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 gravity ?

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

• hep-th/0502160

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.