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| EWT
(Electroweak theory) |
The
Weak Interaction
The weak force (or weak interaction) is a short-range force, and so we do not experience it in every-day
life like we do gravity or electromagnetism. It is short-range because the gauge bosons (see symmetries in
physics) that carry it are massive – they contain quite a lot of mass-energy which means that, according to
Heisenberg’s Uncertainty Principle, these virtual bosons can only exist for a short-time and so they are not
able to travel very far. (Recall that gauge bosons are virtual particles that cause transient violations in
energy-conservation by appearing from nothing but then disappearing again before the violation can be
directly detected!)
Whereas the electromagnetic force is chiefly responsible for the Coulomb force which binds electrons to
nuclei (one caveat will be mentioned below) and the strong force for binding together quarks into nucleons
and (indirectly) nucleons in the nucleus, the weak force governs some less-obvious processes, such as
beta-decay and reactions involving neutrinos. Neutrinos are leptons that carry no electric charge (so they
do not interact or respond to the electromagnetic force) and have almost zero mass. Neutrinos do not
respond to the strong force, but all their reactions are governed by the weak interaction (they only
respond to the weak interaction and to gravity).
There are three generations of leptons, with one pair of leptons, one charged and one neutrino, in each
generation. The generations are numbered in order of increasing mass (of the charged lepton in each
generation) from lightest to heaviest:
The weak force (or weak interaction) is a short-range force, and so we do not experience it in every-day
life like we do gravity or electromagnetism. It is short-range because the gauge bosons (see symmetries in
physics) that carry it are massive – they contain quite a lot of mass-energy which means that, according to
Heisenberg’s Uncertainty Principle, these virtual bosons can only exist for a short-time and so they are not
able to travel very far. (Recall that gauge bosons are virtual particles that cause transient violations in
energy-conservation by appearing from nothing but then disappearing again before the violation can be
directly detected!)
Whereas the electromagnetic force is chiefly responsible for the Coulomb force which binds electrons to
nuclei (one caveat will be mentioned below) and the strong force for binding together quarks into nucleons
and (indirectly) nucleons in the nucleus, the weak force governs some less-obvious processes, such as
beta-decay and reactions involving neutrinos. Neutrinos are leptons that carry no electric charge (so they
do not interact or respond to the electromagnetic force) and have almost zero mass. Neutrinos do not
respond to the strong force, but all their reactions are governed by the weak interaction (they only
respond to the weak interaction and to gravity).
There are three generations of leptons, with one pair of leptons, one charged and one neutrino, in each
generation. The generations are numbered in order of increasing mass (of the charged lepton in each
generation) from lightest to heaviest:
As
usual, the lighter lepton generation containing the electron is
the most important in ordinary matter, since
heavier particles tend to be unstable at ordinary energies and are more important at higher energies.
However, muons (‘heavy electrons’) are an important component of cosmic rays that strike the Earth’s
atmosphere.
The weak interaction is relatively weak, meaning that its reactions are slow on average (that is they occur
with quite low probabilities, though the strength of the interaction increases with energy and these reactions
become more important at high energies). Since neutrinos have very little mass-energy (and so interact with
gravity weakly) their behavior is governed by the weak force, making them very unreactive. Most neutrinos
that strike the Earth will simply pass straight-through it without interacting! Matter is almost transparent to
neutrinos, though in very dense plasmas, a very high neutrino density can have very noticeable effects,
such as in a supernova.
There are actually three gauge bosons that carry the weak interaction, two carry electric charge and are the
so-called charged currents, carried by the W+ and W– bosons (together designated W±) and the other is
the electrically neutral Z0 boson which conveys the so-called neutral currents.
Charged Current reactions
As far as reactions involving leptons are concerned, there are two basic reaction vertices (each comprising 8
basic processes). These two basic vertices are:
heavier particles tend to be unstable at ordinary energies and are more important at higher energies.
However, muons (‘heavy electrons’) are an important component of cosmic rays that strike the Earth’s
atmosphere.
The weak interaction is relatively weak, meaning that its reactions are slow on average (that is they occur
with quite low probabilities, though the strength of the interaction increases with energy and these reactions
become more important at high energies). Since neutrinos have very little mass-energy (and so interact with
gravity weakly) their behavior is governed by the weak force, making them very unreactive. Most neutrinos
that strike the Earth will simply pass straight-through it without interacting! Matter is almost transparent to
neutrinos, though in very dense plasmas, a very high neutrino density can have very noticeable effects,
such as in a supernova.
There are actually three gauge bosons that carry the weak interaction, two carry electric charge and are the
so-called charged currents, carried by the W+ and W– bosons (together designated W±) and the other is
the electrically neutral Z0 boson which conveys the so-called neutral currents.
