8
Probing the Proton: 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 < rp : the wavelength is sufficiently short to resolve
sub-structure. Scattering is from constituent quarks.
• 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: d3�r ρ(�r) = 1. The Fourier Transform of ρ(r) is the form factor, F (q):
�
F (�q ) = d3�r exp{i�q · �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: F1 to describe the distribution
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 K µ (compared to γ µ ):
M(e− p → e− p) =
e2
(ū3 γ µ u1 ) (ū4 Kµ u2 )
(p1 − p3 )2
51
(8.3)
e− (p1 )
e− (p3 )
ieγ µ
q
−ieK µ
p(p2 )
p(p4 )
Figure 8.1: Elastic electron proton scattering. As the proton is a composite object the
vertex factor is K µ .
K µ = γ µ F1 (q 2 ) +
iκp
F2 (q 2 )σ µν qν
2mp
(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 are
functions of the momentum transferred by the photon (q 2 ).
8.4
Elastic Scattering
The elastic scattering in the relativistic limit pe = Ee by the Mott formula:
�
dσ ��
α2
=
dΩ �point 4p2e sin4
θ
2
�
θ
q2
2 θ
cos −
sin
2 2m2p
2
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 ��
α2
=
dΩ �NR 4m2e ve4 sin4
8.4.1
(8.6)
θ
2
Q2 and ν
We define a two new quantities: Q2 and ν, which are useful in describing scattering.
Q2 ≡ −q 2 = (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,
q 2 < 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:
�
�
�
��
dσ ��
α2
E3
q2
κ2 q 2 2
2
2 θ
2 θ
2
=
F cos −
(F1 + κF2 ) sin
(8.8)
F1 −
dΩ �lab 4E12 sin4 2θ E1
4m2p 2
2 2m2p
2
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 +
κq 2
F2
4m2p
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 ��
α2
E3 G2E + τ G2M
2 θ
2
2 θ
=
cos + 2τ GM sin
dΩ �lab 4E12 sin4 2θ E1
1+τ
2
2
(8.10)
where we have used the abbreviation τ = Q2 /4m2p .
The experimental data on the form factors as a function of q 2 are correspond to an
exponential charge distribution:
ρ(r) = ρ0 exp(−r/r0 )
1/r02 = 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, 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 q 2 this is known as deep inelastic scattering (DIS).
e− (p1 )
ieγ
µ
e− (p3 )
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:
x≡
Q2
2p2 · q
(8.12)
where again q 2 is the four-momentum transferred by the photon.
The invariant mass, MX , of the final state hadronic jet is:
MX2 = p24 = (q + p2 )2 = m2p + 2p2 q + q 2
(8.13)
The system X will be a baryon. The proton is the lightest baryon, therefore MX > mp .
Since MX �= mp q 2 and ν, are two independent variables in DIS, and it is necessary to
measure E1 , E3 and θ in the Lab frame to determine the full kinematics.
Futhermore
Q2 = 2p2 · q + Mp2 − MX2 ⇒ 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 scattering in terms of point-like constituents inside the nucleon known as partons with an
effective 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 between
30 GeV electrons and 830 GeV protons.
55
e− (p1 )
e− (p3 )
q
m
m
The parton model restores the elastic scattering relationship between q 2 and ν, with m
(parton mass) replacing mp :
q2
ν+
=0
(8.15)
2m
and the cross section for elastic electron-parton scattering is:
�
�
d2 σ
α2
1
2
2
2 θ
2
2 θ
=
F2 (x, Q ) cos + F1 (x, Q ) sin
(8.16)
dE3 dΩ
2 M
2
4E12 sin4 2θ ν
The structure functions are sums over the charged partons in the proton:
�
2xF1 (x) = F2 (x) =
x e2q q(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� 2
F1 (x) =
e fi (x)
2 i q
(8.18)
xe2q fi (x)
(8.19)
F2 (x) =
�
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: uū, dd̄, ss̄, cc̄, bb̄.
• Gluons, g.
