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, 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
(
u
d
)
{\displaystyle M\left({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,
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).
=== 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 = −(ns − ns̅),
where ns represents the number of strange
quarks (s) and ns̅ 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 = (nc − nc̅), where
nc represents the number of charm quarks (c)
and nc̅ represents the number of charm antiquarks.
The charm quark's value is +1.
Bottomness (or beauty) (B′): Defined as
B′ = −(nb − nb̅), where nb represents
the number of bottom quarks (b) and nb̅ represents
the number of bottom antiquarks.
Topness (or truth) (T): Defined as T = (nt
− nt̅), where nt represents the number
of top quarks (t) and nt̅ represents the
number of top antiquarks. However, because
of the extremely short half-life of the top
quark (predicted lifetime of only 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 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 + 1/2Y (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 ==
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, 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 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.
== History ==
Some of the historical events that led to
the development of flavour symmetry are discussed
in the article on isospin.
== 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
