Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
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5.4 The Strong Interaction 107
r
r
r b
α
s
α
s
b
b
a
r
r
rb
b
b
b c
Fig. 5.2 (a) Illustration of the strong force between a red .r/ quark and a blue .b/ quark through
the exchange of a red-antiblue .r
b/ gluon (˛
S
represents the squared amplitude of the process).
(b), (c) Illustration of the same process with color lines. Note that the line that goes back in time
represents the .
b/ anticolor. The black and white figure does not affect the representation. It would
be difficult to draw the antired or the antiblue. The three colors, in fact, have no relation to the
ordinary colors in the visible electromagnetic spectrum
a molecule. Indeed, the fundamental electromagnetic force occurs between protons
and electrons to form the hydrogen and heavier atoms. The force between atoms
(neutral objects) is a “residual” EM force.
The source of the EM interaction is the electric charge and the interaction is
mediated by the photon. There is only one type of electric charge, and that of
the opposite sign. The theory of strong interaction is quantum chromodynamics
(QCD). It is modeled by analogy to quantum electrodynamics (QED). The source
of the strong force is the color charge. Three degrees of freedom are necessary to
describe the dynamics: the red .r/; blue .b/ and green .g/ colors for quarks and
three anticolors antired .
r/;antiblue .b/ and antigreen .g/ for the corresponding
antiquarks. The color choice is purely conventional.
The strong interaction is mediated by eight massless gluons. Each gluon carries
a color and an anticolor charge: r
b, rg, br, bg, gr, gb, plus two linear combinations.
The Feynman diagram shown in Fig. 5.2 describes the interaction between a red
and a blue quark, with the exchange of a “colored” gluon (red, antiblue). Note
that the color line has a continuous arrow in the same direction as the time; the
arrow directed from right to left corresponds to the anticolor. The strong interaction
changes the quark color, though not the quark flavor (a u or d quark remains
unchanged in the strong interaction). A flavor variation is only possible in weak
interaction processes.
Similarly to what was done with the weak interaction, one can obtain an estimate
of the relationship between the strong and the EM coupling constants using the
lifetimes of strong and EM decays involving two particles with similar masses.
Hadrons decaying via the strong interaction (e.g., N
! N) have lifetimes
S
10
23
s, while particles that decay via the electromagnetic interaction (e.g.,
˙
0
!
0
) have lifetimes of the order of 10
19
s. This yields
˛
S
˛
EM
'
EM
S
1=2
'
10
19
10
23
1=2
' 100 which leads to ˛
S
1: (5.12)
The fact of the matter is, ˛
S
1 has an important consequence, that is, the
condition for the validity of the perturbation theory fails. A perturbation diagram
108 5 First Discussion of the Other Fundamental Interactions
Fig. 5.3 (a) Illustration of a
three-gluon vertex and (b) its
simplest interpretation in
terms of color lines
a b
qq
qq
qq
qq
a
b
c
Fig. 5.4 (a) The interaction between a quark and an antiquark in a meson at a relatively large
distance can be thought of as an elastic force (the spirals of the illustration represent an elastic,
not gluons). (b) An attempt to elongate the elastic to free the quark-antiquark pair results in the
creation (formation) of a new quark-antiquark pair (c)
with the exchange of one gluon, like the one shown in Fig. 5.2, can no longer be the
dominant diagram. The diagrams with the exchange of many gluons are equally
important. The calculation of quantities at low transferred momentum squared
(q
2
) processes is impossible. For high q
2
, i.e., for short distance collisions, it can
be shown (Chaps.9 and 11) that ˛
S
decreases to a value of the order of '0:1
(˛
S
' 0:12 at q
2
D m
2
Z
/. In this case, diagrams with a single gluon exchange
are a good approximation for high q
2
processes.
Since the gluons carry one color and one anticolor charge, the interaction
amongst gluons through three-gluon vertices, as shown in Fig. 5.3a, is possible.
Figure 5.3b shows the interpretation in terms of color lines. The presence of a vertex
with three gluons with a very high probability at low q
2
qualitatively distinguishes
the strong interaction from the other interactions. A vertex with four gluons is also
foreseen. The EM interaction has no vertex with two or more photons.
