triad hmhd
$ T_{S}( k, s_{k}; p, s_{p}; q, s_{q} ; \alpha, \tilde\Omega ) $ = \frac{s_{k} s_{p} s_{q}}{\alpha \mu_{k,s_{k}} \mu_{p,s_{p}} \mu_{q,s_{q}}} \Bigg[ -{\lambda_{k}}^{s_{k}} {\lambda_{p}}^{s_{p}} {\lambda_{q}}^{s_{q}} + \frac{ ( {\lambda_{k}}^{s_{k}} - s_{k} \alpha k ) ( {\lambda_{p}}^{s_{p}} - s_{p} \alpha p ) ( {\lambda_{q}}^{s_{q}} - s_{q} \alpha q ) }{ 2 \alpha \tilde\Omega + 1 } \Bigg]
$ = \frac{s_{k} s_{p} s_{q}}{\alpha \mu_{k,s_{k}} \mu_{p,s_{p}} \mu_{q,s_{q}}} \Bigg[ -{\lambda_{k}}^{s_{k}} {\lambda_{p}}^{s_{p}} {\lambda_{q}}^{s_{q}} + ( 1 - 2 \alpha \tilde\Omega + \cdots ) ( {\lambda_{k}}^{s_{k}} - s_{k} \alpha k ) ( {\lambda_{p}}^{s_{p}} - s_{p} \alpha p ) ( {\lambda_{q}}^{s_{q}} - s_{q} \alpha q ) \Bigg]
$ -{\lambda_{k}}^{s_{k}} {\lambda_{p}}^{s_{p}} {\lambda_{q}}^{s_{q}} + ( 1 - 2 \alpha \tilde\Omega + \cdots ) ( {\lambda_{k}}^{s_{k}} - s_{k} \alpha k ) ( {\lambda_{p}}^{s_{p}} - s_{p} \alpha p ) ( {\lambda_{q}}^{s_{q}} - s_{q} \alpha q )
$ = - 2 \alpha \tilde\Omega {\lambda_{k}}^{s_{k}} {\lambda_{p}}^{s_{p}} {\lambda_{q}}^{s_{q}} - ( s_{k} \alpha k {\lambda_{p}}^{s_{p}} {\lambda_{q}}^{s_{q}} + c.p. )