Two-Loop Superstring Amplitudes

Eric D’Hoker and D.H. Phong this past week finally posted two crucial papers with results from their work on two-loop superstring amplitudes. The first one shows gauge slice independence of the two-loop N-point function, the second shows that, for N less than 3 and for low-order terms at N less than 4, there are no two-loop corrections to the low energy effective action.

D’Hoker and Phong have been studying superstring amplitudes for nearly twenty years, and are justly proud of their recent results, which are a tour de force of careful calculation. Over the years there have been many claims made about two-loop amplitudes, but until their work, no one had managed to really sort out the gauge dependence issues and write down gauge-independent amplitudes. For some comments about some of the issues involved at genus 2 and higher, see postings by Jacques Distler here, here, and here.

I don’t think D’Hoker and Phong will be coming out with complete results for genus 3 anytime soon, so the state of the art is that there is now a finite and well-defined version of the two-loop superstring amplitudes, with the problem of higher loops still open. While claims abound about the finiteness of higher-loop amplitudes, before believing them one should first take a look at the tricky problems that D’Hoker and Phong had to overcome to get well-defined two-loop amplitudes.

Update: Jacques Distler has a new posting about multi-loop amplitudes and potential problems with the Berkovits version of the superstring (he explains in more detail the possible problems with the BRST and picture-changing operators I mentioned). For some mysterious reason Jacques neglects to refer to my posting or comments about this. I encourage those commenters who seemed convinced I didn’t know what I was talking about to now take up their arguments with him.

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21 Responses to Two-Loop Superstring Amplitudes

  1. Peter says:

    What would convince me of finiteness of these perturbative amplitudes would be hearing from the experts in these calculations that they understand theme well-enough to have a solid argument for finiteness. That’s not what those I’ve talked to are telling me.

    I’ve given some detailed technical arguments about this and you’re not addressing them or showing any signs of having looked into the details of what is going on in these two-loop calculations enough to even understand what I am talking about.

    If you talk to people who have done these recent two-loop calculations (d’Hoker-Phong, another group is Zheng-Wu-Zhu) or read their papers they’ll explain to you the subtleties that happen at two loops and that they don’t know what happens at three loops. As for covariant approaches like that of Berkovits, it is simply undeniable that he has yet to calculate a 2-loop amplitude.

    If you don’t believe me, contact any of these experts, or sit down and read some of their papers. Until you’ve done that, you have no business going around making claims that finiteness of multiloop amplitudes is well-established fact. It isn’t and such claims are extremely unfair to people who are working hard to actually understand what is going on, rather than making unsupported claims about the issue.

  2. Anonymous says:

    Ok, you win, it is hard to imagine what will covince you of finiteness of perturbative string theory, you seem to be attached to the idea that it is insufficiently established. On the other hand you are quite convincing to your readers (there is a selection effect in place, of course).

  3. Steve says:

    D’Hoker and Phong are doing superstring theory the way it needs to be done: facing up squarely and honestly to specific and very hard technical problems, sticking with them over time, and getting rigorous results. I think this is the only way one can approach superstring theory because it remains a very wild beast to tame. This is in contrast to the waffling and speculative Landscape and string cosmology papers of recent times. Years ago I came across a review article in Physics Reports I think, written by D’Hoker and Phong on Differential Geometry and it was very readable and illuminating. These guys do know their stuff.

  4. DMS says:


    Many, many thanks for your explanations and for pointing out a nice (very readable!) paper!

    The D=11 supergravity starts having divergences at 2-loops.

    The conclusion seems quite strong (for perturbative supergravity, at least):
    “…there is no local, unitary (ghost/tachyon-free) quantum field
    theory whose action reduces to QGR or classical GR that is also free of infinities; the latter are
    almost certainly there at every order, requiring an infinite number of input parameters to define
    these theories. The conclusion includes all possible SUGRA models, i.e., from D=4 through D=11, as well. Although the presence of new counterterms at all loop orders (or at an infinite set of them) cannot reasonably be rigorously demonstrable, the fact that the ones we did see appeared at lowest permitted order (so that no “hidden” invariances prevented them), is quite convincing evidence.”

  5. Peter says:

    Yes, the general assumption is that string perturbation theory gives an asymptotic series, and so people like to claim that the situation is no worse than in QFT, where the same thing happens. But, at least for non-abelian gauge theories, in that case there is a simple non-perturbative definition of the theory which appears to be well-defined and give finite answers.

    Again, there’s no evidence of a problem that will occur at higher loops in superstring multi-loop amplitudes, there just isn’t a solid argument showing that no problems can occur. I’d have to look up the precise facts about supergravity, I believe they depend on N, the number of supersymmetries. But it has been shown that the known symmetries of the theory are such that at high enough loops you get terms that have no reason to be zero and are presumably divergent. It is true that these problems are at high enough loops that no one has been able to actually compute the diagrams to be sure the infinities are really there.

