Achiral phenomenon is one that is not identical to itsmirror image (see the article onmathematical chirality). Thespin of aparticle may be used to define ahandedness, or helicity, for that particle, which, in the case of a massless particle, is the same as chirality. A symmetry transformation between the two is calledparity transformation. Invariance under parity transformation by aDirac fermion is calledchiral symmetry.

The helicity of a particle is positive ("right-handed") if the direction of itsspin is the same as the direction of its motion. It is negative ("left-handed") if the directions of spin and motion are opposite. So a standardclock, with its spin vector defined by the rotation of its hands, has left-handed helicity if tossed with its face directed forwards. The same standard clock if tossed upwards (with face upwards) would be considered to have left-handed helicity while going up and right-handed helicity while falling back (with face still upwards).

Mathematically,helicity is the sign of the projection of thespinvector onto themomentumvector: "left" is negative, "right" is positive.

Thechirality of a particle is more abstract: It is determined by whether the particle transforms in a right- or left-handedrepresentation of thePoincaré group.^[a]

Formassless particles –photons,gluons, and (hypothetical)gravitons – chirality is the same ashelicity; a given massless particle appears to spin in the same direction along its axis of motion regardless of point of view of the observer.

Formassive particles – such aselectrons,quarks, andneutrinos – chirality and helicity must be distinguished: In the case of these particles, it is possible for an observer to change to areference frame that is moving faster than the spinning particle is, in which case the particle will then appear to move backwards, and its helicity (which may be thought of as "apparent chirality") will be reversed.

• Helicity is aconstant of motion,^[1] but it is notLorentz invariant.^[2]
• Chirality is Lorentz invariant,^[3] but is not a constant of motion: a massive left-handed spinor, when propagating, will evolve into a right handed spinor over time, and vice versa.

Amassless particle moves with thespeed of light, so no real observer (who must always travel at less than thespeed of light) can be in any reference frame in which the particle appears to reverse its relative direction of spin, meaning that all real observers see the same helicity. Because of this, the direction of spin of massless particles is not affected by a change of inertial reference frame (aLorentz boost) in the direction of motion of the particle, and the sign of the projection (helicity) is fixed for all reference frames: The helicity of massless particles is arelativistic invariant (a quantity whose value is the same in all inertial reference frames) and always matches the massless particle's chirality.

The discovery ofneutrino oscillation implies thatneutrinos have mass, leaving thephoton as the only confirmed massless particle;gluons are expected to also be massless, although this has not been conclusively tested.^[b] Hence, these are the only two particles now known for which helicity could be identical to chirality, of which only thephoton has been confirmed by measurement. All other observed particles have mass and thus may have different helicities in different reference frames.^[c]

Particle physicists have only observed or inferred left-chiralfermions and right-chiral antifermions engaging in thecharged weak interaction.^[4] In the case of the weak interaction, which can in principle engage with both left- and right-chiral fermions, only two left-handedfermions interact. Interactions involving right-handed or opposite-handed fermions have not been shown to occur, implying that the universe has a preference for left-handed chirality. This preferential treatment of one chiral realization over another violates parity, as first noted byChien Shiung Wu in her famous experiment known as theWu experiment. This is a striking observation, since parity is a symmetry that holds for all otherfundamental interactions.

Chirality for aDirac fermionψ is defined through theoperatorγ⁵, which haseigenvalues ±1; the eigenvalue's sign is equal to the particle's chirality: +1 for right-handed, −1 for left-handed. Any Dirac field can thus be projected into its left- or right-handed component by acting with theprojection operators⁠1/2⁠(1 −γ⁵) or⁠1/2⁠(1 +γ⁵) onψ.

The coupling of the charged weak interaction to fermions is proportional to the first projection operator, which is responsible for this interaction'sparity symmetry violation.

