Nifty papers I wrote that nobody knows about: (Part 2: Quark-Gluon Plasma)

This is Part 2 of the "Nifty papers" series, talking about papers of mine that I think are cool, but that have been essentially ignored by the community. Part 1 is here.

This is the story of my third paper, still as a graduate student (in my third year) at Stony Brook University, on Long Island.

Here's the title:

Physics Letters B 217 (1989) 5-8

First things first: What on earth is "Charmonium"? To answer this question, I'll give you in what follows a concise introduction to the theory of quarks and gluons, knows as "Quantum Chromodynamics".

Just kidding, of course. The Wiki article I linked above should get you far enough for the purposes of this article right here. But if this is TL;DR for you, here's the synopsis:

There are exactly six quarks in this universe, up (u), down (d), strange (s), bottom (b, also sometimes 'beauty"), and t (top).

These are real, folks. Just because they have weird names doesn't mean you don't carry them in every fibre of your body. In fact you carry only two types of quarks with you, really: the u and d, because they make up the protons and neutrons that make all of you: proton=uud, neutron=udd: three quarks for every nucleon. 

The s, c, and b, quarks exist only to annoy you, and provide work for high-energy experimental physicists! 

Just kidding again, of course. The fact that they (s,c, and b) exist provides us with a tantalizing clue about the structure of matter. As fascinating as this is, you and I have to move on right now. 

For every particle, there's an anti-particle. So there have to be anti-u, and anti-d. They make up almost all of anti-matter. You did know that anti-matter was a real thing, not just an imagination of Sci-Fi movies, right?

The particles that make up all of our known matter (and energy). The stuff that makes you (and almost all of our known universe) is in the first and 4th column. I'm still on the fence about the Higgs. It doesn't look quite right in this graph, does it?  Kind of like it's a bit of a mistake? Or maybe because it really is a condensate of top quarks? Source: Wikimedia

Right. You did. Good thing that. So we can move on then. So if we have u and d, we also must have anti-u and anti-d. And I'm sure you already did the math on charge to figure out that the charge of u better be +2/3, and the charge of d is necessarily -1/3. Because anti-matter has anti-charge, duh. If you're unsure why, contemplate the CPT theorem. 

Yes, quarks have fractional charges. If this blows your mind, you're welcome. And this is how we make one positive charge for the proton (uud), and a neutral particle (the neutron) from udd.

But the tinkerer in you has already found the brain gears in motion: what prevents me from making a (u-anti u), (d-anti d), (u-anti d), (d-anti u ) etc. ?

The answer is: nothing. They exist. (Next time, come up with this discovery when somebody else has NOT YET claimed a Nobel for it, OK?) These things are called mesons. They are important. I wrote my very first paper on the quantization of an effective theory that would predict how nucleons (you remember: protons and neutrons, meaning "we-stuff") interact with pions (a type of meson made only of u,d, anti-u, and anti-d), as discussed here.

What about the other stuff, the "invisible universe" made from all the other quarks, like strange, charm bottom, and top? Well, they also form all kinds of baryons (the world that describes all the heavy stuff, such as protons and neutrons) and mesons (the semi-heavy stuff). But they tend to decay real fast.

But one very important such meson--both in the history of QCD and our ability to understand it-- is the meson called "charmonium".

I did tell yout that it would take me a little time to get you up to date, right? Right. So, Charmonium is a bound state of the charm and anti-charm quark.

(By the way, if there is anybody reading this that still thinks: "Are you effing kidding me with all these quark names and stuff, are they even real?", please reconsider these thoughts, because they are like doubting we landed on the moon. We did, and there really are six quarks, and six anti-quarks. We are discussing their ramifications here. Thank you. Yes, "ramification" is a real word, that's why I linked to a Dictionnary. Yes, those Wiki pages on science are not making things up. Now, let's move on, shall we?)

The reason why we call the \({\rm c}\bar {\rm c}\) meson "charmonium" is because we have a name for the bound state of the electron and positron (also known as the anti-electron): we call it Positronium. Yes, that's a real thing. Not related to the Atomium, by the way. That's a real building, but not a real element.

So why is charmonium important? To understand that, we have to go back to the beginning of the universe.

No, we don't have to do it by time travel. Learning about charmonium might allow you to understand something about what was going on when our universe was really young. Like, less than a second young. Why would we care about these early times? Because they might reveal so us clues about the most fundamental laws of nature. Because the state of matter in the first few milliseconds (even the first few microseconds) might have left clues to decipher for us today.

