Investigating the sources of the weak force

The weak force is responsible for the β decay – the disintegration of the neutron into a proton, an electron and a (anti) neutrino. A recent paper of Eli Comay suggests a new way to explain the forces which create the weak interactions.

This post discusses this discovery, and it includes several equations and some terms which are understood by physicists only. Other readers may skip these equations and terms and still understand the context of this new finding.

A brief history

Just before the end of the 19th century physicists had detected radiation that is emitted from the atomic nucleus. Within few years three different kinds of nuclear radiation have been identified – α, β and γ. It has been found that the particles of the β radiation are electrons and the process also converts a neutron into a proton. It has also been recognized that the α and γ radiations emitted from a specific nucleus have a well-defined energy, whereas in the case of the β radiation, the energy of the electrons has a continuous distribution between zero and a certain maximum.

This evidence has raised the problem of energy conservation in β decay.

Just at the beginning of the quantum theory physicists have recognized the importance of spin of particles and the laws of finding the total spin of a system of several particles have been formulated. It turned out that the β decay violates these laws.

To solve these discrepancies, Wolfgang Pauli proposed in 1930 that alongside the electron, there is another particle which is emitted in a β decay. This particle must be very light, electrically neutral, and, like the electron, it has a spin-1/2. This particle is called neutrino. The first experimental detection of the neutrino was in 1956.

Experiments have shown that the neutrino interacts very weakly with matter. This interaction is so weak that the entire earth (and even much larger stars) is practically transparent to neutrinos.

Neutrino theories

Two kinds of theoretical ideas of the neutrino have been published during the 1930s. Enrico Fermi has proposed a weak interaction theory of four particles. His idea was quite successful and several years later it was extended by Gamow & Teller. This theory uses a new constant, denoted GF, which determines the intensity of the interaction. The role of GF  is analogous to that of the square e2 of the electric charge in electromagnetic processes.

The structure of GF indicates that weak interactions are inherently different from electrodynamics. Indeed, in a well-known system of units the electric charge is a pure number whereas the dimension of GF is [L2], where [L] denotes the dimension of length. A theoretical problem with the Fermi scheme is that it describes a “contact” interaction between the four particles which are involved in the process – neutron, proton, electron and neutrino. This idea is alien to a general physical requirement which says that interaction should be carried out by a mediating field.

At the same decade, theoretical physicists have proposed 3 different quantum equations for the neutrino: Dirac equation, Majorana equation and Weil equation.

The Dirac equation says that the neutrino is a “Dirac particle” which is a kind of an electron that has no electric charge.

The Majorana equation says that the neutrino and the antineutrino are identical.

The Weil equation says that the neutrino is a massless particle that travels at the speed of light.

The decision between these three alternatives is certainly a problem of weak interactions. Evidently, experimental information can help to identify the right equation.

Parity violation

The discovery of new particles has followed the development of accelerators and particle physics has been established as a specific discipline. It was found that weak interactions processes are not restricted to the β decay. In the 1950s an examination of two weak decays yielded a problem called the τ-θ puzzle.

This puzzle can be settled if weak processes do not conserve a kind of symmetry called parity. The parity symmetry can be loosely described as a symmetry between the left and the right directions. Soon after, an experiment that has been carried out in 1956 by a team headed by Wu has shown that weak interactions do not conserve parity.

Later in 1956, G. Sudarshan, who was at that time a graduate student of R. Marshak, analyzed the β decay data that was available at that time. His analysis has shown that this decay can be described by a scheme called V – A.

This outcome provides another experimental evidence for parity non-conservation in weak processes and a specific form of the laws of beta decay.

Less than two years later G. Sudarshan and R. Marshak and separately R. P. Feynman and M. Gell-Mann published papers describing a theoretical scheme for parity violation which uses the factor (1 ± γ5). (γ5 is a 4×4 matrix which is useful in physics). This scheme holds for the massless neutrino of the Weil equation.

Another theoretical progress appeared about ten years later – the electroweak theory. This theory is the sector of the Standard Model of particle physics that describes electrodynamics and weak interactions. The mathematical structure of the electroweak theory is quite complicated. One of its features is that the factor (1 ± γ5) is used by it and that it applies to a massless neutrino.

Experimental and theoretical problems with the electroweak theory

During the 1960s physicists measured the flux of neutrinos emitted from the sun. Only about one third of the expected number of neutrinos has been detected in measurements. A key element of the discrepancy was the assumption of a massless neutrino which is an element of the standard model.

The progress of experimental work has shown that 12 elementary particles can be organized in three generations. Each generation consists of two particles called leptons and two particles called quarks.

The weak interaction is the only interaction capable of changing the flavor of quarks (i.e., of changing one type of quark into another). Experiments show that a change of a quark flavor within the same generation is much easier than a change between different generations. Weak interaction also change flavor of leptons.