56
The parton distribution functions must describe a proton with total fractional momentum x = 1:
� 1
dx x[u(x) + ū(x) + d(x) + d̄(x) + s(x) + s̄(x) + . . .] = 1
(8.20)
0
�
1
dx[d(x) + d̄(x)] = 1
0
�
1
dx[s(x) + s̄(x)] = 0
0
8.8
�1
0
�1
0
dx[u(x) + ū(x)] = 2
(8.21)
dx[c(x) + c̄(x)] = 0
(8.22)
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
F2p (x) =
x e2q q(x) = xu(x) + xd(x)
(8.23)
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
� 1
4
1
4
1
p
F2 (x)dx = fu + fd
F2n (x)dx = fd + fu
(8.24)
9
9
9
9
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
F2n (x)dx = fd + fu
(8.25)
9
9
0
Experimental measurement give:
� 1
F2p (x)dx = 0.18
�
0
0
1
F2n (x)dx = 0.12
(8.26)
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
u
d
ubar
dbar
s
c
0.7
x* f(x, µ=2 GeV)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
X
0.6
0.7
0.8
0.9
1
CT10.00 PDFs (area proportional to momemtum fraction)
1.2
g
u
d
ubar
dbar
s
c
3*x5/3* f(x, µ=85 GeV)
1
0.8
0.6
0.4
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 differences.
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, r̄, ḡ, and b̄. We define three quark eigenstates
r, g, and b:
 
 
 
1
0
0





0
1
0 
r=
g=
b=
(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:






0 1 0
0 −i 0
1 0 0
λ1 =  1 0 0 
λ2 =  i 0 0 
λ3 =  0 −1 0 
(9.2)
0 0 0
0 0 0
0 0 0

0 0 1
λ4 =  0 0 0 
1 0 0




0 0 −i
λ5 =  0 0 0 
i 0 0


0 0 0
1
λ7 =  0 0 −i 
λ8 = √ 
3
0 i 0


0 0 0
λ6 =  0 0 1 
0 1 0

1 0 0
0 1 0 
0 0 −2
(9.3)
(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 ones
that do not change the colour states.
59
The gluon states are:
g1 = √12 (rb̄ + br̄) g2 = √12 (rb̄ − br̄) g3 = √12 (rr̄ − bb̄)
g4 = √12 (rḡ + gr̄) g5 = √12 (rḡ − gr̄) g3 = √12 (bḡ − g b̄)
g7 = √12 (bḡ + g b̄) g8 = √16 (rr̄ + bb̄ − 2gḡ)
(9.5)
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 αS (instead of α): αS ≡ gS /4π.
• αS (q 2 ) decreases rapidly as a function of q 2 . At small q 2 it is large, and QCD is
a non-perturbative theory. At large q 2 QCD becomes perturbative like QED.
• A quark has one of three colour states (replacing electric charge). Antiquarks
have anticolour states.
• A gluon propagator has one of eight colour-anticolour states.
• A quark-gluon vertex has a factor 12 gs λa γ µ .
• 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
a, µ
d, ρ
q2
q1
q3
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 themselves: gluons may undergo three gluon and four gluon interactions as shown in figure 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:
Vqq̄ (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 q q̄ pair
cannot be separated since infinite energy would be required. Instead the flux tube can
break creating an additional q q̄ pair in the middle which combines with the original q q̄
pair to form two separate hadrons.
61
p2 , k
p4 , l
gS
g
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 λji at
the vertex. For the antiquark line we get a term λkl at the vertex.
The matrix element for quark-antiquark scattering is written:
� g
� g
� g µν
�
S
S
M = ūj λaji γ µ ui 2 δ ab v̄k λbkl γ ν vl
2
q
2
(9.7)
where uj , ui , vk , vl are the spinors for the quarks, with the subscripts representing the
colour part of the wavefunction.
gS2 λaji λakl
M= 2
[ūj γ µ ui ] [v̄k γ µ vl ]
q
4
(9.8)
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� a a
f = λaji λakl =
λ λ
(9.9)
4
4 a ji kl
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:
Vqq̄ = −
f αS
r
(9.10)
where f is defined in equation (9.9).
9.5.1
Colour factor for qq̄ → qq̄
For the calculation of the colour factors we can choose particular colours for q and q̄.
