# Conference Amplitudes 2015 – Air on the Superstring

One of the first pieces of Bach ever recorded was August Wilhelmj’s arrangement of the Orchestral Suite in D major. Today the transcription for violin and piano goes by the moniker Air on the G String. It’s an inspirational and popular work in all it’s many incarnations, not least this one featuring my favourite cellist Yo-Yo Ma.

This morning we heard the physics version of Bach’s masterpiece. Superstrings are nothing new, of course. But recently they’ve received a reboot courtesy of Dr. David Skinner among others. The ambitwistor string is an infinite tension version which only admit right-moving vibrations! At first the formalism looks a little daunting, until you realise that many calculations follow the well-trodden path of the superstring.

Now superstring amplitudes are quite difficult to compute. So hard, in fact, that Dr. Oliver Schloterrer devoted an entire talk to understanding particular functions that emerge when scattering just  $4$ strings at next-to-leading order. Mercifully, the ambitwistor string is far more well-behaved. The resulting amplitudes are rather beautiful and simple. To some extent, you trade off the geometrical aesthetics of the superstring for the algebraic compactness emerging from the ambitwistor approach.

This isn’t the first time that twistors and strings have been combined to produce quantum field theory. The first attempt dates back to 2003 and work of Edward Witten (of course). Although hugely influential, Witten’s theory was esoteric to say the least! In particular nobody knows how to encode quantum corrections in Witten’s language.

Ambitwistor strings have no such issues! Adding a quantum correction is easy – just put your theory on a donut. But this conceptually simple step threatened a roadblock for the research. Trouble was, nobody actually knew how to evaluate the resulting formulae.

Nobody, that was, until last week! Talented folk at Oxford and Cambridge managed to reduce the donutty problem to the original spherical case. This is an impressive feat – nobody much suspected that quantum corrections would be as easy as a classical computation!

There’s a great deal of hope that this idea can be rigorously extended to higher loops and perhaps even break the deadlock on maximal supergravity calculations at $7$-loop level. The resulting concept of off-shell scattering equations piqued my interest – I’ve set myself a challenge to use them in the next 12 months!

Scattering equations, you say? What are these beasts? For that we need to take a closer look at the form of the ambitwistor string amplitude. It turns out to be a sum over the solutions of the following equations

$\sum_{i\neq j}\frac{s_{ij}}{z_i - z_j}=0$

The $s_{ij}$ are just two particle invariants – encoding things you can measure about the speed and angle of particle scattering. And the $z_i$ are just some bonus variables. You’d never dream of introducing them unless somebody told you to! But yet they’re exactly what’s required for a truly elegant description.

And these scattering equations don’t just crop up in one special theory. Like spies in a Cold War era film, they seem to be everywhere! Dr. Freddy Cachazo alerted us to this surprising fact in a wonderfully engaging talk. We all had a chance to play detective and identify bits of physics from telltale clues! By the end we’d built up an impressive spider’s web of connections, held together by the scattering equations.

Freddy’s talk put me in mind of an interesting leadership concept espoused by the conductor Itay Talgam. Away from his musical responsibilities he’s carved out a niche as a business consultant, teaching politicians, researchers, generals and managers how to elicit maximal productivity and creativity from their colleagues and subordinates. Critical to his philosophy is the concept of keynote listening – sharing ideas in a way that maximises the response of your audience. This elusive quality pervaded Freddy’s presentation.

Following this masterclass was no mean feat, but one amply performed by my colleague Brenda Penante. We were transported to the world of on-shell diagrams – a modern alternative to Feynman’s ubiquitous approach. These diagrams are known to produce the integrand in planar $\mathcal{N}=4$ super-Yang-Mills theory to all orders! What’s more, the answer comes out in an attractive $d \log$ form, ripe for integration to multiple polylogarithms.

Cunningly, I snuck the word planar into the paragraph above. This approximation means that the diagrams can be drawn on a sheet of paper rather than requiring $3$ dimensions. For technical reasons this is equivalent to working in the theory with an infinite number of color charges, not just the usual $3$ we find for the strong force.

