QCD Meets Gravity 2019

I’m at UCLA this week for QCD Meets Gravity, a conference about the surprising ways that gravity is “QCD squared”.

When I attended this conference two years ago, the community was branching out into a new direction: using tools from particle physics to understand the gravitational waves observed at LIGO.

At this year’s conference, gravitational waves have grown from a promising new direction to a large fraction of the talks. While there were still the usual talks about quantum field theory and string theory (everything from bootstrap methods to a surprising application of double field theory), gravitational waves have clearly become a major focus of this community.

This was highlighted before the first talk, when Zvi Bern brought up a recent paper by Thibault Damour. Bern and collaborators had recently used particle physics methods to push beyond the state of the art in gravitational wave calculations. Damour, an expert in the older methods, claims that Bern et al’s result is wrong, and in doing so also questions an earlier result by Amati, Ciafaloni, and Veneziano. More than that, Damour argued that the whole approach of using these kinds of particle physics tools for gravitational waves is misguided.

There was a lot of good-natured ribbing of Damour in the rest of the conference, as well as some serious attempts to confront his points. Damour’s argument so far is somewhat indirect, so there is hope that a more direct calculation (which Damour is currently pursuing) will resolve the matter. In the meantime, Julio Parra-Martinez described a reproduction of the older Amati/Ciafaloni/Veneziano result with more Damour-approved techniques, as well as additional indirect arguments that Bern et al got things right.

Before the QCD Meets Gravity community worked on gravitational waves, other groups had already built a strong track record in the area. One encouraging thing about this conference was how much the two communities are talking to each other. Several speakers came from the older community, and there were a lot of references in both groups’ talks to the other group’s work. This, more than even the content of the talks, felt like the strongest sign that something productive is happening here.

Many talks began by trying to motivate these gravitational calculations, usually to address the mysteries of astrophysics. Two talks were more direct, with Ramy Brustein and Pierre Vanhove speculating about new fundamental physics that could be uncovered by these calculations. I’m not the kind of physicist who does this kind of speculation, and I confess both talks struck me as rather strange. Vanhove in particular explicitly rejects the popular criterion of “naturalness”, making me wonder if his work is the kind of thing critics of naturalness have in mind.

Life Cycle of an Academic Scientist

So you want to do science for a living. Some scientists work for companies, developing new products. Some work for governments. But if you want to do “pure science”, science just to learn about the world, then you’ll likely work at a university, as part of what we call academia.

The first step towards academia is graduate school. In the US, this means getting a PhD.

(Master’s degrees, at least in the US, have a different purpose. Most are “terminal Master’s”, designed to be your last degree. With a terminal Master’s, you can be a technician in a lab, but you won’t get farther down this path. In the US you don’t need a Master’s before you apply for a PhD program, and having one is usually a waste of time: PhD programs will make you re-take most of the same classes.)

Once you have a PhD, it’s time to get a job! Often, your first job after graduate school is a postdoc. Postdocs are short-term jobs, usually one to three years long. Some people are lucky enough to go to the next stage quickly, others have more postdoc jobs first. These jobs will take you all over the world, everywhere people with your specialty work. Sometimes these jobs involve teaching, but more often you just do scientific research.

In the US system, If everything goes well, eventually you get a tenure-track job. These jobs involve both teaching and research. You get to train PhD students, hire postdocs, and in general start acting like a proper professor. This stage lasts around seven years, while the university evaluates you. If they decide you’re not worth it then typically you’ll have to leave to apply for another job in another university. If they like you though, you get tenure.

Tenure is the first time as an academic scientist that you aren’t on a short-term contract. Your job is more permanent than most, you have extra protection from being fired that most people don’t. While you can’t just let everything slide, you have freedom to make more of your own decisions.

A tenured job can last until retirement, when you become an emeritus professor. Emeritus professors are retired but still do some of the work they did as professors. They’re paid out of their pension instead of a university salary, but they still sometimes teach or do research, and they usually still have an office. The university can hire someone new, and the cycle continues.

