# Anomaly Cancellation

Back in the early 80s, nobody was much interested in string theory. Some wrote it off as inconsistent nonsense. How wrong they were! With a stroke of genius Michael Green and John Schwarz confounded the critics. But how was it done?

First off we’ll need to understand the problem. Our best theory of nature at small scales is provided by the Standard Model. This describes forces as fields, possessing certain symmetries. In particular the mathematical description endows the force fields with an extra redundant symmetry.

The concept of adding redundancy appears absurd at first glance. But it actually makes it much easier to write down the theory. Plus you can eliminate the redundancy later to simplify your calculations. This principle is known as adding gauge symmetry.

When we write down theories, it’s easiest to start at large scales and then probe down to smaller ones. As we look at smaller things, quantum effects come into play. That means we have to make our force fields quantum.

As we move into the quantum domain, it’s important that we don’t lose the gauge symmetry. Remember that the gauge symmetry was just a mathematical tool, not a physical effect. If our procedure of “going quantum” destroyed this symmetry, the fields would have more freedom than they should. Our theory would cease to describe reality as we see it.

Thankfully this problem doesn’t occur in the Standard Model. But what of string theory? Well, it turns out (miraculously) that strings do reproduce the whole array of possible force fields, with appropriate gauge symmetries. But when you look closely at the small scale behaviour, bad things happen.

More precisely, the fields described by propagating quantum strings seem to lose their gauge symmetry! Suddenly things aren’t looking so miraculous. In fact, the string theory has got too much freedom to describe the real world. We call this issue a gauge anomaly.

So what’s the get out clause? Thankfully for string theorists, it turned out that the naive calculation misses some terms. These terms are exactly right to cancel out those that kill the symmetry. In other words, when you include all the information correctly the anomaly cancels!

The essence of the calculation is captured in the image below.

Any potential gauge anomaly would come from the interaction of $6$ particles. For concreteness we’ll focus on open strings in Type I string theory. The anomalous contribution would be given by a $1$-loop effect. Visually that corresponds to an open string worldsheet with an annulus.

We’d like to sum up the contributions from all (conformally) inequivalent diagrams. Roughly speaking, this is a sum over the radius $r$ of the annulus. It turns out that the terms from $r\neq 0$ exactly cancel the term at $r = 0$. That’s what the pretty picture above is all about.

But why wasn’t that spotted immediately? For a start, the mathematics behind my pictures is fairly intricate. In fact, things are only manageable if you look at the $r=0$ term correctly. Rather than viewing it as a $1$-loop diagram, you can equivalently see it as a tree level contribution.

Shrinking down the annulus to $r=0$ makes it look like a point. The information contained in the loop can be viewed as inserting a closed string state at this point. (If you join two ends of an open string, they make a closed one)! The relevant closed string state is usually labelled $B_{\mu\nu}$.

Historically, it was this “tree level” contribution that was accidentally ignored. As far as I’m aware, Green and Schwarz spotted the cancellation after adding the appropriate $B_{\mu\nu}$ term as a lucky guess. Only later did this full story emerge.

My thanks to Sam Playle for an informative discussion on these matters.

# Not a Lentern Fast

Alas – my blog activity has fallen off a cliff recently. I blame the nascent excitement of honing in on my first PhD project. But now that it’s Lent I’m going to buck the trend by making a short post each day.

Expect broad brushstroke paintings of recent papers and lightning reviews of fundamental concepts. Your first instalment is coming right up.

# Maths in Physics – Three Big Ideas

Let me take you back 2400 years to ancient Greece. The philosopher Plato is thinking about the universe. Fundamentally it has to be built of simple objects, he muses. The simplest 3D objects are the most symmetrical ones. So he associates each basic element (fire, water, air, earth and aether) with a regular solid.

Of course, we don’t believe in Plato’s theory today. Nevertheless it’s a fascinating example of the interplay between maths and physics. This dialogue has continued down the ages. For more than two millenia we’ve been striving to arrive at a mathematical understanding of reality.

But what’s the point? On the one hand, there’s the philosophical argument of Plato. Somehow mathematics seems like the fundamental language of the universe. We’d better speak it if we’re to hold a meaningful conversation about physics. But there’s a more prosaic reason too.

Maths is simply the most efficient way of encoding our predictions about experiments. Centuries of science have shown that expressing models mathematically enables swifter and more precise verification of our ideas. It seems then that maths and physics are inextricably linked.

For PhD students and layman alike, this realization is a terrifying one! Pretty much everyone finds mathematics hard. Symbols and equations are easy to muddle up. Calculations are opaque at best, and impossible at worst. Even the basic formula for the current understanding of particle physics is a daunting thicket of letters.

But now I’m going to make a bold assertion. Actually the maths is quite easy. It’s the physics that’s hard. Really maths is just a collection of interesting abstract ideas. Most of these are motivated by some natural intuition. This makes them easy to “get” on a gut-instinct level. The problems arise when you have to select the right mathematical ideas and apply them to the real world.

I’m going to give you three brief examples of big mathematical ideas in modern physics. You’ll see that the concepts themselves are beautifully simple. The scary mess of symbols only rears it’s ugly head once you bolt these ideas together with physics.

## 1. Symmetry

Most of us have an intuitive idea about symmetry. You’d say something’s symmetric if it’s the same after you rotate or reflect it. In many cultures objects and patterns with lots of symmetry are considered beautiful. That’s principally because humans are lazy.

