Spinor Info

Update (September 6, 2013): The analysis in this blog entry is invalid. See my September 6, 2013 blog entry on this topic for an explanation and update.

It has been a while since I last wrote about a pure physics topic in this blog.

A big open question these days is whether or not the particle purportedly discovered by the Large Hadron Collider is indeed the Higgs boson.

One thing about the Higgs boson is that it is a spin-0 scalar particle: this means, essentially, that the Higgs is identical to its mirror image. This distinguishes the Higgs from pseudoscalar particles that “flip” when viewed in a mirror.

So then, one way to distinguish the Higgs from other possibilities, including so-called pseudoscalar resonances, is by establishing that the observed particle indeed behaves either like a scalar or like a pseudoscalar.

Easier said than done. The differences in behavior are subtle. But it can be done, by measuring the angular distribution of decay products. And this analysis was indeed performed using the presently available data collected by the LHC.

Without further ado, here is one view of the data, taken from a November 14, 2012 presentation by Alexey Drozdetskiy:

The solid red line corresponds to a scalar particle (denoted by 0+); the dotted red line to a pseudoscalar (0−). The data points represent the number of events. The horizontal axis represents a “Matrix Element Likelihood Analysis” value, which is constructed using a formula similar to this one (see arXiv:1208.4018 by Bolognesi et al.):

$${\cal D}_{\rm bkg}=\left[1+\frac{{\cal P}_{\rm bkg}(m_{4\ell};m_1,m_2,\Omega)}{{\cal P}_{\rm sig}(m_{4\ell};m_1,m_2,\Omega)}\right]^{-1},$$

where the \({\cal P}\)-s represent probabilities associated with the background and the signal.

So far so good. The data are obviously noisy. And there are not that many data points: only 10, representing 16 events (give or take, as the vertical error bars are quite significant).

There is another way to visualize these values: namely by plotting them against the relative likelihood that the observed particle is 0+ or 0−:

In this fine plot, the two Gaussian curves correspond to Monte-Carlo simulations of the scalar and pseudoscalar scenarios. The position of the green arrow is somehow representative of the 10 data points shown in the preceding plot. The horizontal axis in this case is the logarithm of a likelihood ratio.

On the surface of it, this seems to indicate that the observed particle is indeed a scalar, just like the Higgs. So far so good, but what bothers me is that this second plot does not indicate uncertainties in the data. Yet, judging by the sizable vertical error bars in the first plot, the uncertainties are significant.

However, to relate the uncertainties in the first plot, one has to be able to relate the likelihood ratio on this plot to the MELA value on the preceding plot. Such a relationship indeed exists, given by the formula

$${\cal L}_k=\exp(-n_{\rm sig}-n_{\rm bkg})\prod_i\left(n_{\rm sig}\times{\cal P}^k_{\rm sig}(x_i;\alpha;\beta)+n_{\rm bkg}\times{\cal P}_{\rm bkg}(x_i;\beta)\right).$$

The problem with this formula, from my naive perspective, is that in order to replicate it, I would need to know not only the number of candidate signal events but also the number of background events, and also the associated probability distributions and values for \(\alpha\) and \(\beta\). I just don’t have all the information necessary to reconstruct this relationship numerically.

But perhaps I don’t have to. There is a rather naive thing one can do: and that would be simply calculating the weighted average of the data points in the first plot. When I do this, I get a value of 0.57. Lo and behold, it has roughly the same relationship to the solid red Gaussian in that plot as the green arrow to the 0+ Gaussian in the second.

Going by the assumption that my naive shortcut actually works reasonably well, I can take the next step. I can calculate a \(1\sigma\) error on the weighted average, which yields \(0.57^{+0.24}_{-0.23}\). When I (admittedly very crudely) try the transcribe this uncertainty to the second plot, I get something like this:

Yes, the error is this significant. So while the position of the green arrow is in tantalizing agreement with what one would expect from a Higgs particle, the error bar says that we cannot draw any definitive conclusions just yet.

But wait, it gets even weirder. Going back to the first plot, notice the two data points on the right. What if these are outliers? If I remove them from the analysis, I get something completely different: namely, the value of \(0.43^{+0.26}_{-0.21}\). Which is this:

So without the outliers, the data actually favor the pseudoscalar scenario!

