Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
Подождите немного. Документ загружается.
7.13 The Six Quarks 169
of the symmetry from SU(3)
f
to SU(4)
f
, which is heavily broken since the quark
c mass is much larger than that of the first three quarks. On the other hand, this
symmetry is sufficient to determine the approximate number of baryons and mesons
containing one or more c quarks.
Figure 7.17 shows the J
P
D 1=2
C
and J
P
D 3=2
C
baryons made of u;d;s;c
quarks. The representations are obtained inserting on the third axis (vertical) the
charm quantum number c. The baryons are grouped in 20-plets. The 3=2
C
20-plet
contains the known baryon decuplet of Fig. 7.12 (at c D 0), a sextuplet of baryons
containing one c quark, a triplet with two c quarks and a ccc singlet. The 1=2
C
20-
plet contains the octet of Fig. 7.14, a sextuplet with one c quark, and a triplet with
two c quarks.
Figure 7.18 shows the 0
and 1
mesons 16-plet. Both are made with the u;d;s
quark nonet, two triplets with a c or a
c quark, and a cc singlet with the total charm
quantum number c equal to zero. This singlet state can interfere with the other ;
0
singlet of the 16-plet.
The addition of the b quark leads to the SU(5)
f
symmetry, which is even more
broken than SU(4)
f
due to the very large mass of the b quark.
Table 7.1 summarizes the quantum numbers of the six quark flavors.These
quantum numbers are constrained by the relation
Q D I
3
C
B C S C c C b Ct
2
D I
3
C
Y
2
; (7.57)
which is an extension of Eq. 7.41. The definition of strong hypercharge Y is
extended from Y D B C S to Y D B C S C b C c C t. The letters c, b, t
denote, respectively, the charm, bottom and top quantum numbers.
Ξ
+
c
Ξ
0
Ξ
0
Σ
++
c
Σ
++
c
n
p
c
Ξ
0
udc
dss
dds
uss
uus
uud
uud
udd
uds
dss
dds
uss
uus
udd
uds
ssc
usc
dsc
uuc
udc
usc
uuc
ucc
scc
dcc
ucc
scc
dcc
Ω
+
cc
Ξ
++
cc
Ξ
++
cc
Ξ
+
cc
Ξ
+
cc
Σ
0
c
Σ
0
c
Ξ
−
Σ
−
Σ
−
Σ
+
Σ
+
Λ
,
Σ
0
Λ
,
Σ
0
Σ
+
c
Λ
+
c
,
Ω
0
c
Ω
0
c
ddc
ssc
dsc
ddc
a
Ω
++
ccc
Ω
+
cc
Ξ
+
c
Ξ
0
c
Ω
−
Δ
+
Δ
0
Δ
−
Δ
++
Σ
0
Σ
0
c
Σ
+
ddc
dsc
usc
uuc
uud
uus
uss
dss
udd
dds
ddd
uuu
udc
dcc
ucc
uds
ssc
scc
sss
ddc
dsc
usc
uuc
uud
uus
uss
dss
udd
dds
ddd
uuu
udc
dcc
ucc
uds
ssc
scc
sss
Ω
0
c
b
Fig. 7.17 The SU(4)
f
symmetry for baryonic states made of u;d;s and c quarks. On the x; y;z
axes, one has, respectively, the third component of isospin, the strangeness and the charm quantum
number. (a) 20-plet with J
P
D 1=2
C
containing the known SU(3)
f
octet in the bottom plane
(c D 0). (b) 20-plet with J
P
D 3=2
C
, which contains the known SU(3)
f
decuplet in the bottom
plane [P08]
170 7 Hadron Interactions at Low Energies and the Static Quark Model
0
D
0
K
–
π
+
K
–
K
s
D
−
uc
−
sc
−
dc
−
−
cd
cu
−
cs
−
π
+
D
+
s
D
+
K
0
us
−
ds
−
su
−
sd
−
du
−
η
η
′
η
c
π
0
0
D
ud
−
D
–
a
−
ρ
+
ρ
D
*
0
K
*
−
K
*
+
K
*
0
D
*
−
s
D
*
−
D
*
+
s
D
*
+
−
cd
cu
−
cs
−
us
−
ds
−
su
−
sd
−
ud
−
uc
−
sc
−
dc
−
0
ρ
ω
φ
du
−
D
0
*
K
0
*
ψ
J/
uc
−
sc
−
dc
−
−
cd
cu
−
cs
−
us
−
ds
−
su
−
sd
−
du
−
η
η
′
η
c
π
0
ud
−
−
cd
cu
−
cs
−
us
−
ds
−
su
−
sd
−
ud
−
uc
−
sc
−
dc
−
0
ρ
ω
φ
du
−
ψ
J/
b
c
S
I
z
c
Fig. 7.18 The SU(4)
f
16-plet symmetry for (a) the pseudoscalar (J
P
D 0
) mesons and
(b) the vector (J
P
D 1
) mesons made of the d;u;s;c quarks. (c)Onthex;y;z axes, one
has, respectively, the third component of isospin, the strangeness and the charm quantum number.