At that time (before a millisecond), our universe was very different from how we see it today. No stars, no solar systems. We didn't even have elements. We didn't have nuclei! What we had was a jumble of quarks and gluons, which one charitable soul dubbed the "quark gluon plasma" (aka: QGP). The thing about a plasma is that positive and negative charges mill about unhindered, because they have way too much energy to form these laid-back bound states that we might (a few milliseconds later) find everywhere. 

So, here on Earth, people have been trying to recreate this monster of a time when the QGP reigned supreme, by shooting big nuclei onto other big nuclei. The idea here is that, for a brief moment in time, a quark gluon plasma would be formed that would allow us to study the properties of this very very early universe first hand. Make a big bang at home, so to speak. Study hot and dense matter.

While contemplating such a possibility at the RHIC collider in Brookhaven, NY (not far from where I was penning the paper I'm about to talk to you about), a pair of physicists (Tetsuo Matsui and Helmut Satz [1]) speculated that charmonium (you know, the \(\bar c c\) bound state) might be seriously affected by the hot plasma. In the sense that you could not see the charmonium anymore.

Now, for ordinary matter, the \(J/\psi\) (as the lowest energy state of the charmonium system is called for reasons I can't get into) has well known properties. It has a mass of 3.1 GeV (I still know this by heart), and a short but measurable lifetime. Matsui and Satz in 1986 speculated that this \(J/\psi\) would look very different if it was born in the midst of a quark gluon plasma, and that this would allow us to figure out whether such a state of matter was formed: all you have to do is measure the \(J/\psi\)'s properties: if it is much reduced in appearance (or even absent), then we've created a quark gluon plasma in the lab.

It was a straightforward prediction that many people accepted. The reason why the \(J/\psi\) would disappear in a QGP according to Matsui and Satz was the phenomenon of "color screening". Basically, the energy of the collision would create so many additional \(\bar c c\) pairs that they would provide a "screen" to the formation of a meson. It is as if a conversation shouted over long distances is disrupted by a bunch of people standing in between, whispering to each other. 

For a reason I cannot remember, Ismail Zahed (my Ph.D. advisor) and I came to doubt this scenario. We were wondering whether it was really the "hotness" of the plasma (creating all these screening pairs) that destroyed the \(J/\psi\). Could it instead be destroyed even if a hot plasma was not formed?

Heavy ion collsision in the rest system of the target (above), and in the center of mass rest system (below)

The image we had in our heads was the following. When a relativistically accelerated nucleus hits another nucleus, then in the reference frame of the center of mass of both nuclei each is accelerated (from this center of mass, you see two nuclei coming at you at crazy speed). And when two nuclei are accelerated relativistically, their lateral dimension (in the direction of movement) contracts, while the orthogonal direction remains unchanged. This means that the shapes of the nuclei are affected: instead of spherical nuclei they appear to be squeezed, as the image above suggests.

When looked at from this vantage point, a very different point of view concerning the disappearance of the \(J/\psi\) can be had. Each of the nuclei creates around it a color-electric and color-magnetic field, because of all the gluons exchanged between the flat nuclei. Think of it in terms of electrodynamics as opposed to color dynamics: if the two nuclei would be electrical conductors, they would span between them an electric field. Indeed, a set of conducting plates separated by a small distance is a capacitor. So, could it be that in such a collision, instead of all that hot screening, all that happens is the formation of a color-electric capacitor that simply rips the \(J/\psi\) to pieces?

That's the question we decided to check, by doing an old-fashioned calculation. How do you do this? I recall more or less that if I am going to calculate the fate of a bound state within a color-electric field, I ought to know how to calculate the fate of a bound state in an electric field. Like, for example: who calculated what happens to the hydrogen atom if it is placed between the plates of a capacitor? Today, I would just google the question, but in 1988, you have to really search. But after searching (I spent a lot of time in the library these days) I hit paydirt. A physicist by the name of Cornel Lanczos had done exactly that calculation (the fate of a hydrogen atom in a strong electric field). What he showed is that in strong electric fields, the electron is ripped off of the proton, leading to the ionization of the hydrogen atom.

This was the calculation I was looking for! All I had to do is change the potential (namely the standard Coulomb potential of electrodynamics, and replace it by the effective potential of quantum chromodynamics.

Now, both you and I know that we if don't have a potential, then we can't calculate the energy of the bound state. And the potential for color-electric flux tubes (as opposed to the exchange of photons, which gives rise to the electromagnetic forces as we all know) ought to be notably different from the Coulomb potential. 