I have to emphasize: what I did here is rather naive. The weighted average may not accurately represent the position of the green arrow at all. The coincidence in position could be a complete accident. In which case the horizontal error bar yielded by my analysis is completely bogus as well.

I also attempted to check how much more data would be needed to reduce the size of these error bars sufficiently for a true \(1\sigma\) result: about 2-4 times the number of events collected to date. So perhaps what I did is not complete nonsense after all, because this is what knowledgeable people are saying: when the LHC collected at least twice the amount of data it already has, we may know with reasonable certainty if the observed particle is a scalar or a pseudoscalar.

Until then, I hope I did not make a complete fool of myself with this naive analysis. Still, this is what blogs are for; I am allowed to say foolish things here.

If you were reading newspapers, science blogs, or even some articles written by prominent scientists or announcements by prominent institutions (such as Canada’s Perimeter Institute), you might be under the impression that the Higgs boson is a done deal: it has been discovered. (Indeed, Perimeter’s Web site announces on its home page that “[the] Higgs boson has been found”.

Sounds great but it is not true. Let me quote from a recent New Scientist online article: “Although spotted at last, many properties of the new particle – thought to be the Higgs boson, or at least something similar – have yet to be tested. What’s more, the telltale signature it left in the detectors at the Large Hadron Collider (LHC) does not exactly match what is predicted”.

There, this says it all. We are almost certain that something has been discovered. (This is the 4.9-sigma result). We are not at all certain that it’s the Higgs. It probably is, but there is a significant likelihood that it isn’t, and we will only know for sure one way or another after several more years’ worth of data are collected. At least this is what the experimenters say. And why should you listen to anyone other than the experimenters?

I have been thinking about neutrinos today. No, not about faster-than-light neutrinos. I was skeptical about the sensational claim from the OPERA experiment last year, and my skepticism was well justified.

They may not be faster than light, but neutrinos are still very weird. Neutrinos of one flavor turn into another, a discovery that, to many a particle physicist, had to be almost as surprising as the possibility that neutrinos are superluminal.

The most straightfoward explanation for these neutrino oscillations is that neutrinos have mass. But herein lies a problem. We only ever observed left-handed neutrinos. This makes sense if neutrinos are massless particles that travel at the speed of light, since all observers agree on what left-handed means: the spin of the neutrino, projected along the direction of its motion, is always −1/2.

But now imagine neutrinos that are massive and travel slower than the speed of light. As a matter of fact, imagine a bunch of neutrinos fired by CERN in Geneva in the direction of Gran Sasso, Italy. It takes roughly 2 ms for them to arrive. Now if you can run very, very, very fast (say, you’re the Flash, the comic book superhero) you may be able to outrun the bunch. Looking back, you will see… a bunch of neutrinos with a velocity vector pointing backwards (they’re slower than you, which means they’ll appear to be moving backwards from your perspective) so projecting their spin along the direction of motion, you get +1/2. In other words, you’re observing right-handed neutrinos.

This is just weird. On the surface of it, it means that our fast-running Flash sees the laws of physics change! This is in deep contradiction with the laws of special relativity, Lorentz invariance and all that.

How we can interpret this situation depends on whether we believe that neutrinos are “Dirac” or “Majorana”. Neutrinos are fermions, and fermions are represented by spinor fields. A spinor field has four components: these correspond, in a sense, to a left-handed and a right-handed particle and their respective antiparticles. So if a particle only exists as a left-handed particle, only two of the four components remain; the other two (at least in the so-called Weyl representation) disappear, are “projected out”, to use a nasty colloquialism.

But we just said that if neutrinos are massive, it no longer makes sense of talking about strictly left-handed neutrinos; to the Flash, those neutrinos may appear right-handed. So both left- and right-handed neutrino states exist. Are they mathematically independent? Because if they are, neutrinos are represented by a full 4-component “Dirac” spinor. But there is a possibility that the components are not independent: in effect, this means that the neutrino is its own antiparticle. Such states can be represented by a two-component “Majorana” spinor.

The difference between these two types of neutrinos is not just theoretical. The neutrino carries something very real: the lepton number, in essence the “electronness” (without the electric charge) of an electron. If a neutrino is its own antiparticle, the two can annihilate one another, and two units of “electronness” vanish. Lepton number is not conserved.