A common source of confusion is due to conflating theγ⁵, chirality operator with thehelicity operator. Since the helicity of massive particles is frame-dependent, it might seem that the same particle would interact with the weak force according to one frame of reference, but not another. The resolution to this paradox is thatthe chirality operator is equivalent to helicity for massless fields only, for which helicity is not frame-dependent. By contrast, for massive particles, chirality is not the same as helicity, or, alternatively, helicity is not Lorentz invariant, so there is no frame dependence of the weak interaction: a particle that couples to the weak force in one frame does so in every frame.

A theory that is asymmetric with respect to chiralities is called achiral theory, while a non-chiral (i.e., parity-symmetric) theory is sometimes called avector theory. Many pieces of theStandard Model of physics are non-chiral, which is traceable toanomaly cancellation in chiral theories.Quantum chromodynamics is an example of a vector theory, since both chiralities of all quarks appear in the theory, and couple to gluons in the same way.

Theelectroweak theory, developed in the mid 20th century, is an example of a chiral theory. Originally, it assumed thatneutrinos were massless, and assumed the existence of only left-handedneutrinos and right-handed antineutrinos. After the observation ofneutrino oscillations, which implies that no fewer than two of the threeneutrinos are massive, the revisedtheories of the electroweak interaction now include both right- and left-handedneutrinos. However, it is still a chiral theory, as it does not respect parity symmetry.

The exact nature of theneutrino is still unsettled and so theelectroweak theories that have been proposed are somewhat different, but most accommodate the chirality ofneutrinos in the same way as was already done for all otherfermions.

Vectorgauge theories with massless Dirac fermion fieldsψ exhibit chiral symmetry, i.e., rotating the left-handed and the right-handed components independently makes no difference to the theory. We can write this as the action of rotation on the fields:

${\displaystyle {\psi }_{\mathrm{L}}\to e^{i{\theta }_{\mathrm{L}}}{\psi }_{\mathrm{L}}}$  and  ${\displaystyle {\psi }_{\mathrm{R}}\to {\psi }_{\mathrm{R}}}$

or

${\displaystyle {\psi }_{\mathrm{L}}\to {\psi }_{\mathrm{L}}}$  and  ${\displaystyle {\psi }_{\mathrm{R}}\to e^{i{\theta }_{\mathrm{R}}}{\psi }_{\mathrm{R}}.}$

WithNflavors, we have unitary rotations instead:U(N)_L × U(N)_R.

More generally, we write the right-handed and left-handed states as a projection operator acting on a spinor. The right-handed and left-handed projection operators are

${\displaystyle P_{\mathrm{R}}=\frac{1+{\gamma }^5}{2}}$

and

${\displaystyle P_{\mathrm{L}}=\frac{1-{\gamma }^5}{2}}$

Massive fermions do not exhibit chiral symmetry, as the mass term in theLagrangian,mψψ, breaks chiral symmetry explicitly.

Spontaneous chiral symmetry breaking may also occur in some theories, as it most notably does inquantum chromodynamics.

The chiral symmetry transformation can be divided into a component that treats the left-handed and the right-handed parts equally, known asvector symmetry, and a component that actually treats them differently, known asaxial symmetry.^[5] (cf.Current algebra.) A scalar field model encoding chiral symmetry and itsbreaking is thechiral model.

The most common application is expressed as equal treatment of clockwise and counter-clockwise rotations from a fixed frame of reference.

The general principle is often referred to by the namechiral symmetry. The rule is absolutely valid in theclassical mechanics ofNewton andEinstein, but results fromquantum mechanical experiments show a difference in the behavior of left-chiral versus right-chiralsubatomic particles.