In the middle plane at c D 0, one finds the known nonet consisting of the light u;d;s quarks, to
which the c
c state is added. The mixed neutral mesons made of uu;dd;ss and cc are located in
the center [P08]
7.14 Experimental Tests on the Static Quark Model
There are many possible direct and indirect tests on the static quark model and
on the hadron composition in terms of quarks. Note that the static quark model
is appropriate for explaining the hadron classification, but is not adequate for
representing the quark dynamics, which need the contribution of gluons and q
q
pairs (sea quarks) inside the hadrons. The dynamic quark model shall be presented
in Chap. 10. In this section, some experimental evidence for quarks are presented.
7.14.1 Leptonic Decays of Neutral Vector Mesons
In the static quark model, the
0
;!
0
and
0
neutral vector mesons have the following
structure:
0
D .uu d d/=
p
2mD 769:9 ˙ 0:8 MeV
!
0
D .uu C d d/=
p
2 781:94 ˙ 0:12 MeV
0
D ss 1019:413 ˙ 0:008 MeV:
(7.58)
In the following, the three considered vector mesons are indicated with the
generic symbol V
0
. The decay of a V
0
vector meson in a charged lepton–antilepton
pair, V
0
! `
`
C
, proceeds through the exchange of a virtual photon, as shown in
Fig. 7.19.Thetransition probability W for this decay can be calculated since it is an
electromagnetic decay (see Problem 7.17). This relation is known as the Weisskopf
formula („Dc D 1)
7.14 Experimental Tests on the Static Quark Model 171
Fig. 7.19 Leptonic decay of
a generic V
0
neutral vector
meson
q
q
l
+
l
−
γ
V
0
√αΣQ
i
√αQ
1/q
2
W.V
0
! `
C
`
/ D
16˛
2
EM
.˙
i
Q
i
/
2
m
2
V
j .0/j
2
; (7.59a)
which can be factorized as
W D 16
ˇ
ˇ
ˇ
ˇ
p
˛
EM
˙
i
Q
i
.1=q
2
/
p
˛
EM
Q
ˇ
ˇ
ˇ
ˇ
2
q
2
j .0/j
2
coupling photon coupling phase wave function
q
q propagator `
C
`
space qq at the origin;
(7.59b)
with Q D charge of the ` lepton D˙1, Q
i
= fractional charge of the quarks,
˛
EM
D 1=137. Assuming (for simplicity) that the three vector mesons have the
same mass m
V
and the same .0/ (the wave function of the qq system at the
origin) and assuming that q
2
' m
2
V
, j .0/j
2
=m
2
V
is expected to be roughly constant;
it follows that W j˙
i
Q
i
j
2
. From the quark composition given in (7.58), the
following values for .˙
i
Q
i
/
2
are obtained:
0
D .uu d d/=
p
2 !
1
p
2
2
3
.1/
3
2
D
1
2
!
0
D .uu C d d/=
p
2 !
1
p
2
2
3
1
3
2
D
1
18
0
D ss !
1
3
2
D
1
9
;
(7.60)
from which, one finds that
W.
0
/ W W.!
0
/ W W.
0
/ D 9 W 1 W 2 theory
D .8:8 ˙ 2:6/ W 1 W .1:70 ˙ 0:4/ experiment:
(7.61)
The experimental data confirm the predictions of the static quark model of hadrons.
