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A chiral gauge spectrum has independent left-handed Weyl spinors in complex gauge group representations without an obligatory conjugate pair of left-handed Weyl spinors. The right-handed antiparticles required by the CPT theorem do not themselves make a spectrum vectorlike.

A chiral spinor superfield carries a Spinor representation of the Lorentz group index and satisfies ${\bar{D}}_{\dot{\alpha }}{\Psi }_{\beta }=0$. The chiral field-strength superfield is a fermionic example: its lowest component is the Weyl spinor gaugino. Its spinor index distinguishes it from a scalar chiral superfield, even though both obey the same chirality constraint.

With left Grassmann derivatives, a chiral superfield is a finite expansion in chiral coordinates $y^{\mu }=x^{\mu }+i\theta {\sigma }^{\mu }\bar{\theta }$:Here $\varphi$ is a complex scalar field, $\psi$ a Weyl spinor and $F$ an auxiliary field. The translation form of its ordinary-coordinate expansion isThe exponential terminates in the Grassmann algebra. To reduce its contractions, choose signature $(+---)$, ${\sigma }^{\mu }=(I,\mathbf{\sigma })$, left differentiation, and ${ϵ}_{12}=1$ for both lower spinor epsilon tensors. Define ${\theta }^2={\theta }^{\alpha }{\theta }_{\alpha }$ and ${\bar{\theta }}^2={\bar{\theta }}_{\dot{\alpha }}{\bar{\theta }}^{\dot{\alpha }}$. Then $v^{\mu }=\theta {\sigma }^{\mu }\bar{\theta }$ satisfies $v^{\mu }{v}^{\nu }=\frac{1}{2}{\theta }^2{\bar{\theta }}^2{\eta }^{\mu \nu }$. Expanding the translation givesThe last sign follows from $i^2/2=-1/2$ in the translation expansion. Defining the barred square in the reverse order, ${\bar{\theta }}^{\dot{\alpha }}{\bar{\theta }}_{\dot{\alpha }}$, negates it and writes the same term with a positive quarter coefficient. Spinor contractions and the Grassmann algebra convention must therefore accompany the reduced formula.

With $X^2=0$ and invertible $F_X$, the additional chiral superfield $Y=y+\sqrt{2}\theta \chi +{\theta }^2{F}_Y$ satisfying $XY=0$ has $y=(G\chi )/F_X-(GG)F_Y/(2F_X^2)$. Its Weyl spinor $\chi$ and auxiliary field $F_Y$ remain independent. This eliminates the independent scalar. It does not force $Y^2=0$.

For canonically normalized chiral superfields, the fermion bilinear is $-\frac{1}{2}{W}_{ij}{\psi }_i{\psi }_j+h.c.$ at the chosen vacuum. Its matrix is complex symmetric. The nonnegative physical Weyl spinor masses are its singular values, and their squared sum is $\mathrm{tr}(m^{†}m)$. Using signed eigenvalues of a symmetric mass matrix instead can obscure the physical mass count.

Complex two-by-two matrices of determinant one form a complex three-dimensional Lie group, or a real six-dimensional Lie group. Their congruence action on Hermitian two-by-two matrices gives the Lorentz spinor double cover with kernel $\{I,-I\}$. The two Weyl spinor representations are $A$ and $(A^{†}{)}^{-1}$. Treating the group as real is essential for the second, antiholomorphic representation.

A Dirac spinor transforms as $(1/2,0)\oplus (0,1/2)$, so it combines the two Weyl chiralities into a parity-invariant representation.

In four-dimensional Minkowski spacetime, extended supersymmetry has more than one independent Weyl spinor supercharge, together with their adjoints. The index $A=1,\dots ,\mathcal{N}$ distinguishes these supercharges.

A four-dimensional $\mathcal{N}=4$ supersymmetric gauge theory has one supersymmetric vector multiplet containing a gauge boson, four Weyl spinors, and six real scalar fields in the adjoint group representation. Its massless helicity multiplicities are $1,4,6,4,1$ from helicity $1$ to $-1$.

Four-dimensional $\mathcal{N}=8$ supergravity has eight independent Weyl spinor supercharges. Its supergravity multiplet has helicities from $2$ to $-2$, with multiplicities $1,8,28,56,70,56,28,8,1$.

Canonical gauge interactions preserve the tree-level supertrace mass sum rule at a vacuum with vanishing D-terms. Put $C={\sum }_a{g}_a^2\Vert T_{a}v{\Vert }^2$ for the scalar expectation vector $v$. The extra real-scalar squared-mass trace is $2C$, the extra Weyl spinor squared-mass trace from gaugino mixing is $4C$, and the gauge boson squared-mass trace is $2C$. Their spin-weighted supertrace is $2C-2(4C)+3(2C)=0$. The scalar contribution comes from differentiating $\frac{1}{2}{\sum }_a{D}_a^2$; the fermionic one counts both blocks of the symmetric mixing matrix with entries $\sqrt{2}{g}_a{T}_{a}v$; the vector one follows from the covariant scalar kinetic term. Any zero-mass Goldstone bosons in the unreduced scalar Hessian do not change its trace.