Considerquantum chromodynamics (QCD) with twomasslessquarksu andd (massive fermions do not exhibit chiral symmetry). The Lagrangian reads

${\displaystyle \mathcal{L}=\overline{u}i{\displaystyle \text{⧸}Du+\overline{d}i{\displaystyle \text{⧸}Dd+{\mathcal{L}}_{\mathrm{g}\mathrm{l}\mathrm{u}\mathrm{o}\mathrm{n}\mathrm{s}}.}}}$

In terms of left-handed and right-handed spinors, it reads

${\displaystyle \mathcal{L}={\overline{u}}_{\mathrm{L}}i{\displaystyle \text{⧸}D{u}_{\mathrm{L}}+{\overline{u}}_{\mathrm{R}}i{\displaystyle \text{⧸}D{u}_{\mathrm{R}}+{\overline{d}}_{\mathrm{L}}i{\displaystyle \text{⧸}D{d}_{\mathrm{L}}+{\overline{d}}_{\mathrm{R}}i{\displaystyle \text{⧸}D{d}_{\mathrm{R}}+{\mathcal{L}}_{\mathrm{g}\mathrm{l}\mathrm{u}\mathrm{o}\mathrm{n}\mathrm{s}}.}}}}}$

(Here,i is the imaginary unit and${\displaystyle {\displaystyle \text{⧸}D}}$ theDirac operator.)

Defining

${\displaystyle q=\left[\begin{array}{c}u\\ d\end{array}\right],}$

it can be written as

${\displaystyle \mathcal{L}={\overline{q}}_{\mathrm{L}}i{\displaystyle \text{⧸}D{q}_{\mathrm{L}}+{\overline{q}}_{\mathrm{R}}i{\displaystyle \text{⧸}D{q}_{\mathrm{R}}+{\mathcal{L}}_{\mathrm{g}\mathrm{l}\mathrm{u}\mathrm{o}\mathrm{n}\mathrm{s}}.}}}$

The Lagrangian is unchanged under a rotation ofq_L by any 2×2 unitary matrixL, andq_R by any 2×2 unitary matrixR.

This symmetry of the Lagrangian is calledflavor chiral symmetry, and denoted asU(2)_L × U(2)_R. It decomposes into

${\displaystyle \mathrm{S}\mathrm{U}(2{)}_{\text{L}}\times \mathrm{S}\mathrm{U}(2{)}_{\text{R}}\times \mathrm{U}(1{)}_V\times \mathrm{U}(1{)}_A.}$

The singlet vector symmetry,U(1)_V, acts as

${\displaystyle q_{\text{L}}\to e^{i\theta (x)}{q}_{\text{L}}{q}_{\text{R}}\to e^{i\theta (x)}{q}_{\text{R}},}$

and thus invariant underU(1) gauge symmetry. This corresponds tobaryon number conservation.

The singlet axial groupU(1)_A transforms as the following global transformation

${\displaystyle q_{\text{L}}\to e^{i\theta }{q}_{\text{L}}{q}_{\text{R}}\to e^{-i\theta }{q}_{\text{R}}.}$

However, it does not correspond to a conserved quantity, because the associated axial current is not conserved. It is explicitly violated by aquantum anomaly.

The remaining chiral symmetrySU(2)_L × SU(2)_R turns out to bespontaneously broken by aquark condensate${\displaystyle ⟨{\overline{q}}_{\text{R}}^a{q}_{\text{L}}^b⟩=v{\delta }^{ab}}$ formed through nonperturbative action of QCD gluons, into the diagonal vector subgroupSU(2)_V known asisospin. TheGoldstone bosons corresponding to the three broken generators are the threepions. As a consequence, the effective theory of QCD bound states like the baryons, must now include mass terms for them, ostensibly disallowed by unbroken chiral symmetry. Thus, thischiral symmetry breaking induces the bulk of hadron masses, such as those for thenucleons — in effect, the bulk of the mass of all visible matter.

In the real world, because of the nonvanishing and differing masses of the quarks,SU(2)_L × SU(2)_R is only an approximate symmetry^[6] to begin with, and therefore the pions are not massless, but have small masses: they arepseudo-Goldstone bosons.^[7]

For more "light" quark species,Nflavors in general, the corresponding chiral symmetries areU(N)_L × U(N)_R′, decomposing into

${\displaystyle \mathrm{S}\mathrm{U}(N{)}_{\text{L}}\times \mathrm{S}\mathrm{U}(N{)}_{\text{R}}\times \mathrm{U}(1{)}_V\times \mathrm{U}(1{)}_A,}$

and exhibiting a very analogouschiral symmetry breaking pattern.

