Flavour (particle physics)

Encyclopedia

In particle physics

,

of elementary particle

s. In quantum chromodynamics

, flavour is a 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 oscillation

s.

. Such matrices form a Lie group

called SU(2) (see special unitary group

). This is an example of flavour symmetry.

The term "flavour" was first coined for use in the quark model

of hadron

s in 1968.

s carry a lepton number

,

, muon

and tau) and + for the three associated neutrino

s. Each doublet of a charged lepton and a neutrino consisting of opposite

of leptons. In addition, one defines a quantum number called weak hypercharge

,

leptons. Weak isospin and weak hypercharge are gauged

in the Standard Model

.

Leptons may be assigned the six

. The strength of such mixings is specified by a matrix called the Pontecorvo–Maki–Nakagawa–Sakata matrix

(PMNS matrix).

s carry a baryon number

of quarks.

Quarks have the following flavour quantum numbers:

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:

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.

s have flavour equal in magnitude to the particle but opposite in sign. Hadron

s 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 quark

s.

Quantum chromodynamics

(QCD) contains six flavours of quarkA quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

s. However, their masses differ. 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). Under some circumstances one can take

Under some circumstances, the masses of the quarks can be neglected entirely. In that case, each flavour of quark possesses a chiral symmetryA chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...

. One can then make flavour transformations independently on the left- and right-handed parts of each quark field. The flavour group is then a chiral group .

If all quarks have equal mass, then this chiral symmetry is broken to the

es 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.

for the up, down and strange quarks. The success of chiral perturbation theory

and the even more naive chiral model

s 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.

In some theories, the individual baryon and lepton number conservation can be violated, if the difference between them (

). All other flavour quantum numbers are violated by the electroweak interaction

s. Strong interaction

s conserve all flavours.

.

Particle physics

Particle physics is a branch of physics that studies the existence and interactions of particles that are the constituents of what is usually referred to as matter or radiation. In current understanding, particles are excitations of quantum fields and interact following their dynamics...

,

**flavour**or**flavor**is a quantum numberQuantum number

Quantum numbers describe values of conserved quantities in the dynamics of the quantum system. Perhaps the most peculiar aspect of quantum mechanics is the quantization of observable quantities. This is distinguished from classical mechanics where the values can range continuously...

of elementary particle

Elementary particle

In particle physics, an elementary particle or fundamental particle is a particle not known to have substructure; that is, it is not known to be made up of smaller particles. If an elementary particle truly has no substructure, then it is one of the basic building blocks of the universe from which...

s. In quantum chromodynamics

Quantum chromodynamics

In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

, flavour is a global symmetry

Global symmetry

A global symmetry is a symmetry that holds at all points in the spacetime under consideration, as opposed to a local symmetry which varies from point to point.Global symmetries require conservation laws, but not forces, in physics.-See also:...

. In the electroweak theory, on the other hand, this symmetry is broken, and flavour changing processes exist, such as quark decay

Quark

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

or neutrino oscillation

Neutrino oscillation

Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvowhereby a neutrino created with a specific lepton flavor can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates...

s.

## Overview

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 unitary matrix with a unit determinantDeterminant

In linear algebra, the determinant is a value associated with a square matrix. It can be computed from the entries of the matrix by a specific arithmetic expression, while other ways to determine its value exist as well...

. Such matrices form a Lie group

Lie group

In mathematics, a Lie group is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure...

called SU(2) (see special unitary group

Special unitary group

The special unitary group of degree n, denoted SU, is the group of n×n unitary matrices with determinant 1. The group operation is that of matrix multiplication...

). This is an example of flavour symmetry.

The term "flavour" was first coined for use in the quark model

Quark model

In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons....

of hadron

Hadron

In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

s in 1968.

### Leptons

All leptonLepton

A lepton is an elementary particle and a fundamental constituent of matter. The best known of all leptons is the electron which governs nearly all of chemistry as it is found in atoms and is directly tied to all chemical properties. Two main classes of leptons exist: charged leptons , and neutral...

s carry a lepton number

Lepton number

In particle physics, the lepton number is the number of leptons minus the number of antileptons.In equation form,so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0...

*L*= 1. In addition, leptons carry weak isospinWeak isospin

In particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol T or I with the third component written as Tz, T3, Iz or I3...

