Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
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128 6 Invariance and Conservation Principles
Another test of the conservation of C can be done in reactions that involve
particles and antiparticles which are exchanged under C . By applying the charge
conjugation operation in a typical reaction due to strong and/or electromagnetic
interaction at high energy, one has
C.p
p !
C
C
:::/ D .pp !
C
C
:::/ (6.47a)
C.e
C
e
!
C
C
:::/ D .e
e
C
!
C
C
:::/: (6.47b)
This implies that the average number and energy spectra of
C
and
must be
equal. This was experimentally verified with an accuracy of about 1%.
In the microcosm, at very high energy, one has always observed a symmetry
between particles and antiparticles. These are produced in equal number if the
energies involved are well above given thresholds, such as, very large energies
compared to the masses of the produced particles. In the macrocosm, however,
this particle–antiparticle symmetry is not observed: the universe seems to only
contain matter and not antimatter. This is one of the most fascinating topics
of research interconnecting particle physics, astrophysics and cosmology (see
Chaps.12 and 13).
6.6.2 Violation of C in the Weak Interaction
The strong and electromagnetic interactions are invariant under the C operation, and
C eigenvalues are conserved quantum numbers, while weak interaction processes
are not invariant under C . To check the violation of C in the weak interaction, we
proceed as we did for parity. Applying C to a neutrino, an antineutrino with the
same momentum p and the same spin S is obtained. This state corresponds to a
non-observed left-handed antineutrino:
C.
!
(H
e
/ D .
!
(H
e
/:
One can conclude that the weak interaction involving neutrinos does not conserve
parity nor charge conjugation. By applying to a neutrino the C and P operators
in sequence (e.g., P first and then C or vice versa), an existing right-handed
antineutrino is produced, that is,
CP .
!
(H
e
/ D .
(H
e
/: (6.48)
It follows that the weak interaction may conserve CP (see Fig. 6.3).
6.7 Time Reversal 129
6.7 Time Reversal
The time reversal operator T reverses the time coordinate t:
Tt Dt (6.49a)
T .r;t/ D .r; t/: (6.49b)
In classical mechanics, the systems are invariant under time reversal. For example,
the time reversed condition for a planet moving on a circular orbit around the sun
corresponds to the planet following the same orbit, though in opposite direction.
This is a meaningful situation; the fact that the planet moves in one way depends
on the initial conditions. Similarly, the application of T to a two body scattering
process would reverse the reaction:
=
T
a
b
c
d
a
b
c
d
Table 6.2 summarizes the effects of application of P and T on some physical
quantities. To check the conservation of P and/or T , it is necessary to experi-
mentally study the product of variables given in Table 6.2. One example is the
electron momentum p with respect to its spin . Some studied quantities and their
transformation properties under parity and time reversal are listed in Table 6.3.
If the parity P (or T ) is conserved in a process due to a specific interaction,
the Hamiltonian of the interaction must not contain terms that change sign
after application of the parity or time reversal operators. Referring to Table 6.3,
elementary particles with spin may have a magnetic dipole moment, but not a
static electric dipole moment (as in ordinary matter) because the term
n
E
/ E
is not invariant under T . For this reason, the search for a nonzero electric dipole
moment of the neutron (which can be measured with great precision) is historically
Table 6.2 Effect of the application of the parity .P / and time reversal
.T / operators on some basic physics quantities
Transformation
Quantity PT
r rrPolar vector
p p p Polar vector
Axial vector (like L D r p)
E EERemember that E D@'=@r
B
a
B BBis similar to
a
We can think of B as being due to the current in a coil. Reversing T
means to invert the current direction and therefore the magnetic field
direction
130 6 Invariance and Conservation Principles
Table 6.3 Effect of the application of the parity and time reversal opera-
tors on the products of some fundamental physical quantities
Transformation Physics
Quantity PTquantity
B C B C B Magnetic dipole moment
E E E Electric dipole moment
p p C p Longitudinal polarization
p
1
p
2
C p
1
p
2
p
1
p
2
Transverse polarization
p
1
p
2
p
3
p
1
p
2
p
3
p
1
p
2
p
3
very important. A
n
E
different from zero would imply the violation of both T and
P . Similarly, it is expected that the interaction Hamiltonian does not contain terms
which depend on the longitudinal polarization of the particles (a term p). As
we shall see in Chap. 8, this term is present in the weak interaction Hamiltonian.