We like symmetrical things because they’re simple. A snowflake is easy on the eye because it’s very regular. Look at just a small part of it and you can predict the rest. So symmetry provides us with useful information for free.

You can see why mathematicians and physicists like symmetry – it makes their job easier! The more symmetry you claim to have, the more constrained your answers must be. That’s all very well as an abstract principle perhaps, but where does the physics come in?

Fortunately the real world has lots of symmetry. Look to your left. Have the laws of physics changed? Hopefully not! This tells us that physics has a rotational symmetry. When you walk to the shops reality doesn’t suddenly shift. So there’s a so-called translational symmetry too.

From our experience with the snowflake we should be able to extract some information about the world. We need to get a feel for what that should be.

Imagine that walk to the shops again. You’re pretty meticulous so you always walk at the same speed. And you like some variety so you’ve got a choice of two routes. Each route is exactly a mile long.

It’s obvious that your journey will take the same time whichever route you take. That is providing there’s nothing special to distract you on the way. Perhaps one route has a travelator, so you go faster even though you aren’t pacing quicker. Maybe on the other route you bump into an old friend and stop for a chat.

But in both of these scenarios you’re changing your overall speed. In the absence of such distractions the speed you walk is conserved, precisely because there’s nothing special about either route you take. In other words the translational symmetry gives rise to a speed conservation law.

So spatial symmetry gives rise to speed conservation. More precisely physicists talk about momentum conservation, which takes into account mass as well. What about rotations? Well they lead to conservation of angular momentum. You’ll have felt its effects if you’ve ever been on a rollercoaster!

So we’ve got a double whammy – symmetry exists in the real world, and it makes doing physics easier. It’s no wonder that our best theory of particles (the boringly-named “Standard Model”) is based entirely on symmetries.

Modern physics has all kinds of more general symmetries. Supersymmetry, gauge symmetry, colour symmetry, flavour symmetry – the list goes on! These mathematical principles guide us towards particular models of reality. In fact the terms in the daunting formula above are really just a consequence of choosing appropriate physical symmetries.

Now for a quick tour of the current battle-lines. Lately people have discovered new symmetries related to particle scattering. Others have been able to exploit these to get a better perspective on the theoretical foundations.

The upshot is a symmetrical jewel which encodes a huge amount of information. This amplituhedron is something like Plato’s solids. So maybe the Greek geezer wasn’t so wrong after all!

## 2. Integrability

Unlike symmetry, you’ve probably never heard of integrability. At heart it’s just a posh word for something extremely simple. We say a physical model is integrable if you can get exact answers from it.

This seems a little arcane, until you realise that we only have approximate solutions to most physical problems. Unfortunately the equations that describe reality are pretty hard. So physicists have to estimate the answer using various techniques. Only in exceptional cases can we do a full mathematical derivation.

This malarkey started off with Newton. In his famous Second Law he claimed that force was proportional to acceleration. This is immediately obvious during takeoff in an aeroplane, for example.

Many of you will know that acceleration measures how fast your speed is changing. So if you know the force being applied, Newton is telling you how the speed changes. Mathematicians call this a differential equation since difference is a synonym for change.

The usual way of solving these equations is called integration. Suppose you know the speed at the start of the motion. Newton gives you the change in speed from one moment to the next. Adding up all the changes over these tiny moments gives you the speed at any point in the future!

Sadly, integration gets really tricky for complicated systems. That’s when the physicists have to approximate things. The most famous example dates back to Newton’s time. Suppose you want to understand the motion of three planets interacting gravitationally. Nobody’s ever been able to find an exact solution for this, and there’s strong evidence that one doesn’t exist at all!

The idea of integrability carries over into quantum mechanics too. Basically a quantum theory separates reality out into a spectrum of chunks. In particular, energy comes in discrete lumps called quanta. But finding out what energies are allowed is difficult. Again you’ve got to solve a differential equation!

In general physicists make approximations, usually by perturbing a system slightly from some ideal configuration. This works remarkably well, but can’t give us complete knowledge of a theory. This ignorance is one of the difficulties associated with forming a quantum theory of gravity.

It seems that our mathematical accounts of nature are just not integrable. Neither in Newtonian physics, nor on the quantum level do the complicated interactions admit a pretty solution. In fact, integrability is a rare and magical property.

Typically integrability is associated with high symmetry situations. We saw in the last part that symmetry gives us useful constraints on allowed physics. By adding a healthy tablespoon of symmetry to the mix we raise our chances of finding a neat solution. Such souped-up theories are an active area of research.

But what’s the point? If integrable theories aren’t very realistic, then why should we be looking at them. Quite apart from their mathematical elegance, they give us new insights into the structure of our model. Methods that start off in the highfalutin realms of an integrable theory can percolate down to more realistic descriptions.

That’s just what some people hope will happen for our modern day Platonic amplituhedron. Many regard it’s beauty as a remnant of the integrability of a relevant souped-up theory. But its lessons could well inform our view of the universe.

## 3. Duality

Racing into the final furlong, we’re confronted with another unfamiliar word. But you might have an inkling about what it means. Think of your favourite dynamic duo – two individuals that present different perspectives yet work well together. From Bush and Blair to Ant and Dec, these relationships are common in the celebrity world.

