Classification of Fundamental Particles
The
fundamental particles are classified according to their
spin. Spin is the intrinsic property of the fundamental
particles. It may be thought of as the rotation of a
particle on its axis, just as the Earth rotates on its
axis, hence the name spin.
On the
basis of the spin, the fundamental particles are
classified into two groups:
The
fermions, which are spin-half particles (spin quantum
number s = 1/2).
The
bosons, which are spin-one particles (spin quantum
number s = 1).
note: Due to
the rotation of a fundamental particle on its axis, it
has an intrinsic angular momentum.
A fermion is a
spin half particle means the
magnitude of this
angular momentum (called the z-component of angular
momentum) along the direction of the motion (which is along the +z-axis)
is Sz
= +ħ/2 (spin ‘up’, i.e. the direction of the angular
momentum is along the direction of the motion of the
particle) or Sz
= – ħ/2 (spin ‘down’, i.e. the direction of the
angular momentum is at 1800 with respect to
the direction of the motion of the particle).
There are
no spinless fermions.
The fermions are further classified into two groups: The
quarks and the leptons. There are six quarks and six
leptons.
As per
Einstein’s equation E = mc2, mass of a
particle m = E/c2, and in high energy physics
or particle physics, where energy of the particles is
frequently measured, mass is defined in terms of energy
i.e eV.
1 eV is
the energy acquired by an electron, when accelerated
through a potential difference of 1 volt.
Thus,
mass is defined in terms of ‘electron volt eV/c2’,
‘mega electron volt MeV/c2’, ‘giga electron
volt GeV/c2’ or ‘tera electron
volt TeV/c2’.
(1 MeV =
106 eV, 1 GeV = 1000 MeV = 109 eV,
1 TeV = 1000 GeV = 1012 eV).
Quarks
Six flavours of the quarks (in order of increasing mass)
are: up u, down d, strange s, charm c, bottom b and top
t.
The up
quark u and the down quark d are the lightest quarks
with the rest mass of 2 and 5 MeV/c2
respectively and the strange quark s, the charm quark c,
the bottom quark b are heavier and therefore unstable.
Their rest masses are 100 MeV, 1300 MeV, 4200 MeV
respectively. The top quark t is the heaviest with the
rest mass of 173.3 GeV.
Whenever
during the collisions of electrons, positrons, etc.,
unstable particles having heavy quarks (s, c, b, t) are
produced, they decay into particles having only lighter
quarks (u and d). Most of the visible matter in the
universe is made up of these lighter and stable up and
down quarks, whereas particles having heavier quarks are
either produced during the high energy collisions as in
particle accelerators or they are found in cosmic rays.
Main
properties of the quarks are: mass, spin, parity,
electric charge, baryon number and five flavour quantum
numbers (isospin, charmness, strangeness, topness and
bottomness).
Each of
the up quark u, charm quark c and top quark t has +2/3
unit of electric charge, and each of the down quark d,
strange quark s and bottom quark b has -1/3 unit of
electric charge.
All six
quarks have positive or even intrinsic parity, i.e. P =
+1.
For all
six quarks, baryon number B = +1/3.
The up
and down quarks have isospin +1/2 and -1/2 respectively,
and all other four quarks have isospin zero.
The charm
quark has charmness quantum number C = +1, and all other
five quarks have charmness quantum number C = 0.
The
strange quark has strangeness quantum number S = -1, and
all other five quarks have strangeness quantum number S
= 0.
The top
quark has topness quantum number T = +1, and all other
five quarks have topness quantum number T = 0.
The
bottom quark has bottomness quantum number B` = +1, and
all other five quarks have bottomness quantum number B`
= 0.
note:
Each of
u c
t
quarks has
electric charge: +2/3e.
Each of
d s
b
quarks has
electric charge: -1/3e.
Leptons
Six flavours of the leptons are: electron e−, muon μ−, tauon
τ−, electron-type
neutrino νe, muon-type neutrino νμ
and tauon-type neutrino ντ.
The rest
masses of the electron e−, muon μ−
and tauon τ− are
0.510998928 MeV, 105.6583715 MeV and 1776.82 MeV
respectively, and neutrinos are lighter.
Each of
three charged leptons: electron e−, muon μ−
and tauon τ− carries
-1 unit of electric charge, whereas corresponding
neutrinos are neutral.
Each of
six leptons has the lepton number L = + 1.
Out of
three charged leptons, electron (rest mass 0.511 MeV) is
a stable and well-known particle.
The tauon
(rest mass 1777 MeV) is unstable and decays in τ = 2.9 ×
10−13 seconds.
The muon
(rest mass 105 MeV) is also unstable with mean lifetime
of 2.2 × 10−6 seconds. On the Earth, the
source of the muon is mainly cosmic rays.
note:
The up
quark u and
the down quark d
are the first generation quarks.
The
electron
e− and the electron-type neutrino
νe are the first
generation leptons.
The charm
quark c and
the strange quark
s are the second generation quarks.
