Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
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8.11 Strange Particle Decays 209
Semileptonic decays. Strange mesons and baryons are also subject to ˇ decays.
Examples of the semileptonic decay modes in which both leptons and hadrons are
present in the final state are:
m (MeV) BR D
i
= (s)
S D 0n! pe
e
1.29 1 887
˙
C
!
0
e
C
e
73.7 0:20 10
4
0:80 10
10
˙
!
0
e
e
81.7 0:57 10
4
1:48 10
10
S D 1
0
! pe
e
177.4 8:32 10
4
2:63 10
10
˙
! ne
e
257.8 1:02 10
3
1:48 10
10
0
! ˙
C
e
e
125.5 2:7 10
4
2:90 10
10
!
0
e
e
205.6 5:63 10
4
1:64 10
10
! ˙
0
e
e
128.7 0:87 10
4
1:64 10
10
Antibaryons decay in the charge conjugate states. For all semileptonic decays,
including those listed above, the relation Q D S holds, where Q D Q
f
Q
i
and S D S
f
S
i
are the variation of charge and strangeness of hadrons in the
final versus initial state. This implies that in terms of quarks, the process is
s ! W
u ! .e
e
/u:
Decays that do not satisfy the relation Q D S are largely suppressed. Consider,
for example, the ˇ decays of ˙
and ˙
C
and their interpretation in terms of quarks,
that is,
˙
! ne
e
dds ! ddue
e
S D Q D 1 (8.44a)
˙
C
! ne
C
e
uus ! udde
C
e
S DQ D 1: (8.44b)
The decay branching ratio of the process (8.44a) is BR D 1:0 10
3
, while for
the decay (8.44b), one has BR D 5 10
6
. The second decay is therefore largely
suppressed (by a factor 5 10
3
) compared to the first one. The reason is that in the
decay (8.44b), two quarks must change flavor .u ! d , s ! u) and this can only
occur through higher order weak interaction diagrams (see Fig. 8.16b).
The Sargent rule (8.18) provides a good approximation for the calculation of the
lifetimes of all of these decays. The numerical results obtained using G
F
are almost
correct for S D 0 decays, but wrong by a factor 20 for S D 1 decays. We can
imagine again that in the case of decays without strangeness change, the appropriate
constant is G
d
, while it is G
s
when S D 1. We can estimate the relationship
between the two constants using the ˙
decays
G
2
s
G
2
d
D
.˙
! ne
e
/=m
5
˙n
.˙
!
0
e
e
/=m
5
˙
D 0:057: (8.45)
210 8 Weak Interactions and Neutrinos
d
d
s
d
d
u
u
s
u
e
–
ν
e
d
u
d
e
+
ν
e
Σ
+
W
–
W
+
W
–
n
a
b
Σ
−
n
Fig. 8.16 Feynman diagrams for the semileptonic decays (a) ˙
! ne
e
,(b) ˙
C
! ne
C
e
Fig. 8.17 Feynman diagrams for nonleptonic decays (a)
0
! p
,(b)
0
! n
0
, with the
exchange of a W
boson
Even in the case of semileptonic decays, the appropriate coupling constants to be
used for S D 1; S D 0 transitions are different. The ratio G
2
s
=G
2
d
' 0:05 is
independent from the transition type between hadrons: this must therefore reflect a
property of the constituent quarks.
Nonleptonic decays. Strange particle nonleptonic decays are characterized by the
selection rules S D 1 and I D 1=2 (S is the variation in strangeness, I
is the variation in the strong isospin of the involved hadrons). This selection rule
corresponds to the quark transition s ! u. Here, the appropriate coupling constant
is G
s
.
