Fierz identity for Weyl spinors in tensor currents

Question

Using Fierz identity I found that certain four-fermion operator with left $l_i$ and right-chiral $r_i$ Weyl spinors vanish

$\bar{l}_1\sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4 =$ $ -\frac{3}{2} \bar{l}_1 l_4 \bar{r}_3 r_2 - \frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2$ $ - \frac{3}{2} \bar{l}_1 (-1) l_4 \bar{r}_3 (+1) r_2 =$ $= -\frac{1}{2}\bar{l}_1\sigma_{\mu\nu} l_4 \bar{r}_3 \sigma^{\mu\nu} r_2 = 0$

Why does this operator vanish? Is that true?

Why is the product of tensor currents expressed in tensor and scalar currents only, while for such combinations of Weyl spinors $l_1$, $r_2$, $r_3$ and $l_4$ in the Fierz identity all of them vanish?

This post imported from StackExchange Physics at 2014-07-22 07:50 (UCT), posted by SE-user fen

Comments

If you use the definition of the projectors, $P_+$ and $P_-$, and their commutation with the gamma matrices, you can get the result!

This post imported from StackExchange Physics at 2014-07-22 07:50 (UCT), posted by SE-user Dox

Answer 1

The simplest way to see that all products of your kind vanish is to notice that one of the objects (bilinear invariant) $T_{\mu\nu}$ is a self-dual 2-form while the other $T^{\mu\nu}$ is anti-self-dual, and their contraction without a complex conjugation has to vanish.

A self-dual (anti-self-dual) antisymmetric tensor obeys $$T_{\mu\nu} = \pm \frac i2 \epsilon_{\mu\nu\kappa\lambda} T^{\kappa\lambda}$$ and in $3+1$ dimensions, it has to have complex components.

The objects $\bar \ell \sigma \ell$ are self-dual or anti-self-dual due to the chirality of the spinors. And the inner products are zero because the 6-dimensional complex space of 2-forms may be really decomposed to mutually orthogonal self-dual and anti-self-dual (3-complex-dimensional) parts.

The claims above, when translated to maths, contain lots of signs sometimes convention-dependent signs (and extra signs and flips of chiralities from complex conjugation and so on) that one has to be careful about. But there are also several identities of your kind one may derive.

This post imported from StackExchange Physics at 2014-07-22 07:50 (UCT), posted by SE-user Lubo Motl

Comments

I clearly see, why $\bar{l}l$ or $\bar{l}\sigma_{\mu\nu}l$ vanish, however I am not sure why does $\bar{l}_1 \sigma_{\mu\nu} r_2 \bar{r}_3 \sigma^{\mu\nu} l_4$ vanish. It contains tensor currents with spinors of different chiralities.

This post imported from StackExchange Physics at 2014-07-22 07:50 (UCT), posted by SE-user fen