Braibant S., Giacomelli G., Spurio M. Particles and Fundamental Interactions: An Introduction to Particle Physics
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4.5 Cross-Sections and Lifetime: Theory and Experiment 87
particles with a velocity v
i
relative to the target particle b,theflux˚ (cm
2
s
1
)of
particles incident on the target is
˚ D n
a
v
i
: (4.33)
(If only one particle a, moving with a velocity v
i
and described by the wave function
is considered, the flux is given by ˚ D v
i
j j
2
, whose dimensional units are the
same as in Eq. 4.33). The probability that a beam particle interacts with a target
particle in the time unit is given by the quantity W defined in the previous section.
It depends on ˚ through a constant which has the dimensions of an area
W D ˚ D n
a
v
i
cm
2
cm
3
cm s
1
: (4.34)
(W D j j
2
v
i
D v
i
in the case of only one incoming particle. Remember that
j j has the dimensions of Œvolume
1=2
). If the interaction potential is unknown,
information on the quantity W can be obtained from the cross-section measurement.
This is the case for the strong interaction between hadrons, (discussed in Chap. 7)
and for the weak interaction (described in Chap. 8). In both cases, information
on the unknown interaction potential are inferred from lifetime and cross-section
measurement.
On the opposite side, when the interaction potential is known (as for the
electromagnetic interaction), the quantity W can be computed. In the case of a
single incoming particle, from (4.34), one has
D
W
v
i
: (4.35)
This relation is exactly true if all the involved particles are spinless. As it shall be
shown below, when spins are considered, the number of possible final states given
by (4.35) is multiplied by a factor g
f
D .2s
c
C 1/ .2s
d
C 1/,wheres
c
and s
d
are
respectively the spins of the particles c and d.
To obtain information on the quantity W (4.28), it should be connected to easily
accessible experimental parameters. In the phase space expressed in Cartesian coor-
dinates, the number of states of a particle is given by dN D dxdydzdp
x
dp
y
dp
z
=h
3
.
In spherical coordinates, the number of states in a unit volume v and in the
momentum range between p and .p C dp / in the laboratory frame given by
dN D
d˝
.2/
3
„
3
p
2
dp (4.36)
where d˝ is the differential solid angle; in spherical coordinates .; '/ one has
d˝ D sin dd'. Therefore, the number of available states corresponding to the
total energy E
0
is (using the natural system of units, with „Dc D 1)
dN
dE
0
D
d˝
.2/
3
g
f
p
2
dp
dE
0
: (4.37)
88 4 The Paradigm of Interactions: The Electromagnetic Case
Let us now develop the calculation for the particular situation in which a heavy
particle of mass M undergoes a negligible momentum variation as a result of an
impact process (as is the case for a heavy nucleus in a Coulomb collision). Indicating
respectively with E; p;m, the energy, momentum and mass of the scattered particle,
the total energy in the final state is
E
0
D M C
p
p
2
C m
2
from which dE
0
D
p
E
dp: (4.38)
Making use of the relativistic relations E D mc
2
, p D mv, one obtains (again
using natural units) p=E D v and
dp
dE
0
D
E
p
D
1
v
(4.39)
where v is the velocity of the particle scattered from the diffusion center at rest. It
can be shown (e.g., in [P87]) that (4.39) is also valid in the c.m. system, where v is
the relative velocity between the particles c, d . Equation 4.35 can be rewritten in a
more general form as
d.a C b ! c C d/ D
1
.2/
2
jf.q/j
2
g
f
v
i
v
p
2
d˝: (4.40)
We shall use (4.40) in the next section to obtain the differential cross-section when
f.q/
2
is known; it shall also be used in the following chapters to obtain information
on the interaction potential through the measurement of d=d˝.
4.5.2 Particle Decay and Lifetime
Particle decay is the spontaneous process in which one particle transforms itself
into other lighter particles. Amongst charged particles, only the proton and the
electron are stable (or at least have a lifetime much longer than the age of the
universe, see Chap. 13). A radioactive decay is a process in which an unstable
atomic nucleus disintegrates into a smaller nucleus together with the emission of
particles or radiation (Chap. 14). Particle decays can be described by a simple law
with only one free parameter, that is, the lifetime . This parameter varies from
particle to particle. The lifetime spans from 1 for stable particles, to billions of
years for some nuclei, to 900 s for the neutron, down to 10
23
sformost
of the particles made of quarks (hadrons).