As an example, let us consider the
0
decay shown in Fig. 8.17,forwhichS D
S
f
S
i
D 1.ToverifythatI D 1=2, we need to make some considerations on
the branching ratio of the
0
decays in p
C
and n
0
:
0
! p
BR D .2=3 phase space factor/ D 0:655; Exp D 0:641 ˙ 0:005
! n
0
BR D .1=3 phase space factor/ D 0:345; Exp D 0:367 ˙ 0:005
I D 0 ! I D 1=2:
(8.46)
If the final state has I D 1=2, the respective branching ratios for the decays in p
and n
0
can be derived from the Clebsch–Gordan coefficients (if I D 1=2, one
has 2=3 for p
and 1=3 for n
0
;ifI D 3=2, one has 1=3 for p
and 2=3 for
n
0
), and from a factor which takes into account the small difference in phase space
8.12 Universality of Weak Interactions (II). The Cabibbo Angle 211
between p
and n
0
. The assignment I D 1=2 to the final state is confirmed, and
thus I D 1=2. The rule is not absolute since it is broken by the electromagnetic
interaction; a small “contamination” of I D 3=2 shall be present in the final state.
This selection rule is explained by assuming that in terms of quarks, the following
sequence of “processes” takes place:
0
! p
uds
WI
! udu W
WICSI
! udu C ud
SI
! p
(8.47a)
0
! n
0
uds
WI
! udu W
WICSI
! udd C uu
SI
! n
0
: (8.47b)
The corresponding Feynman diagrams with a W
˙
boson exchange are shown in
Fig. 8.17.
8.12 Universality of Weak Interactions (II). The Cabibbo Angle
The Puppi triangle (Fig.8.5) which involves neutron and muon decays, and the
capture by a proton expresses the universality of weak interactions. The extension
to strange particles led to the Dallaporta tetrahedron where the semileptonic decays
˙
!
0
e
e
,
0
! pe
e
, ˙ ! ne, shown in Fig. 8.5b, are also included.The
“universality” is only approximate because, as we have seen for the K
C
!
C
and
C
!
C
decays, the coupling constants seem quite different, i.e., smaller
for K decay. The same situation occurs for the semileptonic decays of strange
baryons .
0
! pe
e
;˙ ! ne
e
; !
0
e
e
/ compared to the neutron decay.
Accurate measurements show that the coupling constant obtained from the neutron
decay is slightly smaller than that obtained from the muon decay, as can be observed
by comparing the values given in (8.14a, b). It should also be noted that various
empirical selection rules, S D˙1, I D 1=2, Q D S, indicate some
regularity in the decays of strange particles.
The above mentioned experimental facts were brilliantly interpreted by
N. Cabibbo in 1964. The leptons are weak interaction eigenstates and quarks are
eigenstates of the strong interaction. Cabibbo demonstrated that leptons and quarks
are eigenstates of the WI with the following assumptions:
• The coupling of electrons with the weak interaction field is proportional to a weak
charge g
e
;
• The coupling of muons is proportional to g
and is identical to that of electrons:
g
e
D g
;
• The coupling of .u;d/ quarks generates the transitions with S D 0 and is
proportional to g
ud
;
• The coupling of .u;s/ quarks generates transitions with S D 1 and is
proportional to g
us
.
212 8 Weak Interactions and Neutrinos
In a Feynman diagram, the constant corresponding to the particles involved in
a vertex must be inserted. The matrix elements of weak transitions (described by
the Hamiltonian H
W
) involving only leptons (i.e., ji iDjeiIhf jDhej)are
proportional to the Fermi constant
hf jH
W
jii/g
2
e
D G
F
: (8.48a)
The matrix elements of S D 0 semileptonic processes where jiiDj
e
uiIhf jD
he
C
d j or jiiDj
e
d iIhf jDhe
uj (as in Fig. 8.1b) are
hf jH
W
jii
SD0
/ g
e
g
ud
D G
d
: (8.48b)
For S D 1 processes (Fig. 8.16a) where jiiDj
e
siIhf jDhe
uj, one has
hf jH
W
jii
SD1
/ g
e
g
us
D G
s
: (8.48c)
The Cabibbo hypothesis (shown qualitatively in Fig. 8.5c) states that the weak
interaction is truly universal, as thought before the discovery of strange particles.