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. i = 1, k = 1̄ → j = 1, l = 1̄, e.g. rr̄ → rr̄: f1 = 14 a λa11 λa11 The only matrices
that contribute to this sum (elements non-zero) are λ3 and λ8 :
f1 =
1
1
1
1
1� a a 1 3 3
λ11 λ1 = [λ11 λ11 + λ811 λ811 ] = [(1)(1) + ( √ )( √ )] =
4 a
4
4
3
3
3
2. i = 1, k = 2̄ → j = 1, l = 2̄, e.g. rb̄ → rb̄: f2 =
f2 =
�
a
λa11 λa22 :
1� a a
1
1
1
1
1
λ11 λ22 = [λ311 λ322 + λ811 λ822 ] = [(1)(−1) + ( √ )( √ )] = − (9.12)
4 a
4
4
6
3
3
3. i = 1, k = 1̄ → j = 2, l = 2̄, e.g. rr̄ → bb̄: f3 =
f3 =
9.5.2
1
4
(9.11)
1
4
�
a
λa21 λa12 :
1
1� a a
1
1
λ21 λ12 = (λ121 λ112 + λ212 λ221 ) = [(−i)(i) + (1)(1)] =
4 a
4
4
2
(9.13)
Colour factor for Mesons
Mesons are colourless qq̄ states in a ”colour singlet”: rr̄ + bb̄ + gḡ.
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 αS
(9.14)
Vqq̄ = −
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):
Vqq̄ = −
4 αS
+ kr
3 r
(9.15)
This potential describes the allowed states of charmonimum (cc̄) and bottomonium (bb̄)
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 qq̄ 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 experiments
quarks and gluons observed as jets of particles.
Jet events from the ATLAS experiment are illustrated in figure 9.6.
64
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)
Michaelmas 2011
M.A. Thomson
257
Hadronisation and Jets
!Consider a quark and anti-quark produced in electron positron annihilation
q q
i) Initially Quarks separate at
high velocity
ii) Colour flux tube forms
between quarks
q
q
q
iii) Energy stored in the
flux tube sufficient to
produce qq pairs
q
q
q
v) 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 aquarks
quark and
! The main consequence is that
at collider
experiments
andanti-quark
gluons
observed as jets of particles
e+
e–
M.A. Thomson
!
q
q
Michaelmas 2011
258
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Figure 9.6: Jet Events in the ATLAS detector at CERN. Top: two jet event; Bottom:
five jet event.
256
66
jet
jet
Michaelmas 2011
Prof. M.A. Thomson
277
Running of !s
QCD at Colliders
10
QCD
Similar to QED but also have gluon loops
+
+
+
+…
Boson Loops
Fermion Loop
! Remembering adding amplitudes, so can get negative interference and the sum
can 10.1:
be smaller
than theof
original
diagram
alone
Figure
Illustration
processes
contributing
to renormalisation in QCD.
! Bosonic loops “interfere negatively”
10.1
= no. of colours
Strong Coupling
Constant αS
with
= no. of quark flavours
The coupling strength αS is large, which means that higher order diagrams are imporNobel
for Physics,
tant. At low q 2 (where q 2 is!thedecreases
momentum
transferred
by Prize
the gluon)
higher2004
order
with
Q2
S
(Gross, Politzer, Wilczek)
amplitudes are larger than the lowest order diagrams, so the sum of all diagrams does
Michaelmas 2011
278
Prof. M.A. Thomson
not converge, and QCD is non-perturbative. At high q 2 the sum does converge, 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 )
2
α(q ) =
1−
α(µ2 )
3π
log
� 2�
q
µ2
(QED)
(10.1)
In QCD the renormalization is attributed both to the colour screening effect of virtual
q q̄ 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:
αS (q 2 ) =
αS (µ2 )
1+
αS (µ2 )
(11nC
12π
− 2nf ) ln
� 2 � αS (q 2 )
q
µ2
(10.2)
where nC = 3 is the number of colours, and nf ≤ 6 is the number of active quark
flavours which is a function of q 2 .
The anti-screening effect of the gluons dominates, and the strong coupling constant
decreases rapidly as a function of q 2 . The running of the coupling constant is usually
written:
12π
� 2�
αS (q 2 ) =
(10.3)
(11nC − 2nf ) ln Λq 2
where ΛQCD ∼ 217 ± 25 MeV is a reference scale which defines the onset of a strongly
coupled theory where αS ≈ 1.
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 q 2 , the quarks and gluons inside the
proton can be observed as independent partons. This property is known as asymptotic
freedom. It is associated with the small value of αS (q 2 ), which allows the strong
interaction corrections from gluon emission and hard scattering to be calculated using
a perturbative expansion of QCD. The perturbative QCD treatment of high q 2 strong
interactions has been well established over the past 20 years by experiments at high
energy colliders.
