Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
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Chapter 9
Discoveries in Electron-Positron Collisions
9.1 Introduction
The first e
C
e
colliders were completed in the 1960s at the Italian National
Laboratories of Frascati with the small storage ring AdA followed by Adone
(i.e., “big AdA”) at
p
s D 3 GeV. A series of electron-positron colliders were
successively built in the U.S., Europe, Japan, Russia and China. A major motivation
for the construction of higher energy e
C
e
colliders came from the discovery of the
J= particle at SLAC (Stanford, USA) in 1974. This meson is made of a q
q pair,
where q is the fourth type of quark, that is, the charm. Previously, e
C
e
collisions
provided the first experimental indication of the quark color quantum number. In
1977 the third lepton family (with the lepton) was discovered always at SLAC,
immediately after the observation at Fermilab of the third family of quarks (with the
bottom quark). In this chapter, we shall discuss this series of discoveries. Finally,
high-precision measurements of the parameters of the electroweak theory and a
thorough verification of the Standard Model have been performed since 1989 by
the four LEP experiments at CERN (ALEPH, DELPHI, L3 and OPAL) and the
SLD experiment at Stanford. These experiments, together with those at DESY and
at the Fermilab
pp collider, represented a new era in terms of the size, complexity,
accuracy and the number of physicists participating in a single experiment. A further
increase is now observed for the LHC experiments.
Apart from hadrons, which are made of subconstituent quarks, electron and
positron are “fundamental” objects. For this reason, e
C
e
collisions are easier to
analyze compared to hadron-hadron and lepton-hadron collisions. At high energies,
the basic e
C
e
interactions are
e
C
e
! f f (9.1a)
! GG
0
: (9.1b)
S. Braibant et al., Particles and Fundamental Interactions: An Introduction to Particle
Physics, Undergraduate Lecture Notes in Physics, DOI 10.1007/978-94-007-2464-8
9,
© Springer Science+Business Media B.V. 2012
229
230 9 Discoveries in Electron-Positron Collisions
In (9.1a), the fermion-antifermion pair is either a charged lepton pair .e
e
C
;
C
;
C
/, a neutral lepton pair .
e
e
;
;
/, or a quark-antiquark pair
.u
u;dd; ss; cc; bb/;thett pair cannot be produced at current e
C
e
colliders.
Each quark or antiquark hadronizes in a hadron jet which is better identified when
the energy of the colliding e
C
e
is higher. The quark and/or the antiquark can
radiate one (or more) energetic gluon, resulting in a third (or more) jet. The fermion-
antifermion pair production proceeds through the exchange of a or a Z
0
in the
so-called s channel (see Fig.9.2).
In the process e
C
e
! GG
0
, the Gauge boson G is either a ,aW
˙
,oraZ
0
.
Note that for G D or Z
0
, G D G
0
.ForG D W , one has GG
0
D W
C
W
.
At LEP, the case G D and the processes e
C
e
! W
C
W
, e
C
e
! Z
0
Z
0
have been studied in great detail. The weak interaction vertex Z
0
W
C
W
was
studied for the first time at LEP2, i.e., the LEP operating phase in which the c.m.
energy was increased to 209 GeV.
The processes involving the exchange of photons are well described by quantum
electrodynamics (QED). The inclusion of the Z
0
exchange requires the weak
interaction and the interference between the two forces. At energies close to
m
Z
0 , the electromagnetic and weak interactions are unified into the electroweak
interaction (see Chap. 11).
Some processes are due to a single type of interaction. For example, the reaction
e
C
e
! (9.2a)
is only due to the Electromagnetic Interaction (“EM”), while the reaction
e
C
e
! (9.2b)
is only due to the Weak Interaction (“WI”). For the process (` D lepton)
e
C
e
! `
C
`
; (9.2c)
both the weak and the EM interactions must be considered. For e
C
e
WICEM
! qq
SI
!
hadrons,theStrong Interaction (“SI”) must be added for the quark hadronization.
The process e
C
e
! qq ! two hadronic jets can be regarded as one of the best
manifestations of quark existence; this is also true for the reaction e
C
e
! qq !
q
qg ! three hadronic jets for the existence of gluons.
