Flavour (particle physics)
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|Six flavours of leptons|
|Flavour quantum numbers|
|Related quantum numbers|
In particle physics, flavour or flavor refers to a 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 also can be described by some of family symmetries proposed for the quark-lepton generations.
- 1 Quantum numbers
- 2 Flavour symmetry
- 3 Flavour quantum numbers
- 4 Quantum chromodynamics
- 5 History
- 6 See also
- 7 References
- 8 Further reading
- 9 External links
In classical mechanics, forces can only change a particle's momentum. Quantum theory, however, allows forces that can alter other facets of a particle's nature described by quantum numbers.[why?] In particular, the behavior 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 flavor change, or flavor 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, beauty or truth) can characterize the quantum state of quarks (and leptons), 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. For hadrons, these quantum numbers depend on the numbers of constituent quarks of each particular flavour.
All of the various charges discussed above are conserved by the fact that the charge operator can be understood as the generator of a symmetry that commutes with the Hamiltonian. Thus, the eigenvalues of the various charge operators are conserved.
Absolutely conserved flavour quantum numbers are:
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 flavour quantum numbers are violated by the electroweak interactions. Strong interactions conserve all flavours.
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 , 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
All leptons carry a lepton number L = 1. In addition, leptons carry weak isospin, T3, which is −1/ for the three charged leptons (i.e. electron, muon and tau) and +1/ 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, YW, which is −1 for all left-handed leptons. 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).
All quarks carry a baryon number B = 1/. They also all carry weak isospin, T3 = ±1/. 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[clarification needed] the quark flavour quantum numbers (strangeness, charm, topness and bottomness) 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:[clarification needed]
- Isospin (I3), which has value I3 = 1/ for the up quark and I3 = −1/ for the down quark.
- Strangeness (S): Defined as S = −(ns − ns̅), where ns represents the number of strange quarks (
) and ns̅ represents the number of strange antiquarks (
). 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 = (nc − nc̅), where nc represents the number of charm quarks (
) and nc̅ represents the number of charm antiquarks. Is +1 for the charm quark.
- Bottomness (B′): Defined as B′ = −(nb − nb̅), where nb represents the number of bottom quarks (
) and nb̅ represents the number of bottom antiquarks.
- Topness (T): Defined as T = (nt − nt̅), where nt represents the number of top quarks (
) and nt̅ represents the number of top antiquarks. However, because of the extremely short half-life of the top quark (predicted lifetime of only ×10−25 s), by the time it can interact strongly it has already decayed to another flavour of quark (usually to a 5bottom 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 nq − nq̅, 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/Y (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 (| C | = ±1); 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 quark 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 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 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 Cabibbo–Kobayashi–Maskawa matrix (CKM matrix). This matrix is analogous to the PMNS matrix for neutrinos, and defines the strength of flavour changes under 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.
- Flavour symmetry is closely related to chiral symmetry. This part of the article is best read along with the one on chirality.
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[clarification needed], the masses of quarks do not meaningfully contribute to the system's behavior, and can be ignored. 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 SUL(Nf) × SUR(Nf).
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(Nf), which applies the same transformation to both helicities of the quarks. This reduction of symmetry is a form of explicit symmetry breaking. The amount[clarification needed] 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, ΛQCD, 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.
Some of the historical events that led to the development of flavour symmetry are discussed in the article on isospin.
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- 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.
- See table in S. Raby, R. Slanky (1997). "Neutrino Masses: How to add them to the Standard Model" (PDF). Los Alamos Science (25): 64.
- Lessons in Particle Physics Luis Anchordoqui and Francis Halzen, University of Wisconsin, 18th Dec. 2009