Most usually,N = 3 is taken, the u, d, and s quarks taken to be light (theeightfold way), so then approximately massless for the symmetry to be meaningful to a lowest order, while the other three quarks are sufficiently heavy to barely have a residual chiral symmetry be visible for practical purposes.

Intheoretical physics, theelectroweak model breaksparity maximally. All itsfermions are chiralWeyl fermions, which means that the chargedweak gauge bosons W⁺ and W⁻ only couple to left-handed quarks and leptons.^[d]

Some theorists found this objectionable, and so conjectured aGUT extension of theweak force which has new, high energyW′ and Z′ bosons, whichdo couple with right handed quarks and leptons:

${\displaystyle \frac{\mathrm{S}\mathrm{U}(2{)}_{\text{W}}\times \mathrm{U}(1{)}_Y}{{\mathbb{Z}}_2}}$

to

${\displaystyle \frac{\mathrm{S}\mathrm{U}(2{)}_{\text{L}}\times \mathrm{S}\mathrm{U}(2{)}_{\text{R}}\times \mathrm{U}(1{)}_{B-L}}{{\mathbb{Z}}_2}.}$

Here,SU(2)_L (pronounced "SU(2) left") isSU(2)_W from above, whileB−L is thebaryon number minus thelepton number. The electric charge formula in this model is given by

${\displaystyle Q=T_{3\mathrm{L}}+T_{3\mathrm{R}}+\frac{B-L}{2};}$

where${\displaystyle T_{3\mathrm{L}}}$ and${\displaystyle T_{3\mathrm{R}}}$ are the left and rightweak isospin values of the fields in the theory.

There is also thechromodynamicSU(3)_C. The idea was to restore parity by introducing aleft–right symmetry. This is agroup extension of${\displaystyle {\mathbb{Z}}_2}$ (the left-right symmetry) by

${\displaystyle \frac{\mathrm{S}\mathrm{U}(3{)}_{\text{C}}\times \mathrm{S}\mathrm{U}(2{)}_{\text{L}}\times \mathrm{S}\mathrm{U}(2{)}_{\text{R}}\times \mathrm{U}(1{)}_{B-L}}{{\mathbb{Z}}_6}}$

to thesemidirect product

${\displaystyle \frac{\mathrm{S}\mathrm{U}(3{)}_{\text{C}}\times \mathrm{S}\mathrm{U}(2{)}_{\text{L}}\times \mathrm{S}\mathrm{U}(2{)}_{\text{R}}\times \mathrm{U}(1{)}_{B-L}}{{\mathbb{Z}}_6}⋊{\mathbb{Z}}_2.}$

This has twoconnected components where${\displaystyle {\mathbb{Z}}_2}$ acts as anautomorphism, which is the composition of aninvolutiveouter automorphism ofSU(3)_C with the interchange of the left and right copies ofSU(2) with the reversal ofU(1)_B−L. It was shown byMohapatra &Senjanovic (1975)^[8] thatleft-right symmetry can bespontaneously broken to give a chiral low energy theory, which is the Standard Model of Glashow, Weinberg, and Salam, and also connects the small observed neutrino masses to the breaking of left-right symmetry via theseesaw mechanism.

In this setting, the chiralquarks

${\displaystyle (3,2,1{)}_{+\frac{1}{3}}}$

and

${\displaystyle {\left(\overline{3},1,2\right)}_{-\frac{1}{3}}}$

are unified into anirreducible representation ("irrep")

${\displaystyle (3,2,1{)}_{+\frac{1}{3}}\oplus {\left(\overline{3},1,2\right)}_{-\frac{1}{3}}.}$

Theleptons are also unified into anirreducible representation

${\displaystyle (1,2,1{)}_{-1}\oplus (1,1,2{)}_{+1}.}$

TheHiggs bosons needed to implement the breaking of left-right symmetry down to the Standard Model are

${\displaystyle (1,3,1{)}_2\oplus (1,1,3{)}_2.}$

This then provides threesterile neutrinos which are perfectly consistent with current^[update]neutrino oscillation data. Within the seesaw mechanism, the sterile neutrinos become superheavy without affecting physics at low energies.