In the first part of the chapter, we shall analyze the e
C
e
collisions to explain
the cross-sections shown in Fig. 9.1a up to a c.m. energy of about 30 GeV. Then, we
shall focus on the e
C
e
collisions at energies close to the Z
0
peak, using mainly
information from the LEP collider at CERN (LEP phase 1, LEP1). During this
period, the c.m. energy of the colliding particles was around the Z
0
mass (i.e., below
100 GeV). With about 18 million recorded events, the Z
0
peak characteristics were
optimally studied, yielding a systematic and high-precision study of the electroweak
interaction. Finally, we shall present the results obtained in the second phase of LEP
operation (LEP2) with c.m. energies up to 209 GeV.
9.2 Electron-Positron Cross-Section and the Determination of the Number of Colors 231
10
–3
a
b
10
–4
10
–5
10
–6
10
–7
10
–8
10
–2
10
3
10
2
10
1
10
–1
1 10
σ[mb]
ω
ρ
φ
ψ(2S)
Υ
Z
1 10 10
2
10
2
R
ω
ρ
φ
ρ
ρ
J/ψ
ψ(2S)
Υ
Z
√
s [GeV]
J/ψ
Fig. 9.1 (a) Cross-section of the process (e
C
e
! qq ! hadrons) as a function of the center-
of-mass (c.m.) energy. (b) Ratio R D Œ.e
C
e
! hadrons/=Œ.e
C
e
!
C
/ as a
function of the c.m. energy [P08]
9.2 Electron-Positron Cross-Section and the Determination
of the Number of Colors
Figure 9.1a shows the cross-section for the process e
C
e
! hadrons as a function
of the c.m. energy. The experimental points (drawn up to a c.m. energy of about
200 GeV) are the results of a large number of measurements made with different
experiments at various e
C
e
colliders. The purely EM cross-section decreases
regularly with increasing c.m. energy, as expected from the Coulomb interaction
between point-like objects. The peaks are due to the reactions e
C
e
! ! vector
meson ! hadrons. The vector mesons made of quark-antiquark pair and with the
same photon quantum numbers can be easily studied by analyzing the hadronic
cross-section .e
C
e
! hadrons/.
The first two peaks are due to the
0
;!
0
meson resonances. In terms of quarks,
they are resonances of the u
u and d d type. The third peak is due to the D ss
resonance. The fourth peak is also a resonance, the J= D c
c, due to a new quark
flavor. Finally, the last peaks are due to a fifth quark flavor, forming the D b
b and
its excited states.
232 9 Discoveries in Electron-Positron Collisions
Fig. 9.2 Lowest order
Feynman diagrams for
e
C
e
!
C
,(a) with a
virtual photon exchange,
and (b) with a Z
0
boson
exchange
e
+
e
+
μ
–
μ
–
μ
+
μ
+
e
–
e
–
γ
Z
0
ba
Figure 9.1b shows the ratio
R D
.e
C
e
! hadrons/
measured
.e
C
e
!
C
/
calculated
(9.3)
as a function of the c.m. energy in the range from 0.3 to 200 GeV.
9.2.1 The Process e
C
e
! !
C
At the lowest order, the EM cross-section .e
C
e
!
C
/ is explained by the
Feynman diagram of Fig. 9.2a: the e
C
e
pair annihilates into a virtual photon
which materializes as a
C
pair. Outside resonances, the cross-section is given
by Eq.4.62, that is,
.e
C
e
! !
C
/ D
4˛
2
EM
.„c/
2
3
1
s
(9.4)
where ˛
EM
D 1=137 and s D E
2
cm
. Placing the numerical value of ˛
EM
in (9.4),
and using appropriate units (Appendix 2), one obtains
.e
C
e
! !
