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Discovery of Fundamental Particles

Basic experimental tools for the study of the fundamental particles, their interactions, and composite particles produced through those interactions are particle accelerators, colliders and detectors.

In particle accelerators, electrons, positrons, protons, antiprotons, etc. are accelerated to the velocity nearly equal to that of light so they acquire extremely high kinetic energy, then these high energy electrons, positrons, protons, antiprotons are made to collide with other high energy electrons, positrons, protons, antiprotons coming from the opposite direction, and collision may result in the production of massive or high energy particles, i.e. those having rest mass much larger than the rest mass of the electron (0.511 mega electron volt or 0.511 MeV) and the proton (938 MeV). Being massive, these particles are highly unstable unlike electrons & protons, and decay rapidly into lighter or stable particles.

These unstable high energy particles are made to pass through detectors, so that they leave their traces in detectors before decaying or their decay products leave their traces in detectors. Detectors help find mass and other properties of these particles.

Collisions of electrons, positrons, protons, antiprotons, and other particles at high energy resulted in the discovery of the fundamental particles, and hundreds of other particles composed of the fundamental particles.

 

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.

 

Centre-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+epair 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+epair (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+epair 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 Wboson. (Through this emission of the W+ boson or absorption of the Wboson, 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 Wboson 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 Wboson (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 Wboson (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+eannihilates 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 Wboson 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 → 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→ 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 +  We 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 μ → W+ + μ  or  νμ +  W→ μ) and tauon ττ → 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+e` → W+ e+  or  νe` +  W+e+) or antimuon μ+ μ` → W+ μ+  or  νμ` +  W+ → μ+)  or antitauon τ+τ` → 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 Wboson 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 Wboson (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 Wboson 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 Wboson 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 Wboson and therefore transforms into the corresponding electrically neutral lepton that is, a tauon-type neutrino ντ.

The Wboson 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.

 

Coupling and Coupling Constants

The coupling constant for the electromagnetic interaction is αe = e2/4π = 1/137 and specifies the strength of the interaction between ‘a particle having electric charge’ and a photon, i.e. the strength with which a particle having electric charge emits or absorbs a photon.

The coupling constant for an interaction/decay is inversely proportional to the square root of the time taken in that interaction/decay provided the mediator of that interaction/decay is massless.

Typically, the decay through the electromagnetic process occurs in 10−19 s and decay through the strong process occurs in 10−23 s.

Thus, if αs and αe are the strong coupling constant and the electromagnetic coupling constant respectively, then αse = (10−19/10−23)1/2 or αse ~ 100.

Thus, the strong coupling constant is about 100 times greater than the electromagnetic coupling constant. This also implies that the strong interaction is about 100 times stronger than the electromagnetic interaction.

Since the coupling constant for the electromagnetic interaction αe = e2/4π = 1/137, the coupling constant for the strong interaction αs = gs2/4π = 100 × αe = 100 × 1/137 ~ 0.7

Here gs is defined as the strong or colour charge on a quark. Leptons do not carry strong charge.

The coupling constant αs for the strong interaction specifies the strength of the strong interaction between ‘a particle having strong or colour charge’ and a gluon, i.e. the strength with which a particle having strong charge emits or absorbs a gluon, the electrically neutral, massless carrier of the strong force analogous to the photon in electromagnetic interactions.

Typically, the decay through the weak process occurs in 10−10 s and the decay through the electromagnetic process occurs in 10−19 s. Thus, the ratio of the strength of the weak interaction and the electromagnetic interaction is (10−19/10−10)1/2 or ~ 10−5.

Thus, weak interaction is about 105 times weaker than the electromagnetic interaction.

At low energies (say, 1 MeV to 100 MeV), the weak charge gw on a fermion is calculated to be 0.65.

Thus, the coupling constant for the weak interaction is αw = gw2/4π = 0.652/4π = 1/29.5

Here gw is defined as the weak charge on a quark or a lepton.

The coupling constant αw for the weak interaction specifies the strength of the weak interaction between ‘a particle having weak charge’ and a W or Z boson, i.e. the strength with which a particle having weak charge emits or absorbs a W or Z boson, the massive carrier of the weak force analogous to the gluon in QCD (quantum chromodynamics) or the photon in QED (quantum electrodynamics).

At low energies, αw/ αe = 137/29.5 = 4.64, i.e. weak coupling constant is 4.64 times greater than the electromagnetic coupling constant. Thus, the weak interaction is intrinsically 4.64 times stronger the electromagnetic interaction.

αw/ αe = 4.64 also implies gw2/ e2 = 4.64 or gw = (4.64)1/2e = 2.15 e that is, the weak charge is greater than the electric charge.

Now, note that whereas, mediators of the strong and the electromagnetic interactions are massless, the mediators (W and Z bosons) of the weak interactions are very massive, which make the weak coupling much weaker, i.e. in addition to weak charge, weak couping also depends upon the massive propagators. Thus, even though, weak charge and weak coupling constant are greater than the electric charge and the electromagnetic coupling constant, the weak interactions are weaker than the electromagnetic interactions. This is so because the propagators of the weak interactions are very massive, which make the weak interactions much weaker than the electromagnetic interactions.

