# Why I Like Supersymmetry

Supersymmetry can be variously described as beautiful, convenient, unphysical and contrived. The truth is that nobody really knows whether we’re likely to find it in our universe. Like most theoretical physicists I hope we do, but even if we don’t it can still be a useful mathematical tool.

There are tons of reasons to like supersymmetry, as well as a good many arguments against it. I can’t cover all of these in a brief post, so I’m just going to talk about one tiny yet pretty application I glanced at today.

Let’s talk about scattering processes again, my favourite topic of (physics) conversation. These are described by quantum field theory, which is itself based on very general principles of symmetry. In the standard formulation (imaginatively called the Standard Model) these symmetries involve physical motions in spacetime, as well as more abstract transformations internal to the theory. The spacetime symmetries are responsible for giving particles mass, spin and momentum, while the internal ones endow particles with various charges.

At the quantum level these symmetries actually provide some bonus information, in the form of certain identities that scattering processes have to satisfy. These go by the name of Ward identities. For example QED has a both a gauge and a global $U(1)$ symmetry. The Ward identity for the global symmetry tells you that charge must be conserved. The Ward identity for the gauge symmetry tells you that longitudinally polarized photons are unphysical.

If you’re a layman and got lost above then don’t worry. All you need to know is that Ward identities are cool because they tell you extra things about a theory. The more information you have, the more constrained the answer must be, so the less work you have to do yourself! And this is where supersymmetry comes into the picture.

Supersymmetry is another (very special) type of symmetry that pairs up fermions (matter) and bosons (forces). Because it’s a symmetry it has associated Ward identities. These relate different scattering amplitudes. The upshot is that once you compute one amplitude you get more for free. The more supersymmetry you have, the more relations there are, so the easier your job becomes.

So what’s the use if supersymmetry isn’t true then? Well, in general terms it’s still useful to look at these simplified situations because it might help us discover tools that would be hard to uncover otherwise. Take the analogy of learning a language, for example. One way to do it is just to plunge headlong in and try to pick things up as you go along. This way you tend to get lots of everyday phrases quickly, but won’t necessary understand the structure of the language.

Alternatively you can go to classes that break things down into simpler building blocks. Okay spending one hour studying the subjunctive alone might not seem very useful at first, but when you go back to having a real conversation you’ll pick up new subtleties you never noticed before.

If you’re still unconvinced here’s a (somewhat trivial) concrete example. Recall that you can show that purely positive helicity gluon amplitudes must vanish at tree level in QCD. The proof is easy, but requires some algebraic fiddling. The SUSY Ward identity tells us immediately than in a Super-Yang-Mills (SYM) theory this amplitude must vanish to all orders in the loop expansion. So how do we connect back to QCD?

Well the gluon superpartners (gluinos) have quadratic coupling to the gluon, so an all gluon scattering amplitude in SYM can’t include any gluinos at tree level. (Just think about trying to draw the diagram if you’re confused)! In other words, at tree level the SYM amplitude is exactly the QCD amplitude, which proves our result.

Not sure what will be on the menu tomorrow – I’m guessing that either color-ordering or unitarity methods will feature. Drop me a comment if you have a preference.

# Why does Spontaneous Symmetry Breaking Depend on Energy Scale?

In the Standard Model we tend to argue that electroweak symmetry is broken below a certain (large) energy scale, yielding the $W$ and $Z$ bosons and the photon. The usual argument for spontaneously symmetry breaking relies on a Higgs potential of the form

$V(\varphi)= \varphi^4 - \mu^2\varphi^2$

and the argument follows from the degeneracy of the lowest energy state.

Remark: Strictly speaking we really ought to be talking about the effective potential, which takes into account radiative corrections. The lowest energy state would then be the vacuum expectation value of the field. I’ll treat the rigorous foundations of SSB in a future post, a topic which is often overlooked in lectures and textbooks!

Critically the standard argument makes no mention of energy scale. So why should we expect it to play a role. The answer is twofold.

Firstly we must remember that we can never observe a full theory. In fact our observations are at best approximations to our theory. This slightly backwards way of looking at things aids our understanding of spontaneous symmetry breaking. At low energies we see the “Mexican hat” clearly, and observe it’s effects in experiments. But as we go to higher energies we “zoom out” on the $V$ axis. This means that the Mexican hat effect is no longer “visible”. From an experimental perspective it’s effects are dominated by other parameters such that they are unobservable. To all intents and purposes the symmetry remains unbroken.

In some theories there’s something more fundamental which averts the SSB scenario at high energies. Recall that renormalization causes the coupling constants to run. In particular our mass pseudoparameter $\mu^2$ can actually change sign at high energies. This eliminates the vacuum degeneracy in the (effective) potential. To see an example, look at Figure 13 in Ben Allanach’s SUSY notes.

I’m grateful to Zac Kenton for a fruitful discussion over a (much-needed) coffee.