Higgs Particle 10 years (September 2022)
Ten years ago, Higgs particle was "discovered", and a few days ago, Physical Society of Japan sent a notice that a lecture may be given on line in connection to the history of Higgs particle. This is a bit surprising, and, therefore, I started to write this article mainly for young people.
At that time, CERN claimed that they discovered one event of new particle. For this news, I had a doubt as to whether they were sane as physicists. Since then, ten years has already passed. During this period of ten years, they upgraded the machine so that they could confirm their result, but in vain.
Why should the most advanced physics scheme fall into confusion? I believe that the essential point of this problem can be found in the theory of renormalization model. This renormalization scheme was invented by Feynman, Tomonaga and others. They calculated the vertex corrections of electron at the third order of perturbation theory in QED. In this case, they found the logarithmic divergence of their calculation even though the vertex corrections should be physical observables. Therefore, they invented the renormalization scheme in which the logarithmic divergence should be renormalized into the free wave function of electron. In this way, they could reproduce the correct magnetic moment of electron.
For the renormalization scheme that can predict the right (g-2) of electron, people believed that this success must be due to the fact that the QED is a gauge theory. Even though there is no physical foundation why the gauge theory should be very fundamental, people accepted this renormalization scheme as an established theory of physics. Therefore, it is believed that any theoretical model must be constructed from the gauge theory, and thus Weinberg, Salam and others proposed a theoretical model of weak interaction physics in terms of non-abelian gauge field theory.
However, it is, by now, clear that the reason why there appeared a logarithmic divergence in the calculation of vertex corrections must be simply due to the incorrectness of the Feynman propagator. The defect of the Feynman propagator was commonly noticed by experts of quantum field theory since the propagator cannot satisfy the important constraint arising from the polarization sum of photon. Indeed, this point is well explained in the field theory textbooks of Sakurai or Bjorken-Drell, for example.
However, this Feynman propagator can correctly describe the electron-electron scattering T-matrix and thus the experiment as a result. Therefore, people believed that the use of Feynman propagator should be somehow justified. Unfortunately, however, the success of the Feynman propagator when it is employed for describing the electron-electron scattering is accidental. Indeed, we can easily prove that the right description of electron-electron scattering T-matrix using the Feynman propagator must be simply due to the fact that electrons in this scattering process are on the mass shell, and only in this situation, the scattering T-matrix with the Feynman propagator becomes identical to the correct scattering T-matrix. If we consider these important conditions properly in the evaluation of vertex corrections, then we should not find any divergences in the loop diagrams, and therefore, there is no need of renormalization.
In addition, there is a very important discovery in the evaluation of vertex corrections of leptons by Z0 bosons. Namely, the vertex corrections of leptons due to the weak vector boson (Z0 boson) can be calculated, and it is found that the value of this vertex corrections has a finite number. The weak vector bosons are somewhat similar to photon, apart from the finite mass. However, the weak vector bosons do not have any gauge invariances, but still they do not give rise to any divergences in the vertex corrections. This is indeed quite surprising. This strongly suggests that the gauge invariance may not play any important role for the evaluation of vertex corrections.
If these points were properly understood half century ago, then the present confusion in the modern physics might have been well avoided. The discussion of muon (g-2) can be found in
[ Vertex Corrections of Z0 Boson ].
In QED, it is found that the gauge invariance cannot play any important role for the evaluation of physical observables. However, this is quite different from the non-abelian gauge field theory. In the case of non-abelian gauge field theory, the color charge of the constituents should be gauge dependent and, therefore, they are not physically observed. Indeed, QCD is a non-abelian gauge field theory with SU(3) color and thus these constituents of quark and gluon should not be observed at all in nature as free particles. This is called "confinement", and indeed all the experiments indicate that free quarks are not found at all. In this case of non-abelian gauge field theory, the gauge invariance should play a very important role in physics, contrary to the U(1) gauge field. If these points were realized by Weinberg, Salam and others, they would not have proposed their models based on non-abelian gauge field theory.
At this point, I should like to make some extra-comment on the renormalizability of non-abelian gauge field theory. In the middle of 1960's, Faddeev-Popov claimed that they proved that the non-abelian gauge field theory should be renormalizable. This paper was based on the path integral formulation even though it was not clear as to whether the path integral was connected to any physical observables. By now it is proved that the non-abelian gauge field theory has no basis of perturbation theory since the free states of non-abelian gauge field theory are gauge dependent. I believe that their claim must have left some "negative legacy" in the history of science, just like the general relativity which left some serious "negative legacy" in physics.
[Appendix 1] : Higgs Particle Is Not Required From Nature !
