# Scattering Without Scale, Or The S-Matrix In N=4

My research focuses on an unrealistic theory called massless $\mathcal{N}=4$ super Yang-Mills (SYM). This sounds pretty pointless, at least at first. But actually this model shares many features with more complete accounts of reality.  So it’s not all pie in the sky.

The reason I look at  SYM is because it contains lots of symmetry. This simplifies matters a lot. Studying SYM is like going to an adventure playground – you can still have great fun climbing and jumping, but it’s a lot safer than roaming out into a nearby forest.

Famously SYM has a conformal symmetry. Roughly speaking, this means that the theory looks the same at every length scale. (Whether conformal symmetry is equivalent to scale invariance is a hot topic, in fact)! Put another way, SYM has no real notion of length. I told you it was unrealistic.

This is a bit unfortunate for me, because I’d like to use SYM to think about particle scattering. To understand the problem, you need to know what I want to calculate. The official name for this quantity is the S-matrix.

The jargon is quite straightforward. “S” just stands for scattering. The “matrix” part tells you that this quantity encodes many possible scattering outcomes. To get an S-matrix, you have to assume you scatter particles from far away. That’s certainly the case in big particle accelerators – the LHC is huge compared to a proton!

But remember I said that SYM doesn’t have a length scale. So really you can’t get an S-matrix. And without an S-matrix, you can’t say anything about particle scattering. Things aren’t looking good.

Fortunately all is not lost. You can try to define an S-matrix using the usual techniques that worked in normal theories. All the calculations go through fine, unless there are any low energy particles around. Any of these so-called soft particles will cause your S-matrix to blow up to infinity!

But hey, we should expect our S-matrix to be badly behaved. After all, we’ve chosen a theory without a sense of scale! These irritating infinities go by the name of infrared divergences. Thankfully there’s a systematic way of eliminating them.

Remember that I said our SYM theory is massless. All the particles are like photons, constantly whizzing about that the speed of light. If you were a photon, life would be very dull. That’s because you’d move so fast through space you couldn’t move through time. This means that essentially our massless particles have no way of knowing about distances.

Viewed from this perspective it’s intuitive that this lack of mass yields the conformal symmetry. We can remove the troublesome divergences by destroying the conformal symmetry. We do this in a controlled way by giving some particles a small mass.

Technically our theory is now called Coulomb branch SYM. Who’s Coulomb, I hear you cry? He’s the bloke who developed electrostatics 250 years ago. And why’s he cropped up now? Because when we dispense with conformal symmetry, we’re left with some symmetries that match those of electromagnetism.

In Coulomb branch SYM it’s perfectly fine to define an S-matrix! You get sensible answers from all your calculations. Now imagine we try to recover our original theory by decreasing all masses to zero. Looking closely at the S-matrix, we see it split into two pieces – finite and infinite. Just ignore the infinite bit, and you’ve managed to extract useful scattering data for the original conformal theory!

You might think I’m a bit blasé in throwing away these divergences. But this is actually well-motivated physically. The reason is that such infinities cancel in any measurable quantity. You could say that they only appear in the first place because you’re doing the wrong sum!

This perspective has been formalized for the realistic theories as the KLN theorem. It may even be possible to get a rigorous version for our beloved massless $\mathcal{N}=4$ SYM.

So next time somebody tells you that you can’t do scattering in a conformal theory, you can explain why they’re wrong! Okay, I grant you, that’s an unlikely pub conversation. But stranger things have happened.

And if you’re planning to grab a pint soon, make it a scientific one!

# 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?

# Non-Perturbative QFT: A Loophole?

I’ve been musing on yesterday’s post, and in particular a potential loophole in my argument. Recall that the whole shebang hinges on the fact that $e^{1/g^2}$ is smooth but not analytic. We were interpreting $g$ as the coupling constant in some theory, say QED. But hang about, surely we could just do a rescaling $A_\mu \mapsto A_\mu/g^2$ to remove this behaviour? After all the theory is invariant classically under such a transformation.