The universality is expressed by the fact that the WI depends on a single parameter,
the Fermi constant G
F
. The weak interaction couplings with quarks and leptons
occur through the relation
G
F
D g
2
e
D g
2
ud
C g
2
us
! g
ud
D g
e
cos
C
I g
us
D g
e
sin
C
: (8.49)
In the Cabibbo model, this corresponds to the fact that the quarks involved in the
weak interaction are not the flavor eigenstates u;d;s that characterize the strong
interaction. The WI quark eigenstates are a linear combination of u;d;s which can
be denoted as .u
c
;d
c
;s
c
/. By convention, the quark u is not mixed, so u
c
coincides
with u. The other quark pair with charge 1=3 is mixed through an angle
c
,
compared to ordinary quarks. When Cabibbo wrote his paper, two lepton families
and the three quarks u;d;swere known; the weak interaction involves the following
“weak doublets:”
e
e
;
;
u
d
c
D
u
d cos
c
C s sin
c
(8.50)
where
c
D 0:235 rad D 13:5
ı
is the Cabibbo angle. For each lepton doublet,
the weak coupling is specified by the Fermi constant G
F
. For transitions involving
quarks u ! d with S D 0, the equivalent constant corresponds to G
F
cos
c
;for
transitions involving quark s ! u .S D 1/, the equivalent constant corresponds
to G
F
sin
c
. The different value of the effective coupling constants is only due
to the quark mixing process to form the WI eigenstates, see Table 8.2. Since
sin
c
D 0:235 and cos
c
D 0:972, transitions with S D 0 have an effective
coupling constant larger than those with S D 1. This explains why G
n
F
(Eq. 8.14b)
is smaller than G
F
(Eq. 8.14a): indeed, what is measured in the neutron decay is the
quantity G
F
cos
c
.
8.13 Weak Interaction Neutral Current 213
Table 8.2 Couplings for different types of decay .G
F
D G
F
/
Quark Hadronic
Decay transitions J
P
S Coupling
n ! pe
e
d ! ue
e
1=2
C
! 1=2
C
0 G
2
F
cos
2
c
p ! ne
C
e
.in
14
O) u ! de
C
e
0
C
! 0
C
0 G
2
F
cos
2
c
!
0
e
e
d ! ue
e
0
! 0
0 G
2
F
cos
2
c
K
!
0
e
e
s ! ue
e
0
! 0
1 G
2
F
sin
2
c
C
! e
C
e
G
2
F
The second decay is relative to a proton in a
14
O nucleus since free protons do not decay
The value of the Cabibbo angle can be derived by comparing similar S D 1
and S D 0 semileptonic decays. For example, the decays K
C
!
C
./ G
2
F
sin
2
c
/ and
C
!
C
./ G
2
F
cos
2
c
/ can be used. They are very similar,
except for a quark s in K
C
and a quark d in
C
. The ratio of decay fractions gives
K
C
!
C
C
!
C
D
m
2
K
1
m
2
=m
2
K
2
m
2
1
m
2
=m
2
2
tan
2
c
: (8.51)
Using the ; and K masses [P08], the value
c
D .0:235 ˙ 0:006/ is obtained.
Today, three different quark families are known (Sect. 8.14) and different methods
are used to evaluate the mixing angles. The best value of the Cabibbo angle is
c
D
.0:2253 ˙ 0:0007/.
8.13 Weak Interaction Neutral Current
At a fundamental level, the decays and reactions considered thus far are mediated
through the exchange of charged W
C
, W
vector bosons: these processes are weak
charged current (CC) reactions. The reactions
e
!
e
e
!
e
(8.52)
can only occur through the exchange of a neutral boson, the Z
0
(see Fig. 8.18).