If this is indeed the case, it can be observed. The so-called neutrinoless double beta decay is a hypothetical form of radioactive decay in which an isotope that is known to decay by emitting two electrons simultaneously (e.g., potassium-48 or uranium-238) does so without emitting the corresponding neutrinos (because these annihilate each other without going anywhere). Unfortunately, given that neutrinos don’t like to do much interacting to begin with, the probability of a neutrinoless decay occurring at any given time is very small. Still, it is observable in principle, and if observed, it would indicate unambiguously that neutrinos are Majorana spinors. (A prospect that may be appealing insofar as neutrinos are concerned, but I find it nonetheless deeply disturbing that such a fundamental property of a basic building block of matter may turn out to be ephemeral.)

Either way, I remain at a loss when I think about the handedness of neutrinos. If neutrinos are Dirac neutrinos, one may postulate right-handed neutrinos that do not interact the way left-handed neutrinos do (i.e., do not participate in the weak interaction, being so-called sterile neutrinos instead). Cool, but what about our friend, the Flash? Suppose he is observing the same thing we’re observing, a neutrino in the OPERA bunch interacting with something. But from his perspective, that neutrino is a right-handed neutrino that is not allowed to participate in such an interaction!

Or suppose that neutrinos are Majorana spinors, and right-handed neutrinos are simply much (VERY much) heavier, which is why they have not been observed yet (this is the so-called seesaw mechanism). The theory allows us to construct such as mass matrix, but once again having the Flash around leads to trouble: he will observe ordinary “light” neutrinos as right-handed ones!

Perhaps these are just apparent contradictions. In fact, I am pretty sure that that’s what they are, since all this follows from writing down a theory in the form of a Lagrangian density that is manifestly Lorentz (and Poincaré) invariant, hence the physics does not become broken for the Flash. It will just turn weird. But how weird is too weird?

It appears that CERN goofed and as a result, the video of the announcement planned for tomorrow has been leaked. (That is, unless you choose to believe their cockamamie story about multiple versions of the video having been produced.)

The bottom line: there is definitely a particle there with integer spin. Its mass is about 125 GeV. We know it decays into two photons and two Z-bosons. That’s about all we know.

The assessment is that it is either the Higgs or something altogether new.

My blog is supposed to be (mostly) about physics. So let me write something about physics for a change.

John Moffat, with whom I have been collaborating (mostly on his modified gravity theory, MOG) for the past six years or so, has many ideas. Recently, he was wondering: could the celebrated 125 GeV (125 gigaelectronvolts divided the speed of light squared, to be precise, which is about about 134 times the mass of a hydrogen atom) peak observed last year at the LHC (and if rumors are to be believed, perhaps to be confirmed next week) be a sign of something other than the Higgs particle?

All popular accounts emphasize the role of the Higgs particle in making particles massive. This is a bit misleading. For one thing, the Higgs mechanism is directly responsible for the masses of only some particles (the vector bosons); for another, even this part of the mechanism requires that, in addition to the Higgs particle, we also presume the existence of a potential field (the famous “Mexican hat” potential) that is responsible for spontaneous symmetry breaking.

Higgs mechanism aside though, the Standard Model of particle physics needs the Higgs particle. Without the Higgs, the Standard Model is not renormalizable; its predictions diverge into meaningless infinities.

The Higgs particle solves this problem by “eating up” the last misbehaving bits of the Standard Model that cannot be eliminated by other means. The theory is then complete: although it remains unreconciled with gravity, it successfully unites the other three forces and all known particles into a unified (albeit somewhat messy) whole. The theory’s predictions are fully in accordance with data that include laboratory experiments as well as astrophysical observations.

Well, almost. There is still this pesky business with neutrinos. Neutrinos in the Standard Model are massless. Since the 1980s, however, we had strong reasons to suspect that neutrinos have mass. The reason is the “solar neutrino problem”, a discrepancy between the predicted and observed number of neutrinos originating from the inner core of the Sun. This problem is resolved if different types of neutrinos can turn into one another, since the detectors in question could only “see” electron neutrinos. This “neutrino flavor mixing” or “neutrino oscillation” can occur if neutrinos have mass, represented by a mass matrix that is not completely diagonal.

