8 Probing the Proton: Electron - School of Physics and ...

8 Probing the Proton: Electron - Proton Scattering

Scattering of electrons and protons is an electromagnetic interaction. Electron beamshave been used to probe the structure of the proton (and neutron) since the 1960s, withthe most recent results coming from the high energy HERA electron-proton collider atDESY in Hamburg.

These experiments provide direct evidence for the composite nature of protons andneutrons, 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, thescattering 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 extendedcharged object.

• At high electron energies λ < rp: the wavelength is sufficiently short to resolvesub-structure. Scattering is from constituent quarks.

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

8.2 Form Factors

Extended object - like the proton - have a matter density ρ(r), normalised to unitvolume:

d3r ρ(r) = 1. The Fourier Transform of ρ(r) is the form factor, F (q):

F (q) =

d3r expiq · r ρ(r) ⇒ F (0) = 1 (8.1)

Cross section from extended objects are modified by the form factor:

dσ

dΩ

extended

≈ dσ

dΩ

point−like

|F (q)|2 (8.2)

For e−p→ e−p scattering two form factors are required: F1 to describe the distributionof 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 com-posite object the vertex factor is modified by Kµ (compared to γµ):

M(e−p→ e−p) =e2

(p1 − p3)2(u3γ

µu1) (u4Kµu2) (8.3)

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q

p(p2)

e−(p1)

p(p4)

e−(p3)

−ieKµ

ieγµ

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

Kµ = γµF1(q2) +

iκp

2mpF2(q

2)σµνqν (8.4)

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 arefunctions 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

dΩ

point

=α2

4p2e sin4 θ

2

cos2 θ

2− q2

2m2p

sin2 θ

2

(8.5)

For θ and pe are in the Lab frame.

In the non-relativistic limit pe me this reduces to Rutherford scattering:

dσ

dΩ

NR

=α2

4m2ev

4e sin4 θ

2

(8.6)

8.4.1 Q2 and ν

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

Q2 ≡ −q2 = (p1 − p3)2 > 0 ν ≡ p2 · q

Mp(8.7)

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 crosssection for electron-proton scattering becomes:

dσ

dΩ

lab

=α2

4E21 sin4 θ

2

E3

E1

F 2

1 −κ2q2

4m2p

F 22

cos2 θ

2− q2

2m2p

(F1 + κF2)2 sin2 θ

2

(8.8)

For a point-like spin-12 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:

GE = F1 +κq2

4m2p

F2 GM = F1 + κF2 (8.9)

which are referred to as the electric and magnetic form factors, respectively.

The differential cross section can be rewritten as:

dσ

dΩ

lab

=α2

4E21 sin4 θ

2

E3

E1

G2

E + τG2M

1 + τcos2 θ

2+ 2τG2

M sin2 θ

2

(8.10)

where we have used the abbreviation τ = Q2/4m2p.

The experimental data on the form factors as a function of q2 are correspond to anexponential charge distribution:

ρ(r) = ρ0 exp(−r/r0) 1/r20 = 0.71 GeV2 r2 = 0.81 fm2 (8.11)

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, withsimilar form factors for the neutron.

8.5 Deep Inelastic Scattering

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

q

p(p2)

e−(p1)

X

e−(p3)ieγµ

53

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

We introduce a third variable, known as Bjorken x:

x ≡ Q2

2p2 · q(8.12)

where again q2 is the four-momentum transferred by the photon.

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

M2X = p2

4 = (q + p2)2 = m2

p + 2p2q + q2 (8.13)

The system X will be a baryon. The proton is the lightest baryon, therefore MX > mp.

Since MX = mp q2 and ν, are two independent variables in DIS, and it is necessary tomeasure E1, E3 and θ in the Lab frame to determine the full kinematics.

Futhermore

Q2 = 2p2 · q + M2p −M2

X ⇒ Q2 < 2p2 · q (8.14)

Implying the allowed range of x is 0 to 1. 0 < x < 1 represents inelastic scattering,x = 1 represents elastic scattering.

8.6 The Parton Model

The parton model was proposed by Feynman in 1969, to describe deep inelastic scat-tering in terms of point-like constituents inside the nucleon known as partons with aneffective mass m < mp. Nowadays partons are identified as being quarks or gluons.

54

Figure 8.3: The structure function F2 as measured at HERA using collisions between30 GeV electrons and 830 GeV protons.