These are the so-called weak interaction neutral current (NC) processes. A contri-
bution due to the WI neutral current is also present in the reaction
e
C
e
! qq.``/ (8.53)
shown in Fig. 8.18b; this will be discussed in detail in Chap. 9. Historically, the
NC was introduced to remove mathematical divergences. For example, the lowest
214 8 Weak Interactions and Neutrinos
υ
e
υ
e
e
–
e
–
e
–
e
+
Z
0
Z
0
q
–
q
–
ab
Fig. 8.18 Examples of weak interaction neutral current processes. (a)TheZ
0
is exchanged in the
t channel: the electric charge of the leptonic current does not change over time. (b) Exchange in
the s channel: lepton and antilepton annihilate to form the neutral Z
0
boson which decays into a
quark-antiquark pair
Z
0
e
–
W
+
W
–
W
+
W
–
ν
ν
–
ν
ν
–
ab
Fig. 8.19 Reaction ! W
C
W
:(a) diagram with CC interaction; this diagram gives a
divergence that is canceled by the introduction of the NC diagram (b)
order diagram of the reaction ! W
C
W
contains the exchange of an electron
which gives rise to a divergence. The divergence is canceled by the introduction of
a diagram with a Z
0
vector boson exchange (see Fig. 8.19).
The Z
0
, as the virtual , can be exchanged in the t channel (Fig. 8.18a), resulting
in processes where the electric charge of interacting particles does not change. For
example, NC neutrino-nucleon interaction takes place in the t channel through a Z
0
exchange with a quark (or antiquark) of the nucleon (see Fig. 10.1) which remains
unchanged. The interaction can occur with any quark (or antiquark) flavor. If a
neutral boson is exchanged in the s channel (Fig. 8.18b), one has an annihilation
process f
f ! Z
0
followed by the creation of a f
0
f
0
pair. f; f
0
are fermions;
f;f
0
are antifermions.
The neutral current interaction was discovered in 1977 using a heavy liquid
bubble chamber (Gargamelle, CERN) exposed to a beam of high energy muon
neutrinos. The following reactions were observed: (i)
e
!
e
: scattering on
atomic electrons. The electron in the final state is produced by an invisible particle
track. (ii)
N !
C hadrons: interactions on nuclei. Only charged particles
in the final state are visible (hadrons or their decay products). In both cases, the
characteristic signature of a CC interaction, i.e., the presence of a muon, is missing
and therefore, the reactions could only proceed via NC.
8.14 Weak Interactions and Quark Eigenstates 215
The ratio between NC and CC cross-sections for high energy neutrinos is
NC
N
=
CC
N
' 0:25,
NC
N
=
CC
N
' 0:45. The neutral currents are not foreseen in
the Fermi theory and represent one of the reasons for its extension.
8.14 Weak Interactions and Quark Eigenstates
8.14.1 The WI Hamiltonian and the GIM Mechanism
The K
0
, K
0
are hadronic interaction eigenstates. In 1955, Gell–Mann and Pais
found that it was possible to form two linear combinations of the neutral K mesons
which are CP eigenstates; these correspond to particles which decay in two different
CP states
jK
0
1
˛
D 1=
p
2
jK
0
˛
C
ˇ
ˇ
K
0
˛
IjK
0
2
˛
D 1=
p
2
jK
0
˛
ˇ
ˇ
K
0
˛
: (8.54)
K
0
1
and K
0
2
are two distinct states, that is, linear combinations of strong interaction
eigenstates; they have different masses and decay in different ways, for example,
ˇ
ˇ
K
0
1
˛
1
p
2
ˇ
ˇ
d
s
˛
C
ˇ
ˇ
s
d
˛
CP DC1 ! (8.55)
ˇ
ˇ
K
0
2
˛
1
p
2
ˇ
ˇ
d
s
˛
ˇ
ˇ
s
d
˛
CP D1 ! : (8.56)
We will return to this subject in Sect. 12.2. For the moment, it is important to
note that the K
0
e K
0
mesons are charged conjugate states with the same mass.
