# How Exactly Does Linear Regge Trajectories Imply Stability?

+ 6 like - 0 dislike
39 views

(for a more muddled version, see physics.stackexchange: http://physics.stackexchange.com/questions/14020/whats-with-mandelstams-argument-that-only-linear-regge-trajectories-are-stable)

There is a 1974 argument of Mandelstam's that linear Regge trajectories implies stability, from "Dual-Resonance Models" from 1974, sciencedirect.com/science/article/pii/0370157374900349. Expand the Regge trajectory function $\alpha(s)$ in a dispersion relation with two subtractions:

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

The imaginary part of $\alpha(s)$ gives the decay of the string states, since where it hits an integer tells you where the poles are. 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 just as well with two subtractions, three subtractions, etc. Can you conclude that exactly quadratic or exactly cubic Regge trajectories are also stable? What is a quadratic or cubic trajectory?
• The Regge trajectory function appears in the exponent, so you have to take a log to extract it. Why is it clear that it has a representation like the above, without a cut contribution at negative s?
• In string theory, the trajectories are linear when they are long-lived, but the trajectory function doesn't look as fundamental today. Is there a more modern formulation of this, which would tell you which string limits are non-interacting just from a condition on the spectrum?

Mandelstam generously emailed me a short remark, saying essentially that the trajectory function imaginary part is a lifetime, and indeed this is obvious from the fact that it gives the position of the resonances, but I am still confused regarding the questions above.

Even a partial answer would be appreciated.

This post has been migrated from (A51.SE)
retagged Apr 19, 2014

+ 4 like - 0 dislike

I don't have any sharp answer, but the argument seems sketchy. I think we know in the infinite-$N_c$ limit of QCD that we have exactly stable resonances, and some nearly-linear Regge trajectories in some region, but that they're not perfectly linear and they fail badly to be linear at negative s where BFKL describes Regge physics. http://arxiv.org/abs/hep-th/0603115 by Brower, Polchinski, Strassler, and Tan looks at this kind of thing in some detail, and might point to some older literature that has something to say.

This post has been migrated from (A51.SE)
answered Oct 9, 2011 by (1,630 points)
The word "sketchy" is ambiguous, you probably mean "wrong". I know the Brower Polchinski Strassler Tan stuff, and their argument links up perturbative BFKL to nonperturbative pomeron. I believe that the turnaround location depends on N in a way consistent with Mandelstam, so that in the pure large N limit the pomeron trajectory is straight. I am not sure, however, but it is a good thing to verify. Thanks for the answer.

This post has been migrated from (A51.SE)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverf$\varnothing$owThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.