The muon
μ−
and the muon-type neutrino
νμ
are the second
generation leptons.
The top
quark t and
the bottom quark
b are the third generation quarks.
The tauon
τ− and the tauon-type neutrino
ντ are the third
generation leptons.
Antiparticles
The
quarks and the leptons have the corresponding
antiparticles too.
Six flavours of the antiquarks (in order of increasing
mass) are: antiup u`, antidown d`, antistrange s`,
anticharm c`, antibottom b` and antitop t`.
The quark
Q and the corresponding antiquark Q` have the same mass.
The magnitude and the direction of the spin are also
same. However, electric charge, colour charge, weak
charge, baryon number, and flavour quantum numbers for
the antiquark are of opposite signs. For example, the up
and down quarks have electric charge +2/3e and -1/3e
respectively and isospin +1/2 and -1/2 respectively, so
the antiup quark u` and the antidown quark d` have
electric charge -2/3e and +1/3e respectively and isospin
-1/2 and +1/2 respectively. The strange quark has
strangeness S = -1, so the antistrange quark s` has
strangeness S = +1.
Six flavours of antileptons are: positron e+,
antimuon μ+, antitauon τ+,
electron-type antineutrino νe`, muon-type
antineutrino νμ` and tauon-type antineutrino
ντ`.
Each of
three charged antileptons: positron e+,
antimuon μ+ and antitauon τ+
carries +1 unit of electric charge.
Each of
six antileptons has the lepton number L = - 1.
Bosons
The fundamental bosons are gluon, photon, W+− & Z0
bosons and graviton.
The gluon
and the photon are massless. The W+− and Z0
bosons have rest mass 80 and 91 GeV respectively.
The
gluon, the photon, the W+− and Z0
bosons are spin one particles, the graviton is spin two
particle.
The gluon
as well as the photon has negative or odd intrinsic
parity, i.e. P = -1.
The
graviton has positive or even intrinsic parity, i.e. P =
+1.
For W+−
and Z0 bosons, which are spin one particles,
spin Sz = +ħ, 0 or –ħ.
Since
photon is massless, its spin has only two possible
values Sz = +ħ or –ħ.
note:
In the standard model of particle physics, there are
12 fundamental fermions (six quarks and six leptons) and
12 fundamental
bosons (eight gluons, a photon and W+, W−,
Z0 bosons), which are the mediators of the
interactions between the fermions.
In the
standard model, the neutrinos are known to exist in one
helicity state and are assumed therefore to be massless.
However, the solar neutrino deficit and atmospheric
neutrino anomaly are suggestive of neutrino flavour
oscillations and therefore of neutrino masses. (See the
topic: Neutrinos)
Interactions
Each fundamental interaction has an exchanged virtual
boson or gauge boson.
These
bosons are the mediators of the fundamental
interactions, which are of four types:
The
strong interaction, which is mediated by the exchange of
a gluon.
The
electromagnetic interaction, which is mediated by the
exchange of a photon.
The weak
interaction, which is mediated by the exchange of a W+
(W-plus), or a W− (W-minus), or a Z0
(Z-naught or Z-zero) boson.
The
gravitational interaction, which is mediated by the
exchange of a graviton.
The
colour charge gs
in a particle causes it to engage in a strong
interaction/decay.
The
electric charge e
in a particle causes it to engage in an electromagnetic
interaction/decay.
The weak
charge gw
in a particle causes it to engage in a weak
interaction/decay.
The
quarks carry the colour charges, the electric charges
and the weak charges that is, all of these three
charges, hence they take part in all these interactions.
The
charged leptons: electron e−, muon μ−, tauon τ−
do not carry
colour charges, so they cannot take part in the
strong interactions/decays but they have electric and
weak charges so they take part in the electromagnetic
and weak interactions/decays.
The
neutral leptons or
neutrinos do not
carry colour and electric charges, so they cannot
take part in the strong and electromagnetic
interactions/decays but they carry weak charges, so they
take part in the weak interactions/decays.
Just as
electric charges are of two types: positive and
negative, colour charges are of six types: red, green,
blue, antired, antigreen and antiblue. This has nothing
to do with the real colours. The quarks with the same
colour charges repel each other and those with different
colour charges attract each other. The interquark force
is independent of the colours involved.
Each of
six quarks can possess the colour charge in any one of
these three forms: red r, green g or blue b. The
antiquarks possess anticolour charges: antired r`,
antigreen g` and antiblue b`.
The
photon, which is the mediator of the electromagnetic
interaction is uncharged and therefore has no self
interaction. The gluon, however, itself carries a net
colour charge and therefore has self interaction, i.e. a
gluon can couple to the other gluon, i.e. a gluon can
emit or absorb a gluon
Whereas,
the electromagnetic force between electrically charged
particles decreases with the increase of the distance
between the particles,
the strong force
between the quarks, for a very short range, increases as
the distance between them increases.
The
strong interaction is about 100 times stronger than the
electromagnetic interaction and is about 107
times stronger than the weak interaction.
