Some of the most exquisitely precise experimental measurements in all of physics are the masses of the various hadrons.

Consider these examples: the proton mass is known to eleven significant digits. The neutron mass is known to eight. The charged pion mass is known to seven, and so is the neutral pion mass. The neutral D meson mass, the charged D meson mass and the Ds meson mass are known to six. The eta meson mass, the charged kaon mass, the neutral kaon mass, the neutral B meson mass and the charged B meson mass are known to five.

We think we know, in principal, at least, the exact terms of the entire Lagrangian of the Standard Model, and could fit it on a (big) t-shirt. The beta function of each of the Standard Model physical constants can be calculated, in principle, without any experimental input (and usually are calculated that way in practice).

In principle, the hadron masses are largely a function of the quark masses and the strong force coupling constant. The physical constants used to calculate the tweaks to those mass estimates from electromagnetic and weak force effects are also known very precisely (to eight significant digits or more). We can also exploit QCD sum rules, using combinations of existing measurements, including, but not limited to the hadron mass measurements.

But, the up and down quark masses are known to just one significant digit of accuracy, and the strong force coupling constant and the quark masses are known only to three or four significant digits. Also, while these values aren't as precise as we like, we can refine our techniques to reflect the narrow area of parameter space that is allowed by existing measurements of these parameters, in our efforts to determine the values of these physical constants more precisely.

So, why is it that we can't reverse engineer the exactly measured hadron masses, using equations that we know exactly, at least in principle, to get more precise values for the strong force coupling constant and the quark masses? Is this simply a matter of insufficient computational power devoted to the problem? If not, what are the stumbling blocks?

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    $\begingroup$ @CosmasZachos My impression is that we know how, at a brute force level, to do these calculations, but that there are so many components to it, even for say, a five loop computation, that each additional loop takes exponentially more calculation effort. I'm certainly not disparaging the hard work done so far. I'm instead wondering if there is a fairly straightforward route to progress on this front that has been underfunded because it isn't glamorous and doesn't have a big constituency, or if instead there is some other kind of barrier that I don't understand to this approach. $\endgroup$
    – ohwilleke
    Apr 23, 2019 at 19:49
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    $\begingroup$ No, perturbation theory will never capture firmly nonperturbative features, 5 or 37 loops. No, lattice simulations are not underfunded: if anything they are overfunded by misplaced faith in blind computing. Sinking more funds into such deprives science, an ideas enterprise, from diminishing support and harms it irreparably. $\endgroup$ Apr 23, 2019 at 21:04
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    $\begingroup$ I'm not an expert in this particular field, but I do note that many of the powerful resummation methods allowing you to obtain the strong coupling expansion using only a few terms of the weak coupling expansion were not known before the mid 1990s. These often quite simple mathematical methods were not practical in the days before CAS. Many researchers were educated before that time, they don't fully grasp the potential this has to offer. Even a big expert in the field who I talked to several years ago, was skeptical when I suggested applying his methods on a much grander scale. $\endgroup$ Apr 23, 2019 at 21:52
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    $\begingroup$ He said that you'll get a very large number of messy equations, there was no way to handle these properly. But in the years since, I've been able to get good results using such methods, albeit applied to the problems that I work on, not anything related to QCD. $\endgroup$ Apr 23, 2019 at 21:56
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    $\begingroup$ CAS = Computer Algebra System, e.g. Mathematica. $\endgroup$ Apr 23, 2019 at 22:53

1 Answer 1


Perhaps an analogy will help: we can measure the local acceleration of gravity $g$ with exquisite precision. But nobody knows enough about the detailed structure of the Earth to turn that into a precise value of Newton’s $G$.

Although lattice QCD computations have gone a long way, we still lack truly detailed models of the inner structure of hadrons. Those still have large uncertainties relative to what the OP wants to do.

  • $\begingroup$ Are the uncertainties basically in the parton distribution functions or are they elsewhere as well? $\endgroup$
    – ohwilleke
    Nov 20 at 1:29

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