Lubos Motl

2005-06-20 00:46:00 UTC

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http://motls.blogspot.com/2005/06/chaudhuri-on-hagedorn-myth.htmlRaw Message

This short note is closely related to a previous article about the work of

Dienes and Lennek. 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):

hep-th/0506143

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 is the same as in 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.)

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.

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

path-integral rules of QFT, one finds the thermally wound tachyon whose

mass determines the Hagedorn temperature, and the new critical papers

fail.

Feel free to disagree.

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