Center-of-mass Energy
Suppose
an incident particle of mass m1 and total
energy E1, hits a target particle of mass m2
and energy E2. Now, suppose target particle m2
is at rest in the laboratory system, then the
centre-of-mass energy Ecms is approximately
equal to (2m2E1)1/2.
Now, suppose that the incident and the target particles
travel in the opposite directions, as would be the case
in an electron–positron (e+e−) or
proton-antiproton (pp`) collider, then E2cms
is approximately equal to 4E1E2,
if mass m1, m2 are negligible in
comparison with E1, E2. Thus, the
cms energy Ecms available for a new particle
production in a collider with beams of equal energies E
coming from the opposite directions is equal to 2E (E2cms
= 4E2) that is,
the cms energy (2E) of the two colliding particles is equal to the sum
of their energies (E + E), whereas for a fixed
target machine, the cms energy is proportional to the
square root of the incident energy E.
Thus, the
highest possible energies for the production of new
particles are found at colliding-beam accelerators.
These accelerators accelerate two beams, which move in
the opposite directions and then are made to collide.
note:
Collisions of e+e− or pp` in
colliders imply collisions of billions of particles. For
example, a beam consisting of 40 bunches (successive
bunches being separated by say, 5 meters, the spacing
being determined by the radio frequency), each bunch
having 10 billion accelerated particles
(travelling with the velocity nearly equal to that of
light) and each particle having energy of 1000 GeV may
be collided with the similar beam coming from the
opposite direction in a real experiment in laboratories
to discover a massive particle.
Virtual Photons
The
electromagnetic interaction takes place only between
particles that possess electric charges. Electrically
charged particle may be thought of as continuously
emitting photons and then reabsorbing them. If another
charged particle is nearby, then the photon can be
absorbed by it. Photons, which we usually see have zero
mass. However, the exchanged photons cannot have zero
mass. Thus, they are called virtual photons. Exchanged
bosons are often called gauge bosons.
Virtual photons
are the exchanged bosons or gauge bosons that mediate
the electromagnetic interactions. Virtual photons
cannot exist as free particles and therefore are
absorbed immediately.
The
annihilation of an electron e− and a positron
e+ to a muon μ− and an antimuon μ+
is an example of the electromagnetic interaction. Here,
during the collision of the electron e− and
the positron e+, e+e−
pair transforms into a virtual photon. The virtual
photon then decays into a μ+μ−
pair.
When an
electron and a positron collide head-on (i.e. at 180
degree while coming from the opposite directions), a
virtual photon may be produced with all the energy of
the annihilation going into the virtual photon. That is,
this virtual photon will have energy Eγ equal
to the cms energy Ecms of the colliding e+e−
pair (Eγ = Ecms). The
momentum p of this photon is zero as it is produced in a
head-on collision. Thus, from the equation E2
= p2 + m2, Eγ2
= 0 + Mγ2 (where Mγ is
the mass of this photon). Thus, the mass of the photon
is equal to its energy (Mγ = Eγ =
Ecms) that is, this mass is not zero, so it
is a virtual photon (the real photon has zero mass).
This virtual photon survives for less than say, 10-25
seconds and transforms into a charged
particle-antiparticle pair.
Just as a
real photon can transform into an electron-positron
pair, a virtual photon can also transform into an
electron-positron pair but as these virtual photons may
have extremely high energy too, they can produce
muon-antimuon pairs or quark-antiquark pairs (which are
very massive) too.
Electromagnetic Interactions
Basically, the electromagnetic interactions involve
electrons, positrons and photons: real as well as
virtual ones. The
electric charge in a particle causes it to engage in an
electromagnetic interaction through a virtual photon.
As electric charges are carried by quarks too, they can
also involve in the electromagnetic interactions.
Moller Scattering or electron–electron scattering (e− + e−
→
e− + e−):
In this case, when two electrons coming from the
opposite directions collide, one of them
emits a virtual photon and remains itself as an electron. The other
electron absorbs
that virtual photon and remains itself as an
electron. After the exchange of a virtual photon between
them, these two electrons move in the opposite
directions with respect to each other but at an angle
with respect to the path of these electrons before the
collision.
Bhabha Scattering or electron–positron scattering (e− + e+
→ e− + e+): In this case, an electron and a positron collide
and annihilate to
a virtual photon, which subsequently decays into a
new electron-positron pair. These new electron and
positron move in the opposite directions with respect to
each other but at an angle with respect to the path of
the original electron-positron pair before the
collision.
note:
A virtual photon may also decay into
virtual
electron-positron pair, which may then again
annihilate to a virtual photon, then this virtual photon
may decay into a real electron-positron pair. This is
equivalent to vacuum polarization.
OR
Like the
Moller scattering, the electron and the positron coming
from the opposite directions collide. Then, the electron
(or it may be positron too)
emits a virtual
photon and remains itself as an electron. The
positron subsequently
absorbs that
virtual photon and remains itself as a positron.