Table 6.4 summarizes some invariance properties of fundamental interactions.
Time Reversal in Macroscopic Processes. At the microscopic level, all
processes (except those that violate CP , Sect. 6.8) are reversible and therefore
invariant under time reversal. For complex systems at the macroscopic level,
a well defined arrow for the time exists. The time reversal applied to a human
being does not make sense: the person should die first, then rejuvenate and
finally is born. The time arrow is also established for a gas in a container
under pressure, which expands through a hole into a second empty container.
The system evolves increasing its entropy, as in all irreversible processes. A
movie of the gas expansion gives us a realistic situation; if the film is shown
in reverse, it gives us an unrealistic view.
The time reversal symmetry is valid for microscopic systems, though not
for complex systems. Let us better understand the situation by considering
a man smoking a pipe (though it is a serious detriment to health). In the
“normal” time direction, the smoke exits the pipe; in the “reverse” direction,
it enters the pipe. The movie seems realistic in the normal direction, while in
the reverse direction, it seems absurd. Suppose we can do an incredible zoom
that allows us to view the individual collisions of the molecules of smoke.
At that level, we observe molecules moving and colliding; the situation is
invariant under time reversal and when we reverse the film, the situation
seems completely normal. Let us decrease the zoom to see aggregates of
smoke particles: also in this case, the dynamics of the aggregates do not
appear strange when the film is shown in reverse. By gradually decreasing
the magnification, we can begin to see puffs of smoke forming clots, that is,
something that condenses rather than expands. Finally, we have a situation
where it becomes clear if the molecules are expanding or condensing.
Changing from the microscopic to the macroscopic scale, i.e., from
molecules to large sets of molecules (by removing the motion of individual
6.8 CP and CPT 131
Table 6.4 Conservation laws and their validity in strong, weak and electromagnetic processes.
The quantum numbers N may be additive (A) or multiplicative (M)
Interaction
Conservation of Strong Electromagnetic Weak N
Energy–Momentum E; p yes yes yes A
Angular momentum J yes yes yes A
Parity P yes yes no M
Baryonic number B yes yes yes A
Leptonic numbers
b
L
e
;L
;L
yes yes yes A
Electric charge Q yes yes yes A
Charge conjugation C yes yes no M
Time reversal T yes yes yes
a
M
CP yes yes yes
a
M
CPT yes yes yes M
Strong isospin I yes no no A
3
rd
isospin component I
z
yes yes no A
Strangeness S yes yes no A
Lifetime 10
23
s 10
20
s 10
12
s–
Interaction range 10
13
cm infinite <10
15
cm –
a
Except for some meson decays, such as, K
0
; K
0
and (B
0
; B
0
/
b
Except for neutrino oscillations, Sect. 12.6
molecules), we lose information. By averaging the microscopic description
and eliminating details, we get a situation that is not symmetric with respect
to time and violates the invariance under time reversal. Probably, it is our
exact request of a macroscopic description that introduces the arrow of time.
However, why do we need to perform averages? This is probably associated
to the process of biological evolution, and to the fact that organisms develop
sensors that receive average quantities, for example, the temperature. Biolog-
ical selection leads to the development of more adapted organisms. Evolution
is dependent on success, which is nothing but a mechanism of variations, of
trials and errors, based on sensitivity to averaged properties.
6.8 CP and CPT
Although the weak interaction violates C and P , it was believed that it conserves
CP. Figure 6.3 illustrates the effect of applying C , P and CP to the electron
neutrino. In 1964, Christenson and other physicists discovered a rare decay of the
K
0
L
meson that violated the conservation of CP (Chap. 12). The violation of CP is a
small effect which only involves a small part of the weak interaction. It was initially
132 6 Invariance and Conservation Principles
Fig. 6.3 The application of
the operator C ,orP to the
electron neutrino state
generates a state that does not
exist in nature. The
application of CP generates
the antineutrino with the
correct helicity
ν
e
ν
e
P
P
C
C
ν
e
ν
e
CP
only found in the neutral kaon system, which acts as a very sensitive interferometer.
Recently, it was also found in other mesons.
