# How does the exchange of pions result in the strong force?

I understand that the residual strong force is a result of an exchange of pions. But I fail to understand how this exchange results in a force that holds nuclei together! May this query please be answered?

• Do you understand how the electromagnetic force is mediated by the exchange of (virtual) photons? – PM 2Ring Nov 5 '18 at 4:35
• Your question is three-in-one: (1) Why does particle exchange cause a force? (2) Why is the force attractive? (3) Why pions? Which issue are you really after? – Bert Barrois Nov 6 '18 at 14:18

It does not "result",it models the strong force

That short-range nucleon-nucleon interaction can be considered to be a residual color force extending outside the boundary of the proton or neutron. That strong interaction was modeled by Yukawa as involving an exchange of pions, and indeed the pion range calculation was helpful in developing our understanding of the strong force.

Once quantum electrodynamics had produced the picture of the electromagnetic force as a process of exchanging photons, the question of whether or not the other forces were also exchange forces was a natural one. In 1935, Hideki Yukawa reasoned that the electromagnetic force was infinite in range because the exchange particle was massless. He proposed that the short range strong force came about from the exchange of a massive particle which he called a meson. By observing that the effective range of the nuclear force was on the order of a fermi, a mass for the exchange particle could be predicted using the uncertainty principle. The predicted particle mass was about 100 MeV. It did not receive immediate attention since no one knew of a particle which fit that description.

.....

In 1947, Lattes, Muirhead, Occhialini and Powell conducted a high altitude experiment, flying photographic emulsions at 3000 meters. These emulsions revealed the pion, which met all the requirements of the Yukawa particle.

We now know that the pion is a meson, a composite particle, and the current view is that the strong interaction is an interaction between quarks, but the Yukawa theory stimulated a major advance in the understanding of the strong interaction.

So the model followed the successful electromagnetic models and proposed the pion was the carrier of the strong interaction, and was confirmed.

Lattice QCD is the tool to study strong forces and fairly recently this has been done for the Yukawa exchanges:

However, due to high complexities of the dynamics governing the quarks, the quantum Chromodynamics (QCD), it has been extremely difficult to study the origin of the strong nuclear force from QCD.

Very recently, Dr. N. Ishii (Univ. of Tsukuba), Dr. S. Aoki (Univ. of Tsukuba) and Dr. T. Hatsuda (Univ. of Tokyo) have succeeded, for the first time, in unraveling the nature of the nuclear force on the basis of lattice gauge theory originally formulated by Dr. K. Wilson (Nobel Prize Laureate, Physics, 1982). By carrying out massive numerical simulations using the IBM supercomputer "BlueGene/L" in High Energy Accelerator Research Organization (KEK) in Japan, they could prove not only the validity of the Yukawa’s meson theory from QCD but also the existence of a strong repulsive core of the nuclear force at short distance

For the feynman diagram that describes nucleon nucleon scattering using the yukawa potential see this.

How Does The Exchange Of Pions Result In The Strong Force?

In the same way that when one calculates the Feynman diagrams for the electromagnetic exchanges, the Coulomb force appears, calculating the Feynman diagrams for pion exchange gives the strong nuclear force.

Force is dp/dt , the change in momentum, the Feynman diagrams organize this momentum transfer from the initial particles to the final, and this after the integration of the implied integrals, results in the force experienced in the interaction.

• This looks rather outdated to me. Pion exchange probably did originate as a model, but within QCD it can almost certainly be obtained as an effective picture in the low-energy limit. (That requires an expert to confirm, though.) And, in addition, it doesn't answer the core of the question, namely how a pion exchange results in a force. – Emilio Pisanty Nov 5 '18 at 7:10
• @EmilioPisanty I gave a link of lattice QCD deriving the pion exchange. Of course it is outdated, as is the vector meson dominance model. QCD changed the way one models strong interactions. This does not mean that the one complex particle exchange is not usedul in nuclear phenomenology, it is a good approximation for certain calculations imo. – anna v Nov 5 '18 at 7:35
• The "why* a force is answered by the feynman diagram link. It is the same as why the photon exchange results in a force for electromagnetic interactions. It is contributions of feynman diagrams, positive or negative dp/dt – anna v Nov 5 '18 at 7:37

Saying that the residual strong force between nucleons is due to pion exchange is an oversimplification. Pion exchange may give a good approximation to the longest-range part of the force, but not the shorter-range part. We can get a better model by including the exchange of other more-massive mesons, and even that is still only an approximation.

More importantly for your question, the words "pion exchange" refer to one way of arranging the calculation, much like the words "carrying the $$1$$" refer to a step in the usual manual procedure for adding two integers: \begin{align*} \begin{matrix} \phantom{+}\scriptsize{1}\phantom{0} \\ \phantom{+}25 \\ +17 \\ \hline \phantom{+}42 \end{matrix} \end{align*} "Carrying the $$1$$" is important for getting the right answer when the calculation is done this way, but this isn't the only way to do the calculation, and it's not a good way to think about the meaning of addition. The type of calculation to which the words "pion exchange" refer is a type of manual calculation that is mostly useful for describing scattering experiments, where the nucleons are close to each other only for a brief instant. To understand the force between nucleons, a different computational approach is more appropriate, one that is not accurately described by the words "pion exchange".

This can be illustrated using classical field theory instead of quantum field theory. A pion is a spin-$$0$$ particle, so if we're going to construct a model of the pion-mediated force in quantum field theory, we should use a scalar field. If we're going to consider the analogous classical field theory, we should again use a scalar field. An interaction mediated by a classical scalar field leads to an attractive force, even though it does not involve the exchange of any kind of particle. (The classical field theory doesn't have any particles!) The calculation that demonstrates this is analogous to the calculation we would use in classical electrodynamics to determine the sign of the force between two electrostatically charged objects. In the scalar-mediated case (the classical-field version of the pion-mediated case), the force between like "charges" turns out to be attractive. The reason for this sign-difference between the electromagnetic-mediated and scalar-mediated forces is outlined in another post.

By the way, we could express the classical-field calculation in terms of Feynman diagrams, like we often do in quantum field theory. In classical field theory, we only get tree diagrams, which are Feynman diagrams with no loops. If we look at the (tree) diagram that represents the classical calculation just described, it does indeed consist of a single "pion" (scalar-field) line connecting the two nucleons, so we could describe the force as being due to "pion exchange" — but that would be just as unenlightening as describing the meaning of addition using phrases like "carrying the $$1$$".

The math is harder in quantum field theory, but the conclusion is similar: the force mediated by a scalar field between "charges" with the same sign is attractive. In particular, the pion-mediated force between two protons is attractive, and so is the pion-mediated force between two neutrons. As far as the strong force is concerned, protons and neutrons have the same "charge", so the pion-mediated force between protons and neutrons is also attractive. When I say "pion-mediated" here, I am not thinking of the exchange of pion particles; I am thinking of the field. Quantum field theory is formulated in terms of fields, not particles; so when questions like this come up, it can be helpful to return to the original formulation.