,

*T*_{3}, which is − for the three charged leptons (i.e. electronElectron

The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...

, muon

Muon

The muon |mu]] used to represent it) is an elementary particle similar to the electron, with a unitary negative electric charge and a spin of ½. Together with the electron, the tau, and the three neutrinos, it is classified as a lepton...

and tau) and + for the three associated neutrino

Neutrino

A neutrino is an electrically neutral, weakly interacting elementary subatomic particle with a half-integer spin, chirality and a disputed but small non-zero mass. It is able to pass through ordinary matter almost unaffected...

s. Each doublet of a charged lepton and a neutrino consisting of opposite

*T*_{3}are said to constitute one generationGeneration (particle physics)

In particle physics, a generation is a division of the elementary particles. Between generations, particles differ by their quantum number and mass, but their interactions are identical....

of leptons. In addition, one defines a quantum number called weak hypercharge

Weak hypercharge

The weak hypercharge in particle physics is a conserved quantum number relating the electrical charge and the third component of weak isospin, and is similar to the Gell-Mann–Nishijima formula for the hypercharge of strong interactions...

,

*Y*_{W}, which is −1 for all left-handedChirality (physics)

A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...

leptons. Weak isospin and weak hypercharge are gauged

Gauge theory

In physics, gauge invariance is the property of a field theory in which different configurations of the underlying fundamental but unobservable fields result in identical observable quantities. A theory with such a property is called a gauge theory...

in the Standard Model

Standard Model

The Standard Model of particle physics is a theory concerning the electromagnetic, weak, and strong nuclear interactions, which mediate the dynamics of the known subatomic particles. Developed throughout the mid to late 20th century, the current formulation was finalized in the mid 1970s upon...

.

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 electromagnetic interactions, but violated by weak interactions. Therefore, such*flavour*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 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 flavourNeutrino oscillation

Neutrino oscillation is a quantum mechanical phenomenon predicted by Bruno Pontecorvowhereby a neutrino created with a specific lepton flavor can later be measured to have a different flavor. The probability of measuring a particular flavor for a neutrino varies periodically as it propagates...

. The strength of such mixings is specified by a matrix called the Pontecorvo–Maki–Nakagawa–Sakata matrix

Pontecorvo–Maki–Nakagawa–Sakata matrix

In particle physics, the Pontecorvo–Maki–Nakagawa–Sakata matrix , Maki–Nakagawa–Sakata matrix , lepton mixing matrix, or neutrino mixing matrix, is a unitary matrixThe PMNS matrix is not unitary in the seesaw model which contains information on the mismatch of quantum states of leptons when they...

(PMNS matrix).

### Quarks

All quarkQuark

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

s carry a baryon number

*B*= . In addition they carry weak isospin,*T*_{3}= ±. The positive*T*_{3}quarks (up, charm, and top quarks) are called*up-type quarks*and negative*T*_{3}ones are called*down-type quarks*. Each doublet of up and down type quarks constitutes one generationGeneration (particle physics)

In particle physics, a generation is a division of the elementary particles. Between generations, particles differ by their quantum number and mass, but their interactions are identical....

of quarks.

Quarks have the following flavour quantum numbers:

- IsospinIsospinIn physics, and specifically, particle physics, isospin is a quantum number related to the strong interaction. This term was derived from isotopic spin, but the term is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number...

which has value*I*_{3}= for the up quark and value*I*_{3}= − for the down quark. - StrangenessStrangenessIn particle physics, strangeness S is a property of particles, expressed as a quantum number, for describing decay of particles in strong and electromagnetic reactions, which occur in a short period of time...

(*S*): a quantum number introduced by Murray Gell-MannMurray Gell-MannMurray Gell-Mann is an American physicist and linguist who received the 1969 Nobel Prize in physics for his work on the theory of elementary particles...

. The strange quark is defined to have strangeness −1. - Charm (
*C*) number which is +1 for the charm quark. - BottomnessBottomnessIn physics, bottomness also called beauty, is a flavour quantum number reflecting the difference between the number of bottom antiquarks and the number of bottom quarks that are present in a particle: B^\prime = -Bottom quarks have a bottomness of −1 while bottom antiquarks have a...