The CP violation probably played an important role in the earliest moments
of the Universe. It is believed that at the beginning, all quantum numbers of the
Universe were equal to zero, with an equal number of particles and antiparticles.
Probably after t ' 10
35
s, a phase transition took place. After this transition, the
particles began to decay with a small CP violation, producing a slight predominance
of particles with respect to antiparticles less than one part per billion). When later
particle–antiparticle annihilated, that little excess of particles produced the matter-
dominated Universe which we observe today. The small amount of CP violation
observed in the weak interaction therefore seems not enough to fully explain this
scenario.
The CP violation involves a violation of T as well because all interactions
are invariant under CPT in any order the three transformations are applied. The
CPT invariance is one of the fundamental properties of quantum field theories.
It is often called CPT theorem (from C. L¨uders): any quantum theory that (1)
obeys the postulates of special relativity, (2) admits a state with minimum energy
and (3) respects the microcausality is invariant under CPT. The microcausality
requires that the fields obey commutation or anticommutation relations, implying
the correct statistics according to the spin of the particles (the Fermi–Dirac statistics
for fermions, the Bose–Einstein statistics for bosons).
The consequence of the CPT theorem is that a particle and its corresponding
antiparticle must have the same mass and lifetime; the magnetic moments must
be equal, but with opposite sign. Let us check this: if CPT is conserved, one has
ŒCPT;H D 0; on the other hand, one has .CPT/
2
D 1. If the particle state jai is
an eigenstate for the Hamiltonian (mass m and lifetime ):
hajH jaiDhajH.CPT/
2
jaiDhajCP THCP T jaiDhajH jai!m
a
D m
a
:
The mass, lifetime and magnetic moment equality is experimentally well verified
[P08] for particle and antiparticle. For example,
6.9 Electric Charge and Gauge Invariance 133
ˇ
ˇ
ˇ
ˇ
q
p
m
p
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
ˇ
q
p
m
p
ˇ
ˇ
ˇ
ˇ
< 0:99999999991 ˙ 0:00000000009 (6.50a)
Œm
e
C
m
e
=m
e
<8 10
9
(6.50b)
C
=
< 1:00002 ˙ 0:00008 (6.50c)
Œj
e
C
jj
e
j=j
e
j <.0:5 ˙ 2:1/ 10
12
: (6.50d)
6.9 Electric Charge and Gauge Invariance
The last subject presented in this chapter refers to a particular invariance of quantum
field theories called gauge invariance. This invariance was discovered in the EM
interaction and was found to be deeply correlated to electric charge conservation.
The gauge invariance mechanism can be extended in quantum mechanics to a “local
invariance,” as discussed in Chap. 11.
In classical electrostatics, the potential ' is defined up to an arbitrary constant.
Measurable quantities (as the electric field E) depend on potential difference, and
not on its absolute value. The fields E, B can be expressed in terms of a scalar and
a vector potential (see Appendix A.3). E, B are invariant under a transformation of
the scalar and vector potential of the type:
A
! A
0
.x
0
/ D A
.x
/ C
@
@x
A
0
D A C r
'
0
D '
1
c
@
@t
defining the gauge invariance. The existence of this symmetry is linked to the
absence of a mass term in the electromagnetic field equations (Chap. 11). As a
consequence, the photon mass is zero.
The conservation of electric charge leads to the invariance for a local group of
gauge transformations. An electromagnetic field A
, also called gauge field, with
many massless quanta (photons) coupled to the electric charge is needed. In this
way, it is clear how a gauge invariance became the basis of electromagnetism and of
its quantization.
The success of this procedure suggests that other forces may have a similar origin
with a different gauge invariance.
Chapter 7
Hadron Interactions at Low Energies
and the Static Quark Model
7.1 Hadrons and Quarks
The current vision of the submicroscopic world is based on a relatively small
number of constituents that interact through three fundamental interactions: the
electromagnetic, the weak and the strong interactions. The behavior of matter at
the smallest distances currently accessible (about 10
17
m) is explained in terms of
fundamental point-like and indivisible fermions (quarks and leptons) and of vector
bosons which mediate the interactions between fermions.
The list of “elementary particles” is much longer and not as well defined as the
elementary constituents. In this chapter, some simple schemes for the classification
of particles made of quarks is presented. As soon as the 1960s, the growing number
of hadrons and the repetition of some characteristics led some to believe that hadrons
were not “elementary particles,” but that they are made up of smaller objects,
the quarks. Initially, it was thought that only three quark types .u;d;s/ plus their
antiquarks existed. Today, six quark flavors u;d;s;c;b;t, are known.
