Does the renormalization group apply to string theory?

Question

Can we implement a scale dependent cutoff Λ to string theory? Can we perform a renormalization group analysis of string theory consistently?

Answer 1

Good question. The short answer is no, cutoff scales have no relevance to string theory.

Cutoff scales are given by maximum or minimum energies or distances where the given theory may be applied. This concept is only useful because in quantum field theory, such cutoffs are natural regulators to get rid of short-distance divergences. These divergences reflect the fact that field theories should only be thought of as approximate theories that are only valid at distances longer than the cutoff distance.

In the case of renormalizable quantum field theories, the cutoff distance may be made arbitrarily short in principle; in the case of non-renormalizable quantum field theories, it may not; in those cases, the theory is literally ill-behaved at short enough distances. However, in both cases, some cutoff is useful because only if this characteristic scale is finite, one may assume that the fundamental parameters of the Lagrangian are finite.

String theory doesn't suffer from any short-distance problems. It's an almost defining property of string theory, one that can really be demonstrated even in its simplest "toy models" such as bosonic string theory. That means that it is incorrect to think of string theory as an approximate theory that only works at long enough distances. Quite on the contrary, the very purpose of string theory - one of its main advantages over quantum field theory - is that it is valid up to arbitrarily short distances, despite the fact that the gravitational interaction is one of the major predictions.

So the integrals in string theory that replace the divergent integrals in quantum field theory are convergent; there are no short-distance divergences we get in string theory at all. For this reason, there is no reason to cutoff the integrals i.e. to introduce cutoffs in general.

The characteristic scale of perturbative string theory is the string scale - and at the non-perturbative level, various Planck scales may be more universal typical distance scales - but that doesn't mean that string theory breaks down at the string scale. Quite on the contrary, the bulk of computations in string theory is exactly about the exact physics that takes place at the string scale energies or slightly larger ones. To describe (much) lower energies than the string scale, an effective quantum field theory would be enough.

So string theory incorporates all the effects from particles that are 2 times or 10 times heavier than the string scale automatically. It's how its formalism works. There is no reason to have cutoffs. They exist in effective field theories - which are assumed to be long-distance approximations of whatever more detailed theory that is relevant at all scales. String theory is such a theory that is relevant at all scales.

However, descriptions of stringy physics in terms of effective field theories are omnipresent. Those universally depend on a cutoff, much like effective field theories in any other context.

String field theory: level truncation

Just an addition. There is an approach to string theory called string field theory - which leads to the same results as more standard calculations in string theory but that is formulated to be as similar to quantum field theory as possible. In particular, string field theory is a quantum field theory with infinitely many fields that are associated with the individual string vibration patterns.

In string field theory, people often do numerical calculations - e.g. solutions with D-branes and tachyon condensation - and they often do so numerically (although analytic solutions became common after the groundbreaking advances by Martin Schnabl and others a few years ago). When numerical calculations are being done, one can't really deal with infinitely many fields. In that case, people only deal with fields corresponding to "light enough particles", e.g. those that are lighter than $\sqrt{13} m_{s}$. This scheme looks like a refined cutoff - and it is known as the level truncation scheme.

But the reason why the levels are truncated is somewhat different from effective field theory. In effective field theory, a cutoff is needed for finite results. In string field theory, level truncation is just a way to make numerical calculations more tractable; however, the exact results that include the whole infinite tower of states are still totally finite.