    A quick search turned up this reference

  6. DMS says:


    It is pretty sad to see (some) string theorists being unreliable even about technical aspects of the subject. Prior to this, I thought string theory had already been “proven finite” to all orders: thanks for pointing out D’Hoker-Phong’s remarakable paper.

    A couple of questions:
    “…even if the multiloop amplitudes are finite, their sum isn’t”

    Does that mean string perturbation theory series is an asymptotic series?

    “Maybe someone will find that, just like in supergravity, if you go to high enough order, you run into trouble. ”

    Surprising result to me. Is it easy to understand why this happens? A reference perhaps?

  7. Peter says:

    I think there are still potential problems with singularities in superstring amplitudes at the boundaries in moduli space, not just the interior. But sure, the reason you give was a good reason for Schwarz et. al to look into superstring theory as a theory of quantum gravity 30 years ago.

    But for the last 20 years this idea has been beaten to death by thousands of theorists writing tens of thousands of papers on the subject. The bottom line of all this work is that, even if the multiloop amplitudes are finite, their sum isn’t, so this idea doesn’t give a well-defined, finite theory of quantum gravity. There’s a lot of wishful thinking about hypothetical M-theory and non-perturbative versions of string theory, but wishful thinking is all it is right now. The main achievement of this whole program, the supposed calculation of the entropy of certain special black-hole like configurations using branes, is pretty underwhelming.

    You can’t just keep repeating the initial motivation for a speculative idea, long after a huge amount of effort has gone into working on it, with extremely disappointing results. At some point you have to admit failure and move on, something that most string theorists seem incapable of.

  8. Anonymous says:

    OK, just to summarize the situation in my mind: there is a powerful physical mechanism that cancels the divergences that cause real problems for any other attempts to quantize gravity. This is the old insight that caused many, who were not born string theorists, to be attracted to string theory as a quantum theory of gravity (if not as a theory of particle physics), and what is referred to by them as finiteness. The potential problems with the interior of moduli space have to be sorted through, but are as far as we can tell they are just a technicality. Regardless of the hype, the semantics, etc., if you were interested in quantum gravity, is it not a good enough reason to work on the field?

  9. Anonymous says:

    OK, I think we do agree on the facts, though I seem to give much more weight to that UV finiteness and unitarity, again since it is not acheivable in any other context. As for your points:

    1. The particle like limit of the string corresponds to various boundaries of moduli space. I was under the impression that these limits are simple and one is able to make general statements about them, though I am far from being an expert. The tricky parts have to do with the interior of moduli space. If there were some problems there they would not be called UV or IR diveregences, they would be inherently stringy. No reason to suspect they are there, but in any event I am more impressed with cancelling the divergences we already knew about, not with the elimination of all logical possibilities for problems.

    2. Divergences are wonderful since they give rise to proper understanding of field theories via the renormalization group. This very understanding tells you GR cannot be treated as QFT, as it is not renormalizable. This is a concrete and straightforward problem, not an interpretational issue or an aesthetic displeasure (as in “background independence”), one simply cannot calculate finite and unambigous results.

  10. Peter says:


    What’s his argument that divergences in the interior of moduli space can’t occur in his formalism? Do you mean the footnote where he says that in conformal gauge, there are no obvious potential sources of divergences since the amplitudes are independent (up to surface terms) of the locations of picture-changing operators?

    This isn’t a very solid argument. How do I know there isn’t a “non-obvious” potential source of divergences? And how do I know that the picture-changing operators are really well-defined and do exactly what they are supposed to? If Berkovits could explicitly work out what happens at two loops so one could be sure that at least in that case he really had well-defined amplitudes and there were no “non-obvious” problems, that would be a lot more convincing.

    That’s about all I have to say about this, if you really are so interested in this topic, you should read D’Hoker and Phong’s papers carefully. I think if you do this you’ll see that this is an exceedingly tricky business and that it is far from obvious that Berkovits has a solid argument.

  11. Peter says:

    To whoever my non-abusive commenter is:

    I think we’re agreed about the bosonic string and the fact that at one loop it doesn’t have the same problems that come about at one loop in a field theory due to integration over arbitrarily high momentum going around the loop. Of course it has other problems.

    But two points about this:

    1. The superstring is much trickier. If you look at D’Hoker and Phong’s two-loop calculation it involves very subtle cancelations of singularities in order to get a finite result. It’s not a matter of combining together well-behaved bosonic amplitudes with signs to get something well-behaved. There are singularities that have to cancel precisely. They are able to show that this happens at two-loops, but aren’t able to do this for more than two loops. Whether these singularities have an interpretation as “IR” or “UV” singularities I don’t know. I was under the impression that you couldn’t cleanly separate which was which.

