You may go to the main Strings 2008 page on this blog; that page includes the live webcast

**Lance Dixon**starts the morning session (the only one today) by a talk about the structure of gauge-theoretical and gravitational scattering (on-shell) amplitudes: PDF here. (See also a text about his talk on transcendentality and about his SUGRA puzzle on this blog.)

Jester at Resonaances dedicates a text to Dixon's talk, too.

A ten-minute delay was caused by repeated trouble with the Mac. Couldn't they train the technology in advance? Dixon's title page has various Feynman diagrams and boxes divided to smaller boxes and his first sentences were compliments to CERN and the conference. He plans to talk about QCD for the LHC (with many external particles and many loops), N=4 and AdS/CFT, and finiteness of N=8 SUGRA.

He explains that on-shell amplitudes are more physical, independent of field redefinitions and gauge choices. They may have hidden structures that are only understood later, e.g. the MHV amplitudes (clarified by twistor string theory). Infrared (physical) divergences are often a part of the story, making things complicated but doable (describe them by jet functions).

His big message is that the amplitudes are "plastic" - he means various factorizations well-known to string theorists (with Riemann surfaces). Unitarity determines discontinuities across branch cuts, too. These features - together with the helicity approach - may be enough to determine the amplitudes. Mantis shrimp have different reflections and sensitivities for left-handed and right-handed photons: they are able to communicate via helicity formalism and only theorists understand these shrimp (other animals and experimenters are blind). ;-)

He reviews 2-spinors and their products, including the <12> and [34] notation we know from the twistor minirevolution. It's important to know how to continue momenta to complex values because for complex momenta, one can make various polynomials of momenta vanish without being pushed to singular collinear configurations. Britto et al. (BCFW) 2005, shifting the spinors around and deriving the recursion relations for amplitudes from some residues in contour integrals, is reviewed. It's an environmentally, politically correct way to calculate because the trees are recycled. ;-)

For 6 gluons, instead of 220 Feynman diagrams, you have 3 terms, one of them vanishes and the other two are mirror images of each other: really 1 contribution is here. Like in Olympics he watches, difficulty increases with the number of loops and legs. ;-) The style also matters but it is more subjective - for example, he cares about the proximity to experiments. The difficulty index decreases with the number of supersymmetries. He draws many papers into this 2D graph spanned by (legs, loops).

At the LHC, we should understand QCD with many partons and a couple of loops: complicated SM backgrounds - as Oliver Buchmüller was explaining in the context of the "SUSY" signals with cascade decays. One-loop amplitudes are then decomposed into a rational term and a singular one (with cuts). The singular term is decomposed into a combination of standardized functions with poles. A useful concept is generalized unitarity - with cuts where not 2 but 3 or 4 particles are on-shell. Britto, Cachazo, Feng 2004 were able to split a "box" coefficient in the decomposition into an integrated product of four tree amplitudes. One can explain why no pentagons appear.

Besides boxes, one must look at triangles and bubbles. Nice pictures but I don't think he explained the machinery - the reorganization of the diagrams - too well. The next question are the rational terms of the 1-loop amplitudes. Unlike a dozen of other papers, his strategy is to use plasticity, recycling trees and loops again. CutTools, Rocket, and Blackhat are three state-of-the-art programs to calculate 1-loop amplitudes. Rocket goes up to 20 gluons.

Now he gets to multi-loop & multi-leg diagrams, pointing out that things are similar for N=4 SYM and N=8 SUGRA, due to the stringy KLT relations. So he's dissolving very complex diagrams now. After he finishes a procedure, he talks about some "missing diagrams" (near-maximal cuts etc.). It's not clear whether their procedure is systematic and leads to a full answer or whether it is just heuristics to guess some singular parts of the answer.

Some elementary facts about AdS/CFT and planar limit of N=4 SYM follow. Interesting related things occurred in N=4 SYM: exponentiation of finite terms, dual superconformal invariance, and a MHV-Wilson_line equivalence. Exponentiation: the ration of an amplitude and its tree part is naturally an exponential of a simpler expression. The simpler expression, a sum, only depends on the kinematic variables through some one-loop amplitudes. This surprising Ansatz was shown to work for 4 gluons, indirectly for 5 gluons, but it is known to fail above 5 gluons. If some promising conjectures are right, one could explicitly write the full 4-gluon amplitude.