Charged Current reactions
As far as reactions involving leptons are concerned, there are two basic reaction vertices (each comprising 8
basic processes). These two basic vertices are:
Remember
that a vertex alone does not represent a complete reaction, but
only the creation or annihilation
of the gauge boson, which is virtual and so must be created and destroyed in a reaction in order to meet the
requirements of energy-conservation. We need to combine at least two vertices together to represent a
complete reaction. Remember that in these Feynman diagrams, the arrow points right for a particle, left for
an anti-particle. Each vertex represents 8 processes, for the first vertex these processes are:
of the gauge boson, which is virtual and so must be created and destroyed in a reaction in order to meet the
requirements of energy-conservation. We need to combine at least two vertices together to represent a
complete reaction. Remember that in these Feynman diagrams, the arrow points right for a particle, left for
an anti-particle. Each vertex represents 8 processes, for the first vertex these processes are:
The
second basic vertex can also be expanded into 8 basic processes
which are the same as those shown
above for the first vertex, but with all particles replaced by their anti-particles, including replacing
anti-particles by particles (in a transformation process known as charge conjugation).
Charged current reaction examples:
above for the first vertex, but with all particles replaced by their anti-particles, including replacing
anti-particles by particles (in a transformation process known as charge conjugation).
Charged current reaction examples:
Remember
that time travels from left to right in these diagrams, so note
the order in which the W bosons are
exchanged. We could use a single non-time-ordered diagram with a vertical W± squiggly line to represent
both processes together.
A higher-order process in which more than one W boson is exchanged between the electron and muon, is
shown below. In this reaction the W boson exchange causes the electron and muon to transform alternately
between neutrinos and charged leptons. Higher-order diagrams are less likely to occur and usually add
minor corrections to the rates of reaction and cross-sections (probabilities) of reaction.
exchanged. We could use a single non-time-ordered diagram with a vertical W± squiggly line to represent
both processes together.
A higher-order process in which more than one W boson is exchanged between the electron and muon, is
shown below. In this reaction the W boson exchange causes the electron and muon to transform alternately
between neutrinos and charged leptons. Higher-order diagrams are less likely to occur and usually add
minor corrections to the rates of reaction and cross-sections (probabilities) of reaction.
Beta
Decay
In beta-decay a neutron in the nucleus of an atom (which has too many neutrons, or too high a neutron :
proton ratio, and so is unstable) decays into a proton and a beta-particle (and also an anti-electron
neutrino). The beta-particle is a high-energy electron (emitted from the nucleus) and beta-particles make
beta-radiation. A neutron consists of three quarks (ddu) two down and one up quark and a proton (uud)
consists of one down and two up quarks, so the process is a conversion of a d quark into a u quark:
In beta-decay a neutron in the nucleus of an atom (which has too many neutrons, or too high a neutron :
proton ratio, and so is unstable) decays into a proton and a beta-particle (and also an anti-electron
neutrino). The beta-particle is a high-energy electron (emitted from the nucleus) and beta-particles make
beta-radiation. A neutron consists of three quarks (ddu) two down and one up quark and a proton (uud)
consists of one down and two up quarks, so the process is a conversion of a d quark into a u quark:
Due to colour confinement (see QCD)
quarks can not exist as isolated particles, and in this case the
quarks remain bound to the nucleons (neutron and proton) and so we can also represent this reaction as:
quarks remain bound to the nucleons (neutron and proton) and so we can also represent this reaction as:
Free
neutrons (those outside atoms) are also unstable and decay by the
same process.
Positron decay
In positron decay, a proton in an atom decays into a neutron whilst emitting a positron (anti-electron) as
radiation:
Positron decay
In positron decay, a proton in an atom decays into a neutron whilst emitting a positron (anti-electron) as
radiation:
In
contrast to neutrons, free protons are stable (at least their
half-life is expected to be greater than the age
of the Universe!).
of the Universe!).
Neutral
Current Reactions
The Z0 boson can mediate any reaction that is mediated by a virtual photon. In other words it can mediate
electromagnetic-like interactions! (See QED for electromagnetic interactions). However, at low energies its
contributions are generally small, but at high energies Z0 exchange dominates over photon-exchange.
(Presumably this means that the Z0 boson becomes important in mediating the Coulomb interaction between
electrons and nuclei in heavy atoms in which the electrons may become relativistic, that is they may
approach speeds that are a significant fraction of the speed of light.)
The Z0 boson was originally introduced theoretically, since higher-order contributions to this
electron-positron reaction (that is those reactions involving the exchange of more than one photon) were
found experimentally to have a very small contribution, but in the original theory there contribution was
infinite! Introducing the Z0 boson introduced additional infinities, but these cancelled the first infinite
contributions and produced results that agreed with experiment. There is now much experimental evidence
for the existence of the Z0 boson, whose existence is well-established.
This interplay between electromagnetism and the weak interaction suggested that they might in fact be
aspects of the same unified electroweak force. A quantum theory was developed, much like those of
QCD and QED, which explained the elctroweak field in terms of quantum mechanics and the theory does
indeed match experimental evidence and predicts that the electroweak force is mediated by four gauge
bosons, the photon, the W+, W- and Z0 bosons. The photon mediate electromagnetic interactions which are
now a part of the electroweak force.
