Spinor Info

I am reading a very interesting paper by Christian Beck, recently published in Physical Review Letters.

Beck revives the proposal that at least some of the as yet unobserved dark matter in the universe may be in the form of axions. But he goes further: he suggests that a decade-old experiment with superconducting Josephson-junctions that indicated the presence of a small, unexplained signal may in fact have been a de facto measurement of the axion background in our local galactic neighborhood.

If true, Beck’s suggestion has profound significance: not only would dark matter be observable, but it can be observed with ease, using a tabletop experiment!

What is an axion? The Standard Model of particle physics (for a very good comprehensive review, I recommend The Standard Model: A Primer by Cliff Burgess and Guy Moore, Cambridge University Press, 2007) can be thought of as the most general theory based on the observed particle content in the universe. By “most general”, I mean specifically that the Standard Model can be written in the form of a Lagrangian density, and all the terms that can be present do, in fact, correspond to physically observable phenomena.

All terms except one, that is. The term, which formally reads

\begin{align}{\cal L}_\Theta=\Theta_3\frac{g_3^2}{64\pi^2}\epsilon^{\mu\nu\lambda\beta}G^\alpha_{\mu\nu}G_{\alpha\lambda\beta},\end{align}

where \(G\) represents gluon fields and \(g_3\) is the strong coupling constant (\(\epsilon^{\mu\nu\lambda\beta}\) is the fully antisymmetric Levi-Civita pseudotensor), does not correspond to any known physical process. This term would be meaningless in classical physics, on account of the fact that the coupling constant \(\Theta_3\) multiplies a total derivative. In QCD, however, the term still has physical significance. Moreover, the term actually violates charge-parity (CP) symmetry.

The fact that no such effects are observed implies that \(\Theta_3\) is either 0 or at least, very small. Now why would \(\Theta_3\) be very small? There is no natural explanation.

However, one can consider introducing a new scalar field into the theory, with specific properties. In particular this scalar field, which is called the axion and usually denoted by \(a\), causes \(\Theta_3\) to be replaced with \(\Theta_{3,{\rm eff}}=\Theta_3 + \left<a\right>/f_a\), where \(f_a\) is some energy scale. If the scalar field were massless, the theory would demand \(\left<a\right>/f_a\) to be exactly \(-\Theta_3\). However, if the scalar field is massive, a small residual value for \(\Theta_{3,{\rm eff}}\) remains.

As for the Josephson-junction, it is a superconducting device in which two superconducting layers are separated by an isolation layer (which can be a normal conductor, a semiconductor, or even an insulator). As a voltage is introduced across a Josephson-junction, a current can be measured. The peculiar property of a Josephson-junction is the current does not vanish even as the voltage is reduced to zero:

(The horizontal axis is voltage, the vertical axis is the current. In a normal resistor, the current-voltage curve would be a straight line that goes through the origin.) This is the DC Josephson effect; a similar effect arises when an AC voltage is applied, but in that case, the curve is even more interesting, with a step function appearance.

The phase difference \(\delta\) between the superconductors a Josephson-junction is characterized by the equation

\begin{align}\ddot{\delta}+\frac{1}{RC}\dot{\delta}+\frac{2eI_c}{\hbar C}\sin\delta&=\frac{2e}{\hbar C}I,\end{align}

where \(R\) and \(C\) are the resistance and capacitance of the junction, \(I_c\) is the critical current that characterizes the junction, and \(I\) is the current. (Here, \(e\) is the electron’s charge and \(\hbar\) is the reduced Planck constant.)

Given an axion field, represented by \(\theta=a/f_a\), in the presence of strong electric (\({\bf E}\)) and magnetic (\({\bf B}\)) fields, the axion field satisfies the equation

\begin{align}\ddot{\theta}+\Gamma\dot{\theta}+\frac{m_a^2c^4}{\hbar^2}\sin\theta=-\frac{g_\lambda c^3e^2}{4\pi^2f_a^2}{\bf E}{\bf B},\end{align}

where \(\Gamma\) is a damping parameter and \(g_\lambda\) is a coupling constant, while \(m_a\) is the axion mass and of course \(c\) is the speed of light.

