Wednesday, June 20, 2018

Measuring The Electromagnetic Force Coupling Constant

Jester at Resonaances has a new post on a new ultraprecision measurement of the electromagnetic force coupling constant based upon a two month old paper that missed headlines when it came out because of the way that it was published and tagged. He notes:
What the Berkeley group really did was to measure the mass of the cesium-133 atom, achieving the relative accuracy of 4*10-10, that is 0.4 parts par billion (ppb). . . . the measurement of the cesium mass can be translated into a 0.2 ppb measurement of the fine structure constant: 1/α=137.035999046(27). One place where precise knowledge of α is essential is in calculation of the magnetic moment of the electron. Recall that the g-factor is defined as the proportionality constant between the magnetic moment and the angular momentum. For the electron we have:

Experimentally, ge is one of the most precisely determined quantities in physics, with the most recent measurement quoting ae = 0.00115965218073(28), that is 0.0001 ppb accuracy on ge, or 0.2 ppb accuracy on ae. In the Standard Model, ge is calculable as a function of α and other parameters. In the classical approximation ge=2, while the one-loop correction proportional to the first power of α was already known in prehistoric times thanks to Schwinger. The dots above summarize decades of subsequent calculations, which now include O(α^5) terms, that is 5-loop QED contributions! . . . the main theoretical uncertainty for the Standard Model prediction of ge is due to the experimental error on the value of α. The Berkeley measurement allows one to reduce the relative theoretical error on ae down to 0.2 ppb: ae = 0.00115965218161(23), which matches in magnitude the experimental error and improves by a factor of 3 the previous prediction based on the α measurement with rubidium atoms. . . .  
it also provides a powerful test of the Standard Model. New particles coupled to the electron may contribute to the same loop diagrams from which ge is calculated, and could shift the observed value of ae away from the Standard Model predictions. In many models, corrections to the electron and muon magnetic moments are correlated. The latter famously deviates from the Standard Model prediction by 3.5 to 4 sigma, depending on who counts the uncertainties. Actually, if you bother to eye carefully the experimental and theoretical values of ae beyond the 10th significant digit you can see that they are also discrepant, this time at the 2.5 sigma level. So now we have two g-2 anomalies! 
FWIW, I calculate the discrepancy to be 2.43 sigma, and not 2.5.

Jester has a pretty chart that illustrates the discrepancies, but it does more to obscure than reveal what is going on to the uninitiated. Words, which I will paraphrase below for even greater clarity, are more clear in this case.

As Jester explains, the direction of the discrepancy is important. 

New physics fixes that treat electrons and muons the same, in general, don't work, because the electron g-2 calls for a negative contribution to the theoretically calculated value, while the muon g-2 needs a positive contribution to the theoretically calculated value.

So, new physics can't solve both discrepancies without violating lepton universality, which is tightly constrained by other measurements that seem to contradict evidence that this is violated in B meson decays, so this isn't possible without some sort of elaborate theoretical structure that cause it to be violated sometimes, but not others.

On the other hand, discrepancies in the opposite directions in measurements of two quantities that are extremely analogous to each other in the Standard Model, and in different magnitudes, are exactly what you would expect to see if there is theoretical or experimental error in either of the measurement. If you assume that lepton universality is not violated and pool the results for electron g-2 and muon g-2 in a statistically sound way, the discrepancies tend to cancel each other other producing a global average that is closer to the Standard Model prediction.

More experimental data regarding these measurements is coming soon.
The muon g-2 experiment in Fermilab should soon deliver first results which may confirm or disprove the muon anomaly. Further progress with the electron g-2 and fine-structure constant measurements is also expected in the near future. The biggest worry is that, if the accuracy improves by another two orders of magnitude, we will need to calculate six loop QED corrections...

It is also worth pausing for just a moment to compare the state of QED (the Standard Model theory of the electromagnetic force) with QCD (the Standard Model theory of the strong force).

The strong force coupling constant discussed in my previous post at this blog is known with a precision of 7 parts per 1000, which may be overestimated and actually be closer to 4 parts per 1000. This is based on NNLO calculations (i.e. three loops).

The electromagnetic force coupling constant, which is proportionate to the fine structure constant, α, is known with a precision of 0.2 parts per billion, and the electron g-2 is calculated to five loops. So, we know the electromagnetic coupling constant to a precision 2-4 million times greater than we know the strong force coupling constant.