Thus, historically, electricity and magnetism were realised to be aspects of the same electromagnetic force
(which is explained by special relativity) and this accounted for light. Later the weak and electromagnetic
forces were found to be part of the same electroweak force, which thus combines at least three distinct
forces. In particular, the electromagnetic and weak forces become unified at high energies. This leads to an
obvious question, can the electroweak and strong force also be unified? The strength of the weak
interaction increases at higher energies, whilst the strength of the strong interaction decreases at higher
energies, suggesting that at some very high energy the two forces may have the same strength and may
behave like the same force, or different facets of the same force. Theories that work on the assumption that
the electroweak and strong force can be unified at high energies are called Unification Theories. These
high energies could exist in a powerful particle accelerator and would have existed very early on the
formation of the Universe, when the Universe was in a very dense, hot and highly energetic state, before it
expanded and cooled. This leads to the interesting possibility that the early universe was either perfectly
symmetric, or nearly so, and that the forces and physics came about as these symmetries broke, creating
the asymmetries that make a complex Universe with life possible.
The Z0 boson can mediate any reaction that is mediated by a virtual photon. In other words it can mediate
electromagnetic-like interactions! (See QED for electromagnetic interactions). However, at low energies its
contributions are generally small, but at high energies Z0 exchange dominates over photon-exchange.
(Presumably this means that the Z0 boson becomes important in mediating the Coulomb interaction between
electrons and nuclei in heavy atoms in which the electrons may become relativistic, that is they may
approach speeds that are a significant fraction of the speed of light.)
The Z0 boson was originally introduced theoretically, since higher-order contributions to this
electron-positron reaction (that is those reactions involving the exchange of more than one photon) were
found experimentally to have a very small contribution, but in the original theory there contribution was
infinite! Introducing the Z0 boson introduced additional infinities, but these cancelled the first infinite
contributions and produced results that agreed with experiment. There is now much experimental evidence
for the existence of the Z0 boson, whose existence is well-established.
This interplay between electromagnetism and the weak interaction suggested that they might in fact be
aspects of the same unified electroweak force. A quantum theory was developed, much like those of
QCD and QED, which explained the elctroweak field in terms of quantum mechanics and the theory does
indeed match experimental evidence and predicts that the electroweak force is mediated by four gauge
bosons, the photon, the W+, W- and Z0 bosons. The photon mediate electromagnetic interactions which are
now a part of the electroweak force.
Thus, historically, electricity and magnetism were realised to be aspects of the same electromagnetic force
(which is explained by special relativity) and this accounted for light. Later the weak and electromagnetic
forces were found to be part of the same electroweak force, which thus combines at least three distinct
forces. In particular, the electromagnetic and weak forces become unified at high energies. This leads to an
obvious question, can the electroweak and strong force also be unified? The strength of the weak
interaction increases at higher energies, whilst the strength of the strong interaction decreases at higher
energies, suggesting that at some very high energy the two forces may have the same strength and may
behave like the same force, or different facets of the same force. Theories that work on the assumption that
the electroweak and strong force can be unified at high energies are called Unification Theories. These
high energies could exist in a powerful particle accelerator and would have existed very early on the
formation of the Universe, when the Universe was in a very dense, hot and highly energetic state, before it
expanded and cooled. This leads to the interesting possibility that the early universe was either perfectly
symmetric, or nearly so, and that the forces and physics came about as these symmetries broke, creating
the asymmetries that make a complex Universe with life possible.
Lepton-Quark
Symmetry
Quarks interact by the strong force, which is why three quarks are so tightly bound together in each proton
and neutron. This strong force involves the exchange of gluons between the quarks, and the gluons
themselves carry strong force charge and so gluons interact one-another by further gluon exchange.
Quarks also carry electric charges, and so also interact by the electromagnetic force. Perhaps, then not
surprisingly when we consider the electroweak unified force, quarks also interact via the weak force. Thus,
in contrast to neutrinos, quarks respond to all four fundamental forces.
Quarks can emit and absorb W+ and W- bosons. Quarks can take part in reactions involving the leptons
(like electrons and neutrinos) in so-called semileptonic processes, and they may also partake in pure
hadronic processes.
Quarks interact by the strong force, which is why three quarks are so tightly bound together in each proton
and neutron. This strong force involves the exchange of gluons between the quarks, and the gluons
themselves carry strong force charge and so gluons interact one-another by further gluon exchange.
Quarks also carry electric charges, and so also interact by the electromagnetic force. Perhaps, then not
surprisingly when we consider the electroweak unified force, quarks also interact via the weak force. Thus,
in contrast to neutrinos, quarks respond to all four fundamental forces.
Quarks can emit and absorb W+ and W- bosons. Quarks can take part in reactions involving the leptons
(like electrons and neutrinos) in so-called semileptonic processes, and they may also partake in pure
hadronic processes.
Hadrons
consist of baryons and mesons. Baryons are made-up of three
quarks, like the proton and neutron
and the lambda particle, whilst mesons consist of a quark and an antiquark, like the pion. Most hadrons are
composed of quarks taken from the first two generations. (Recall that each generations consists of heavier
and less stable particles, so the lower generations are more important in our world). According to
lepton-quark symmetry, the first two generations of leptons have identical weak interactions to the first two
generations of quarks. (We shall not consider the third generation).
and the lambda particle, whilst mesons consist of a quark and an antiquark, like the pion. Most hadrons are
composed of quarks taken from the first two generations. (Recall that each generations consists of heavier
and less stable particles, so the lower generations are more important in our world). According to
lepton-quark symmetry, the first two generations of leptons have identical weak interactions to the first two
generations of quarks. (We shall not consider the third generation).