The formal similarity between these two equations is striking. Now Beck suggests that the similarity is more than formal: that in fact, under the right circumstances, the axion field and a Josephson-junction can form a coupled system, in which resonance effects might be observed. The reason Beck gives is that the axion field causes a small CP symmetry perturbation in the Josephson-junction, to which the junction reacts with a small response in \(\delta\).

Indeed, Beck claims that this effect was, in fact, observed already, in a 2004 experiment by Hoffman, et al., who attempted to measure the noise in a certain type of Josephson-junction. In their experiment, a small, persistent peak appeared at a voltage of approximately 0.055 mV:

If Beck is correct, this observation corresponds to an axion with a mass of 0.11 meV (that is to say, the electron is some five billion times heavier than this axion) and the local density of the axion field would be about one sixth the presumed dark matter density in this region of the Galaxy.

I don’t know if Beck is right or not, but unlike most other papers about purported dark matter discoveries, this one does not feel like clutching at straws. It passes the “smell test”. I’d be very disappointed if it proved to be true (I am not very much in favor of the dark matter proposal) but if it is true, I think it qualifies as a Nobel-worthy discovery. It is also eerily similar to the original discovery of the cosmic microwave background: it was first observed by physicists who were not at all interested in cosmology but instead, were just trying to build a low-noise microwave antenna.

Last December, I wrote a blog entry in which I criticized one aspect of the LHC’s analysis of the scalar particle discovered earlier this year, which is believed to be the long sought-after Higgs boson.

The Higgs boson is a scalar. It is conceivable that the particle observed at the LHC is not the Higgs particle but an “impostor”, some composite of known (and perhaps unknown) particles that behaves like a scalar. Or, I should say, almost like a scalar, as the ground state of such composites would likely behave like a pseudoscalar. The difference is that whereas a scalar-valued field remains unchanged under a reflection, a pseudoscalar field changes sign.

This has specific consequences when the particle decays, apparent in the angles of the decay products’ trajectories.

Several such angles are measured, but the analysis used at the ATLAS detector of the LHC employs a method borrowed from machine learning research, called a Boosted Decision Tree algorithm, that synthesizes a single parameter that has maximum sensitivity to the parity of the observed particle. (The CMS detector’s analysis uses a similar approach.)

The result can be plotted against scalar vs. pseudoscalar model predictions. This plot, shown below, does not appear very convincing. The data points (which represent binned numbers of events) are all over the place with large errors. Out of a total of only 43 events (give or take), more than 25% are the expected background, only 30+ events represent an actual signal. And the scalar vs. pseudoscalar predictions are very similar.

This is why, when I saw that the analysis concluded that the scalar hypothesis is supported with a probability of over 97%, I felt rather skeptical. And I thought I knew the reason: I thought that the experimental error, i.e., the error bars in the plot above, was not properly accounted for in the analysis.

Indeed, if I calculate the normalized chi-square per degree of freedom, I get \(\chi^2_{J^P=0^+} = 0.247\) and \(\chi^2_{J^P=0^-} = 0.426\), respectively, for the two hypotheses. The difference is not very big.

Alas, my skepticism was misplaced. The folks at the LHC didn’t bother with chi-squares, instead they performed a likelihood analysis. The question they were asking was this: given the set of observations available, what are the likelihoods of the scalar and the pseudoscalar scenarios?

At the LHC, they used likelihood functions and distributions derived from the actual theory. However, I can do a poor man’s version myself by simply using the Gaussian normal distribution (or a nonsymmetric version of the same). Given a data point \(D_i\), a model value \(M_I\), and a standard deviation (error) \(\sigma_i\), the probability that the data point is at least as far from \(M_i\) as \(D_i\) is given by

\begin{align}
{\cal P}_i=2\left[1-\Psi\left(\frac{|D_i-M_i|}{\sigma_i}\right)\right],
\end{align}

where \(\Psi(x)\) is the cumulative normal distribution.