Dual conformal invariance is shown using a box diagram with 4 gluons. Write the Feynman integral over momentum. Now, the point is to look for conformal transformations acting on the momentum space and not x-space as usual. Among conformal transformations, the spherical inversion is a sufficient generator (with translations). Cutely enough, straightforward rules are known to decide whether this invariance under inversion holds. And it does hold 4-gluon amplitudes at 1-loop, 2-loop, 3-loop, 4-loop, 5-loop level. ;-)

This symmetry sounds crazy. Where does it come from? String theory, of course, namely a T-duality of AdS5 x S5 and Wilson line calculations (that seem helicity-blind, however).

The Ansatz should work not only for 4 but also for 5 gluons. It is known to fail for 6 or more gluons - there are new invariant "double ratios" - but the dual conformal symmetry and Wilson lines are still there.

Finally, he jumped to hidden cancellations in N=8 SUGRA. Up to 3 loops, it has been shown that, as hinted by KLT relations, the N=8 SUGRA is as finite as N=4 SYM. The 3-loop integrals include 9 Feynman diagram topologies (2 of which are new). Full calculations may look tough but the "equal finiteness" as N=4 SYM is manifest. Whether or not the theory is finite to all orders is an open question, he says. Conclusions repeat some points above.

A question asked him about the Ansatz extending 4 gluons to 6 gluons or more. They're working on it: not yet. Very nice talk.

**Michael Green**is following with a talk about the constraining power of supersymmetry combined with dualities in (type IIB) string theory and supergravity: PDF here. See also Two roads from N=8 SUGRA to string theory. He will talk about the perturbative finiteness, too.

First part: he shows a derivative expansion of low-energy actions in string theory and reviews the type IIB massless fields. He explains why the moduli space is SL(2,R) \ SL(2,Z) / U(1); it can also be seen from a U(1) anomaly in the SUGRA framework. Only SL(2,Z) is the symmetry of the quantum (string) theory. Terms in the action, F, are not holomorphic. How are they constrained by SUSY? The constraints are hard to illuminate without a superspace formalism.

The closure of SUSY schematically requires "DF = F + FF + FFF ..." with all kinds of indices and coefficients. It is also useful to add "Dbar" to the equation, to obtain a Laplace "eigenvalue" equation - that can be solved by "F = Eisenstein series", generalizing the Riemann zeta function. By SUSY, he can conclude that the R^4 term only comes from the 1-loop term, multiplied by E_{3/2} (Eisenstein). Similar for other terms at other loop orders.

The four-graviton amplitude is now expanded in alpha', starting with the beautiful Virasoro-Shapiro amplitude that can be expanded. The expansions of higher-loop amplitudes are tougher. Green expanded various amplitudes, showing the coefficients being products of zeta functions.

Finally, he jumps to the relationship between SUGRA (M-theory on tori) and type IIA, IIB string theory: the radii must go to zero in various ways to get the limits. Green kind of assumes that SUGRA may need perturbative counterterms. However, he says that some of the terms may be determined from string theory. The elegant mapping is explained, for a 2-loop case, by mapping a real torus to the auxiliary torus.

Green skipped a few pages of details and gets to the loops in SUGRA. To study the divergence structure, he wants to look at the powers of S at various loop levels. The exponent is known up to 3 loops; higher powers of S make the expressions more convergent. Because 3-loops begin as S^3 R^4, S^2 R^4 can't be renormalized beyond 2 loops - e.g. a five-loop counterterm of this kind has to vanish. Similarly for other terms with other number of loops.

To summarize, one of the conclusions is that they argue that there's no divergence up to 9 loops. Green suggests that it might be that it holds to all orders, too - having the same degree of divergence as N=4 SYM.

If that's finite, why were we doing string theory? So he reviews his work with Ooguri and Schwarz: you can't decouple SUGRA from string theory because in any limit, there is a tower of states that is light. Perturbative finiteness doesn't imply finiteness. The final slide is a joke that the going to higher orders is speeding up, and one can resum it to show that we will get to "L=infinity" loops by 2014. ;-)

Juan asked about some relationship between the divergence structure in 4D and 10D. In 10D, things are very divergent. Both in 4D and 10D, there's an anomaly that prevents U(1) to be the exact symmetry, Green says. Nice.

