In Bootstrap and Amplitudes: A Hike in the Landscape of Quantum Field Theory there are few statements about analytical structure of amplitudes.

I want to understand statement:

In a local theory of massless scalars, they can have simple poles and the residue of such a pole is, by unitarity, a product of lower-point amplitudes.

I understand this property from Feynman rules for tree level amplitude, but I didn't get role of unitarity.

I wanna understand, how unitarity related to such property? How residues will change, if we lose unitarity?


Unitarity ($SS^\dagger = 1$) dictates that \begin{equation}\label{key} T-T^\dagger = iTT^\dagger \end{equation} For tree-level $2\rightarrow 2$ scattering, we have then that $$ \langle p_1,p_2|T|p_3,p_4\rangle - \langle p_1,p_2|T|p_3,p_4\rangle^* = i\langle p_1,p_2|TT^\dagger|p_3,p_4\rangle. $$ Using $\langle p_1,p_2|T|p_3,p_4\rangle = (2\pi)^4\delta^{(4)}(p_1 + p_2 - p_3 - p_4)\mathcal{A}[12\rightarrow 34]$, and inserting a complete set of one particle states (to stay at tree-level), we can write this as \begin{align} 2\text{Im}(\mathcal{A}[12\rightarrow 34]) &= 2\pi\sum_k\int \frac{d^3k}{2E_k}\delta^{(4)}(p_1+p_2 - k)\mathcal{A}[12\rightarrow k]\mathcal{A}^*[k\rightarrow 34]\\ &= 2\pi\sum_k\int d^4k\delta(k^2)\delta^{(4)}(p_1+p_2 - k)\mathcal{A}[12\rightarrow k]\mathcal{A}^*[k\rightarrow 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 $p_i^2 + i\epsilon$, where $i$ labels the ways we can arrange a particle exchange (the $s,t,u$ channels). Thus, we have \begin{equation} 2\text{Im}\left(\sum_k\frac{n_k}{k^2+i\epsilon} + \text{contact}\right) = 2\pi\sum_k\int d^4k\delta(k^2)\delta^{(4)}(p_1+p_2 - k)\mathcal{A}[12\rightarrow k]\mathcal{A}^*[k\rightarrow 34], \end{equation} In a local theory of massless scalars, we can write the imaginary part of the propagator as $$ \text{Im}\left(\frac{1}{p^2 + i\epsilon}\right) = \frac{1}{2i}\left(\frac{1}{p^2 + i\epsilon} - \frac{1}{p^2 - i\epsilon}\right) = \frac{-\epsilon}{p^4 + \epsilon^2}. $$ 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 $$ \lim_{\epsilon\rightarrow 0}\frac{-\epsilon}{p^4 + \epsilon^2} = \pi\delta(p^2). $$

Plugging this in, we find that, as the propagator goes on shell, we have \begin{equation} 2\pi\sum_kn_k\delta(k^2) = 2\pi\sum_k\int d^4k\delta(k^2)\delta^{(4)}(p_1+p_2 - k)\mathcal{A}[12\rightarrow k]\mathcal{A}^*[k\rightarrow 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^4k\delta^{(4)}(p_1+p_2 - k)\mathcal{A}[12\rightarrow k]\mathcal{A}^*[k\rightarrow 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 $p_i\cdot p_j = 0$ 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.

  • $\begingroup$ Thank you,but I didn't understand, how from residues we can see such factorisation. Also my question about tree level amplitudes, but your discussion is general. Maybe there are some simplifications in tree level? $\endgroup$ – Nikita Oct 9 '20 at 21:44
  • $\begingroup$ I've updated the answer to specifically focus on tree-level. $\endgroup$ – Akoben Oct 10 '20 at 11:58
  • $\begingroup$ Beautiful answer! Why numerator is real? $\endgroup$ – Nikita Oct 10 '20 at 19:25
  • 1
    $\begingroup$ The numerator is a Lorentz invariant that can only be made out of products of momenta or products of momenta with polarization tensors. For this argument, it is assumed to be real, although in reality you could consider complex momenta (as in BCFW, for example). $\endgroup$ – Akoben Oct 11 '20 at 20:40

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