55

q

m

e−(p1)

m

e−(p3)

The parton model restores the elastic scattering relationship between q2 and ν, with m(parton mass) replacing mp:

ν +q2

2m= 0 (8.15)

and the cross section for elastic electron-parton scattering is:

d2σ

dE3dΩ=

α2

4E21 sin4 θ

2

1

νF2(x, Q2) cos2 θ

2+

2

MF1(x, Q2) sin2 θ

2

(8.16)

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

2xF1(x) = F2(x) =

q

x e2qq(x) (8.17)

where eq is the charge of the parton (+2/3 or −1/3 for quarks). Essentially the structurefunctions 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 thatto find a parton in the proton the carries energy between x and x + dx.

The structure functions can then be written:

F1(x) =1

2

i

e2qfi(x) (8.18)

F2(x) =

i

xe2qfi(x) (8.19)

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: uu, dd, ss, cc, bb.

• Gluons, g.

56

The parton distribution functions must describe a proton with total fractional momen-tum x = 1:

1

0

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

1

0

dx[d(x) + d(x)] = 1 1

0 dx[u(x) + u(x)] = 2 (8.21)

1

0

dx[s(x) + s(x)] = 0 1

0 dx[c(x) + c(x)] = 0 (8.22)

8.8 Structure Functions and Quarks in the Nucleon

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

The structure function F2(x) (equation 8.19) for a proton, consisting of just uud valancequarks is:

F p2 (x) =

q=u,d

x e2qq(x) =

4

9xu(x) +

1

9xd(x) (8.23)

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

1

0

F p2 (x)dx =

4

9fu +

1

9fd

1

0

F n2 (x)dx =

4

9fd +

1

9fu (8.24)

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

0

F n2 (x)dx =

4

9fd +

1

9fu (8.25)

Experimental measurement give:

1

0

F p2 (x)dx = 0.18

1

0

F n2 (x)dx = 0.12 (8.26)

Implyingfu = 0.36 fd = 0.18 (8.27)

The quarks constitute only 54% of the proton rest mass! The remainder is carried bygluons which must also be considered as partons.

Understanding the gluon component of the proton presents a problem, since the lowestorder scattering processes of electrons and neutrinos only couple to the quarks.

57

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x* f(

x, µ

=2 G

eV)

X

CT10.00 PDFs

g/5ud

ubardbar

sc

0

0.2

0.4

0.6

0.8

1

1.2

.9.8.7.6.5.4.3.2.1.0510-210-310-5

3*x5/

3 * f(

x, µ

=85

GeV

)

X

CT10.00 PDFs (area proportional to momemtum fraction)

gud

ubardbar

sc

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

58

9 Quantum Chromodynamics

Quantum Chromodynamics or QCD is the theory of strong interactions between quarksand 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 intrinsiccharge, a quark always carries a colour charge (just as an electron always carries anelectric charge).

Antiquarks carry the anticolour states, r, g, and b. We define three quark eigenstatesr, g, and b:

r =

100

g =

010

b =

001

(9.1)

The quark wavefunction is describes by a spinor (see section 5) plus the above colourwavefunction.

The colour states are related by an SU(3) symmetry group which defines rotationsin colour. Strong interactions are invariant under SU(3) colour rotations. The couplingis 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 betweenquarks and are therefore themselves coloured. The allowed gluon states are describedby the eight Gell-Mann matrices which describe some properties (are the generators of)the SU(3) group:

λ1 =

0 1 01 0 00 0 0

λ2 =

0 −i 0i 0 00 0 0

λ3 =

1 0 00 −1 00 0 0

(9.2)

λ4 =

0 0 10 0 01 0 0

λ5 =

0 0 −i0 0 0i 0 0

λ6 =

0 0 00 0 10 1 0

(9.3)

λ7 =

0 0 00 0 −i0 i 0

λ8 =1√3

1 0 00 1 00 0 −2

(9.4)

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 onesthat do not change the colour states.

59

The gluon states are:

g1 = 1√2(rb + br) g2 = 1√

2(rb− br) g3 = 1√

2(rr − bb)

g4 = 1√2(rg + gr) g5 = 1√

2(rg − gr) g3 = 1√

2(bg − gb)

g7 = 1√2(bg + gb) g8 = 1√

6(rr + bb− 2gg)

(9.5)

You do not have to learn these matrices or the allowed gluon colour states, just how touse 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 followingchanges compared to QED:

• The coupling constant is αS (instead of α): αS ≡ gS/4π.

• αS(q2) decreases rapidly as a function of q2. At small q2 it is large, and QCD isa non-perturbative theory. At large q2 QCD becomes perturbative like QED.

• A quark has one of three colour states (replacing electric charge). Antiquarkshave anticolour states.

• A gluon propagator has one of eight colour-anticolour states.

• A quark-gluon vertex has a factor 12gsλaγµ.

• As a consequence of the gluon colour states, gluons can self-interact in three orfour gluon vertices.

