Hideki Yukawa and the Meson: How a Force’s Range Predicted a New Particle

Gravity and electromagnetism reach across the universe. The force that holds protons and neutrons together in a nucleus barely reaches past its neighbours. In 1934 a young Japanese physicist, Hideki Yukawa, turned that difference into a prediction of a new particle. Our Hideki Yukawa sticker sums up his idea in four words, “Heavy Ball. Short Game.” This post expands our explainer What Holds the Nucleus Together?

What problem was Yukawa trying to solve?

The nucleus is packed with positively charged protons that repel each other electrically. Something stronger must hold them, but that something cannot have a long reach, or nuclei would pull on one another across ordinary distances. The 1949 Nobel presentation speech describes Yukawa’s approach: he “used the electromagnetic field as a model” and found that the field “could be modified so as to give forces which like the nuclear forces have a short range” (Nobel presentation speech 1949^[1]).

Why does a heavy carrier mean a short range?

In quantum field theory, the electromagnetic, weak and strong forces are carried by particles. A force carried by a massless particle, like the photon of electromagnetism, can reach arbitrarily far. Yukawa “discovered that there is a simple relation between the range of the forces and the mass of the corresponding particles” (Nobel presentation speech 1949^[1]). The relation is

λ = ħ/(mc)

λ is the range in metres, ħ = 1.054 571 817 × 10⁻³⁴ J·s is the reduced Planck constant, m is the carrier’s mass in kilograms and c = 299 792 458 m/s is the speed of light. One heuristic way to see it (a picture, not a derivation): creating a carrier of mass m borrows energy mc², and quantum mechanics only tolerates such a loan for a short time, so the carrier cannot travel far before it must be reabsorbed. The heavier the ball, the shorter the game.

The potential energy between two nucleons then falls off faster than the familiar 1/r of electricity:

V(r) ∝ −e^(−r/λ)/r

Here r is the distance between the two particles. For r much smaller than λ it looks like an ordinary 1/r force; beyond a few λ the exponential factor e^(−r/λ) kills it off.

How heavy did Yukawa predict the meson would be?

Yukawa ran the relation backwards. He “estimated the range from known experimental data and found that the new particles should be about 200 times heavier than the electrons” (Nobel presentation speech 1949^[1]). The Nobel Foundation dates the prediction to 1934 and his estimate of about 200 electron masses (NobelPrize.org^[2]). The theory was published in 1935 as “On the Interaction of Elementary Particles. I” in the Proceedings of the Physico-Mathematical Society of Japan, volume 17, pages 48–57 (Yukawa 1935^[3]; Nobel biography^[4]).

We can check his logic with today’s numbers. Using ħc = 197.326 980 4 MeV·fm, an exact value in the SI shown here to ten significant figures (NIST CODATA 2022^[5]), and the charged pion mass of 139.570 39 MeV/c² from the Particle Data Group (PDG 2024^[6]):

λ = 197.327 MeV·fm ÷ 139.570 MeV ≈ 1.41 fm

One femtometre (fm) is 10⁻¹⁵ m, about the size of a proton. The electron’s mass is 0.510 999 MeV/c² (PDG 2024^[7]), so the pion is about 273 electron masses: the same order as Yukawa’s “about 200”.

Why did the first “meson” turn out to be the wrong particle?

Yukawa suggested looking in cosmic rays, and the presentation speech says that “this evidence was given in 1937 by Anderson and Neddermeyer and other American physicists” (Nobel presentation speech 1949^[1]; Neddermeyer & Anderson 1937^[8]). The particle they found had roughly the right mass, about 207 electron masses (today’s muon, 105.658 MeV/c² (PDG 2024^[7])), but, as later work showed, it interacted far too weakly with nuclei to be the carrier of the nuclear force.

The resolution came about a decade later. Using the photographic method on cosmic-ray particles at high altitudes, Cecil Powell’s group, with César Lattes, Hugh Muirhead and Giuseppe Occhialini, reported in Nature in 1947 “evidence of mesons which, at the end of their range, produce secondary mesons” (Lattes et al. 1947^[9]). There were two particles. The speech puts it plainly: Powell and collaborators “found that there exist two kinds of mesons,” and “the heavier mesons, but not the lighter ones, have an interaction with the nucleons about as strong as Yukawa had postulated” (Nobel presentation speech 1949^[1]). The heavier one is what we now call the pion; the lighter one, the muon, is not a meson at all in modern terms; the Nobel Foundation describes the muon and the tau as the electron’s “two much heavier and short-lived relatives” (NobelPrize.org^[10]).