The quasistatic potential between two quarks within a hadron can be parameter-
ized in the following form:
V
S
D
4
3
˛
S
r
C Kr : (5.13)
The Coulomb-type term dominates at small distances; it is not divergent, as
˛
S
is not constant, but decreases with decreasing distance (asymptotic freedom).
This makes the radius of hadrons finite, as an equilibrium position is reached. The
second term, which linearly increases with the distance r between the two quarks,
gives rise to an elastic-type force. It is related to the interaction between gluons
5.5 Particle Classification 109
Table 5.1 Main properties of the four fundamental interactions at low energy
Mass Spin
Parity
Strength Exchanged of the of the
(adimensional Range Subjected (boson) exchanged exchanged
Force constant) (cm) particles particle bosons bosons
Strong 0:1
short
distances
10
13
Quarks, 8 gluons 0 1
1
large
distances
gluons
Electro- 1/137 1 Charged Photon 0 1
magnetic particles
Weak 1:027 10
5
<10
15
Leptons, Vector 80.6 GeV 1
C
;1
a
quarks intermediate 91.2 GeV
bosons
(W
˙
,Z
0
)
Gravitation 5:9 10
39
1 All Graviton 0 2
C
a
In weak interaction, parity is violated
and manifests itself by the confinement of quarks within hadrons. The effect of this
term is illustrated in Fig. 5.4: trying to free quarks, extending the elastic “spring of
gluons” that keeps them tied together, leads to the rupture of the elastic with the
creation of a quark-antiquark pair (a system which is more stable from the energetic
point of view).
The three “coupling constants” of the electromagnetic, weak and strong inter-
actions are slightly energy dependent. In Chap. 13, we shall see that at extremely
high energies, of the order of
p
s 10
15
GeV, the “constants” are believed to have
approximately the same value. This corresponds to the unification of fundamental
interactions. Table 5.1 summarizes the main properties of the four fundamental
interactions.
5.5 Particle Classification
Each particle is subject to one or more fundamental interactions. All fermions
interact through the weak interaction; electrically charged particles “feel” the
EM interaction and particles made of quarks (hadrons) are subject to the strong
interaction through their color charge. From an experimental point of view, it is
useful to classify particles in several categories.
110 5 First Discussion of the Other Fundamental Interactions
5.5.1 Classification According to Stability
A first classification of particles with masses below 3 GeV can be done in terms of
their stability.
Stable particles are: the photon ./, the electron (e
) and the corresponding
antiparticle; in the Standard Model (SM), neutrinos and antineutrinos are also stable.
Among hadrons, only the proton and the antiproton are stable. In models beyond the
Standard Model, the proton and the neutrinos may be unstable.
Many particles are unstable and can be classified according to their lifetime.
Remember that the initial number N
0
of unstable particles at the time t D 0 is
reduced after a time t to N D N
0
exp.t=/,where is the lifetime measured at
rest (t
1=2
D ln 2 D 0:693 is the half-life).
Particles with masses in the range 0.1–3 GeV/c
2
and with lifetimes ranging from
10
6
to 10
12
s decay via the weak interaction. This group includes the
˙
,
˙
,the
d;s;c;b quarks and the
˙
, K
˙
, K
0
, K
0
,
0
, ˙
˙
,
, D, F ,
c
, B, ::: hadrons.
Particles with lifetimes ranging from 10
16
to 10
20
s decay via the electromag-
netic interaction; these are, for instance, the hadrons
0
,
0
, ˙
0
.
Hadrons with lifetimes of the order of 10
23
s decay via the strong interaction.
These are the so-called resonances, e.g., , !, K
, N
, , Y
,etc.
The W
C
;W
;Z
0
bosons, which are the mediators of the weak interaction,
have lifetimes of the order of 10
25
s. These lifetimes are very short due to the
vector boson large masses with respect to the relatively small mass of the final state
particles. The phase space factor dN=dE
0
(4.37) is huge and the decay is very rapid,
even if it is caused by the weak interaction.
In most cases, particles with lifetimes longer than about 10
8
s are considered to
be “almost stable” since they can be used in an accelerator as secondary beams of
particles.
5.5.2 Classification According to the Spin
One of the most important quantum numbers is the spin, that is, the intrinsic angular
momentum of each particle.