This isn’t the only path scientists take. Some work in a national lab instead. These don’t usually involve teaching duties, and the path to a permanent job is a bit different. Some get teaching jobs instead of research professorships. These teaching jobs are usually not permanent, instead universities are hiring more and more adjunct faculty who have to string together temporary contracts to make a precarious living.

I’ve mostly focused on the US system here. Europe is a bit different: Master’s degrees are a real part of the system, tenure-track doesn’t really exist, and adjunct faculty don’t always either. Some particular countries, like Germany, have their own quite complicated systems, other countries fall in between.

The Teaching Heuristic for Non-Empirical Science

Science is by definition empirical. We discover how the world works not by sitting and thinking, but by going out and observing the world. But sometimes, all the observing we can do can’t possibly answer a question. In those situations, we might need “non-empirical science”.

The blog Slate Star Codex had a series of posts on this topic recently. He hangs out with a crowd that supports the many-worlds interpretation of quantum mechanics: the idea that quantum events are not truly random, but instead that all outcomes happen, the universe metaphorically splitting into different possible worlds. These metaphorical universes can’t be observed, so no empirical test can tell the difference between this and other interpretations of quantum mechanics: if we could ever know the difference, it would have to be for “non-empirical” reasons.

What reasons are those? Slate Star Codex teases out a few possible intuitions. He points out that we reject theories that have “unnecessary” ideas. He imagines a world where chemists believe that mixing an acid and a base also causes a distant star to go supernova, and a creationist world where paleontologists believe fossils are placed by the devil. In both cases, there might be no observable difference between their theories and ours, but because their theories have “extra pieces” (the distant star, the devil), we reject them for non-empirical reasons. Slate Star Codex asks if this supports many-worlds: without the extra assumption that quantum events randomly choose one outcome, isn’t quantum mechanics simpler?

I agree with some of this. Science really does use non-empirical reasoning. Without it, there’s no reason not to treat the world as a black box, a series of experiments with no mechanism behind it. But while we do reject theories with unnecessary ideas, that isn’t our only standard. We also need our theories to teach us about the world.

Ultimately, we trust science because it allows us to do things. If we understand the world, we can interact with it: we can build technology, design new experiments, and propose new theories. With this in mind, we can judge scientific theories by how well they help us do these things. A good scientific theory is one that gives us more power to interact with the world. It can do this by making correct predictions, but it can also do this by explaining things, making it easier for us to reason about them. Beyond empiricism, we can judge science by how well it teaches us.

This gives us an objection to the “supernova theory” of Slate Star Codex’s imagined chemists: it’s much more confusing to teach. To teach chemistry in that world you also have to teach the entire life cycle of stars, a subject that students won’t use in any other part of the course. The creationists’ “devil theory” of paleontology has the same problem: if their theory really makes the right predictions they’d have to teach students everything our paleontologists do: every era of geologic history, every theory of dinosaur evolution, plus an extra course in devil psychology. They end up with a mix that only makes it harder to understand the subject.

Many-worlds may seem simpler than other interpretations of quantum mechanics, but that doesn’t make it more useful, or easier to teach. You still need to teach students how to predict the results of experiments, and those results will still be random. If you teach them many-worlds, you need to add more discussion much earlier on, advanced topics like self-localizing uncertainty and decoherence. You need a quite extensive set of ideas, many of which won’t be used again, to justify rules another interpretation could have introduced much more simply. This would be fine if those ideas made additional predictions, but they don’t: like every interpretation of quantum mechanics, you end up doing the same experiments and building the same technology in the end.

I’m not saying I know many-worlds is false, or that I know another interpretation is true. All I’m saying is that, when physicists criticize many-worlds, they’re not just blindly insisting on empiricism. They’re rejecting many-worlds, in part, because all it does is make their work harder. And that, more than elegance or simplicity, is how we judge theories.

