# What is meant by a dispersion relation when complex integration in the Cutkosky rules' derivation or the computation of form factors $F(q^2)$ is used?

In the context of calculations of loop diagrams with Cutkosky-rules very often the concept of dispersion relation is mentioned. For instance this technique is used in Vol. 4 of Landau-Lifshitz for the computation of the polarization tensor in paragraph 110. It also exists a note of B.A.Kniehl "Dispersion Relations in Loop Calculations" hep-ph/9607255 that is devoted to the subject and uses as an example the computation of the photon vacuum polarization.

I will give here as an example the integral representation of the photon self-energy tensor(also to be found in Peskin& Schroeder):

$$\Pi_{\mu\nu} = \Pi(k^2)\left(g_{\mu\nu} - \frac{k_\mu k_\nu}{k^2}\right)$$

defined in Landau-Lifshitz paragraph 101 and further discussed in paragraph 108: There the function $$\Pi(t)$$ is represented as an integral along a branch cut:

$$\Pi(t) = \int_0^\infty \frac{\rho(t') dt}{t-t' +i\epsilon}$$

where $$\rho(t)$$ is called the spectral density function. More below it can be found the the spectral density function can be expressed by the imaginary part of $$\Pi(t)$$:

$$Im(\Pi(t)) = -\pi \rho(t)$$

so that one can write:

$$\Pi(t) = \frac{1}{\pi}\int_0^\infty \frac{Im(\Pi(t')}{t'-t-i\epsilon} dt'$$

This formula is quoted by Landau & Lifshitz as dispersion relation. Actually, similar formulas can be found in more modern books as Peskin& Schroeder in the sections on the optical theorem and Cutkosky rules, however, the concept dispersion relation is not mentioned.

Elder books like the 2nd volume of Bjorken & Drell dedicate a whole chapter of 75 pages on the subject. They start talking about impinging and scattered wave amplitudes (but not about a relation between frequency and the wave vector) and mention at the end of the chapter's first paragraph the Kramer-Kronig relations.

I even found the concept dispersion relation mentioned in the A.Chao's book on "Physics of collective beam instabilities in High-Energy Accelerators" in section 5.2 when the author discusses the damping effect (also called Landau damping) of a whole ensemble of particles undergoing oscillations driven by wake fields if the oscillation frequency of the particle ensemble has spectral density $$\rho(\omega)$$ (different from a delta function), i.e. particles can oscillate at different frequencies. For a consistent oscillatory motion a self-consistency condition has to be fulfilled:

$$1 = - \frac{N r_0 {\cal {W}}}{2\omega_\beta \gamma T_0}\left[ P.V. \int \frac{\rho(\omega)}{\omega-\Omega}d\omega + i \pi \rho(\Omega)\right]$$

which Chao also calls a dispersion relation. $${\cal{W}}$$ stands for the wake field, the rest are some normalizing factors which are here of little interest.

I absolutely see the similarity in the problems mentioned, however, why would one call such a relation dispersion relation ? I would call such a relation as corollary of Cauchy's integral theorem, or Hilbert transform, or Kramer-Kronig relation, but why dispersion relation ? Actually, on Wikipedia a dispersion relation is defined as relation between frequency and wavelength respectively wave vector. So actually I my question is mainly about terminology. But I guess there is some physics behind why this type of relations got the name dispersion relation.

But it seems that nowadays this technique is a kind of old-fashioned as it was used a lot in S-matrix theory (bootstrap etc.) which seems to have only a few adepts today.

You are correct that these relations are what we would today call Kramers-Kronig relations, but the place where physicists, especially early quantum theorists, most likely knew the concept of Kramers-Kronig relations from is from the notion of complex refractive indices, where it relates the absorption and dispersion to each other, and one form of the Kramers-Kronig relations for optics says that $$n(\omega) = 1 + \frac{c}{\pi}\int \frac{\alpha(\omega')}{\omega'^2 - \omega^2} \mathrm{d}\omega'$$ for $$n(\omega)$$ the refractive index and $$\alpha(\omega)$$ the absorption coefficient. This is a dispersion relation since it's an equation for the refractive index and it does look a lot like the equations you've quoted if we squint, and so the name probably stuck to the HEP application of Kramers-Kronig relations that you have found even though they are not literally related to dispersion of waves.