Curiously this kind of friendship crops up mathematically too! Here’s an easy example. Suppose you draw two random lines on a piece of paper. If your paper is big enough they’ll meet at a point. Similarly if you put two dots on your paper there’s a single line that joins them.

Writing this down as a formula we can roughly say

$\displaystyle \textrm{two lines} = \textrm{one point}\\ \textrm{two points} = \textrm{one line}$

So if you prove something about points, you can immediately translate it into a statement about lines and vice versa. This is a simple example of a mathematical duality. You’ve got two different perspectives on geometry that work together to enrich your understanding.

You’d be forgiven for finding that example unremarkable. There’s nothing very exciting about lines on bits of paper. So let me convince you that duality is important in the real world.

Suppose you’re sitting at your computer, listening to your favourite song. Chances are that it’s been compressed into MP3 format. Have you ever wondered how that’s done? You’ve got to take the wave encoding the song and smush it together somehow.

It’s hard to work out a procedure at first. Just putting all the wave peaks closer together won’t work because that’ll just raise the pitch of the whole song! The key is to come at the issue from a different perspective. You decompose the song into different frequencies and then remove pitches that a human can’t hear.

The trick we used was to exploit a duality between time and frequency. Mathematically the act of decomposing the song is called a Fourier transform. This provides a new perspective on the original wave. You can transform things back and forth as you wish depending on which viewpoint is most convenient.

Duality is convenient because it gives us several mathematical models reproducing the same physics. You’re free to analyse the theory from whatever perspective is the most appealing. We’ve seen that our models are generally very hard to solve. Duality can help to simplify the situation.

Physicists exploit duality in two main ways. Many dualities allow you to improve your approximations of a system. Others give you further details of hidden solutions or symmetries. Those which do both are especially highly prized.

We’ll finish with a modern duality which remains a hot topic today. It has been instrumental in improving our knowledge of several physical models. More specifically it relates a souped-up quantum theory to a stringy theory of gravity. We call it the AdS/CFT duality.

This duality is a mathematical chocolate ganache – decadent, moreish and extremely rich. Many of these qualities arise through large amounts of symmetry. Each of the AdS and CFT duo are exquisitely symmetrical and perhaps even integrable.

The heady mixture of duality, symmetry and integrability has inspired much research over the past 15 years. In many ways, the amplituhedron is the grandchild of this cross-fertilisation. And the story is similar elsewhere in physics. From cosmologists to materials scientists, everyone’s using this powerful brew.

So get out there and impress your friends! I’ve given you a toolbox of ideas from the forefront of theoretical research. In a sense, you are right at the cutting edge. Though you might need a bigger instruction set before you try to build anything.

# Journal Club

It’s been a while since I last posted – research life has become rather hectic! This week has been one of firsts. I’ve given my first PhD talk (on twistors), marked my first set of undergrad work (on QED) and taught my first class (on rotating frames). Meanwhile, I’ve been trying to fathom the mysterious amplituhedron that’s got many physicists excited. I hope you’ll forgive my extended blog absence.

To tide you over until the next full article, take a look at our new journal club website. The PhD students at CRST get together once a week to discuss recent advances or fundamental principles. It’s a great opportunity for us to keep up to date with work in diverse areas of theory. We’ve decided to make resources from our meetings publicly available.

Why? I hear you ask. I believe that science is best done openly. A quick browse of our website gives the layman a taste of what researchers get up to day to day. Moreover our compilation of topics, ideas and resources may be thought-provoking for other students. Keeping an online record of our meetings is the best way to benefit others outside the confines of an individual institution.

I promise I’ll be back this weekend with another popsci article. I’ll talk about five big mathematical ideas in modern physics and explain why they’re cool!

Finally, a shout out to my colleague Brenda Penante who has a paper out on the arXiv today. The paper focusses on an interesting generalization of scattering amplitudes, known as form factors. These can be used for experimental approximation but also give a deep insight into the structure of QFT.

The paper today pushes back the boundaries of our knowledge about form factors. As we chip away at the coal face of QFT, who knows what jewels we might find?

# A Tale of Two Calculations

This post is mathematically advanced, but may be worth a skim if you’re a layman who’s curious how physicists do real calculations!

Recently I’ve been talking about the generalized unitarity method, extolling its virtues for $1$-loop calculations. Despite all this hoodoo, I have failed to provide a single example of a successful application. Now it’s time for that to change. I’m about to show you just how useful generalized unitarity can be, borrowing examples from $\mathcal{N}=4$ super-Yang-Mills (SYM) and $SU(3)$ Yang-Mills (YM).

We’ll begin by revising the general form of the generalized unitarity method. In picture form

What exactly does all the notation mean? On the left hand side, I’m referring to the residue of the integrand when all the loop momenta $l_i$ for $i = 1,2,3,4$ are taken on-shell. On the right hand side, I take a product of tree level diagrams with external lines as shown, and sum over the possible particle content of the $l_i$ lines. Implicit in each of the blobs in the equation is a sum over tree level diagrams.

We’d like to use this formula to calculate $1$-loop amplitudes. But hang on, doesn’t it only tell us about residues of integrands? Naively, it seems like that’s too little information to reconstruct the full result.

Fear not, however – help is at hand! Back in 1965, Don Melrose published his first paper. He presciently observed that loop diagrams in $D$ dimensions could be expressed as linear combinations of scalar loop diagrams with $\leq 4$ sides. Later Bern, Dixon and Kosower generalized this result to take account of regularization.

