Tag Archives: quantum field theory

Nonperturbative Methods for Conformal Theories in Natal

I’m at a conference this week, on Nonperturbative Methods for Conformal Theories, in Natal on the northern coast of Brazil.

Where even the physics institutes have their own little rainforests.

“Nonperturbative” means that most of the people at this conference don’t use the loop-by-loop approximation of Feynman diagrams. Instead, they try to calculate things that don’t require approximations, finding formulas that work even for theories where the forces involved are very strong. In practice this works best in what are called “conformal” theories, roughly speaking these are theories that look the same whichever “scale” you use. Sometimes these theories are “integrable”, theories that can be “solved” exactly with no approximation. Sometimes these theories can be “bootstrapped”, starting with a guess and seeing how various principles of physics constrain it, mapping out a kind of “space of allowed theories”. Both approaches, integrability and bootstrap, are present at this conference.

This isn’t quite my community, but there’s a fair bit of overlap. We care about many of the same theories, like N=4 super Yang-Mills. We care about tricks to do integrals better, or to constrain mathematical guesses better, and we can trade these kinds of tricks and give each other advice. And while my work is typically “perturbative”, I did have one nonperturbative result to talk about, one which turns out to be more closely related to the methods these folks use than I had appreciated.

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Hexagon Functions V: Seventh Heaven

I’ve got a new paper out this week, a continuation of a story that has threaded through my career since grad school. With a growing collaboration (now Simon Caron-Huot, Lance Dixon, Falko Dulat, Andrew McLeod, and Georgios Papathanasiou) I’ve been calculating six-particle scattering amplitudes in my favorite theory-that-does-not-describe-the-real-world, N=4 super Yang-Mills. We’ve been pushing to more and more “loops”: tougher and tougher calculations that approximate the full answer better and better, using the “word jumble” trick I talked about in Scientific American. And each time, we learn something new.

Now we’re up to seven loops for some types of particles, and six loops for the rest. In older blog posts I talked in megabytes: half a megabyte for three loops, 15 MB for four loops, 300 MB for five loops. I don’t have a number like that for six and seven loops: we don’t store the result in that way anymore, it just got too cumbersome. We have to store it in a simplified form, and even that takes 80 MB.

Some of what we learned has to do with the types of mathematical functions that we need: our “guess” for the result at each loop. We’ve honed that guess down a lot, and discovered some new simplifications along the way. I won’t tell that story here (except to hint that it has to do with “cosmic Galois theory”) because we haven’t published it yet. It will be out in a companion paper soon.

This paper focused on the next step, going from our guess to the correct six- and seven-loop answers. Here too there were surprises. For the last several loops, we’d observed a surprisingly nice pattern: different configurations of particles with different numbers of loops were related, in a way we didn’t know how to explain. The pattern stuck around at five loops, so we assumed it was the real deal, and guessed the new answer would obey it too.

Yes, in our field this counts as surprisingly nice

Usually when scientists tell this kind of story, the pattern works, it’s a breakthrough, everyone gets a Nobel prize, etc. This time? Nope!

The pattern failed. And it failed in a way that was surprisingly difficult to detect.

The way we calculate these things, we start with a guess and then add what we know. If we know something about how the particles behave at high energies, or when they get close together, we use that to pare down our guess, getting rid of pieces that don’t fit. We kept adding these pieces of information, and each time the pattern seemed ok. It was only when we got far enough into one of these approximations that we noticed a piece that didn’t fit.

That piece was a surprisingly stealthy mathematical function, one that hid from almost every test we could perform. There aren’t any functions like that at lower loops, so we never had to worry about this before. But now, in the rarefied land of six-loop calculations, they finally start to show up.

We have another pattern, like the old one but that isn’t broken yet. But at this point we’re cautious: things get strange as calculations get more complicated, and sometimes the nice simplifications we notice are just accidents. It’s always important to check.

Deep physics or six-loop accident? You decide!

This result was a long time coming. Coordinating a large project with such a widely spread collaboration is difficult, and sometimes frustrating. People get distracted by other projects, they have disagreements about what the paper should say, even scheduling Skype around everyone’s time zones is a challenge. I’m more than a little exhausted, but happy that the paper is out, and that we’re close to finishing the companion paper as well. It’s good to have results that we’ve been hinting at in talks finally out where the community can see them. Maybe they’ll notice something new!


Amplitudes in String and Field Theory at NBI

There’s a conference at the Niels Bohr Institute this week, on Amplitudes in String and Field Theory. Like the conference a few weeks back, this one was funded by the Simons Foundation, as part of Michael Green’s visit here.