60

q1q3

q2

a, µ b, ν

c, λa, µ

b, ν

d, ρ

c, λ

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 them-selves: 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 exchangebetween the gluons will be attracted to each other, forming a colour flux tube. Thepotential to separate the quarks grows as the distance between the quarks grows. Atlong distances the potential grows as the distance between the quarks:

Vqq(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 andbaryons. 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) connectingthe 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 twoquarks, due to the interactions of the gluons with other gluons.

As a consequence of the positive term in the strong interaction potential, a qq paircannot be separated since infinite energy would be required. Instead the flux tube canbreak creating an additional qq pair in the middle which combines with the original qqpair to form two separate hadrons.

61

g

p1, i

p2, k

p3, j

p4, l

gS

gS

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

9.5 Quark–Anti-quark Scattering

Quark–anti-quark scattering is shown in figure 9.3. Recall, to write the matrix elementwe follow the fermion arrows backwards. For the quark line j → i we get a term λji atthe vertex. For the antiquark line we get a term λkl at the vertex.

The matrix element for quark-antiquark scattering is written:

M =uj

gS

2λa

jiγµui

gµν

q2δab

vk

gS

2λb

klγνvl

(9.7)

where uj, ui, vk, vl are the spinors for the quarks, with the subscripts representing thecolour part of the wavefunction.

M =g2

S

q2

λajiλ

akl

4[ujγ

µui] [vkγµvl] (9.8)

This looks very similar to the matrix element for electron-positron scattering, exceptthat e (EM coupling strength) is replaced by gs (strong coupling strength) and there isa colour factor f :

f =1

4λa

jiλakl =

1

4

a

λajiλ

akl (9.9)

where sum is over all eight λ matrices.

At the lowest order we can describe this t-channel scattering similar to electromagneticscattering in term of a Coulomb-like potential:

Vqq = −fαS

r(9.10)

where f is defined in equation (9.9).

9.5.1 Colour factor for qq → qq

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 willgive us the same answer.

There are three colour options:

62

1. i = 1, k = 1 → j = 1, l = 1, e.g. rr → rr: f1 = 14

a λa

11λa11 The only matrices

that contribute to this sum (elements non-zero) are λ3 and λ8:

f1 =1

4

a

λa11λ

a1 =

1

4[λ3

11λ311 + λ8

11λ811] =

1

4[(1)(1) + (

1√3)(

1√3)] =

1

3(9.11)

2. i = 1, k = 2 → j = 1, l = 2, e.g. rb→ rb: f2 = 14

a λa

11λa22:

f2 =1

4

a

λa11λ

a22 =

1

4[λ3

11λ322 + λ8

11λ822] =

1

4[(1)(−1) + (

1√3)(

1√3)] = −1

6(9.12)

3. i = 1, k = 1 → j = 2, l = 2, e.g. rr → bb: f3 = 14

a λa

21λa12:

f3 =1

4

a

λa21λ

a12 =

1

4(λ1

21λ112 + λ2

12λ221) =

1

4[(−i)(i) + (1)(1)] =

1

2(9.13)

9.5.2 Colour factor for Mesons

Mesons are colourless qq states in a ”colour singlet”: rr + bb + gg.

When the quarks inside a meson are heavy (e.g. charm or bottom quarks) the maininteractions will take place due to t-channel scattering as we have just calculated. Thereare 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). Thereare two ways for this second contribution e.g. the an initial red–anti-red pair, they canturn into green–anti-green or blue–anti-blue. This have the same colour factor, this isguaranteed by the SU(3) symmetry. Note that f2 does not contribute as the quark andanti-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 potentialis therefore:

Vqq = −4

3

αS

r(9.14)

This is the short-distance contribution to the potential (before we have to considergluon self-interactions).

9.5.3 QCD Potential for Mesons

The overall QCD potential is shown in figure 9.4 for mesons containing heavy quarksis the sum of terms in equation (9.14) and equation (9.6):

Vqq = −4

3

αS

r+ kr (9.15)

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

63

Figure 9.4: QCD potential. The dotted line shows the short distance potential; thesolid 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 infigure 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 qq pairs

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

This process is called hadronisation. It is not (yet) calculable. At collider experimentsquarks 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

Prof. M.A. Thomson Michaelmas 2011 257

! Gluon self-interactions are believed to give rise to colour confinement

! Qualitative picture:•Compare QED with QCD

e+

e-

q

q•In QCD “gluon self-interactions squeeze

lines of force into a flux tube”

q q! What happens when try to separate two coloured objects e.g. qq

•Form a flux tube of interacting gluons of approximately constantenergy 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 258

Hadronisation and Jets!Consider a quark and anti-quark produced in electron positron annihilation

i) Initially Quarks separate athigh velocity

ii) Colour flux tube formsbetween quarks

iii) Energy stored in theflux tube sufficient to produce qq pairs

q q

q q

q qq q

iv) Process continuesuntil quarks pairup into jets ofcolourless hadrons

! This process is called hadronisation. It is not (yet) calculable.! The main consequence is that at collider experiments quarks and gluons

observed as jets of particles

e–

e+!

q

q

Figure 9.5: Hadronisation of a quark and anti-quark

65

Prof. M.A. Thom

son

Michaelm

as 2011

255

!So w

e might expect 9 physical gluons:

OCTET:

SINGLET:

!BUT, colour confinem

ent hypothesis:Colour singlet gluonw

ould be unconfined.

