Why we’re teaching the Standard Model all wrong

In any description if the Standard Model of Particle Physics, from the serious graduate-level lecture course to the jolly outreach chat for Joe Public, you pretty soon come up against a graphic like this.

“Particles of the Standard Model”

It appears on mugs and on T shirts, on posters and on websites. The colours vary, and sometimes bosons are included. It may be – somewhat pretentiously – described as “the new periodic table”. We’ve all seen it many times. Lots of us have used it – I have myself.

And it’s wrong.

Fundamentally wrong. And we’ve known about it since the 1990’s.

The problem lies with the bottom row: the neutrinos. They are shown as the electron, mu and tau neutrinos, matching the charged leptons.

But what is the electron neutrino? It does not exist – or at least if it does exist, it cannot claim to be a ‘particle’. It does not have a mass. An electron neutrino state is not a solution of the Schrödinger equation: it oscillates between the 3 flavours. Anything that changes its nature when left to itself, without any interaction from other particles, doesn’t deserve to be called an ‘elementary particle’.

That this changing nature happened was a shattering discovery at the time, but now it’s been firmly established over 20 years of careful measurement of these oscillations: from solar neutrinos, atmospheric neutrinos, reactors, sources and neutrino beams.

There are three neutrinos. Call them 1, 2 and 3. They do have definite masses (even if we don’t know what they are) and they do give solutions of the Schrödinger equation: a type 1 neutrino stays a type 1 neutrino until and unless it interacts, likewise 2 stays 2 and 3 stays 3.

So what is an ‘electron neutrino’? Well, when a W particle couples to an electron, it couples to a specific mixture of ν1, ν2, and ν3, That specific mixture is called νe. The muon and tau are similar. Before the 1990s, when the the only information we had about neutrinos came from their W interactions, we only ever met neutrinos in these combinations so it made sense to use them. And they have proved a useful concept over the years. But now we know more about their behaviour – even though that is only how they vary with time – we know that the 1-2-3 states are the fundamental ones.

By way of an analogy: the 1-2-3 states are like 3 notes, say C, E and G, on a piano. Before the 1990s our pianist would only play them in chords: CE, EG and CG (the major third, the minor third and the fifth, but this analogy is getting out of hand…) As we only ever met them in these combinations we assumed that these were the only combinations they ever occurred in which made them fundamental. Now we have a more flexible pianist and know that this is not the case.

We have to make this change if we are going to be consistent between the quarks in the top half of the graphic and the leptons in the bottom. When the W interacts with a u quark it couples to a mixture of d, s and b. Mostly d, it is true, but with a bit of the others. We write d’=Uudd+Uuss+Uubb and introduce the CKM matrix or the Cabibbo angle. But we don’t put d’ in the “periodic table”. That’s because the d quark, the mass eigenstate, leads a vigorous social life interacting with gluons and photons as well as Ws, and it does so as the d quark, not as the d’ mixture. This is all obvious. So we have to treat the neutrinos in the same way.

So if you are a bright annoying student who likes to ask their teacher tough questions (or vice versa), when you’re presented with the WRONG graphic, ask innocently “Why are there lepton number oscillations among the neutral leptons but not between the charged leptons?”, and retreat to a safe distance. There is no good answer if you start from the WRONG graphic. If you start from the RIGHT graphic then the question is trivial: there are no oscillations between the 1-2-3 neutrinos any more than there are between e, mu and tau, or u, c, and t. If you happen to start with a state which is a mixture of the 3 then of course you need to consider the quantum interference effects, for the νe mixture just as you do for the d’ quark state (though the effects play out rather differently).