Let’s express those words mathematically. We have

$\displaystyle \mathcal{A}_n^{1\textrm{-loop}} = \sum_i D_i I_4(K^i) + \sum_j C_j I_3 (K^j) + \sum_m B_m I_2 (K^m) + R_n + O(\epsilon)\qquad (*)$

where $I_a$ are integrals corresponding to particular scalar theory diagrams, $K_a^i$ indicate distribution of momenta on external legs, $R_n$ is a rational function and $\epsilon$ a regulator.

The integrals $I_4$, $I_3$ and $I_2$ are referred to as box, triangle and bubble integrals respectively. This is an obvious homage to their structure as Feynman diagrams. For example a triangle diagram looks like

where $K_1$, $K_2$, $K_3$ label the sums of external momenta at each of the vertices. The Feynman rules give (in dimensional regularization)

$\displaystyle I_3(K_1, K_2, K_3) = \mu^{2 \epsilon}\int \frac{d^{4-2\epsilon}l}{(2\pi)^{4-2\epsilon}}\frac{1}{l^2 (l-K_1)^2 (l+K_3)^2}$

We call result $(*)$ above an integral basis expansion. It’s useful because the integrands of box, triangle and bubble diagrams have different pole structures. Thus we can reconstruct their coefficients by taking generalized unitarity cuts. Of course, the rational term cannot be determined this way. Theoretically we have reduced our problem to a simpler case, but not completely solved it.

Before we jump into a calculation, it’s worth taking a moment to consider the origin of the rational term. In Melrose’s original analysis, this term was absent. It appears in regularized versions, precisely because the act of regularization gives rise to extra rational terms at $O(\epsilon^0)$. Such terms will be familiar if you’ve studied anomalies.

We can therefore loosely say that rational terms are associated with theories requiring renormalization. (This is not quite true; see page 44 of this review). In particular we know that $\mathcal{N}=4$ SYM theory is UV finite, so no rational terms appear. In theory, all $1$-loop amplitudes are constructible from unitarity cuts alone!

Ignoring the subtleties of IR divergences, let’s press on and calculate an $\mathcal{N}=4$ SYM amplitude using unitarity. More precisely we’ll tackle the $4$-point $1$-loop superamplitude. It’s convenient to be conservative and cut only two propagators. To get the full result we need to sum over all channels in which we could make the cut, denoted $s = (12)$, $t = (13)$ and $u=(14)$.

To make our lives somewhat easier, we’ll work in the planar limit of $\mathcal{N}=4$ SYM. This means we can ignore any diagrams which would be impossible to draw in the plane, in particular the $u$-channel ones. We make this assumption since it simplifies our analysis of the color structure of the theory. In particular it’s possible to factor out all $SU(3)$ data as a single trace of generators in the planar limit.

Assuming this has been done, we’ll ignore color factors and calculate only the color-ordered amplitudes. We’ve got two channels to consider $s$ and $t$. But since the trace is cyclic we can cyclically permute the external lines to equate the $s$ and $t$ channel cuts. Draw a picture if you are skeptical.

So we’re down to considering the $s$-channel unitarity cut. Explicitly the relevant formula is

where $\mathcal{A}_4$ is the tree level $4$-particle superamplitude. Now observe that by necessity $\mathcal{A}_4$ must be an MHV amplitude. Indeed it is only nonvanishing if exactly two external particles have $+$ve helicity. Leaving the momentum conservation delta function implicit we quote the standard result

$\displaystyle \mathcal{A}_4(-l_1, 1, 2, l_2) = \frac{\delta^{(8)}(L)}{\langle l_1 1\rangle\langle 1 2\rangle\langle 2l_2\rangle\langle l_2 l_1 \rangle}$

where $\delta^{(8)}(L)$ is a supermomentum conservation delta function. We get a similar result for the other tree level amplitude, involving a delta function $\delta^{(8)}(R)$. Now by definition of the superamplitude, the sum over states can be effected as an integral over the Grassman variables $\eta_{l_1}$ and $\eta_{l_2}$. Under the integral signs we may write

$\displaystyle \delta^{(8)}(L) \delta^{(8)}(R) = \delta^{(8)}(L+R)\delta^{(8)}(R) = \delta^{(8)}(\tilde{Q})\delta^{(8)}(R)$

where $\delta^{(8)}(\tilde{Q})$ is the overall supermomentum conservation delta function, which one can always factor out of a superamplitude in a supersymmetric theory. The remaining delta function gives a nonzero contribution in the integral. To evaluate this recall that the Grassman delta function for a process with $n$ external particles has the form

$\displaystyle \delta^{(8)}(R) = \prod_{A=1}^4 \sum_{i

We know that Grassman integration is the same as differentiation, so

$\displaystyle \int d^4 \eta_{l_1} d^4 \eta_{l_2} \delta^{(8)}(R) = \langle l_1 l_2 \rangle ^4$

Now plugging this in to the pictured formula we find the $s$-channel residue to be

$\displaystyle \textrm{Res}_s = \frac{\delta^{(8)}(\tilde{Q})\langle l_1 l_2 \rangle^2}{\langle 12 \rangle\langle 34 \rangle \langle l_1 1 \rangle \langle 2 l_2 \rangle \langle l_2 4 \rangle \langle 3 l_1 \rangle} \qquad (\dagger)$

Now for the second half of our strategy. We must compare this to the residues from scalar box, triangle and bubble integrands. We aim to pull out a kinematic factor depending on the external momenta, letting the basis integrand residue absorb all factors of loop momenta $l_1$ and $l_2$. But which basis integrands contribute to the residue from our unitarity cut?

