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Gravitons as squared gluons & twistors

In December 2003, Edward Witten has started a new industry - the application of Penrose's idea of twistors to the scattering amplitudes of four-dimensional gauge theory. What is the status of this field more than one year later?

Witten's twistors 1 year later

The gauge-theoretical amplitudes at the tree level - the objects that have a very simple form in the twistorial variables - have been kind of understood; various prescriptions (disconnected and connected) have been found and relations between them have been identified; recursion relations have been proved, and so forth. I feel it's fair to say that at the classical level, the power of the twistor formalism has been almost fully revealed. The quantum loops in gauge theory are much more difficult, and the original prescription using topological string theory has not been terribly useful, as far as I can say - and the same is probably true about other stringy incarnations of the formulae. The unwanted conformal gravity states appear in the loops and give undesired contributions, but it does not seem to be the only problem. However, we can ask a simple question:

What about ?

There exists one fundamental reason why the twistor methods should be more natural in the case of gravity, namely:

  • The scattering amplitudes in gauge theory are not terribly natural objects; we prefer the off-shell correlators of gauge-invariant operators - the latter are the quantities relevant for the AdS/CFT correspondence and other applications
  • On the other hand, gravity is naturally defined on-shell - the scattering amplitudes are naturally the only simple gauge-invariant quantities, and because it's only the scattering amplitudes that the twistors give us, the twistor language may seem efficient and sufficient
However, there is one elementary theoretical counter-argument and one experimental argument that show that the twistorial gravity math class is tough:
  • The real power of twistors emerges when the twistors are applied to scale-invariant or conformal physical systems such as the N=4 gauge theory in d=4. Gravity is not conformal - it has a priviliged distance scale (the Planck length). The only gravity that is conformal is conformal gravity ;-), whose Lagrangian is essentially the squared Weyl tensor - and conformal gravity is not a physically appealing theory because of the ghosts and other defects
  • The LHC will be measuring the scattering amplitudes involving up to 8 gluons or so and therefore there is an experimental motivation to learn new efficient methods to calculate and new patterns underlying multi-gluon scattering; on the other hand, an experimental observation of multi-graviton scattering belongs to science fiction, and the experimental motivation to study complicated gravitons' amplitudes does not exist
These are the reasons why the marriage of gravity and twistors should be happy as well as unhappy. But what do the twistors actually tell us about gravity? How do the gravitational scattering amplitudes look like in the twistorial variables? You should look at an up-to-date paper, for example the paper by Cachazo and Svrček
This paper derives some recursion relations for the gravity amplitudes in four dimensions - relations analogous to the recursion relations mentioned (and linked) at the beginning of the article. Such recursion relations may eventually be useful to prove a new full twistorial prescription for the amplitudes. What do the amplitudes look like?

KLT relations: squaring the amplitudes

I believe that the most powerful relations are the Kawai-Lewellen-Tye (KLT) relations from 1986 that essentially say
  • Closed string (or gravity) amplitude equals open string (or gluon) amplitude squared
That was too rough, was not it? We should be a little bit more concrete. The gravitational scattering amplitudes are the low-energy limit of a special type of scattering amplitudes for the closed string. At the tree level, the relevant stringy diagram is a sphere. On the other hand, the gluon scattering is the low-energy limit of a special type of open string scattering amplitudes that arise from the disk diagram at the tree level. (Similar relations for the loop amplitudes can't be derived easily because of the complexities of the closed string moduli of Riemann surfaces.)

The sphere or disk diagram is evaluated as the correlator of the vertex operators associated with the external states - and these vertex operators must be integrated over all positions. There is a simple qualitative relation between the closed and open vertex operators:
  • Closed(z,ž) = Open(z) Open(ž)
The closed string vertex operator is typically a product of two vertex sub-operators: one of them comes from the holomorphic sector (z) and the other from the antiholomorphic sector (ž) where the caron should be replaced by a bar. Moreover, both factors of the vertex operator look much like the open string vertex operators, except for the detail that the holomorphic (or antiholomorphic) derivative in the closed string case replaces the tangent derivative in the open string case with respect to the real parameter "tau".