Which Nobel Prizes came from the meson?

Yukawa received the 1949 Nobel Prize in Physics, unshared, “for his prediction of the existence of mesons on the basis of theoretical work on nuclear forces” (NobelPrize.org^[2]). One year later Cecil Powell received the 1950 prize “for his development of the photographic method of studying nuclear processes and his discoveries regarding mesons made with this method” (NobelPrize.org^[11]).

Does the same rule explain the weak force?

Yes, and it is the clearest modern use of Yukawa’s formula. The weak force is carried by the W and Z bosons. With the W mass of 80.369 GeV/c² (PDG 2024^[12]), λ = 197.327 MeV·fm ÷ 80 369 MeV ≈ 0.0025 fm, roughly 580 times shorter than the pion’s range. That is why the weak force is so feeble at everyday distances; see our post Beta Decay Unveiled. The strong force between quarks is a different story again, governed by gluons, which we cover in Why Can’t Quarks Be Isolated?

FAQ

Is the pion the “real” carrier of the strong force?

Yukawa’s pion exchange describes the force between whole protons and neutrons. The underlying strong force between quarks is carried by gluons, a theory developed decades after Yukawa.

Why is it called a meson?

Its mass sits in the middle, between the electron and the proton; the speech notes the name “was not introduced until later” (Nobel presentation speech 1949^[1]).

Where can I find Yukawa on a sticker sheet?

He also appears on our Fine Print sticker sheet, and the Forces collection has the nucleus characters.

References

1. NobelPrize.org, Presentation speech by I. Waller, Nobel Prize in Physics 1949. https://www.nobelprize.org/prizes/physics/1949/ceremony-speech/
2. NobelPrize.org, Hideki Yukawa, Physics 1949, facts. https://www.nobelprize.org/prizes/physics/1949/yukawa/facts/
3. H. Yukawa (1935), “On the Interaction of Elementary Particles. I,” Proc. Phys.-Math. Soc. Japan 17, 48–57 (DOI 10.11429/ppmsj1919.17.0_48, J-STAGE). https://doi.org/10.11429/ppmsj1919.17.0_48
4. NobelPrize.org, Hideki Yukawa – Biographical. https://www.nobelprize.org/prizes/physics/1949/yukawa/biographical/
5. NIST CODATA 2022 recommended values: reduced Planck constant times c in MeV fm. https://physics.nist.gov/cgi-bin/cuu/Value?hbcmevf
6. S. Navas et al. (Particle Data Group) (2024), Review of Particle Physics, Phys. Rev. D 110, 030001: Meson summary table. https://pdg.lbl.gov/2024/tables/rpp2024-sum-mesons.pdf
7. S. Navas et al. (Particle Data Group) (2024), Review of Particle Physics, Phys. Rev. D 110, 030001: Lepton summary table. https://pdg.lbl.gov/2024/tables/rpp2024-sum-leptons.pdf
8. S. H. Neddermeyer & C. D. Anderson (1937), “Note on the Nature of Cosmic-Ray Particles,” Phys. Rev. 51, 884–886 (DOI 10.1103/PhysRev.51.884). https://journals.aps.org/pr/abstract/10.1103/PhysRev.51.884
9. C. M. G. Lattes, H. Muirhead, G. P. S. Occhialini & C. F. Powell (1947), “Processes Involving Charged Mesons,” Nature 159, 694–697 (DOI 10.1038/159694a0). https://www.nature.com/articles/159694a0
10. NobelPrize.org, Popular information: The Nobel Prize in Physics 2015 (“Metamorphosis in the particle world”). https://www.nobelprize.org/prizes/physics/2015/popular-information/
11. NobelPrize.org, The Nobel Prize in Physics 1950 (Cecil Powell). https://www.nobelprize.org/prizes/physics/1950/summary/
12. S. Navas et al. (Particle Data Group) (2024), Review of Particle Physics, Phys. Rev. D 110, 030001: Gauge and Higgs boson summary table. https://pdg.lbl.gov/2024/tables/rpp2024-sum-gauge-higgs-bosons.pdf

Written by Pixelated Physics. Every factual claim is linked to the source we checked; points that are still debated are labelled open.