But it turns out that this kind of rescaling is anomalous in (most) quantum field theories. Recall that renormalization endows quantum field theories with a $\beta$ function, which determines the evolution of coupling constants as energy changes. The rescaling will only remain a symmetry if $\beta(g)$ is globally zero. Otherwise the rescaling only superficially eliminates the non-perturbative effect – it will reappear at different energies!

This raises a natural question: can you have instantons in finite quantum field theories? By definition these have $\beta$ function zero. Naively we might expect scale invariance to kill non-perturbative physics. A popular finite theory is $\mathcal{N}=4$ SYM, which crops up in AdS/CFT. A quick google suggests that my naive thinking is wrong. There are plenty of papers on instantons in this theory!

There must be a still deeper level to non-perturbative understanding. Sadly most physics papers gloss over the details. Let’s keep half an eye out for an explanation!

# Non-Perturbative Effects in Quantum Field Theory

In elementary QFT we only really know how to solve free theories. These require quantizing infinitely many harmonic oscillators, which is an easy problem. But the real world is described by interacting theories with Lagrangians like

$\mathcal{L}_{\textrm{QED}} = \overline{\psi}(i\not D-m)\psi - \frac{1}{4}F^2$

in which the fields are mixed up together. The standard way of dealing with these is to write them as

$\mathcal{L} = \mathcal{L}_{\textrm{free}}+\mathcal{L}_{\textrm{int}}$

and then split the path integral as

$\exp(i\int d^4x \mathcal{L}) = \exp(i\int d^4 x \mathcal{L}_{\textrm{free}})(1+i\int d^4x\mathcal{L}_{\textrm{int}}+\dots)$

The first factor explicitly yields free theory propagators. The second factor may now be expanded as a Taylor series in some coupling constant $g$. Often this is trivial since $g$ appears in $\mathcal{L}_{\textrm{int}}$ precisely to first order. For example in QED we have

$\mathcal{L}_{\textrm{int}} = g\overline{\psi}\gamma^{\mu}\psi$

with $g$ being the electric charge. This means that the perturbation expansion is exactly the usual exponential series.

At present it looks like perturbation theory must capture all information in a QFT, provided you sum up all the terms. However, we’ve missed a crucial mathematical point. We’ll view the scattering amplitude $\Gamma$ for a given process as a complex-valued function of a real coupling $g$; that is to say $\Gamma \equiv \Gamma(e)$. We construct this function in (somewhat) trivial way from smooth functions, so it should be smooth.

However, smoothness does not guarantee analyticity! Put another way, we cannot be sure that the Taylor series for $g$ will converge for $g\neq 0$. Even if it does, it’s not guaranteed to converge to the function $\Gamma$ itself. This means that there are effects that lie outside the realm of perturbation theory, even if you could sum every term.

Let’s take an example. Suppose for instance that $\mathcal{L}_{\textrm{int}}$ has a $1/g^2$ dependence. If you’re balking at this “unphysical” choice, then rest assured that such Lagrangians do crop up. Now try to compute the Taylor expansion around $g = 0$ for $exp(-1/x^2)$. A little thought shows that this is identically zero.

Such non-perturbative effects require new approaches. One fruitful method involves solving the full theory classically then then considering quantum perturbations around such solutions. For Yang-Mills theories in particular this yields the notion of an instanton.

Finally let’s go back to QED and work out whether we should see non-perturbative effects there. Although not immediately as obvious as the simple $1/g^2$ example, one can argue that the perturbation series has $0$ radius of convergence. We follow Dyson and observe that if the radius of convergence were not zero, we’d be able to “reverse” the electric field by flipping the sign of $g$. This would render the vacuum unstable against decays into electrons and positrons, since these would repel rather than annihilate within a “Heisenberg allowed” time period. To rule out this option, we must take $g = 0$ formally.

This argument is not mathematically rigorous, and I don’t know whether one can make it so. Please comment if you know! Thankfully there’s a more modern perspective that sheds new light. We can view the sickness in QED in the context of Wilsonian renormalization. In particular, the theory has no well-defined high energy limit, as the coupling constants flow to infinite values at finite energy. This Landau pole behaviour is perhaps evidence that non-perturbative effects rule high energy QED. Again, I don’t whether this argument can be made watertight, so please take a rain-cheque on that one!