This is quite easy to spot. Suppose we consider the residue of a loop integrand after a generic unitarity cut. Any remaining dependence on loop momentum $l$ appears as factors of $(l-K)^{-2}$. These may be immediately matched with uncut loop propagators in the basis diagrams. Simple counting then establishes which basis diagram we want. As an example

$\displaystyle \textrm{factor of }(l-K_1)^{-2}(l-K_2)^{-2}\Rightarrow 2 \textrm{ uncut propagators} \Rightarrow \textrm{box diagram}$

We’ll momentarily see that this example is exactly the case for our calculation of $\mathcal{A}_4^{1\textrm{-loop}}$. To accomplish this, we must express the residue $(\dagger)$ in more familiar momentum space variables. Our tools are the trusty identities

$\displaystyle \langle ij \rangle [ij] =(p_i + p_j)^2$

$\displaystyle \sum_i \langle ri \rangle [ik] = 0$

The first follows from the definition of the spinor-helicity formalism. Think of it as a consequence of the Weyl equation if you like. The second encodes momentum conservation. We’ve in fact got three set of momentum conservation to play with. There’s one each for the left and right hand tree diagrams, plus the overall $(1234)$ relation.

To start with we can deal with that pesky supermomentum conservation delta function by extracting a factor of the tree level amplitude $\mathcal{A}_4^{\textrm{tree}}$. This leaves us with

$\displaystyle \textrm{Res}_s = \mathcal{A}_4^{\textrm{tree}} \frac{\langle 23 \rangle \langle 41 \rangle \langle l_1 l_2 \rangle^2}{ \langle l_1 1 \rangle \langle 2 l_2 \rangle \langle l_2 4 \rangle \langle 3 l_1 \rangle}$

Those factors of loop momenta in the numerator are annoying, because we know there shouldn’t be any in the momentum space result. We can start to get rid of them by multiplying top and bottom by $[l_2 2]$. A quick round of momentum conservation leaves us with

$\displaystyle \textrm{Res}_s = \mathcal{A}_4^{\textrm{tree}} \frac{\langle 23 \rangle \langle 41 \rangle [12] \langle l_1 l_2 \rangle}{(l_2 + p_2)^2\langle l_2 4 \rangle \langle 3 l_1 \rangle}$

That seemed to be a success, so let’s try it again! This time the natural choice is $[3l_1]$. Again momentum conservation leaves us with

$\displaystyle \textrm{Res}_s = \mathcal{A}_4^{\textrm{tree}} \frac{\langle 23 \rangle \langle 41 \rangle [12] [34]}{(l_2 + p_2)^2 (l_1+p_3)^2}$

Overall momentum conservation in the numerator finally leaves us with

$\displaystyle \textrm{Res}_s = -\mathcal{A}_4^{\textrm{tree}} \frac{\langle 12 \rangle [12] \langle 23 \rangle [23]}{(l_2 + p_2)^2 (l_1+p_3)^2} = -\mathcal{A}_4^{\textrm{tree}} \frac{st}{(l_2 + p_2)^2 (l_1+p_3)^2}$

where $s$ and $t$ are the standard Mandelstam variables. Phew! That was a bit messy. Unfortunately it’s the price you pay for the beauty of spinor-helicity notation. And it’s a piece of cake compared with the Feynman diagram approach.

Now we can immediately read off the dependence of the residue on loop momenta. We have two factors of the form $(l-K)^{-2}$ so our result matches only the box integral. Therefore the $4$-point $1$-loop amplitude in $\mathcal{N}=4$ SYM takes the form

$\displaystyle \mathcal{A}_4^{1\textrm{-loop}} = DI_4(p_1,p_2,p_3,p_4)$

We determine the kinematic constant $D$ by explicitly computing the $I_4$ integrand residue on our unitarity cut. This computation quickly yields

$\displaystyle \mathcal{A}_4^{1\textrm{-loop}} = st \mathcal{A}_4^{\textrm{tree}}I_4(p_1,p_2,p_3,p_4)$

Hooray – we are finally done. Although this looks like a fair amount of work, each step was mathematically elementary. The entire calculation fits on much less paper than the equivalent Feynman diagram approach. Naively you’d need to draw $1$-loop diagrams for all the different particle scattering processes in $\mathcal{N}=4$ SYM, including possible ghost states in the loops. This itself would take a long time, and that’s before you’ve evaluated a single integral! In fact the first computation of this result didn’t come from classical Feynman diagrams, but rather as a limit of string theory.

A quick caveat is in order here. The eagle-eyed amongst you may have spotted that my final answer is wrong by a minus sign. Indeed, we’ve been very casual with our factors of $i$ throughout this post. Recall that Feynman rules usually assign a factor of $i$ to each propagator in a diagram. But we’ve completely ignored this prescription!

Sign errors and theorists are best of enemies. So we’d better confront our nemesis and find that missing minus sign. In fact it’s not hard to see where it comes from. The only place in our calculation where extra factors of $i$ wouldn’t simply cancel comes from the cut propagators. Look back at the very first figure and observe that the left hand side has four more factors of $i$ than the right.