It would behave like a strongly interacting

photon infinite range Strong force.

only colour singlet statescan exist as free particles

!Em

pirically, the strong force is short range and therefore know that the physical

gluons are confined. The colour singlet state does not existin nature !

NOTE:this is not entirely ad hoc. In the context of gauge field theory (see m

inor

option) the strong interaction arises from a fundam

ental SU(3)symm

etry.

The gluons arise from the generators of the sym

metry

group (the

Gell-M

ann matrices). There are 8 such m

atrices 8 gluons.

Had nature “chosen”a

U(3)symm

etry, would have 9 gluons, the additional

gluon would be the colour singlet state and Q

CD would be an unconfined

long-range force. NO

TE:the“gauge sym

metry”

determines the exact nature of the interaction

FEYNMAN RULES

Prof. M.A. Thom

son

Michaelm

as 2011

256

Gluon-G

luon Interactions!

InQ

EDthe

photondoes not carry the charge of the EM

interaction (photons are

electrically neutral) !

In contrast, in QCD

thegluons

do carry colour chargeG

luon Self-Interactions

!Tw

o new vertices (no Q

ED analogues) triple-gluon

vertex

quartic-gluonvertex

!In addition to quark-quark scattering, therefore can have gluon-gluon scattering

e.g. possiblew

ay of arrangingthe colour flow

jetjet

Figure 9.6: Jet Events in the ATLAS detector at CERN. Top: two jet event; Bottom:five jet event.

66

10 QCD at Colliders

Prof. M.A. Thomson Michaelmas 2011 277

! In QED, running coupling increasesvery slowly•Atomic physics:

•High energy physics:

OPAL Collaboration, Eur. Phys. J. C33 (2004)

! Might worry that coupling becomes infinite at

i.e. at• But quantum gravity effects would come

in way below this energy and it ishighly unlikely that QED “as is” wouldbe valid in this regime

Prof. M.A. Thomson Michaelmas 2011 278

Running of !sQCD Similar to QED but also have gluon loops

+ + + +…

Fermion Loop Boson Loops

! Bosonic loops “interfere negatively”

with

!S decreases with Q2 Nobel Prize for Physics, 2004(Gross, Politzer, Wilczek)

= no. of colours= no. of quark flavours

! Remembering adding amplitudes, so can get negative interference and the sumcan be smaller than the original diagram alone Figure 10.1: Illustration of processes contributing to renormalisation in QCD.

10.1 Strong Coupling Constant αS

The coupling strength αS is large, which means that higher order diagrams are impor-tant. At low q2 (where q2 is the momentum transferred by the gluon) higher orderamplitudes are larger than the lowest order diagrams, so the sum of all diagrams doesnot converge, and QCD is non-perturbative. At high q2 the sum does converge, andQCD 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(q2) =α(µ2)

1− α(µ2)3π log

q2

µ2

(QED) (10.1)

In QCD the renormalization is attributed both to the colour screening effect of virtualqq pairs and to anti-screening effects from gluons since they have colour-anticolourstates, see figure 10.1. This leads to a running of the strong coupling constant:

αS(q2) =αS(µ2)

1 + αS(µ2)12π (11nC − 2nf ) ln

q2

µ2

αS(q2) (10.2)

where nC = 3 is the number of colours, and nf ≤ 6 is the number of active quarkflavours which is a function of q2.

The anti-screening effect of the gluons dominates, and the strong coupling constantdecreases rapidly as a function of q2. The running of the coupling constant is usuallywritten:

αS(q2) =12π

(11nC − 2nf ) ln

q2

Λ2

(10.3)

where ΛQCD ∼ 217± 25 MeV is a reference scale which defines the onset of a stronglycoupled theory where αS ≈ 1.

67

Figure 10.2: The strong coupling constant as a function of the energy scale at which itis measured. Note that at low energy scales the coupling is large!

There are many independent determinations of αS at different scales µ, which aresummarised in the figure 10.2. The results are compared with each other by adjustingthem 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 theproton can be observed as independent partons. This property is known as asymptoticfreedom. It is associated with the small value of αS(q2), which allows the stronginteraction corrections from gluon emission and hard scattering to be calculated usinga perturbative expansion of QCD. The perturbative QCD treatment of high q2 stronginteractions has been well established over the past 20 years by experiments at highenergy colliders.

