While thinking about how to answer a "describe string theory" question, I remembered an old argument of Stanley Mandelstam's that linear Regge trajectories implies stability. I never fully understood the exact argument, or its limitations. (The article is "Dual-Resonance Models" from 1974, sciencedirect.com/science/article/pii/0370157374900349, thanks for digging it up)

Here is the argument as I remember it: consider the Regge trajectory function $\alpha(s)$, and expand it in a dispersion relation with two subtractions:

$$ \alpha(s) = b + as + {1\over i\pi} \int_0^\infty {\mathrm{Im}(\alpha(s'))\over s-s'} ds'$$

Where a,b are constants. Mandelstam says that the imaginary part of $\alpha(s)$ is a measure of some sort of instability or decay of the string states, so if the string resonances are exactly stable, then the imaginary part is zero, and the trajectory is linear.

This argument bugged me for these reasons

  • It seems to work with no subtractions, with one subtraction, with two subtractions, etc. Can you conclude that exactly constant, exactly linear, exactly quadratic Regge trajectories are also stable? Maybe exactly constant trajectories at special values can be interpreted as free stable point particles of a given spin and mass, but what are quadratic trajectories exactly? Does he mean to argue that nearly linear trajectories are necessarily stable when they are exactly linear? What is the precise conclusion?
  • The Regge trajectory function appears in an exponent in the scattering amplitude, so you have to take a log to extract it. Why you don't get a cut contribution from taking the log of negative values which has nothing to do with physics, just from the Regge scattering ansatz?
  • Actual scattering amplitudes are combinations of different trajectory contributions, so why does each individual trajectory have to be separately analytic, with singularities that are determined by physics? Is this a heuristic assumption?
  • Ok, even if you don't have cuts from the log, the expansion suggests that an imaginary part of the trajectory function, like the imaginary part of a two-point function, has some sort of physical interpretation that allows you to interpret it immediately as some sort of decay rate. What is that physical interpretation? Does it work far away from linear trajectories?
  • How does this argument translate to modern string theory? I have never seen this kind of argument anywhere else.

Mandelstam is still around, but I don't know him, and I think that someone else who is better versed in Regge theory than I am will know the answer immediately, because Mandelstam states it in two lines, without explanation, so it must be obvious.

Later Comment: Having received a short cryptic answer from Mandelstam regarding this, I started thinking about it again, and I gave a partial answer below. But I am not really satisfied with it yet. But one of my complaints, though, that separating out the individual trajectory and expanding it by analytic dispersion relations might fail, is not really serious. The trajectory exchange amplitude is the sum of the resonances on the trajectory, so it is always separately analytic. Why it's log should also be analytic to justify the dispersion expansion is probably equally easy to see, but I don't see it

  • $\begingroup$ I wish I knew enough to answer this... as it stands I'll be interested to see what anyone else can come up with. $\endgroup$
    – David Z
    Aug 27, 2011 at 22:12
  • $\begingroup$ I asked Mandelstam by email just now, maybe he'll respond. I already know that this is true in string perturbation theory, but I want to understand how he knew this before having a theory. $\endgroup$
    – Ron Maimon
    Aug 27, 2011 at 22:16
  • $\begingroup$ Enlosing your main equation in \$\$...\$\$ to give it a line by itself would make it more readable $\endgroup$ Aug 28, 2011 at 13:45
  • $\begingroup$ I got this answer from Mandelstam: "The reason is tht the decay time of a resonance is inversely proportional to the imaginary part of the pole position. With linear trajectories the imaginary part is zero, so the decay time is infinite, i. e., the resonance is stable." Well, I knew that already, and it doesn't help at all. $\endgroup$
    – Ron Maimon
    Aug 28, 2011 at 14:40
  • $\begingroup$ @Ron: maybe you could point him to this site. It'd be awesome, if someone like him would leave an answer here. $\endgroup$
    – luksen
    Aug 29, 2011 at 15:59

1 Answer 1


I will try to expand Mandelstam's cryptic comment into a full answer, because I think I am starting to understand his argument. I have very little intuition for the analytic structure of trajectory functions, and when I read Mandelstam, I get the feeling of a great lost world of analytic constraints which have been dropped into the memory hole.

Anyway, if resonances are exactly stable, their pole position is real. This means that the scattering amplitude is infinitesimal, and concentrated right near mass-shell, because a particle is produced and propagates very far before being absorbed. So the scattering amplitude is real to zeroeth infinitesimal order.

The assumption Mandelstam is making, as I see it, is analyticity of the most radical kind--- every trajectory function is separately analytic and its contribution is described by the exchange of its resonances only. When you take the log in s, you get the trajectory function, and he is also assuming that this function has a convergent dispersion integral after two subtractions, meaning that once you start with an approximation of straight-line trajectories, any correction is perturbative, and this is justified by the statement that the trajectories are approximately linear, not approximately quadratic, or approximately cubic.

Then if the resonances have no imaginary part, the trajectory function has no imaginary part either, because whenever it hits an integer angular momentum, it becomes a resonance. The argument then follows.

This argument is a convincing argument that there exists a perturbation expansion which starts with linear Regge trajectories, but it is not at the level of rigor of a matheamtical proof, nor should it be expected to be.


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