S) are produced in e e− collisions.

12.3 Decays of Hadrons

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 third component I3 are conserved. • Strong interactions always lead to hadronic ﬁnal states.

For electromagnetic decays total isospin I is not conserved, but I3 is. There are • often photons or charged lepton-antilepton pairs in the ﬁnal state.

For weak decays I and I3 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 neu- • trino in the final state is a clear signature of a weak interaction.

The lightest hadrons cannot decay to other hadrons via strong interactions. The π0, 0 0 η and Σ decay electromagnetically. The π±, K±, K , n,Λ,Σ±,Ξand Ω− decay via ﬂavour-changing weak interactions, with long lifetimes.

82 12.4 Pion and Kaon Decays

12.4.1 Charged Pion Decay π+ µ+ν → µ + u µ (p3)

fπ W

d¯ νµ(p4) Charged pions decay mainly to a muon and a neutrino:

+ + π µ ν π− µ−ν¯ (12.3) → µ → µ This occurs through the annihilation of the ud¯quark-antiquark into a charged W boson, which is described in terms of a pion decay constant, fπ. The matrix element for the decay is: 1 = v¯(d)¯ g V f γµ(1 γ5)u(u) u¯(ν )g γµ(1 γ5)v(µ+)(12.4) M W ud π − q2 m2 µ W − − W 2 gW ¯ µ 5 µ 5 + Vudfπ 2 v¯(d)γ (1 γ )u(u) u¯(νµ)γ (1 γ )v(µ ) (12.5) ≈ mW − − 2 =4G2 V 2f 2 m2 [p p ] (12.6) |M| F | ud| π µ µ · ν and the total decay rate is: 2 1 V 2G2 m2 Γ= = | ud| F f 2m m2 1 µ (12.7) τ 8π π π µ − m2 π π From the charged pion lifetime, τπ+ = 26 ns, the pion decay constant can be deduced, fπ = 131 MeV. Note that this is very similar to the charged pion mass, mπ+ = 139.6 MeV.

+ + The decay of a charged pion to an electron and a neutrino π e νe, is helicity 2 2 → suppressed. This can be seen from the factor mµ/mπ in the decay rate. Replacing mµ 4 with me reduces the decay rate by a factor 10− . 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 right-handed µ−.

12.4.2 Charged Kaon Decays

The charged Kaon has a mass of 494 MeV and a lifetime τK+ = 12 ns. Its main decay modes are:

Purely leptonic decays, where thesu ¯ annihilate, similar to π+ decay: • BR(K+ µ+ν )=63.4% → µ

83 + + Semileptonic decays, wheres ¯ u¯ and the W boson couples to either e νe or • + → µ νµ: BR(K+ π0+ν )=8.1% → Hadronic decays, where ¯s ¯uand the W + boson couples to ud:¯ • → BR(K+ π+π0) = 21.1% → + + + BR(K π π π−)=5.6% →

12.5 The Cabibbo-Kobayashi-Maskawa (CKM) Matrix

Mass eigenstates and weak eigenstates of quarks are not identical. Decay properties measure mass eigenstates with a deﬁnite lifetime and decay width, wheras the weak force acts on the weak eigenstates. (The weak eigenstates couple to the W boson.) Weak eigenstates are admixture of mass eigenstates, conventionally described using CKM matrix a mixture of the down-type quarks:

d Vud Vus Vub d s = V V V s (12.8)    cd cs cb    b Vtd Vts Vtb b       Where d, s, b are the mass eigenstate; d, s, b are the weak eigenstates. The matrix is unitary, and its elements satisfy:

2 2 Vij =1 Vij = 1 (12.9) i j

VijVik =0 VijVkj = 0 (12.10) i j By considering the above unitarity constraints it can be deduced that the CKM matrix can be written in terms of just four parameters: three angles si = sin θi, ci = cos θi, and a complex phase δ:

c1 s1c3 s1s3 s c c c c s s eiδ c c s + s c eiδ (12.11)  1 3 1 2 3 2 3 1 2 3 2 3  −s s c s c − c s eiδ c s s + c c eiδ 1 2 − 1 2 3 − 2 3 − 1 2 3 2 3   More about the CKM matrix in Chapter 14!