The first day featured a two-part talk by Michael Green and Congkao Wen. They are looking at the corrections that string theory adds on top of theories of supergravity. These corrections are difficult to calculate directly from string theory, but one can figure out a lot about them from the kinds of symmetry and duality properties they need to have, using the mathematics of modular forms. While Michael’s talk introduced the topic with a discussion of older work, Congkao talked about their recent progress looking at this from an amplitudes perspective.

Francesca Ferrari’s talk on Tuesday also related to modular forms, while Oliver Schlotterer and Pierre Vanhove talked about a different corner of mathematics, single-valued polylogarithms. These single-valued polylogarithms are of interest to string theorists because they seem to connect two parts of string theory: the open strings that describe Yang-Mills forces and the closed strings that describe gravity. In particular, it looks like you can take a calculation in open string theory and just replace numbers and polylogarithms with their “single-valued counterparts” to get the same calculation in closed string theory. Interestingly, there is more than one way that mathematicians can define “single-valued counterparts”, but only one such definition, the one due to Francis Brown, seems to make this trick work. When I asked Pierre about this he quipped it was because “Francis Brown has good taste…either that, or String Theory has good taste.”

Wednesday saw several talks exploring interesting features of string theory. Nathan Berkovitz discussed his new paper, which makes a certain context of AdS/CFT (a duality between string theory in certain curved spaces and field theory on the boundary of those spaces) manifest particularly nicely. By writing string theory in five-dimensional AdS space in the right way, he can show that if the AdS space is small it will generate the same Feynman diagrams that one would use to do calculations in N=4 super Yang-Mills. In the afternoon, Sameer Murthy showed how localization techniques can be used in gravity theories, including to calculate the entropy of black holes in string theory, while Yvonne Geyer talked about how to combine the string theory-like CHY method for calculating amplitudes with supersymmetry, especially in higher dimensions where the relevant mathematics gets tricky.

Thursday ended up focused on field theory. Carlos Mafra was originally going to speak but he wasn’t feeling well, so instead I gave a talk about the “tardigrade” integrals I’ve been looking at. Zvi Bern talked about his work applying amplitudes techniques to make predictions for LIGO. This subject has advanced a lot in the last few years, and now Zvi and collaborators have finally done a calculation beyond what others had been able to do with older methods. They still have a way to go before they beat the traditional methods overall, but they’re off to a great start. Lance Dixon talked about two-loop five-particle non-planar amplitudes in N=4 super Yang-Mills and N=8 supergravity. These are quite a bit trickier than the planar amplitudes I’ve worked on with him in the past, in particular it’s not yet possible to do this just by guessing the answer without considering Feynman diagrams.

Today was the last day of the conference, and the emphasis was on number theory. David Broadhurst described some interesting contributions from physics to mathematics, in particular emphasizing information that the Weierstrass formulation of elliptic curves omits. Eric D’Hoker discussed how the concept of transcendentality, previously used in field theory, could be applied to string theory. A few of his speculations seemed a bit farfetched (in particular, his setup needs to treat certain rational numbers as if they were transcendental), but after his talk I’m a bit more optimistic that there could be something useful there.

Hadronic Strings and Large-N Field Theory at NBI

One of string theory’s early pioneers, Michael Green, is currently visiting the Niels Bohr Institute as part of a program by the Simons Foundation. The program includes a series of conferences. This week we are having the first such conference, on Hadronic Strings and Large-N Field Theory.

The bulk of the conference focused on new progress on an old subject, using string theory to model the behavior of quarks and gluons. There were a variety of approaches on offer, some focused on particular approximations and others attempting to construct broader, “phenomenological” models.

The other talks came from a variety of subjects, loosely tied together by the topic of “large N field theories”. “N” here is the number of colors: while the real world has three “colors” of quarks, you can imagine a world with more. This leads to simpler calculations, and often to connections with string theory. Some talks deal with attempts to “solve” certain large-N theories exactly. Others ranged farther afield, even to discussions of colliding black holes.

Changing the Question

I’ve recently been reading Why Does the World Exist?, a book by the journalist Jim Holt. In it he interviews a range of physicists and philosophers, asking each the question in the title. As the book goes on, he concludes that physicists can’t possibly give him the answer he’s looking for: even if physicists explain the entire universe from simple physical laws, they still would need to explain why those laws exist. A bit disappointed, he turns back to the philosophers.

Something about Holt’s account rubs me the wrong way. Yes, it’s true that physics can’t answer this kind of philosophical problem, at least not in a logically rigorous way. But I think we do have a chance of answering the question nonetheless…by eclipsing it with a better question.