Elementary particles are classified in bosons and fermions depending on whether
they have integer or semi-integer spin, respectively. Fermions follow the Fermi–
Dirac statistics and the Pauli exclusion principle. A system of identical fermions
is described by an antisymmetric wave function for the exchange of any fermion
pair. Bosons follow the Bose–Einstein statistics and the wave function of a system
of identical bosons is symmetric for the exchange of any two bosons. As a result,
identical bosons produced in high energy collisions tend to assume the same
quantum numbers and have similar energies and momenta, similar to lasers.
5.5 Particle Classification 111
Bosons are divided into fundamental bosons, mediators of interactions, and
mesons, which are hadrons. Fermions are divided into baryons, which feel the strong
interaction, and leptons, which are not subject to the strong interaction.
5.5.3 Classification According to the Baryon
and Lepton Numbers
We have already seen that one can assign a baryon number B DC1 to hadrons
as the proton and neutron, B D1 to antibaryons and B D 0 to mesons. We
shall see in Chap. 7 that baryons are made of three quarks, and mesons consist of a
quark-antiquark pair. The conservation of the baryon number means that a baryon
cannot decay into a system without baryons. It is equivalent to state that an unstable
baryon is a hadron that, after a series of decays, ends up as the lightest baryon, that
is, a proton. Proton decay into lighter particles, such as, mesons or leptons, has been
searched for, but it has never been experimentally observed.
As for the baryon number, a lepton number can be defined. Its conservation
prohibits the conversion of leptons into bosons or baryons. Three types of lepton
numbers, corresponding to the three known families of leptons, can be defined as
the electron lepton number L
e
DC1 is attributed to the electron, e
, and to its
neutrino
e
; L
e
D1 for their antiparticles (e
C
;
e
). L
e
D 0 for the leptons and
antilepton of the other two families. The muonic lepton number is L
DC1 for
and
.Thetau lepton number L
DC1 for the
lepton and its neutrino,
.
The baryon and lepton numbers are always conserved in any process resulting
from any kind of interaction.
2
Other quantum numbers are conserved only in strong
interaction processes, or in electromagnetic and strong interactions. These quantum
numbers are related to approximate conservation principles.
2
For the family lepton number, this is not completely true since the discovery of neutrino
oscillations, Chap. 12.
Chapter 6
Invariance and Conservation Principles
6.1 Introduction
Two important aspects in physics are the invariance (or symmetry) of the equations
describing the behavior of a system under a given transformation (for example, a
translation of the system frame) and the conservation of some physical quantities
(e.g., the momentum). The invariance properties are abstract features of the
mathematical formalism. These invariance properties are intimately connected to
conservation laws. For example, the conservation of angular momentum is related
to the invariance under spatial rotations; the conservation of the linear momentum is
related to the homogeneity of space. Noether’s theorem formally expresses the fact
that a physical conserved quantity corresponds to each invariant, and vice versa.
The equations which rule the dynamic evolution of a system (e.g., the Schr
¨
odinger
or the Lagrange equations) are first order differential equations in time and second
order in space coordinates. Every first integral of motion gives rise to a conser-
vation law. Each fundamental interaction (as the electromagnetic one described
by Maxwell’s equations) obeys various conservation laws. The formalism of the
considered interaction must obey several invariance requirements, which limit its
mathematical description.
Transformations can be continuous or discrete. In the first case, the transforma-
tion can be achieved by applying successive infinitesimal transformations. This is
not possible for a discrete transformation. A rotation is an example of continuous
transformation; the mirror reflection in space is an example of discrete transforma-
tion. The conservation laws related to these transformations are respectively additive
and multiplicative.
1
1
A given quantum number q is additive if in a reaction the sum of the q-values of the interacting
particles is the same before and after the reaction. Most conserved quantum numbers, as the electric
charge, are additive in this sense. Instead, for a multiplicative quantum number q, the corresponding
product, rather than the sum, is conserved
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
6,
© Springer Science+Business Media B.V. 2012
113
114 6 Invariance and Conservation Principles
In this chapter, we shall discuss invariance principles and conservation laws
in classical and quantum mechanics. Some examples will then be presented
and analyzed.