7.14.2 Lepton Pair Production
A second test of the quark model is given by the production of lepton pairs,
especially muons, in pion-nucleon collisions. The mechanism is illustrated in
Fig. 7.20: the antiquark of the meson annihilates with a quark of the nucleon,
giving rise to a virtual photon that becomes a
C
pair (Drell–Yan model). The
172 7 Hadron Interactions at Low Energies and the Static Quark Model
Fig. 7.20 Illustration of the
lepton pair production in a
pion-nucleon collision. The
basic mechanism is the
quark-antiquark annihilation
into a virtual photon which
gives rise to the
lepton–antilepton pair
q
q
q
q
q
π
μ
+
μ
−
N
γ
noninteracting quarks act as spectators. The process cross-section is proportional to
the square of the quark charges.
In particular, consider the
.D ud/ interaction with the isoscalar nucleus
12
C
.D 6p C 6n D 18u C 18d/. An isoscalar nucleus contains the same number of
protons and neutrons. Only the u
u annihilation can occur in this case; the cross-
section is proportional to the square of the quark charges and
.
C !
C
:::/ / 18Q
2
u
D 18 4=9 D 8: (7.62)
On the other side, in the interaction of a
C
.D d u/ with
12
C, only the dd
annihilation can occur, that is,
.
C
C !
C
:::/ / 18Q
2
d
D 18 1=9 D 2: (7.63)
Outside the energy region corresponding to resonances, one must have
.
C !
C
X/=.
C
C !
C
X/ D 4;
which was experimentally verified, thus confirming the validity of the quark model
and especially the electric charge assignment of u;d quarks.
7.14.3 Hadron-Hadron Cross-Sections at High Energies
Let us assume that in the interaction of two high energy hadrons, the collision
occurs between a quark of the first hadron and a quark of the second hadron,
independently of the other quarks. This is, of course, an approximation that does
not take into account gluons and sea quarks within hadrons. Let us further assume
that at high energies, one has .uu/ D .ud/ D .dd/ (corresponding to the
isospin invariance) and even .q
q/ D .qq/, and finally that the collisions between
different quarks are additive. Under these conditions it follows that the ratio of the
total N and NN cross-sections is given by the ratio of the numbers of possible
combinations between quark pairs, that is,
.N/
.NN/
D
2 3
3 3
D
2
3
;
7.14 Experimental Tests on the Static Quark Model 173
which is experimentally verified (see Figs. 7.1 and 7.2). For example, at E
D
60 GeV, one has .
C
p/ ' .
p/ ' 24 mb, and .pp/ ' .pn/ ' 38 mb;
the cross-section ratio is therefore Œ.N /=.NN /
exp
' 24/38 ' 2/3.
7.14.4 Baryon Magnetic Moments
In the Dirac theory, a point-like fermion with electric charge q,massm and spin
1=2 has a magnetic dipole moment. The magnetic dipole operator is defined as
D
q„
2mc
„DcD1
!
q
2m
; (7.64a)
where are the Pauli matrices (see Appendix A.4). The value of the magnetic dipole
moment of a fermion f is the expectation value (defined by Eq. 6.3) of the operator
(7.64a):
f
hf jjf i: (7.64b)
In the case of the electron and the muon, the comparison between prediction and
measurement of the magnetic dipole moment is one of the successes of the Dirac
theory and of QED. Moreover, it is a clear confirmation that leptons are really
“elementary particles.” Following the same argument, the proton and neutron are
not “elementary” because their magnetic dipole moments are very different from
those predicted by the Dirac theory. In particular, since the neutron is electrically
neutral, the theory predicts
n
D 0.
Now, assume that each quark is a Dirac particle with an expectation value for the
magnetic moment given by (7.64b). In an explicit form for the for u and d quarks,
u
DhujjuiD
q
u
2m
u
I
d
Dhd jjd iD
q
d
2m
d
: (7.64c)
Since m
u
' m
d
and q
u
D2q
d
, one obtains
u
'2
d
: (7.64d)
We can now assume that the baryon magnetic moment is equal to the vector sum
of quark magnetic moments, and that there is no contribution from orbital angular
moments, i.e., ` D `
0
D 0. Furthermore, gluons and sea quarks are not taken into
account.