The decay law can be determined as follows. The probability P.t/ that a
particle at time t decays in the following time interval t is equal to the product
of t by a constant 1=, which depends on the particle
P.t/ D
t
:
4.5 Cross-Sections and Lifetime: Theory and Experiment 89
For a large number N of identical particles, the number of those decaying in the
time interval t is NP.t/. These decays decrease the number of initial particles
by an amount of t.dN=dt/,thatis,
NP.t/ D
Nt
Dt
dN
dt
!
dN
dt
D
N
: (4.41)
Indicating with N
0
the number of particles present at the initial time, the solution of
Eq. 4.41 is
N.t/ D N
0
e
t=
: (4.42)
Equation 4.42 is the law of radioactive decay.Since
R
tN.t/dt
R
N.t/dt
D , the constant is
called the lifetime. It is conventionally defined in the reference system in which the
decaying particle is at rest. Equation 4.42 is a phenomenological law in agreement
with experimental observations and quantum mechanics theory. The objective of
a fundamental interaction theory is to predict the value of . A decay process is
nothing more than a transition from a defined physics state into one of the possible
final states with a smaller mass, taking into account conservation laws. During this
process, an intermediate particle (such as the photon in EM decays or the W
˙
;Z
0
bosons in weak decays) is exchanged between the initial and final states. If the
transition probability W (units: s
1
) is large, the particle lifetime is small, and vice
versa. Therefore, the transition probability W and the particle lifetime are related as
W D
1
(4.43)
4.5.2.1 Decay Fraction and Branching Ratio
In general a particle has several decay modes. The decay fraction is the fraction
of particles which decay through a particular decay mode with respect to the total
number of possible decays; the decay fraction is measured by the relative amount
of this decay
i
with respect to the total amount of decays:
i
= . This ratio is
also called branching ratio (BR). Obviously,
P
i
i
= D 1.Agivendecay mode
of a particle usually has a characteristic lifetime, i.e., the average time between
decays in the same specific channel. For example, the muon decay from the Particle
Data Book [P10] (the biennial report on progress in particle physics and on particle
properties) gives:
Mass Lifetime Decay Decay
Particle (MeV) (s) Mode Fraction (
i
= )
˙
105:65837.˙1/ 2:19703.˙4/ 10
6
e 100%
90 4 The Paradigm of Interactions: The Electromagnetic Case
From the table, we learn that the decays into an electron plus two neutrinos in
100% of the cases. The number in brackets represents the uncertainty on the last
significant digit. In the case of the pion, one instead has
Mass Lifetime Decay Decay
Particle (MeV) (s) Mode Fraction (
i
= )
˙
139:5702.˙4/ 2:6033.˙5/ 10
8
' 99.98770%
e 1:230 10
4
2:0 10
4
e 7:39 10
7
0
e 1:036 10
8
ee
C
e
3:2 10
9
This means that the pion has a main decay mode (into ), but there are other more
unique decays, each measured by a given decay fraction
i
= . We shall explain
in Sect. 8.10 the reason why the pion decay into e is suppressed by a factor 10
4
compared to the decay into .
Different decay channels
i
= are often caused by different interaction mecha-
nisms (only the weak interaction is involved in the pion case, as a neutrino is always
present in the final state); each decay channel lifetime
i
is obtained from Eq. 4.43
and is given by
W
i
D
.
i
= /
: (4.44)
4.6 Feynman Diagrams
The Feynman diagrams are graphical representations of a perturbation contribution
to the transition amplitude. They represent mathematical expressions defined by
a set of well-defined rules which are not discussed in detail here. A thorough
description of the rule codification can be found in theoretical physics texts (see,
for instance, the review on the CERN website [4T73]). Feynman diagrams offer
a simple visualization of the interaction mechanism between particles and provide
rules for calculating the transition amplitude. A physics quantity is expressed as a
series development of terms with increasing power of a dimensionless parameter
(˛
EM
in the EM case). As the parameter is <1, the sum is composed of terms
decreasing with increasing order. The first order approximation gives a better
approximation if the coupling constant of the considered interaction is small.