(*B*′) number which is −1 for the 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:

- HyperchargeHyperchargeIn particle physics, the hypercharge Y of a particle is related to the strong interaction, and is distinct from the similarly named weak hypercharge, which has an analogous role in the electroweak interaction...

(*Y*):*Y*=*B*+*S*+*C*+*B*′ +*T* - Electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

:*Q*=*I*_{3}+*Y*(see Gell-Mann–Nishijima formulaGell-Mann–Nishijima formulaThe Gell-Mann–Nishijima formula relates the baryon number B, the strangeness S, the isospin I3 of hadrons to the charge Q. It was originally given by Kazuhiko Nishijima and Tadao Nakano in 1953, and lead to the proposal of strangeness as a concept, which Nishijima originally called "eta-charge"...

)

A quark of a given flavour is an eigenstate of the weak interaction

Weak interaction

Weak interaction , is one of the four fundamental forces of nature, alongside the strong nuclear force, electromagnetism, and gravity. It is responsible for the radioactive decay of subatomic particles and initiates the process known as hydrogen fusion in stars...

part of the Hamiltonian

Hamiltonian (quantum mechanics)

In quantum mechanics, the Hamiltonian H, also Ȟ or Ĥ, is the operator corresponding to the total energy of the system. Its spectrum is the set of possible outcomes when one measures the total energy of a system...

: it will interact in a definite way with the W and Z bosons

W and Z bosons

The W and Z bosons are the elementary particles that mediate the weak interaction; their symbols are , and . The W bosons have a positive and negative electric charge of 1 elementary charge respectively and are each other's antiparticle. The Z boson is electrically neutral and its own...

. On the other hand, a fermion

Fermion

In particle physics, a fermion is any particle which obeys the Fermi–Dirac statistics . Fermions contrast with bosons which obey Bose–Einstein statistics....

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

CP violation

In particle physics, CP violation is a violation of the postulated CP-symmetry: the combination of C-symmetry and P-symmetry . CP-symmetry states that the laws of physics should be the same if a particle were interchanged with its antiparticle , and left and right were swapped...

if there are at least three generations.

### Antiparticles and hadrons

Flavour quantum numbers are additive. Hence antiparticleAntiparticle

Corresponding to most kinds of particles, there is an associated antiparticle with the same mass and opposite electric charge. For example, the antiparticle of the electron is the positively charged antielectron, or positron, which is produced naturally in certain types of radioactive decay.The...

s have flavour equal in magnitude to the particle but opposite in sign. Hadron

Hadron

In particle physics, a hadron is a composite particle made of quarks held together by the strong force...

s inherit their flavour quantum number from their valence quarks: this is the basis of the classification in the quark model

Quark model

In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons....

. The relations between the hypercharge, electric charge and other flavour quantum numbers hold for hadrons as well as quark

Quark

A quark is an elementary particle and a fundamental constituent of matter. Quarks combine to form composite particles called hadrons, the most stable of which are protons and neutrons, the components of atomic nuclei. Due to a phenomenon known as color confinement, quarks are never directly...

s.

## Quantum chromodynamics

*Flavour symmetry is closely related to chiral symmetry*Chirality (physics)A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...

. This part of the article is best read along with the one on chiralityChirality (physics)A chiral phenomenon is one that is not identical to its mirror image . The spin of a particle may be used to define a handedness for that particle. A symmetry transformation between the two is called parity...

.

Quantum chromodynamics

Quantum chromodynamics

In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

(QCD) contains six flavours of quark

Quark

s. However, their masses differ. 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

Isospin

In physics, and specifically, particle physics, isospin is a quantum number related to the strong interaction. This term was derived from isotopic spin, but the term is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number...

symmetry). Under some circumstances one can take

*N*_{f}flavours to have the same masses and obtain an effective SU(*N*_{f}) flavour symmetry.Under some circumstances, the masses of the quarks can be neglected entirely. In that case, each flavour of quark possesses a chiral symmetry

Chirality (physics)

. One can then make flavour transformations independently on the left- and right-handed parts of each quark field. The flavour 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 flavour 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 massCurrent quark mass

The current quark mass is also called the mass of the 'naked' quarks.The mass of the current quark is reduced by the term of the constituent quark covering mass.The current quark mass is a logical consequence of the mathematical formalism of the QFT,...

es in QCD.

