Parity violation of primordial magnetic fields in the CMB bispectrum

We study the parity violation in the cosmic microwave background (CMB) bispectrum induced by primordial magnetic fields (PMFs). Deriving a general formula for the CMB bispectrum generated from not only non-helical but also helical PMFs, we find that helical PMFs produce characteristic signals, which disappear in parity-conserving cases, such as the intensity-intensity-intensity bispectra arising from $\sum_{n=1}^3 \ell_n = {\rm odd}$. For fast numerical calculation of the CMB bispectrum, we reduce the one-loop formula to the tree-level one by using the so-called pole approximation. Then, we show that the magnetic anisotropic stress, which depends quadratically on non-helical and helical PMFs and acts as a source of the CMB fluctuation, produces the local-type non-Gaussianity. Comparing the CMB bispectra composed of the scalar and tensor modes with the noise spectra, we find that assuming the generation of the nearly scale-invariant non-helical and helical PMFs from the grand unification energy scale ($10^{14} {\rm GeV}$) to the electroweak one ($10^{3} {\rm GeV}$), the intensity-intensity-intensity bispectrum for $\sum_{n=1}^3 \ell_n = {\rm odd}$ can be observed by the WMAP experiment under the condition that $B_{1 \rm Mpc}^{2/3} {\cal B}_{1 \rm Mpc}^{1/3} > 2.7 - 4.5 {\rm nG}$ with $B_{1 \rm Mpc}$ and ${\cal B}_{1 \rm Mpc}$ being the non-helical and helical PMF strengths smoothed on 1 Mpc, respectively.