Of course we’ve only cut two propagators to obtain the amplitude. This means that we should pick up an extra factor of $(1/i)^2 = -1$. This precisely corrects the sign error than pedants (or experimentalists) would find irritating!

I promised an $SU(3)$ YM calculation, and I won’t disappoint. This will also provide a chance to show off generalized unitarity in all it’s glory. Explicitly we’re going to show that the NMHV gluon four-mass box coefficients vanish.

To start with, let’s disentangle some of that jargon. Remember that an $n$-particle NMHV gluon amplitude has $3$ negative helicity external gluons and $n-3$ positive helicity ones. The four-mass condition means that each corner of the box has more than two external legs, so that the outgoing momentum is a massive $4$-vector.

The coefficient of the box diagram will be given by a generalized unitarity cut of four loop propagators. Indeed triangle and bubble diagrams don’t even have four propagators available to cut, which mathematically translates into a zero contribution to the residue. The usual rules to compute residues tell us that we’ll always have a zero numerator factor left over in for bubble and triangle integrands.

Now the generalized unitarity method tells us to compute the product of four tree diagrams. By our four-mass assumption, each of these has at least $4$ external gluons. We must have exactly $4$ negative helicity and $4$ positive helicity gluons from the cut propagators since all lines are assumed outgoing. We have exactly $3$ further negative helicity particles by our NMHV assumption, so $7$ negative helicity gluons to go round.

But tree level diagrams with $\geq 4$ legs must have at least $2$ negative helicity gluons to be non-vanishing. This is not possible with our setup, since $7 < 8$. We conclude that the NMHV gluon four-mass box coefficients vanish.

Our result here is probably a little disappointing compared with the $\mathcal{N}=4$ SYM example above. There we were able to completely compute a $4$ point function at $1$-loop. But for ordinary YM there are many more subcases to consider. Heuristically we lack enough symmetry to constrain the amplitude fully, so we have to do more work ourselves! A full analysis would consider all box cases, then move on to nonzero contributions from triangle and bubble integrals. Finally we’d need to determine the rational part of the amplitude, perhaps using BCFW recursion at loop level.

Don’t worry – I don’t propose to go into any further detail now. Hopefully I’ve sketched the mathematical landscape of amplitudes clearly enough already. I leave you with the thought-provoking claim that the simplest QFTs are those with the most symmetry. As Arkani-Hamed, Cachazo and Kaplan explain, this is at odds with our childhood desire for simple Lagrangians!

# What Can Unitarity Tell Us About Amplitudes?

Let’s start by analysing the discontinuities in amplitudes, viewed as a function of external momenta. The basic Feynman rules tell us that $1$-loop processes yield amplitudes of the form

$\displaystyle \int d^4 l \frac{A}{l^2(p+q-l)^2}$

where $A$ is some term independent of $l$. This yields a complex logarithm term, which thus gives a branch cut as a function of a Mandelstam variable $(p+q)^2$.

It’s easy to get a formula for the discontinuity across such a cut. Observe first that amplitudes are real unless some internal propagator goes on shell. Indeed when an internal line goes on shell the $i\epsilon$ prescription yields an imaginary contribution.

Now suppose we are considering some process as a function of an external momentum invariant $s$, like a Mandelstam variable. Consider the internal line whose energy is encoded by $s$. If $s$ is lower than the threshold for producing a multiparticle state, then the internal line cannot go on shell. In that case the amplitude and $s$ are both real so we may write

$\displaystyle \mathcal{A}(s) = \mathcal{A}(s^*)^*$

Now we analytically continue $s$ to the whole complex plane. This equation must still hold, since each side is an analytic function of $s$. Fix $s$ at some real value greater than the threshold for multiparticle state production, so that the internal line can go on shell. In this situation of course we expect a branch cut.

Our formula above enforces the relations

$\displaystyle \textrm{Re}\mathcal{A}(s+i\epsilon) = \textrm{Re}\mathcal{A}(s-i\epsilon)$

$\displaystyle \textrm{Im}\mathcal{A}(s+i\epsilon) = -\textrm{Im}\mathcal{A}(s-i\epsilon)$

Thus we must indeed have a branch cut for $s$ in this region, with discontinuity given by

$\displaystyle \textrm{Disc}\mathcal{A}(s) = 2\textrm{Im}\mathcal{A}(s) \qquad (*)$

Now we’ve got a formula for the discontinuity across a general amplitude branch cut, we’re in a position to answer our original question. What can unitarity tell us about discontinuities?

When I say unitarity, I specifically mean the unitarity of the $S$-matrix. Remember that we compute amplitudes by sandwiching the $S$-matrix between incoming and outgoing states defined at a common reference time in the far past. In fact we usually discard non-interacting terms by considering instead the $T$-matrix defined by

$\displaystyle S = \mathbf{1}+iT$

The unitarity of the $S$-matrix, namely $S^\dagger S = \mathbf{1}$ yields for the $T$-matrix the relation

$\displaystyle 2\textrm{Im}(T) = T^\dagger T$

Okay, I haven’t quite been fair with that final line. In fact it should make little sense to you straight off! What on earth is the imaginary part of a matrix, after all? Before you think to deeply about any mathematical or philosophical issues, let me explain that the previous equation is simply a notation. We understand it to hold when evaluated between any incoming and outgoing states. In other words