QCD and Reductionism: Stranger Than You’d Think

Earlier this year, I made a list of topics I wanted to understand. The most abstract and technical of them was something called “Wilsonian effective field theory”. I still don’t understand Wilsonian effective field theory. But while thinking about it, I noticed something that seemed weird. It’s something I think many physicists already understand, but that hasn’t really sunk in with the public yet.

There’s an old problem in particle physics, described in many different ways over the years. Take our theories and try to calculate some reasonable number (say, the angle an electron turns in a magnetic field), and instead of that reasonable number we get infinity. We fix this problem with a process called renormalization that hides that infinity away, changing the “normalization” of some constant like a mass or a charge. While renormalization first seemed like a shady trick, physicists eventually understood it better. First, we thought of it as a way to work around our ignorance, that the true final theory would have no infinities at all. Later, physicists instead thought about renormalization in terms of scaling.

Imagine looking at the world on a camera screen. You can zoom in, or zoom out. The further you zoom out, the more details you’ll miss: they’re just too small to be visible on your screen. You can guess what they might be, but your picture will be different depending on how you zoom.

In particle physics, many of our theories are like that camera. They come with a choice of “zoom setting”, a minimum scale where they still effectively tell the whole story. We call theories like these effective field theories. Some physicists argue that these are all we can ever have: since our experiments are never perfect, there will always be a scale so small we have no evidence about it.

In general, theories can be quite different at different scales. Some theories, though, are especially nice: they look almost the same as we zoom in to smaller scales. The only things that change are the mass of different particles, and their charges.


One theory like this is Quantum Chromodynamics (or QCD), the theory of quarks and gluons. Zoom in, and the theory looks pretty much the same, with one crucial change: the force between particles get weaker. There’s a number, called the “coupling constant“, that describes how strong a force is, think of it as sort of like an electric charge. As you zoom in to quarks and gluons, you find you can still describe them with QCD, just with a smaller coupling constant. If you could zoom “all the way in”, the constant (and thus the force between particles) would be zero.

This makes QCD a rare kind of theory: one that could be complete to any scale. No matter how far you zoom in, QCD still “makes sense”. It never gives contradictions or nonsense results. That doesn’t mean it’s actually true: it interacts with other forces, like gravity, that don’t have complete theories, so it probably isn’t complete either. But if we didn’t have gravity or electricity or magnetism, if all we had were quarks and gluons, then QCD could have been the final theory that described them.

And this starts feeling a little weird, when you think about reductionism.

Philosophers define reductionism in many different ways. I won’t be that sophisticated. Instead, I’ll suggest the following naive definition: Reductionism is the claim that theories on larger scales reduce to theories on smaller scales.

Here “reduce to” is intentionally a bit vague. It might mean “are caused by” or “can be derived from” or “are explained by”. I’m gesturing at the sort of thing people mean when they say that biology reduces to chemistry, or chemistry to physics.

What happens when we think about QCD, with this intuition?

QCD on larger scales does indeed reduce to QCD on smaller scales. If you want to ask why QCD on some scale has some coupling constant, you can explain it by looking at the (smaller) QCD coupling constant on a smaller scale. If you have equations for QCD on a smaller scale, you can derive the right equations for a larger scale. In some sense, everything you observe in your larger-scale theory of QCD is caused by what happens in your smaller-scale theory of QCD.

But this isn’t quite the reductionism you’re used to. When we say biology reduces to chemistry, or chemistry reduces to physics, we’re thinking of just a few layers: one specific theory reduces to another specific theory. Here, we have an infinite number of layers, every point on the scale from large to small, each one explained by the next.

Maybe you think you can get out of this, by saying that everything should reduce to the smallest scale. But remember, the smaller the scale the smaller our “coupling constant”, and the weaker the forces between particles. At “the smallest scale”, the coupling constant is zero, and there is no force. It’s only when you put your hand on the zoom nob and start turning that the force starts to exist.