Let us consider the EM interaction of electrons e
and positrons e
C
with the
presence of both real and/or virtual photons ; these particles are represented (see
Fig. 4.3) respectively by
• e
: a solid line with an arrow pointing in the same direction as the time (always
assumed from left to right);
4.6 Feynman Diagrams 91
e
–
e
+
γ
time
abc
Fig. 4.3 Representation of free particles in Feynman diagrams. (a) The electron e
moves with
constant velocity v <cand is denoted by an arrow in the same direction as the time arrow;(b)the
positron e
C
moves with constant velocity v <cin the direction opposite to the direction of time;
(c) the photon moves with speed c
Fig. 4.4 Basic vertex of
quantum electrodynamics.
The solid line with an arrow
represents an electron and the
wavy line represents a photon
γ
e
–
e
–
γ
e
–
e
–
e
–
e
–
e
+
e
+
e
+
e
+
e
+
e
+
e
–
e
–
γ
γ
γ
γ
γ
an electron
radiates a photon
an electron
absorbs a photon
a positron
radiates a photon
a positron
absorbs a photon
a photon
materializes in
an e
+
e
–
pair
an e
+
e
–
pair
annihilates in
a photon
Fig. 4.5 Six examples of vertex diagrams oriented in time in order to describe six different basic
QED processes
• e
C
: a solid line with an arrow opposite to the time direction;
• : a wavy line (sometimes indicated by an arrow, if the photon is real).
The particle is real or on mass shell if the relation E
2
D m
2
c
4
C p
2
c
2
is valid. For
a virtual particle (or off mass shell), this energy-momentum relation is not verified.
Particles connecting two interaction vertices are virtual (see below). In particular, a
real photon has m
D 0; a virtual photon can have E>pc(atime-like photon), or
E<pc(a space-like photon). A virtual particle cannot become free, it exists only
for a time allowed by the uncertainty principle. The virtual photon is the messenger
of the interaction between electrically charged particles (see Fig. 4.1).
The basic element of Feynman diagrams is the vertex (consisting of two
fermionic lines plus one bosonic line), as shown in Fig. 4.4. The time is assumed
on the horizontal axis. If the “legs” of the diagram are rotated in various ways with
respect to the vertex point, the six diagrams of Fig. 4.5 are obtained. Each of them
represents six different basic QED processes.
At each vertex, the energy, momentum, electric charge and other quantities (e.g.,
the baryon and lepton numbers introduced in the next chapter) must always be
92 4 The Paradigm of Interactions: The Electromagnetic Case
e
+
ab c
e
+
e
+
e
+
e
+
e
+
e
–
e
–
e
–
e
–
e
–
e
–
γ
γ
Z
0
Fig. 4.6 (a, b) Feynman diagrams contributing to the lowest order elastic collision e
C
e
!
e
C
e
due to the EM interaction only. (c) Contribution of the Z
0
boson of the weak interaction
Fig. 4.7 Lowest order
Feynman diagrams for the
emission of a photon by an
electron in the Coulomb field
of a nucleus (bremsstrahlung)
γ
ab
Ze
Ze
γ
e
–
e
–
e
–
e
–
e
–
e
–
γ
conserved. As discussed in Eq.4.3, not one of the six processes shown in Fig.4.5 is
possible if the three particles are all real.
Let us use the basic QED vertex to build more complicated diagrams. For
example, the diagram of Fig. 4.1 for the elastic e
e
! e
e
collision (Møller
scattering) is obtained using twice the vertex of Fig. 4.4 . Note that electrons are
real, while the photon is virtual. This diagram is actually composed of two diagrams,
each specifying which electron emits or absorbs the exchanged photon, as shown in
Fig. 4.1a,b. These diagrams are the simplest combinations of the vertex of Fig. 4.4
(the leading order diagrams). Other examples of the lowest order diagrams are
shown in Fig. 4.6a,b for the elastic e
C
e
! e
C
e
collision (Bhabha scattering).
Note that the diagram of Fig. 4.6a is similar than that of Fig. 4.1c (and should be
treated as Fig. 4.1a,b), while that of Fig. 4.6b(theannihilation diagram) is typical
of an annihilation process. Figure 4.6c shows the contribution of the Z
0
boson (a
neutral current weak interaction which is discussed later).
Figure 4.7 shows the emission of a photon by an electron in the Coulomb field
of a nucleus of charge Ze (bremsstrahlung). Compared to the diagrams of Figs. 4.5
and 4.6, there is an additional vertex represented by the nucleus connected by the
virtual photon. In addition to the real photon and the initial and final electrons, a
virtual electron is also present (the e
line between the two vertices).
