Modern physicists are used to a perturbative way to solve or understand problems in modern physics. In particular, since the invention of the powerful Feynman diagram technique by Richard Feynman, particle physicists are so fond of this perturbation tool and can seldom talk about physics without showing some Feynman diagrams.

However, there indeed exist some fundamental physical processes that can not be described by Feynman diagrams. These processes are typically called nonperturbative or topological transitions that have been studied since the discovery of “instanton” about half a century ago.

Unfortunately, perturbation theory is planted in the minds of a lot of particle physicists so firmly that they could not think in other possible topological ways. This has to be part of the reasons why some editors and reviewers have been so easy to dismiss my works. It may also be causing other physicists jumping on and off the bandwagon of my theory.

The earliest example of a topological process is called “instanton” first studied by Gerald ‘t Hooft in 1970s. It originates from the non-trivial vacuum structure of the Standard Model of particle physics. “Instanton” describes the nonperturbative tunneling effect between the topologically inequivalent θ-vacua in a Yang-Mills quantum field theory where each vacuum configuration is labeled by an integer index N_{cs}.

In early 1980s, Klinkhamer and Manton first proposed another kind of topological transitions, as they named it, “sphaleron” as a saddle point gauge field solution under the electroweak gauge theory. In standard model, this process is related to a topological transition involving nine quarks and three leptons from each of the three generations of elementary particles. Such a transition rate could be very high at high temperatures like about 100 GeV of the electroweak phase transition. This work has stimulated a lot of studies for its baryon-number (B-) violation property to solve the baryon asymmetry problem in the universe around the energy scale of electroweak phase transition.

Instanton is more like for the zero-temperature transitions (ΔN_{cs}=1) between θ-vacua that are extremely suppressed while Sphaleron describes the middle-point unstable solution (N_{cs }= half integer) at the energy barrier of two vacuum configurations. A similar sphaleron solution was calculated by Klinkhamer et al. later for SU(3) gauge theory as well. These sphaleron processes are discussed at finite temperatures, typically close to the height of the energy barrier.

In my mirror matter model, similar ideas are applied to staged quark condensation and corresponding spontaneous symmetry breaking for both mirror and chiral symmetries. A similar topological transition, coined “quarkiton”, is proposed due to such global anomalies for each of the quark condensation processes. A quarkiton transition involves three quarks of the same flavor (but with three different colors) and three leptons within the same generation. It works like follows: the three quarks and three leptons are excited in one vacuum configuration (labeled with N_{cs}), then turn into a quarkiton at the energy barrier of N_{cs}+1/2, and finally decay into their anti-counterparts in the adjacent vacuum configuration (labeled with N_{cs}+1).

The early proposed electroweak sphaleron coincides with the top-quarkiton at the same energy scale of electroweak phase transition (about 100 GeV). But the strange-quarkiton process occurs at much lower energy scale (100-200 MeV) during strange quark condensation and therefore it has a very fast rate. At the same time, it provides the necessary B-violation for baryon asymmetry where strange quarks can be quickly converted into their anti-partners, and vice versa.

As a matter of fact, these instanton-like (instanton, sphaleron, and quarkiton) processes are not the only type of known topological transitions. They are similar, in the sense that they all have to overcome a space-like energy barrier. But experimentally confirmed neutrino oscillations show another type of topological processes. Such neutrino oscillations exist due to the mass differences between different neutrino species, or in a more figurative way, they have to overcome a time-like barrier.

There is no creditable way to obtain neutrino oscillations from perturbation theory, i.e., from an underlying perturbative interaction with Feynman diagrams. Instead, neutrino oscillations are naturally treated in a topological way, i.e., using the plain quantum mechanics. The reason is because the global flavor/generation symmetry is broken leading to such oscillations. It is intriguing to note that global symmetry anomalies and topological processes are closely related.

Similarly in our mirror matter model, particle-mirror particle oscillations become possible when the mirror symmetry is spontaneously broken. To be exact, we should observe neutral hadron oscillations under the new mirror matter model. Similar to neutrino oscillations, these oscillations have to be topological as well since there is no other interaction between the particles of ordinary and mirror sectors other than gravity.

The two most important processes are n-n’ and K^{0}-K^{0}‘ oscillations that are critical for understanding the origin of dark/mirror matter and matter-antimatter imbalance in the universe, respectively. Unitarity means probability is conserved in quantum theory. If some of the particle mixing strength is lost topologically, the convectional 3×3 CKM matrix for perturbation theory can not be unitary any more. The current data have already shown evidence of it. However, no matter how I pointed this out, some physicists just choose to believe otherwise.

Unfortunately and fortunately, nature chooses to present her most beautiful scenes in topological ways. To unveil it, we have to think topologically.