After the exchange of a virtual photon between them,
these electron and positron move in the opposite
directions with respect to each other but at an angle
with respect to the path of this electron-positron pair
before the collision.
Pair annihilation (e−
+ e+
→
γ + γ): In this case, an electron and a positron
collide and
annihilate to a virtual photon, which subsequently
decays into a real photon pair.
If the energy is sufficiently high,
then the virtual photon produced through the
annihilation of the electron-positron pair would be of
high energy and such a virtual photon may decay into a
pair of muons: e− + e+ → γ → μ−
+ μ+. If the cms energy of colliding e+e−
pair is even larger, then the virtual photon would
be even more massive and may decay into a pair of tauons
or a pair of quarks: e− + e+ → γ →
τ− + τ+
or
e− + e+ → γ → Q
+ Q`.
OR
The
electron emits a
virtual electron and therefore emits one unit of
negative charge, and transforms into a real photon
γ (which has
zero electric charge). The positron having one unit of
positive charge then absorbs this virtual electron and
therefore absorbs one unit of negative charge, and
transforms into a real photon
γ having zero
electric charge.
Pair production (γ + γ
→
e−
+ e+): Here, a photon
emits a virtual
electron and therefore emits one unit of negative
charge, and transforms into a real positron
e+
having one unit of positive charge. The other photon
then absorbs this virtual electron and therefore absorbs
one unit of negative charge, and transforms into a real
electron e−
having one unit of negative charge.
Compton Scattering (e−
+ γ
→ e− + γ): Here, the electron
emits a virtual electron
e−
and therefore emits one unit of negative charge, and
transforms into a real photon
γ having zero
electric charge. The virtual electron is absorbed by the
photon (which had collided with the electron) and in
this way, this photon (which has zero electric charge)
absorbs one unit of negative charge and transforms into
a real electron e−
having one unit of negative charge.
Also,
electron and photon may annihilate to a
virtual
electron,
which may then decay into an electron and a photon.
note:
Compton Scattering is the
inelastic
scattering of a photon by an electron that is, energy of
the photon decreases as part of the energy of the photon
is transferred to the electron. The collision causes the
electron to recoil and a new photon of lesser energy
emerges at an angle from the photon’s incoming path).
note:
When an electron and a positron collide, they can
scatter elastically: e− + e+ → e−
+ e+. The kinetic energy is conserved
before and after the elastic collision. The word
‘elastic collision’ in particle physics means that the
same particles came out as went in, i.e. same particles,
which collide, exist after the collision. In the elastic
collision, rest energies and masses of the colliding
particles are conserved before as well as after the
collision.
In QFT
(quantum field theory), the electric field is quantized
(in the form of photons), and the interaction may be
thought of as consisting of a stream of photons passing
back and forth between two electric charges, each
electric charge continually emitting photons and
continually absorbing them.
Virtual
particles can have any mass. We say, virtual particles
do not lie on their mass shell.
Gluons and Colour Charges
In the strong interactions, gluons are exchanged between
quarks.
The
gluons are bicoloured, carrying one unit of positive
colour charge and one unit of negative colour charge.
There are
eight types of gluons: rb`(red, antiblue), rg`(red,
antigreen), bg`, br`, gr`, gb`, (rr`- bb`)/21/2,
(rr` + bb` - 2gg`)/61/2.
The
gluon may decay
into an up quark u and an antiup quark u`.
The gluon may decay into a down quark d and an antidown
quark d`.
Similarly, a gluon may decay into a strange quark s and
an antistrange quark s`, into a charm quark c and an
anticharm quark c`, into a bottom quark b and an
antibottom quark b`, into a top quark t and an antitop
quark t`.
If
rg`(red antigreen) gluon (electrically neutral) decays into an up
quark u (electric charge: +2/3e) and an antiup quark
u`(electric charge: -2/3e), then the up quark will have
red colour charge
and the antiup quark will have
antigreen colour
charge, so that colour charge remained conserved in
the decay of a gluon into a quark-antiquark pair.
Similarly, if
gb`(green, antiblue) gluon decays into a strange
quark s (charge: -1/3e) and an antistrange quark s`
(charge: +1/3e), then the strange quark will have
green colour
charge and the antistrange quark will have
antiblue colour charge.
Similarly, an up quark u (charge: +2/3e) carrying
blue colour
charge and an antiup quark u` (charge: -2/3e)
carrying antired colour charge may
annihilate to a
br`(blue, antired) gluon (electric charge: zero), so
that colour
charge and electric charge remained conserved in the
annihilation. The
br` (blue, antired) gluon may then materialize into,
say a down quark d (charge: -1/3e) having
blue colour
charge and an antidown quark d` (charge: +1/3e)
having antired
colour charge.
Whereas,
many particles carry electric charge, no naturally
occurring particles carry colour charge.
The only colourless combinations of the quarks are the
baryons and the only colourless combinations of the
quarks and the antiquarks are the mesons.
By colourless, we mean, either the total amount of each
colour is zero, as in mesons or all these three colours
are present in equal amounts, as in baryons.