C
/ '
86:8 [nb]
s [GeV
2
: (9.5)
The cross-section is expressed in [nb] = (10
33
cm
2
)ifE
2
cm
D s is expressed in
[GeV
2
]. The contribution due to the weak interaction, with a Z
0
exchange as shown
in Fig. 9.2b, is only important around
p
s m
Z
90 GeV.
9.2.2 The Color Quantum Number
The e
C
e
! hadrons process for
p
s<30GeV proceeds through the exchange of
a virtual photon which subsequently produces a quark-antiquark pair, i.e., e
C
e
!
! q
q (see Fig. 9.3b). Since free quarks cannot be produced, the qq pair is thus
a virtual pair: the quark and antiquark give rise to two jets of hadrons that are well
9.2 Electron-Positron Cross-Section and the Determination of the Number of Colors 233
e
+
e
+
e
–
e
–
γ
γ
q
–
q
hadrons
hadrons
resonance
qq
–
ab
Fig. 9.3 For
p
s<30GeV, the hadron production proceeds through the annihilation e
C
e
! ;
(a) at the energy corresponding to a q
q resonance, with spin-parity J
P
D 1
,the directly
couples to the resonance which then decays in hadrons; (b) in the continuum regions of Fig. 9.1a,
the gives rise to a q
q pair which subsequently produces hadrons (two well defined jets at high
energies)
defined and clearly visible at higher energies. For each type of quark “flavor” with
charge Q
q
, the cross-section outside the resonances is equal to (9.4) multiplied by
the considered quark charge Q
q
squared, that is,
.e
C
e
! ! qq ! hadrons/ D
4˛
2
EM
Q
2
q
3
.„c/
2
s
: (9.6)
As seen in Sect. 7.8.1, Fermi statistics require that baryons are described by
antisymmetric wave functions. This means the introduction in the wave function of a
new quantum number, the so-called “color” quantum number. The first experimental
confirmation came precisely from the study of (9.6): the theoretical prediction
differed by a factor of three compared to experimental data. The quarks produced
in the final state each have three different degrees of freedom, corresponding to the
three colors. As a result, a multiplicative factor N
C
D 3 must be introduced in (9.6).
If there are N
f
different types of quarks, i.e, N
f
flavors, for energies higher than
their production threshold, one has
.e
C
e
! ! qq ! hadrons/ D N
C
4˛
2
EM
3
.„c/
2
s
N
f
X
nD1
Q
2
n
: (9.7)
Therefore, the expected ratio R is
R D
.e
C
e
! ! qq ! hadrons/
.e
C
e
! !
C
/
D N
C
N
f
X
nD1
Q
2
n
; (9.8)
where .e
C
e
!
C
/ is the predicted cross-section obtained in Eq. 9.5.
Let us study the expected ratio R (9.8) as a function of the c.m. energy:
• 1:5 <
p
s<3GeV. The three lower mass quarks are kinematically accessible,
the quarks u with charge C2=3, d and s with charge 1=3; therefore,
R
3q
D N
C
.Q
2
u
C Q
2
d
C Q
2
s
/ D N
C
C
2
3
2
C
1
3
2
C
1
3
2
D
2
3
N
C
:
(9.9)
234 9 Discoveries in Electron-Positron Collisions
The measured value is R D 2, as shown in Fig.9.1b. The agreement between
data and (9.9) is obtained only if N
C
D 3, which confirms the existence of three
different degrees-of-freedom (colors) for the quarks.
• 3<
p
s<9GeV . After
p
s 3 GeV, the measured ratio R increases to 10/3.
This requires an additional term in (9.9). This is explained by the existence of a
fourth quark (the charm c) and the fact that the c
c pair production threshold is
reached. The quark c has a charge C2=3;thevalue3.2=3/
2
D 4=3 must therefore
be added to the value obtained in (9.9) which gives 2 C 4=3 D 10=3.
• 9<
p
s<30GeV. For
p
s 9:5 GeV, the bb (the fifth quark) production
threshold is exceeded. Since Q
b
D1=3, there is an additional contribution of
C3.1=3/
2
D 3=9 D 1=3, which added to 10=3 gives 11=3, very close to the
value measured for energies between 10 and 30 GeV.
Note that the process e
C
e
! f f cannot be experimentally disentangled from
e
C
e
! f ffor low energy . These radiative processes (which explain the small
differences between experimental data and the values from Eq. 9.8) must therefore
be included in the calculations.