For sake of completeness, we know the weak force coupling constant (which is proportional to the Fermi coupling constant) to a precision of about 2 parts per million. This is about 5,000 times less precise than the electromagnetic coupling constant, but about 2000 times more precisely than the strong force coupling constant.

We know the gravitational coupling constant (i.e. Newton's constant G) which isn't strictly analogous to the three Standard Model coupling constants since it doesn't run with energy scale in General Relativity and isn't dimensionless, to a precision of about 2 parts per 10,000. This is about 20 times more precise than the precision with which we have measured the strong force coupling constant (even incorporating my conjecture that the uncertainty in the strong force coupling constant's global average value is significantly overestimated), is about 100 times less precise than our best measurement of the weak force coupling constant, and is about 500,000 times less precise than our best measurement of the electromagnetic coupling constant.

Tuesday, June 19, 2018

Measuring The Strong Force Coupling Constant

The Latest Measurements Of The Strong Force Coupling Constant

The strong force coupling constant has a global average best fit value with a precision of a little less than 0.7% which is slowly but steadily improving over time according to the linked preprint released today. After considering the latest available data, the strength of the coupling constant at the Z boson mass momentum transfer scale is as follows:


This means, roughly, that there is a 95% chance that the true value of the strong force coupling constant is between 0.1168 and 0.1199.

Why Does This Matter?

This roughly 27% precision increase in the precision of the global average, from plus or minus 0.0011 until the latest measurements, matters because the strong force coupling constant is a key bottleneck limiting the accuracy of calculations made throughout the Standard Model in phenomena that have significant QCD (i.e. strong force physics) contributions.

The Example Of Muon g-2

For example, 98.5% of the uncertainty involved in calculating the theoretically expected value of the muon magnetic moment, i.e. muon g-2, in the Standard Model comes from uncertainties in the strong force physics part of that calculation, even though the strong force contribution to muon g-2 accounts for only about one part in 16,760 of the final value of the muon g-2 calculation.

The remaining 1.5% of the uncertainty comes from weak force and electromagnetic force physics uncertainties, with 92.6% of that uncertainty, in turn, coming from weak force physics as opposed to electromagnetic force physics uncertainties, even though the weak force component of the calculation has only about a one part per 759,000 impact on the final value of the muon g-2 calculation.

One part of the strong force contribution to the muon g-2 calculation (hadronic light by light) has a 25% margin of error. The other part of the strong force contribution to the muon g-2 calculation (hadronic vacuum polarization) has a a 0.6% margin of error. The weak force component of the calculation has a 0.7% margin of error. The electromagnetic component of the calculation, in contrast, has a mere 1 part in 1.46 billion margin of error.

The overall discrepancy between a 2004 measurement of muon g-2 and the Standard Model prediction was one part in about 443,000. So, a 0.7% imprecision in a Standard Model constant necessary to make every single QCD calculation seriously impedes the ability of the Standard Model to make more precise predictions.

Implications For Beyond The Standard Model Physics Searches

Imprecision makes it hard to prove or disprove beyond the Standard Model physics theories, because with a low level of precision, both possibilities are often consistent with the Standard Model.

Even "numerology" hypothesizing ways to calculate the strong force coupling constant from first principles is pretty much useless due to this imprecision, because coming up with first principles combinations of numbers that can match a quantity with a margin of error of plus or minus 0.7% is trivially easy to do in myriad ways that are naively sensible, and hence not very meaningful.

Is The Margin Of Error In The Strong Force Coupling Constant Overstated?

Given the relative stability of the global average over the last couple of decades or so, during which many experiments would have been expected to have tweaked this result more dramatically than they actually have if the stated margin of error was accurate, my intuition is also that the margin of error that is stated is probably greater than the actual difference between the global average value and the true value of this Standard Model constant.

I suspect that the actual precision is closer to plus or minus 0.0004 and is overstated due to conservative estimates of systemic error by experimental high energy physicists. This would put the true "two sigma" error bars that naively mean that there is a 95% chance that the true value is within them at 0.1175 to 0.1191.

A review of the relevant experimental data can be found in Section 9.4 of a June 5, 2018 review for the Particle Data Group.

Why Is The Strong Force Coupling  Constant So Hard To Measure?

The strong force coupling constant can't be measured directly.

