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Hideki Yukawa: an anniversary

Today, several mathematicians and physicists would celebrate their birthday or deathday. (Some cosmologists are still confused why people don't celebrate their deathdays too often: such an asymmetry shamefully breaks the politically correct equivalence between the different arrows of time! Well, it indeed does: the breaking comes from the so-called "logical arrow of time".)

Marin Mersenne was born in 1588, Joseph Liouville died in 1882, Hermann von Helmholtz died in 1894. But let us look at this guy.

Hideki Yukawa was born in Tokyo on January 23th, 1907 and died in Kyoto on September 8th, 1981. Just like the death is the time reversal of the birth, Kyo-To is the time reversal of To-Kyo, so it makes sense in this case.

When he was 26, he was hired as an assistant professor in Osaka which was a great choice because two years later, in 1935, he published his theory of mesons. The pion was observed in 1947 and Yukawa received his Nobel prize in 1949: that was the first Japanese Nobel prize. He also predicted the K-capture, i.e. the absorption of a low-lying, "n=0" electron by the nucleus of a complicated atom.

Let us return to the Yukawa theory of the strong force and discuss its philosophical implications and lessons.

The Yukawa force and its lessons

People knew that there were protons and neutrons in the nuclei. An urgent question was: What was the force that was holding them together? The protons - "like charges" - should be repelled by the electrostatic force and the nuclei should explode. There had to be something stronger and something attractive. But it only acted on short distances because we don't observe a new long-range force between the protons that would exceed their electric repulsion.

Yukawa was able to see what it meant that the new force should be a short-range interaction. If electromagnetism were a good analogy, one thing had to be modified: the messenger particle had to become massive. The photon had to be replaced by a pion.

The electrostatic potential around a charged particle should satisfy "Δ phi(r) = delta function", which is solved by "phi(r) = C/r". Because the Klein-Gordon equation was known, Yukawa was able to guess that the relevant equation for the strong force would have to include an extra mass term in the operator:

(Δ + m2) phi(x,y,z) = Q delta3(x,y,z).
Surprisingly, the solution to this tougher equation in 3 spatial dimensions is simple: the "1/r" is multiplied by another exponentially decreasing factor, "exp(-m.r)". Their product, "exp(-m.r)/r", is called the Yukawa potential.

The simplicity comes from the fact that the three-dimensional Laplacian can be written as "1/r.d^2/dr^2.r" in this order (the things on the right side act first), plus the angular part, and the "r" at the end cancels against the "1/r" in the potential. The new exponentially dropping factor makes the range of the force short.

Also, the force had to attract "like charges", like two protons, instead of repelling them. Yukawa realized that a spin-one messenger, e.g. the photon, repels "like charges". But even spins - such as a spin-zero scalar particle or the spin-two graviton - attract "like charges" (or two positive masses). "Knowing" in advance that the only consistent spin-two messenger has to produce gravity (which satisfies the equivalence principle), he postulated his new scalar particle responsible for the attractive force between protons and neutrons. It was later identified with the pion.

In terms of Feynman diagrams, the interaction vertex has one scalar external legs and two proton (or two neutron) legs. This cubic vertex - with one spin-zero legs and two spin-1/2 legs - is called the Yukawa interaction. However, in the 21st century, we almost exclusively use this term for three elementary particles, according to the state-of-the-art effective theories. In their optics, neither pions nor protons or neutrons are elementary. The Yukawa interactions are discussed in 35 articles on this blog.

The pions were extended to an SU(2) triplet - pi(plus), pi(zero), pi(minus) - that was appropriate to attract the members of the isospin SU(2) doublet, the proton and the neutron. The charged pions were also observed and they allow vertices with one proton and one neutron. But I want to return to the question whether these particles were elementary.

Well, they were not. Pions and nucleons were learned to be made out of quarks and gluons. But that doesn't really invalidate the logic behind Yukawa's precious guesses. In a more modern and detailed treatment, the pions are the pseudo-Nambu-Goldstone bosons. The approximate SU(2) symmetry is broken and any broken symmetry generator has to create a new spin-zero particle species. That's true even for effective theories.

The usefulness of this way of looking at the phenomena depends on the hierarchy between various numbers. For example, the difference between the up-quark and down-quark bare masses, something that breaks the SU(2) symmetry, is around 4 MeV and is much smaller than the 150 MeV scale that is characteristic for the strong force.

Whenever some numbers of the "same kind" are so much smaller than others, a good physicist must be able to figure out how the physics simplifies when he completely neglects the smaller number. The smaller number itself may be later added as a perturbation and its effects may be systematically calculated. But a good physicist simply "can't deny" that "4 MeV / 150 MeV" is close to zero and something important follows from this fact. If a physicist (or a layman) is disgusted by these "dirty methods" or by the very perturbation techniques, he almost certainly ends up knowing nothing about the real world because almost all of our knowledge about the world depends on such successive approximations.

Also, the theory including the pions is nonlinear: the pions are scalars that parameterize a curved manifold, essentially a quotient of two non-Abelian Lie groups. Because the interactions in this model are non-polynomial, you shouldn't be shocked that the model is non-renormalizable. In fact, it inevitably breaks at energy scales that are still pretty close to the QCD scale.

Today, we know that this very fact means that the pions couldn't have been the final word. If you go just a little bit higher along the energy scale, you inevitably find new particles or forces - or something "morally equivalent to it" which we call "new physics" - that clarifies all the divergences from the non-renormalizable theory and picks the right, privileged values of the divergent integrals that appear in loop calculations involving the non-linear sigma model, the non-renormalizable theory with the pions.

This kind of progress "by scale" is well understood and standard today. We understand that certain "effective theories" can be still morally correct at low energies. And we kind of understand what about their validity can be trusted or extrapolated and what cannot be trusted or extrapolated. This intuition of the "Renormalization Group" is almost certainly universally valid up to the fundamental Planck scale, the maximum energy scale where the intuition can be used in the ordinary, field-theoretical edition.

When you summarize all these things, Yukawa was "morally right" about all these things, even though he couldn't guess that the particles had a substructure. His insights were important because he forced the people to consider the effect of a modified spin; and a nonzero mass of the messenger particles.

Without Yukawa's ideas, many people would think that the spin-one massless photon can be the only intermediate particle in Nature. They wouldn't understand the other options. Good theoretical physicists always try to generalize any notion in all the "equally consistent" and "equally symmetric" ways they can imagine - different masses, spins, and other properties are the simplest ways to a generalization - and see what happens under different circumstances. They always try to "sketch a map" of all similar ideas.

This is a point that most of the laymen - and lousy physicists - don't understand. They don't understand how a physicist can get an idea about a "whole map of ideas" which includes many concepts that "have not yet been measured". But that's exactly what is possible in physics. If you validate a theory or a calculational framework for one mass or one value of the spin, you may be pretty much certain that you know how to calculate these things even for different masses and spins, even if you can't observe any system that has these different quantities.

In the case of the strong force, people could observe these new phenomena because nuclear physics is pretty accessible. But it's important to realize that in theoretical physics in general, and in quantum gravity in particular, we have to deal with many new hypothetical particle species that have not yet been observed. But that doesn't mean that we can't calculate their properties and/or decide whether they're compatible with our experience or not!

And that's the memo.

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