We calculate the CMB mu-distortion, , and the angular power spectrum of its cross-correlation with the temperature anisotropy, , in the presence of the non-Gaussian neutrino isocurvature density (NID) mode. While the pure Gaussian NID perturbations give merely subdominant contributions to and do not create , we show large can be realized in case where, especially, the NID perturbations S (x) are proportional to the square of a Gaussian field g (x), i.e. S (x) proportional to g(2) (x). Such Gaussian-squared perturbations contribute to not only the power spectrum, but also the bispectrum of CMB anisotropies. The constraints from the power spectrum is given by Pss (k(0)) similar to P-g(2) (k(0)) less than or similar to 10(-10) at k(0) = 0.05Mpc(-1). We also forecast constraints from the CMB temperature and E-mode polarisation bispectra, and show that P-g (k(0)) less than or similar to 10(-5) would be allowed from the Planck data. We find that and broken vertical bar l(l broken vertical bar 1) C-l(mu T)broken vertical bar can respectively be as large as 10(-9) and 10(-14) with uncorrelated scale-invariant NID perturbations for P-g (k(0)) = 10(-5). When the spectrum of the Gaussian field is blue-tilted (with spectral index n(g) similar or equal to 1.5), can be enhanced by an order of magnitude.
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