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.