Lepton-quark
symmetry groups the generations of leptons and
quarks together. It also means that in the reactions we have
looked at, it would be possible to replace an electron-neutrino
with a u-quark, an electron with a d-quark, a mu-neutrino with a
c-quark and a muon with an s-quark. (Note that the particles in
each generation have to be in the correct order for this
correspondence to work). Additionally, the coupling constant at
each vertex is assumed equal. These substitutions give rise to
the following reaction vertices for charged weak-current
reactions involving quarks:
quarks together. It also means that in the reactions we have
looked at, it would be possible to replace an electron-neutrino
with a u-quark, an electron with a d-quark, a mu-neutrino with a
c-quark and a muon with an s-quark. (Note that the particles in
each generation have to be in the correct order for this
correspondence to work). Additionally, the coupling constant at
each vertex is assumed equal. These substitutions give rise to
the following reaction vertices for charged weak-current
reactions involving quarks:
This
symmetry scheme works well for some reactions, like pion-decay,
but is violated by kaon-decay.
From the diagrams below it can be seen that we so far lack the necessary vertex for the kaon decay:
From the diagrams below it can be seen that we so far lack the necessary vertex for the kaon decay:
The pion
decay utilises the vertex for the ud quarks (udW vertex) and their
antiquarks (exchanging one or
more quark for an antiquark is permitted). However, we have no usW vertex.
The existence of the usW vertex is explained by a phenomenon called quark mixing (a theory developed
by Cabibbo). According to this theory d quarks participate in weak interactions as (linear) combinations or
mixtures of the d and s quark. Similarly, s quarks also participate as mixtures of the d and s quark. By
mixture, I do not mean that there is more than one quark present, but rather we have a quark that is a
hybrid between the s and d quarks. This 'schizophrenic' behaviour is quite common in quantum mechanics,
in which a particle can be in a hybrid state, neither in one pure state nor another. This is possible because
quantum mechanics considers wave-particle duality, such that particles are types of wave that behave as
particles. Waves have the property that two or more waves can be added together (superposed) to form a
new hybrid wave with a different pattern. (This is the Principle of Superposition).
Quark mixing allows the 's-quark' in the kaon to undergo d-quark like reactions, since this s-quark can
actually be in the hybrid state (as an s' quark) and possess both s and d quark characteristics!
Quarks can also participate in reactions involving Z0 bosons. The z0 vertices for leptons and quarks are
shown below:
more quark for an antiquark is permitted). However, we have no usW vertex.
The existence of the usW vertex is explained by a phenomenon called quark mixing (a theory developed
by Cabibbo). According to this theory d quarks participate in weak interactions as (linear) combinations or
mixtures of the d and s quark. Similarly, s quarks also participate as mixtures of the d and s quark. By
mixture, I do not mean that there is more than one quark present, but rather we have a quark that is a
hybrid between the s and d quarks. This 'schizophrenic' behaviour is quite common in quantum mechanics,
in which a particle can be in a hybrid state, neither in one pure state nor another. This is possible because
quantum mechanics considers wave-particle duality, such that particles are types of wave that behave as
particles. Waves have the property that two or more waves can be added together (superposed) to form a
new hybrid wave with a different pattern. (This is the Principle of Superposition).
Quark mixing allows the 's-quark' in the kaon to undergo d-quark like reactions, since this s-quark can
actually be in the hybrid state (as an s' quark) and possess both s and d quark characteristics!
Quarks can also participate in reactions involving Z0 bosons. The z0 vertices for leptons and quarks are
shown below:
Again
with quark mixing, the d and s quarks can operate as mixtures of
one-another. (S quarks participate
in several strange reactions, and so it seems appropriate to call them strange quarks). Recall also that
photons can substitute for Z0 bosons (and vice versa) in reactions between quarks and charged leptons
(electron, muon, tauon) (the 2nd and 3rd vertex above) but NOT in reactions between neutrinos only (the
first vertex above) since neutrinos do not experience the electromagnetic force (they have no electric
charge).
Weak Reactions and Physical Symmetries
The importance of symmetry in physics has been discussed in another article. This article discusses parity
(P) and charge conjugation (C) and their conservation. Weak interactions, however, violate parity
conservation and also charge conjugation.
Parity is a transformation in which all x-coordinates in a system are replaced by negative x-coordinates
and all y-coordinates by negative y-coordinates, and likewise negative coordinates become positive (all
coordinates are inverted through the origin). If parity is conserved, then such a change leaves the
fundamental physics unchanged. This was expected to be the case in all physical reactions, after all a ball
moves according to the same physics regardless of what direction you throw it in! This is known as
P-invariance. The parity transformation may seem abstract, since we can not use a device to simply
invert a system through its origin, however, we could set-up a system, ascertain its physics and then set
the system up again with the directions reversed and expect the physics to be identical. With a
mathematical parity transform we could simply convert our equations for the first system and expect them
to work for the inverted system. This is generally the case, however, a little thought will reveal obvious
exceptions. Consider a corkscrew, it has to be turned in a particular direction in order to work, and a
left-handed corkscrew does not work the same as a right-handed one, because they have to be turned in
opposite directions!