It’s reductionism, perhaps, but not as we know it.

Now that I understand this a bit better, I get some of the objections to my post about naturalness a while back. I was being too naive about this kind of thing, as some of the commenters (particularly Jacques Distler) noted. I believe there’s a way to rephrase the argument so that it still works, but I’d have to think harder about how.

I also get why I was uneasy about Sabine Hossenfelder’s FQXi essay on reductionism. She considered a more complicated case, where the chain from large to small scale could be broken, a more elaborate variant of a problem in Quantum Electrodynamics. But if I’m right here, then it’s not clear that scaling in effective field theories is even the right way to think about this. When you have an infinite series of theories that reduce to other theories, you’re pretty far removed from what most people mean by reductionism.

Finally, this is the clearest reason I can find why you can’t do science without an observer. The “zoom” is just a choice we scientists make, an arbitrary scale describing our ignorance. But without it, there’s no way to describe QCD. The notion of scale is an inherent and inextricable part of the theory, and it doesn’t have to mean our theory is incomplete.

Experts, please chime in here if I’m wrong on the physics here. As I mentioned at the beginning, I still don’t think I understand Wilsonian effective field theory. If I’m right though, this seems genuinely weird, and something more of the public should appreciate.

Guest Post: On the Real Inhomogeneous Universe and the Weirdness of ‘Dark Energy’

A few weeks ago, I mentioned a paper by a colleague of mine, Mohamed Rameez, that generated some discussion. Since I wasn’t up for commenting on the paper’s scientific content, I thought it would be good to give Rameez a chance to explain it in his own words, in a guest post. Here’s what he has to say:

In an earlier post, 4gravitons had contemplated the question of ‘when to trust the contrarians’, in the context of our about-to-be-published paper in which we argue that accounting for the effects of the bulk flow in the local Universe, there is no evidence for any isotropic cosmic acceleration, which would be required to claim some sort of ‘dark energy’.

In the following I would like to emphasize that this is a reasonable view, and not a contrarian one. To do so I will examine the bulk flow of the local Universe and the historical evolution of what appears to be somewhat dodgy supernova data. I will present a trivial solution (from data) to the claimed ‘Hubble tension’.  I will then discuss inhomogeneous cosmology, and the 2011 Nobel prize in Physics. I will proceed to make predictions that can be falsified with future data. I will conclude with some questions that should be frequently asked.

Disclaimer: The views expressed here are not necessarily shared by my collaborators. 

The bulk flow of the local Universe:

The largest anisotropy in the Cosmic Microwave Background is the dipole, believed to be caused by our motion with respect to the ‘rest frame’ of the CMB with a velocity of ~369 km s^-1. Under this view, all matter in the local Universe appear to be flowing. At least out to ~300 Mpc, this flow continues to be directionally coherent, to within ~40 degrees of the CMB dipole, and the scale at which the average relative motion between matter and radiation converges to zero has so far not been found.

This is one of the most widely accepted results in modern cosmology, to the extent that SN1a data come pre ‘corrected’ for it.

Such a flow has covariant consequences under general relativity and this is what we set out to test.

Supernova data, directions in the sky and dodgyness:

Both Riess et al 1998 and Perlmutter et al 1999 used samples of supernovae down to redshifts of 0.01, in which almost all SNe at redshifts below 0.1 were in the direction of the flow.