After the coffee break and a few announcements,

**Freddie Cachazo**speaks about the simplest quantum field theory, which is their new label for N=8 SUGRA (work with Kaplan and Arkani-Hamed): PDF here. The main "simple" feature of the S-matrix of this theory are amazing convergent properties of the amplitudes at an infinite complex momentum. Tree level amplitudes are determined by Lorentz symmetry (3 particles). One-loop and higher-loop amplitudes are all determined by the leading singularities.

Be careful: asymptotic/nonperturbative corrections are still large at trans-Planckian energies, E7 must be broken to the discrete subgroup, and so on.

So Freddie returns to the S-matrix programs of the 1960s. It was all hard - but only because they looked at hard theories. They should have looked at N=8 SUGRA, back in the 1960s. ;-) At least, we can do it today. Freddie begins with the 1-complex-parameter deformation of the amplitudes: momenta are shifted by a multiple (z times) a momentum "q" which is (0,0,1,i). Here, q is null. The amplitude "M" becomes a rational function of "z". By contour integrals, it goes like 1/z^J for spins J=0,1,2, so to say. Gravity falls faster than scalars and Yang-Mills, in this complex direction!

Why? In this novel complex infinite momentum limit, one finds a new symmetry - the so-called "spin symmetry" - that decouples Lorentz indices from the derivatives. Gravity has twice as many indices relatively to gauge theory, so it goes like 1/z^2. If gravity is so nice, why do they add SUSY? Because without SUSY, unitarity leads you to combine the "very convergent" amplitudes with other, "divergent" z^2 amplitudes. With SUSY, all of them are related. So "everything" is smooth and nice in "z".

In fact, he can define the N=8 SUGRA S-matrix without any Lagrangian, just with some Grassmannian coherent states. Scalars didn't vanish in the past. So he modifies the 1-parameter transformation of BCFW: also the fermionic variable is shifted by "z" times something, bringing the decreasing behavior of gravity to the whole multiplet.

Now, he wants to see nice things like the E7(7) symmetry from their S-matrix that was produced from Lorentz symmetry and the complex tricks only. Of course, the moduli space is seen from a soft behavior of single soft emission of scalars. (Gravitons are divergent.) The double soft emissions seems to show a paradox. It is explained by seeing that a limit is finite but depends on a regulator, the direction, and the structure of it encodes the E7(7) symmetry. I haven't quite understood it yet.

At the tree level, the only singularities are poles. However, cuts appear at one loop. The log^2 functions here have a discontinuity of the log form which is itself discontinuous (and appears at quadrupole cuts). This discontinuity of discontinuity is thus seen near the highest-codimension branch cuts and the whole amplitude is fully determined by the behavior near this special point, i.e. by the leading singularity.

He finds that triple cuts are determined by quadruple cuts, and double cuts are determined by triple cuts (no new singularities). So everything at one-loop is really determined by the quadruple cuts. The no-triangle hypothesis is rephrased in a new way here. Everything is included in the scalar boxes.

At L-loop level, one cuts 4L propagators instead (very many!). Sounds crazy but anyway, the whole perturbative N=8 S-matrix is determined by the leading singularities. This is equivalent to the UV finiteness: if a part of the S-matrix is not determined by the leading singularity, there must be a divergence.

To summarize, he wants the leading singularity conjecture to be proven, find the explicit action of E7(7), and try to go above d=4.

Simeon Hellerman complains that it can't be true that SUSY and Lorentz determine everything because one could add any N=8 invariant counterterm. Freddie says that such new terms wouldn't spoil the behavior at infinity. Lance Dixon thought that some singularities could still appear in the denominator but Freddie insisted they would cancel. Another question was about the relation of E7(7) and its continuity beyond the tree level and Freddie said "Yes, yes" but didn't quite clarify the situation.