Leptons have a similar property to our corkscrews. They have spin, the electron has a spin, s = 1/2. The
spin is a measure of the electron's intrinsic angular momentum. Angular momentum is due to circular
motion. A ball may have angular momentum because it is moving in a circle (orbital angular momentum)
but it may have intrinsic angular momentum of its own if it also spins on its axis. Quantum mechanical spin
is similar though not identical to the spin of a ball, it is analogous, but we must not assume that an electron
is really a tiny spinning ball, because it is not! The physics of atoms and particles is simply different to that
of large objects. (Of course, both are described by the same equations, since they belong to the same
universe, but these equations have size-dependent effects built-in to them). The spin of a particle can be
depicted in an arrow which points up for a particle spinning to the right (anticlockwise) and down for a
particle spinning to the left (clockwise):
in several strange reactions, and so it seems appropriate to call them strange quarks). Recall also that
photons can substitute for Z0 bosons (and vice versa) in reactions between quarks and charged leptons
(electron, muon, tauon) (the 2nd and 3rd vertex above) but NOT in reactions between neutrinos only (the
first vertex above) since neutrinos do not experience the electromagnetic force (they have no electric
charge).
Weak Reactions and Physical Symmetries
The importance of symmetry in physics has been discussed in another article. This article discusses parity
(P) and charge conjugation (C) and their conservation. Weak interactions, however, violate parity
conservation and also charge conjugation.
Parity is a transformation in which all x-coordinates in a system are replaced by negative x-coordinates
and all y-coordinates by negative y-coordinates, and likewise negative coordinates become positive (all
coordinates are inverted through the origin). If parity is conserved, then such a change leaves the
fundamental physics unchanged. This was expected to be the case in all physical reactions, after all a ball
moves according to the same physics regardless of what direction you throw it in! This is known as
P-invariance. The parity transformation may seem abstract, since we can not use a device to simply
invert a system through its origin, however, we could set-up a system, ascertain its physics and then set
the system up again with the directions reversed and expect the physics to be identical. With a
mathematical parity transform we could simply convert our equations for the first system and expect them
to work for the inverted system. This is generally the case, however, a little thought will reveal obvious
exceptions. Consider a corkscrew, it has to be turned in a particular direction in order to work, and a
left-handed corkscrew does not work the same as a right-handed one, because they have to be turned in
opposite directions!
Leptons have a similar property to our corkscrews. They have spin, the electron has a spin, s = 1/2. The
spin is a measure of the electron's intrinsic angular momentum. Angular momentum is due to circular
motion. A ball may have angular momentum because it is moving in a circle (orbital angular momentum)
but it may have intrinsic angular momentum of its own if it also spins on its axis. Quantum mechanical spin
is similar though not identical to the spin of a ball, it is analogous, but we must not assume that an electron
is really a tiny spinning ball, because it is not! The physics of atoms and particles is simply different to that
of large objects. (Of course, both are described by the same equations, since they belong to the same
universe, but these equations have size-dependent effects built-in to them). The spin of a particle can be
depicted in an arrow which points up for a particle spinning to the right (anticlockwise) and down for a
particle spinning to the left (clockwise):
If you
imagine an electron spinning (I am not saying that it really does
spin, but it does something similar)
whilst it moves through space, then it is essentially tracing out a helix. We say that the electron has
helicity. Furthermore, there are two possible helicities an electron can have, right-handed or left-handed.
Just as with our corkscrew, we might imagine that reversing the spatial coordinates in a parity transform
will change the behaviour of the electron in an important way, and indeed it does. Whole atoms may have
spin if they have an unpaired electron in their outermost (valency) shell, in which case the electron
imparts its spin to the atom. Similarly, nuclei can possess spin if they have an unpaired nucleon. (When
electrons and nucleons pair they tend to spin in opposite directions and the effects of their spins then
cancel). The important point is that some nuclei also have spin.
Neutrinos also have spin, s = 1/2, and if the spin is positive (spin-up in our diagram above) or
right-handed then the neutrino is an antineutrino.
Neutrinos have left-handed (LH) spin/helicity, antineutrinos right-handed (RH).
Charge conjugation is when the electric charges of particles are reversed in sign, positive charges
become negative and positive charges become negative. More generally, it is replacing all the particles in
a system with their anti-particles and all the anti-particles with their particles. Again, we have no magic
wand with which to do this in reality, but we could set-up two systems, one with particles and one with
anti-particles, and then the physics of these systems would be related by charge conjugation, or a C
transform. For a long time, since antimatter and matter were seen as exact mirror images of one-another,
it was assumed that a C transform would have no effect on the physics, the so-called C-invariance,
however, once again helicity is an exception. Antiparticles have helicities opposite in polarity to their
corresponding particles, so once again the sense of our corkscrew changes.