In contrast, at low q 2 the quarks and gluons are tightly bound into hadrons. This is
known as 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 qq̄ pair.
Bottom: 3 jets produced from an initial qq̄g state.
In the electromagnetic process e+ e− → q q̄ (figure 10.4) the flavour and colour charge of
the quark and anti-quark q and q̄ 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− → q q̄ 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, nC = 3:
e2
[v̄(e+ )γ µ u(e− )][v(µ+ )γ µ ū(µ− )]
q2
e eq
M(e+ e− → qq̄) =
[v̄(e+ )γ µ u(e− )][v(q̄)γ µ ū(q)]
q2
M(e+ e− → µ+ µ− ) =
(10.4)
(10.5)
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:
R≡
�
σ(e+ e− → hadrons)
=
3
e2q
σ(e+ e− → µ+ µ− )
q
69
(10.6)
e+ (p2)
q(p4)
γ
e− (p1)
q̄(p3)
Figure 10.4: e+ e− → hadrons
Where the
�is over the number of quarks that can be produced at the centre of mass
√ sum
energy s = q 2 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− → q q̄g. 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:
�
� �
αS (q 2 )
2
R=3
eq 1 +
π
q
(10.7)
This is used to measure the strong coupling constant as a function of q 2 .
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
10
J/ψ
ψ(2S)
-4
10
σ [mb]
φ
-3
ρ
ρ
Υ
!
Z
-5
10
-6
10
-7
10
-8
10
1
10
10
R
10
10
2
Υ
3
J/ψ
ψ(2S)
Z
2
φ
ω
10
ρ!
1
ρ
10
-1
1
√
10
10
2
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
+ “Quantum
−
(green) is a naive quark-parton model prediction, and the solid one (red) is 3-loop pQCD prediction (see
Chromodynamics” section of
this Review, Eq. (9.7) or, for more details, K. G. Chetyrkin et al., Nucl. Phys. B586, 56 (2000) (Erratum ibid. B634, 413 (2002)). Breit-Wigner
parameterizations of J/ψ, ψ(2S), and Υ(nS), n = 1, 2, 3, 4 are also shown. The full list of references to the original data and the details of
the R ratio extraction from them can be found in [arXiv:hep-ph/0312114]. Corresponding computer-readable data files are available at
http://pdg.lbl.gov/current/xsect/. (Courtesy of the COMPAS (Protvino) and HEPDATA (Durham) Groups, May 2010.) See full-color
version 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 GeV
bottom 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
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 textbooks (e.g. [13]). Instead of giving an exhaustive description of those techniques here, we
focusHadron
on thoseCollider
aspects ofDictionary*
the calculations employed most frequently in the experimental
10.5.1
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 at3high q 2 between partons (gluons, quarks,
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
energy E
(� T ) 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 information
about the jet:
72
• Jet energy E =
jet axis.
�
i
Ei and momentum p� =
�
i
p�i The direction of p� defines the
• Jet invariant mass W 2 = E 2 − |�p|2
�
• Transverse energy flow within the jet |pT |2 = i |�
pi − p�|2
�
• Transverse jet momentum pt = (p2x + p2y ), 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− → q q̄(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:
Ri2 = (yi − yjet )2 + (φi − φjet )2
A typical value used to form jets is R = 0.7.
73
(10.8)
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 = √ (rr̄ + gḡ + bb̄)
(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, ms ≈ 100 MeV.
Most of the mass of hadrons is due to the QCD interactions between the quarks. Therefore in discussing hadron masses and magnetic moments we use constituent quark
masses 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 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, 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 same
value of I = +1/2, they are differentiated by their value of I3 (isospin up or isospin
down):
1
1
1
1
d : I = , I3 = −
(11.3)
u : I = , I3 = +
2
2
2
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:
π + [I = 1, I3 = +1] = ud̄
1
π 0 [I = 1, I3 = 0] = √ (uū − dd̄)
2
π − [I+1, I3 = −1] = dū
(11.4)
There is also a singlet state with I = 0, the eta meson:
1
η [I = 0, I3 = 0] = √ (uū + dd̄)
2
(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:
1
1
p [I = , I3 = + ] = uud
2
2
1
1
n [I = , I3 = − ] = ddu
2
2
(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, 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
3
h!!!!!h
1+
-i 2
u
1/3
1/2
!!i!h 7
h!6!+-
-2/3
1/3
1/2
I3
-1/3
5
-1/3
I3
h!!
