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Ab initio calculation of the neutron-proton mass difference

Sz. Borsanyi,1 S. Dürr,1,2 Z. Fodor,1,2,3 C. Hoelbling,1 S. D. Katz,3,4 S. Krieg,1,2 L. Lellouch,5 T. Lippert,1,2 A. Portelli,5,6 K. K. Szabo,2 B.C. Toth1

Science 347 (2015) 1452

Supplementary materials

1 Department of Physics, University of Wuppertal, D-42119 Wuppertal, Germany
2 Jülich Supercomputing Centre, Forschungszentrum Jülich, D-52428 Jülich, Germany
3 Institute for Theoretical Physics, Eötvös University, H-1117 Budapest, Hungary
4 Lendület Lattice Gauge Theory Research Group, Magyar Tudományos Akadémia–Eötvös Loránd University, H-1117 Budapest, Hungary
5 CNRS, Aix-Marseille Université, Université de Toulon, CPT UMR 7332, F-13288, Marseille, France
6 School of Physics and Astronomy, University of Southampton, Southampton SO17 1BJ, UK
The existence and stability of atoms rely on the fact that neutrons are more massive than protons. The measured mass difference is only 0.14% of the average of the two masses. A slightly smaller or larger value would have led to a dramatically different universe. Here, we show that this difference results from the competition between electromagnetic and mass isospin breaking effects. We performed lattice quantum-chromodynamics and quantum-electrodynamics computations with four nondegenerate Wilson fermion flavors and computed the neutron-proton mass-splitting with an accuracy of 300 kilo–electron volts, which is greater than 0 by 5 standard deviations. We also determine the splittings in the Σ, Ξ, D, and Ξcc isospin multiplets, exceeding in some cases the precision of experimental measurements.

Mass splittings of hadrons (here particles composed of quarks and/or antiquarks) in channels that are stable under the strong and electromagnetic interactions. Both of these interactions are fully unquenched in our 1+1+1+1 flavor calculation. The horizontal lines are the experimental values, and the gray shaded regions represent the experimental error. Our results are shown by red dots with their uncertainties. The error bars are the squared sums of the statistical and systematic errors. The results for the neutron-proton, Σ--Σ+ and D+-D0 meson mass splittings are post-dictions, in the sense that their values are known experimentally with higher precision than from our calculation. On the other hand, our calculations yield results for the Ξ--Ξ0, and Ξcc++-Ξcc+, as well as the Coleman-Glashow difference (CG), which have either not been measured in experiment or are measured with less precision than obtained here. This feature is represented by a blue shaded region around the label.

Dependence of the neutron-proton mass difference on two combinations of fundamental parameters. On the horizontal axis, the fine structure constant, proportional to the square of the elementary unit of charge is varied from zero to twice its measured value. Along the vertical axis, it is the difference in mass between down and up quarks that is varied. The blue cross indicates the point corresponding to our universe. The black curves correspond to contours of constant neutron-proton mass difference. Each has a label indicating the value of the neutron-proton mass difference to which it corresponds. Any point in the blue region corresponds to a very different universe from ours, in which the proton could decay into a neutron through the weak interaction. Units are mega-electron-volts (MeV). The mass of a hydrogen atom is approximately 1000 MeV.


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