What is the Renormalization Group Flow of Gravitons in String Theory?

[Haelfix]

I can't seem to get an answer on sps, so maybe someone here knows.

Typically in Quantum gravity, people deal with the truncated graviton series (since its not renormalizable) following Wilson's ideas. Its usually cut off at 1 loop. It still gives a value, and people take it fairly seriously, at least they did back in the 70s.

Now, in string theory the graviton is renormalizable and also free of anomalies, sooo has anyone calculated Stringy Graviton diagrams to say two loops and compared numbers? Better yet, what does the renormalization group flow look like?

Forgive the ignorance if I'm asking a stupid question, I know little about Stringy mechanics. I know in other problems where some dumb particle theorist asks for something he knows about, computed in String calculations, he often gets poopoo'd since fields don't seem to be seperable like in naive qft and giving an answer seems to be non trivial. Is this also the case for Gravitons.

Make whatever simplifications and assumptions you like, (eg perhaps deal with the bosonic string) I just want to look at the differences between series.

 

[sol2]

http://wc0.worldcrossing.com/WebX?14@187.PFGybxg9W7t.8@.1dde4729/6 Here is a representation that includes images for consideration.


http://superstringtheory.com/basics/anims/photon2.gif

I wanted to look at what you were saying to "try," and understand.

http://www.maths.soton.ac.uk/relativity/GRExplorer/buttons/Grav_Waves.gif

http://www.sukidog.com/jpierre/strings/gravmove.gif

This mode is characteristic of a spin-2 massless graviton (the particle that mediates the force of gravity). This is one of the most attractive features of string theory. It naturally and inevitably includes gravity as one of the fundamental interactions.

http://www.sukidog.com/jpierre/strings/scatmov.gif

http://www.sukidog.com/jpierre/strings/basics.htm
----------------------------------------------------

By looking at the quantum mechanics of the relativistic string normal modes, one can deduce that the quantum modes of the string look just like the particles we see in spacetime, with mass that depends on the spin according to the formula

http://superstringtheory.com/equations/regge.gif

Remember that boundary conditions are important for string behavior. Strings can be open, with ends that travel at the speed of light, or closed, with their ends joined in a ring.

http://superstringtheory.com/basics/basic4a.html

 

[Javier]

Now, in string theory the graviton is renormalizable and also free of anomalies, sooo has anyone calculated Stringy Graviton diagrams to say two loops and compared numbers?
...fields don't seem to be seperable like in naive qft and giving an answer seems to be non trivial...

The field theory of linearized gravity is a truncation of the full string description. If you expand the string Lagrangian out in terms of graviton fields, you get linearized gravity Lagrangian (which you can write in terms of the Ricci scalar...this is the Einstein-Hilbert Lagrangian)+ higher order corrections in powers of the curvature. If you are going to truncate this expansion so that you are no longer doing full string theory, but rather a field theory, then you must truncate all the way to the linearized theory.
Now, as you said, doing quantum field theory with linearized gravity (i.e. described by a graviton field) leads to a theory that is not very useful: whether matter is coupled or not, each higher order term after a couple of loops diverges hopelessly.
But string theory says that GR is modified at more fundamental scales and that the Lagrangian should really be an infinite series of corrections: the "limit of this series" is simply the string Lagrangian for our graviton. The calculation of graviton scattering in string theory was done in the early eighties, and this is the reason for the initial burst of interest: the scattering amplitude at each order (indexed by the genus of the worldsheet) is finite. You can find calculations in the original papers, or in Green, Schwarz, and Witten's "Superstring theory" volume 2. Overall, the nice behavior of string amplitudes at each order in a perturbative expansion is a sign that generalizing Yang-Mills field theories and general relativity in this way is on a right track.
However, the full series for a scattering amplitude diverges in general. Therefore, perturbative string theory seems to be an effective theory for a more fundamental theory.

 

[selfAdjoint]

However, the full series for a scattering amplitude diverges in general. Therefore, perturbative string theory seems to be an effective theory for a more fundamental theory.

Javier, could you expand a little on this last sentence? By "known to diverge" do you mean from the theory or by calculating successive orders? Is there a pole like the Landau pole of QED? (I would suspect not, since no renormalisation, but does the math of limit points apply anyway?)

 

[Haelfix]

In field theory, one typically can see that even a renormalizable theory in 4dimensions is divergent, as it is not Borel Resummable.

One can see this fairly trivially, as a general series typically has one or two graph counting features that manifest in the form of powers of factorials. Even if there is exponential damping, that won't kill the factorials fast enough. (Alternatively, you can think of one of the factorials in the form of instanton contributions)

In nonrenormalizable theories, its worse as you get an infinite amount of counterterms in order to make things finite.

What I don't understand, is precisely why you have truncate all the way down to linearized gravity in String theory. Surely perturbative string theory is valid at scales that exceed naive quantum gravity, you should be able to get a few more good terms, even if there is an ultimate cutoff.