6.2 Invariance Principle Reminder
6.2.1 Invariance in Classical Mechanics
Lagrange equations. In classical mechanics, the state of a system with n degrees
of freedom is described by a Lagrangian L D T V D kinetic energy
potential energy. The Lagrangian contains n generalized coordinates q
i
from which
n conjugated momenta p
i
D @L=@ Pq
i
are derived. The motion of a system is
described, for each degree of freedom, by the Lagrange equation:
dp
i
dt
@L
@q
i
D 0: (6.1)
Suppose that for a particular system, the Lagrangian L does not dependent on the
q
k
coordinate. In this case, L is independent (or symmetric) with respect to any
transformation of this coordinate, which is called ignorable.IfL does not depend
on q
k
, one has @L=@q
k
D 0 and then, from (6.1), one has dp
k
=dt D 0, i.e.,
the conjugated momentum p
k
is constant. The momentum p
k
, conjugated to the
ignorable variable q
k
, is therefore conserved.
Translation along x. Let us consider the system Lagrangian L D T V D
.1=2/m Px
2
. In this case, L does not depend on x and is invariant under translations
along the x axis. From the Lagrange Eq. 6.1, one has p
x
D @L=@ Px D m Px D
constant, i.e., the linear momentum along x.p
x
D m Px/ is conserved.
Rotations. The Lagrangian L D T V D .1=2/m P'
2
r
2
where P'r D v, does
not depend on '; this implies that L is invariant under spatial rotations. From
(6.1) follows that p
'
D @L=@ P' D m P'r
2
D mvr= constant, i.e., that the angular
momentum is conserved.
Noether’s theorem. The conservation laws considered above are examples of
Noether’s Theorem: in a Lagrangian field theory, a conserved quantity is associated
to a continuous symmetry (and vice versa) [G91]. A symmetry condition constrains
the Lagrangian equation (note that the Lagrangian density L D L=vwherevis
the volume is sometimes used). For example, the assumption of time-translation
invariance requires that the Lagrangian does not depend on t. The hypothesis of
Poincare invariance (that is, the invariance under Lorentz transformation and space-
time translations) requires that the Lagrangian relativistically transforms as a scalar.
The inverse is also true: the symmetries of the motion equations can be deduced
from the Lagrangian analysis.
6.2 Invariance Principle Reminder 115
Hamilton equations. The motion of a classical system can also be described in
terms of the Hamilton equations:
Pq
i
D @H =@p
i
IPp
i
D@H =@q
i
(6.2)
where the Hamiltonian H is given by H D T C V . In this description, an
additional relation exists between the invariance principles and the conservation
laws: invariance means that H does not change under a given transformation.
Space translations. Consider an infinitesimal translation along the x axis, i.e., a
transformation x ! x C dx. In this case, the Hamiltonian varies by the quantity
dH D dx.@H=@x/ Ddx Pp
x
.Ifp
x
remains constant during the transformation, we
have Pp
x
D 0 and then dH D 0. In other words, if p
x
is constant, the Hamiltonian is
invariant for space translation along the x axis.
Similarly, it can be shown that the energy conservation implies time-translation
invariance and that the angular momentum conservation implies invariance under
space rotations.
6.2.2 Invariance in Quantum Mechanics
In quantum mechanics, the state of a system is described by a wave function .The
average of the results of a measurement on the system corresponds to the average
value hqi of an operator Q associated with the observable quantity, which acts on
the wave function . The average value (or the expectation value)isgivenby( D
visthevolume)
hqiD
Z
Q d h jQj i : (6.3)
In the last equality, the Dirac notation, using the state vector “ket” j i and its
conjugate transpose “bra” h j, is introduced. The operator Q should be Hermitian,
i.e., Q
C
D Q. This condition ensures that the average values (6.3) are real
numbers corresponding to measurable quantities. If Q is represented by a matrix
with elements Q
ij
, the operator Q
C
has elements Q
ji
.
The time evolution of hqican be described by either the time evolution of , D
.t/, or by the time evolution of Q; Q D Q.t/. The first case is the Schr¨odinger
representation; the second is the Heisenberg representation.