So don’t use the WRONG Standard model graphic. Change those subscripts on the bottom row, and rejoice in the satisfaction of being right. At least until somebody shows that neutrinos are Majorana particles and we have to re-think the whole thing…

Antineutrinos and the failure of Occam’s Razor

William Of Ockham is one of the few medieval theologian/philosophers whose name survives today, thanks to his formulation of the principle known as Occam’s Razor. In the original latin, if you want to show off, it runs Non sunt multiplicanda entia sine necessitate, or Entities are not to be multiplied without necessity, which can be loosely paraphrased as The simplest explanation is the best one, an idea that is as attractive to a  21st century audience as it was back in the 14th.

 William of Ockham

Now fast forward a few centuries and let’s try and apply this to the neutrino. People talk about the “Dirac Neutrino” but that’s a bit off-target. Paul Dirac produced the definitive description not of the neutrino but of the electron. The Dirac Equation shows – as explained in countless graduate physics courses – that there have to be 2×2=4 types of electron: there are the usual negatively charged ones and the rarer positively charged ones (usually known as positrons), and for each of these the intrinsic spin can point along the direction of motion (‘right handed’) or against it (‘left handed’). The charge is a basic property that can’t change, but handedness depends on the observer (if you and I observe and discuss electrons while the two of us are moving, we will agree about their directions of spin but not about their directions of motion.)

Paul Dirac, 1933

Dirac worked all this out to describe how the electron experienced the electromagnetic force.  But it turned out to be the key to describing its behaviour in the beta-decay weak force as well. But with a twist. Only the left handed electron and the right handed positron  ‘feel’ the weak force. If you show a right handed electron or a left handed positron to the W particle that’s responsible for the weak force then it’s just not interested.   This seems weird but has been very firmly established by decades of precision experiments.

(If you’re worried that this preference appears to contradict the statement earlier that handedness is observer-dependent then well done! Let’s just say I’ve oversimplified a bit, and the mathematics really does take care of it properly. Give yourself a gold star, and check out the difference between ‘helicity’ and ‘chiralilty’ sometime.) 

Right, that’s enough about electrons, let’s move on to neutrinos. They also interact weakly, very similarly to the electron: only the left-handed neutrino and the right-handed antineutrino are involved, and the right-handed neutrino and left-handed antineutrino don’t.

But it’s worse than that. The left handed neutrino and right handed antineutrino don’t interact weakly: they also don’t interact electromagnetically because the neutrino, unlike the electron, is neutral. And they don’t interact strongly either. In fact they don’t interact full stop.  

And this is where William comes in wielding his razor. Our list of fundamental particles includes this absolutely pointless pair that don’t participate at all. What’s the point of them? Can’t we rewrite our description in a way that leaves them out?

And it turns out that we can.

Ettore Majorana

Ettore Majorana, very soon after Dirac published his equation for the electron, pointed out that for neutral particles a simpler outcome was possible. In his system the ‘antiparticle’ of the left-handed neutrino is the right-handed neutrino. The neutrino, like the photon, is self-conjugate. The experiments that showed that neutrinos and antineutrinos were distinct (neutrinos produce electrons in targets: antineutrinos produce positrons) in fact showed the difference between left-handed and right-handed neutrinos. There are only 2 neutrinos and they both interact, not 2×2 where two of the foursome just play gooseberry.

So hooray for simplicity. But is it?

The electron (and its heavier counterparts, the mu and the tau) is certainly a Dirac particle. So are the quarks, both the 2/3 and the -1/3 varieties. If all the other fundamental fermions are Dirac particles, isn’t it simpler that the neutrino is cut to the same pattern, rather than having its own special prescription? If we understand electrons – which it is fair to say that we do – isn’t it simpler that the neutrino be just a neutral version of the electron, rather than some new entity introduced specially for the purpose?

And that’s where we are. It’s all very well advocating “the simple solution” but how can you tell what’s simple? The jury is still out. Hopefully a future set of experiments (on neutrinoless double beta decay) will give an answer on whether a neutrino can be its own antiparticle, though these are very tough and will take several years. After which we will doubtless see with hindsight the simplicity of the answer, whichever it is, and tell each other that it should have been obvious thanks to William.   But at the moment he’s not really much help.