Also, the integral over the position of the closed string vertex operator
  • int d^2 z = int dz int dž
looks like a product of two independent integrals over "z" and over "ž". Now if you treat "z" and "ž" as independent variables and replace the contours "z = ž*" by the contours over the independent real values of "z" and "ž", you will see that the integral that defines the closed string amplitude looks like the product of two open string integrals. You must be careful about the contours. The results are subtle. But if you do things properly, the closed string Virasoro-Shapiro amplitude is related to the bilinear expression involving the Veneziano amplitudes, for example. For the special case of the graviton and gluon amplitudes, you obtain relations such as
  • A(gravity; 1,2,3) = A(gluons; 1,2,3) A(gluons; 1,2,3)
  • A(gravity; 1,2,3,4) = s_{12} A(gluons; 1,2,3,4) A(1,2,4,3)
  • A(gravity; 1,2,3,4,5) = s_{12} s_{34} A(gluons; 1,2,3,4,5) A(gluons; 2,1,4,3,5) + s_{13} s_{24} A(gluons; 1,3,2,4,5) A(gluons; 3,1,4,2,5)
and so on. You see that that the gravitational scattering amplitude is bilinear in the gluon amplitudes - but it involves various sums over the permutations of the gluons while the amplitudes are multiplied by the (n-3)-rd power (where "n" is the number of external particles) of the Mandelstam invariants constructed from the momenta via
  • s_{ij} = (p_i + p_j)^2

I wonder whether someone, such as Peter Woit, would be able to derive these relations between the gravitational and gauge-theoretical amplitudes without string theory.

The precise structure and permutations are obtained from a careful treatment of the contours in the "closed=open squared" argument outlined previously. Note that the closed string (graviton) amplitudes don't have a preferred ordering of the external closed strings while the open string (gluon) amplitudes are defined with a fixed cyclical ordering of the gluons around the boundary of the disk.

The KLT relations sketched above are absolutely independent of the twistor language, but they can also be combined with the twistorial tools. This implies, for example, that the more-than-maximally helicity violating (more-than-MHV) gravitational amplitudes also vanish simply because they would have to involve the vanishing more-than-MHV gluon amplitudes on the right hand side. The MHV amplitudes (those with two negatively-handed gravitons and the rest positively-handed) for gravity look like a particular bilinear expression involving the gluons' MHV amplitudes, and so forth.

I am afraid that the form of the gravitational MHV vertices, as defined by the KLT relations, is the final answer, and no further significant simplification is possible. It's because there are simply many different permutations with different poles contributing. Is there some deeper answer waiting to be uncovered? I am pretty skeptical. I've also tried to find a natural target space for a topological string theory that could be relevant for "N=8" supergravity, for example, and I failed. It's pretty clear that many other people have tried the same thing and they have failed, too. This is not yet a no-go theorem, and even if it were a no-go theorem, almost everyone knows the theorem that almost every no-go theorem may be circumvented. ;-)

But at any rate, the analysis of the situation does not look too encouraging.

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snail feedback (18) :

reader Anonymous said...

By working on twistor gravity, Euclidean quantum gravity, the Wheeler-deWitt equation and other competing quantum gravity models, aren't you essentially admiting the failure of string theory as a theory of quantum gravity? I'm still waiting for the time when you will write a positive article on loop quantum gravity. Judging from your previous articles, which were mostly on competing theories, I don't think I have to wait very long...

"Gravity is naturally defined on-shell"

I beg to differ...

reader Quantoken said...

Oh no no, Anonymous. Super string theory is never wrong and it never fails. And it never dies. It only fades away:-) After a whole you will get bored when it doesn't seems to produce anything useful in making falsifiable predictions.

And what makes you think Lubos would say anything nice about LQG, even if he changes his mind on string theory. LQG is not doing much better, either.

As he said, he looks some where else because Witten, the big guy, create a whole new industry. And it is worth for all theoretical physicists to spend another 20 years in that direction, regardless of the outcome. Even if they still come up empty handed, it is at least not worse than what they already get here:-)


reader Anonymous said...

Nobody has ever seen even a single graviton, so who cares about multi-graviton scattering?

reader Luboš Motl said...

Let me admit that I would prefer a different, more technical type of comments, but reality is reality.

Jean-Paul: yes, it was one of my points. There is no experimental motivation to study multigraviton scattering. Of course, there are lots of theoretical reasons to do so.

