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The Balmer-Like Formula for Mass Distribution of Elementary Particle Resonances

Elementary particle resonances have been systematically analysed using all available experimental data. We have come to the conclusion that the resonance decay prodact momenta and masses of resonances are to be quantized. The Balmer- like formula for masses of elementary particle resonances was obtained. These observations allow us to formulate a strategy of experimental searches for new resonances and systematize the known one.

The merits of contemporary quark models (the Standard Model) in describing properties of elementary particles are impressive. However, it is well known that [1] the only uncertain aspect of the Standard Model is the mechanism that gives elementary particles their masses. In the simplest version of the model, these masses depend on constants that specify the strength of the interaction of various elementary particles with a new kind of field but these constants are just free parameters of the theory. It seems that the mass distribution of elementary particle resonances is the fundamental problem of modern theory until we have a final theory of force and matter, particles and fields. Therefore, the systematic analysis of existing experimental data is desirable to establish simple phenomenological rules for the mass distribution of resonances.
One of the most remarkable groundstones for foundation of the quantum theory was the Balmer formula for the spectra of a hydrogen atom. It is our purpose here to demonstrate that the Balmer-like formula for the mass distribution of elementary particle resonances can be obtained from a systematic analysis of all available experimental data.
Some of the resonances have a dominant decay channel, and we suggest that the momentum of this channel should manifest itself in properties of decays through other channels. For example, it is known that the pion π ± decays through the muon and neutrino with a probability about unity and asymptotic momentum P 1 = 29.7918M eV /c. We are unable to explain this property of the pion; we only suggest that it is fundamental for comprehension of the structure of some resonances. To establish some common properties of the mass distribution of elementary particle resonances, we check the usefulness of a commensurable principle [2] of decay asymptotic momenta. In other words, we check the following ratio: where P 1 = 29.7918 Mev/c. The masses of resonances are calculated by the formula where m a and m b are the masses of decay products of the resonance to be considered. The results and corresponding experimental data [3] are given in Table I. Note that we present here the results of calculations for a few resonances only due to restriction of place. A more complete analysis has been made in [4] containing a few hundred resonances. Our complete analysis containing almost all data from [3] can be sent by requirement to everybody. Of course, there arises a question: what is the role of resonance decay channels with a very small probability? We know from nuclear physics (see, for example, [5]) that basic vectors with small (large) weights in the main states of nuclei become the dominant (small) ones in highly excited states. The same phenomenon is expected to be present in the physics of resonances. Let us consider as basic the following decay channel π 0 → µ ± e ∓ with P 1 =26.1299 MeV/c and fraction ≈ 10 −8 %. The results of calculations are presented in Table II. Tables I, II contain rich information, and it is possible to make many fundamental conclusions based on them: We think that the results contained in the tables convincingly demonstrate the empirical fact of commensurability of resonance decay product momenta within the accuracy of existing experimental data. In other words, resonance decay product momenta are quantized. It is clear from the Tables that commensurability of momenta does not depend on the type of interaction between resonance decay products, quantum numbers of resonances and type of particles. Moreover, commensurability of momenta is justified for all considered resonances. It seems to be a universal property of resonances, and many periodic structures are just governed by this property. A main question arises here: is this commensurability precise or approximate? We do not know the answer but think that it is as precise as the one in solids and crystals. If so, the common structure of elementary particles must be analogous to that of solids and crystals. An excellent possibility for prediction of new resonances and verification of masses of existing ones arises in any case.
The systematics of resonance decay chains and decay products [3] in turn suggests a simple idea about the structure of elementary particle resonances: all heavy particles decay into lighter ones in a tree-like manner, and the final decay products are electrons, photons, neutrinos, and protons. Resonance decay is of a self-similar cascade character and, moreover, the relevant decay channel momenta are commensurable.
Obviously, the accuracy of resonance masses to be predicted depends on the accuracy of decay product masses used in calculations. Therefore, our predictions of resonance masses are to be considered as preliminary ones. We ask readers to send information about more accurate contemporary experimental resonance mass data. For this we will be indebted. The obtained results coincided with existing experimental data within the accuracy of decay product masses.
We have established that experimental values of masses of P P and P P resonances [6] coincide within the accuracy of experimental data. This remarkable observation was highlighted in our earlier works [7,8]. But the systematic analysis of experimental data brings us to the more general conclusion. The masses of resonances coincide within the accuracy of experimental data decaying through the same channels: particle-particle, particle-antiparticle and antiparticle-antiparticle. This observation is valid for any resonances independent of the type of particles, interaction and so on. If so, good perspectives for a combined systematic analysis of this type of resonances are opened.
So, we come to the conclusion that resonances have an extremely rich structure. They are complex wave systems. All the motions of subsystems (constituents) are commensurable according to the laws of wave mechanics. Thus, there is observed self-organization of constituents of matter between each other and also with the whole system irrespective of the type of interaction between the constituents. Moreover, the constituents of matter (clusters) are similar to each other and to the whole system. Actually, the resonances decay in sequence in a tree-like manner forward for lighter ones so that the branches of the tree are similar to each other and to the whole tree.
Our systematic analysis of resonance decay products shows that the corresponding motions of the constituents do not exist independently. The motions of constituents in one resonance are consistent with those of constituents in the others. Each resonance itself plays a three-fold role: has a complex structure, is included into other resonances, and takes part in the exchange between components of a subsystem keeping up the unity of the structure. Thus, a harmonic unity of motions of all particles is established.
To understand how an unstable particle (resonance) can communicate with some open channels we have found that it is necessary to develop an analogy between the behavior of unstable particles and resonant cavities such as organ pipes and electromagnetic resonators. Chew, Gell-Mann, and Rosenfeld were the first who paid attention to this analogy in 1964 [9]. They discussed in detail the possibility of representing elementary particle resonances as resonant cavities in full analogy with our approach. But the 1964 paper [9] remained unnoticed.
As the final conclusion we can say that we have established the Balmer-like formula for masses of elementary particle resonances which is supported by a systematic analysis of experimental data.
The authors would like to thank the Russian Foundation for Basic Research for financial support of our work.