At the fundamental level, the strong interaction occurs between quarks. Nuclei
(which are discussed in Chap. 14) are formed through the interaction between the
constituent protons and neutrons (the nucleons); this interaction can be considered
as a “residual” interaction of the strong interaction. The same occurs in atomic
physics, where the force acting between atoms to form molecules is a “residual”
electromagnetic interaction. The basic electromagnetic interaction occurs between
the nucleus and electrons to form the atom. The interaction between nucleons inside
nuclei is a complicated many-body problem.
A “normal” hadron is made of quarks and has dimensions of about 1 fm. Hadrons
with integer spin are called mesons, those with semi-integer spin are called baryons.
The hyperons are “strange” baryons, i.e., composed of at least one s quark. Hadron
“spectroscopy” can be explained by using the simple static quark model of hadrons,
which is described in this chapter with some experimental verifications and model
limits.
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
7,
© Springer Science+Business Media B.V. 2012
135
136 7 Hadron Interactions at Low Energies and the Static Quark Model
The constituent quarks, i.e., the valence quarks, explain the regularities of the
hadron spectra. At first, the quarks were considered as a mathematical fiction
because free quarks have never been observed. The evidence for quarks occurred
via lepton-hadron collisions with high transferred momentum (deep inelastic
scattering), consisting of direct collision between two point-like constituents (the
lepton and one of the quarks in the hadron). The dynamic quark structure of hadrons
shall be presented in Chap. 10. Deep inelastic scattering experiments demonstrated
that hadrons also contain other particles, namely, the gluons and virtual q
q pairs,
rapidly created and annihilated. These virtual q
q pairs are called sea quarks.
7.1.1 The Yukawa Model
The first attempt to explain the interaction between nucleons in nuclei with a
quantum mechanical model was developed by Yukawa in the 1930s. For each
particle, particularly for a boson of mass m, the relation between its energy E and its
momentum p is E
2
p
2
c
2
Dm
2
c
4
. The corresponding quantum equation is obtained
by replacing E, p with the corresponding operators, E ! i„@=@t , p !i„5,
acting on a wave function :
„
2
@
2
@t
2
C„
2
c
2
5
2
D m
2
c
4
:
This is the Klein–Gordon equation, already presented in Sect. 4.2.2. In the static
case (i.e., independent of time), the following solution is obtained:
5
2
D
mc
„
2
! D
K
r
e
r=a
(7.1)
with a D„=mc. Yukawa applied Eq. 7.1 to the case of nucleon interactions in nuclei.
He imagined that a nucleon could interact via a bosonic field with a dependence on
the distance from the center of the nucleon within a range a. He required that the
static potential U.r/ between two nucleons follows the same law, namely,
U.r/ D
K
r
e
r=a
I a D„=mc: (7.2)
If the range of the nuclear force is a ' 2 fm, the bosonic field must have m '
100 MeV/c
2
. It follows that the nuclear force mediator predicted by the Yukawa
model is a boson with a mass of about 100 MeV/c
2
. The mediator was later identified
as the meson, see Problem 7.1. For some time, it was believed that the muon
1
found in cosmic rays was the meson. An important experiment performed in
1
The muon was originally called meson. However, it is not a meson according to the definition
stating that a meson is a particle made of a q
q pair; therefore, it is not correct to call “meson” the
lepton.
7.2 Proton-Neutron Symmetry and the Isotopic Spin 137
Rome during World War II by Conversi, Pancini, and Piccioni provedthat the does
not interact strongly. Therefore, it could not be the meson predicted by Yukawa.
Today, we know that the strong interaction is much more complicated, and that
at the fundamental level, the strong interaction occurs between quarks, and not
between nucleons.
