Revisiting the Exact Dynamical Structure Factor of the Heisenberg Antiferromagnetic Model

We revisit our initial derivations of the exact 2-spinon $S_2$ and 4-spinon $S_4$ dynamical structure factors (DSF). First we show that the latter derivations had normalization factors that are twice and quadriple the correct ones respectively. This means that $S_2$ contributes not 72% as was previously thought but 36% to the total DSF. We also calculated the contribution of $S_4$ to be between 18% and 20% and not 27% as was calculated by Caux and Hagemans. In fact we show that the latter reference had also the normalization factor twice the correct value and had it done the numeric integrations correctly it would have obtained a contribution between 36% and 40% for $S_4$. Furthermore, we prove that its claim that our initial derivation of $S_4$ was also incorrect in its dependency on the spectral parameters is incorrect because fixing the momentun transfer $k$ up to $2\pi$ as the latter reference did to justify its claim only amounts to mutltiplying the overall factor by 2 because as we will prove in this paper $S_4$ is periodic in $k$ with period $2\pi$. Also in this paper we derive $S_n$ for general $n$ by following a different approach compared to our initial derivation of $S_4$. Although for $S_4$ both the new derivation and the initial one lead to equivalent formulas that are expressed differently, the new form presented in this paper is much more elegant and compact and also reveals new hidden and nontrivial symmetries which substantially simplify the numeric evaluation of $S_4$ and its sum rules. Moreover based on the results of this paper we propose a simple approximation to the total DSF of the Heisenberg model. Finally we comment on how our work might resolve the discrepancy between the exact theoretic results and experiemntal data as reported by Zaliznyak et al.