Flavour

In particle physics, flavour or flavor is a quantum number of elementary particles. In quantum chromodynamics flavor is a global symmetry. In the electroweak theory, on the other hand, this symmetry is broken, and flavor changing processes exist.

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 particles give the same physics, as long as they are orthogonal or perpendicular to each other. In other words, the theory possesses symmetry transformations such as , where u and d are the two fields, 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 flavor symmetry.

The term "flavor" was first coined for use in the quark model of hadrons in 1968.

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

Quarks

All quarks carry a baryon number B = ¹⁄₃. In addition they carry weak isospin, T₃ = ±¹⁄₂. The positive T₃ quarks (up, charm, and top quarks) are called up-type quarks and negative T₃ ones are called down-type quarks. Each doublet of up and down type quarks constitutes one generation of quarks.

Quarks have the following flavor quantum numbers:

• Isospin which has value I₃ = ¹⁄₂ for the up quark and value I₃ = −¹⁄₂ for the down quark.
• Strangeness (S): a quantum number introduced by Murray Gell-Mann. The strange quark is defined to have strangeness −1.
• Charm (C) number which is +1 for the charm quark.
• Bottomness (B) number which is −1 bottom quark.
• Topness (T) quantum number which is +1 for the top quark.

These are useful quantum numbers since they are conserved by both the electromagnetic and strong interactions (but not the weak interaction). Out of 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)

A quark of a given flavor is an eigenstate of the weak interaction part of the Hamiltonian: it will interact in a definite way with the W and Z bosons. On the other hand, a fermion of a fixed mass (an eigenstate of the kinetic and strong interaction parts of the Hamiltonian) is normally a superposition of various flavors. As a result, the flavor content of a quantum state may change as it propagates freely. The transformation from flavor to mass basis for quarks is given by the so-called Cabibbo–Kobayashi–Maskawa matrix (CKM matrix). This matrix is analagous to the MNS matrix for quarks, and defines the strength of flavor changes under weak interactions of quarks.

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

Antiparticles and hadrons

Flavor quantum numbers are additive. Hence antiparticles have flavor equal in magnitude to the particle but opposite in sign. Hadrons inherit their flavor 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 flavor quantum numbers hold for hadrons as well as quarks.

Quantum chromodynamics

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

Quantum chromodynamics contains six flavors of quarks. However, their masses differ. As a result, they are not strictly interchangeable with each other. The up and down flavors are close to having equal masses, and the theory of these two quarks possesses an approximate SU(2) symmetry (isospin symmetry). Under some circumstances one can take N_f flavors to have the same masses and obtain an effective SU(N_f) flavor symmetry.

Under some circumstances, the masses of the quarks can be neglected entirely. In that case, each flavor of quark possesses a chiral symmetry. One can then make flavor transformations independently on the left- and right-handed parts of each quark field. The flavor group is then a chiral group .

If all quarks have equal mass, then this chiral symmetry is broken to the vector symmetry of the "diagonal flavor group" which applies the same transformation to both 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 flavor 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 flavors of quarks are much smaller than the QCD scale, Λ_QCD, hence chiral flavor 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 flavor symmetries in other ways.

Conservation laws

Absolutely conserved flavor quantum numbers are

• electric charge (Q)
• baryon number (B)
• lepton number (L)

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

History

Some of the historical events that lead to the development of flavor symmetry are discussed in the article on isospin.

See also

• Field theoretical formulation of the standard model
• Weak interactions, flavor changing processes and CP violation
• Quantum chromodynamics, strong CP problem and chirality (physics)
• Chiral symmetry breaking and quark matter
• Quarks, leptons and hadrons.
• Quark flavor tagging, such as B-tagging, is an example of particle identification in experimental particle physics.

External links

• The particle data group.

References

1. ^ See table in Raby, S.; Slanky, R. (1997). "Neutrino Masses: How to add them to the Standard Model". Los Alamos Science (Los Alamos Laboratory Neutrino Book) (25): 64. http://theta13.lbl.gov/neutrino_book_pdfs/04_03_Neutrino_Masses.pdf.