Thus, for
example, if the up quark u (charge: +2/3e) of the
positive pion π+ (ud`) has
blue colour
charge, then the antidown quark d` (charge: +1/3e) of
the positive pion must have
antiblue colour charge, so that the net colour charge is zero.
Similarly, if the down quark d of a proton p (uud) has
red colour
charge, then one up quark u of the proton must have
blue colour charge and the other up quark u of the proton must have
green colour
charge.
A red
quark (u, d, s, c, b or t), i.e. a quark having red
colour charge carries
one unit of redness, zero blueness, and zero greenness. The
corresponding antiquark (u`, d`, s`, c`, b` or
t`), i.e. antired quark carries
minus one unit of
redness, zero blueness, and zero greenness.
Example of decay through a gluon:
The decay of the neutral delta baryon Δ0
(udd) into a proton p (uud) and a negative pion π−
(u`d):
Δ0 (udd)
→
p (uud) + π−
(u`d)
Here, one down quark of the neutral delta baryon
emits a gluon
and remains itself as a
down quark d.
The
gluon then decays into an up quark u and an antiup quark u`. This
antiup quark u`
combines with the
down quark d
(which had emitted the gluon) of the neutral delta
baryon to produce a negative pion π− (u`d).
The
up quark u (produced by the decay of the gluon) combines with the
remaining up quark and the down quark of the neutral
delta baryon to produce a
proton p (uud).
note:
The mass (1232 MeV) of the neutral delta baryon
Δ0
(udd) >
mass (938 MeV) of the proton
p (uud) +
mass (139 MeV) of the negative pion
π−
(u`d). Thus, this decay is kinematically allowed.
W
Boson and Weak Processes
As the
leptons and the quarks carry weak charges too, they
continually emit and absorb virtual W+− and Z0
bosons, which could be absorbed by a nearby lepton or
quark. The exchanged virtual W+− and Z0
bosons behave like exchanged virtual photons of the
electromagnetic interactions but the real W+−
and Z0 bosons are very massive, which make
the interactions between the quarks and the leptons
involving W+− and Z0 bosons much
weaker, hence it is called the weak interaction. Virtual
W+− and Z0 bosons can exist for
3.1 × 10−25 and 2.6 × 10−25
seconds respectively, and the range of the force is only
10−17 meters.
In
charged current weak interactions/decays, the up quark (charge:
+2/3e) can transform into a down quark (charge: -1/3e)
by emitting a W+ boson or by absorbing a W−
boson. (Through this emission of the W+
boson or absorption of the W− boson, the up
quark emits one unit of positive charge or absorbs one
unit of negative charge). Similarly, the down quark
(charge: -1/3e) can transform into an up quark (charge:
+2/3e) by emitting a W− boson or by absorbing
a W+ boson. (Through this emission of the W−
boson or absorption of the W+ boson, the down
quark emits one unit of negative charge or absorbs one
unit of positive charge).
For
example, the neutron (udd) can transform into a proton
(uud) by emitting a W− boson through its one
down quark d (charge: -1/3e), which will become the up
quark u (charge: +2/3e) and then that W−
boson may decay into an electron e− and
electron-type antineutrino νe`.
W−
→ e−
+ νe`
The
strange quark s (charge: -1/3e) may emit one unit of negative charge
by emitting a W− boson (which carries one
unit of negative charge) and may
transform into an
up quark u
(charge +2/3e).
The W−
boson may decay into an antiup quark u` (charge:
-2/3e) and a down quark d (charge: -1/3e).
W−
→ u`d
The
antistrange quark s` (charge: +1/3e) may emit one unit of positive
charge by emitting a W+ boson (which carries
one unit of positive charge) and may
transform into an
antiup quark u`
(charge -2/3e).
The W+
boson may decay into an up quark u (charge: +2/3e) and
an antidown quark d`(charge: +1/3e).
W+
→
ud`
Lepton
conservation requires that the total lepton number must
be the same on both sides, hence both a neutral and a
charged lepton of the same flavour (e.g. electron type
antineutrino νe` with electron
e−, muon
type antineutrino νμ` with muon μ−)
to appear together. Thus, when a W− boson
(lepton number L = 0) decays into an electron
e- (lepton number L = +1), an electron-type
antineutrino νe` (lepton number L= -1) is
produced too, so that the lepton number is conserved.
W−
→ e−
+ νe`
When a W−
boson (lepton number L = 0) decays into a muon μ−
(lepton number L = +1), a muon-type antineutrino νμ`
(lepton number L= -1) is produced too, so that the
lepton number is conserved:
W−→
μ−
+ νμ`
When a W+ boson
(lepton
number L = 0)
decays into an antimuon μ+ (lepton number L= -1),
a
muon-type neutrino νμ (lepton
number L= +1) is produced too.
