Unitarity and amplitudes

Unitarity ($SS^dagger = 1$) dictates that begin{equation}label{key} T-T^dagger = iTT^dagger end{equation} For tree-level $2rightarrow 2$ scattering, we have then that $$ langle p_1,p_2|T|p_3,p_4rangle - langle p_1,p_2|T|p_3,p_4rangle^* = ilangle p_1,p_2|TT^dagger|p_3,p_4rangle. $$ Using $langle p_1,p_2|T|p_3,p_4rangle = (2pi)^4delta^{(4)}(p_1 + p_2 - p_3 - p_4)mathcal{A}[12rightarrow 34]$, and inserting a complete set of one particle states (to stay at tree-level), we can write this as begin{align} 2text{Im}(mathcal{A}[12rightarrow 34]) &= 2pisum_kint frac{d^3k}{2E_k}delta^{(4)}(p_1+p_2 - k)mathcal{A}[12rightarrow k]mathcal{A}^*[krightarrow 34]\ &= 2pisum_kint d^4kdelta(k^2)delta^{(4)}(p_1+p_2 - k)mathcal{A}[12rightarrow k]mathcal{A}^*[krightarrow 34] end{align} Now, the left hand side is the imaginary part of the 4pt amplitude, which we will take to have numerators $n_i$ and propagators $p_i^2 + iepsilon$, where $i$ labels the ways we can arrange a particle exchange (the $s,t,u$ channels). Thus, we have begin{equation} 2text{Im}left(sum_kfrac{n_k}{k^2+iepsilon} + text{contact}right) = 2pisum_kint d^4kdelta(k^2)delta^{(4)}(p_1+p_2 - k)mathcal{A}[12rightarrow k]mathcal{A}^*[krightarrow 34], end{equation} In a local theory of massless scalars, we can write the imaginary part of the propagator as $$ text{Im}left(frac{1}{p^2 + iepsilon}right) = frac{1}{2i}left(frac{1}{p^2 + iepsilon} - frac{1}{p^2 - iepsilon}right) = frac{-epsilon}{p^4 + epsilon^2}. $$ This seems like it vanishes for $epsilon rightarrow 0$, which, by the optical theorem, means that your amplitude must be zero for real external momenta. However, this is misleading and only true when the propagator is off-shell. Recognising the fact that the last term above is the nascent dirac delta function, we learn that $$ lim_{epsilonrightarrow 0}frac{-epsilon}{p^4 + epsilon^2} = pidelta(p^2). $$

Plugging this in, we find that, as the propagator goes on shell, we have begin{equation} 2pisum_kn_kdelta(k^2) = 2pisum_kint d^4kdelta(k^2)delta^{(4)}(p_1+p_2 - k)mathcal{A}[12rightarrow k]mathcal{A}^*[krightarrow 34]. end{equation} Or, in other words, the numerators of the tree-level amplitudes factorize into two lower-point amplitudes (the residues) as the propagator goes on-shell begin{equation} n_k = int d^4kdelta^{(4)}(p_1+p_2 - k)mathcal{A}[12rightarrow k]mathcal{A}^*[krightarrow 34] end{equation} We note now a major problem: the right hand side is actually zero due to momentum conservation, and this is probably the reason most books don't discuss the optical theorem at tree-level. This is due to the fact that Lorentz-invariant three-particle amplitudes vanish on-shell by virtue of the fact that $p_icdot p_j = 0$ for all $i,j$ due to momentum conservation. However, this is not true if we use spinor helicity variables and assume complex momentum, which is exactly what the bootstrapping amplitudes program does.