A left-handed massless Weyl spinor has a positive-energy plane wave transforming by $e^{i\alpha /2}$ under a rotation about its momentum. Its particle has helicity $-1/2$. The negative-frequency term multiplies an antiparticle fermionic creation operator, so its corresponding one-particle state carries the conjugate rotation phase and helicity $+1/2$. Thus a left-handed field has one particle and one antiparticle helicity; the right-handed field interchanges the signs. A full massless Dirac field contains both chiral fields and hence both particle helicities.

A higgsino is the Weyl spinor partner of a Higgs scalar field in a chiral superfield. After a supersymmetric Higgs mechanism, its combination along a broken gauge direction mixes with the gaugino in the resulting massive N=1 vector multiplet.

A Majorana mass term pairs a Weyl field with itself and violates its continuous phase symmetry. Gauge invariance therefore permits it only when the field is neutral under every unbroken gauge charge.

A Majorana spinor equals its charge conjugate. In the Weyl representation it can be written ${\psi }_M=(u_L,i{\sigma }^2{u}_L^{\ast }{)}^T$ from one left-handed Weyl spinor.

A Majorana-Weyl spinor satisfies both a Majorana spinor reality condition and a Weyl spinor chirality condition when dimension and signature permit them. In ten-dimensional Lorentzian spacetime it has sixteen real components off shell and eight physical massless spin-one-half polarizations.

For positive real $M,\mu ,\lambda$ with $M^2>2{\lambda }^2{\mu }^2$, minimizing $V={\lambda }^2|z^2-{\mu }^2{|}^2+M^2|z{|}^2+|2\lambda xz+My{|}^2$ gives $z=y=0$ with arbitrary $x$. Around $x=0$, the real scalar squared masses are $0,0,M^2,M^2,M^2-2{\lambda }^2{\mu }^2,M^2+2{\lambda }^2{\mu }^2$, while the three Weyl spinor masses are $0,M,M$. Their weighted supertrace vanishes. The massless ${\psi }_X$ is the goldstino; the two real $x$ modes form a tree-level pseudomodulus. The stability bound follows from $|z^2-{\mu }^2{|}^2\ge (|z{|}^2-{\mu }^2{)}^2$, after minimizing the final square over $y$.

In the MSSM, each fermion generation has chiral superfields $Q:(3,2,1/6)$, $U^c:(\bar{3},1,-2/3)$, $D^c:(\bar{3},1,1/3)$, $L:(1,2,-1/2)$, and $E^c:(1,1,1)$. This uses hypercharge normalized by $Q_{\mathrm{electric}}=T^3+Y$ and only left-handed Weyl spinors. Two Higgs chiral doublets have charges $\pm 1/2$. The three vector superfields transform as $(8,1,0)$, $(1,3,0)$ and $(1,1,0)$, supplying the gauge bosons and gauginos.

With $K_i=M_{0i}$ and $M_{\mu \nu }=i{\gamma }_{\mu \nu }/2$ in the supplied mostly-minus index convention, the two Weyl spinor blocks have equal rotation generators $J_i={\sigma }_i/2$ and opposite boost generators. A complex-linear intertwiner must commute with every SU(2) rotation and hence be scalar by the Schur lemma, but no nonzero scalar intertwines the opposite boosts. This proves inequivalence of the two complex representations; parity exchanges them.

Chirality distinguishes the two inequivalent Weyl spinor representations of the proper Lorentz group. For a Dirac spinor, define ${\gamma }^5=i{\gamma }^0{\gamma }^1{\gamma }^2{\gamma }^3$ and the chiral projectorsThey satisfy $P_L{P}_R=0$ and $P_L+P_R=I$. Proper Lorentz transformations preserve each chiral representation, whereas parity exchanges them. Helicity instead measures the projection of spin angular momentum along momentum. For a massive particle it can change under a boost that overtakes the particle, so it must not be identified with chirality. For massless positive-energy particles the left-chiral field annihilates negative-helicity particles and creates positive-helicity antiparticles.

The Standard Model is a chiral gauge theory: left- and right-chiral fields have different gauge group representations. In the convention $Q=T_3+Y$, one generation containsThe minimal field content has no right-handed neutrino. A sterile right-handed field is a possible extension, not a required member of that minimal list. The table displays weak isospin, hypercharge and color charge assignments, not assignments of massive-particle helicity.

The weak charged current contains $\bar{\nu }{\gamma }^{\mu }{P}_{L}e{W}_{\mu }^{+}$ and its Hermitian conjugate, with coupling $g/\sqrt{2}$. Only the left-chiral lepton doublet participates, giving parity violation. Both chiralities can couple to the Z boson, but their coefficients differ:Here $T_3$ is that of the left-chiral component. For the electron, it is $-1/2$. The photon and gluon couplings of a charged massive fermion are vectorlike; this does not make the full electroweak interaction vectorlike.

A bare Dirac mass term $m\bar{\psi }\psi =m({\bar{\psi }}_L{\psi }_R+{\bar{\psi }}_R{\psi }_L)$ joins the two chiral representations. Their different electroweak quantum numbers forbid that term before electroweak symmetry breaking. A Yukawa interaction, such as $-y_e{\bar{L}}_{L}H{e}_R+h.c.$, is gauge invariant; the Higgs doublet expectation value then gives $m_e=y_{e}v/\sqrt{2}$. Up-type quarks use the conjugate doublet. Thus chirality explains the weak charge assignments, their parity violation, and why masses arise through the Higgs mechanism. The consistency of these chiral assignments also requires Standard Model anomaly cancellation.