Obviously, it would be helpful to move beyond this limit. Brenda explained a decisive step in this direction, providing a mechanism for computing all leading singularities of non-planar amplitudes. By examining specific examples the collaboration uncovered new structure invisible in the planar case.

Technically, they observed that the boundary operation on a reduced graph identified non-trivial singularities which can’t be understood as the vanishing of minors. At present, there’s no proven geometrical picture of these new relations. Amazingly they might emerge from a 1,700-year-old theorem of Pappus!

Bootstraps were back on the agenda to close the session. Dr. Agnese Bissi is a world-expert on conformal field theories. These models have no sense of distance and only know about angles. Not particularly useful, you might think! But they crop up surprisingly often as approximations to realistic physics, both in particle smashing and modelling materials.

Agnese took a refreshingly rigorous approach, walking us through her proof of the reciprocity principle. Until recently this vital tool was little more than an ad hoc assumption, albeit backed up by considerable evidence. Now Agnese has placed it on firmer ground. From here she was able to “soup up” the method. The supercharged variant can compute OPE coefficients as well as dimensions.

Alas, it’s already time for the conference dinner and I haven’t mentioned Dr. Christian Bogner‘s excellent work on the sunrise integral. This charmingly named function is the simplest case where hyperlogarithms are not enough to write down the answer. But don’t just take it from me! You can now hear him deliver his talk by visiting the conference website.

Conversations

I’m very pleased to have chatted with Professor Rutger Boels (on the Lagrangian origin of Yang-Mills soft theorems and concerning the universality of subheading collinear behaviour) and Tim Olson (about determining the relative sign between on-shell diagrams to ensure cancellation of spurious poles).

Note: this post was originally written on Thursday 9th July but remained unpublished. I blame the magnificent food, wine and bonhomie at the conference dinner!

# Conference Amplitudes 2015 – Integration Ahoy!

I recall fondly a maths lesson from my teenage years. Dr. Mike Wade – responsible as much an anyone for my scientific passion – was introducing elementary concepts of differentiation and integration. Differentiation is easy, he proclaimed. But integration is a tricky beast.

That prescient warning perhaps foreshadowed my entry into the field of amplitudes. For indeed integration is of fundamental importance in determining the outcome of scattering events. To compute precise “loop corrections” necessarily requires integration. And this is typically a hard task.

Today we were presented with a smorsgasbord of integrals. Polylogarithms were the catch of the day. This broad class of functions covers pretty much everything you can get when computing amplitudes (provided your definition is generous)! So what are they? It fell to Dr. Erik Panzer to remind us.

Laymen will remember logarithms from school. These magic quantities turn multiplication into addition, giving rise to the ubiquitous schoolroom slide rules predating electronic calculators. Depending on your memory of math class, logarithms are either curious and fascinating or strange and terrifying! But boring they most certainly aren’t.

One of the most amusing properties of a logarithm comes about from (you guessed it) integration. Integrating $x^{a-1}$ is easy, you might recall. You’ll end up with $x^a/a$ plus some constant. But what happens when $a$ is zero? Then the formula makes no sense, because dividing by zero simply isn’t allowed.

And here’s where the logarithm comes to the rescue. As if by witchcraft it turns out that

$\displaystyle \int_0^x x^{-1} = -\log (1-x)$

This kind of integral crops when you compute scattering amplitudes. The traditional way to work out an amplitudes is to draw Feynman diagrams – effectively pictures representing the answer. Every time you get a loop in the picture, you get an integration. Every time a particle propagates from A to B you get a fraction. Plug through the maths and you sometimes see integrals that give you logarithms!

But logarithms aren’t the end of the story. When you’ve got many loop integrations involved, and perhaps many propagators too, things can get messy. And this is where polylogarithms come in. They’ve got an integral form like logarithms, only instead of one integration there are many!