The proton consists of uud in antisymmetric combination of spin and flavor.
In this case, because the flavor antisymmetry is irrelevant, we can rewrite (7.45),
avoiding to report the permutations amongst flavors:
jpiDj2u
"
u
"
d
#
u
"
u
#
d
"
u
#
u
"
d
"
i=
p
6: (7.65)
174 7 Hadron Interactions at Low Energies and the Static Quark Model
Table 7.2 Predicted and observed [P10] magnetic moments of “stable” baryons
of the J
P
D 1=2
C
octet, in units of nuclear magnetons (n.m.)
N
D e„=2m
p
c.
The p and n magnetic moments are measured with a relative precision of better
than 10
7
Baryon
Magnetic moment
(quark model) Prediction (n.m.) Observed (n.m.)
p
4
3
u
1
3
d
2:793 2:793
n
4
3
d
1
3
u
1:913 1:913
s
0:613 0:613 ˙ 0:004
˙
C
4
3
u
1
3
s
2:68 2:46 ˙ 0:01
˙
0
2
3
.
u
C
d
/
1
3
s
0:791
˙
4
3
d
1
3
s
1:09 1:160 ˙ 0:003
0
4
3
s
1
3
u
1:43 1:250 ˙ 0:014
4
3
s
1
3
d
0:49 0:651 ˙ 0:003
The corresponding magnetic dipole moment is obtained from the calculation of the
expectation value (7.64b)appliedtothejpi state:
p
DhpjjpiD
Œ4.
u
C
u
d
/ C .
u
u
C
d
/ C .
u
C
u
C
d
/
6
D
.8
u
2
d
/
6
: (7.66)
The signs in the calculation take into account the relative orientation of the spin of
the quarks. Recalling (7.64d), one has
p
' Œ4
u
.
u
=2/=3 D .3=2/
u
: (7.67)
Similarly, for the neutron, one has
jniDj2d
"
d
"
u
#
d
"
d
#
u
"
d
#
d
"
u
"
i=
p
6; (7.68)
and thus
n
D Œ4.
d
C
d
u
/ C
u
C
u
=6 D .4
d
u
/=3 '
u
: (7.69)
The model predicts
n
=
p
D2=3, which is in reasonable agreement with the
observed value (D0:685). Table 7.2 shows all the magnetic moments predicted
by the static quark model and those measured for the hadrons of the J
P
D 1=2
C
baryonic octet.
The static quark model predicts with some accuracy the
n
=
p
ratio. The
predicted values of the hadron magnetic moments (in units of nuclear magnetons
(n.m.)
N
D e„=2m
p
c) can be expressed in terms of
u
;
d
;
s
. The computation
of the quark magnetic moments resulting from (7.64c) requires the quark mass
and electric charge values. Using the “bare” quark masses (m
d
' m
u
' 5 MeV;
7.14 Experimental Tests on the Static Quark Model 175
Table 7.3 Quark contents of “quasi-stable” hadrons (i.e., not decaying through the strong
interaction with a lifetime 10
23
s). The masses are in MeV. * See Sect. 12.2
Mesons Baryons
Quark Mass(MeV) (s) Quark Mass(MeV) (s)
ud
C
.139:57/ 2:6 10
8
uud p (938.272) stable
u
u;dd
0
.134:97/ 8:4 10
17
udd n (939.566) 886
d
u
.139:57/ 2:6 10
8
uds
0
.1115:63/ 2:6 10
10
uu;dd; ss .548:8/ D 1:18 keV uus ˙
C
.1189:37/ 0:8 10
10
uu;dd; ss
0
.957:5/ D 2:2MeV uds ˙
0
.1192:55/ 7:4 10
20
usK
C
.493:65/ 1:2 10
8
dds ˙
.1197:43/ 0:8 10
10
dsK
0
.497:67/ *
s
d K
0
.497:67/ * uss
0
.1314:9/ 2:9 10
10
su K
.493:65/ 1:2 10
8
dss
.1321:3/ 1:6 10
10
cdD
C
.1869:3/ 1:0 10
12
sss ˝
.1672:4/ 0:82 10
10
cu D
0
.1864:5/ 4:1 10
13
udc
C
c
.2285:0/ 2:0 10
13
uc D
0
.1864:5/ :1 10
13
usc
C
c
.2467/ 1:4 10
13
dcD
.1869:3/ 1:0 10
12
udb
0
b
.5500/ 1:2 10
12
csD
C
s
.1969/ 4:9 10
13
scD
s
.1969/ 4:9 10
13
ubB
C
.5278/ 1:7 10
12
dbB
0
.5279/ 1:5 10
12
bd B
0
.5279/ 1:5 10
12
bu B
.5278/ 1:7 10
12
m
s
D m
d
C 150 MeV), the values computed using the formulae given in the
second column of Table 7.2 completely disagree with the measured values. Using
the “effective” quark masses (m
d
' m
u
D m
p
=3; m
s
D m
d
C 150 MeV), the
values of quark magnetic moments are
u
D 2
N
I
d
D1
N
I
s
D0:67
N
;
these values are in much better agreement with the observed ones. If the measured
p; n and moments are used as input, the quark moments are