Even if quarks are massless, chiral flavour symmetry can be spontaneously broken

Spontaneous symmetry breaking

Spontaneous symmetry breaking is the process by which a system described in a theoretically symmetrical way ends up in an apparently asymmetric state....

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 QCDQuantum chromodynamics

In theoretical physics, quantum chromodynamics is a theory of the strong interaction , a fundamental force describing the interactions of the quarks and gluons making up hadrons . It is the study of the SU Yang–Mills theory of color-charged fermions...

for the up, down and strange quarks. The success of chiral perturbation theory

Chiral perturbation theory

Chiral perturbation theory is an effective field theory constructed with a Lagrangian consistent with the chiral symmetry of quantum chromodynamics , as well as the other symmetries of parity and charge conjugation. ChPT is a theory which allows one to study the low-energy dynamics of QCD...

and the even more naive chiral model

Chiral model

In nuclear physics, the chiral model is a phenomenological model describing mesons in the chiral limit where the masses of the quarks go to zero . It's a nonlinear sigma model with the principal homogeneous space of the Lie group SU as its target manifold where N is the number of quark flavors...

s spring from this fact. The valence quark masses extracted from the quark model

Quark model

In physics, the quark model is a classification scheme for hadrons in terms of their valence quarks—the quarks and antiquarks which give rise to the quantum numbers of the hadrons....

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.

## Conservation laws

Absolutely conserved flavour quantum numbers are- electric chargeElectric chargeElectric charge is a physical property of matter that causes it to experience a force when near other electrically charged matter. Electric charge comes in two types, called positive and negative. Two positively charged substances, or objects, experience a mutual repulsive force, as do two...

(*Q*) - weak isospinWeak isospinIn particle physics, weak isospin is a quantum number relating to the weak interaction, and parallels the idea of isospin under the strong interaction. Weak isospin is usually given the symbol T or I with the third component written as Tz, T3, Iz or I3...

(*I*)_{3} - baryon number (
*B*) - lepton numberLepton numberIn particle physics, the lepton number is the number of leptons minus the number of antileptons.In equation form,so all leptons have assigned a value of +1, antileptons −1, and non-leptonic particles 0...

(*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 anomalyChiral anomaly

A chiral anomaly is the anomalous nonconservation of a chiral current. In some theories of fermions with chiral symmetry, the quantization may lead to the breaking of this chiral symmetry. In that case, the charge associated with the chiral symmetry is not conserved.The non-conservation happens...

). All other flavour quantum numbers are violated by the electroweak interaction

Electroweak interaction

In particle physics, the electroweak interaction is the unified description of two of the four known fundamental interactions of nature: electromagnetism and the weak interaction. Although these two forces appear very different at everyday low energies, the theory models them as two different...

s. Strong interaction

Strong interaction

In particle physics, the strong interaction is one of the four fundamental interactions of nature, the others being electromagnetism, the weak interaction and gravitation. As with the other fundamental interactions, it is a non-contact force...

s conserve all flavours.

## History

Some of the historical events that lead to the development of flavour symmetry are discussed in the article on isospinIsospin

In physics, and specifically, particle physics, isospin is a quantum number related to the strong interaction. This term was derived from isotopic spin, but the term is confusing as two isotopes of a nucleus have different numbers of nucleons; in contrast, rotations of isospin maintain the number...

.

## See also

- Standard Model (mathematical formulation)
- Cabibbo–Kobayashi–Maskawa matrix
- Strong CP problem and chirality (physics)Chirality (physics)
- Chiral symmetry breakingChiral symmetry breakingIn particle physics, chiral symmetry breaking is an example of spontaneous symmetry breaking affecting the chiral symmetry of gauge theory such as Quantum Chromodynamics. The origin may be described as a fermion condensate...

and quark matter - Quark flavour tagging, such as B-taggingB-taggingb-tagging is an example of a jet flavor tagging method used in modern high-energy particle physics experiments. It is the identification of jets originating from bottom quarks .-Importance:...

, is an example of particle identificationParticle identificationParticle identification is the process of using information left by a particle passing through a particle detector to identify the type of particle. Particle identification reduces backgrounds and improves measurement resolutions, and is essential to many analyses at particle detectors.-Charged...

in experimental particle physics.