In contrast, at low q2 the quarks and gluons are tightly bound into hadrons. This isknown as colour confinement. For large αS QCD is not a perturbative theory anddifferent mathematical methods have to be used to calculate the properties of hadronicsystems. 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 thecontributions from three-gluon and four-gluon vertices which do not have an analoguein 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 qq pair.Bottom: 3 jets produced from an initial qqg state.

In the electromagnetic process e+e− → qq (figure 10.4) the flavour and colour charge ofthe quark and anti-quark q and q must be the same. The final state quark and antiquarkeach form a jets as described in section 9.6. The two jets follow the directions of thequarks, and are back-to-back in the center-of-mass system. An illustration of an eventfrom 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 issimilar to e+e− → µ+µ− apart from the final state charges and a colour factor, nC = 3:

M(e+e− → µ+µ−) =e2

q2[v(e+)γµu(e−)][v(µ+)γµu(µ−)] (10.4)

M(e+e− → qq) =e eq

q2[v(e+)γµu(e−)][v(q)γµu(q)] (10.5)

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

R ≡ σ(e+e− → hadrons)

σ(e+e− → µ+µ−)= 3

q

e2q (10.6)

69

γ

e−(p1)

e+(p2)

q(p3)

q(p4)

Figure 10.4: e+e− → hadrons

Where the sum is over the number of quarks that can be produced at the centre of massenergy

√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 beproduced, 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 Rshows 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 thequark energy, e+e− → qqg. In this process there are three jets, with one coming fromthe gluon. This provided the first direct evidence for the gluon in 1979. The jets areno longer back-to-back, and the gluon jet has a different pT distribution from the quarkjets, as illustrated in figure 10.3. The production rate of three-jet versus two-jet eventsis proportional to αS.

Including higher order processes, the ratio R is:

R = 3

q

e2q

1 +

αS(q2)

π

(10.7)

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 inthe 1980s, and the top quark at the Tevatron (Fermilab) in 1995. In 2010 the LargeHadron Collider (LHC) began operation at CERN. This collides two proton beams withan initial CM energy of 7 TeV. This is half the eventual design energy, but is alreadythe world’s highest energy collider.

70

6 41. Plots of cross sections and related quantities

σ and R in e+e− Collisions

10-8

10-7

10-6

10-5

10-4

10-3

10-2

1 10 102

σ[m

b]

ω

ρ

φ

ρ′

J/ψ

ψ(2S)Υ

Z

10-1

1

10

10 2

10 3

1 10 102

R ω

ρ

φ

ρ′

J/ψ ψ(2S)

Υ

Z

√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) = 4πα2(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 ofthis Review, Eq. (9.7) or, for more details, K. G. Chetyrkin et al., Nucl. Phys. B586, 56 (2000) (Erratum ibid. B634, 413 (2002)). Breit-Wignerparameterizations 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 ofthe R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable data files are available athttp://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS (Protvino) and HEPDATA (Durham) Groups, May 2010.) See full-colorversion on color pages at end of book.

Figure 10.5: Top: measurements of the cross section for e+e− → hadrons. Bottom:the ratio R as defined in equation 10.6. Note the jumps at

√s ≈ 4 and 10 GeV. At√

s ≈ 4 GeV charm quark pairs can be made in addition to u, d and s. At√

s ≈ 10 GeVbottom quark pairs can be made in addition to u, d, s and c. The peaks labelled J/ψ,ψ, Υ are resonances (see section 12.1) and Z is due to Z →jets (chapter 17).

71

10.5.1 Hadron Collider Dictionary*

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

focus on those aspects of 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

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).

3

Figure 10.6: Production of Jets at a Hadron Collider

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,antiquarks).

• The underlying event is the interactions of what is left of the protons after partonscattering.

• Initial and final state radiation (ISR and FSR) are high energy gluon emissionsfrom the scattering partons.

• Fragmentation is the process of producing final state particles from the partonproduced 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 eventwith high transverse momentum (pT ) jets or isolated leptons, is a signature for theproduction of high mass particles (W, Z, H,t). An event with missing transverseenergy (ET ) is a signature for neutrinos, or other missing neutral particles.

10.5.2 Description of Jets*

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

72

• Jet energy E =

i Ei and momentum p =

i pi The direction of p defines thejet axis.

• Jet invariant mass W 2 = E2 − |p|2

• Transverse energy flow within the jet |pT |2 =

i |pi − p|2

• Transverse jet momentum pt =

(p2x + p2

y), relative to beam axis.