The weak interaction violates P and C invariance because of the spin-dependence of its
reactions.
An historic experiment demonstrated C and P invariance violation in the beta-decay of cobalt-60 nuclei, a
weak-force reaction:
whilst it moves through space, then it is essentially tracing out a helix. We say that the electron has
helicity. Furthermore, there are two possible helicities an electron can have, right-handed or left-handed.
Just as with our corkscrew, we might imagine that reversing the spatial coordinates in a parity transform
will change the behaviour of the electron in an important way, and indeed it does. Whole atoms may have
spin if they have an unpaired electron in their outermost (valency) shell, in which case the electron
imparts its spin to the atom. Similarly, nuclei can possess spin if they have an unpaired nucleon. (When
electrons and nucleons pair they tend to spin in opposite directions and the effects of their spins then
cancel). The important point is that some nuclei also have spin.
Neutrinos also have spin, s = 1/2, and if the spin is positive (spin-up in our diagram above) or
right-handed then the neutrino is an antineutrino.
Neutrinos have left-handed (LH) spin/helicity, antineutrinos right-handed (RH).
Charge conjugation is when the electric charges of particles are reversed in sign, positive charges
become negative and positive charges become negative. More generally, it is replacing all the particles in
a system with their anti-particles and all the anti-particles with their particles. Again, we have no magic
wand with which to do this in reality, but we could set-up two systems, one with particles and one with
anti-particles, and then the physics of these systems would be related by charge conjugation, or a C
transform. For a long time, since antimatter and matter were seen as exact mirror images of one-another,
it was assumed that a C transform would have no effect on the physics, the so-called C-invariance,
however, once again helicity is an exception. Antiparticles have helicities opposite in polarity to their
corresponding particles, so once again the sense of our corkscrew changes.
The weak interaction violates P and C invariance because of the spin-dependence of its
reactions.
An historic experiment demonstrated C and P invariance violation in the beta-decay of cobalt-60 nuclei, a
weak-force reaction:
Where
the asterisk indicates that the nickel nucleus is produced in an
energized or excited state (and can
de-excite by emitting gamma-radiation). In this reaction, a neutron in the nucleus of cobalt has decayed
into a proton, a high-energy electron (beta-particle) and an anti-electron neutrino. When the cobalt was
chilled to 0.01K the thermal motions of the atoms became so feeble that magnetic forces became
dominant and the atoms all aligned with the direction of a magnetic field, that is their spins aligned, since a
spinning electric charge acts like a magnetic dipole (like a bar magnet) and will align in a magnetic field. At
higher temperatures, the atoms jostle about so much that their spins become randomly aligned. By
measuring the angles at which the electrons were emitted, from this cold cobalt, it was found that the
electrons were emitted in the 'forward' direction. A P transform would reverse the direction of electron
emission, but parity does not transform angular momentum including spin, so the spin vector is left
unchanged (only linear momentum gets reversed). This is illustrated below:
de-excite by emitting gamma-radiation). In this reaction, a neutron in the nucleus of cobalt has decayed
into a proton, a high-energy electron (beta-particle) and an anti-electron neutrino. When the cobalt was
chilled to 0.01K the thermal motions of the atoms became so feeble that magnetic forces became
dominant and the atoms all aligned with the direction of a magnetic field, that is their spins aligned, since a
spinning electric charge acts like a magnetic dipole (like a bar magnet) and will align in a magnetic field. At
higher temperatures, the atoms jostle about so much that their spins become randomly aligned. By
measuring the angles at which the electrons were emitted, from this cold cobalt, it was found that the
electrons were emitted in the 'forward' direction. A P transform would reverse the direction of electron
emission, but parity does not transform angular momentum including spin, so the spin vector is left
unchanged (only linear momentum gets reversed). This is illustrated below:
Notice
that parity must be violated, since it would lead to the unnatural
result that the electrons are
emitted backwards, which they are not as beta-decay is a weak reaction and us spin-dependent.
Similarly for neutrinos, a P transform is unphysical, since it would change a normal left-handed neutrino
into a right-handed neutrino, and neutrinos are not normally right-handed. Similarly, charge conjugation
produces a left-handed antineutrino, but neutrinos are generally right-handed! One operation is still
physically feasible, however, and that is applying a parity trasnform and a charge conjugation transform
(in either order), a CP transform. This successfully converts a neutrino into an antineutrino:
emitted backwards, which they are not as beta-decay is a weak reaction and us spin-dependent.