No, I'm not known to be sidetracked by engaging in a celebration of the pioneers of quantum mechanics. But the career of Lanczos should give you pause. The guy was obviously brilliant (another one of the Hungarian diaspora) but he is barely remembered now. Spend some time with his biography on Wikipedia: there are others besides Schrödinger, Heisenberg, Planck, and Einstein that advanced our understanding of not just physics, but in the case of Lanczos, computational physics as well.

So I was sidetracked after all. Moving on. So, I take Lanczos's calculation, and just replace the Coulomb potential by the color-electric potential. Got it? 

Easier said than done! The Coulomb potential is, as everybody knows, \(V(r)=-\frac er\). The color-electric potential is (we decided to ignore color-magnetic forces for reasons that I don't fully remember, but that made perfect sense then)

$$V(r)=-\frac43\frac{\alpha_s}r+\sigma r.      (1)$$

"All right", you say, "what's all this about?"

Well, I respond, you have to understand that when it comes to color-electric (rather than just electric) effects, the coupling constant is not the electron charge, but 4/3 of the strong coupling constant \(\alpha_s\).

"But why 4/3?"

Ok, the 4/3 is tricky. You don't want to see this, trust me. It's not even in the paper. You do want to see it? OK. All others, skip the colored text.

How to obtain the quark-antiquark Coulomb potential

To calculate the interquark potential you have to take the Fourier transform of the Feynman diagram of quark-anti-quark scattering:

The solid lines are quarks or anti-quarks, and the dashed line is the gluon exchanged between them. Because the gluon propagator \(D^{-1}_{ab}\) is diagonal, the amplitude of the process is given mostly by the expectation value of \(\vec T^2\). \(T^{(a)}\) is the generator of the symmetry group of quarks SU(3), given by \(T^{(a)}=\frac12\lambda^a\). And \(\lambda^a\) is a Gell-Mann matrix. There are eight of them. What the value of \(\langle \vec T^2\rangle\) is depends on the representation the pair of quarks is in. A little calculation shows that for a quark-antiquark pair in a singlet state, \(\langle \vec T^2\rangle=-4/3\). If the pair is in an octet state, this same expectation value gives you 1/6, meaning that the octet is unbound. 

More interesting than the Coulomb term is the second term in the potential (1), the one with the wavy thing in it. Which is called a "sigma", by the way.

"What of it?"

Well, \(\sigma\) is what is known as the "string tension". As I mentioned earlier, quarks and anti-quarks can't just run away from each other (if you gave them enough energy). In strong interactions, the force between a quark and an anti-quark increases in proportion to their distance. In the lingo of strong interaction physics, this is called "asymptotic freedom", because it means that at short distances, quarks get to be free. Not so if they attempt to stray, I'm afraid.

So suppose we insert this modified potential, which looks just like a Coulomb potential but has this funny extra term, into the equations that Lanczos wrote down to show that the hydrogen atoms gets ripped apart by a strong electric field?

Well, what happens is that (after a bit of a song and dance that you'll have to read about by yourself), it turns out that if the color-electric field strength just marginally larger than the string tension \(\sigma\), then this is sufficient to rip apart the charmonium bound state. Rip apart, as in disintegrate. The color-electric field generated by these colliding nuclei will kill the charmonium, but it is not because a hot quark gluon plasma creates a screening effect, it is because the cold color-electric field rips the bound state apart!

The observable result of these two very different mechanisms might look the same, though: the number of \(J/\psi\) particles you would expect is strongly reduced.

So what have we learned here? One way to look at this is to say that a smoking gun is only a smoking gun if there are no other smoking guns nearby. A depletion of \(J/\psi\)s does not necessarily signal a quark gluon plasma.

But this caveat went entirely unheard, as you already know because otherwise I would not be writing about this here. Even though we also published this same paper as a conference proceeding, nobody wanted to hear about something doubting the holy QGP.

Is the controversy resolved today? Actually, it is still in full swing, almost 30 years after the Matsui and Satz paper [1], and 25 years after my contribution that was summarily ignored. How can this still be a mystery? After all, we have had more and more powerful accelerators attempt to probe the elusive QGP. At first it was CERN's SPS, followed by RHIC in Brookhaven (not far away from where I wrote the article in question). And after RHIC, there was the LHC, which after basking in the glory of the Higgs discovery needed something else to do, and turned its attention to.... the QGP and \(J/\psi\) suppression!