For the first person: you misunderstood the science of this article if your conclusion is that the ideas described in this text are not derived from string theory.

The whole rule "gravity equals gauge theory squared" is a result of string theory, for example, and the twistors are another tool to investigate physics of these vacua and limits of string theory.

However, if you expect that every article I write must be labeled "for string theory" or "against string theory", I am afraid that my blog is not for you. You should switch to a different kind of blogs for less demanding readers, by a few orders of magnitude, for example "Not even wrong". This is a blog where every post is defined to be another dumb attack against string theory, and this is the type of approach that you may find more satisfactory.

I describe and investigate interesting and/or relevant ideas and I don't care a single bit whether someone thinks that they too strongly related to string theory or too weakly related to string theory. Of course that at the end, all the good ideas in theoretical physics are facets of string/M-theory, but my goal is definitely not to preach this rather trivial observation in every text.

reader Anonymous said...

Guess: Why does Lubos hero-worship Witten?

reader Anonymous said...

Because, unlike you, Lubos is smart and Witten even smarter.

reader Anonymous said...

that's a really prefound comment. and the first person is really confused by/or don't care about the difference between a string model with twistor target space and twistor gravity?

string theory may not be falsifiable just yet, but some theories don't even need to wait to be falsified by experiments. they're falsified by their incosistent math?

reader Anonymous said...

Mathematically inconsistent theories don't even deserve the name theory. Any conclusion from a false statement is true by definition. So in any such theory 2 apples can be proven to be precisely equivalent to 17 cantaloupes ;))

But more seriously, imgagine you live in a universe, that is described by phi^4 theory, and your friends scatter the phi at low energies and tell you about cross sections. Could you make a statement about the existence of new physics (beyond phi^4) at much higher energies? Remarkably, the answer is yes. Just from the low energy scattering cross sections you can infer what the running of the coupling constant is. And you find that at very large energies the effective coupling will diverge (it hits a Landau pole). At this energy scale, therefore, new physics *must* take over. Of course, new physics could occur much earlier. But the Landau pole gives you a strict upper bound for where it must happen at the latest.

This is an example of a theory that is perfectly consistent -- though not to arbitrarily high energies -- but can never be all there is. The status of the standard model is very much like this.

Best, Michael

reader Anonymous said...


I guess I kind of follow your explanation and I have a naive/non-technical question:

A tree-level diagram which contains gluons and gravitons (e.g. two gluons exchanging one graviton) seems to be similar to
1-loop digrams (for the gluons) connected
by a graviton tube.
Would it make sense then to search for
recursion relations like gluon + graviton
tree diagram <-> 1-loop gluon diagrams + graviton propagator?

This is probably nonsense, but I thought I should ask,

Wolfgang Beirl

reader Anonymous said...

I was hoping that from the 26 comments on the "Quantum Foam and Melting Calibi Yau Crystal" thread, that a good discussion had occured on this very interesting stuff but it seems all the comments were just complete off-topic crap. So I will take the liberty of discussing a bit here since this blog seems to move on at quite a pace:) I will admit I have always been fascinated by the quantum foam and I very much like the term "Calibi Yau crystal". I also think analogies with condensed matter and statistical mechanics is a good thing too. So to see if I have this right the classical geometry would correspond to a macroscopic crystal and quantum geometry to the underlying discrete or atomic description.? And clearly the topological string is used to make things tractable. So identifying the temperature of the crystal with the inverse string coupling the probability of an atom leaving the crystal lattice is the same as that for splitting of strings. Thus you can describe melting of a crystal and splitting of strings in the same way. Essentially dual descriptions! At high temperatures the crystal melts, and at low temperatures you have large coupling in the dual description so the crystal is a discrete (unmelted) structure. In other words the CY space is discrete.
This is rather neat stuff. It is fair to say that one of the limitations of string theory to date has been that the strings are quantised but the background geometry is always a fixed and classical manifold. This seems to be an interesting step to describing string propagation on a quantum geometry or substructure, and also is compatible with the idea of topological foam. I am very much a believer in the idea that classical spacetime/geometry and gravity are emergent from a much deeper picture of what is going on. In this sense GR is simply not fundamental and can't be "quantised" in the usual sense as in lqg. Classical geometry and gravity have to emerge in a continuum limit somehow. GR is an infrared continuum approximation with the Einstein equations being more like the Navier-Stokes equations for a fluid (which are also nonlinear and emergent)or equations for continuum mechanics of elastic solids say (or a crystal)on macroscopic scales where you can forget about the underlying discrete/molecular structure. But on these smaller scales a new picture must take over. Some condensed matter systems can mimic the kinematics of GR so this is farther evidence too I feel. Anyway, I like the kinds of ideas in this Vafa paper. One can always expect difficulties but what would be the next step though? I am not surprised to see the name "Vafa" here. The other idea idea I liked a lot were Brandenberger-Vafa cosmological scenarios about why do 3 dimensions decompactify and 6 remain compact?(Do you know if Cumrun is still enthusiastic about this idea? )Finally, graviton scattering is excellent and should be thought about even if gravitons don't exist! You know theoretical physics is as much art as it is science. Thats my opinion. Obviously the main goal is to connect to nature, but even if some (or many) of these ideas ultimately only exist in our minds they can still be beautiful and are worth developing for their own sake. Anyway, just my 2c on the matter.
Dekuji vam!