Now \({\cal P}_i\) also happens to be the likelihood of the model value \(M_i\) given the data point \(D_i\) and standard distribution \(\sigma_i\). If we assume that the data points and their errors are statistically indepdendent, the likelihood that all the data points happen to fall where they fell is given by

\begin{align}
{\cal L}=\prod\limits_{i=1}^N{\cal P}_i.
\end{align}

Taking the data from the ATLAS figure above, the value \(q\) of the commonly used log-likelihood ratio is

\begin{align}
q=\ln\frac{{\cal L}(J^P=0^+)}{{\cal L}(J^P=0^-)}=2.89.
\end{align}

(The LHC folks calculated 2.2, which is “close enough” for me given that I am using a naive Gaussian distribution.)

Furthermore, if I choose to believe that the only two viable hypothesis for the spin-parity of the observed particle are the scalar and pseudoscalar scenarios (e.g., if other experiments already convinced me that alternatives, such as intepreting the result as a spin-2 particle, can be completely excluded) I can normalize these two likelihoods and interpret them as probabilities. The probability of the scalar scenario is then \(e^{2.89}\simeq 18\) times larger than the probability of the pseudoscalar scenario. So if these probabilities add up to 100%, that means that the scalar scenario is favored with a probability of nearly 95%. Not exactly “slam dunk” but pretty darn convincing.

As to the validity of the method, there is, in fact, a theorem called the Neyman-Pearson lemma that states that the likelihood-ratio test is the most powerful test for this type of comparison of hypotheses.

But what about my earlier objection that the observational error was not properly accounted for? Well… it appears that it was, after all. In my “poor man’s” version of the analysis, the observational error was used to select the appropriate form of the normal distribution, through \(\sigma_i\). In the LHC’s analysis, I believe, the observational error found its was into the Monte-Carlo simulation that was used to develop a physically more accurate probability distribution function that was used for the same purpose.

Even clever people make mistakes. Even large groups of very clever people sometimes succumb to groupthink. But when you bet against clever people, you are likely to lose. I thought I spotted an error in the analysis performed at the LHC, but all I really found were gaps in my own understanding. Oh well… live and learn.

Physics blog sites are abuzz about Eric Weinstein and his Amazing New Theory of Everything. For a moment, I actually confused him with Eric Weisstein, well known in physics and math circles as the founder of Mathworld, which, in the pre-Wikipedia days, was the Internet’s pre-eminent mathematics encyclopedia (only to be hijacked for a while by an unscrupulous CRC press). No, Weinstein is someone else: he is a mathematical physicist turned economist. In any case, he is no dummy, nor does he appear to be a crackpot. He is outside of academia, but, well, so am I, so who am I to complain?

So Weinstein gets invited to Oxford to give a public lecture, and he talks, for the first time, about ideas he has been working on for the past twenty years, about unifying physics.

This is greeted by a headline in The Guardian that reads, “Roll over Einstein: meet Weinstein“. Others follow suit, and soon physics news and blog sites far and wide discuss… what, exactly? Well, no-one really knows.

That is because Weinstein has not published anything yet. Not even a non-peer reviewed manuscript on arxiv.org. This is pointed out in one of the few sensibly skeptical blog posts, written by Jennifer Ouellette on Scientific American’s blog site. Ouellette actually quotes a tweet by Sean Carroll: “Pretty sure Einstein actually wrote research papers, not just gave interviews to newspapers.”

Ouellette goes on to quote Oxford cosmologist Andrew Pontzen, who observes that these “shenanigans” have “short-circuited science’s basic checks and balances”. I couldn’t agree more. This is true even if Weinstein turns out to be right in the end.

Which is conceivable, since Weinstein is no crackpot. But it is much more likely that his theory will join many others, including Garrett “surfer dude” Lisi’s aesthetically beautiful E8 theory, that just don’t have much to do with observable reality.

In a post a few days ago, I expressed my skeptical views concerning the interpretation of some of the recent Higgs results from CERN. I used a simple analogy, an example in which measuring the average height of the inhabitants in a set of buildings is used to determine which of them may house the Harlem Globetrotters.