Subsequently in Astier et al 2006, Kowalsky et al 2008, Amanullah et al 2010 and Suzuki et al 2011, it is reported that a process of outlier rejection was adopted in which data points >3\sigma from the Hubble diagram were discarded. This was done using a highly questionable statistical method that involves adjusting an intrinsic dispersion term \sigma_{\textrm{int}} by hand until a \chi^2/\textrm{ndof} of 1 is obtained to the assumed \LambdaCDM model. The number of outliers rejected is however far in excess of 0.3% – which is the 3\sigma expectation. As the sky coverage became less skewed, supernovae with redshift less than ~0.023 were excluded for being outside the Hubble flow. While the Hubble diagram so far had been inferred from heliocentric redshifts and magnitudes, with the introduction of SDSS supernovae that happened to be in the direction opposite to the flow, peculiar velocity ‘corrections’ were adopted in the JLA catalogue and supernovae down to extremely low redshifts were reintroduced. While the early claims of a cosmological constant were stated as ‘high redshift supernovae were found to be dimmer (15% in flux) than the low redshift supernovae (compared to what would be expected in a \Lambda=0 universe)’, it is worth noting that the peculiar velocity corrections change the redshifts and fluxes of low redshift supernovae by up to ~20 %.

When it was observed that even with this ‘corrected’ sample of 740 SNe, any evidence for isotropic acceleration using a principled Maximum Likelihood Estimator is less than 3\sigma , it was claimed that by adding 12 additional parameters (to the 10 parameter model) to allow for redshift and sample dependence of the light curve fitting parameters, the evidence was greater than 4\sigma .

As we discuss in Colin et al. 2019, these corrections also appear to be arbitrary, and betray an ignorance of the fundamentals of both basic statistical analysis and relativity. With the Pantheon compilation, heliocentric observables were no longer public and these peculiar velocity corrections initially extended far beyond the range of any known flow model of the Local Universe. When this bug was eventually fixed, both the heliocentric redshifts and magnitudes of the SDSS SNe that filled in the ‘redshift desert’ between low and high redshift SNe were found to be alarmingly discrepant. The authors have so far not offered any clarification of these discrepancies.

Thus it seems to me that the latest generation of ‘publicly available’ supernova data are not aiding either open science or progress in cosmology.

A trivial solution to the ‘Hubble tension’?

The apparent tension between the Hubble parameter as inferred from the Cosmic Microwave Background and low redshift tracers has been widely discussed, and recent studies suggest that redshift errors as low as 0.0001 can have a significant impact. Redshift discrepancies as big as 0.1 have been reported. The shifts reported between JLA and Pantheon appear to be sufficient to lower the Hubble parameter from ~73 km s^-1 Mpc^-1 to ~68 km s^-1 Mpc^-1.

On General Relativity, cosmology, metric expansion and inhomogeneities:

In the maximally symmetric Friedmann-Lemaitre-Robertson-Walker solution to general relativity, there is only one meaningful global notion of distance and it expands at the same rate everywhere. However, the late time Universe has structure on all scales, and one may only hope for statistical (not exact) homogeneity. The Universe is expected to be lumpy. A background FLRW metric is not expected to exist and quantities analogous to the Hubble and deceleration parameters will vary across the sky.  Peculiar velocities may be more precisely thought of as variations in the expansion rate of the Universe. At what rate does a real Universe with structure expand? The problems of defining a meaningful average notion of volume, its dynamical evolution, and connecting it to observations are all conceptually open.

On the 2011 Nobel Prize in Physics:

The Fitting Problem in cosmology was written in 1987. In the context of this work and the significant theoretical difficulties involved in inferring fundamental physics from the real Universe, any claims of having measured a cosmological constant from directionally skewed, sparse samples of intrinsically scattered observations should have been taken with a grain of salt.  By honouring this claim with a Nobel Prize, the Swedish Academy may have induced runaway prestige bias in favour of some of the least principled analyses in science, strengthening the confirmation bias that seems prevalent in cosmology.

This has resulted in the generation of a large body of misleading literature, while normalizing the practice of ‘massaging’ scientific data. In her recent video about gravitational waves, Sabine Hossenfelder says “We should not hand out Nobel Prizes if we don’t know how the predictions were fitted to the data”. What about when the data was fitted (in 1998-1999) using a method that has been discredited in 1989 to a toy model that has been cautioned against in 1987, leading to a ‘discovery’ of profound significance to fundamental physics?