It has to be inferred from physics results involving hadrons (composite particles made up of quarks) and from top force physics measurements that infer the properties of top quarks from their decay products.

So, once you have your raw experimental data, you then have to fit that data to a set of very long and difficult to calculation equations that include the strong force coupling constant as one of the variables, containing multiple quantities that have to be approximated using infinite series that a truncated to a manageable level, to convert the experimental data into an estimated value of the strong force coupling constant that can be inferred from the results.

The main holdup on getting a more precise measurement of the strength of the strong force coupling constant is the difficulty involved in calculating what value for the constant is implied by an experimental result, not, for the most part, the precision of the experimental data itself.

For example, the masses and properties of the various hadrons that are observed (i.e. of composite particles made of quarks) have been measured experimentally to vastly greater precision than they can be calculated from first principles, even though a first principles calculation is, in principle, possible with enough computing power and enough time, and almost nobody in the experimental or theoretical physics community thinks that the strong force part of the Standard Model of particle physics in incorrect at a fundamental level.

This math is hard mostly because QCD calculations are very hard to do in practice to sufficient precision, which is mostly because the infinite series involved in calculating them converge so much more slowly than those in other parts of quantum physics, so far more terms must be calculated to get comparable levels of precision.

The Running Of The Strong Force Coupling Constant

Another probe of beyond the Standard Model physics is to look at how the strength of the strong force coupling constant varies with momentum transfer scale. 

Like all Standard Model empirically determined constants, the strength of the strong force coupling constant varies with energy scale, which is why the global average has to be reported in a manner normalized for energy scale, something called the "running" of the strong force coupling constant.

At low energies, in the Standard Model (illustrated in the chart below from a source linked in the linked blog post), as confirmed experimentally, the strong force coupling constant gets close to or near zero at zero energy, peaks at about 216 MeV, and then gradually decreases as the energy scale increases beyond that point. There is considerable debate over whether it goes to zero, or instead to a finite value close to zero, at zero energy, which is important for a variety of theoretical reasons and has not been definitively resolved.

The running of the strong force coupling constant in beyond the Standard Model theories like Supersymmetry (a.k.a. SUSY) is materially different at high energies than it is in the Standard Model (as shown in the charge below from the following linked post where the inverse of the strength of the strong force coupling constant at increasing energies on a logarithmic scale shown on the X-axisi is the SU(3) line) and the differences might be possible to distinguish with maximal amounts of high energy data from the LHC, is progress can be made in the precision of those measurements, as I explained in the linked blog post from January 28, 2014:

The strong force coupling constant, which is 0.1184(7) at the Z boson mass, would be about 0.0969 at 730 GeV and about 0.0872 at 1460 GeV, in the Standard Model and the highest energies at which the strong force coupling constant could be measured at the LHC is probably in this vicinity. 
In contrast, in the MSSM [minimal supersymmetric standard model], we would expect a strong force coupling constant of about 0.1024 at 730 GeV (about 5.7% stronger) and about 0.0952 at 1460 GeV (about 9% stronger). 
Current individual measurements of the strong force coupling constant at energies of about 40 GeV and up (i.e. without global fitting or averaging over multiple experimental measurements at a variety of energy scales), have error bars of plus or minus 5% to 10% of the measured values. But, even a two sigma distinction between the SM prediction and SUSY prediction would require a measurement precision of about twice the percentage difference between the predicted strength under the two models, and a five sigma discovery confidence would require the measurement to be made with 1%-2% precision (with somewhat less precision being tolerable at higher energy scales).
The same high energy running without a logarithmic scale and an inverse function plotted looks like this in the range where it has been experimentally measured:

In a version of SUSY where supersymmetric particles are very heavy (in tens of TeV mass range, for example), however, the discrepancies in the running of the strong force coupling constant between the Standard Model and SUSY crop up sufficiently to be distinguished only at significantly higher energy scales than those predicted for the MSSM version of SUSY.

The paper linked above doesn't discuss the latest measurements of the running of the strong force coupling constant, however. 

So far, the running of the strong force coupling constant is indistinguishable from the Standard Model prediction in all currently available data that I have seen, while monitoring new experimental results regarding this matter fairly closely since my last comprehensive review of it four and a half years ago. Of course, as always, I welcome comments reporting any new data that I have missed regarding this issue.