Similarly for neutrinos, a P transform is unphysical, since it would change a normal left-handed neutrino
into a right-handed neutrino, and neutrinos are not normally right-handed. Similarly, charge conjugation
produces a left-handed antineutrino, but neutrinos are generally right-handed! One operation is still
physically feasible, however, and that is applying a parity trasnform and a charge conjugation transform
(in either order), a CP transform. This successfully converts a neutrino into an antineutrino:
CP
invariance
is almost exactly conserved and very few reactions are known to
violate it. For example it
is satisfied by muon decays. Decay of the negative muon is very similar to that of the positive muon,
except all particles are swapped for their antiparticles:
is satisfied by muon decays. Decay of the negative muon is very similar to that of the positive muon,
except all particles are swapped for their antiparticles:
Clearly
such a reaction will violate charge conjugation and parity, when
only one of these transforms is
applied, for one thing it would produce forbidden neutrino/antineutrino polarisations. However, these two
reactions are related by CP symmetry, one can be converted into the other by a CP transform. We would
expect the rate of each reaction to be the same, since CP transformation should not change this
physics, and indeed the two rates are the same!
Neutral kaons (K0) are a different story. These particles allow very accurate measurements of the
effects of CP transformation and there is found to be a slight violation of CP invariance. However, this
apparent violation can be explained (at least in large part) by kaon mixing, in which each state of the
kaon is actually a mixture (linear combination) of the neutral kaon and neutral anti-kaon particle states.
This means that the kaon and anti-kaon particles can not be distinguished, but both blend into a hybrid
state (as if the particle was part matter and part antimatter or oscillated between the two). Experiment
actually reveals that there are two such neutral kaon states, one is a short kaon, as it is very short-lived
and typically decays in about one ten billionth of a second (half-life) the other is a long kaon, which
decays in about one ten millionth of a second (half-life) and so is longer lived. Why kaons? Well this
strange behaviour arises from the fact that the neutral kaon is a meson consisting of a d quark and an
anti-s quark. Yes, it is those strange quarks again!
This allows reactions forbidden by the need for CP invariance (that is reactions bound by the need for a
CP transform to leave the physics unchanged) to proceed at very low rates (with very low probabilities).
The kaon works around this forbidden 'barrier' by using its anti-kaon character to complete reactions
without violating CP conservation! Such reactions include the decay of the mixed kaon states into pairs
or triplets of pions. Some of these reactions are allowed and other forbidden, according to CP
invariance. However, experiment has shown that some of the forbidden reactions occur at low rates,
which can be explained by assuming the appropriate mixing of kaon and anti-kaon character in each of
the two neutral kaon types (short and long) and so CP only appears to be violated. However, a very
small actual violation of CP invariance is still not ruled out, however combining CP transforms with a
time-reversal transform (T) in which the direction of time is reversed (or in real terms a reaction is set-up
to run backwards without actually reversing time!) to form a combined CPT transformation (in any order)
is thought to be inviolable, and is referred to as CPT invariance.
applied, for one thing it would produce forbidden neutrino/antineutrino polarisations. However, these two
reactions are related by CP symmetry, one can be converted into the other by a CP transform. We would
expect the rate of each reaction to be the same, since CP transformation should not change this
physics, and indeed the two rates are the same!
Neutral kaons (K0) are a different story. These particles allow very accurate measurements of the
effects of CP transformation and there is found to be a slight violation of CP invariance. However, this
apparent violation can be explained (at least in large part) by kaon mixing, in which each state of the
kaon is actually a mixture (linear combination) of the neutral kaon and neutral anti-kaon particle states.
This means that the kaon and anti-kaon particles can not be distinguished, but both blend into a hybrid
state (as if the particle was part matter and part antimatter or oscillated between the two). Experiment
actually reveals that there are two such neutral kaon states, one is a short kaon, as it is very short-lived
and typically decays in about one ten billionth of a second (half-life) the other is a long kaon, which
decays in about one ten millionth of a second (half-life) and so is longer lived. Why kaons? Well this
strange behaviour arises from the fact that the neutral kaon is a meson consisting of a d quark and an
anti-s quark. Yes, it is those strange quarks again!
This allows reactions forbidden by the need for CP invariance (that is reactions bound by the need for a
CP transform to leave the physics unchanged) to proceed at very low rates (with very low probabilities).
The kaon works around this forbidden 'barrier' by using its anti-kaon character to complete reactions
without violating CP conservation! Such reactions include the decay of the mixed kaon states into pairs
or triplets of pions. Some of these reactions are allowed and other forbidden, according to CP
invariance. However, experiment has shown that some of the forbidden reactions occur at low rates,
which can be explained by assuming the appropriate mixing of kaon and anti-kaon character in each of
the two neutral kaon types (short and long) and so CP only appears to be violated. However, a very
small actual violation of CP invariance is still not ruled out, however combining CP transforms with a
time-reversal transform (T) in which the direction of time is reversed (or in real terms a reaction is set-up
to run backwards without actually reversing time!) to form a combined CPT transformation (in any order)
is thought to be inviolable, and is referred to as CPT invariance.
Strangeness
Strangeness, S, is a measure of the s-quark content of a particle or group of particles and is given a
value of +1 for every anti-s quark present and -1 for every s quark. Thus, S = +1 for the neutral kaon
(d, anti-s) and -1 for the neutral anti-kaon (s, anti-d).
Strangeness is conserved (left unchanged) by the strong reaction but strangeness is not conserved (its
value can change) in weak interactions.