A prediction with future cosmological data:

With the advent of high statistics cosmological data in the future, such as from the Large Synoptic Survey Telescope, I predict that the Hubble and deceleration parameters inferred from supernovae in hemispheres towards and away from the CMB dipole will be found to be different in a statistically significant (>5\sigma ) way. Depending upon the criterion for selection and blind analyses of data that can be agreed upon, I would be willing to bet a substantial amount of money on this prediction.

Concluding : on the amusing sociology of ‘Dark Energy’ and manufactured concordance:

Of the two authors of the well-known cosmology textbook ‘The Early Universe’, Edward Kolb writes these interesting papers questioning dark energy while Michael Turner is credited with coining the term ‘Dark Energy’.  Reasonable scientific perspectives have to be presented as ‘Dark Energy without dark energy’. Papers questioning the need to invoke such a mysterious content that makes up ‘68% of the Universe’ are quickly targeted by inane articles by non-experts or perhaps well-meant but still misleading YouTube videos. Much of this is nothing more than a spectacle.

In summary, while the theoretical debate about whether what has been observed as Dark Energy is the effect of inhomogeneities is ongoing, observers appear to have been actively using the most inhomogeneous feature of the local Universe through opaque corrections to data, to continue claiming that this ‘dark energy’ exists.

It is heartening to see that recent works lean toward a breaking of this manufactured concordance and speak of a crisis for cosmology.

Questions that should be frequently asked:

Q. Is there a Hubble frame in the late time Universe?

A. The Hubble frame is a property of the FLRW exact solution, and in the late time Universe in which galaxies and clusters have peculiar motions with respect to each other, an equivalent notion does not exist. While popular inference treats the frame in which the CMB dipole vanishes as the Hubble frame, the scale at which the bulk flow of the local Universe converges to that frame has never been found. We are tilted observers.

Q. I am about to perform blinded analyses on new cosmological data. Should I correct all my redshifts towards the CMB rest frame?

A. No. Correcting all your redshifts towards a frame that has never been found is a good way to end up with ‘dark energy’. It is worth noting that while the CMB dipole has been known since 1994, supernova data have been corrected towards the CMB rest frame only after 2010, for what appear to be independent reasons.

Q. Can I combine new data with existing Supernova data?

A. No. The current generation of publicly available supernova data suffer from the natural biases that are to be expected when data are compiled incrementally through a human mediated process. It would be better to start fresh with a new sample.

Q. Is ‘dark energy’ fundamental or new physics?

A. Given that general relativity is a 100+ year old theory and significant difficulties exist in describing the late time Universe with it, it is unnecessary to invoke new fundamental physics when confronting any apparent acceleration of the real Universe. All signs suggest that what has been ascribed to dark energy are the result of a community that is hell bent on repeating what Einstein supposedly called his greatest mistake.

Digging deeper:

The inquisitive reader may explore the resources on inhomogeneous cosmology, as well as the works of George Ellis, Thomas Buchert and David Wiltshire.

Academia Has Changed Less Than You’d Think

I recently read a biography of James Franck. Many of you won’t recognize the name, but physicists might remember the Franck-Hertz experiment from a lab class. Franck and Hertz performed a decisive test of Bohr’s model of the atom, ushering in the quantum age and receiving the 1925 Nobel Prize. After fleeing Germany when Hitler took power, Franck worked on the Manhattan project and co-authored the Franck Report urging the US not to use nuclear bombs on Japan. He settled at the University of Chicago, which named an institute after him.*

You can find all that on his Wikipedia page. The page also mentions his marriage later in life to Hertha Sponer. Her Wikipedia page talks about her work in spectroscopy, about how she was among the first women to receive a PhD in Germany and the first on the physics faculty at Duke University, and that she remained a professor there until 1966, when she was 70.

Neither Wikipedia page talks about two-body problems, or teaching loads, or pensions.

