# 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.

# Complex Scattering For Beginners

Quantum field theory is a description of interacting particles. These are the fundamental constituents of our universe. They are real, in the sense that their properties are described by real numbers. For example, a typical particle has a momentum through spacetime described by a vector with four real components.

In my line of work we’re generally interested in finding out what happens when particles scatter. There are various rules that enable you to determine a numerical amplitude from diagrams of the process. These so-called “Feynman rules” combine the real quantities in different ways depending on the structure of the theory.

Trouble is, it can be quite tricky to do the exact calculations from these rules. It’s a bit like trying to put together a complicated piece of Ikea furniture with no idea what the end product is meant to look like! In a sense the task is possible, but you’d be hard pressed not to go wrong. Plus it would take you ages to finish the job.

What we really need is some extra pointers that tell us what we’re trying to build. Turns out that we can get that kind of information by performing a little trick. Instead of keeping all of our particle properties real, we bring in the complex numbers.

The complex numbers are like a souped up version of the real numbers. The extra ingredient is a new quantity $i$ which squares to $-1$. This might all sound rather contrived at the moment, but in fact mathematically the complex numbers are a lot nicer behaved. By bringing them into play you can extract more information about your original scattering process for free!

Let’s go back to our Ikea analogy. Suppose that you get a mate in to help with the job. Your task is still the same as ever, but now as you construct it you can share tips. This makes everything easier. Moreover you can pool your guesses about what piece of furniture you’re building. The end result is still the same (hopefully!) but the extra input helped you to get there.

The Kallen-Lehmann Representation

Enough waffle, let’s get into some maths. Warning: you might find this hard going if you’re a layman! Consider the propagator of a generic (interacting) quantum field theory

$\displaystyle \mathcal{A}(x,y) = \langle 0 | T \phi (x)\phi (y) | 0 \rangle$

We’d like to look at it’s analytic properties as a function of the momentum $p$ it carries. The first step is to use the standard completeness relation for the quantum states of the theory

$\displaystyle \mathbf{1} = | 0 \rangle \langle 0 | + \sum_{\lambda} \int\frac{d^3p}{2(2\pi)^3E_{\mathbf{p}}(\lambda)} |\lambda_{\mathbf p}\rangle \langle \lambda_{\mathbf p} |$

where $|\lambda_{\mathbf{p}}\rangle$ is a general (possibly multiparticle) eigenstate of the Hamiltonian with momentum $\mathbf{p}$. Inserting this in the middle of the propagator we get

$\displaystyle \mathcal{A}(x,y) = \sum_{\lambda} \int\frac{d^3p}{2(2\pi)^3E_{\mathbf{p}}(\lambda)} \langle 0 | \phi(x) |\lambda_{\mathbf p}\rangle \langle \lambda_{\mathbf p} | \phi(y) | 0 \rangle$

where we have assumed that the VEV $\langle \phi(x) \rangle$ vanishes, which is equivalent to no interactions at $\infty$. This is a very reasonable assumption, and in fact is a key assumption for scattering processes. (It’s particularly important in the analysis of spontaneous symmetry breaking, for example).

Now a little bit of manipulation (exercise: use the transformation of the quantities under the full Poincare group) gives us that

$\displaystyle \langle 0 | \phi(x) |\lambda_{\mathbf p}\rangle = \langle 0 | \phi(0) | \lambda_{0}\rangle e^{-ip.x}|_{p_0 = E_{\mathbf{p}}}$

Now substituting and introducing an integration over $p_0$ we get

$\displaystyle \mathcal{A}(x,y) = \sum_{\lambda} D(x-y, m_{\lambda}^2)Z$

where $D(x-y,m_{\lambda}^2)$ is a Feynman propagator and $Z = |\langle0|\phi(0)|\lambda_0\rangle|^2$ is a renormalization factor. We’ll safely ignore $Z$ for the rest of this post, since it doesn’t contribute to the analytic behaviour we’re interested in.

So why is this useful? One natural way to extract information from this formula might be to distinguish one-particle states. Let’s see how that helps. Recall that our states $|\lambda_{\mathbf{p}}\rangle$ are eigenvalues of the energy-momentum operator $(H, \mathbf{P})$. Generically we get one-particle states of mass $m$ arranged along a hyperboloid in energy-momentum space, due to special relativity. We also have multiparticle states of mass at least $2m$ forming a continuum at higher energy and momenta. (This is obvious if you consider possible vector addition of one-particle states).

Now we can use this newfound knowledge to rewrite the sum over $\lambda$ as an isolated one-particle term, plus an integral over the multiparticle continuum as follows

$\displaystyle \mathcal{A}(x,y) = D(x-y,m^2)Z + \int_{4m^2}^\infty dM^2 D(x-y, M^2)$

This is starting to look promising. Transforming to momentum space is the last step we need to extract something useful. We find

$\displaystyle \mathcal{A}(p^2) = \frac{iZ}{p^2-m^2} + \int_{4m^2}^{\infty}dM^2 \frac{iZ}{p^2 - M^2}$

Considering the amplitude as an analytic function of the “Mandelstam variable” $p^2$ we find an isolated simple pole from an on-shell single particle state, plus a branch cut from multiparticle states.

It’s easy to generalize this to all Feynman diagrams. The key point is that all the analytic structure of an amplitude is encoded by the propagators. Indeed, the vertices and external legs merely contribute polarization vectors, internal symmetry factors and possibly positive factors of momentum. Singularities and branch cuts can only arise from propagators.

So what’s the big deal?

We’ve done a lot of work to extract some seemingly abstract information. But now it’s time for a substantial payoff! The analytic structure of Feynman diagrams can help us to determine their values. I won’t go into details here, but I will briefly mention one important application.

Remember that the scattering matrix in any sensible theory must conserve probabilities, and so be unitary. This requirement, coupled with our observations about Feynman diagrams tells us a lot about perturbative results. The result is usually known as the optical theorem and allows you to extract information about the discontinuities of higher loop diagrams from those at lower loops.

Still this seems rather esoteric, until you turn the whole procedure on it’s head. Suppose you are trying to guess a $1$-loop amplitude. You know it’s general form perhaps, but need to fix some constants. Well from the $S$-matrix unitarity we know it has a branch cut and that the discontinuity is encoded by some tree level diagrams. These diagrams are essentially given by “cutting” the loop diagram.

So go ahead and compare the discontinuity you have with the product of the relevant tree diagrams. This will give you constraints on the constants you need to fix. Do this enough times, for different “cuts” and you will have fixed your $1$-loop amplitude. Simple!

This method is known as generalized unitarity. It’s a vital tool in the modern amplitudes box, and has been used successfully to attack many difficult loop calculations. I’ll return to it more rigorously later, and promise to show you a genuine calculation too.