# 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!

# Recursion Relations for Amplitudes

So yesterday’s discussion was all about how simple some gluon scattering processes (or amplitudes) look. In particular the maximal helicity violating (MHV) ones are special, because hundreds of terms cancel down to give a single neat result. Today I’ve been looking at how to prove this result, so I can now sketch the main ingredients for you.

If you’re from a mathsy background you won’t be surprised to learn that the $n$-gluon Parke-Taylor identity is proved using induction. For the uninitiated there’s a simple analogy with climbing stairs. If you can get up the first one, and you can get from every one to the next one then you can get to the top!

With this in mind, our first task is to prove the simplest case, which turns out to be $n=3$. Why exactly? Well the Feynman rules for QCD have 3 and 4 point vertices at tree level, so there’s no tree level 2 point amplitudes! Turns out that the $n = 3$ case is neatly dealt with using spinor helicity formalism. Roughly speaking this takes into account the special helicity structure of the MHV amplitudes to lock in simplifications right from the start of the calculation! Add in momentum conservation and hey presto the $n=3$ Parke-Taylor identity drops right out.

So now we need to climb from one stair to the next. This is where recursion relations come in handy. Nine years ago, a group of theoreticians spotted a cunning way to break apart tree level gluon amplitudes into smaller, more manageable pieces. Mathematically they spotted that the $n$-gluon scattering amplitude factorized into the product of two distinct on-shell amplitudes, each with a complementary subset of the original external legs plus an extra leg with momentum $\hat{P}$. The only added ingredient needed was a factor of $1/P^2$ corresponding to a propagator between the two diagrams.

Woah – hold up there! What’s all this terminology all of a sudden. For the uninitiated I’m guessing that on-shell sounds a bit confusing. But it’s no cause for alarm. In general an on-shell quantity is one which obeys the equations of motion of the system involved. Here the relevant equation is the Weyl equation, which implies that $\hat{P}^2 = 0$.

Why are these recursion relations so useful? Well, they give us exactly the ingredient we needed for the induction step. And we’re done – the Parke-Taylor identity is proved, with a little bit of algebra I’ve shoved under the rug.

There’s one more point I’ve neglected to mention. How do you go about finding these mythical on-shell recursion relations? The answer comes from doing some subtle complex analysis, transforming momenta into the complex plane. It might not sound very physical to do that, but in fact the method opens up oodles of new possibilities. One reason is that complex integration is both more powerful and easier than its real counterpart, so it can be used to extract valuable identities from the world of scattering processes.

I’ll leave you with this great basic article clarifying the subtleties of the on-shell/off-shell distinction. It goes a bit deeper than that too, so is worth a read even if you’re more of an expert!

Tomorrow I hope I’ll bring you something more supersymmetrical in nature. $\mathcal{N}=4$ SUSY is a favourite playground for scattering enthusiasts because it is finite (no renormalization needed) and very simple (no free parameters). We’ll encounter its stark beauty in due course.