However, I came to realize (thanks in part to some helpful criticism that my post received) that I left out one possibility. What if the buildings are too small? Or the ‘Trotters are just too, hmm, tired after a long party and end up in the wrong building? In that case, a measurement may look like this:

If we have an a priori reason to believe that, for whatever reason, the players are indeed spread out across several buildings, then we can indeed not expect to see a sharp peak at #4 (or whichever building is assigned to the Globetrotters); instead, we should see a broad excess, just what the CMS experiment is seeing when it measured the decay of the presumed Higgs boson into a τ⁺τ⁻ pair.

So is there an a priori reason for the data to be spread out like this? I believe there is. No instrument detects τ leptons directly, as their lifetime is too short. Instead, τ events are reconstructed from decay products, and all forms of τ decay involve at least one neutrino, which may carry away a significant portion of the lepton’s energy. So the final uncertainty in the total measured energy of the τ⁺τ⁻ pair can be quite significant.

In other words, many of the Globetrotters may indeed be sleeping in the wrong building.

Nonetheless, as my copy of the venerable 20-year old book, The Higgs Hunter’s Guide suggests, the decay into τ leptons can be a valuable means of confirmation. Which is perhaps why it is troubling that for now, the other major detector at the LHC, ATLAS, failed to see a similar broad excess of τ⁺τ⁻ events near the presumed Higgs mass.

A few weeks ago, I exchanged a number of e-mails with someone about the Lanczos tensor and the Weyl-Lanczos equation. One of the things I derived is worth recording here for posterity.

The Lanczos tensor is an interesting animal. It can be thought of as the source of the Weyl curvature tensor, the traceless part of the Riemann curvature tensor. The Weyl tensor, together with the Ricci tensor, fully determine the Riemann tensor, i.e., the intrinsic curvature of a spacetime. Crudely put, whereas the Ricci tensor tells you how the volume of, say, a cloud of dust changes in response to gravity, the Weyl tensor tells you how that cloud of dust is distorted in response to the same gravitational field. (For instance, consider a cloud of dust in empty space falling towards the Earth. In empty space, the Ricci tensor is zero, so the volume of the cloud does not change. But its shape becomes distorted and elongated in response to tidal forces. This is described by the Weyl tensor.

Because the Ricci tensor is absent, the Weyl tensor fully describes gravitational fields in empty space. In a sense, the Weyl tensor is analogous to the electromagnetic field tensor that fully describes electromagnetic fields in empty space. The electromagnetic field tensor is sourced by the four-dimensional electromagnetic vector potential (meaning that the electromagnetic field tensor can be expressed using partial derivatives of the electromagnetic vector potential.) The Weyl tensor has a source in exactly the same sense, in the form of the Lanczos tensor.

The electromagnetic field does not uniquely determine the electromagnetic vector potential. This is basically how integrals vs. derivatives work. For instance, the derivative of the function \(y=x^2\) is given by \(y’=2x\). But the inverse operation is not unambiguous: \(\int 2x~ dx=x^2+C\) where \(C\) is an arbitrary integration constant. This is a recognition of the fact that the derivative of any function in the form \(y=x^2+C\) is \(y’=2x\) regardless of the value of \(C\); so knowing only the derivative \(y’\) does not fully determine the original function \(y\).

In the case of electromagnetism, this freedom to choose the electromagnetic vector potential is referred to as the gauge freedom. The same gauge freedom exists for the Lanczos tensor.

Solutions for the Lanczos tensor for the simplest case of the Schwarzschild metric are provided in Wikipedia. A common characteristic of these solutions is that they yield a quantity that “blows up” at the event horizon. This runs contrary to accepted wisdom, namely that the event horizon is not in any way special; a freely falling space traveler would never know that he is crossing it.

But as it turns out, thanks to the gauge freedom of the Lanczos tensor, it is easy to construct a solution (an infinite family of solutions, as a matter of fact) that do not behave like this at the horizon.

Well, it was a fun thing to compute anyway.