Dirac mass term ($m\bar{\psi }\psi$)

A Dirac mass term pairs left- and right-handed Weyl fields in conjugate Lorentz representations. Gauge invariance requires the two chiral fields to carry matching internal quantum numbers, directly or after multiplication by a scalar field that acquires a vacuum expectation value.

A spatial change of the sign of a Dirac mass coefficient. In the two-dimensional model, a wall along one direction can support a mode exponentially confined across the wall and propagating chirally along it. The existence and orientation can be obtained directly from the first-order wave equation without assigning a bulk invariant to an unregularized continuum model.

For $H=c(p_x{\sigma }_x+p_y{\sigma }_y)+V(y){\sigma }_z$, $c>0$, and a mass changing from negative to positive as $y$ increases, $e^{i{k}_{x}x}\exp [-{\int }_0^{y}V(s)\mathrm{d}s/(\hslash c)](1,-1{)}^T$ has energy $-\hslash c{k}_x$. The derivative and mass terms cancel. Normalizable confinement additionally requires the envelope to be square-integrable, as for nonzero asymptotic masses. Reversing the wall reverses the propagating direction.

A branch $E=-\hslash c{k}_x$ gives a packet $f(y)ug(x+ct)$: it translates rigidly in the negative longitudinal direction while remaining transversely localized. Curvature of a more complete physical dispersion can restore packet spreading; the rigid result belongs to the exactly linear model.

When the mass is constant $V_{+}>0$ above the wall and $V_{-}<0$ below it, amplitude decay lengths are ${\xi }_{+}=\hslash c/V_{+}$ and ${\xi }_{-}=\hslash c/|V_{-}|$. A scalar transverse envelope multiplying a unit spinor has normalization $\sqrt{2/({\xi }_{+}+{\xi }_{-})}$. Density decays twice as fast as amplitude.