The strong force coupling constant is a dimensionless constant that tells you how strongly gluons and quarks couple with each other which runs with the energy scale of the interaction in quantum chromodynamics (QCD), according to its beta function, whose Standard Model terms are known exactly in the high energy "ultraviolet" regime.
If you plot the strong force coupling constant's strength against energy scale, near its peak it looks like a bell curve (with a linear strength scale and logarithmic energy scale), with a peak close to 1 (maybe even as much as 1.25) at roughly q2=0.5 GeV2, that declines in a very long tail towards higher energies, and in a much shorter tail towards lower energies. It isn't entirely resolved whether the strong force coupling constant takes a value that is 0 (a value sometimes called "trivial") or some fixed value greater than zero in the limit at an energy scale of q2=0 (called an "infrared fixed point"). The value that the strong force coupling constant takes at the Z boson mass of 91.1876(21) GeV on the ultraviolet side of the peak is roughly the same as the value it takes at a hair over q2=0.01 GeV2 on the infrared side of the peak.
The most accurate world average measurement of this quantity (as of 2014) at an energy scale equal to the Z boson mass is 0.1185 +/- 0.0006. By comparison, twenty years ago, back in 1994, the most accurate world average measurement was 0.123 +/- 0.006. So, since then, the best fit value has fallen by about 3.8% and the precision of the measurement has improved by a factor of ten (the values are consistent with each other to within the margin of error). We've improved, but only a little. At this rate, we'll have a measurement as accurate as the current electromagnetic coupling constant measurement in another ten million years, and will reach the accuracy with which we have measured the Newton's constant of gravitation in about 250 years.
This is about 20 times less precise than the most accurate combined measurement of the W boson mass. It about 50 times less precise than the most accurate combined measurement of the Newton's constant in general relativity. It is about 1,200,000 times less precise than the the most accurate combined measurement coupling constant for electromagnetism, and it is about 100,000 times less precise than the most accurate combined measurement of the coupling constant for the weak force.
The only constants in the Standard Model known less precisely than the strong force coupling constant are some of the quark masses, some of the tiniest CKM matrix entries, and some of the neutrino physics constants.
The latest ATLAS measurement of this quantity as 7 TeV is 0.1173 +/- 0.0046, with a theoretical error of about four times as much as the experimental error. This is consistent with previous measurements at the one sigma (i.e. one standard deviation) level.
This ATLAS experiment in question was primarily measuring something else and wasn't designed to produce a world class strong force coupling constant measurement, so the 4% margin of error doesn't reflect badly on ATLAS and confirms once again that Standard Model constants measured in many different ways have the same values within error bars, confirming that it is robust and that the quantity defined in theory corresponds to a meaningful quantity in the real world.
The great inaccuracy with which the strong force coupling constant is known does, however, does represent a fundamental barrier to accurate QCD calculations. For example, the proton mass can be calculated from QCD first principles to a precision of only about 1%, about half of which is attributable to uncertainty in the strong force coupling constant, even though the proton mass has been directly measured to a precision of eight significant digits.
And, while this would seem like it is the fault of imprecise experiments, it really isn't. We have measured observable properties of all manner of hadrons (particle made of quarks that are bound together by the strong force such as protons, neutrons and pions), which are much more precise. But, there is so much theoretical uncertainty in how to correctly calculate those measured observables that it isn't possible to reverse engineer them into very precise measurements of quark masses and the strong force coupling constant.
This isn't for want of trying. Theoretical physicists like the author of the 4gravitons blog devote the lion's share of their professional efforts every day to "come up with mathematical tricks to make particle physics calculations easier." Physicist have been working steadily on this problem for half a century and have made only modest progress, although steady advances in computers computational power over that time period have helped a great deal. But, given that computation power has increasing according to Moore's law by a factor of about 2.6 billion over that time period since QCD was invented, progress has still been painstakingly slow. The math really is just that hard.
