Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
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7.5 Breit–Wigner Equation for Resonances 149
The shape of the BW curve can be obtained from the formalism of the amplitudes
and phases of matter waves. However, it is instructive to derive it from the general
property that the resonance is an unstable particle, with lifetime , and that time and
energy are variables correlated by the uncertainty principle. The energy dependence
of the amplitude of Fig. 7.6 is the Fourier transform of a wave function that describes
a survival probability decreasing exponentially over time, with lifetime .
Let us imagine the elastic formation process of a generic resonance R,which
decays with lifetime into the same initial particles. The presence of a interaction
process is demonstrated by the different directions and momenta of the particles in
the final state, that is,
a C b ! R ! a
0
C b
0
: (7.18)
The unstable resonance R is described by the free particle wave function (4.11)
multiplied by a real function describing its decay probability as a function of time,
that is,
.t/ D .0/e
i!
R
t
e
t
2
D .0/e
iE
R
„
t
e
2„
t
; (7.19)
where the relations !
R
D E
R
=„ and D„= have been inserted in the last
equality. The probability of finding the particle at a time t is
I.t/ D
D .0/
2
e
t=
D I.0/e
t=
; (7.20)
corresponding to the radioactive decay law (Sect. 4.5.2). The Fourier transform of
(7.19), in the natural unit system („Dc D 1), is
.E/ D
Z
.t/e
iEt
dt D .0/
Z
e
tŒ.=2/CiE
R
iE
dt D
D
K
.E
R
E/ i=2
: (7.21)
The constant K must be determined with the normalization properties of wave
functions. Since the square of the wave function .E/ represents the probability
of finding the particle in the energy state E, it must be proportional to the process
cross-section, that is,
.E/ D
0
.E/.E/ D
0
K
2
Œ.E
R
E/
2
C
2
=4
: (7.22)
Since Eq. 7.22 presents a maximum at E D E
R
, one can determine K as
1 D
.E
R
/.E
R
/ D 4K
2
=
2
! K
2
D
2
=4: (7.23)
The proportionality constant
0
must be related to the wavelength of the incident
particle, as mentioned at the beginning of the paragraph. From a dimension analysis,
one has
o
' ¯
2
. Detailed calculations [P87] show that
0
D .2¯/
2
D 4¯
2
: (7.24)
150 7 Hadron Interactions at Low Energies and the Static Quark Model
0 100 200
250
1100 1200 1300 1400
200
150
100
50
300 400
s (MeV)
Kinetic energy (MeV)
σ (mb)
8π λ
2
Fig. 7.7 Total cross-section
C
p as a function of the kinetic energy of the incident pion (the scale
indicated below the x-axis) in the region of the
CC
.1; 232/ resonance. The maximum value of
the cross-section is consistent with the value 8¯
2
predicted for an elastic resonance with spin 3/2.
The abscissa scale at the top refers to the c.m. energy, corresponding to the effective mass of the
psystem
Let us now consider the formation and decay of a resonance of total angular
momentum J by the collision of two particles a; b, with spin s
a
, s
b
. In this case, the
cross-section must be averaged over the number of spin states of the incoming par-
ticles and multiplied (Sect. 4.5) by a factor .2J C1/. Taking this into account, from
Eqs. 7.23 and 7.24, the elastic cross-section (7.22) as a function energy becomes
el
.EIJ/ D 4¯
2
.2J C 1/
.2s
a
C 1/.2s
b
C 1/
2
=4
.E
R
E/
2
C
2
=4
:
(7.25)
This equation corresponds to the Breit–Wigner equation. The formula must be fur-
ther modified to describe the production of nonelastic resonances (see Sect.9.3.2).
7.5.1 The
CC
.1232/ Resonance
The total cross-section
C
p at low energy, see Figs. 7.1 and 7.7, shows a large peak
at the kinetic energy of the incident pion T
lab
D 191 MeV. This is equivalent to a
total energy in the c.m. system E
cm
:
7.5 Breit–Wigner Equation for Resonances 151
E
cm
D 1232 MeV D 120 MeV: (7.26)
The width at half-maximum of the peak is obtained from the data shown in
Fig. 7.7. This resonance peak corresponds to the .1232/ which is the most famous
baryonic resonance. Because it is produced in
C
p interactions, it is a baryon state
(B DC1/, with isospin I D 3=2 and total angular momentum J D 3=2 (the “spin”
of the resonance).