The linear combinations K
0
1
and K
0
2
are two different particles with different mass.
The mass difference m Djm
K
0
2
m
K
0
1
j can be calculated in the framework of the
quark model, and is proportional to the matrix element that describes the K
0
$ K
0
transition probability. This is a second order transition, with S D 2: the K
0
has
strangeness S D 1, while the
K
0
has S D –1. When computed considering only the
u;d;s quark contributions, the obtained m is much larger than the experimental
value. There must be some new phenomenon that prevents transitions that change
the quark flavor, but do not change the electric charge. This was pointed out in
1970 by Glashow, Iliopoulos and Maiani in the so-called GIM mechanism where
they proposed the existence of a fourth quark. According to this hypothesis, the
properties of the fourth quark (called c, charm) are
1. Electric charge 2/3, isospin I D 0, strangeness S D 0, baryon number B D 1=3
2. “Charm” quantum number c D 1. This is a new quantum number that, like
strangeness, is conserved in strong interactions and violated in weak interactions
216 8 Weak Interactions and Neutrinos
3. It is a weak interaction eigenstate and forms a second quark doublet with the
second “rotated” state of the Cabibbo theory: s
c
D s cos
c
d sin
c
. With this
assumption, a new doublet for the weak interactions is obtained, that is,
c
s
c
D
c
s cos
c
d sin
c
: (8.57)
Let us again consider the weak Hamiltonian H
W
appearing in (8.48); it is an
operator acting on particle wave functions and consisting of four terms:
• A universal dimensional constant G
F
(the same for all particles)
• A combination of (dimensionless) operators O
i
which assigns the helicity
state preferences of the involved fermions and defines the behavior under the
parity operator. The mathematical structure of the O
i
operators present in H
W
(described in detail in Sect. 8.16) is deduced from the experimental results
described in previous sections
• A multiplicative term that takes into account the multiplicity of spin and isospin
states available in the final state (the term m
i;s
defined in Sect. 8.6)
• A (dimensionless) term equal to one if the interaction is purely leptonic; if the
interaction is nonleptonic, its value depends on the involved quark flavor
Using the same notations used in (8.48), the semileptonic c quark decay (c D 1;
Q D 1) can be described using the GIM mechanism hypothesis:
c
g
cd
(g
cs
)
g
eν
e
+
ν
e
W
+
d
(s)
sc h.
e
e
C
/sjH
W
jci/g
e
g
cs
D G
F
cos
c
(8.58a)
dc h.
e
e
C
/d jH
W
jci/g
e
g
cd
D G
F
sin
c
: (8.58b)
Note the shortening
sc for the transition probability from a quark c to a quark s
(8.58a)and
dc for the transition from a quark c to d (8.58b). Up to Eq. 8.62c the
symbol, e.g., c stands for the wave operator which destroys the quark c, while
s is
the operator that creates the quark s, and similarly for other quarks. Here, do not
confuse the object
s with the strange antiquark.
In c D 1 charmed meson decays, the c ! s transitions with coupling / cos
2
c
are dominant with respect to c ! d transitions, with coupling / sin
2
c
.
Let us, for example, consider the transition from an initial d
c
or s
c
quark into a
final
u or c,plusaW
state. With two quark families, the corresponding term in the
Hamiltonian H
W
can be written using (8.50)and(8.57)as
8.14 Weak Interactions and Quark Eigenstates 217
.u; c/
d
c
s
c
D .
u; c/
d cos
c
C s sin
c
s cos
c
d sin
c
D .
u; c/
cos
c
sin
c
sin
c
cos
c
d
s
D .
u; c/V
C
d
s
: (8.59)
For the first two quark families, the 2 2V
C
mixing matrix only contains
one free parameter: the Cabibbo angle. Experimental data on weak transitions of
u;d;s;cquarks are consistent with the same value of
c
. The weak Hamiltonian H
W
eigenstates of charge 1=3 quarks do not coincide with the d;s strong interaction
eigenstates. The charge 2/3 quarks are unchanged by convention.
