I haven’t said much since this year’s start-up of the LHC, but there have been some interesting developments, so I’ll add one last update. If you haven’t been following the LHC status, it has been exponentially increasing in collision rate while maintaining a fixed collision energy (about 3 pb^{-1} of 7 TeV collisions have been collected by the LHC experiments, which is a few thousand times less than the Tevatron’s 9 fb^{-1} of 1.96 TeV collisions, collected since 2001). My “particle body counts” are now completely obsolete: nearly all known particles have been re-discovered in the LHC experiments. And today, the first *unexpected* effect has been presented by an LHC experiment: “Long-range, near-side angular correlations,” which is presented in detail on the CMS public page. Below the cut here, I’ll explain what this means.

The new result is not a search for a new type of particle or a decay chain that would indicate beyond-the-Standard Model physics; it’s a statistical correlation in the events that would be considered uninteresting backgrounds in most searches. When you collide protons, most of the time (by many orders of magnitude in rate), they interact via the strong force according to the rules of Quantum ChromoDynamics (QCD). This has only been demonstrated in certain limits of QCD, because the theory is highly coupled: quarks are attracted to other quarks by exchanging a gluon, in analogy with electromagnetism (replace “charged particles” for “quarks” and “photons” for “gluons”), but gluons are attracted to other gluons by exchanging yet more gluons: the result is a sticky mess. I would put QCD/the strong force in the same category as turbulent flows of water: we believe that the underlying physics is understood, but exact results are incalculable because of complexity. The result that has just been announced is a qualitatively new feature in the angular correlations of QCD interactions.

QCD interactions all take place on very short distance scales, about meters, so the products of the reaction all emerge from a single point. A useful way to look at them is in the projective space of azimuthal () and polar () angles relative to the beamline, with the polar angle usually represented as pseudorapidity (). The angles are illustrated on a typical QCD event in the CMS detector below (the detector has a rough cylindrical symmetry around the beamline; the beamline points into the page in the figure on the right).

QCD interactions produce a lot of particles, as you can see in the event display above (the yellow lines are tracks from charged particles; the pink and green bars are energy deposits in the calorimeters). We can quantify the correlation between pairs of particles as a function of angular differences and and plot what we see (see the paper, Eq. 1, for an exact definition of the correlation function). The plot below is the observed pairwise correlation (vertical axis) versus each angular difference (horizontal axes).

The sharp peak at , is expected: a quark or gluon (collectively called “partons“) emerging from the collision must make a collimated “jet” of particles from partons begetting more partons and finally forming the final-state hadrons that we see. (“Hadron” collectively refers to any particles made of quarks and held together by gluons, such as protons.) At , there’s a second peak from the fact that partons usually emerge from the collision in pairs. Momentum conservation forces the two jets to be back-to-back in , so when you select one final-state hadron from one jet and another from the other jet, they’ll be close to . The fact that they don’t have close to zero— in other words, momentum conservation in the component parallel to the beamline— is a feature of hadron collisions: most of this component of momentum is carried by the proton’s two out of three quarks that *didn’t* collide (and usually aren’t observed because their deflection angles are too small to reach the detector).

Now here’s the new feature: in events with large numbers of particles () with moderate momenta ( GeV/c), there’s an excess of final-state hadrons correlated with , .

These are particles that aren’t close enough in polar angle to be in the same jet, but they would nearly line up in an azimuthal view of the collision. That’s a strange feature, and none of the existing simulations of QCD (all approximations to the full theory) reproduce it.

What makes it especially interesting is its passing resemblance to similar correlations seen in heavy ion collisions. The LHC is currently colliding single protons, but other experiments such as RHIC collide whole nuclei of heavy elements. The figure on the right shows the same particle-pair correlation function from gold-gold collisions, and you can see a , ridge like the one observed at the LHC.

In heavy ion collisions, this correlation is interpreted as part of the evidence that high-energy, high-density collisions produce a new state of matter called a quark-gluon plasma, in which the nuclei literally melt into a dense, zero-viscosity fluid of constituent quarks and gluons. The high density is key: a gold-gold collision, in which each of the gold nuclei has 197 protons and neutrons, is like a bloodbath of QCD interactions. Partons produced in one interaction can’t get away from the partons produced by the other interactions, so they interact many times, like water molecules bumping into each other enough to become a fluid with an equilibrium temperature. This is what makes it possible to apply thermodynamic terms to the process: concepts such as “fluid” and “temperature” only make sense in the context of a large number of particles undergoing many interactions.

In proton-proton collisions like those at the LHC, the usual picture is that one parton from each proton interacts and the others get away with only a minor deflection (called “spectators”). It would be surprising if the collision products interact with themselves enough to make any kind of thermodynamic fluid. The reason this interpretation is conceivable is that the LHC collision energies are the highest ever explored, and multiplicity (number of particles emerging from the collision) rises with collision energy. Could 7 TeV be high enough to start melting single protons, the way that large nuclei are melted at, say, 0.2 TeV?

The onset of a quark-gluon plasma in proton-proton collisions is not the only interpretation, and in fact, it would be a highly contraversial interpretation. Remember that the QCD-like simulations that tell us what to expect are only approximations to the full QCD theory; it may be possible to explain this new feature without invoking quark-gluon plasmas. I’ve heard people talking about specific effects in single-interaction QCD that might account for it. The experimental paper is careful to describe only the observed effect and leave the interpretation to theorists— I’m only describing the quark-gluon plasma interpretation here to explain why the result could be exciting. At minimum, if this only amounts to a tuning of the QCD-like simulations, I don’t think anyone expected having to add a qualitatively new feature to the simulations, rather than just tweaking the numerical values of their parameters. In that sense, it is an unexpected discovery however you look at it.

(Remember that the CMS public page has more details. It’s also worth noting that the LHC will begin a 7 TeV heavy ion run later this year, which ought to produce quark-gluon plasmas well above the phase transition.)

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By perturbations of problem A by means problem B

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Let be a guaranteed error estimate for the gradient algorithm in some unperturbed (perturbed) discrete optimization problem. As usual (see. [3]), we say that the gradient algorithm is stable if , where as .

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, (1)

where

– is the set of all maximal elements of the partially ordered set .

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the last inequality yields

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Therefore

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Where