The momentum corresponding to the c.m. energy value mc
2
DE
cm
D1232 MeV
is p
lab
D 300 MeV/c in the laboratory system and p
cm
D 228 MeV/c in the c.m.
system. The corresponding total and kinetic energies in the laboratory system are
E
lab
D
q
p
2
C m
2
D
p
300
2
C 139:6
2
D 330:9 MeV
T
lab
D E
lab
m
D 330 139:6 D 191:3 MeV:
Let us check the resonance spin assignment using the experimental values (7.26).
Recalling that s
p
D 1=2 and s
D 0, the Breit–Wigner (7.25)forthe
CC
.1232/
resonance can be written as (energy in GeV)
el
.EIJ/ D 4¯
2
.2J C 1/
.2s
p
C 1/.2s
C 1/
2
=4
.E
R
E/
2
C
2
=4
D 2¯
2
.2J C 1/
0:120
2
=4
.1:232 E/
2
C 0:120
2
=4
: (7.27)
At the maximum of the cross-section E D E
R
and assuming J D 3=2, the formula
gives
max
D
el
.E D E
R
IJ D 3=2/ D 8¯
2
D
8
p
2
cm
D
8.„c/
2
.0:228 GeV/
2
' 188 mb
(recall that 1 mb D10
27
cm
2
; „c D197 MeV fm, .„c/
2
D0:388 GeV
2
mb; 1 fm
=10
15
m/. The computed value of
max
corresponds to that experimentally shown
in Fig. 7.7. It is also worth stressing that any other assignment of the particle spin
different from J D 3=2, leads to a different value of
max
. For example, for J D
1=2, one would obtain
el
.E D E
R
IJ D 1=2/ D 94 mb, which is half of the
measured value.
7.5.2 Resonance Formation and Production
The
CC
.1232/ resonance as studied in elastic
C
p !
C
p collisions corre-
sponds to the process of resonance formation in the s-channel,seeFig.7.8a. For
the s-channel shown in Fig. 7.8a, it is assumed that time runs from left to right,
corresponding to the reaction
C
p !
CC
!
C
p;inthet-channel, time
152 7 Hadron Interactions at Low Energies and the Static Quark Model
formation in terms of quarks
π
+
π
+
Δ
++
Δ
++
(a)
p
p
u
u
u
u
u
u
d
d
u
u
u
d
d
formation
π
+
π
+
Δ
++
pp
production
π
+
π
+
π
0
p
p
abc
Fig. 7.8 Illustration of the (a)
CC
(1232) resonance formation in the s-channel from
C
p
interactions; (b)
CC
(1232) production in inelastic collisions:
C
p !
CC
0
!
C
p
0
.In
(c), the
CC
resonance formation is shown in terms of constituent quarks
goes from the bottom to the top of the same figure, corresponding to the reaction
pp !
C
.
The resonance can also be produced through reactions of the type
C
p !
CC
0
!
C
p
0
: (7.28)
This process is called resonance production,seeFig.7.8b. The case of the ˙
.1385/ discussed in Sect. 7.4.2 corresponds to a resonance production. The
CC
(1232) lifetime is so short (10
23
s) that the resonance remains confined in
the interaction region. Its traveling distance is of the order of c D 3 10
8
10
23
D
3 10
15
m=3fm.
In terms of constituent quarks, the proton consists of three quarks (p D uud ),
the meson
C
is made of a quark and an antiquark pair (
C
D ud), and the
CC
of three quarks (
CC
D uuu). The formation of the
CC
resonance in elastic
C
p
collisions in terms of constituent quarks is illustrated in Fig. 7.8c.
7.5.3 Angular Distribution of Resonance Decay Products
In this subsection, which can be skipped during an initial reading, the Clebsh–
Gordan coefficients [P08], known from the spin combinations in atomic physics,
are used. Our primary interest is based on the experimental fact that the angular
distribution of emitted particles can be used to determine the spin of the resonance.
For example, the assignment J D 3=2 to the
CC
may be confirmed by the study
of the angular distribution of the outgoing
C
meson at the resonance energy.
Consider the situation in the c.m. system before the collision, as shown in
Fig. 7.9. Taking the quantization z axis in the direction of the incident pion, one has
m
`
D 0. The spin of the proton can be directed towards the z axis or in the opposite
direction. The initial state has a wave function with J D 3=2; since the is spinless
and m
`
D 0, the third component of m
j
depends on the proton polarization:
i
.J D 3=2; m
i
D˙1=2/ D .`; 0/ X.1=2; ˙1=2/: (7.29)
.`; 0/ is the orbital wave function of the p system and X.1=2; ˙1=2/ is the
proton spin wave function. To obtain J D 3=2, ` must be equal to one or two. Let
us assume ` D 1 and m DC1=2.