Because the left–right symmetry is spontaneously broken, left–right models predictdomain walls. This left-right symmetry idea first appeared in thePati–Salam model (1974)^[9] and Mohapatra–Pati models (1975).^[10]

Chirality in other branches of physics is often used for classifying and studying the properties ofbodies and materials under external influences. Classification by chirality, as a special case ofsymmetry classification, allows for a better understanding offirst-principles construction ofmolecules,crystals,quasicrystals, and more. An example is thehomochirality ofamino acids in all known forms oflife,^[11] which can be reproduced in physical experiments under external influence.^[12]Optical activity (includingcircular dichroism^[13] andmagnetic circular dichroism^[13]) ofmaterials is determined by their chirality.

Chiralphysical systems are characterized by the absence ofinvariance under theparity operator. An ambiguity arises^[14] in defining chirality in physics depending on whether one compares directions of motion using thereflection orspatialinversion operation. Accordingly, one distinguishes^[14][15] between "true" chirality (which isinvariant under thetime-reversal operation) and "false" chirality (non-invariant under time reversal).

Manyphysical quantities change sign under thetime-reversal operation (e.g.,velocity,power,electric current,magnetization). Accordingly, "false" chirality is so typical in physics that the term can be misleading, and it is clearer to speak ofT-invariant andT-non-invariant chirality.^[15] Effects related to chirality are described usingpseudoscalar oraxial vector physical quantities in general, and particularly, in magnetically ordered media, are described^[16][17] using time-direction-dependent chirality. This approach is formalized usingdichromatic symmetry groups.T-invariant chirality corresponds to the absence in the symmetry group of any symmetry operations that includespatial inversion${\displaystyle \overline{1}}$ orreflection m, according tointernational notation. The criterion forT-non-invariant chirality is the presence of these symmetry operations, but only when combined withtime reversal${\displaystyle 1^{\prime }}$,^[17] such as operations m′ or${\displaystyle {\overline{1}}^{\prime }}$.

At the level of atomic structure of materials, one distinguishes^[18] vector, scalar, and other types of chirality depending on the direction/sign oftriple andvector products ofspins.

• Electroweak theory
• Chirality (chemistry)
• Chirality (mathematics)
• Chiral symmetry breaking
• Handedness
• Spinors
• Fermionic field § Dirac fields
• Sigma model
• Chiral model

• Walter Greiner; Berndt Müller (2000).Gauge Theory of Weak Interactions. Springer.ISBN 3-540-67672-4.
• Gordon L. Kane (1987).Modern Elementary Particle Physics. Perseus Books.ISBN 0-201-11749-5.
• Kondepudi, Dilip K.; Hegstrom, Roger A. (January 1990). "The Handedness of the Universe".Scientific American.262 (1):108–115.Bibcode:1990SciAm.262a.108H.doi:10.1038/scientificamerican0190-108.
• Winters, Jeffrey (November 1995)."Looking for the Right Hand".Discover. Retrieved12 September 2015.

• "History of Science: Parity Violation". Archived fromthe original on 3 April 2005.
• "Helicity, Chirality, Mass, and the Higgs".Quantum Diaries.
• Robert D. Klauber."Chirality vs Helicity".

• Quantum field theory
• Quantum chromodynamics
• Symmetry
• Chirality

• All articles with bare URLs for citations
• Articles with bare URLs for citations from July 2025
• Articles with PDF format bare URLs for citations
• Articles with short description
• Short description is different from Wikidata
• Articles containing potentially dated statements from 2005
• All articles containing potentially dated statements