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For a number of years now the Particle Data Group (pdg.lbl.gov) has listed the measured mass difference between the charged and neutral pion as being equal to 4.5936 (with an experimental uncertainty = .0005) MeV. While this mass-difference is attributed to the added electric component of the charged state, one theoretically can equivalently derive the mass of the neutral pion by adding an up quark mass m(u) to the charged pion mass (~139.57 MeV) and subsequently subtracting a down quark mass m(d).
But just as a quark's mass can't be individually measured directly, one can't in any way infer the down or up quark mass from this differential, though it again must exactly correspond to this measurable difference between the charged and neutral pion. Which somewhat naively might seem to preclude knowing the mass of either an up or down quark per se.
However, the mass differential basically remains an independent issue, as the up and down quark masses are respectively derived from two different fine-structured sets of equations But it follows that we're certainly free to give the reader a tangible value for, say, the down mass, where m(d) = 7.763258 MeV. While we could go on to just as directly give the up mass, we'll instead let you compute the up from the calculated d-u mass differential
m(d) - m(u) = 4.593453 MeV = (pi)± - (pi)o
[m(u+2d)o - m(2u+d)+] = {m(d+d)o - m(u+d)± = m(u+d)± - m(u+u)o}.
These equations and value allow me to share one of the strongest and first proofs discovered when I initially derived the whole spectrum of similarly precise quark masses - within a couple of weeks of solving the central problem of a fully dimensionless mapping of physics (see http://www.isnare.com/?aid=184377&ma=members&ca=Education). For comparing the measured PDG value of 4.5936 MeV with the derived d-u quark mass differential of 4.593453 MeV differs my a mere .000147 MeV: clearly well within the bounds of the .0005 MeV experimental uncertainty - an empirical proof of no small order. Likewise, inspection of the lower set of equivalences give further theoretical support to the uppermost equations - where the square-bracketed relation on the bottom left refers to the quark content of the neutron minus that of the proton, while the lowermost parenthetical bracket on the right represents the differences between the respective quark contents of the charged and neutral pi mesons per se - as pions mediate quark transformations between nucleon matter - being the, or at least amongst, the most important processes in physics.
Adding to this significance is the fact that though direct measurement of quarks is impossible, they clearly are hardly incalculable, but even more precisely so than the mass differential between pions themselves! And although perhaps more spectacular 'pudding proofs' employing much heavier particle states exist that are open to similar empirical and theoretic confirmation (see http://www.isnare.com/?aid=190170&ma=members&ca=Education), I still consider this the Best 'Proof in Pudding' or Theory. It follows that we accordingly try not to make a habit of giving away more unprecedented precise data beyond the two most important and fundamental (quark) masses in the universe. In any case, the above equations and values are proof positive of what can be called 'Kaluza-Klein Mumbers' aren't constrained by such obvious elusiveness; and positive proof, both from and that, standard theory and experiment is incapable of giving a precise prediction for either quark at the core of the basic baryon matter in a neutron and proton - lacking an explanation for material baryogenesis as well.
Sean Sheeter is an independent theorist, geometer, and author of 241-Mumbers: The Definitive Data for Fundamental Physics and Cosmology. This essay concerns the most important of the four examples of the Sample Data and Proofs at http://www.241mumbers.com/page2.html
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