8.14.2 Hints on the Fourth Quark from WI Neutral Currents
A fourth quark was also necessary to solve some problems associated with WI
neutral currents. The observed NC processes are characterized by the selection rule
S D 0. Neutral currents with S D 1 were never observed (Problem 8.18).
However, assuming the existence of only the u;d;s quarks, the mechanism which
rotates the .d; s/ quarks in the NC includes a S D 1 term, that is,
u; d
c
u
d
c
D .
u; d cos
c
C s sin
c
/
u
d cos
c
C s sin
c
D u
u C .d d cos
2
c
C ss sin
2
c
/
„
ƒ‚ …
SD0
C.sd C sd/ sin
c
cos
c
„ ƒ‚ …
SD1
:
Each term u
u, d d , ss represents the probability density for the transition from a
quark of a given flavor in the initial state toward the same quark flavor plus a Z
0
boson in the final state. Although sd and ds processes were never observed, the
mechanism foresees their existence. With the new doublet (8.57), the neutral current
can be rewritten in the form
u; d
c
u
d
c
C .
c;s
c
/
c
s
c
D u
u Ccc C .d d C ss/ cos
2
c
C .ss Cd d/sin
2
c
„ ƒ‚ …
SD0
C .sd C sd sd sd/sin
c
cos
c
„ ƒ‚ …
SD1
(8.60)
and the term with S D 1 is automatically canceled.
218 8 Weak Interactions and Neutrinos
8.14.3 The Six Quarks and the Cabibbo–Kobayashi–Maskawa
Matrix
Today, we are aware of six different quarks; the discovery of the heaviest quark is
described in Chap.9. The generalization of (8.59) leads to the following form of the
weak interaction current in the quark sector, namely,
.
u; c; t/ V
CKM
0
@
d
s
b
1
A
(8.61)
where the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix) V
CKM
is a 3 3
matrix. In the literature, several parametrizations of the CKM matrix can be found.
The form originally proposed by Kobayashi and Maskawa is .c
i
D cos
i
;s
i
D
sin
i
/:
0
@
d
c
s
c
b
c
1
A
D V
CKM
0
@
d
s
b
1
A
V
CKM
D
0
B
@
c
1
c
3
s
1
s
1
s
3
c
2
s
1
c
1
c
2
c
3
s
2
s
3
e
iı
c
1
c
2
s
3
C c
3
s
2
e
iı
s
1
s
2
c
1
c
3
s
2
c
2
s
3
e
iı
c
1
s
2
s
3
C c
2
c
3
e
iı
1
C
A
(8.62a)
where
1
;
2
;
3
are three mixing angles and ı is a phase angle (note that some
authors use the form s
1
!s
1
). V
CKM
can be regarded as a three-dimensional
rotation matrix with three Euler angles. In the limit
2
'
3
' 0, the matrix only
contains the Cabibbo angle. Another standard parametrization is
V
CKM
D
0
B
B
@
c
12
c
13
s
12
c
13
s
13
e
iı
13
s
12
c
23
c
12
s
23
s
13
e
iı
13
c
12
c
23
s
12
s
23
s
13
e
iı
13
s
23
c
13
s
12
c
23
c
12
s
23
s
13
e
iı
13
c
12
s
23
s
12
c
23
s
13
e
iı
13
Cc
23
c
13
1
C
C
A
: (8.62b)
Experimentally, s
12
' 0:23, s
13
' 0:003, s
23
' 0:04. The phase angle ı
13
,if
different from zero, leads to the CP violation in the weak interaction (see Chap. 12).
Nowadays, the following parametrization is preferred, that is,
0
@
d
0
s
0
b
0
1
A
D
0
B
B
@
V
ud
V
us
V
ub
V
cd
V
cs
V
cb
V
td
V
ts
V
tb
1
C
C
A
0
@
d
s
b
1
A
(8.62c)