The coupling constants are not in fact constant but depend (logarithmically) on the energy scale, at which measurements are being made, hence coupling constants are called running coupling constants.

The effective coupling for the electromagnetic interaction increases (very slowly) as energy-momentum transfer increases. If we take αe = 1/137 at 1 MeV, then near the Z boson masses (q2 = 91 GeV), the effective value of the electromagnetic coupling constant is αe(q2) = αe (91 GeV) = 1/128.

That is, for electron-positron pair colliding at cms energies of 30 GeV, 50 GeV, 60 GeV, 91 GeV, if the value of electromagnetic coupling is αe (30), αe (50), αe (60), αe (91) respectively, then αe (30)  <  αe (50)  <  αe (60)  <  αe (91).

The effective coupling for the strong interaction decreases as energy-momentum transfer increases. For example, in electron-positron collisions at cms energies of 14 GeV, 22 GeV, 35 GeV, 44 GeV, values of strong coupling constants have been measured as αs (14 GeV) = 0.170, αs (22 GeV) = 0.151, αs (35 GeV) = 0.145, αs (44 GeV) = 0.139 respectively, i.e. αs (14 GeV)  >  αs (22 GeV)  >  αs (35 GeV)  >  αs (44 GeV)

The strong or colour coupling constant αs and therefore the value of strong or colour charge gs decreases with the increase of the energy-momentum transfer (q2).

The electromagnetic coupling constant αe and therefore the value of electric charge e increases with the increase of the energy-momentum transfer (q2).

The weak coupling constant αw and therefore the value of weak charge gw increases with the increase of the energy-momentum transfer (q2) but much more slowly than the electromagnetic coupling constant αe.  

 

Hadrons

As the quarks possess colour charges, they engage in the strong interactions through the exchange of the gluons and the attraction between different quarks due to the strong interactions causes the production of the composite particles, which are called the hadrons. Inside hadrons, a quark emits a gluon, which is absorbed by the other quark and vice versa. This continuous emission and absorption of the gluons inside the hadrons gives rise to the strong force. There are two types of the hadrons: the baryons and the mesons.

The force between the two quarks (which is responsible in the first instance for binding the quarks together to make the hadrons) is the strong force, which is mediated by the exchange of the gluons between the quarks.

 

The binding of quarks into hadrons increase their effective mass by about 340 MeV.

Whereas, the bare masses of u, d, s, c, b, t quarks are 2, 5, 100, 1300, 4200, 173300 MeV respectively, the effective masses are 336, 340, 486, 1550, 4730, 177000 MeV respectively.

Why do the bare quarks have the particular masses they do, no one knows, where all those masses come from.

Isolated quarks cannot be produced because of the quark confinement that is, if the separation between the two quarks increases, their strong interaction causes a quark to radiate a gluon, which subsequently decays into a quark-antiquark pair. Original quarks then combine with them to produce the hadrons instead of being isolated.

 

note:

The gluons themselves carry colour charges and therefore (like the quarks) cannot exist as isolated particles. We can detect gluons only within the hadrons or in colourless combinations with other gluons (glueballs).

The direct gluon-gluon coupling may result in the formation of the glueballs, which are bound states of interacting gluons without any quark.

 

Baryons and Mesons

A baryon QQQ consists of three quarks and a meson QQ` consists of a quark and an antiquark.

Examples of the baryons are protons and neutrons. A proton p (uud) consists of two up quarks and one down quark and a neutron n (udd) consists of one up and two down quarks.

Examples of the mesons are positive pion π+ (ud`), negative pion π(u`d), positive Kaon K+ (us`), negative Kaon K(u`s) and upsilon meson ϒ (bb`).

Each antibaryon is composed of three antiquarks.

note: ud` may be read as u anti-d, u`d as anti-u d, us` as u anti-s, u`s as anti-u s, bb` as b anti-b and so on.

 

As baryons are composed of three valence quarks, i.e. odd number of quarks, they have odd half-integer spin, hence the baryons are fermions.

A baryon will have spin 1/2, if two of the three quarks are spinning (rotating) in the same direction, and hence they have spin in the same direction and the third quark is spinning in the opposite direction, and hence it has spin in the opposite direction (1/2 + 1/2 - 1/2 = 1/2).

A baryon will have spin 3/2, if all the three quarks are spinning (rotating) in the same direction, and hence they have spin in the same direction (1/2 + 1/2 + 1/2 = 3/2).

 

As mesons are composed of quark-antiquark and quark has spin components +1/2, it means mesons have spins +1/2 +1/2, i.e. 0, -1, +1, i.e. integral spin, hence the mesons are bosons. Thus, mesons are not subject to the exclusion principle, which means that there is no limit to the number of mesons that can be squeezed into a small space.

 

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