$\displaystyle \textrm{Li}_{\sigma_1,\dots \sigma_n}(x) = \int_0^z \frac{1}{z_1- \sigma_1}\int_0^{z_1} \frac{1}{z_2-\sigma_2} \dots \int_0^{z_{n-1}}\frac{1}{z_n-\sigma_n}$

It’s easy to check that out beloved $\log$ function emerges from setting $n=1$ and $\sigma_1=0$. There’s some interesting sociology underlying polylogs. The polylogs I’ve defined are variously known as hyperlogs, generalized polylogs and Goncharov polylogs depending on who you ask. This confusion stems from the fact that these functions have been studied in several fields besides amplitudes, and predictably nobody can agree on a name! One name that is universally accepted is classical polylogs – these simpler functions emerging when you set all the $\sigma$s to zero.

So far we’ve just given names to some integrals we might find in amplitudes. But this is only the beginning. It turns out there are numerous interesting relations between different polylogs, which can be encoded by clever mathematical tools going by esoteric names – cluster algebras, motives and the symbol to name but a few. Erik warmed us up on some of these topics, while also mentioning that even generalized polylogs aren’t the whole story! Sometimes you need even wackier functions like elliptic polylogs.

All this gets rather technical quite quickly. In fact, complicated functions and swathes of algebra are a sad corollary of the traditional Feynman diagram approach to amplitudes. But thankfully there are new and powerful methods on the market. We heard about these so-called bootstraps from Dr. James Drummond and Dr. Matt von Hippel.

The term bootstrap is an old one, emerging in the 1960s to describe methods which use symmetry, locality and unitarity to determine amplitudes. It’s probably a humorous reference to the old English saying “pull yourself up by your bootstraps” to emphasise the achievement of lofty goals from meagre beginnings. Research efforts in the 60s had limited success, but the modern bootstrap programme is going from strength to strength. This is due in part to our much improved understanding of polylogarithms and their underlying mathematical structure.

The philosophy goes something like this. Assume that your answer can be written as a polylog (more precisely as a sum of polylogs, with the integrand expressed as $\prod latex d \log(R_i)$ for appropriate rational functions $R_i$). Now write down all the possible rational functions that could appear, based on your knowledge of the process. Treat these as alphabet bricks. Now put your alphabet bricks together in every way that seems sensible.

The reason the method works is that there’s only one way to make a meaningful “word” out of your alphabet bricks. Locality forces the first letter to be a kinematic invariant, or else your answer would have branch cuts which don’t correspond to physical particles. Take it from me, that isn’t allowed! Supersymmetry cuts down the possibilities for the final letter. A cluster algebra ansatz also helps keep the possibilities down, though a physical interpretation for this is as yet unknown. For $7$ particles this is more-or-less enough to get you the final answer. But weirdly $6$ particles is smore complicated! Counter-intuitive, but hey – that’s research. To fix the six point result you must appeal to impressive all-loop results from integrability.

Next up for these bootstrap folk is higher loops. According to Matt, the $5$-loop result should be gettable. But beyond that the sheer number of functions involved might mean the method crashes. Naively one might expect that the problem lies with having insufficiently many constraints. But apparently the real issue is more prosaic – we just don’t have the computing power to whittle down the options beyond 5-loop.

With the afternoon came a return to Feynman diagrams, but with a twist. Professor Johannes Henn talked us through an ingenious evaluation method based on differential equations. The basic concept has been known for a long time, but relies heavily on choosing the correct basis of integrals for the diagram under consideration. Johannes’ great insight was to use conjectures about the dlog form of integrands to suggest a particularly nice set of basis integrals. This makes solving the differential equations a cinch – a significant achievement!

Now the big question is – when can this new method be applied? As far as I’m aware there’s no proof that this nice integral basis always exists. But it seems that it’s there for enough cases to be useful! The day closed with some experimentally relevant applications, the acid test. I’m now curious as to whether you can link the developments in symbology and cluster algebras with this differential equation technique to provide a mega-powerful amplitude machine…! And that’s where I ought to head to bed, before you readers start to worry about theoretical physicists taking over the world.

Conversations

It was a pleasure to chat all things form factors with Brenda Penante, Mattias Wilhelm and Dhritiman Nandan at lunchtime. Look out for a “on-shell” blog post soon.

I must also thank Lorenzo Magnea for an enlightening discussion on soft theorems. Time to bury my head in some old papers I’d previously overlooked!