u
DC1:852
N
I
d
D0:972
N
I
s
D0:613
N
[P10], from which the predicted values
inserted in the table are derived.
The static quark model, although it qualitatively reproduces some observed
features of hadron magnetic moments, fails to give precise agreements with
experimental data. This can only be achieved through a “dynamics” description of
the hadron constituents which includes gluons and quark of the sea, as discussed in
Chap. 10.
7.14.5 Relations Between Masses
Quark masses cannot really be measured because quarks are never free; they can
only be determined indirectly through their influence on the hadron properties
(Table 7.3). The mass differences between hadrons of the same isospin multiplet
176 7 Hadron Interactions at Low Energies and the Static Quark Model
u
ab
u
d
Fig. 7.21 (a) In the simplest static quark model of hadrons, the proton is made of three valence
quarks (u; u;d) which are much lighter and smaller than the global proton mass and size. (b)Inthe
dynamic quark model of hadrons, in addition to the three valence quarks, many gluons and many
quark-antiquark pairs contribute to the proton mass
are attributed both to the m
u
m
d
mass difference and to the electromagnetic
interaction. The latter introduces mass differences of a few MeV. The mass
differences between particles of different isospin multiplets are mainly due to the
mass difference between quarks.
It is often assumed that m
u
' m
d
' few MeV and m
s
' m
u
C 150 MeV.
Mass differences between u;d and s quarks explain many of the mass differences
of normal and strange hadrons. They are however not sufficient enough to explain
the observed mass differences between the members of the 1=2
C
baryonic octet and
the members of the 3=2
C
decuplet. These must be interpreted in terms of differences
between quark–quark interaction, with the largest contribution given by the part of
the strong interaction which depends on the spin (Chaps.11 and 14).
The masses of heavier quarks are m
c
' 1;550 MeV and m
b
' 4;300 MeV.
Hadrons and mesons made of the t quarks cannot be formed for the reasons
explained in Sect. 10.2. The quark t is free for a very short time.
Atomic energy levels and electromagnetic properties are derived from QED. The
calculation of the hadron mass spectrum and of the basic hadron properties from
fundamental QCD principles is a fundamental theoretical problem. The dynamic
range of strong interaction between quarks when forming hadrons made of u;d
and s quarks corresponds to large values of the ˛
S
coupling constant (Sect. 11.9.4).
As a consequence, the contributions of the diagrams of increasing order do not
decrease and a perturbation developmentis not possible. The problem is now studied
with numerical methods (lattice QCD): powerful theoretical techniques using large
computer farms have been developed. The proton mass is the sum of the small
masses of the constituent quarks, plus the work that must be done on the system
to bring the constituents into this minimum energy configuration. The sum of the
proton valence quark masses is less than 2% of the proton mass: the rest must
come from the gluons and virtual particles, Fig. 7.21. The scale of the energy
7.15 Searches for Free Quarks and Limits of the Model 177
corresponding to a quark confined within 1 fm is 200–300MeV (see Sect. 7.8); this
is the order of magnitude of the work needed to bring one quark into a noninteracting
configuration. For three quarks, it corresponds to 1GeV.