Schr
¨
odinger representation. The Schr
¨
odinger equation describes the time evolu-
tion of the wave function
S
:
i„
@
@t
S
.t/ D H
S
S
.t/ (6.4)
where H is the Hamiltonian and the status
S
is known. For stationary states,
the eigenvalue equation is H D E . The time development of
S
can also be
described in terms of an operator U applied to
S
S
.t/ D U.t; t
0
/
S
.t
0
/ (6.5)
116 6 Invariance and Conservation Principles
where U must be a unitary operator, i.e., U
1
D U
C
, since only this condition
allows one to maintain the normalization of
S
.t/. A unitary operator can be written
as
U.t;t
0
/ D e
i.tt
0
/H=„
: (6.6)
For the complex conjugated wave function, one has
S
.t/ D
S
.t
0
/U
1
.t; t
0
/: (6.7)
Heisenberg representation. The Heisenberg equation describes the time evolution
of Q:
i„
dQ
dt
D i„
@Q
@t
C ŒQ; H : (6.8a)
The commutation parenthesis is defined as ŒQ; H D QH HQ. Note that if
@Q=@t D 0, one has:
i„
dQ
dt
D ŒQ; H : (6.8b)
If ŒQ; H D 0,thendQ=dt D 0, and therefore Q is constant. The quantity
corresponding to the Q operator is conserved (and consequently, the corresponding
quantum numbers are conserved) if Q commutes with H.
Let us now derive a relation between the two representations. A mean value hqi
(eigenvalue) must be the same in the two representations, i.e., one must have
hqiD
Z
S
.t
0
/
Q
S
.t
0
/d D
Z
S
.t/
Q
0
S
.t/ d (6.9)
Heisenberg Schr
¨
odinger
where d D d v is the volume element. As the volume is arbitrary, the equation is
always true only if the two integrands are equal, that is,
S
.t
0
/
Q
S
.t
0
/ D
S
.t/
Q
0
S
.t/: (6.10)
Since
S
.t/
D
S
.t
0
/
U
1
and
S
.t/ D U
S
.t
0
/, the second member of
Eq. 6.10 can be written as
S
.t
0
/
U
1
Q
0
U
S
.t
0
/;
which is equal to the left side of (6.10)if
Q D U
1
Q
0
U: (6.11)
The time derivative of (6.11), multiplied by i„,gives(if@Q
0
=@t D 0)
i„
dQ
dt
D i„
dU
1
dt
Q
0
U C i„U
1
Q
0
dU
dt
:
6.2 Invariance Principle Reminder 117
Using the form (6.6)forU , one has
i„
dQ
dt
D i„
iH
„
e
i.tt
0
/H=„
Q
0
U C i„
iH
„
U
1
Q
0
e
i.tt
0
/H=„
DHU
1
Q
0
U C U
1
Q
0
UH
DHQ C QH D ŒQ; H ;
(6.12)
that is, i„
dQ
dt
D ŒQ; H . This is the Heisenberg equation when Q does not explicitly
depend on time. We obtained the Heisenberg equation starting from (6.9) and using
the transformations of the wave function (6.5 and 6.7). Equation 6.12 can then be
generalized to Eq. 6.8a.
6.2.3 Continuous Transformations: Translations and Rotations
Translations. Let us consider an infinitesimal translation dx along the x axis: x
0
D
x C dx. Its effect on the wave function .x/ is
.x
0
/ D .x Cdx/ D .x/ C dx
@ .x/
@x
D
1 C dx
@
@x
.x/ D dD
x
.x/
(6.13)
where the operator dD
x
D 1 Cdx@=@x is the operator which generates an infinites-
imal translation. Recall that the linear momentum operator is p
x
D .„=i/@=@x.The
operator dD
x
can therefore be written in the form
dD
x
D 1 C .i=„/p
x
dx: (6.14)
A finite translation x can be thought of as a series of infinitesimal translations,
x D ndx, with n !1; it follows that .dx D x=n/
D
x
D lim
n!1
1 C
i
„
p
x
dx
n
D exp
i
„
p
x
x
: (6.15)
D
x
is a unitary operator, i.e., D
C
x
D
x
D 1. The momentum p
x
is the generator
of the operator D
x
, which in turn is associated with spatial translations along x.
If the Hamiltonian H is invariant under space translation along the x axis, one has
ŒD
x
;HD 0.FromEq.6.14, it follows that p
x
commutes with H , i.e., Œp
x
;HD 0.
The momentum p
x
is a Hermitian operator; from Eq. 6.12, one finds that Pp
x
D 0
and consequently, p
x
is conserved. One can therefore conclude that the following
statements are equivalent:
1. The Hamiltonian is invariant under space translations
2. The operator p commutes with the Hamiltonian
3. The momentum p is conserved