• “Pseudorapidity” y = ln [(E + pz)/(E − pz)]/2], related to cos θ. This goes to ∞along the beam axis, and is zero at 90.

• Azimuthal angle φ = tan−1(py/px)

In the case of e+e− → qq(g) it is straightforward to decide which hadrons belong towhich jets. At a hadron collider a minimum bias event has at least two jets from thehard scatter, as well as two jets from the underlying event. With large numbers of jetsit is hard to decide which hadrons belong to which jets. What is done is to define aradial distance from the jet axis for each hadron:

R2i = (yi − yjet)

2 + (φi − φjet)2 (10.8)

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+Swhere 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 quarksare observed as free particles, known as hadrons. They are colour singlets, with gluonexchange between the quarks inside the hadrons, but no colour field outside.

Mesons are formed from a quark and an antiquark with colour and anticolour stateswith a symmetric colour wavefunction:

χc =1√3(rr + gg + bb) (11.1)

Baryons are formed from three quarks, all with different colour states, with an anti-symmetric colour wavefunction:

χc =1√6(rgb− rbg + gbr − grb + brg − bgr) (11.2)

11.2 Quark and Hadron Masses

There are several definitions of the quark masses, depending on how renormalisation isdefined. 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 ofthe s quark, ms ≈ 100 MeV.

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 quarkmasses mu = md ≈ 300 MeV and ms ≈ 500 MeV.

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 theweak isospin we will meet later). Isospin, like spin, has two quantum numbers: totalisospin (I) and the third component of isospin (I3). I can take on positive integer andhalf-integer values, I3 can take value I3 = −I,−I + 1, . . . , I.

The u and d quarks are assigned to an ”isospin doublet”, as they both have the samevalue of I = +1/2, they are differentiated by their value of I3 (isospin up or isospindown):

u : I =1

2, I3 = +

1

2d : I =

1

2, I3 = −1

2(11.3)

74

11.3.1 Pions

The lowest lying meson states are the pions, with spin J = 0 (↑↓). These states areknown as ”pseudoscalars”, as they appear to be scalars with J = 0, but are built oftwo objects with J = 0.

Pions form an triplet of states with I = 1:

π+ [I = 1, I3 = +1] = ud π0 [I = 1, I3 = 0] =1√2(uu− dd) π− [I+1, I3 = −1] = du

(11.4)There is also a singlet state with I = 0, the eta meson:

η [I = 0, I3 = 0] =1√2(uu + dd) (11.5)

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:

p [I =1

2, I3 = +

1

2] = uud n [I =

1

2, I3 = −1

2] = ddu (11.6)

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, dand s quarks. This symmetry is broken by the strange quark mass, which is muchlarger than the up and down quark masses. However the strong interactions are almostinvariant under this SU(3) symmetry. A strangeness quantum number is assigned tothe strange quark. A strange quark has S = 1 and I = 0. The u and d quarks haveS = 0 and I = 1/2. (This SU(3) symmetry was first proposed by Gell-Mann in 1961as 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, dand 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 beused 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 spinof the individual quarks is ↑↓. They are illustrated in figure 11.2. Three of the nine

75

+- i1 2!!!!!

+-i4

5

!!!!!!+- i6 !!!!!

7

+- i1 2!!!!!

+-i6!!!!!

7

s

+- i4

5!!!!!!

-3d u

-1/3

2/3

-1/2 1/2I3

-2/3

-1/3I3

-2/3 s

du

2/3YY

-1/2 1/21/31/3

3

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

J = 0 mesons are neutral with I3 = 0. These states are π0 (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 ↑↑. Theyare illustrated in figures 11.3. Three neutral states that are observed are ρ0, ω and φ,where the φ is purely an ss state.

The lowest lying baryon octet (8 states) with J = 1/2 (↑↑↓) contains p, n, Λ, Σ+, Σ0,Σ−, Ξ0 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√6(dus + uds + sud + sdu + dsu + usd) (11.7)

The ∆++ (uuu) has symmetric (S) flavour, spin and orbital wavefunctions, and anantisymmetric (A) colour wavefunction. It is overall antisymmetric, as it must be for asystem of identical fermions.

The wavefunctions for the lowest baryon octet have a combined symmetry of the flavourand 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 Ap A S S A

Note that there are no J = 1/2 states uuu, ddd, sss because the flavour symmetric partwould have to be combined with an antisymmetric spin part.