What’s wrong with introducing such a matrix, one might ask? Two things. First, this matrix necessarily contains dimensionless quantities that are very small. While there is no a priori reason to reject them, dimensionless numbers in a theory that are orders of magnitude bigger or smaller than 1 are always suspect. But the second problem is perhaps the bigger one: massive neutrinos make the Standard Model non-renormalizable again. This can only be resolved by either exotic mechanisms or the introduction of new elementary particles.

This challenge to the Standard Model perhaps makes the finding of the Higgs particle less imperative. Far from turning a nearly flawless theory into a perfect one, it only addresses some problems in an otherwise still flawed, incomplete theory. Conversely, not finding the Higgs particle is less devastating: it does invalidate a theory that would have been perfect otherwise, it simply prompts us to look for solutions elsewhere.

In light of that, one may wish to take a second look at the observations reported at the LHC last fall. The Higgs particle, if it exists, can decay in several ways. We already know that the Higgs particle cannot be heavier than 130 GeV, and this excludes certain forms of decay. One of the decays that remains is the decay into a quark and its antiparticle that, in turn, decay into two photons. Photons are easy to observe, but there is a catch: when the LHC collides large numbers of protons with one another at high energies, a huge number of photons are created as a background. It is against this background that two-photon events with a signature specific to the Higgs particle must be observed.

Diphoton results from the Atlas detector at the LHC, late 2011.

And observed they have been, albeit not with a resounding statistical significance. There is a small excess of such two-photon events indicating a possible Higgs mass of 125 GeV. Many believe that this is because there is indeed a Higgs particle with this mass, and its discovery will be confirmed with the necessary statistical certainty once more data are collected.

Others remain skeptical. For one thing, that 125 GeV peak is not the only peak in the data. For another, it is a peak that is a tad more pronounced than what the Higgs particle would produce. Furthermore, there is no corresponding peak in other “channels” that would correspond to other forms of decay of the Higgs particle.

This is when Moffat’s idea comes in. John had in mind the many “hadronic resonances”, all sorts of combinations of quarks that appear at lower energies, some of which still befuddle particle physicists. What if, he asks, this 125 GeV peak is due to just another such resonance?

Easier said than done. At low energies, there are plenty of quarks to choose from and combine. But 125 GeV is not a very convenient energy from this perspective. The heaviest quark, the top quark has a mass of 173 GeV or so; far too heavy for this purpose. The next quark in terms of mass, the bottom quark, is much too light at around 4.5 Gev. There is no obvious way to combine a reasonably small number of quarks into a 125 GeV composite particle that sticks around long enough for it to be detected. Indeed, the top quark is so heavy that “toponium”, a hypothetical combination of a top quark and its antiparticle, is believed to be undetectable; it decays so rapidly, it really never has time to form in the first place.

But then, there is another possibility. Remember how neutrinos oscillate between different states? Well, composite particles can do that, too. And as to what these “eigenstates” are, that really depends on the measurement. One notorious example is the neutral kaon (also known as the neutral K meson). It has one eigenstate with respect to the strong interaction, but two quite different eigenstates with respect to the weak interaction.

So here is John’s perhaps not so outlandish proposal: what if there is a form of quarkonium whose eigenstates are toponium (not observed) and bottomonium with respect to some interactions, but two different mixed states with respect to whatever interaction is responsible for the 125 GeV resonance observed by the LHC?

Such an eigenstate requires a mixing angle, easily calculated as 20 degrees. This mixing also results in another eigenstate, at 330 GeV, which is likely so heavy that it is not stable enough to be observed. This proposal, if valid, would explain why the LHC sees a resonance at 125 GeV without a Higgs particle.

Indeed, this proposal can also explain a) why the peak is stronger than what one would predict for the Higgs particle, b) why no other Higgs-specific decay modes were observed, and perhaps most intriguingly, c) why there are additional peaks in the data!

That is because if there is a “ground state”, there are also excited states, the same way a hydrogen atom (to use a more commonplace example) has a ground state and excited states with its lone electron in higher energy orbits. These excited states would show up in the plots as additional resonances, usually closely bunched together, with decreasing magnitude.

Could John be right? I certainly like his proposal, though I am not exactly the unbiased observer, since I did contribute a little to its development through numerous discussions. In any case, we will know a little more next week. An announcement from the LHC is expected on July 4. It is going to be interesting.