Spinor Bilinears

Chiral projectors reveal which components a spinor bilinear connects. Vector currents pair a field of one chirality with its own adjoint; scalar mass terms pair opposite chiralities. This distinction is useful when translating left–right couplings into vector–axial notation or checking an interaction term. The complete Lorentz classification and parity table are on Bilinear Covariants; this page applies them without repeating that classification.

Required background. Bilinear Covariants fixes the covariants; Dirac Spinors explains the chiral decomposition and scalar pairing.

The adjoint changes the projector

Let $\Psi_L=P_L\Psi$ and $\Psi_R=P_R\Psi$, with $P_{L,R}=(1\mp\gamma^5)/2$. Since $\gamma^0$ anticommutes with $\gamma^5$,

$$\overline{\Psi_L}=\bar\Psi P_R,\qquad \overline{\Psi_R}=\bar\Psi P_L.$$

The subscript on $\overline{\Psi_L}$ means “the adjoint of the left-chiral field.” It does not mean $\bar\Psi P_L$. This distinction controls every selection rule below.

For a matrix $\Gamma$ commuting with $\gamma^5$, insertion gives $\overline{\Psi_L}\Gamma\Psi_L =\bar\Psi P_R\Gamma P_L\Psi=0$. It can instead connect opposite chiralities. If $\Gamma$ anticommutes with $\gamma^5$, it reverses the projector in that algebra and allows the same-chirality pairing.

Matrix in the bilinear	Relation to $\gamma^5$	Nonzero chiral pairings allowed
$I$, $i\gamma^5$	Commutes	$\overline{\Psi_L}\Gamma\Psi_R$ and $\overline{\Psi_R}\Gamma\Psi_L$
$\gamma^\mu$, $\gamma^\mu\gamma^5$	Anticommutes	$\overline{\Psi_L}\Gamma\Psi_L$ and $\overline{\Psi_R}\Gamma\Psi_R$
$\sigma^{\mu\nu}$	Commutes	Opposite-chirality pairings

“Allowed” is an algebraic statement; a particular state or internal symmetry can still make the bilinear vanish. Chirality here labels field components, not an arbitrary massive state’s helicity.

Left–right currents and vector–axial couplings

Define

$$j_L^\mu=\bar\Psi\gamma^\mu P_L\Psi,\qquad j_R^\mu=\bar\Psi\gamma^\mu P_R\Psi.$$

With $V^\mu=\bar\Psi\gamma^\mu\Psi$ and $A^\mu=\bar\Psi\gamma^\mu\gamma^5\Psi$, one has

$$V^\mu=j_L^\mu+j_R^\mu,\qquad A^\mu=j_R^\mu-j_L^\mu.$$

Therefore a coupling to an external vector quantity $B_\mu$ can be written in either form:

$$\begin{aligned} B_\mu(g_Lj_L^\mu+g_Rj_R^\mu) &=B_\mu\bar\Psi\gamma^\mu(g_V+g_A\gamma^5)\Psi,\\ g_V&=\frac{g_L+g_R}{2},\qquad g_A=\frac{g_R-g_L}{2}. \end{aligned}$$

Some sources write $g_V-g_A\gamma^5$ instead. Their axial coefficient has the opposite sign; the physical left and right couplings are unchanged after translation. Check the written operator before importing a numerical $g_A$.

For an ordinary polar vector $B_\mu$, equal real couplings $g_L=g_R$ give a parity-even vector interaction. Unequal couplings contain an axial component and generally violate parity for that fixed field assignment. If $B_\mu$ is assigned pseudovector transformation properties, the parity test changes. Lorentz covariance under the connected group alone does not decide invariance under parity.

Complex mass coefficients pair opposite chiralities

For real constants $m_S,m_P$, a Hermitian expression invariant under proper Lorentz transformations is

$$\bar\Psi(m_S+i m_P\gamma^5)\Psi =M\overline{\Psi_L}\Psi_R +M^*\overline{\Psi_R}\Psi_L, \qquad M=m_S+i m_P.$$

The overall minus sign convention in a Lagrangian mass term is separate from this identity. A real scalar mass corresponds to $m_P=0$; a pseudoscalar coefficient multiplies $i\bar\Psi\gamma^5\Psi$.

For a single free Dirac field, a constant chiral rephasing can make a nonzero $M$ real. In a larger theory, other masses and interactions transform too, and an axial redefinition can affect the quantum measure. One cannot declare every mass phase unobservable by inspecting this bilinear in isolation.

Internal charges impose an additional condition. If $\Psi_L\mapsto e^{iq_L\alpha}\Psi_L$ and $\Psi_R\mapsto e^{iq_R\alpha}\Psi_R$, then $\overline{\Psi_L}\Psi_R$ acquires $e^{i(q_R-q_L)\alpha}$. A constant bare mass preserves this $U(1)$ only when $q_L=q_R$. A scalar field with a compensating transformation can instead make a Yukawa coupling invariant. This is a transformation test, not yet a model of symmetry breaking.

Tensor couplings provide a second selection-rule check

Since $\sigma^{\mu\nu}$ contains two gamma matrices, it commutes with $\gamma^5$. A dipole-like expression $\bar\Psi\sigma^{\mu\nu}\Psi F_{\mu\nu}$ therefore connects opposite chiralities in the same projector sense as a mass. The ordinary vector coupling $\bar\Psi\gamma^\mu\Psi A_\mu$ preserves them at the vertex.

This statement does not imply that the full propagation of a massive particle preserves chirality, or that a massive helicity amplitude vanishes whenever a chiral pairing does. The mass term, external spinors, and the requested observable must all be included. It also does not derive a dipole coefficient: such a coefficient can be supplied by an effective theory matched to more detailed dynamics.

Exercises

1. Translate a purely left-current coupling $B_\mu j_L^\mu$ into the stated vector–axial convention.

Solution

$g_L=1$, $g_R=0$ gives $g_V=1/2$ and $g_A=-1/2$. Thus the operator is $\frac12B_\mu\bar\Psi\gamma^\mu(1-\gamma^5)\Psi$. The factor one half is part of the chiral projector.

2. Show that $\overline{\Psi_L}\sigma^{\mu\nu}\Psi_L=0$, without choosing explicit gamma matrices.

Solution

Two anticommutations show $[\gamma^5,\sigma^{\mu\nu}]=0$. Hence $\bar\Psi P_R\sigma^{\mu\nu}P_L\Psi =\bar\Psi\sigma^{\mu\nu}P_RP_L\Psi=0$. No equation of motion or massless approximation was required.

3. A complex scalar $\phi$ transforms as $\phi\mapsto e^{iq_\phi\alpha}\phi$. What charge makes $\phi\overline{\Psi_L}\Psi_R$ invariant?

Solution

Its total phase is $e^{i(q_\phi-q_L+q_R)\alpha}$, so $q_\phi=q_L-q_R$. Its Hermitian conjugate is then invariant too. Lorentz invariance of the spinor contraction alone would not have supplied this internal-charge condition.

References

• H. K. Dreiner, H. E. Haber, and S. P. Martin, “Two-component spinor techniques and Feynman rules for quantum field theory and supersymmetry,” Physics Reports 494, 1–196, 2010, doi:10.1016/j.physrep.2010.05.002; corrected arXiv:0812.1594v6, 2022 — chiral bilinears and mass couplings.
• M. D. Schwartz, Quantum Field Theory and the Standard Model, Cambridge University Press, 2014, doi:10.1017/9781139540940 — chiral currents, gauge representations, and field redefinitions.