Pauli moment of a massless electron gas ($\mathcal{M}=3N{\mu }^{2}B/(c{p}_F)+O(B^3)$)

For noninteracting spin-half particles with dispersion $E=cp$, the two spin populations at common chemical potential $\epsilon$ are $V(\epsilon \pm \mu B{)}^3/(6{\pi }^2{\hslash }^3{c}^3)$. Fixed total number changes $\epsilon$ only at quadratic order in $B$. Their population difference is therefore $3N\mu B/(c{p}_F)$ to first order, giving the displayed magnetic moment. Only the specified spin-energy splitting is included, not orbital magnetic effects.