A second nontrivial problem is connected to the nucleon radius. Fundamental
fermions (as quark and leptons) are point-like particles to a distance <10
16
cm
[7B79]. Hadrons have dimensions of the order of 10
13
cm because of the QCD
properties. As for the case of the electrons in the atomic system, the three quarks
choose their average distances in order to minimize the energy.
7.15 Searches for Free Quarks and Limits of the Model
Quarks were introduced in order to explain the regularity and the symmetry
properties of hadron spectroscopy. In Chap. 10, we shall study the inelastic lepton-
nucleon and nucleon-nucleon collisions at high energies. These reveal a spatial
structure for the proton and neutron that is easily explainable in terms of point-
like constituents (quarks, gluons and quark-antiquark pairs). The hadron production
in high energy e
C
e
collisions is also explained in terms of quarks and gluons.
Since the first formulation of Gell-Mann and Zweig in 1964, the question
regarding the possible existence of free quarks fuels continuing interest. Many
searches for free quarks have been performed [7P04], all without success (although
some have indicated possible, but not confirmed, signals). QCD is consistent with
the hypothesis of quark confinement within hadrons. However, the search for free
quarks continues at increasing levels of precision. The research is based on the
fact that any free quark or quark tied in nuclei corresponds to a fractional charge.
Two research lines are followed: (1) the search for free quarks in terrestrial and
extraterrestrial stable matter and (2) the search for fractional charged particles
produced in high energy collisions or in the penetrating cosmic radiation.
Examples of the type (1) method are experiments like that used by Millikan for
the measurement of the electron q=m ratio or experiments dealing with magnetic
levitation. These experiments use microscopic samples of matter. The best limits
obtained are less than one free quark over 10
22
nucleons of stable matter.
Particles with fractional electric charge were searched for amongst the products
of hadron-hadron, lepton-nucleon and e
C
e
inelastic collisions. The searches are
based on the fact that particles with ˙1=3 and ˙2=3 electric charge ionize 1/9 and
4/9 compared to a muon with the same momentum. The best limits obtained are
at the level of less than one quark for many millions of ordinary particles. Some
studies were devoted to the search for free quarks after a possible deconfinement
phase in nucleus–nucleus collisions at high energies.
In addition to the presented multiplets, other baryon multiplets with higher spin
were observed. They can be interpreted as combinations of the three u;d;s quarks
with orbital angular momenta `, `
0
different from zero, in order to obtain the
observed spin of baryons. Remember that the separate quantization of spin and
orbital angular momentum is possible only in the nonrelativistic approximation.
178 7 Hadron Interactions at Low Energies and the Static Quark Model
The spin internal structure of the nucleons was studied through the inelastic
collisions of polarized electrons and muons on polarized targets. An electron
(or muon) only interacts with one of the quarks (valence or sea) of a proton by
exchanginga virtual photon. If the electron is polarized, also photon the is polarized,
and it interacts differently with quarks of different polarization. The photons do not
interact directly with gluons. Therefore, the information obtained from the study of
the deep inelastic electron and muon collisions with the proton constituents only
refer to the spin of the quark and antiquark. Measurements show that less than half
of the spin of the proton is due to the valence quarks and sea quarks, the other half
would be due to the gluons (Chap. 10).
No resonances were observed in
C
C
,
C
C
C
, K
K
meson states. If
existing, they would be exotic meson resonances. In the static quark model, they
would require an exotic qq
qq structure. Similarly, baryon states of the form K
C
N
would be exotic baryon states which require a quark composition of the type qqqq
q.
Resonant states of di-baryons, like pp, were also hypothesized; this would require
a more complex quark composition.
There could also be resonant states composed of only gluons (the glueballs) with
spin parity of the type 0
C
(scalar mesons), 2
C
(tensor mesons), etc. There are some
tentative experimental indications not confirmed, regarding their existence. There
mayalsoexisthybrid states made, e.g., of a quark, an antiquark and an “effective”
gluon; they would have spin parity of the type 1
C
,etc.[P10].