4 -!+!!i
!h
-1/2
-1/2
s
h5
!+!!i!
h!4! -
d
2/3
h!!+
6- !!i!
h
2/3
Y
7
Y
3
u
h!!!!!h
1+
-i 2
d
-2/3
s
Figure 11.1: Illustration of SU(3) flavour symmetry.
J = 0 mesons are neutral with I3 = 0. These states are π 0 (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 ss̄ 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
√ (dus + uds + sud + sdu + dsu + usd)
6
(11.7)
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
∆++
p
χc
A
A
χf
S
χS
S
S
χL
S
S
ψ
A
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
+1
0
✻
K0 (ds̄)
π−
(dū)
−1
K+ (us̄)
π0
η8 η1
−1/2
√
π 0 = (dd̄ − uū)/ 2
√
η8 = (dd̄ + uū − 2ss̄)/ 6
√
η1 = (dd̄ + uū + ss̄)/ 3
π+
(ud̄)
0
K− (sū)
−1
Pseudoscalar mesons
J =0
K (sd̄)
0
+1/2
✲
I3
+1
Figure 11.2: The lightest pseudoscalar mesons with J = 0. The y-axis represents
strangeness, S and the x-axis represents I3 .
S
+1
0
✻
K∗0 (ds̄)
ρ−
(dū)
−1
K∗+ (us̄)
ρ0
ω φ
−1/2
√
ρ0 = (dd̄ − uū)/ 2
√
ω = (dd̄ + uū)/ 2
φ = ss̄
ρ+
(ud̄)
∗0
K∗− (sū)
−1
Vector mesons
J =1
K (sd̄)
0
+1/2
✲
I3
+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
ddd
−1
−2
∆0
ddu
∆+
duu
Σ�0
dus
Σ�−
dds
∆++
uuu
1232
Σ�+
uus
1385
Ξ�0
uss
Ξ�−
dss
1533
Ω−
sss
−3
1672
✲I
3
−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 ✻
n
ddu
0
−1
p
uud
Σ0
Λ
uds
Σ−
dds
1193
1116
Σ+
uus
Ξ−
dss
−2
938.9
Ξ0
uss
1318
✲
−1
−1/2
0
+1/2
I3
+1
Figure 11.5: The lightest baryons with J = 1/2.
79
12
12.1
Decays of Hadrons
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 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 interactions in which a second quark-antiquark pair is produced, e.g. ρ0 → 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):
D+ (cd̄)
D0 (cū)
Ds+ (cs̄)
(12.1)
and four possible B states:
B + (b̄u)
B 0 (b̄d)
Bs0 (b̄s)
Bc+ (b̄c)
(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
mc ≈ 1.5 GeV and mb ≈ 4.6 GeV.
12.2.1
Charmonium and Bottomonium
Charmonium and bottomonium are bound states of cc̄ and bb̄ states respectively.
The J/ψ meson is identified as the lowest J = 1 bound state of cc̄. The width of this
state is narrow because MJ/ψ < 2MD , so it cannot decay into a DD̄ 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
41. Plots of cross sections and related quantities
!"#$$%$&'()#*%+,-.
14
10
2
π + p total
⇓
10
π + p elastic
Plab GeV/c
10
-1
1
10
2010 — Subatomic:
Particle Physics
πp
1.2
√s GeV
πd
2
2.2
3
3
4
4
5
10
5
6
7
6
2
7 8 9 10
20
8 9 10
20
30
30
40
40
2
50 60
+
++√
+
+
2.Figure
Draw12.1:
a Feynman
diagram
the process
→ ∆at
→
Cross section
for pπfor
scattering.
The pπ
resonance
s ∼pπ
1.2 .GeV is due
⇓
to the production and decay of the ∆+ + baryon.
∓
10
2
!"#$$%$&'()#*%+,-.
π d total
⇓
π − p total
10
π − p elastic
Plab GeV/c
10
-1
1
10
10
2
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+ d̄u → uuu → uud+ d̄u.
2007/08
— Physics
3: Particlediagram
Physics
5
Figure
12.2:
Feynman
for pπ + → ∆++ → pπ + scattering.
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
10−24 s (any) one of the quarks emits a gluon which turns into a dd̄ 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 dd̄, and because the minimum energy
state is favoured. The mass of the pion and the proton is smaller than the mass of
the ∆++ .