7.2 Proton-Neutron Symmetry and the Isotopic Spin
The neutron and proton behave similarly with respect to the strong interaction. In
1932, Heisenberg proposed to consider the neutron and the proton as two states of
a single particle, the nucleon N . In analogy with spin 1/2 states, which can have
two orientations along the third component, i.e., s
z
DC1=2 and s
z
D1=2,the
strong isotopic spin (or isospin) I D 1=2 is assigned to the nucleon, with the third
component I
z
DC1=2 for the proton and I
z
D1=2 for the neutron. The isotopic
spin is denoted by I and the three components with I
x
, I
y
, I
z
(or I
1
;I
2
;I
3
). The
isotopic spin can be visualized as a vector in the three-dimensional space of the
isospin, a hypothetical space with axis I
x
;I
y
;I
z
. The strong interaction depends on
I , not on I
z
: the third component of the strong isotopic spin behaves like the electric
charge. The strong isotopic spin is not conserved in decays induced by weak and
electromagnetic interactions (the latter conserves I
z
).
The isotopic spin has the same mathematical behavior as the spin. The spin is
a physical quantity measured in units of „, and with the dimension [Energy
Time]. The isotopic spin is a dimensionless quantity. The introduction of
the isotopic spin helps to classify the hadrons into multiplets. See Problems
from 7.11 to 7.16.
If the Hamiltonian H is invariant for all operations in the abstract isospin space,
the energy levels of the system are degenerate and can be classified according to the
total isospin I . The eigenvalues of the operator I
2
D I
2
x
C I
2
y
C I
2
z
is I.I C 1/.
Possible values of I are integers or semi-integer 0; 1=2; 1; 3=2; : : : For each value
of I , there is a multiplet with .2I C 1/ eigenstates of H with the same energy but
with different values of I
z
. For a fixed value of I , the possible values of I
z
are I ,
.I 1/;:::;I .
Isospin in nuclear physics. In nuclear physics, the conservation of I is connected
to the observation that nuclear states with the same number of nucleons but with a
different number of protons have the same energy, spin and parity. The invariance
of the Hamiltonian of nuclear interactions corresponds to the nuclear interaction
independence from the electric charge.
For example, the ground states of
7
Be and of
7
Li (which differ by the presence of
a pp pair in the
7
Be state and a nn pair in the
7
Li state) have the same energy, spin
138 7 Hadron Interactions at Low Energies and the Static Quark Model
and parity. The value I D 1=2 can be attributed to these two states with I
z
DC1=2
assigned to
7
Be, and I
z
D1=2 to
7
Li. Excluding electromagnetic effects, this
implies the equality of forces between the nn and pp pairs.
A second example of the nuclear interaction independence from the nucleon
electric charge comes from the observation that the three nuclei with A D 14 have
J
P
D 0
C
and practically the same energy:
8
<
:
14
C;I
z
D1; with a nn pair
14
N;I
z
D 0; with a np pair
14
O;I
z
DC1; with a pp pair:
A third example comes from the reaction
d C d !
4
He C
0
I0 0 0 1 I is not conserved
I
z
00 0 0 I
z
is conserved.
(7.3)
This reaction is prohibited for the strong interaction, though it is allowed for the
electromagnetic one. The process (7.3) must therefore occurs with a production
cross-section typical of the electromagnetic interaction. Under similar kinematic
conditions, it was experimentally confirmed that the measured cross-section was
almost one hundred times smaller than a typical “strong interaction” cross-section,
confirming the hypothesis that the process is indeed due to the electromagnetic
interaction.
Isotopic spin of the meson. The three
C
;
0
;
mesons have almost identical
properties, with the exception of the electric charge. For the strong interaction, it
can be hypothesized that the triplet corresponds to a single particle, the meson.
This is similar to the case of the nucleon, with the difference that there are three
instead of two degenerate states. Since the number of degenerate states associated
with the isotopic spin I is N
I
D .2I C1/,thevalueI
D 1 is attributed to the pion,
so that 2I
C 1 D 3. The three pions form an isospin triplet with I
z
DC1; 0; 1,
respectively for the
C
;
0
;
.
Conservation of isotopic spin in strong interaction. The relation between electric
charge Q, isospin I and baryon number B (B D 0 for the meson, B D 1 for the
nucleon) is
Q D I
z
C B=2: (7.4)
Let us generalize the above observations with the two following assumptions:
1. The strong interaction involving mesons and nucleons depends only on the total
isospin I and is independent of I
z
and Q.
2. The total isotopic spin I is conserved in strong interaction processes.
The conservation of the isospin leads to selection rules and precise relations
between the production cross-sections of related processes. Consider for example