7.5 Breit–Wigner Equation for Resonances 153
π
+
π
+
p
p
a
π
+
π
+
θ
∗
θ
p
p
Lab.
c.m.
b
Fig. 7.9 Elastic
C
p collision: (a) before and (b) after the collision in the laboratory and in the
center-of-mass system
Two options are possible for the final state wave function; the first one cor-
responds to the configuration in which the proton spin is oriented as before the
collision; in the second one, the spin is flipped with respect to its original orientation.
Since the orbital wave functions are the spherical harmonic functions Y.`;m/,and
using the Clebsh–Gordan coefficients (the radial wave function does not contribute
to the angular distribution), one has
f
.3=2; C1=2/ D
p
2=3 Y.1; 0/X.1=2; C1=2/ C
p
1=3 Y .1; 1/X
1=2; 1=2
(7.30)
(the notation X.1=2;C1=2/ D X
"
;X.1=2;1=2/ D X
#
is also used). Note that,
due to the choice of the reference system, the scattering angle coincides with the
zenith angle
that appears in Y
m
`
. The spherical harmonic functions are
(
Y.1; 0/ D
p
3=4 cos
Y.1;C1/ D
p
3=8 sin
e
i'
:
(7.31)
The angular distribution of the pion, I.
/, is given by the probability density
f
f
. The spin part of the wave functions are orthonormal functions: X
2
"
D X
2
#
D
1, X
"
X
#
D 0. Therefore, one has
I.
/ D
f
f
D
2
3
jY.1; 0/j
2
hX
"
jX
"
iC
1
3
jY.1; 1/j
2
hX
#
jX
#
i
D
2
3
3
4
cos
2
C
1
3
3
8
sin
2
D
1
2
cos
2
C
1
8
sin
2
D
1
8
.1 C 3 cos
2
/: (7.32)
154 7 Hadron Interactions at Low Energies and the Static Quark Model
25
20
15
10
5
090
180
T
π
= 190 MeV
T
π
=100
T
π
=307
θ
∗
(degree)
Ι (θ
∗
)
1 + 3 cos
2
θ
∗
Fig. 7.10 Angular distribution in the c.m. system of the scattered pion with respect to the incident
pion direction in the elastic
C
p collision at the
CC
(1232) resonance energy and two nearby
energies. Note that the shape of the distribution I.
/ continuously changes with energy
Figure 7.10 shows the experimental angular distribution of the scattered pion with
respect to the incident pion direction in the c.m. system. The results for
C
p elastic
collisions at the energy of the
CC
(1232) resonance and at two nearby energies are
shown. At the
CC
energy, the angular distribution follows (7.32), confirming the
assignment of J D 3=2, ` D 1 to the resonance. The value ` D 2 would produce a
different angular distribution.
7.6 Production and Decay of Strange Particles
In this section, we introduce the so-called strange particles. This term does not
mean that these particles are strange in the etymological sense of the word. It means
that when they were discovered their behavior seemed strange. These particles were
produced abundantly, which means that their production was through the strong
interaction. As subjected to strong interaction, one would have expected a very
short lifetime (10
23
s). Instead, they had relatively long lifetimes of the order of
10
8
–10
13
s, corresponding to typical lifetimes of weak interaction decays. Why
are their lifetimes so long even though they are abundantly produced? In addition,
why are they produced in pairs? The name “strange” particles was derived from
these very dilemmas. The solution was to invent the conservation of a new quantum
number: the “strangeness.” Due to their long lifetimes yielding path lengths up to
several centimeters, the bubble chamber was well suited to study the strange particle
properties. In Fig. 7.11, an incoming
meson from the left reaches the liquid
hydrogen bubble chamber. In addition to the
, two pairs of tracks are visible.
Each pair has a V-shape. The two arms of the V correspond to two opposite sign
7.6 Production and Decay of Strange Particles 155
π
–
π
–
π
–
Λ
0
Κ
0
π
+
p
A
B
C
Fig. 7.11 Example of a bubble chamber picture showing an associated production of two strange
particles (the K
0
meson and the
0
baryon) and their decay products. The associated production
p !