The reason why this is not yet solved is that the signal of \(J/\psi\) suppression is tricky. What you want to do is compare the number of \(J/\psi\) produced in a collision of really heavy nuclei (say, lead on lead) with those produced when solitary protons hit other protons, scaled by the number of nucleons in the lead-on-lead collision. Except that in the messy situation of lead-on-lead, \(J/\psi\) can be produced at the edge rather than the center, be ripped apart, re-form, etc. Taking all these processes into account is tedious and messy.

So the latest news is: yes, \(J/\psi\) is suppressed in these collisions. But whether it is due to "color-screening" as the standard picture of the QGP suggests, or whether it is because a strong color-electric field rips apart these states (which could happen even if there is no plasma present at all as I have shown in the paper you can download from here), this we do not yet know. After all this time.

[1] T. Matsui and H. Satz, “J/ψ Suppression by Quark-Gluon Plasma Formation,” Phys. Lett. 178 (1986) 416.

Now, move over to Part 3, where I awkwardly explain the meaning of the word "Prolegomena".

Nifty papers I wrote that nobody knows about (Part I: Solitons)

I suppose this happens even to the best of us: you write a paper that you think is really cool and has an important insight in it, but nobody ever reads it. Or if they read it, they don't cite it. I was influenced here by the blog post by Claus Wilke, who argues that you should continue writing papers even if nobody reads them. I'm happy to do that, but I also crave attention. If I have a good idea, I want people to notice. 

The truth is, there are plenty of papers out there that are true gems and that should be read by everybody in the field, but are completely obscure for one reason or another. I know this to be true but I have little statistical evidence because, well, the papers I am talking about are obscure. You can actually use algorithms to detect these gems, but they usually only find papers that are already fairly well known. 

In fact, this is just common sense: once in a while a paper just "slips by". You have a bad title, you submitted to the wrong journal, you wrote in a convoluted manner. But you had something of value. Something that is now, perhaps, lost. One of my favorite examples of this sort of overlooked insight is physicist Rafael Sorkin's article: "A Simple Derivation of Stimulated Emission by Black Holes", familiar to those of you who follow my efforts in this area. The article has 10 citations. In my view, it is brilliant and ground-breaking in more than one way. But it was summarily ignored. It still is, despite my advocacy.

I was curious how often this had happened to me. In the end the answer is: not so much, actually. I counted five four papers that I can say have been "overlooked". I figured I would write a little vignette about each of them, why I like them (as opposed to the rest of the world), and what may have gone wrong--meaning--why nobody else likes them.

Here are my criteria for a paper to be included into the list:

1.) Must be older than ten years. Obviously, papers written within the last decade may not have had a significant amount of time to "test the waters". (But truthfully, if a paper does not get some citations within the first 5, it probably never will. )

2.) Must have had fewer than 10 citations on Google Scholar (excluding self-citations).

3.) Must not be a re-hash of an idea published somewhere else (by me) where it did get at least some attention.

4.) Must not be a commentary about somebody else's work (obvious, this one). 

5.) Must be work that I'm actually proud of. 

When going through my Google Scholar list, I found exactly four papers that meet these criteria. 

(Without taking into account criterion 5, the list is perhaps twice as long, mind you. But some of my work is just not that interesting in hindsight. Go figure.)

These are the four papers in the final list:

1. Soliton quantization in chiral models with vector mesons, C Adami, I Zahed (1988)
2. Charmonium disintegration by field-ionization, C Adami, M Prakash, I Zahed (1989)
3. Prolegomena to a non-equilibrium quantum statistical mechanics, C Adami, NJ Cerf (1999)
4. Complex Langevin equation and the many-fermion problem, C Adami, SE Koonin (2001).

I will publish a blog post about one of these each of the coming weeks.

I'll start in reverse chronological order:

Physics Letters B 215 (1988) 387-391. Number of citations: 10

This is actually my first paper ever, written at the tender age of 25. But it didn't get cited nearly as much as the follow-up paper, which was published a few months earlier: Physics Letters B 213 (1988) 373-375.

How is this possible, you ask? 

Well, the editors at Physics Letters lost my manuscript after it was accepted, is how it happened! 

You have to remember that this was "the olden days". We had computers all right. But we used them to make plots, and send Bitnet messages. You did not send electronic manuscripts to publishers. These were sent around in large manila envelopes.  And one day I get the message (after the paper was accepted): "Please send another copy, we lost ours". Our triplicates, actually, because each reviewer gets a copy that you send in, of course. I used to keep all the correspondence about manuscripts from these days, but I guess after moving offices so many times, at one point stuff gets lost. So I can't show you the actual letter that said this (I looked for it).  Of course, after that mishap the editorial office used a new "received" date, just so that it doesn't look so embarrassing. And arxiv wouldn't exist for another 4 years to prove my point.