The calculation of the transition probability and of the cross-sections is based
on the use of the perturbation expansion. The bosonic propagator (4.32) can be
interpreted as the exchange of a boson between two particles connected by two
vertices and with a probability proportional to .q
2
C m
2
c
2
/
1
. Each vertex is
4.7 A Few Examples of Electromagnetic Processes 93
ab c d e
Fig. 4.8 Second order Feynman diagrams for the elastic collision e
e
! e
e
characterized by the presence of a vertex factor g
0
;g describing the coupling of
the boson with the particles. The cross-section (4.40) is the product of jf.q/j
2
with
quantities depending on the phase space and on factors related to the particle spins.
For the exchange of a photon (which is massless), one has m D 0 and the propagator
is 1=q
2
; therefore, the cross-section is proportional to 1=q
4
D 1=t
2
where (t D q
2
).
The addition of higher order diagrams slightly changes the results obtained with
lower order calculation. In terms of Feynman diagrams, in addition to the lowest
order diagram (leading order), other diagrams with additional lines and vertices are
added. The next order (next-to-leading-order) involves two more vertices, as shown
in Fig. 4.8 in the case of the elastic e
e
! e
e
collision. Since each vertex
contributes with
p
˛
EM
to the matrix element jM
if
j, their contribution is about
˛
EM
times lower (137 times lower).
The Feynman diagrams are important for high precision calculations of QED
processes (see specialized texts for details, e.g., [A89]). The diagrams are more
easily interpretable in the momentum space than in the coordinate space (see
comment after (4.32)).
4.7 A Few Examples of Electromagnetic Processes
4.7.1 Rutherford Scattering
The interaction between two particles is described in terms of a cross-section,that
is, a quantity representing the likelihood that a reaction occurs. Let us consider the
simple case of particles that collide against a single massive target particle at rest
in the origin of a Cartesian reference frame. In the following, the description is
limited to the specific case of the elastic scattering of point-like and spinless particles
(the kinetic energy of the incident particles is conserved, the momentum changes),
assuming that the collision takes place through a spherically symmetric potential
(independent from the azimuth angle ').
As shown in Fig. 4.9a, the diffusion angle is determined by the value of the
impact parameter b. Particles with an impact parameter between b and b Cdb reach
the annular region between .b; b C db/ and are scattered within the angular region
.; d/, corresponding to a solid angle d˝. For a larger impact parameter b,
94 4 The Paradigm of Interactions: The Electromagnetic Case
ab
θ
θ
Δ
ψ
Fig. 4.9 Elastic scattering of charged particles reaching the region of surface 2b db around a
fixed massive center of diffusion (generally, a heavy nucleus) which produces a Coulomb type
potential. (a)The incident particles are elastically scattered in the angular range .; d/.
(b) Relation between the impact parameter b and the deflection angle in the Coulomb elastic
scattering
the diffusion occurs at a smaller angle . For this reason, a negative d corresponds
to a positive db. The number of incident particles elastically scattered per time unit
in the interval .; d/ is
dN D 2N
0
bdbD N
0
d (4.45)
where N
0
is the number of incident particles per area and time units and d D
2bdb is the surface of the annular ring crossed by the incident particles scattered
in the angular range .; d/. Note that if one considers dN as the difference
between the number of initial and final particles, the variation of the number of beam
particles is written as a negative term. The elastic differential cross-section d=d˝
is defined as
d./ D
d
d˝
d˝ D
d
d˝
2 sin dD2b db; (4.46)
(d˝ D 2 sin d is the differential solid angle) from which one gets
d
d˝
./ D
b
sin
db
d
: (4.47)
In the general case, without assuming spherical symmetry, Eq. 4.46 is
d.;'/ D bdbd'D
d
d˝
.; '/ d˝ D
d
d˝
.; '/ sin dd': (4.48)
The total elastic cross-section is obtained by integrating Eq. 4.47 in :
el
tot
D
Z
d
d˝
./d˝ D2
Z
0
sin
d
d˝
./d (4.49)
as
R
d' D 2. We can consider that the total cross-section represents the actual size
of the target, or rather the “transverse” area offered by the diffusion center to the
4.7 A Few Examples of Electromagnetic Processes 95
incident particles. This is only qualitatively true: the effective area depends on the
considered process and the incident particle energy. The cross-section .d=d˝/
el
is the fraction of cross-section due to elastic scattering at an angle in the range
.; C d/.