We have explained the continuous behavior of the cross-section .e
C
e
! !
hadrons/ showninFig.9.1a, and corresponding to the steps visible in Fig.9.1bup
to
p
s ' 30 GeV. The peaks observed in .e
C
e
! ! hadrons/ of Fig. 9.1a
must now be explained. These peaks are due to resonances in the q
q systems, as
shown in Fig. 9.3a.
9.3 The Discovery of Charm and Beauty Quarks
9.3.1 Mesons with c, c Quarks
In 1974, a new vector meson (qq state with spin D1) with a large mass was
discovered at the Stanford Linear Accelerator Center (SLAC) e
C
e
collider. The
vector meson with a mass of 3,097 MeV was observed by the research group headed
by Burton Richter (who named the new particle ) through both leptonic decays
e
C
e
! ! e
C
e
;
C
and hadronic decays e
C
e
! ! hadrons.
It was also independently observed in hadron-hadron collisions at the AGS
proton accelerator at the Brookhaven National Laboratory (BNL) by a group headed
by Samuel Ting (who named the new particle J ): p Be ! JX; J ! e
C
e
.Richter
and Ting (Nobel laureate in 1976) announced their discoveries on the same day,
and the new particle was named J= .TheJ= is regarded as a pure c
c state as
the .1;020/ meson is considered to be a pure s
s state. The c quark is the fourth
quark predicted by the GIM mechanism (see Sect. 8.14). The J= meson has a very
small width . D 93 keV, see Problem 9.7) and thus a relatively long lifetime
( D„= ' 10
20
s).
The excited state of the J= were found at SLAC, and the first one was called
0
. It is now called .2S / or .3686/, respectively indicating its quantum state or
9.3 The Discovery of Charm and Beauty Quarks 235
Table 9.1 Main characteristics and decay modes of the “charmed” J= and .3686/ vector
mesons
State Mass (MeV) J
P
;I
tot
(keV) Branching ratios
J= .3100/ 3;096:916 ˙0:011 1
; 0 93:2 ˙ 2:1 hadrons 88%
[mostly .2n C1/
e
C
e
6%
C
6%
.3686/ 3;686:09 ˙ 0:04 1
; 0 317 ˙ 9 J= 2 49:4%
26:5%
e
C
e
0:8%
C
0:8%
mass in MeV. Other cc vector mesons are denoted similarly with and the quantum
state (if known) or the mass. The name charmonium is used for the J= and
other charm-anticharm bound states, Sect. 9.4. This is analogous with positronium,
which consists of a bound electron-positron state. Table 9.1 summarizes the main
characteristics of the J= and .3686/ vector mesons.
9.3.2 The J= Resonance Properties
Let us consider the cross-section .e
C
e
! ! qqresonance! hadrons/ at
energies close to a q
q resonance, producing the peaks observed in Fig. 9.3a, b. Close
to a resonance, the cross-section is given by the usual Breit–Wigner formula (7.25).
We need to modify it in order to introduce the resonance parameters for a nonelastic
process. In particular, the factor
2
appearing at the numerator of (7.25) must be
substituted with
ee
h
to take into account that the production takes place through
a resonant annihilation of an electron-positron pair and a decay into hadrons.
ee
is the resonance formation width in e
C
e
! resonance. It is equal to the decay
probability resonance ! e
C
e
, which is proportional to the branching ratio BR, see
Sect. 4.5.2. At the denominator, represents the resonance total width (in MeV).
h
is the width (BR) for the hadronic decay resonance ! hadrons. The Breit–
Wigner formula for the formation of a spin J D 1 resonance from .e
C
;e
/ with
spin s
1
D s
2
D 1=2 becomes
had
D 4¯
2
.2J C 1/
.2s
1
C 1/.2s
2
C 1/
ee
h
=4
Œ.E E
R
/
2
C
2
=4
(9.10)
where ¯ D„=p is the e
C
;e
de Broglie wavelength in the c.m.