How would that work? Let’s consider a historical example.

Does the Earth go around the Sun, or does the Sun go around the Earth? We learn in school that this is a solved question: Copernicus was right, the Earth goes around the Sun.

The details are a bit more subtle, though. The Sun and the Earth both attract each other: while it is a good approximation to treat the Sun as fixed, in reality it and the Earth both move in elliptical orbits around the same focus (which is close to, but not exactly, the center of the Sun). Furthermore, this is all dependent on your choice of reference frame: if you wish you can choose coordinates in which the Earth stays still while the Sun moves.

So what stops a modern-day Tycho Brahe from arguing that the Sun and the stars and everything else orbit around the Earth?

The reason we aren’t still debating the Copernican versus the Tychonic system isn’t that we proved Copernicus right. Instead, we replaced the old question with a better one. We don’t actually care which object is the center of the universe. What we care about is whether we can make predictions, and what mathematical laws we need to do so. Newton’s law of universal gravitation lets us calculate the motion of the solar system. It’s easier to teach it by talking about the Earth going around the Sun, so we talk about it that way. The “philosophical” question, about the “center of the universe”, has been explained away by the more interesting practical question.

My suspicion is that other philosophical questions will be solved in this way. Maybe physicists can’t solve the ultimate philosophical question, of why the laws of physics are one way and not another. But if we can predict unexpected laws and match observations of the early universe, then we’re most of the way to making the question irrelevant. Similarly, perhaps neuroscientists will never truly solve the mystery of consciousness, at least the way philosophers frame it today. Nevertheless, if they can describe brains well enough to understand why we act like we’re conscious, if they have something in their explanation that looks sufficiently “consciousness-like”, then it won’t matter if they meet the philosophical requirements, people simply won’t care. The question will have been eaten by a more interesting question.

This can happen in physics by itself, without reference to philosophy. Indeed, it may happen again soon. In the New Yorker this week, Natalie Wolchover has an article in which she talks to Nima Arkani-Hamed about the search for better principles to describe the universe. In it, Nima talks about looking for a deep mathematical question that the laws of physics answer. Peter Woit has expressed confusion that Nima can both believe this and pursue various complicated, far-fetched, and at times frankly ugly ideas for new physics.

I think the way to reconcile these two perspectives is to know that Nima takes naturalness seriously. The naturalness argument in physics states that physics as we currently see it is “unnatural”, in particular, that we can’t get it cleanly from the kinds of physical theories we understand. If you accept the argument as stated, then you get driven down a rabbit hole of increasingly strange solutions: versions of supersymmetry that cleverly hide from all experiments, hundreds of copies of the Standard Model, or even a multiverse.

Taking naturalness seriously doesn’t just mean accepting the argument as stated though. It can also mean believing the argument is wrong, but wrong in an interesting way.

One interesting way naturalness could be wrong would be if our reductionist picture of the world, where the ultimate laws live on the smallest scales, breaks down. I’ve heard vague hints from physicists over the years that this might be the case, usually based on the way that gravity seems to mix small and large scales. (Wolchover’s article also hints at this.) In that case, you’d want to find not just a new physical theory, but a new question to ask, something that could eclipse the old question with something more interesting and powerful.

Nima’s search for better questions seems to drive most of his research now. But I don’t think he’s 100% certain that the old questions are wrong, so you can still occasionally see him talking about multiverses and the like.

Ultimately, we can’t predict when a new question will take over. It’s a mix of the social and the empirical, of new predictions and observations but also of which ideas are compelling and beautiful enough to get people to dismiss the old question as irrelevant. It feels like we’re due for another change…but we might not be, and even if we are it might be a long time coming.

Made of Quarks Versus Made of Strings

When you learn physics in school, you learn it in terms of building blocks.

First, you learn about atoms. Indivisible elements, as the Greeks foretold…until you learn that they aren’t indivisible. You learn that atoms are made of electrons, protons, and neutrons. Then you learn that protons and neutrons aren’t indivisible either, they’re made of quarks. They’re what physicists call composite particles, particles made of other particles stuck together.

Hearing this story, you notice a pattern. Each time physicists find a more fundamental theory, they find that what they thought were indivisible particles are actually composite. So when you hear physicists talking about the next, more fundamental theory, you might guess it has to work the same way. If quarks are made of, for example, strings, then each quark is made of many strings, right?

Nope! As it turns out, there are two different things physicists can mean when they say a particle is “made of” a more fundamental particle. Sometimes they mean the particle is composite, like the proton is made of quarks. But sometimes, like when they say particles are “made of strings”, they mean something different.