    2. In non-abelian gauge theories, the fact that loop integrations diverge as you take the momentum cutoff to infinity is not a bad thing, but a good thing. For one thing, it tells you about asymptotic freedom. For another, via dimensional transmutation, in pure gauge theory you end up with a theory with zero free parameters. If all your Feynman diagrams were finite and no renormalization was required you would have a theory with a free parameter (and in the pure gauge theory case would be conformally invariant). So the fact that your diagrams require renormalization leads to a proper understanding of the theory’s short distance behavior, and as a final result a very non-trivial theory with a complicated spectrum and zero free parameters. What’s bad about that?
    OK, this doesn’t work for the Einstein action, but it indicates the problem is not necessarily inherent in the divergent integrations in Feynman diagrams.

  12. Lubos Motl says:

    Just to explain more clearly why I wrote that Peter was ridiculous.

    Peter Woit wrote:

    Berkovits himself says his argument only gives finiteness if you assume “there are no unphysical divergences in the interior of moduli space”. But this absence of “unphysical divergences” is a crucial part of what one needs to prove.


    This is very funny because the reason why Nathan wrote this is *exactly* because this problem (unphysical divergences inside the moduli space) can *not* arise in his formalism.

  13. Anonymous says:

    I hope nothing I writen here qualifies as abusive, though I have to admit I find the sociological points less than fascinating.

    I am not saying anything that you don’t know. Take the bosonic string in flat non-compact space and calculate (for example) one loop scattering, you get results that are UV finite. I am explicitely talking about the bosonic string because:

    1. It has severe IR problems, which is not what we are discussing.

    2. It has no technical problems due to fermions.

    3. One cannot claim miracles due to SUSY, which should be irrelevant for UV phenomena anyhow.

    What I mean by UV finite is very well defined: you look at the contribution to the amplitudes from the limit where the string is particle-like, and that particle moves at very high momentum. This is the region which leads to all the celebrated UV problems of gravity. The contribtions to the string amplitudes from this corner are finite, and one can look for specific mechanical reasons for this. These are covered in all standard textbooks, and have to do with duality , or more physically the exponential softness of tree level string amplitudes, which suppresses the UV region when performing loop amplitudes.

    These properties cannot hold in any local QFT with finitely many degrees of freedom (consequently they do not look very natural in string field theory, and are lost in any truncation thereof). Any other attempt to make these amplitudes finite, a subject with a very long history, involves some ad-hoc cutoff and the expected problems with unitarity. I am not aware of any other physical mechanism to cut-off the problematic region of integration without loss of unitarity.

    This is maybe less than a complete finiteness, and I am not claiming that’s the most impressive statement one can make, but this is already plenty impressive, at least for me.

  14. Peter says:

    I agree with your statement that it is reasonable to let the multi-loop problem go and think about other things. But when string theorists do this, they should stop using multi-loop finiteness as their big selling point for the theory (or thuggishly abusing anyone who mentions the problem to them)

    This is not an argument about semantics, but about a very specific factual question. Are there well-defined, finite, multi-loop amplitudes for the superstring? Right now the answer is yes up to two loops. For three loops and higher the answer to this is yes or no, but we don’t know which yet. Maybe Berkovits will develop his formalism to the point where he really can explicitly construct such amplitudes. Maybe someone will find that, just like in supergravity, if you go to high enough order, you run into trouble. Nobody has a specific proposal for what this trouble would be, so it is reasonable to guess that it doesn’t occur, but who knows?

    I’m not at all sure what you mean by “UV finite” or that there is any sensible meaning to such a phrase. What fact is true for what specific reasons special to the extended nature of the string? Remember, in the case of strings you can’t disentangle UV and IR phenomena (think about T duality and the implications of modular invariance). Multi-loop superstring amplitudes are certainly different kinds of objects (for one thing, they’re a couple orders of magnitude more complicated…)than Feynman diagrams. They don’t have the same problems with integrating momenta running around loops that Feynman diagrams have. But that doesn’t mean they don’t have potential problems with singularities, some of which one could even interpret as coming from short distances.

  15. Anonymous says:

    Ok, another way of saying things is that the difference in opinion is about semantics, not about the facts. Maybe a better formulation is to state that string theory is UV finite, a statement which refers to the type of infinities one finds in QFT, but does not cover absolutely everyhting one may think of. This fact is true for very specific and well-understood reasons to do with the extended nature of the string. This is a precise statement which is correct, I believe.

    It is not proven that there are no other mysterious divergences that will be specific for strings and will somehow conspire to appear at 3-loops (usually potential problems appear as soon as you include quantum corrections), but there is no reason to suspect it. Given that, and that nobody has specific reasons to be interested in the result of the multiloop amplitudes, it is a reasonable attitude to just let it go and think about other things.