76

K0(ds) K+(us)

π−(du)

π0

η8 η1

π+

(ud)

K−(su) K0(sd)

S+1

0

−1

I3

−1 −1/2 0 +1/2 +1

Pseudoscalar mesons

J = 0

π0 = (dd− uu)/√

2

η8 = (dd + uu− 2ss)/√

6

η1 = (dd + uu + ss)/√

3

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

K∗0(ds) K∗+(us)

ρ−(du)

ρ0

ω φρ+

(ud)

K∗−(su) K∗0

(sd)

S+1

0

−1

I3

−1 −1/2 0 +1/2 +1

Vector mesons

J = 1

ρ0 = (dd− uu)/√

2

ω = (dd + uu)/√

2

φ = ss

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

77

∆−

ddd∆0

ddu∆+

duu∆++

uuu

Σ−

ddsΣ0

dusΣ+

uus

Ξ−

dssΞ0

uss

Ω−

sss

I3

−3/2 −1 −1/2 0 +1/2 +1 +3/2

S

−3

−2

−1

0

MMeV/c2

1232

1385

1533

1672

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

78

nddu

puud

Σ−

dds

Σ0

Λuds

Σ+

uus

Ξ−

dssΞ0

uss

S

0

−1

−2

I3

−1 −1/2 0 +1/2 +1

M

MeV/c2

938.9

11931116

1318

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

79

12 Decays of Hadrons

12.1 Resonances

Hadrons which decay due to the strong force have a very short lifetime, e.g. τ ∼ 10−24 s.Evidence for the existence of these hadrons can be found through resonances producedin scattering. Resonances are broad mass peaks in the combinations of their daughterparticles, see figure 12.1.

The J = 1 vector mesons decay to J = 0 pseudoscalar mesons through strong interac-tions in which a second quark-antiquark pair is produced, e.g. ρ0 → π+π−. Similarlythe J = 3/2 baryons (except for the Ω−), decay to the J = 1/2 baryons through stronginteractions, e.g. ∆++ → π+p (figure 12.1).

The Feynman diagram for resonance production can be drawn in reduced form showingjust 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 tothe 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):

D+ (cd) D0 (cu) D+s (cs) (12.1)

and four possible B states:

B+ (bu) B0 (bd) B0s (bs) B+

c (bc) (12.2)

Similarly there are heavy baryons such as Λc and Λb, where a heavy quark replaces oneof the light quarks.

The masses of heavy quark states are dominated by the heavy quark constituent massesmc ≈ 1.5 GeV and mb ≈ 4.6 GeV.

12.2.1 Charmonium and Bottomonium

Charmonium and bottomonium are bound states of cc and bb states respectively.

The J/ψ meson is identified as the lowest J = 1 bound state of cc. The width of thisstate is narrow because MJ/ψ < 2MD, so it cannot decay into a DD meson pair. Itdecays to light hadrons by quark-antiquark annihilation into gluons. The quark linediagram is drawn as being disconnected as shown in figure 12.3.

80

14 41. Plots of cross sections and related quantities

10

102

10-1

1 10 102

⇓

Plab

GeV/c

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

10

102

10-1

1 10 102

⇓

⇓

Plab

GeV/c

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

√s GeVπd

πp1.2 2 3 4 5 6 7 8 9 10 20 30 40

2.2 3 4 5 6 7 8 9 10 20 30 40 50 60

π+ p total

π+ p elastic

π∓ d total

π− p total

π− p elastic

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 totalcenter-of-mass energy. Corresponding computer-readable data files may be found at http://pdg.lbl.gov/current/xsect/. (Courtesy of theCOMPAS Group, IHEP, Protvino, August 2005)

Figure 12.1: Cross section for pπ+ scattering. The resonance at√

s ∼ 1.2 GeV is dueto the production and decay of the ∆++ baryon.

2010 — Subatomic: Particle Physics 2

2. Draw a Feynman diagram for the process pπ+ → ∆++ → pπ+.

You have to think of the constituents quarks. We have uud+du→ uuu→ uud+du.

The down and anti-down quarks annihilate by producing a gluon. The gluon canbe absorbed by any of the quarks. Remember as long as the flavour of the quarkdoesn’t change, a gluon can be emitted or absorbed by any quark. After about10−24 s (any) one of the quarks emits a gluon which turns into a dd pair. It decaysso quickly as the strong coupling constant is large, so the quarks are very likely toemit an energetic enough gluon to make a dd, and because the minimum energystate is favoured. The mass of the pion and the proton is smaller than the mass ofthe ∆++.

This looks like a pretty silly Feynman diagram, but the uuu state actually lives forslightly longer than just three quarks would if they didn’t form a bound state.

3. The φ meson decays via the strong force as follows:

φ→ K+K− 49%

→ K0K0 34%

→ π+π−π0 17%

What is the amount of kinetic energy produced in the decay, for thesedecays (also know as the Q-value of the decay).