To understand what this “something different” is, let’s go back to quarks for a moment. You might have heard there are six types, or flavors, of quarks: up and down, strange and charm, top and bottom. The different types have different mass and electric charge. You might have also heard that quarks come in different colors, red green and blue. You might wonder then, aren’t there really eighteen types of quark? Red up quarks, green top quarks, and so forth?

Physicists don’t think about it that way. Unlike the different flavors, the different colors of quark have a more unified mathematical description. Changing the color of a quark doesn’t change its mass or electric charge. All it changes is how the quark interacts with other particles via the strong nuclear force. Know how one color works, and you know how the other colors work. Different colors can also “mix” together, similarly to how different situations can mix together in quantum mechanics: just as Schrodinger’s cat can be both alive and dead, a quark can be both red and green.

This same kind of thing is involved in another example, electroweak unification. You might have heard that electromagnetism and the weak nuclear force are secretly the same thing. Each force has corresponding particles: the familiar photon for electromagnetism, and W and Z bosons for the weak nuclear force. Unlike the different colors of quarks, photons and W and Z bosons have different masses from each other. It turns out, though, that they still come from a unified mathematical description: they’re “mixtures” (in the same Schrodinger’s cat-esque sense) of the particles from two more fundamental forces (sometimes called “weak isospin” and “weak hypercharge”). The reason they have different masses isn’t their own fault, but the fault of the Higgs: the Higgs field we have in our universe interacts with different parts of this unified force differently, so the corresponding particles end up with different masses.

A physicist might say that electromagnetism and the weak force are “made of” weak isospin and weak hypercharge. And it’s that kind of thing that physicists mean when they say that quarks might be made of strings, or the like: not that quarks are composite, but that quarks and other particles might have a unified mathematical description, and look different only because they’re interacting differently with something else.

This isn’t to say that quarks and electrons can’t be composite as well. They might be, we don’t know for sure. If they are, the forces binding them together must be very strong, strong enough that our most powerful colliders can’t make them wiggle even a little out of shape. The tricky part is that composite particles get mass from the energy holding them together. A particle held together by very powerful forces would normally be very massive, if you want it to end up lighter you have to construct your theory carefully to do that. So while occasionally people will suggest theories where quarks or electrons are composite, these theories aren’t common. Most of the time, if a physicist says that quarks or electrons are “made of ” something else, they mean something more like “particles are made of strings” than like “protons are made of quarks”.

Assumptions for Naturalness

Why did physicists expect to see something new at the LHC, more than just the Higgs boson? Mostly, because of something called naturalness.

Naturalness, broadly speaking, is the idea that there shouldn’t be coincidences in physics. If two numbers that appear in your theory cancel out almost perfectly, there should be a reason that they cancel. Put another way, if your theory has a dimensionless constant in it, that constant should be close to one.

(To see why these two concepts are the same, think about a theory where two large numbers miraculously almost cancel, leaving just a small difference. Take the ratio of one of those large numbers to the difference, and you get a very large dimensionless number.)

You might have heard it said that the mass of the Higgs boson is “unnatural”. There are many different physical processes that affect what we measure as the mass of the Higgs. We don’t know exactly how big these effects are, but we do know that they grow with the scale of “new physics” (aka the mass of any new particles we might have discovered), and that they have to cancel to give the Higgs mass we observe. If we don’t see any new particles, the Higgs mass starts looking more and more unnatural, driving some physicists to the idea of a “multiverse”.

If you find parts of this argument hokey, you’re not alone. Critics of naturalness point out that we don’t really have a good reason to favor “numbers close to one”, nor do we have any way to quantify how “bad” a number far from one is (we don’t know the probability distribution, in other words). They critique theories that do preserve naturalness, like supersymmetry, for being increasingly complicated and unwieldy, violating Occam’s razor. And in some cases they act baffled by the assumption that there should be any “new physics” at all.

Some of these criticisms are reasonable, but some are distracting and off the mark. The problem is that the popular argument for naturalness leaves out some important assumptions. These assumptions are usually kept in mind by the people arguing for naturalness (at least the more careful people), but aren’t often made explicit. I’d like to state some of these assumptions. I’ll be framing the naturalness argument in a bit of an unusual (if not unprecedented) way. My goal is to show that some criticisms of naturalness don’t really work, while others still make sense.