  16. Peter says:

    I understand well the difference between math and physics cultures about proof, since my Ph.D was in a physics department and I work in a math department. That’s not what’s at issue here.

    The situation is not like in QCD, where there’s lots of evidence from computer calculations for confinement. There’s also the fact that whenever you can check QCD against experiment it works precisely, and you never observe free quarks.

    For superstring amplitudes there’s nothing like this. There’s no experimental evidence concerning them and the only existing actual calculation of a well-defined multi-loop amplitude is the calculation at two-loops. No one has any kind of approximate or computer calculation of an amplitude beyond two loops. There’s not much known about these things beyond formal expressions, expressions you still need to do a lot of work with to make sense of. If Berkovits did this in his formalism for two loops and things worked out as he expects, it would provide some evidence that things would work at higher loops, but he hasn’t done this yet.

    The only “physical” argument for finiteness of multiloop superstring amplitudes is the bogus one I’ve seen a couple hundred times now at the beginning of every general string theory talk. This is the one where you draw a world sheet and a Feynman graph and point out that while the interaction takes place at a point on the graph, there is no well-defined interaction point on the worldsheet. Then the speaker claims this shows string theory doesn’t have singularity problems. This ignores all kinds of problems, many of them associated with the fact that there is a moduli space of metrics on the worldsheet. As you go to the boundaries of this moduli space there is all sorts of singular behavior to worry about. In addition there are lots of other possible sources of trouble. Berkovits uses BRST and picture-changing operator techniques involving composite operators. When you multiply operators together in QFT, there are plenty of new singularity problems to worry about. Whether all of these problems can be handled beyond two loops (or even at two-loops in the Berkovits formalism) is still an open problem.

    I don’t have any problem with string theorists hoping for the best, assuming that superstring multi-loop amplitudes are finite, and then going on to see if they can get a sensible TOE out of the theory. But what has happened is that this has failed, superstring theory has not lead to a sensible TOE. Instead of abandoning the theory, they keep promoting how wonderful it is because of its finite multi-loop amplitudes. It’s only fair to point out to them that this is an assumption, not a result. Just because you would like something to be true, doesn’t mean it is.

  17. Anonymous says:

    A bit of a clash of cultures here, physicists usually do not care if something is proven beyond mathematical doubt, anyone you will ask these days will believe in confinement of quarks for example.
    To convince a physicist something is false, after enough evidence is presented in favor of it, a counter-argument is needed. The mathematician’s line “you did not prove it” cannot be convincing- almost nothing is proven rigorously in any non-trivial dynamical system (which is reason to suspect that if you are after rigorous proofs you will end up studying trivial dynamical systems such as TFT).

  18. Peter says:

    Hi Lubos,

    It’s kind of pathetic the way you just start spewing insults when you don’t have an argument.

    What is in the D’Hoker-Phong papers and in the Berkovits papers, together with what is not in the Berkovits papers (a two-loop calculation) speaks for itself.

    If I had to bet, I’d bet that at three loops superstring amplitudes are finite. But I don’t know for a fact that this is true, and neither does anyone else at this point.

  19. Luboš Motl says:

    You’re ridiculous, Peter. You have obviously searched the paper for something that could be argued to be a loophole, so that you can make your simple readers happier. Well, it’s your decision that you don’t care if the physicists think that you are about as stupid as your readers.

    It has never happened that string theory would suddenly display some inconsistency, and your belief that something like that could suddenly occur at 3 loops or anywhere else is silliness.

  20. Peter says:

    Berkovits himself says his argument only gives finiteness if you assume “there are no unphysical divergences in the interior of moduli space”. But this absence of “unphysical divergences” is a crucial part of what one needs to prove.

    If you look at the history of the two-loop calculation before D’Hoker/Phong, there were many people claiming to have written these amplitudes down and to have proofs of finiteness. But their amplitudes were gauge dependent, and once D’Hoker/Phong finally sorted things out, it turns out the correct amplitudes have a different structure than what earlier people had assumed in their “proofs”. Berkovits has yet to write down explicitly a two-loop amplitude in his formalism and see if it really has the properties he would like.

  21. Luboš Motl says:

    D’Hoker and Phong are doing very difficult stuff. On the other hand, the finiteness is easier to prove in formalisms with manifest spacetime supersymmetry where the vanishing of the surface terms is easier to prove.

    The only known working covariant formalism of this type is Berkovits’ pure spinor formalism. Nathan’s calculations are of course extremely impressive, but they were doable and finite for him, and he finished the proof of the finiteness of stringy perturbative expansion as far as we’re able to say (although I don’t follow all the details):

    This kind of humiliates the rest of us, especially the skeptics who have been saying for years that it is impossible to prove the finiteness at all orders, or who – the more weird ones among them – were even suggesting that it does not have to be true.

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