Draw Feynman diagrams of the above decays and explain why the decaysto 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 thephenomena where quarks are only found in hadrons. The gluons exchanged betweenthe quarks self interact. The potential between two quarks, due to the gluon inter-actions is V (r) = kr. The energy stored between the quarks increases as they arepulled apart!

Figure 12.2: Feynman diagram for pπ+ → ∆++ → pπ+ scattering.2007/08 — Physics 3: Particle Physics 5

The cross section is therefore proportional to α6S. This give a longer lifetime than if

only one gluon was exchanged.

This makes the matrix elementM(J/ψ → π+π−π0) small compared to most strongdecays. Therefore there is relatively small probability per unit time that this decayhappens, giving the J/ψ a relatively long lifetime.

In fact it’s a long enough lifetime to give the J/ψ → µ+µ− decay a chance tocontribute as well.

7. 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 twoquarks, due to the gluon interactions is V (r) = kr. The energy stored between thequarks increases as they are pulled apart!

Figure 12.3: Illustration of charmonium decay. Note the charm and anti-charm quarksannihilate with each other, either via a photon or an odd number, n ≥ 3 gluons.

81

There is a complete spectroscopy of cc charmonium states with angular momentumstates equivalent to atomic spectroscopy. Only the states with M(cc) < 2MD havenarrow widths.

A similar spectroscopy is observed for bb bottomonium states.

(2S)

c(2S)

c(1S)

hadrons

hadrons hadrons

hadrons

radiative

hadronshadrons

c2(1P)

c0(1P)

(1S)J/

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

c1(1P)

0

hc(1P)

, 0

hadrons

Figure 12.4: Spectroscopy of lightest charmonium states with M(cc) < 2MD. Notethat 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 areoften 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 canchange 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,η and Σ0 decay electromagnetically. The π±, K±, K0, n, Λ, Σ±, Ξ and Ω− decay viaflavour-changing weak interactions, with long lifetimes.

82

12.4 Pion and Kaon Decays

12.4.1 Charged Pion Decay π+ → µ+νµ

W

d

u

νµ(p4)

µ+(p3)

fπ

Charged pions decay mainly to a muon and a neutrino:

π+ → µ+νµ π− → µ−νµ (12.3)

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 thedecay is:

M =v(d)gW Vudfπγµ(1− γ5)u(u)

1

q2 −m2W

u(νµ)gW γµ(1− γ5)v(µ+)

(12.4)

≈ Vudfπg2

W

m2W

v(d)γµ(1− γ5)u(u)

u(νµ)γµ(1− γ5)v(µ+)

(12.5)

|M|2 = 4 G2F |Vud|2f 2

π m2µ [pµ · pν ] (12.6)

and the total decay rate is:

Γ =1

τπ=|Vud|2G2

F

8πf 2

πmπm2µ

1−

m2µ

m2π

2

(12.7)

From the charged pion lifetime, τπ+ = 26 ns, the pion decay constant can be de-duced, 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 helicitysuppressed. This can be seen from the factor m2

µ/m2π in the decay rate. Replacing mµ

with me reduces the decay rate by a factor ≈ 10−4. The suppression is associated withthe helicity states of neutrinos, which force the spin-zero π+ to decay to a left-handedneutrino and a left-handed µ+, or the π− to decay to a right-handed antineutrino anda 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 decaymodes are:

• Purely leptonic decays, where the su annihilate, similar to π+ decay:

BR(K+ → µ+νµ) = 63.4%

83

• Semileptonic decays, where s → u and the W+ boson couples to either e+νe orµ+νµ:

BR(K+ → π0+ν) = 8.1%

• Hadronic decays, where s → u and 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 propertiesmeasure mass eigenstates with a definite lifetime and decay width, wheras the weakforce acts on the weak eigenstates. (The weak eigenstates couple to the W boson.)

Weak eigenstates are admixture of mass eigenstates, conventionally described usingCKM matrix a mixture of the down-type quarks:

d

s

b

=

Vud Vus Vub

Vcd Vcs Vcb

Vtd Vts Vtb

dsb

(12.8)

Where d, s, b are the mass eigenstate; d, s, b are the weak eigenstates.

The matrix is unitary, and its elements satisfy:

i

V 2ij = 1

j

V 2ij = 1 (12.9)

i

VijVik = 0

j

VijVkj = 0 (12.10)

By considering the above unitarity constraints it can be deduced that the CKM matrixcan be written in terms of just four parameters: three angles si = sin θi, ci = cos θi,and a complex phase δ:

c1 s1c3 s1s3

−s1c3 c1c2c3 − s2s3eiδ c1c2s3 + s2c3eiδ

s1s2 −c1s2c3 − c2s3eiδ −c1s2s3 + c2c3eiδ

(12.11)

More about the CKM matrix in Chapter 14!

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