I’d like to state the naturalness argument as follows:

  1. The universe should be ultimately described by a theory with no free dimensionless parameters at all. (For the experts: the theory should also be UV-finite.)
  2. We are reasonably familiar with theories of the sort described in 1., we know roughly what they can look like.
  3. If we look at such a theory at low energies, it will appear to have dimensionless parameters again, based on the energy where we “cut off” our description. We understand this process well enough to know what kinds of values these parameters can take, starting from 2.
  4. Point 3. can only be consistent with the observed mass of the Higgs if there is some “new physics” at around the scales the LHC can measure. That is, there is no known way to start with a theory like those of 2. and get the observed Higgs mass without new particles.

Point 1. is often not explicitly stated. It’s an assumption, one that sits in the back of a lot of physicists’ minds and guides their reasoning. I’m really not sure if I can fully justify it, it seems like it should be a consequence of what a final theory is.

(For the experts: you’re probably wondering why I’m insisting on a theory with no free parameters, when usually this argument just demands UV-finiteness. I demand this here because I think this is the core reason why we worry about coincidences: free parameters of any intermediate theory must eventually be explained in a theory where those parameters are fixed, and “unnatural” coincidences are those we don’t expect to be able to fix in this way.)

Point 2. may sound like a stretch, but it’s less of one than you might think. We do know of a number of theories that have few or no dimensionless parameters (and that are UV-finite), they just don’t describe the real world. Treating these theories as toy models, we can hopefully get some idea of how theories like this should look. We also have a candidate theory of this kind that could potentially describe the real world, M theory, but it’s not fleshed out enough to answer these kinds of questions definitively at this point. At best it’s another source of toy models.

Point 3. is where most of the technical arguments show up. If someone talking about naturalness starts talking about effective field theory and the renormalization group, they’re probably hashing out the details of point 3. Parts of this point are quite solid, but once again there are some assumptions that go into it, and I don’t think we can say that this point is entirely certain.

Once you’ve accepted the arguments behind points 1.-3., point 4. follows. The Higgs is unnatural, and you end up expecting new physics.

Framed in this way, arguments about the probability distribution of parameters are missing the point, as are arguments from Occam’s razor.

The point is not that the Standard Model has unlikely parameters, or that some in-between theory has unlikely parameters. The point is that there is no known way to start with the kind of theory that could be an ultimate description of the universe and end up with something like the observed Higgs and no detectable new physics. Such a theory isn’t merely unlikely, if you take this argument seriously it’s impossible. If your theory gets around this argument, it can be as cumbersome and Occam’s razor-violating as it wants, it’s still a better shot than no possible theory at all.

In general, the smarter critics of naturalness are aware of this kind of argument, and don’t just talk probabilities. Instead, they reject some combination of point 2. and point 3.

This is more reasonable, because point 2. and point 3. are, on some level, arguments from ignorance. We don’t know of a theory with no dimensionless parameters that can give something like the Higgs with no detectable new physics, but maybe we’re just not trying hard enough. Given how murky our understanding of M theory is, maybe we just don’t know enough to make this kind of argument yet, and the whole thing is premature. This is where probability can sneak back in, not as some sort of probability distribution on the parameters of physics but just as an estimate of our own ability to come up with new theories. We have to guess what kinds of theories can make sense, and we may well just not know enough to make that guess.

One thing I’d like to know is how many critics of naturalness reject point 1. Because point 1. isn’t usually stated explicitly, it isn’t often responded to explicitly either. The way some critics of naturalness talk makes me suspect that they reject point 1., that they honestly believe that the final theory might simply have some unexplained dimensionless numbers in it that we can only fix through measurement. I’m curious whether they actually think this, or whether I’m misreading them.

There’s a general point to be made here about framing. Suppose that tomorrow someone figures out a way to start with a theory with no dimensionless parameters and plausibly end up with a theory that describes our world, matching all existing experiments. (People have certainly been trying.) Does this mean naturalness was never a problem after all? Or does that mean that this person solved the naturalness problem?

Those sound like very different statements, but it should be clear at this point that they’re not. In principle, nothing distinguishes them. In practice, people will probably frame the result one way or another based on how interesting the solution is.

If it turns out we were missing something obvious, or if we were extremely premature in our argument, then in some sense naturalness was never a real problem. But if we were missing something subtle, something deep that teaches us something important about the world, then it should be fair to describe it as a real solution to a real problem, to cite “solving naturalness” as one of the advantages of the new theory.

If you ask for my opinion? You probably shouldn’t, I’m quite far from an expert in this corner of physics, not being a phenomenologist. But if you insist on asking anyway, I suspect there probably is something wrong with the naturalness argument. That said, I expect that whatever we’re missing, it will be something subtle and interesting, that naturalness is a real problem that needs to really be solved.