0
K
0
occurs in A. The K
0
meson travels from A to B, where it decays into two charged
particles .K
0
!
C
/;the
0
baryon travels from A to C, where it decays into two charged
particles (
0
! p
)
charged tracks starting from a single point, called the vertex.InA,the
interacts
with a proton, giving rise to:
p ! two strange neutral particles: (7.33)
Because of electric charge conservation, neither the
collision on an electron nor
a
decay can be considered. The process
Ce
would produce two negatively
charged particles.The
decay would produce an odd number of visible charged
tracks. Two neutral particles are then originated from the
interaction on proton
in A. Each unknown neutral state was originally denoted by the symbol V
0
.
If at the vertex B or C, one of the two V
0
interacts with a proton, only one charged
particle track would be visible. As a pair of charged particles can be seen at each
vertex, the simplest interpretation is that each pair corresponds to the decay of both
the neutral V
0
particle, with conservation of the electric charge:
V
0
! positively charged particle C negatively charged particle: (7.34)
The explanation of the series of events in the three vertices of Fig. 7.11 is as follows:
in A, two neutral strange particles are produced, the K
0
meson and the
0
baryon
according to the reaction
p ! K
0
0
: (7.35)
The K
0
meson (with mass m
K
0 D 497:7 MeV) travels from A to B. In B, it decays
into two charged mesons:
K
0
!
C
: (7.36)
The
0
baryon (m
D 1115:7 MeV) travels from A to C; in C, it decays into p
and
:
0
! p
: (7.37)
These statements can be verified by measuring all the parameters in the bubble
chamber pictures and using conservation principles.
156 7 Hadron Interactions at Low Energies and the Static Quark Model
The “strange” production of associated particles is explained assuming that the
0
and the K
0
produced in A have opposite strangeness quantum number; the
strangeness must be conserved in strong and electromagnetic interactions, but can
be violated in weak interaction processes.
The strangeness quantum number is explained in terms of the constituents of
hadrons, the quarks. An “ordinary” (not strange) hadron is made of u;d quarks.
A “strange” hadron contains at least one strange s quark; a “strange” antihadron
contains at least one strange antiquark
s. A hadron with one strange quark has
strangeness 1 (e.g.,
0
D sdu), those with two strange quarks have strangeness
2 (e.g.,
D ssd), those with three strange quarks have strangeness 3 (e.g.,
˝
D sss).
The production of
0
, K
0
is abundant and caused by the strong interaction, which
conserves strangeness
C p ! K
0
C
0
(7.38)
strangeness 00C1 1:
The associated production phenomenon (strange particles are produced in pairs)
is explained by the conservation of strangeness.
0
and K
0
have relatively long
lifetimes, typical of the weak interaction. The strangeness is not conserved in weak
interaction decays, that is,
0
! p C
(7.39)
strangeness 100:
The system of K mesons has the following strangeness assignment: the K
and K
0
have S D1;theK
C
and K
0
have S DC1. Another example of production and
successive decay of strange particles is
K
C p ! ˝
C K
C
C K
0
(7.40)
strangeness 103 C 1 C 1:
7.7 Classification of Hadrons Made of u;d;sQuarks
Before quarks were hypothesized, the increasing number of discovered hadronic
particles led one to suspect the existence of some inner symmetry law that could
explain the particle proliferation. The quark model solved the problem of hadron
classification. Until the discovery of the heaviest quarks (the c; b and t, see Chap. 9),
only the quarks u;d and s were known. From the regularity of discovered hadrons
(including many resonances with lifetime of the order of 10
23
s), it was derived
that quarks have fractional electric charge with respect to that of the proton.
7.7 Classification of Hadrons Made of u;d;s Quarks 157
Σ (1384), Ι = 1
Ω (1672), Ι = 0
3
2
Δ (1232), Ι =
1
2
Ξ (1533), Ι =
Ι
3
0
–2
–3
Δ
−
Σ
−
Σ
+
Σ
0
Ξ
−
Ξ
0
Δ
0
Δ
+
Δ
++
Ω
−
S
–1
Ι
3
0
–2
–3
ddd
dds
dss uss
sss
dus
uus
ddu
duu
uuu
S
–3/2 –1/2 +1/2 +3/2
ab
Fig. 7.12 The J
P
D 3=2
C
baryonic decuplet. (a) Assignment of the observed baryons and
(b) interpretation in terms of quarks. The isotopic spin and the average mass of each observed
baryon multiplet are indicated on the right. The third component of isospin is indicated as I
3
It was found that hadrons having the same spin and parity could be grouped in
families, graphically represented in a S; I
z
diagram. The charge Q is in units of the
proton charge and the baryon number is B DC1=3 for each quark. According to
the static model, quarks are grouped in order to form particles with integer electric
charge in two different ways:
•Thebaryons are made of three quarks (called valence quarks), the antibaryons
of three antiquarks.