Let us calculate the cross-section for the formation of the J= resonance in
e
C
e
! J= ! hadrons; the value obtained can be compared with the results
shown in Fig. 9.1,thatis,
236 9 Discoveries in Electron-Positron Collisions
.e
C
e
! J= ! hadrons/ D
¯
2
.2J C 1/
ee
h
.2s
1
C 1/.2s
2
C 1/Œ.E E
R
/
2
C
2
=4
D
3¯
2
h
ee
4Œ.E 3097/
2
C
2
=4
(9.11)
where E
R
D 3;097 MeV is the energy corresponding to the resonance, ¯ is the
de Broglie wavelength (in the e
C
;e
collider, p ' E
e
D
p
s=2 D 3;097=2 D
1548 MeV):
¯ D„=p D„c=pc ' 197 MeV fm=1548 MeV ' 0:127 fm: (9.12)
' 93 keV is the resonance total width. The ratio
ee
h
=
2
' .0:05/ .0:88/ '
0:04: in 5% of the cases, the J= decays in (or it is formed by) an e
C
e
pair and
87.7% of the time, it decays in hadrons [P08]. At the resonance energy (E D E
R
D
3097 MeV), the cross-section (9.11) is equal to
.e
C
e
! J= ! hadrons/ D 3¯
2
ee
h
2
D 0:07 mbarn: (9.13)
This value must be summed to the EM cross-section. At 3 GeV, the (9.7)givesa
value of 20 nb. The cross-section for the resonance production is about four orders
of magnitude larger. The photon directly couples to resonance with J
P
equal to that
of the photon, J
P
D 1
. This explains the major peaks appearing in the hadron
production cross-section for energies up to 12 GeV. Note that the vector meson
resonances can decay into e
C
e
and
C
with more equal probability than in
hadrons. Peaks are also therefore expected in the reactions e
C
e
! e
C
e
;
C
.
Finally, let us explain the structures observed in .e
C
e
! hadrons/ between
4 and 4.5 GeV and in the ratio R between 3.5 and 4.5GeV. The structures around
4.1 GeV are connected to the energy threshold corresponding to 2m
c
. Above this
threshold, the pair production of charmed D mesons
1
, for example, e
C
e
! !
c
c ! D
C
D
, is now kinematically accessible.
A situation similar to that of the charmed mesons was later observed in the region
around 9–10 GeV.
9.3.3 Mesons with b, b Quarks
In 1977, 3 years after the J= discovery, the (9880) and other similar vector
mesons were observed. They were discovered by the Fermilab E288 experiment
team headed by Leon M. Lederman in the reaction pBe ! X !
C
X.
These mesons require a fifth quark, the bottom/beauty quark b, with a mass m
b
'
4;300 MeV . D b
b/.
1
The composition in terms of the quark of these mesons is D
C
D cd; D
D cd.
9.4 Spectroscopy of Heavy Mesons and ˛
S
Estimate 237
Table 9.2 Main parameters of the vector mesons
.1
3
S/
0
.2
3
S/
00
.3
3
S/
000
.4
3
S/
Mass (MeV) 9,460.30 ˙ 0.26 10,023.26 ˙ 0.31 10,355.2 ˙ 0.5 10,579.4 ˙ 1.2
e
C
e
(keV) 1.340˙ 0.018 0.612 ˙ 0.011 0.443 ˙ 0.008 0.272 ˙ 0.029
tot
(keV) 54.02 ˙ 1.25 31.98 ˙ 2.63 20.32 ˙ 1.85 20,500 ˙ 2,500
The neutral vector mesons are formed by a quark-antiquark bb pair and have
masses of 9.46, 10.02, 10.35 and 10.58 GeV. The discovery was confirmed and the
mesons were analyzed in more detail in e
C
e
collisions. Table 9.2 gives the various
states for the b
b system (the mesons formed from a bottom quark and its antiquark
with different total or orbital angular momentum are called “bottomonium”).