Classical calculation. The calculation can be applied to the elastic Coulomb
scattering. For example, Rutherford used ˛ particles (that is,
4
He nuclei) with a
few MeV of energy, colliding against heavy gold (Z D 79) nuclei. The Coulomb
potential of the nucleus with charge Ze is
U.r/ D
Ze
r
: (4.50)
If ze is the charge of the incident particle (z D 2 for helium nuclei), the Coulomb
energy potential is V.r/ D zeU.r/.
Classically (see Fig. 4.9b), the relation between the scattering angle ,the
incident particle kinetic energy E
c
D .p
2
=2m/ and the impact parameter b is given
by (see Problem 4.3)
tan
2
D
zZe
2
2E
c
b
D
Potential energy at a distance 2b
Kinetic energy
; (4.51)
from which one has
b D
Zze
2
2E
c
cot
2
: (4.52)
Then, it follows (d.cotx/ D.sin x/
2
dx):
db
d
D
Zze
2
2 4E
c
1
sin
2
=2
: (4.53)
Placing this relation into Eq. 4.47, one obtains
d
d˝
./ D
b
sin
db
d
D
.Zze
2
/
2
.4E
c
/
2
1
sin
4
.=2/
: (4.54)
This corresponds to Rutherford’s classical formula for elastic scattering between
two spinless charged particles. The integral of (4.54) gives the total elastic cross-
section,
el
tot
D
Z
d
d˝
./d˝ D 8
Zze
2
4E
c
2
Z
1
0
d.sin.=2//
sin
3
.=2/
: (4.55)
Equation 4.55 diverges for ! 0. This typical divergence of the Coulomb
interaction is due to the potential spatial dependence in (1=r
2
), corresponding to
an infinite range of the force. For very large r, a very small deflection is allowed.
96 4 The Paradigm of Interactions: The Electromagnetic Case
To allow a comparison with experimental data, Eq. 4.55 can be integrated up to
an angle D
0
>0
ı
, representing, for example, the angular resolution of the
experimental apparatus. A second effect should be considered: the target scattering
centers are affected by the screening action of the atomic electrons. Incoming
particles with impact parameters larger than the distance between the nucleus and
the inner electrons of the gold atom feel an “effective” charge smaller than Ze.
The quantity e
2
=E
c
in (4.55) has the dimensions of a length; in particular,
e
2
=m
e
c
2
D 2:8 10
13
cm D r
e
represents the classical electron radius. The
term .e
2
=4E
c
/
2
in (4.55)atE
c
D 1 MeV ' 2m
e
corresponds to r
2
e
=64.
Calculation in perturbation theory. It is instructive to consider the same process
in the context of perturbation theory, using the bosonic propagator. As seen in
Sect. 4.4, the matrix element jM jDf.q/ is given by Eq. 4.32 which includes the
special case of the photon (m D 0). The matrix element for the transition probability
of a particle with “charge” g
0
D .ze/ on a nucleus with charge g=4 D .Ze/ is
f.q/ D 4
Zze
2
q
2
(4.56)
where q is the transferred momentum between the initial and final states in the
interaction (Fig. 4.9b)
3
q
2
D .p p
0
/
2
D p
2
C p
02
2p p
0
' 2p
2
.1 cos/ D 4p
2
sin
2
=2: (4.57)
From Eq. 4.40, assuming relativistic velocities before and after the collision and
using „Dc D 1, one has
d D
1
.2/
2
jf.q/j
2
p
2
v
i
v
d˝ (4.58)
that is, as
p
2
v
i
v
' m
2
:
d
d˝
D
4
Zze
2
4p
2
sin
2
=2
2
m
2
.2/
2
: (4.59)
In the approximation of a nucleus with very high mass, spinless and non-relativistic
incident particles (p
2
=.2m/ D E
c
), one finally has
d
d˝
R
D
Z
2
z
2
e
4
m
2
4p
4
sin
4
=2
D
Z
2
z
2
e
4
.4E
c
/
2
sin
4
=2
(4.60)
3
The quantity q
2
defined above is not relativistically invariant. In Sect. 10.3.1, it shall be redefined
to make it independent of the choice of the reference system.