reader Luboš Motl said...

Hi Wolfgang, thanks for your comment. Is not your proposed relation something more like the usual unitarity relations between the tree diagrams and the loop diagrams? You seem to be trading external particles for propagators, so it seems so...

These things also work, but it's something different than the relation between closed string vertex operators and open string vertex operators.

All the best

reader Anonymous said...

Interesting comments. Although someone (not necessarily a person) may live in say a conformal field theory, maybe even a 4 dimensional one:))

reader Wolfgang said...


what do you think of the paper hep-th/0502227 ?
If one could find a (relatively simple) matrix- or lattice-model in the same universality class as string theory,
non-perturbative, numerical simulations would be possible and eventually replace the "landscape" with something more predictive ...
This is my bias as you knnow already.
I notice taht this is OT, but I would appreciate your comment.

Thank you,

reader Luboš Motl said...

Dear Wolfgang,

I think it's a very nice paper. It's good that they study the matrix models with the group-manifold-valued matrices U because they may be relevant.

As far as I understand, it's still just a small subsector describing a particular (nearly thermal?) regime of string theory in one particular background. I don't know why you think that it may be "in the same universality class as the whole string theory", so that it "could replace the landscape". Of course that what you say would be great, but I don't see hints of it in their paper.

Because I have not read the whole text, there are still many basic questions open for me from the abstract. For example, I used to think that the Horowitz-Polchinski transition is smooth - while they map it to the Gross-Witten phase transition - well, it seems to be a 3rd order phase transition in their picture.

Their paper is probably a very useful toy model for the phase diagram of AdS/CFT, but that's a very different thing than the landscape of vacua, Wolfgang.

All the best

reader Luboš Motl said...

Incidentally, concerning the previous discussion - many people in that discussion would probably discard the paper 0502227 immediately because it uses the methods of Euclidean quantum gravity - which is also usual for the study of the Hawking-Page transition.

I would find such an approach highly unconstructive - and these particular methods involving Euclidean quantum gravity are trustworthy, I think.

reader Wolfgang said...


I do NOT think that their model is in the universality class of full string theory.
Indeed I used "if" in my previous post:
IF one could find a simplified matrix- and/or lattice model to the full string theory, (as they claim to have found to AdS/CFT) it would make the "landscape" talks obsolete.

I consider their paper as a 1st step in the right direction so to speak.

Thank you,

reader Luboš Motl said...

I understand, Wolfgang. You propose to extend their model so that it contains all of string theory including the realistic backgrounds.

I don't see whether enough has been done to make this step now realistic. The matrix model is strongly confined to anti de Sitter space once again. When you add enough new stuff to this simple model, you will return back to N=4 super Yang-Mills which describes everything, but only on the AdS5 x S5 background.

Different backgrounds, like dS space with Calabi-Yaus etc., are probably very different, but I certainly don't want to discourage you if you think that you have an idea how to transfer their model outside AdS5 x S5.

reader Wolfgang said...

Ok, let me think about it 8-)
Seriously, I think this could be 1 or 2 really good ideas away from doable.
But it would probably need a younger brain than mine.

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