So that's probably the reason why the paper didn't get cited: people cited the second one that was published first, instead. But what is this paper all about?

It is about solitons, and how to quantize them. Solitons were my first exposure to theoretical physics in a way, because I had to give a talk about topological solitons called "Skyrmions" in a theoretical physics seminar at Bonn University in, oh, 1983. Solitons are pretty cool things: they are really waves that behave like particles. You can read a description of how they were discovered by John Scott Russell riding his horse alongside a canal in Scotland, and noticing this wave that just... wouldn't... dissipate, here. 

Now, there is a non-linear field theory due to T.H.R. Skyrme that has such soliton solutions, and people suggested that maybe these Skyrmions could describe a nucleon. You know, the thing you are made of, mostly? A nucleon is a proton or a neutron, depending on charge. Nuclei are are made from them. Your are all nucleons and electrons really. Deal with it. 

Skyrme incidentally is the one who died just days after I submitted the very manuscript I'm writing about, which started the rumour that my publications are lethal. Irrelevant fact, here. 

Skyrme's theory was a classical one, and so the question arose what happens if you quantize that theory. This is an interesting question because usually, if you quantize a field you create fluctuations of that field, and if these fluctuations were of the right kind, they should (if they fluctuate around a nucleon) describe pions. And voilà: we would have a theory that describes how nucleons have to interact with pions. 

What are pions, you ask? Go read the Wiki page about them. But really, they are the stuff you get if you bang a nucleon and an anti-nucleon together. They have a quark and an anti-quark in them, as opposed to the nucleons, that have three quarks: Three quarks for Muster Mark! 

Now, people actually already knew at the time what such an interaction term was supposed to look like: the so-called pion-nucleon coupling. But if the term that comes out of quantizing Skyrme's theory did not look like this, well then you could safely forget about that theory being a candidate to describe nucleons. Water waves maybe, just not the stuff we are made out of.  

So I started working this out, using the theory of quantization under constraints that Paul Dirac developed, because we (my thesis advisor Ismail Zahed and I) had stabilized the Skyrmion using another meson, namely the ω-meson. You don't have to know what this is, but what is important here is that the components of the ω field are not independent, and therefore you have to quantize under that constraint.

You very quickly run into a problem: you can't quantize the field because there are fluctuation modes that have zero energy. Indeed, because in order to do the quantization you have to take the inverse of the matrix of fluctuations, these zero modes create a matrix that cannot be inverted (its determinant vanishes). What to do?

The answer is: you find out what those zero modes are, and quantize them independently. It turns out that those zero modes were really rotations in "isospin-space", and they naturally have zero energy because you can rotate that soliton in iso-space and it costs you nothing. I figured out how to quantize those modes by themselves (you just get the Hamiltonian for a spinning top out of that), then project out these zero modes from the Skyrmion fluctuations, and quantize only those modes that are orthogonal to the zero modes. And that's what I proceeded to do. Easy as pie.

And the result is fun too, because the resulting interaction term looks almost like the one we should have gotten, and then we realized that the "standard" term of chiral theory comes out in a particular limit, known as the "strong coupling" limit. Even better, using this interaction I could calculate the mass of the first excitation of the nucleon, the so-called Δ resonance. That would be the content of the second paper, which you now know actually got published first, and stole the thunder of this pretty calculation.  

So what did we learn in this paper in hindsight? Skyrmions are actually very nice field-theoretic objects, and the effective theory (while obviously not the full underlying theory that should describe you, namely the theory of quarks and gluons called Quantum Chromodynamics, or QCD), this approximate theory can give you very nice predictions about low energy hadronic physics, where QCD actually is not at all predictive. That's because we can only calculate QCD in the high-energy limit (for example what happens when you shoot quarks at quarks with lots of energy, for example). Research on Skyrmions (and low-energy effective theories in general) is still going on strong, it turns out. And perhaps even more surprising is this: there is now a connection (uncovered by my former advisor), between these Skyrmions and the holographic principle! 

So even old things turn out to be new sometimes, and old calculations can still teach you something today. Also we learn: electronic submissions aren't as easily lost behind file cabinets. So there is that.

Next up:  Charmonium Disintegration by Field-Ionization [Physics Letters B 217 (1988), 5-8]. A story involving the quark-gluon plasma, and how an old calculation by Cornel Lanczos from 1930 can shed light on what happens to the \(J/\psi\), when suitably modernized. All of 5 citations on Google Scholar this one got. But what a fun calculation! Read on here.