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Six flavours of leptons

In particle physics, flavour or flavor refers to the species of an elementary particle. The Standard Model counts six flavours of quarks and six flavours of leptons. They are conventionally parameterized with flavour quantum numbers that are assigned to all subatomic particles. They can also be described by some of the family symmetries proposed for the quark-lepton generations.

Quantum numbers

In classical mechanics, a force acting on a point-like particle can only alter the particle's dynamical state, i.e., its momentum, angular momentum, etc. Quantum field theory, however, allows interactions that can alter other facets of a particle's nature described by non dynamical, discrete quantum numbers. In particular, the action of the weak force is such that it allows the conversion of quantum numbers describing mass and electric charge of both quarks and leptons from one discrete type to another. This is known as a flavour change, or flavour transmutation. Due to their quantum description, flavour states may also undergo quantum superposition.

In atomic physics the principal quantum number of an electron specifies the electron shell in which it resides, which determines the energy level of the whole atom. Analogously, the five flavour quantum numbers (isospin, strangeness, charm, bottomness or topness) can characterize the quantum state of quarks, by the degree to which it exhibits six distinct flavours (u, d, s, c, b, t).

Composite particles can be created from multiple quarks, forming hadrons, such as mesons and baryons, each possessing unique aggregate characteristics, such as different masses, electric charges, and decay modes. A hadron's overall flavour quantum numbers depend on the numbers of constituent quarks of each particular flavour.

Conservation laws

All of the various charges discussed above are conserved by the fact that the corresponding charge operators can be understood as generators of symmetries that commute with the Hamiltonian. Thus, the eigenvalues of the various charge operators are conserved.

Absolutely conserved flavour quantum numbers are:

• 
  electric charge (Q)


• 
  weak isospin (I₃)


• 
  baryon number (B)


• 
  lepton number (L)

In some theories, such as the grand unified theory, the individual baryon and lepton number conservation can be violated, if the difference between them (B − L) is conserved (see chiral anomaly). All other flavour quantum numbers are violated by the electroweak interactions. Strong interactions conserve all flavours.

Flavour symmetry

If there are two or more particles which have identical interactions, then they may be interchanged without affecting the physics. Any (complex) linear combination of these two particles give the same physics, as long as the combinations are orthogonal, or perpendicular, to each other.

In other words, the theory possesses symmetry transformations such as M(ud){displaystyle Mleft({u atop d}right)}, where u and d are the two fields (representing the various generations of leptons and quarks, see below), and M is any 2×2 unitary matrix with a unit determinant. Such matrices form a Lie group called SU(2) (see special unitary group). This is an example of flavour symmetry.

In quantum chromodynamics, flavour is a conserved global symmetry. In the electroweak theory, on the other hand, this symmetry is broken, and flavour changing processes exist, such as quark decay or neutrino oscillations.

Flavour quantum numbers

Leptons

All leptons carry a lepton number L = 1. In addition, leptons carry weak isospin, T₃, which is −1/2 for the three charged leptons (i.e. electron, muon and tau) and +1/2 for the three associated neutrinos. Each doublet of a charged lepton and a neutrino consisting of opposite T₃ are said to constitute one generation of leptons. In addition, one defines a quantum number called weak hypercharge, Y_W, which is −1 for all left-handed leptons.^[1] Weak isospin and weak hypercharge are gauged in the Standard Model.

Leptons may be assigned the six flavour quantum numbers: electron number, muon number, tau number, and corresponding numbers for the neutrinos. These are conserved in strong and electromagnetic interactions, but violated by weak interactions. Therefore, such flavour quantum numbers are not of great use. A separate quantum number for each generation is more useful: electronic lepton number (+1 for electrons and electron neutrinos), muonic lepton number (+1 for muons and muon neutrinos), and tauonic lepton number (+1 for tau leptons and tau neutrinos). However, even these numbers are not absolutely conserved, as neutrinos of different generations can mix; that is, a neutrino of one flavour can transform into another flavour. The strength of such mixings is specified by a matrix called the Pontecorvo–Maki–Nakagawa–Sakata matrix (PMNS matrix).

Quarks

All quarks carry a baryon number B = 1/3. They also all carry weak isospin, T₃ = ±1/2. The positive-T₃ quarks (up, charm, and top quarks) are called up-type quarks and negative-T₃ quarks (down, strange, and bottom quarks) are called down-type quarks. Each doublet of up and down type quarks constitutes one generation of quarks.

For all the quark flavour quantum numbers listed below, the convention is that the flavour charge and the electric charge of a quark have the same sign. Thus any flavour carried by a charged meson has the same sign as its charge. Quarks have the following flavour quantum numbers:

• The third component of isospin (sometimes simply isospin) (I₃), which has value I₃ = 1/2 for the up quark and I₃ = −1/2 for the down quark.


• 
  Strangeness (S): Defined as S = −(n_s − n_s̅), where n_s represents the number of strange quarks (
  s
  ) and n_s̅ represents the number of strange antiquarks (
  s
  ). This quantum number was introduced by Murray Gell-Mann. This definition gives the strange quark a strangeness of −1 for the above-mentioned reason.


• 
  Charm (C): Defined as C = (n_c − n_c̅), where n_c represents the number of charm quarks (
  c
  ) and n_c̅ represents the number of charm antiquarks. The charm quark's value is +1.


• 
  Bottomness (or beauty) (B′): Defined as B′ = −(n_b − n_b̅), where n_b represents the number of bottom quarks (
  b
  ) and n_b̅ represents the number of bottom antiquarks.