W+
→ μ+ + νμ
Similarly,
W+
→ e+ +
νe
W−
→
τ−
+ ντ`
W+
→
τ+
+ ντ
W+−
and Z0 bosons (rest mass 80 and 91 GeV) being very massive
bosons have a very short lifetime. This means
W+−
bosons and Z0
bosons are not directly observed
in general. Instead, their decay products are measured.
The
W+
boson decays into a charged antilepton and corresponding neutrino
or
a
quark-antiquark pair, e.g. W+
→ μ+ + νμ or W+
→ ud`.
The W−
boson decays into a charged lepton and corresponding antineutrino
or
a quark-antiquark pair, e.g.
W−
→ e− + νe` or W−
→ u`d.
Neutral current weak interactions/decays are
mediated by the neutral Z0 boson.
For
example, electron-positron pair e+e−
annihilates to a Z0 boson and the Z0
boson subsequently decays into muon-antimuon pair μ+μ−:
e− + e+ → Z0 → μ− + μ+).
The Z0
boson (rest mass 91 GeV) being highly unstable
immediately decays after its production into
fermion-antifermion pair. For example, the Z0
boson may decay into electron-positron pair e+e−
or muon-antimuon pair μ+μ−: Z0
→ e− + e+
or Z0
→ μ− + μ+.
Not only
is the the lepton number L conserved in an
interaction/decay, the electron number Le,
the muon number Lμ, and the tauon number Lτ
are also conserved.
The
electron e− and the electron-type neutrino νe are
assigned the electron number Le= +1.
The muon
μ− and the
muon-type neutrino νμ are assigned the muon
number Lμ = +1.
The tauon
τ− and the tauon-type neutrino ντ are
assigned the tauon number Lτ = +1
Corresponding antiparticles are assigned Le=
-1 (e+ and νe`),
Lμ = -1 (μ+ and νμ`)
and Lτ = -1 (τ
+ and ντ`).
Thus, for
example, W− boson cannot decay into
e−
and νμ` (W− →
e− + νμ`
is forbidden) that is, a W− boson will always
emit an electron-type antineutrino νe` with
an electron
e− (W−
→
e− + νe`).
Thus, the electron number Le = 0 before the
decay of W− boson and Le = +1 – 1
= 0 after the decay too.
Similarly, if an electron
e−
emits a W− boson or absorbs
a W+ boson, it will always convert into an
electron-type neutrino νe (e−
→ W− + νe
or
e−
+
W+ → νe)
that is, electron
e−
cannot convert into a muon-type neutrino νμ (e−
→ W− + νμ as
well as
e− +
W+ → νμ
is forbidden) or tauon-type neutrino ντ (e−
→ W− + ντ as
well as
e−
+
W+ → ντ
is forbidden).
Similarly, if a muon μ− emits a W− boson or absorbs a W+
boson, it will always convert into a muon-type neutrino
νμ (μ−
→ W− + νμ
or μ−
+
W+ → νμ)
and if a tauon τ− emits a W-
boson or absorbs a W+ boson, it will always
convert into a tauon-type neutrino ντ (τ−→
W− + ντ
or τ−
+
W+ → ντ).
The
electron-type neutrino νe neither can emit a W−
boson nor absorb a W+ boson to convert
into a positron
e+ i.e.
the processes: νe → W− +
e+ or
νe
+
W+ →
e+ are
forbidden as the lepton number is not conserved. L = +1
and -1 before and after these processes.
However,
electron-type
neutrino νe can emit a W+ boson or
absorb a W−
boson to
convert into an electron e−, i.e. the
processes: νe → W+ +
e− or
νe
+
W− →
e− are
allowed as the lepton number is conserved. L = +1 before
and after these processes.
Similarly,
muon-type neutrino νμ and tauon-type neutrino
ντ can emit a W+ boson or absorb a
W− boson, to convert into a muon μ−
(νμ → W+ + μ−
or νμ
+
W− → μ−)
and tauon τ− (ντ → W+ + τ−
or ντ
+
W− → τ−)
respectively.
Similarly, an
electron-type antineutrino νe` or muon-type
antineutrino νμ` or tauon-type antineutrino ντ`
can emit a W− boson or absorb a W+
boson, to convert into a positron e+
(νe` → W− +
e+
or νe` +
W+→
e+) or antimuon μ+
(νμ` → W− + μ+
or νμ` +
W+ → μ+)
or antitauon τ+
(ντ` → W− + τ+
or ντ` +
W+ → τ+)
respectively.
The first
experimental test of the separate conservation of the
electron number and the muon number was conducted at
Brookhaven in 1962. About 1014 muon-type
antineutrinos νμ` were collided with the
protons p, then 29 antimuons μ+
were detected but no positron
e+ was detected, i.e. the
interaction νμ`
+ p → μ+
+ n occurred
(muon number Lμ = -1 before as well as after
this interaction). However, the interaction νμ`
+ p → e+
+ n did not occur that is, the
interaction: νμ` + p → e+
+ n is forbidden by muon and electron
number conservation as the muon number Lμ =
-1 and 0 before and after this interaction and the
electron number Le = 0 and -1 before and
after this interaction, that is, Le and Lμ
are not conserved in this interaction.
