S-matrices and bi-linear sum rules of conserved charges in affine Toda field theories

The exact quantum $S$-matrices and conserved charges are known for affine Toda field theories(ATFTs). In this note we report on a new type of bi-linear sum rules of conserved quantities derived from these exact $S$ matrices. They exist when there is a multiplicative identity among $S$-matrices of a particular ATFT. Our results are valid for simply laced as well as non-simply laced ATFTs. We also present a few explicit examples.