With respect to flavor, there are three kinds of neutrino, called electron neutrino, muon neutrino and tau neutrino.

At about the end of the millennium experiments have determined that the neutrino has mass. As a result, a free neutrino oscillates between the three possible neutrino flavors. This outcome resolves the problem of the neutrinos that come from the sun. The measurements detect only the electron neutrinos, which, due to the oscillations between flavors, make only one third of the total neutrinos.

Thus, the neutrino mass has resolved one problem. However, another result of this evidence is that the electroweak sector of the Standard Model faces a serious problem because it is based on a massless neutrino. Here, a massive neutrino is inconsistent with the electroweak’s factor (1 ± γ5), which is used for explaining parity violation in weak interactions. A proof of this claim can be found in section 2 here.

It turns out that contradictions exist not only in the match between the theory and the experimental data, but also in the mathematical structure of two electroweak particles, W and Z bosons. See section 2 of that article and references therein.

Looking for a correct weak interaction theory

This development puts forward the following problem: should the electroweak theory be amended or be replaced by another theory?

If we look on electrodynamics, which is considered as a very successful theory, we see that the theory describes electrically charged particles, electromagnetic field, and provides 3 formulae:

The fundamental expression describing the behavior of a free elementary charged particle (say, an electron) is:


The corresponding expression that describes the behavior of free electromagnetic fields is:


Eq. (1) yields the Dirac equation of a free electron and eq. (2) yields the Maxwell equations of free electromagnetic fields. The interaction between electric charges and electromagnetic fields is described by the following term of the Lagrangian density:


The entire electromagnetic theory can be mathematically developed from these three expressions.

Is it possible to have a similar structure of equations to describe the weak interactions? Evidently, weak interactions effects are quite different from those of electrodynamics. For example, the above-mentioned parity violation and flavor violation are inherently alien to electrodynamics. Therefore, one may be inclined to try constructing a weak interaction theory that is completely different from electrodynamics.

On the other hand, there is one weak interactions process that is analogous to electrodynamics. It is the elastic scattering of a neutrino on an electron, in which a neutrino hits an electron and the particles exchange energy and momentum. Thus, no new particles are created and no particle is destroyed.

On this basis, a reasonable first step in the construction of a weak interactions theory is to find the term that is analogous to the electromagnetic interaction term (3). Namely, explaining weak interactions processes where no new particles are created. Comay’s answer to this problem is here:


The meaning of these symbols is explained in his article.

Analysis of this equation shows that weak interaction is a sum of two terms, one is proportional to V and the other is proportional to A. It means that according to this equation weak interactions do not conserve parity. This outcome is consistent with the experimentally confirmed V – A property of the β decay.

The result of this discussion, is that this theory has two huge advantages over the electroweak theory:

(a) In the electroweak theory, the parity violation was a postulate of the theory, while in Comay’s theory the parity violation is a result of the fundamental formulas.

(b) The electroweak theory is built on the erroneous assumption that the neutrino doesn’t have mass. In accordance with experimental data, Comay’s theory assumes that neutrinos are massive Dirac particles.

Why this development was not obvious

After Eli Comay found the theoretical contradictions to the equations of the Z, W (and also the Higgs) bosons, he tried to construct a coherent weak interaction theory.

He decided to start with the simplest weak interaction effect of a neutrino-electron scattering. Then, he tried to find the interaction term of the Lagrangian density. This problem is analogous to that of the well known elastic electron-electron scattering in electrodynamics.

At this point, Eli decided to take the following experimentally related issues as elements that should be satisfied:

  1. The theory should be consistent with the Fermi constant and its dimension.
  2. The theory cannot take the same form as electrodynamics. A convincing argument for this statement is that the cross section of electron scattered from an electrically charged particle decreases with the increase of energy whereas in the case of a neutrino the cross section increases with the increase of energy.
  3. The theory should be in accordance with the parity violation property of weak interaction. Hence, it must be a linear combination of a vector and an axial vector. In particular, the data prove that the tensor operator is inadequate.

The task is to define two expressions, called Lagrangian density and Hamiltonian density, where the Hamiltonian density is derived from the Lagrangian density by a well-known procedure. There is a certain difference in the structure of the interaction term of the Lagrangian density and of its associated term in the Hamiltonian density. It turns out that in electrodynamics this difference does not make a significant change. In particular, a vector in the 3-dimensional space remains a vector of this kind. On the basis of this evidence Comay concluded that the parity violating combination of the V,A terms should be in the Lagrangian density and in the Hamiltonian density as well.

This was an incorrect conclusion! It took several years until Comay realized this matter. As a result, he has found that a tensor interaction term in the Lagrangian density yields a parity violating combination of the two V, A terms in the Hamiltonian density.

Comay’s wrong perception (before he found his mistake) is probably shared by the entire physical community.

The scientific paper that describes this discovery was published in Open Access Library Journal 2016, Volume 3, Issue 12.