Examples of Charged current weak interactions:
1. Collision between a muon-type antineutrino νμ`
and a proton p (uud) may result in the production of an
antimuon μ+ and a neutron n (udd):
νμ` + p (uud)
→
μ+
+ n (udd)
In this
case, the muon-type antineutrino νμ`
with zero electric charge emits one unit of negative
electric charge (-1e) by emitting a
W−
boson and
transforms into the corresponding charged antilepton,
i.e. antimuon μ+
with electric charge: +1e.
Then, one
of the two up quarks (charge: +2/3e) of the proton (uud)
absorbs that W− boson and transforms into a
down quark (charge: -1/3e) and the proton becomes a
neutron n (udd).
2.
Collision between
a muon-type neutrino νμ and a neutron n (udd)
may result in the production of a muon μ−
and a proton p (uud):
νμ + n (udd)
→
μ−+
p (uud)
In this
case, the muon-type neutrino νμ with zero
electric charge emits one unit of positive electric
charge (+1e) by emitting a
W+
boson and transforms into the corresponding charged
lepton, i.e. muon
μ− with electric charge: -1e.
Then, one
of the two down quarks (charge: -1/3e) of the neutron
(udd) absorbs that W+ boson and transforms
into an up quark (charge: +2/3e) and the neutron becomes
a proton p (uud).
Through a
W boson, quark having up flavour u can transform into a
quark having down flavour d (u → d) and similarly, a
lepton having electron flavour
e− can change into a lepton having electron-type neutrino
flavour νe (e− → νe). However, photon, gluon, Z0
boson cannot change quark or lepton flavour. For
example, if an up quark emits a photon or a gluon or Z
boson, it will remain the up quark that is, no change in
the flavour of the quark. Similarly, if an electron
emits a photon or a Z0 boson, it will remain
the electron that is, no change in the flavour of the
lepton.
Thus, only charged current weak interaction/decay can
change the flavour of a fermion (quark or lepton).
The quarks generations are ‘skewed’ in weak
interactions/decays, i.e. ‘cross-generational’
transitions could occur for the quarks in the weak
interactions/decays. In other words, unlike
electron number (which is related to the first
generation leptons and antileptons: e−, νe
, e+, νe`),
upness-plus-downness (which is related to the first
generation quarks and antiquarks: u, d, u`,d`) is not
conserved in the weak interactions/decays.
Similarly, unlike muon number,
strangeness-plus-charmness is not conserved in the weak
interactions/decays.
That is,
each of the
up quark u (charge: +2/3e), charm quark c (charge:
+2/3e), top quark t (charge: +2/3e) can convert into
any of the
down quark d (charge: -1/3e) or strange quark s (charge:
-1/3e) or bottom quark b (charge: -1/3e) by emitting a W+
boson or by absorbing a W− boson.
Similarly, each
of the down quark d (charge: -1/3e), strange quark s
(charge: -1/3e), bottom quark b (charge: -1/3e) can
convert into any
of the up quark u (charge: +2/3e) or charm quark c
(charge: +2/3e) or top quark t (charge: +2/3e) by
emitting a W− boson or by absorbing a W+
boson.
The
probability with which an up quark u (charge: +2/3e)
can transform
into a down quark d (charge: -1/3e), by emitting a W+
boson or by absorbing a W− boson as well as
the probability
with which a down quark d (charge: -1/3e)
can transform
into an up quark u (charge: +2/3e), by emitting a W−
boson or by absorbing a W+ boson, is
|Vud |2 = 0.9742
= 0.948
Similarly, the
probability with which a charm quark c (charge:
+2/3e) can
transform into a bottom quark b (charge: -1/3e), by
emitting a W+ boson or by absorbing a W−
boson as well as the
probability with which a bottom quark b (charge: -1/3e)
can transform
into a charm quark c (charge: +2/3e), by emitting a
W− boson or by absorbing a W+
boson, is |Vcb|2 = 0.0412
= 0.0016
The
amplitudes with which u, c, t quarks can convert into d,
s, b quarks are given below:
Vud = 0.974
Vus = 0.225
Vub = 0.004
Vcd = 0.225
Vcs = 0.986
Vcb = 0.041
Vtd = 0.008
Vts = 0.040
Vtb = 0.999
Note that
|Vud |2 +
|Vus |2 +
|Vub |2
= 1, |Vcd
|2 +
|Vcs |2 +
|Vcb |2 = 1, |Vtd
|2 +
|Vts |2 +
|Vtb |2 = 1
|Vud
|2 +
|Vus |2 +
|Vub |2
= 1 implies if an up quark u (charge: +2/3)
emits a W+ boson or absorbs a W−
boson, it must
have to transform into a down quark d (charge: -1/3e) or
a strange quark s (charge: -1/3e) or a bottom quark
(charge: -1/3e), as the
sum of the
separate probabilities: |Vud |2
(which is the probability of an up quark to
transform into a down quark), |Vus |2
(which is the probability of an up quark to transform
into a strange quark) and |Vub |2
(which is the probability of an up quark to transform
into a bottom quark)
is one.
|Vud
|2 = 0.9742 = 0.948 is nearly
equal to 1. This implies
most of the time,
in the weak interactions/decays, an up quark will
transform into a down quark by emitting a W+
boson or by absorbing a W− boson.