For young and/or budding researchers, I should explain and stress that the existence of Higgs particle has nothing to do with nature, but it is simply a prediction within theoretical frameworks. In fact, most of the experimental observations in the weak interaction processes can be well described by the Conserved Vector Current (CVC) model which is an excellent theory. However, this CVC theory has an intrinsic problem which is connected to the quadratic divergences in the second order perturbation theory. Therefore, in order to avoid this defect, people considered that there must exist very heavy vector bosons that mediate weak processes involving fermions. In fact, in 1970's, people confirmed that the mass of weak vector bosons must be heavier than 30 GeV. In addition, CERN discovered the neutral current following the suggestion of SU(2) currents in the standard model. Afterwards, in 1980's, CERN discovered the weak vector bosons of W and Z0. The discoveries of heavy weak bosons together with the neutral current should be very important, and they are the great achievements which are made by CERN. In this sense, almost all of the experiments in the weak interaction processes can be understood by the CVC theory, and there is no room left for the Higgs particle.
In this case, why did people believe that the Higgs particle was necessary for weak interactions? This is related to the basic problem (defect) of the structure of standard model. In this model, they made use of the non-abelian gauge fields in which the gauge particle must be massless by its construction. Therefore, this starting model should not have been appropriate for the application to the weak interaction physics.
However, people have employed a magic trick by which they gave a heavy mass to the gauge particle. The secret of this magic trick was a Higgs particle. Unfortunately, however, this trick has nothing to do with nature, and therefore, the existence of Higgs particle is scientifically meaningless. The problem of this standard model is discussed and explained in the textbook of [ Fundamental Problems in Quantum Field Theory ( Bentham Publishers, 2013) ].
In order to ensure fairness, I should like to make one extra-comment on the standard model. In this model, they introduced SU(2) for the weak vector bosons, and this is quite important. The idea of SU(2) suggests the existence of neutral current while the CVC theory only considered the charged currents. Indeed, people confirmed the existence of the neutral current, and this is a very important progress in science history of weak interactions.
As is clear by now, if they did not make use of gauge theory, then their model would have become a real standard model. In addition, these heavy vector bosons do not give rise to any divergences in the evaluation of vertex corrections, and thus the new standard model should be quite a sane scheme in quantum field theory.
[Appendix 2] : Scalar Particle Should Not Exist, But Gravity Is Scalar Field !
The following comments should be added so as to help budding researchers understand the theory of gravity. Half century ago, most of the experts in quantum field theory may well have had some doubts on the existence of Higgs particle. This is because the Higgs particle is defined as a scalar particle which cannot exist in nature as a result of quantum field theoretical evaluation.
On the other hand, gravity is defined as a massless scalar field. But this field of gravity is never quantized due to the experimental as well as theoretical requirements. From the theoretical point, the gravity should not be quantized due to its consistency within the theoretical framework. In this sense, the gravity field is just the same as Coulomb field which is never quantized.
In this case, why should the gravity be a scalar field? This is simply because only the scalar field can always give rise to attractive forces. For example, gauge fields may give attractive as well as repulsive forces depending on the charges of particles involved in the interactions.
There should be two more important points for the gravitational field. The first point is concerned with the gravitational potential in the Dirac equation. Namely, the Dirac equation should include the gravitational potential, and this is absolutely necessary. This is simple as explained in the following. In order to obtain the Newton equation, we first make the FoldyーWouthuysen transformation which should give the non-relativistic Hamiltonian properly. From this Hamiltonian, we obtain the Newton equation with the use of Ehrenfest theorem. In this case, if the gravitational potential is not included in the Dirac equation, then the Newton equation does not have any gravity, and this is not acceptable at all. Therefore, the inclusion of the gravitational potential in the Dirac equation is absolutely necessary, and there is no other choice. Thus, it was the most important task to extend the quantum electrodynamics to the new quantum field theory that includes the gravity.
As the second point, there is a very important experimental evidence that the inertial mass agrees with the gravitational mass. In order to construct the gravity model, it is definitely necessary to consider that the model should satisfy this experimental fact. Here, the inertial mass is just the mass of a particle with its mass m. Then, what should be the gravitational mass ? For this, we should look at the gravitational potential U(r). This U(r) can be written as U(r) =-G(Mm/r), and the gravitational mass corresponds to the mass m in this potential. However, it is not so trivial that this mass m should be the same as the inertial mass m. Therefore, we should take into account this point properly in the new gravity model.
The new gravity model satisfies all the conditions mentioned above. If you want to learn this theoretical scheme in detail, please see the following text book [ Cosmology and Field Theory ] [ Fundamental Problems in Quantum Field Theory ( Bentham Publishers, 2013) ] .
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