•Themesons are made of a quark and an antiquark.
Ordinary matter only consists of u;d quarks .p D uud;n D ddu/.Thevalence
quarks explain the hadron spectroscopy. Hadrons are considered eigenstates of a
system of quarks interacting via the strong interaction in a similar way as the hydro-
gen atom energy levels are eigenstates of the p e system in the Coulomb field.
Here, the hadrons are classified on the basis of the static quark model. Originally,
the formalism of group theory was used. The unitary group of matrices with three
rows and three columns in the u;d;s space of quarks was called SU(3) (sometimes
written as SU(3)
uds
or SU(3)
f
,wheref stands for flavor).
After the discovery that the strong interaction dynamic is ruled by the “color”
quantum number, this classification became obsolete. However, it is still useful (with
its extension to four and five quarks) for the nomenclature of particles made of
quarks.
The baryons (and similarly, the antibaryons) can either have the three quark
spins aligned ("""), or one quark with a spin opposite to that of the other two
quarks (""#). In the first case, the baryons have spin
parity
J
P
D 3=2
C
and form
a decuplet, graphically represented by a triangle, see Fig. 7.12. If one quark has its
spin in the opposite direction of the two others, the baryons have J
P
D 1=2
C
and
form an octet (which includes protons and neutrons), as shown in Fig. 7.14.
158 7 Hadron Interactions at Low Energies and the Static Quark Model
The mesons may be formed either with two quarks with spins in opposite
directions ("#) or with two spins aligned in the same direction (""). In the first
case, the mesons have spin zero and negative parity, J
P
D 0
, forming a nonet
graphically represented by the hexagon of Fig.7.15a. In the case of spin aligned in
the same direction, the mesons form the nonet of particles shown in Fig. 7.15b.
All the multiplets would be degenerate states for the strong interaction if the
masses of all quark flavors were the same. The strong interaction depends on an
inner quark quantum number, the color. The color force is independent of the
flavor. The mass of the hadrons made of the u;d;s quarks is larger than the sum
of the masses of their quarks. Notably m
u
' m
d
, very small mass in comparison
to the nucleon mass. The masses of the nonstrange hadrons consist predominantly of
the color field energy. This explains why isotopic spin is a good symmetry. Particles
with the same J
P
and differing only by the presence of a u or d quark have almost
the same characteristics. They are isospin multiplets indicated with the same letter
(the only notably known exception is that of p and n). Small differences in mass
between members of a multiplet are due to the electromagnetic interaction. The case
of neutral mesons in the center of the hexagon of Fig. 7.15 is more complicated.
The larger mass m
s
of the strange quark with respect to m
u
;m
d
corresponds to a
larger mass of the particles obtained from the substitution of a u;d quark with a s.
In the following sections, these multiplets are illustrated in detail.
7.8 The J
P
=3/2
C
Baryonic Decuplet
Consider the particles that have J
P
D 3=2
C
; they form a decuplet. Figure 7.12a
shows the ten baryon with J
P
D 3=2
C
and lower masses (states with higher orbital
angular momentum have higher masses). These baryons are
•The(1232) with isospin I D 3=2. It has four different charged states
CC
,
C
,
0
,
.
• The hyperon ˙ (1385) with strangeness 1,(namely˙
(1385)), with I D 1
and the three states ˙
C
, ˙
0
, ˙
.
• The hyperon (1533) with strangeness 2,(namely,
(1533)), with I D 1=2
and the two states
0
,
.
• The hyperon ˝
(1672) with strangeness 3, with I D 0.
The masses given in the figure are the average mass of each isospin multiplet. The
graph shows the third component of isospin along the x-axis, and the strangeness
S along the y-axis. The ten baryons are arranged in a regular pattern forming an
inverted triangle. The mass difference between two neighboring isospin multiplets
is m
˙
m
D 152 MeV, m
m
˙
D 149 MeV, m
˝
m
D 139 MeV, a
difference that is almost constant, with an average of about 147 MeV. On can recall
the empirical relation m D a C bY ,whereY D B C S is the strong hypercharge.