The Strong Coupling Constant and BSM Theories
You might think that people crafting beyond the Standard Model physics theories would be swarming to produce theories that would predict slightly higher or slightly lower strong force coupling constants because this is an area where there is considerable room to make contradictory predictions within the existing margin of error. But, in fact, there is almost no activity on this front and beyond the Standard Model physics proposals tend to leave the strong force coupling constant and the QCD equations of the strong force, more or less unchanged apart from the number of strongly interacting particles. There isn't even a great deal of work being done that makes differing predictions regarding the quark masses.
Supersymmetry theories, generically, predict that the running of the strong force coupling constant will take place at a different rate (i.e. that it will have a different "beta function") than in the Standard Model, which may be possible to compare to the Standard Model prediction at LHC Run II. Basically, the strong force coupling constant gets weaker with higher energies in the Minimal Supersymmetric Model (MSSM) at about half the rate that it does in the Standard Model (the weak force coupling constant gets weaker at higher energies in the Standard Model but stronger at higher energies in the MSSM, and the electromagnetic force coupling constant which gets stronger at higher energies in both theories gets 30% stronger at given higher energies in the MSSM than in the Standard Model).
The actual value of the strong force coupling constant, however, is actually harder to measure at high energies than at low energies, because using the Standard Model beta function for the strong force coupling constant, a comparatively wide range of low energy values at lower energies when run according to the strong force coupling constant beta function of the Standard Model, all converge at points which are extremely close to each other at much high energies. So high energy measurements of the strong force coupling constant must be much more precise at higher energies than at lower energies to make a measurement of the same precision. As a result, LHC Run II probably won't do much to improve the accuracy with which the strong force coupling constant is measured.
Many beyond the Standard Model physics proposals either don't answer these questions at all, leaving them as experimentally measured physical constants, just as the Standard Model does, or assumes relationships between these valuable that are right at an order of magnitude level but impossible to calculate more precisely at this time. Phenomenologists try to look for patterns in the currently measured values of these constants, but even there, the results have thus far been underwhelming.
The three coupling constants in superysmmetric theories have one less degree of freedom than in the Standard Model (i.e. two instead of three). This is because, in principle, in supersymmetric theories, the strong force coupling constant can be calculated exactly from an ultraviolet fixed point where the strength of the strong, weak and electromagnetic force coupling constants are the same at the "GUT" scale of about 1016 GeV, using the supersymmetric strong force coupling constant beta function. The coupling constant strength and the fixed point energy scale could be calculated using supersymmetric beta functions for the weak force coupling constant and electromagnetic force coupling constant whose values are known much more precisely than the strong force coupling constant. This fixed point, calculated from much more precise experimental inputs, could in turn be used to back out a strong force coupling constant at a measurable energy which should be much more precise than the experimentally measured version. The result should be almost 10,000 times as precise as the current experimentally measured values, since the beta functions, in both the Standard Model and supersymmetry, in principle, ought to be possible to determine exactly without experimental inputs from the theory.
I have not seen a paper making the calculation this way, although it should be straightforward to do once the supersymmetric beta functions are established. This calculation wouldn't be useful from a predictive perspective, because there are so many different variations on the supersymmetry theme out there. But, it would make it possible to use the measured strong force coupling constant value at low energy scales to discriminate experimentally between the possibilities, thus narrowing "theory space" and the parameter space of theories. Perhaps existing supersymmetric parameter fitting software does this in a way that isn't transparent to a casual observer like me.
My own conjecture (meaning that I am someone who thinks that this might be true, not that I am the first or only person to have come up with it) is that the three Standard Model force coupling constants do actually converge at a GUT scale, but that this happens only once quantum gravity effects on the ultraviolet running of these constants are considered, which the Standard Model beta functions do not.
Newton's constant is generally a running coupling constant in quantum gravity theories, something which greatly impacts the physics of the very early universe, and a number of beta functions, such as this one, have been proposed.