That’s why I was surprised when the biography covered Franck’s later life. Until Franck died, he and Sponer would travel back and forth, he visiting her at Duke and she visiting him in Chicago. According to the biography, this wasn’t exactly by choice: they both would have preferred to live together in the same city. Somehow though, despite his Nobel Prize and her scientific accomplishments, they never could. The biography talks about how the university kept her teaching class after class, so she struggled to find time for research. It talks about what happened as the couple got older, as their health made it harder and harder to travel back and forth, and they realized that without access to their German pensions they would not be able to support themselves in retirement. The biography gives the impression that Sponer taught till 70 not out of dedication but because she had no alternative.

When we think about the heroes of the past, we imagine them battling foes with historic weight: sexism, antisemitism, Nazi-ism. We don’t hear about their more everyday battles, with academic two-body problems and stingy universities. From this, we can get the impression that the dysfunctions of modern academia are new. But while the problems have grown, we aren’t the first academics with underpaid, overworked teaching faculty, nor the first to struggle to live where we want and love who we want. These are struggles academics have faced for a long, long time.

*Full disclosure: Franck was also my great-great-grandfather, hence I may find his story more interesting than most.

Rooting out the Answer

I have a new paper out today, with Jacob Bourjaily, Andrew McLeod, Matthias Wilhelm, Cristian Vergu and Matthias Volk.

There’s a story I’ve told before on this blog, about a kind of “alphabet” for particle physics predictions. When we try to make a prediction in particle physics, we need to do complicated integrals. Sometimes, these integrals simplify dramatically, in unexpected ways. It turns out we can understand these simplifications by writing the integrals in a sort of “alphabet”, breaking complicated mathematical “periods” into familiar logarithms. If we want to simplify an integral, we can use relations between logarithms like these:

\log(a b)=\log(a)+\log(b),\quad \log(a^n)=n\log(a)

to factor our “alphabet” into pieces as simple as possible.

The simpler the alphabet, the more progress you can make. And in the nice toy model theory we’re working with, the alphabets so far have been simple in one key way. Expressed in the right variables, they’re rational. For example, they contain no square roots.

Would that keep going? Would we keep finding rational alphabets? Or might the alphabets, instead, have square roots?

After some searching, we found a clean test case. There was a calculation we could do with just two Feynman diagrams. All we had to do was subtract one from the other. If they still had square roots in their alphabet, we’d have proven that the nice, rational alphabets eventually had to stop.


So we calculated these diagrams, doing the complicated integrals. And we found they did indeed have square roots in their alphabet, in fact many more than expected. They even had square roots of square roots!

You’d think that would be the end of the story. But square roots are trickier than you’d expect.

Remember that to simplify these integrals, we break them up into an alphabet, and factor the alphabet. What happens when we try to do that with an alphabet that has square roots?

Suppose we have letters in our alphabet with \sqrt{-5}. Suppose another letter is the number 9. You might want to factor it like this:

9=3\times 3

Simple, right? But what if instead you did this:

9=(2+ \sqrt{-5} )\times(2- \sqrt{-5} )

Once you allow \sqrt{-5} in the game, you can factor 9 in two different ways. The central assumption, that you can always just factor your alphabet, breaks down. In mathematical terms, you no longer have a unique factorization domain.

Instead, we had to get a lot more mathematically sophisticated, factoring into something called prime ideals. We got that working and started crunching through the square roots in our alphabet. Things simplified beautifully: we started with a result that was ten million terms long, and reduced it to just five thousand. And at the end of the day, after subtracting one integral from the other…

We found no square roots!

After all of our simplifications, all the letters we found were rational. Our nice test case turned out much, much simpler than we expected.

It’s been a long road on this calculation, with a lot of false starts. We were kind of hoping to be the first to find square root letters in these alphabets; instead it looks like another group will beat us to the punch. But we developed a lot of interesting tricks along the way, and we thought it would be good to publish our “null result”. As always in our field, sometimes surprising simplifications are just around the corner.