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Flavour (particle physics)

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Flavour in particle physics

Flavour quantum numbers

Y=B+S+C+B'+T
Q=I_z+Y/2
Q=T_z+Y_W/2
B-L conserved in the standard model

Definition

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 give the same physics, as long as they are orthogonal to each other. In other words, the theory possesses symmetry transformations such as $M\left({u\atop d}\right)$ , where $u$ and $d$ are the two fields, and $M$ is any $2\times2$ matrix with an unit determinant. Such matrices form a Lie group called SU(2). This is an example of flavour symmetry.

This symmetry is global for strong interactions, and gauged for weak interactions.

The term "flavour" was first coined for use in the quark model of hadrons in 1968. A name for the set of quantum numbers related to isospin and hypercharge is said to have been found on the way to lunch by Murray Gell-Mann and Harald Fritzsch when they passed a Baskin-Robbins advertising 31 flavours.

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Weak interactions

Quarks come in 6 flavours:

up u (B=1/3, Q=2/3, I_z=1/2, Y=1/3, Y_W=1/3)
down d (B=1/3, Q=-1/3, I_z=-1/2, Y=1/3, Y_W=1/3)
strange s (B=1/3, Q=-1/3, I=0, Y=-2/3, Y_W=1/3)
charm c (B=1/3, Q=2/3, C=1, Y_W=1/3)
bottom (also called beauty) b (B=1/3, Q=-1/3, B'=-1, Y_W=1/3)
top (was sometimes called truth) t (B=1/3, Q=2/3, T=1, Y_W=1/3)

Here B is the baryon number, Q the electric charge, I is the isospin, Y the hypercharge, S, C, B', T are the flavour charges strangeness, charm, bottomness and topness. Y_W is the weak hypercharge.

Hadrons inherit their flavour quantum number from their valence quarks: this is the basis of the classification called the Eightfold way or the quark model. The following relations hold for hadrons as well as quarks: Y=B+S+C+B'+T and Q=I_z+Y/2.

Leptons occur in six other flavours:

electron e, muon μ, tau τ (L=1, Q=-1, Y_W=-1)
electron neutrino ν_e, muon neutrino ν_μ, tau neutrino ν_τ (L=1, Q=0, Y_W=-1)

Here L is the lepton number.

Antiparticles have the opposite quantum number of the corresponding particle. For example, the positron (which is the anti-electron) has L=-1 and Q=1.

The six flavours of quarks are grouped into three generations: (u,d), (c,s), (t,b). The first member of each generation has charge Q=2/3 and second has Q=-1/3. The leptons are grouped into three generations: (e,ν_e), (μ,&nu_μ), (τ,ν_τ). The first member has Q=-1 and the second has Q=0. The number of generations of quarks and leptons must match in order to cancel the chiral anomaly.

A fermion of a given flavour is an eigenstate of the weak interaction part of the Hamiltonian: it will interact in a definite way with the W⁺, W^- and Z bosons. On the other hand, a fermion of a fixed mass (an eigenstate of the kinetic part of the Hamiltonian) is normally a superposition of various flavours. As a result, the flavour content of a quantum state may change as it propagates freely. The transformation from flavour to mass basis for quarks is given by the so-called Cabbibo-Kobayashi-Maskawa matrix (CKM matrix). The equivalent for neutrinos is the MNS matrix.

The CKM matrix allows for CP violation if there are at least three generations. The connection with the strong CP problem is explored in a separate article.

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Quantum Chromodynamics

(Flavour symmetry is closely related to chiral symmetry. This part of the article is best read along with the one on chirality (physics).)

Quantum chromodynamics contains six flavours of quarks. However, their masses differ. As a result, they are not strictly interchangeable with each other. Two of the flavours, called up and [[[down]], are close to having equal masses, and the theory of these two quarks possesses an approximate SU(2) symmetry. Under some circumstances one can take N_f flavours to be nearly degenerate and obtain an effective SU(N_f) flavour symmetry.

Under some circumstances, the masses of the quarks can be neglected. In that case, each flavour of quarks possesses a chiral symmetry. One can then make flavour transformations independently on the left and right handed quarks. The flavour group is then a chiral group $SU_L(N_f)\times SU_R(N_f)$ .

If all quarks have equal mass, then this chiral symmetry is broken to the vector symmetry of the diagonal flavour group which involves the same transformation to the two helicities of the quarks. Such a reduction of the symmetry is called explicit symmetry breaking. The amount 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 for some reason the vacuum of the theory contains a chiral condensate. This gives rise to an effective mass for the quarks, often identified with the valence quark mass in QCD.

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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. 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.