These kaons can react with baryons in matter, namely protons and neutrons, by strong-force mediated
reactions, producing a mixture of mesons and baryons, for example:
Strangeness, S, is a measure of the s-quark content of a particle or group of particles and is given a
value of +1 for every anti-s quark present and -1 for every s quark. Thus, S = +1 for the neutral kaon
(d, anti-s) and -1 for the neutral anti-kaon (s, anti-d).
Strangeness is conserved (left unchanged) by the strong reaction but strangeness is not conserved (its
value can change) in weak interactions.
These kaons can react with baryons in matter, namely protons and neutrons, by strong-force mediated
reactions, producing a mixture of mesons and baryons, for example:
The
quark composition and strangeness of each particle is shown
beneath each equation. Notice
that both reactions conserve strangeness: the total strangeness on the left-hand side of the
equation equals the total strangeness on the right=hand side. Baryons have a strangeness of either
zero (when they contain no s quarks, like the proton) or less than one (when they contain one or
more s quarks) but never greater than 1. This means that there are more strangeness-conserving
reactions between anti-kaons and protons and neutrons than there are for neutral kaons. (For
example a lambda baryon with S = +1 can be produced). This means that anti-kaons react more
readily with matter than kaons do, since they have more reaction pathways that conserve
strangeness.
If we produced a beam of kaons, containing an equal number of K-short and K-long particles, then
as the K-short decay more rapidly, after some time the beam will be found to contain mostly K-long
particles only. Interestingly, when we pass such a K-long beam through a slab of matter, in addition
to the beam losing intensity (as expected as some of the kaons collide with atoms and are removed
from the beam) we find that the emergent beams is once again a mixture of K-short and K-long
particles! It has become enriched in K-short particles in a process called K-short regeneration.
This regeneration occurs because kaon strong-interactions with matter conserve strangeness, but
since the kaons are mixed states of so-much kaon and so-much anti-kaon, they do not have definite
values of strangeness! It can be shown that K-short regeneration occurs because the anti-kaon
character of the kaon beam is absorbed more than the kaon character (since anti-kaons react more
with matter) and it works-out that K-long character is more readily absorbed by matter.
Another interesting property of kaons is that no matter what their initial state or purity, e.g.
supposing a reaction produces anti-kaon particles with S = -1, they acquire mixed character over
time and cease to have a definite value of strangeness, but instead they oscillate with the amount of
S = +1 character and the amount of S = -1 character varying over time. These are called
strangeness oscillations. Interestingly they enable the masses of kaons and anti-kaons to be
measured with extreme accuracy. This is an important test of the theory that anti-particles have
exactly the same mass as particles. These measurements confirm this, with the masses of the
kaon and anti-kaon being equal to very high precision. Furthermore, this mass identity is predicted
by CPT invariance, which thus far appears to hold (pending more exact measurements).
that both reactions conserve strangeness: the total strangeness on the left-hand side of the
equation equals the total strangeness on the right=hand side. Baryons have a strangeness of either
zero (when they contain no s quarks, like the proton) or less than one (when they contain one or
more s quarks) but never greater than 1. This means that there are more strangeness-conserving
reactions between anti-kaons and protons and neutrons than there are for neutral kaons. (For
example a lambda baryon with S = +1 can be produced). This means that anti-kaons react more
readily with matter than kaons do, since they have more reaction pathways that conserve
strangeness.
If we produced a beam of kaons, containing an equal number of K-short and K-long particles, then
as the K-short decay more rapidly, after some time the beam will be found to contain mostly K-long
particles only. Interestingly, when we pass such a K-long beam through a slab of matter, in addition
to the beam losing intensity (as expected as some of the kaons collide with atoms and are removed
from the beam) we find that the emergent beams is once again a mixture of K-short and K-long
particles! It has become enriched in K-short particles in a process called K-short regeneration.
This regeneration occurs because kaon strong-interactions with matter conserve strangeness, but
since the kaons are mixed states of so-much kaon and so-much anti-kaon, they do not have definite
values of strangeness! It can be shown that K-short regeneration occurs because the anti-kaon
character of the kaon beam is absorbed more than the kaon character (since anti-kaons react more
with matter) and it works-out that K-long character is more readily absorbed by matter.
Another interesting property of kaons is that no matter what their initial state or purity, e.g.
supposing a reaction produces anti-kaon particles with S = -1, they acquire mixed character over
time and cease to have a definite value of strangeness, but instead they oscillate with the amount of
S = +1 character and the amount of S = -1 character varying over time. These are called
strangeness oscillations. Interestingly they enable the masses of kaons and anti-kaons to be
measured with extreme accuracy. This is an important test of the theory that anti-particles have
exactly the same mass as particles. These measurements confirm this, with the masses of the
kaon and anti-kaon being equal to very high precision. Furthermore, this mass identity is predicted
by CPT invariance, which thus far appears to hold (pending more exact measurements).
Beta-decay
of certain isotopes, which produces beta-radiation consisting of
beta-particles
(which are energetic electrons emitted from the nucleus) is governed by the weak force (weak
interaction).
(which are energetic electrons emitted from the nucleus) is governed by the weak force (weak
interaction).