**Nathan Berkovits**explains one trick that has emerged in his pure-spinor analysis of AdS_5 x S^5: namely the fermionic T-duality: PDF here. He found it with Juan Maldacena. A dense slide with formalism follows immediately. ;-)

A T-duality of the AdS5 x S5 maps the NS-NS fields well but the 5-form field strength is not behaving well. However, the latter is subleading for large radii. Moreover, it can be completely fixed by adding the fermionic T-duality: then it works for all, not just large, 't Hooft coupling. Shockingly enough, this thing hasn't been considered before even though it is a direct extension of the bosonic case.

Nathan reviews the bosonic T-duality, following the Buscher rules. The action is invariant under a shift of a boson - that only appears with derivatives in the action. One can add Lagrange multipliers to get an equivalent theory if one variable is integrated out; if you integrate A instead, you obtain the T-dual theory.

For worldsheets with cycles, the "A" can have "Wilson lines" and one must be careful about the compactness of X, Xdual. The periodicities are inversely related. For open worldsheets (and open strings), the Neumann and Dirichlet conditions are interchanged.

To get a fermionic counterpart, the "isometry" fermion must be a worldsheet scalar. So the RNS picture is no good but the Green-Schwarz, pure spinor, and hybrid formalisms are fine. The fermionic Buscher procedure is almost identical and the differences can't be easily written without equations. Again, two ways to integrate things out exist.

This T-duality works on simply connected surfaces. At worldsheets with cycles, one can make the fermion "multi-valued" but the details are subtle. The fermionic periodicity must be a zero mode that must be integrated over. That's really strange. Is that integration added ad hoc to the stringy rules for the S-matrix? That would probably spoil unitarity. The Dirichlet/Neumann boundary conditions are now not switched.

Now, he would like to expand the superfields in the components (spacetime fields) and see how they transform under the fermionic T-duality separately. The T-duality transforms a SUGRA-equations solving background to another background solving them, and replaces the original fermionic Abelian symmetry by another one.

Example 1 involves 4D Minkowski space times a Calabi-Yau 3-fold in the hybrid formalism. He needs a coefficient of (d theta)^2 to be nonzero which can be achieved by a harmless surface term. Not clear what the physical interpretation of this new duality is. Too much formalism here.

Example 2 is the AdS5 x S5 as a GS sigma-model, with 4 bosonic and 8 fermionic translations ready to be T-dualized. The dilaton doesn't transform: the fermionic contributions to the gradient cancel against the bosonic ones. The dual conformal symmetry that should be explained by the fermionic T-duality is sketched - too quickly to learn it. To summarize, one has a new symmetry, one that works for the tree (meaning genus 0 = planar) amplitudes only. He wants to check other backgrounds and extend the transformation to a fermionic U-duality. I am confused what it means if this operation only works at the tree level while U-duality relates strong couplings with other things.

I had almost no chance to understand the first question. ;-) But the answer is that one can make a sequence of several fermionic T-dualities and they commute in all his examples: he only knows what to do with Abelian supergroups. Another question was answered that he doesn't know whether one can go beyond genus 0. The last question had no known answer. Interesting stuff, indeed.

**Emery Sokatchev**has a related talk about the dual superconformal symmetry of the N=4 scattering amplitudes, a topic mentioned by Nathan, Juan, Lance, and others: PDF here. A review similar to a slide by Dixon (about IR divergences etc.) appears at the beginning, together with a few words on MHV amplitudes.

The dual conformal symmetry only works at the planar level - it must therefore be dynamical, invisible at the Lagrangian level. It effectively acts on the momentum space - more precisely on a fictitious x_{i,i+1}-space obtained by a field redefinition in the normal momentum space (no Fourier transform). Here, this "x" is inserted in between the external momenta p_i and p_{i+1}.

The vev of the Wilson loop with carefully engineered cusps, according to the momenta, is proportional to the scattering amplitude.

The dual conformal symmetry is directly relevant for MHV amplitudes - unfortunately, I completely missed what the relevance is. Did he say it? But he already asks about the non-MHV amplitudes. I am getting lost because the difference between things that are new and things we normally know is not highlighted too clearly. I suppose that he is now reviewing some well-known scalings and degrees from the helicity formalism. The MHV amplitude is factorized, to keep track of all the infrared divergences. Fermionic coordinates are added to the dual space, to extend the dual conformal symmetry to dual superconformal symmetry.

Moreover and however, I have to go now, sorry.

Monday, Tuesday, Wednesday, Thursday, Friday

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