$\displaystyle 2 \textrm{Im} \langle \mathbf{p}_1 \dots \mathbf{p}_n | T | \mathbf{k}_1 \dots \mathbf{k}_m\rangle = \langle \mathbf{p}_1 \dots \mathbf{p}_n | T^\dagger T | \mathbf{k}_1 \dots \mathbf{k}_m\rangle$

But there’s still a problem: how do you go about evaluating the $T^\dagger T$ term? Thinking back to the heady days of elementary quantum mechanics, perhaps you’re inspired to try inserting a completeness relation in the middle. That way you obtain a product of amplitudes, which are things we know how to compute. The final result looks like

$\displaystyle 2 \textrm{Im} \langle \mathbf{p}_1 \dots \mathbf{p}_n | T | \mathbf{k}_1 \dots \mathbf{k}_m\rangle = \sum_l \left(\prod_{i=1}^l \int\frac{d^3 \mathbf{q}_i}{(2\pi)^3 2E_i}\right) \langle \mathbf{p}_1 \dots \mathbf{p}_n | T^\dagger | \{\mathbf{q_i}\} \rangle \langle \{\mathbf{q_i}\} | T | \mathbf{k}_1 \dots \mathbf{k}_m\rangle$

Now we are in business. All the matrix elements in this formula correspond to amplitudes we can calculate. Using equation $(*)$ above we can then relate the left hand side to a discontinuity across a branch cut. Heuristically we have the equation

$\displaystyle \textrm{Disc}\mathcal{A}(1,\dots m \to 1,\dots n) = \sum_{\textrm{states}} \mathcal{A}(1,\dots m \to \textrm{state})\mathcal{A}(1,\dots n \to \textrm{state})^* \qquad (\dagger)$

Finally, after a fair amount of work, we can pull out some useful information! In particular we can make deductions based on a loop expansion in powers of $\hbar$ viz.

$\displaystyle \mathcal{A}(m,n) = \sum_{L=0}^\infty \hbar^L \mathcal{A}^{(L)}(m,n)$

where $\mathcal{A}^{(L)}(m,n)$ is the $L$-loop amplitude with $m$ incoming and $n$ outgoing particles. Expanding equation $(\dagger)$ order by order in $\hbar$ we obtain

$\displaystyle \textrm{Disc}\mathcal{A}^{(0)}(m,n) = 0$

$\displaystyle \textrm{Disc}\mathcal{A}^{(1)}(m,n) = \sum_{\textrm{states}} \mathcal{A}^{(0)}(m,\textrm{state})\mathcal{A}^{(0)}(n,\textrm{state})^*$

and so forth. The first equation says that tree amplitudes have no branch cuts, which is immediately obvious from the Feynman rules. The second equation is more interesting. It tells us that the discontinuities of $1$-loop amplitudes are given by products of tree level amplitudes! We can write this pictorially as

Here we have specialized to $m=2$, $n=3$ and have left implicit a sum over the possible intermediate states. This result is certainly curious, but it’s hard to see how it can be useful in its current form. In particular, the sum we left implicit involves an arbitrary number of states. We’d really like a simpler relation which involves a well-defined, finite number of Feynman diagrams.

It turns out that this can be done, provided we consider particular channels in which the loop discontinuities occur. For each channel, the associated discontinuity is computed as a product of tree level diagrams obtained by cutting two of the loop propagators. By momentum conservation, each channel is uniquely determined by a subset of external momenta. Thus we label channels by their external particle content.

How exactly does this simplification come about mathematically? To see this we must take a more detailed look at Feynman diagrams, and particularly at the on-shell poles of loop integrands. This approach yields a pragmatic method, at the expense of obscuring the overarching role of unitarity. The results we’ve seen here will serve as both motivation and inspiration for the pedestrian perturbative approach.

We leave those treats in store for a future post. Until then, take care, and please don’t violate unitarity.

# The Calculus of Particle Scattering

Quantum field theory allows us to calculate “amplitudes” for particle scattering processes. These are mathematical functions that encode the probability of particles scattering through various angles. Although the theory is quite complicated, miraculously the rules for calculating these amplitudes are pretty easy!

The key idea came from physicist Richard Feynman. To calculate a scattering amplitude, you draw a series of diagrams. The vertices and edges of the diagram come with particular factors relevant to the theory. In particular vertices usually carry coupling constants, external edges carry polarization vectors, and internal edges carry functions of momenta.

From the diagrams you can write down a mathematical expression for the scattering amplitude. All seems to be pretty simple. But what exactly are the diagrams you have to draw?

Well there are simple rules governing that too. Say you want to compute a scattering amplitude with $2$ incoming and $2$ outgoing particles. Then you draw four external lines, labelled with appropriate polarizations and momenta. Now you need to connect these lines up, so that they all become part of one diagram.

This involves adding internal lines, which connect to the external ones at vertices. The types and numbers of lines allowed to connect to a vertex is prescribed by the theory. For example in pure QCD the only particles are gluons. You are allowed to connect either three or four different lines to each vertex.

Here’s a few different diagrams you are allowed to draw – they each give different contributions to the overall scattering amplitude. Try to draw some more yourself if you’re feeling curious!

Now it’s immediately obvious that there are infinitely many possible diagrams you could draw. Sounds like this is a problem, because adding up infinitely many things is hard! Thankfully, we can ignore a lot of the diagrams.