• 
  Topness (or truth) (T): Defined as T = (n_t − n_t̅), where n_t represents the number of top quarks (
  t
  ) and n_t̅ represents the number of top antiquarks. However, because of the extremely short half-life of the top quark (predicted lifetime of only 6975500000000000000♠5×10⁻²⁵ s), by the time it can interact strongly it has already decayed to another flavour of quark (usually to a bottom quark). For that reason the top quark doesn't hadronize, that is it never forms any meson or baryon.

These five quantum numbers, together with baryon number (which is not a flavour quantum number), completely specify numbers of all 6 quark flavours separately (as n_q − n_q̅, i.e. an antiquark is counted with the minus sign). They are conserved by both the electromagnetic and strong interactions (but not the weak interaction). From them can be built the derived quantum numbers:

• 
  Hypercharge (Y): Y = B + S + C + B′ + T
  


• 
  Electric charge: Q = I₃ + 1/2Y (see Gell-Mann–Nishijima formula)

The terms "strange" and "strangeness" predate the discovery of the quark, but continued to be used after its discovery for the sake of continuity (i.e. the strangeness of each type of hadron remained the same); strangeness of anti-particles being referred to as +1, and particles as −1 as per the original definition. Strangeness was introduced to explain the rate of decay of newly discovered particles, such as the kaon, and was used in the Eightfold Way classification of hadrons and in subsequent quark models. These quantum numbers are preserved under strong and electromagnetic interactions, but not under weak interactions.

For first-order weak decays, that is processes involving only one quark decay, these quantum numbers (e.g. charm) can only vary by 1, that is, for a decay involving a charmed quark or antiquark either as the incident particle or as a decay byproduct, ΔC = ±1; likewise, for a decay involving a bottom quark or antiquark ΔB′ = ±1. Since first-order processes are more common than second-order processes (involving two quark decays), this can be used as an approximate "selection rule" for weak decays.

A special mixture of quark flavours is an eigenstate of the weak interaction part of the Hamiltonian, so will interact in a particularly simple way with the W bosons (charged weak interactions violate flavor). On the other hand, a fermion of a fixed mass (an eigenstate of the kinetic and strong interaction parts of the Hamiltonian) is an eigenstate of flavour. The transformation from the former basis to the flavor-eigenstate/mass-eigenstate basis for quarks underlies the Cabibbo–Kobayashi–Maskawa matrix (CKM matrix). This matrix is analogous to the PMNS matrix for neutrinos, and quantifies flavour changes under charged weak interactions of quarks.

The CKM matrix allows for CP violation if there are at least three generations.

Antiparticles and hadrons

Flavour quantum numbers are additive. Hence antiparticles have flavour equal in magnitude to the particle but opposite in sign. Hadrons inherit their flavour quantum number from their valence quarks: this is the basis of the classification in the quark model. The relations between the hypercharge, electric charge and other flavour quantum numbers hold for hadrons as well as quarks.

Quantum chromodynamics

Quantum chromodynamics (QCD) contains six flavours of quarks. However, their masses differ and as a result they are not strictly interchangeable with each other. The up and down flavours are close to having equal masses, and the theory of these two quarks possesses an approximate SU(2) symmetry (isospin symmetry).

Chiral symmetry description

Under some circumstances (for instance when the quark masses are much smaller than the chiral symmetry breaking scale of 250MeV), the masses of quarks do not meaningfully contribute to the system's behavior, and can be ignored to zeroth approximation. The simplified behavior of flavour transformations can then be successfully modeled as acting independently on the left- and right-handed parts of each quark field. This approximate description of the flavour symmetry is described by a chiral group SU_L(N_f) × SU_R(N_f).

Vector symmetry description

If all quarks had non-zero but equal masses, then this chiral symmetry is broken to the vector symmetry of the "diagonal flavour group" SU(N_f), which applies the same transformation to both helicities of the quarks. This reduction of symmetry is a form of explicit symmetry breaking. The strength of explicit symmetry breaking is controlled by the current quark masses in QCD.

Even if quarks are massless, chiral flavour symmetry can be spontaneously broken if the vacuum of the theory contains a chiral condensate (as it does in low-energy QCD). This gives rise to an effective mass for the quarks, often identified with the valence quark mass in QCD.

Symmetries of QCD

Analysis of experiments indicate that the current quark masses of the lighter flavours of quarks are much smaller than the QCD scale, Λ_QCD, hence chiral flavour symmetry is a good approximation to QCD for the up, down and strange quarks. The success of chiral perturbation theory and the even more naive chiral models spring from this fact. The valence quark masses extracted from the quark model are much larger than the current quark mass. This indicates that QCD has spontaneous chiral symmetry breaking with the formation of a chiral condensate. Other phases of QCD may break the chiral flavour symmetries in other ways.

History

Some of the historical events that led to the development of flavour symmetry are discussed in the article on isospin, the eightfold way (physics) and chiral symmetry.

See also

• Standard Model (mathematical formulation)


• Cabibbo–Kobayashi–Maskawa matrix


• 
  Strong CP problem and chirality (physics)
  


• 
  Chiral symmetry breaking and quark matter
  


• Quark flavour tagging, such as B-tagging, is an example of particle identification in experimental particle physics.

References

Further reading

• 
  Lessons in Particle Physics Luis Anchordoqui and Francis Halzen, University of Wisconsin, 18th Dec. 2009

External links

• The particle data group.