If a left handed electron and a right handed antimatter electron were to meet, would they still annihilate?
In the same way, if a left handed electron and a left handed antimatter electron meet, will they still annihilate?
Is matter antimatter annihilation a physical process that cares about particle parity?
My question is for all fermions in the standard model including neutrinos.
The dominant force in the Standard Model is electromagnetism, by far, a vectorlike interaction which preserves parity, with interaction vertex $$ eA_\mu (\overline {e_L } ~\gamma^\mu e_L + \overline {e_R } ~\gamma^\mu e_R), $$ so L-chiral electrons annihilate with R-chiral positrons, and R-chiral electrons annihilate with L-chiral positrons.
The secondary interaction is the charged weak current, which does not involve R-chiral leptons, violating parity maximally, $$ gW^-_\mu \overline {e_L } ~\gamma^\mu \nu_L + \mathrm {h.c.}, $$ so a L neutrino annihilates against a R positron, and a L electron against a R antineutrino, only.
The tertiary interaction is the P-violating, but not maximally, neutral current interaction, $$ \propto {-g\over 2\cos\theta_w}Z_\mu \overline {e_L } ~\gamma^\mu e_L +e\tan\theta_W \sin\theta_W Z_\mu \overline {e } ~\gamma^\mu e , $$ plus more obscure terms annihilating L-neutrinos with R-chiral antineutrinos. This term often confuses students, as less memorable; a mnemonic is taking the vanishing Weinberg angle limit. Electrons without subscript mean a sum of L and R-chiral components, which may now be involved in the weak annihilation into Zs, as in electromagnetism, albeit at a lower rate.
Be careful not to confuse parity, helicity, and chirality. Parity is the transformation which inverts a system’s coordinate axes, which has the same effect as a mirror reflection (plus a rotation). Because linear and angular momentum transform differently under inversion, a parity transformation changes a particle with left-handed spin polarization into a particle with right-handed spin polarization. In the relativistic limit, left-polarized particles become equivalent to left-chiral particles. Contrary to another answer, you can generate right-chiral fermions just by polarizing a relativistic beam. However, chirality is frame-dependent. In its rest frame, a massive particle is equal parts left- and right-chiral, thanks to the term $m\psi_L\psi_R$ in the Lagrangian.
The electromagnetic and strong interactions are symmetric under parity transformations, so the overall parity is a conserved quantum number in annihilations mediated by those interactions. For example, positronium may annihilate into two or three photons, depending on which is required to conserve the parity of the initial state.
Weak interactions are irrelevant for annihilation if a strong or electromagnetic annihilation process is also available, except perhaps for a tiny correction to the electromagnetic or strong annihilation. The weak charged current mediated by the $W$ bosons, interacts only with left-handed particles and right-handed antiparticles. But we usually think of charged-current processes as “decays,” rather than “annihilations.” The neutral current, mediated by the $Z$, has a parity-mixing term related to the Weinberg angle. But $Z$-mediated annihilation is only important for neutrinos, and we don’t know whether a neutrino and its antiparticle are different or not.
