Monday, June 20, 2005 ... //

Chaudhuri on the Hagedorn "myth"

This short note is closely related to a previous article about the work of Dienes and Lennek.

IMPORTANT NOTE ADDED LATER: Although the topic and the character of the conclusions of the paper by Dienes and Lennek on one side and Chaudhuri on the other side may look similar, there are very important differences that can make one paper completely correct, if they're (or, on the contrary, she's) lucky, and the other incorrect. Please don't assume that the validity of these two papers is equivalent. And please be aware that I believe that these are kind of interesting and well-done papers, otherwise they would not be discussed here.

Back to the original text.

Tonight, it is Shyamoli Chaudhuri who is "dispelling the Hagedorn myth" (incidentally, it was already in 1965 when Hagedorn suggested that at high enough temperatures, open strings merge into a gas of chaotic long closed strings):

She calculates the thermal free energy - apparently in a different way than we are used to (from Atick and Witten and related works) - to conclude that the exponential growth of the states with the energy does not exist. In section 2.1 she argues that the growth of the number of states with the level does not imply the same growth of free energy as a function of temperature (or the density of states with the total energy). The true growth is slower, she says, making the full expression convergent. Nevertheless, she finds a first order phase transition at the T-self-dual temperature.

Her basic argument similar to the Dienes and Lennek's paper: the correct one-loop torus path integral only goes over the fundamental region of the modular group which removes the dangerous region with small "Im(tau)" and makes, according to her beliefs, the integral convergent for any temperature.

I encourage everyone for whose research and thinking the Hagedorn behavior is important to decide about the fate of the transition without any prejudices. After checking various things, I personally believe that the Hagedorn "folklore" will survive and both of the recent anti-Hagedorn papers are misled. (Chaudhuri is more radical because she seems to believe that the transition would be absent even in type 0 and other strings. It is much harder to isolate an error in the Dienes and Lennek's paper.)

The integral over the fundamental region combined with the summation over the two winding numbers that count how both circles of the worldsheet torus wind around the thermal circle in spacetime may be replaced by a full integral over the upper "tau" half-plane, which re-introduces the dangerous region with small "Im(tau)" and revives the "Hagedorn myth".

Technically, I think that her error is the step from (15) to (16) in her paper where she uses the Hardy-Ramanujan formula, assuming that the excitation of the string is very large, which removes by hand the actual divergence that would, in this calculational procedure, emerge from the thermal tachyon (the ground state of the winding sector "w=1" around the thermal circle in spacetime - in this sector the GSO projection is reversed) - a contribution that she neglects because the Hardy-Ramanujan formula is definitely not applicable for low-lying states such as this thermal tachyon.

This is an error in the approximate calculation; in the exact calculation, as a reader pointed out, there is a wrong factor of "(-1)^w" in equation (25) that breaks modular invariance.

Note that once you admit that the relevant CFT has a thermal tachyon, the discussion simply ends. With a thermal tachyon, the Hagedorn divergence arises from the region with large values of "Im(tau)", not small ones. And this "infrared" region is definitely not removed in string theory. To summarize, I now believe that if one defines the thermal stringy amplitudes in the most obvious stringy extension of the thermal rules of QFT, one finds the thermally wound tachyon whose mass determines the Hagedorn temperature, and at least one of the recent papers is not quite right.