(u → d)
|Vus|2
= 0.2252
= 0.050 is although near to zero but still has some
value. This implies
sometimes, in the weak interactions/decays, an up quark will
transform into a strange quark by emitting a W+
boson or by absorbing a W− boson.
(u →
s)
|Vub|2
= 0.0042 = 0.000016 is almost zero.
This implies very
rarely, in the weak interactions/decays, an up quark
will transform into a bottom quark by emitting a W+
boson or by absorbing a W− boson.
(u → b)
Similarly, Vcd = 0.225, Vcs =
0.986, Vcb = 0.041 imply
most of the time, in the weak interactions/decays, a charm quark
will transform into a strange quark
(c →
s), sometimes into a down quark
(c → d), and rarely into a
bottom quark (c
→ b) by
emitting a W+ boson or by absorbing a W−
boson.
Similarly, Vtd = 0.008, Vts =
0.040, Vtb = 0.999 imply
most of the time, in the weak interactions/decays, a top quark will
transform into a bottom quark
(t →
b), rarely into a strange quark
(t → s), and very
rarely into a
down quark (t
→ d) by emitting a W+ boson or by absorbing a W−
boson.
Above
values show, the transition between quarks of different
generations are suppressed, which is called
CKM
suppression (after Cabibbo, Kobayashi, and Maskawa). The transition
between the first and the second generation quarks is
less suppressed: e.g (u → s),
but the
transition between the first and the third generation
quarks is highly
suppressed: e.g.
(u → b).
There does not seem any cross-generational mixing among
the leptons,
so electron number, muon number, tauon number should be
separately conserved but neutrino oscillations indicate,
this conservation is not absolute. The transition
between leptons of different generations rarely exists.
Examples of the decays through W bosons:
1. The decay of the
neutron n (udd) into a proton p (uud), an
electron e− and an electron-type antineutrino
νe` (β-decay):
n (udd)
→ p (uud) + e−
+ νe`
This is a
weak decay as the antineutrino is involved in this
decay, which carries only the weak charge.
In this
case, one down
quark d (charge: -1/3e) of the neutron emits a W−
boson (and therefore emits charge: -1e) and
transforms into
an up quark u (charge: -1/3e
+1e = +2/3e) and the neutron transforms into a
proton p (uud).
The W− boson subsequently decays into an
electron e− and an
electron-type antineutrino νe`.
note:
The neutron n (udd) can decay into a proton p (uud), an
electron e− and an electron-type antineutrino
νe` because the mass of the neutron n (939.6
MeV) > mass of the proton p (938.2 MeV) + mass of the
electron e− (0.511 MeV) + mass of the
electron-type antineutrino νe` (~ 0 MeV) )
The neutron n (udd) cannot decay into a proton p (uud)
and a negative pion π− (u`d), since the mass
of the neutron n (939.6 MeV) < mass of the proton p
(938.2 MeV) + mass of the negative pion π−
(139.5 MeV). Thus, this decay is kinematically
forbidden. That is, the W− boson cannot decay
into an antiup quark u` and a down quark d to produce a
negative pion π− (u`d).
2. The decay of the
tauon τ− into an electron e−, an electron-type
antineutrino νe` and a tauon-type neutrino
ντ or into a muon μ−, a muon-type
antineutrino νμ` and a tauon-type neutrino
ντ or into a negative
pion π− (u`d) and a tauon-type neutrino
ντ:
τ−
→
e− + νe`
+ ντ
or
τ− → μ−
+ νμ`+ ντ
or
τ− → π− (u`d)
+
ντ
In this case, the tauon τ− with one unit of
negative electric charge emits that one unit of negative
electric charge by emitting a W− boson and
therefore transforms into the corresponding electrically
neutral lepton that is, a
tauon-type
neutrino ντ.
The W− boson subsequently decays into an
electron e− and the corresponding antineutrino, i.e.
electron-type
antineutrino νe` or decays into a
muon and a
muon-type
antineutrino ν`μ or decays into an
antiup quark u`
(charge: -2/3e) and a
down quark d
(charge: -1/3e), which combine to produce a
negative pion π− (u`d).
note:
In this decay, the lepton number is conserved. The tauon
τ-
having lepton number L = +1 decays into
e−
(L= +1), νe`
(L=-1) and
ντ
(L= +1) or decays into
μ− (L= +1), ν`μ
(L= -1) and
ντ
(L= +1) or decays into
π−
(L= 0)
and ντ (L= +1).
note: The tauon is the only lepton that can decay into the
hadrons (baryons and mesons) because its mass (1777 MeV)
> masses of many hadrons.
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