So why’s that? Well it transpires that each loop in the diagram contributes an extra factor of Planck’s constant $\hbar$. This is a very small number, so the effect on the amplitude from diagrams with many loops is negligable. There are situations in which this analysis breaks down, but we won’t consider them here.

So we can get a good approximation to a scattering amplitude by evaluating diagrams with only a small number of loops. The simplest have $0$-loops, and are known as tree level diagrams because they look like trees. Here’s a QCD example from earlier

Next up you have $1$-loop diagrams. These are also known as quantum corrections because they give the QFT correction to scattering processes from quantum mechanics, which were traditionally evaluated using classical fields. Here’s a nice QCD $1$-loop diagram from earlier

If you’ve been reading some of my recent posts, you’ll notice I’ve been talking about how to calculate tree level amplitudes. This is sensible because they give the most important contribution to the overall result. But the real reason for focussing on them is because the maths is quite easy.

Things get more complicated at $1$-loop level because Feynman’s rules tell us to integrate over the momentum in the loop. This introduces another curve-ball for us to deal with. In particular our arguments for the simple tree level recursion relations now fail. It seems that all the nice tricks I’ve been learning are dead in the water when it comes to quantum corrections.

But thankfully, all is not lost! There’s a new set of tools that exploits the structure of $1$-loop diagrams. Back in the 1950s Richard Cutkosky noticed that the $1$-loop diagrams can be split into tree level ones under certain circumstances. This means we can build up information about loop processes from simpler tree results. The underlying principle which made this possible is called unitarity.

So what on earth is unitarity? To understand this we must return to the principles of quantum mechanics. In quantum theories we can’t say for definite what will happen. The best we can do is assign probabilities to different outcomes. Weird as this might sound, it’s how the universe seems to work at very small scales!

Probabilities measure the chances of different things happening. Obviously if you add up the chances of all possible outcomes you should get $1$. Let’s take an example. Suppose you’re planning a night out, deciding whether to go out or stay in. Thinking back over the past few weeks you can estimate the probability of each outcome. Perhaps you stay in $8$ times out of $10$ and go out $2$ times out of ten. $8/10 + 2/10 = 10/10 = 1$ just as we’d expect for probability!

Now unitarity is just a mathsy way of saying that probabilities add up to $1$. It probably sounds a bit stupid to make up a word for such a simple concept, but it’s a useful shorthand! It turns out that unitarity is exactly what we need to derive Cutkosky’s useful result. The method of splitting loop diagrams into tree level ones has become known as the unitarity method.

The nicest feature of this method is that it’s easy to picture in terms of Feynman diagrams. Let’s plunge straight in and see what that looks like.

At first glance it’s not at all clear what this picture means. But it’s easy to explain step by step. Firstly observe that it’s an equation, just in image form. On the left hand side you see a loop diagram, accompanied by the word $\textrm{Disc}$. This indicates a certain technical property of a loop diagram that it’s useful to calculate. On the right you see two tree diagrams multiplied together.

Mathematically these diagrams represent formulae for scattering amplitudes. So all this diagram says is that some property of $1$-loop amplitudes is produced by multiplying together two tree level ones. This is extremely useful if you know about tree-level results but not about loops! Practically, people usually use this kind of equation to constrain the mathematical form of a $1$-loop amplitude.

If you’re particularly sharp-eyed you might notice something about the diagrams on the left and right sides of the equation. The two diagrams on the right come from cutting through the loop on the left in two places. This cutting rule enables us to define the unitarity method for all loop diagrams. This gives us the full result that Cutkosky originally found. He’s perhaps the most aptly named scientist of all time!

We’re approaching the end of our whirlwind tour of particle scattering. We’ve seen how Feynman diagrams give simple rules but difficult maths. We’ve mentioned the tree level tricks that keep calculations easy. And now we’ve observed that unitarity comes to our rescue at loop-level. But most of these ideas are actually quite old. There’s just time for a brief glimpse of a hot contemporary technique.

In our pictorial representation of the unitarity method, we gained information by cutting the loop in two places. It’s natural to ask whether you could make further such cuts, giving more constraints on the form of the scattering amplitude. It turns out that the answer is yes, so long as you allow the momentum in the loop to be a complex number!

You’d be forgiven for thinking that this is all a bit unphysical, but in the previous post we saw that using the complex numbers is actually a very natural and powerful mathematical trick. The results we get in the end are still real, but the quickest route there is via the complex domain.

So why do the complex numbers afford us the extra freedom to cut more lines? Well, the act of cutting a line is mathematically equivalent to taking the corresponding momentum $(l-K)$ to be on-shell; that is to say $(l-K)^2 =0$. We live in a four-dimensional world, so $l$ has four components. That means we can solve a maximum of four equations $(l-K)^2 =0$ simultaneously. So generically we should be allow to cut four lines!

However, the equations $(l-K)^2 =0$ are quadratic. This means we are only guaranteed a solution if the momentum $l$ can be complex. So to use a four line cut, we must allow the loop momentum to be complex. With our simple $2$ line cuts there was enough freedom left to keep $l$ real.

The procedure of using several loop cuts is known as the generalized unitarity method. It’s been around since the late 90s, but is still actively used to determine scattering amplitudes. Much of our current knowledge about QCD loop corrections is down to the power of generalized unitarity!

That’s all for now folks. I’ll be covering the mathematical detail in a series of posts over the next few days.

My thanks to Binosi et al. for their excellent program JaxoDraw which eased the drawing of Feynman diagrams.