9.4 Spectroscopy of Heavy Mesons and ˛
S
Estimate
Atomic physics: ˛
EM
analogy. The discovery of the c and b heavy quarks, and
of the particles they form, led to the possibility of studying all mesons made of
the same quark-antiquark pair, but with different spin states. In analogy with the
atomic system, which gives precise information on the EM interaction, a sort of
“spectroscopy” of states composed of heavy quarks was used to obtain information
on the potential energy that binds quarks and on the strong interaction coupling
constant ˛
S
[P95].
A positron and an electron at a relative distance r, under the Coulomb potential
V
em
D˛
EM
=r; (9.14)
can form bound states (positronium) similar to those of the hydrogen atom. The
energy levels, as for the Bohr atom, are
E
n
D
˛
2
EM
m
e
c
2
4n
2
(9.15)
where n is the total quantum number (9.15) differs by a factor 1/2 with respect to the
Balmer formula due to the fact that the electron reduced mass m
e
=2 is introduced
(in the hydrogen atom the proton mass is much higher than m
e
). The spin-orbit
interaction splits the levels: for each n,thereare` D 0, 1, ..., n 1 possible values
(fine structure). The spin-spin interaction produces an additional splitting in a triplet
("", with three sub-states) and in a singlet ."#/ (hyperfine structure).
Figure 9.4a shows the positronium quantum numbers and energy levels. The
lower level corresponds to the state n
2SC1
L
S
D 1
1
S
0
, with n D1; ` D0; J D0;
S D0 (spin singlet; orthopositronium). The first excited level is the state 1
3
S
1
, with
n D1; ` D0; J D1; S D1 (spin triplet; parapositronium). The transition 1
3
S
1
!
1
1
S
0
produces photons with a frequency of 203,286MHz, in good agreement with
238 9 Discoveries in Electron-Positron Collisions
1
3
S
1
1
3
S
1
ψ
2
3
S
1
3
3
S
1
2
3
S
1
ψ'
2
1
P
1
2
1
P
1
2
3
P
1
2
3
P
1
χ
1
2
3
P
0
2
3
P
0
χ
0
2
3
P
2
2
3
P
2
χ
2
1
1
S
0
1
1
S
0
η
c
2
1
S
0
2
1
S
0
η'
c
1
S
3
S
1
P
3
PSTATES
1
S
3
S
1
P
3
PSTATES
n = 1
n = 2
x10000x1000
DISSOCIATION
ENERGY
POSITRONIUM CHARMONIUM
7
6
5
4
3
2
1
0
RELATIVE ENERGY (MeV)
BOUND STATES QUASI-BOUND STATES
1000
900
800
700
600
500
400
300
200
100
0
–100
ab
RELATIVE ENERGY (eV)
Fig. 9.4 Energy levels for (a)thepositronium and (b)thecharmonium. Note that the scale on the
ordinate of the figure on the left is in eV, while it is in MeV for the figure on the right [9B82]
the theoretical prediction. The state 1
1
S
0
has a charge conjugation C DC1 and
gives rise to a fast annihilation . D 1:25 10
10
s) into two photons. The state 1
3
S
1
has C D1 and produces a slower annihilation rate . D 1:4 10
7
s) in three
photons.
The coupling constant ˛
S
. Figure 9.4b shows the energy levels of the “charmo-
nium” (c
c states). Note the great similarity between these levels and those of the
positronium. The levels of the q
q systems shown in Fig. 9.4b can be calculated with
good approximation assuming that the potential produced by a quark (or by the
antiquark) on the other is (natural units, „Dc D 1):
V
QCD
D
4
3
˛
s
r
C Kr: (9.16)
The factor 4/3 is the color factor, see Sect. 11.9.3. At small r values (r<1fm), only
the Coulomb-like potential plays a role. To reproduce the experimental sequence
shown in Fig. 9.4, the strong coupling constant must be ˛
s
' 0:3, see Problem 9.1.
At large distances (r>1fm), the elastic-type potential Kr (with K ' 1 GeV/fm)
is dominant.
9.5 The Lepton
The known charged leptons are the electron e
, the muon
and the tau
,
with respective masses of 0.5110, 105.66, and 1,777.1MeV. The third lepton
was first observed in a series of experiments between 1974 and 1977 by M.L.