6
Figure
ofsilly
charmonium
decay.
andthan
anti-charm
quarks
cross section
is therefore
proportional
to αNote
give
acharm
longer
if
S . This
This 12.3:
looks Illustration
like The
a pretty
Feynman
diagram,
butthe
the
uuulifetime
state
actually
lives
for
only
one
gluon
was
exchanged.
annihilate
with
each
other,
either
via
a
photon
or
an
odd
number,
n
≥
3
gluons.
slightly longer than just three quarks would if they didn’t form a bound state.
3. The φ
This makes the matrix element M(J/ψ → π + π − π 0 ) small compared to most strong
decays. Therefore there is relatively small probability per unit time that this decay
giving
the J/ψ
lifetime.as follows:
mesonhappens,
decays
via
thea relatively
stronglongforce
In fact it’s a long enough lifetime to give the J/ψ → µ+ µ− decay a chance to
81
contribute as well.
7. What is meant by colour
confinement?
φ → K +K −
49%
Colour confinement describes the phenomena where quarks are only found in hadrons.
0
0
The gluons exchanged
the quarks self interact. The potential
→ K between
K̄
34% between two
quarks, due to the gluon interactions is V (r) = kr. The energy stored between the
+ − 0
There is a complete spectroscopy of cc̄ charmonium states with angular momentum
states equivalent to atomic spectroscopy. Only the states with M (cc̄) < 2MD have
narrow widths.
A similar spectroscopy is observed for bb̄ bottomonium states.
s (2S)
dc(2S)
a
a›
a
rc1(1 P)
hadrons
r (1 P)
c0
hadrons
a
d,/0
//
dc(1S)
J PC !
hadrons
a
hc(1 P)
r (1 P)
c2
hadrons
hadrons / 0
a
a
J/s (1S)
a
hadrons
hadrons a› radiative
0 <"
1 <<
0 ""
1 ""
1 "<
2 ""
Figure 12.4: Spectroscopy of lightest charmonium states with M (cc̄) < 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 neutrino in the final state is a clear signature of a weak interaction.
The lightest hadrons cannot decay to other hadrons via strong interactions. The π 0 ,
η and Σ0 decay electromagnetically. The π ± , K ± , K 0 , n, Λ, Σ± , Ξ and 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+ νµ
µ+ (p3 )
u
fπ
W
d¯
νµ (p4 )
Charged pions decay mainly to a muon and a neutrino:
π − → µ− ν¯µ
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 the
decay is:
�
�
�
�
1
M = v̄(d̄)gW Vud fπ γ µ (1 − γ 5 )u(u) 2
ū(νµ )gW γ µ (1 − γ 5 )v(µ+ ) (12.4)
2
q − mW
��
�
g2 �
≈ Vud fπ W2 v̄(d̄)γ µ (1 − γ 5 )u(u) ū(νµ )γ µ (1 − γ 5 )v(µ+ )
(12.5)
mW
|M|2 = 4 G2F |Vud |2 fπ2 m2µ [pµ · pν ]
(12.6)
and the total decay rate is:
�
�2
m2µ
1
|Vud |2 G2F 2
2
Γ=
=
fπ mπ mµ 1 − 2
τπ
8π
mπ
(12.7)
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
suppressed. This can be seen from the factor m2µ /m2π in the decay rate. Replacing mµ
with me reduces the decay rate by a factor ≈ 10−4 . 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 the s̄u annihilate, similar to π + decay:
BR(K + → µ+ νµ ) = 63.4%
83
• Semileptonic decays, where s̄ → ū and the W + boson couples to either e+ νe or
µ+ νµ :
BR(K + → π 0 �+ ν� ) = 8.1%
• Hadronic decays, where s̄ → ū and the W + boson couples to ud̄:
BR(K + → π + π 0 ) = 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�  =  Vcd Vcs Vcb   s 
(12.8)
�
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:
�
�
Vij2 = 1
Vij2 = 1
i
�
(12.9)
j
�
Vij Vik = 0
i
Vij Vkj = 0
(12.10)
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
s1 c 3
s1 s3
 −s1 c3 c1 c2 c3 − s2 s3 eiδ
c1 c2 s3 + s2 c3 eiδ 
(12.11)
iδ
iδ
s1 s2 −c1 s2 c3 − c2 s3 e −c1 s2 s3 + c2 c3 e
More about the CKM matrix in Chapter 14!
84