Thanks for Wilhelmus de Wilde for a link to a popular article in Schitechdaily telling about completely unexpected finding by by a team led by professors Tacemichi Okui and Kohsaku Tobioka. The decay of longlived kaon KL suggests the existence of new longlived particle with quantum numbers of axion - or equivalently pion. The finding is published in Physical Review Letters (see this). Standard model cannot explain this kind of particle.
A rough estimate for mass is not far from pion mass. There exists earlier evidence that pion has mass spectrum. Could an excitation of pion be involved?
This is actually not new. The experimental claim of Tatischeff and Tomasi-Gustafsson (see this ) is that pion is accompanied by pion like states organized on Regge trajectory and having mass 60, 80, 100, 140, 181, 198, 215, 227.5, and 235 MeV means that besides pion also other pion like states should be there. Similar satellites have been observed for nucleons with ground state mass 934 MeV: the masses of the satellites are 1004, 1044, 1094 MeV. Also the signal cross sections for Higgs to gamma pairs at LHC suggest the existence of several pion and spion like states, and this was the reason why I decided to again the search for data about this kind of states. Their possible interpretation in TGD framework is discussed in here.
One explanation could be that the states correspond to "infrared Regge trajectories" of pion related to the structure of its magnetic body. Genuine Regge trajectories would have slope of about GeV. IR trajectories could be associated with the electromagnetic body and ordinary Regge trajectories with the color magnetic body. One can also consider p-adically scaled down variant of color interactions.
It is interesting to look the situation quantitatively.
- It is clear that the masses in question do not fit to a single Regge trajectory. One can however restrict the consideration to Regge trajectory M2=M02 + nT(π), where T(π) denotes string tension. Since the masses obey approximately linear formula one can assume linear approximation Δ M2= 2MΔM at pion mass M1= m(π)= .140 GeV and consider the mass squared difference for pion and its predecessor with M0=.100 GeV so that one has Δ M=.040 GeV.
One obtains Δ M2= M12-M02= T(π). This would give for the string tension T(π)= 0.96×10-2 TH≃ .96× 10-2 GeV2, where TH≈ GeV2 is hadronic string tension assignable to color interactions.
- What about the value of M02? In string models it tends to be negative but one can assume that the values of mass squared for physical states are negative. Also in TGD the value is negative in p-adic mass calculations. One must require that several values for pion mass below m(π) are possible. The formula m(π)2 =M02+nT(π) gives formula M02=
m(π)2-nT(π). For n(π)=2, which looks rather reasonable guess, one has M02=.04 GeV2, which corresponds to M0= 20 MeV.
- In early models hadronic string tension was taken to be 1 GeV. Much smaller values for the string tension smaller by a factor or order x×10-2 GeV2, x in the range 2-11.1 for mesons and in the range 2.2-4.55 for baryons are however suggested by the study of hadronic spectrum (see this). Intriguingly, the lower bounds is twice the above estimate T(π)≃ .01 GeV2 obtained above. Does this mean that the p-adic prime involved is about 2 times smaller or is this factor due to a numerical factor 1/2 related to the difference between N-S and Ramond type representations of Super-Virasoro algebra.
- The reason for the confusion about string tension could be simple: besides the string tension 1 GeV assignable to color flux tubes there are string tensions assignable to possible scaled down color flux tubes and possible elecromagnetic and even weak flux tubes. Several p-adic length scales could be associated coming in powers of 2 by p-adic length scales hypothesis are involved.
The basic objection against the model is that the harmonic oscillator model for nuclear works surprisingly well. The justification for this model is that one can reasonably well describe nucleus as motion of nucleons in an effective nuclear potential, which in linearization becomes harmonic. Nucleons themselves have no mutual interactions in this approximation.
Could nuclear string model allow to understand harmonic oscillator model of nuclei as an approximation?
- It is best to start from the problems of the harmonic oscillator model. The first problem is that the description of nuclear binding energies is poorly understood. For instance, nuclear binding energies have scale measured in MeVs. The scale is much smaller than energy scale of hadronic strong interactions for which pion mass is a natural scale. Rather remarkably, the ratio of the scales is roughly the ratio of fine structure constant to color coupling strength. Could one imagine that electromagnetic interactions somehow determine the energy scale of nuclear binding energies and excitations?
- As noticed, also nucleons are reported to have IR Regge trajectories. The first guess is that the trajectories have same string tension as in the case of pion. TGD suggests a model of nuclei as three nucleons connected by color flux tubes characterized by hadronic string tension TH≈ 1 GeV2. Besides color flux tubes hadrons are expected to have also electromagnetic and perhaps also weak flux tubes with a smaller value of string tension. Em flux tubes should give a contribution to the energy, which is of the order of Coulomb energy of nucleon about α/Lc(p) ≈ 7.5 MeV. Intriguingly, this is of same order of magnitude as nuclear binding energy: could IR Regge trajectories correspond to em interaction so that the spectrum of nuclear binding energies and excitation energies would be determined by electromagnetic interactions?
- If the value of p-adic prime p ≃ 2k corresponds to k=113 assumed to characterize nuclei in nuclear string model, hadronic string tension would be scaled down by factor 2107-113= 1/64 to TH/64, which corresponds to a mass of 125 MeV, which is somewhat larger than the value about 96 MeV obtained from the above estimate. For Δ M2 ≃ 2MΔ M = T(π) this gives ΔM ≈ 7.8 MeV for Δ n=1, which corresponds to the maximal nuclear binding energy per nucleon. This string tension is naturally assignable to em flux tubes assignable nuclei as 3 -quark states. Color flux tubes would be responsible for the hadronic string tension TH.
Remark: Flux tubes carry all classical gauge fields, which are induced from the spinor connection of CP2 but it seems that one can assign to given flux tube quanta of particular interaction.
- In the case of baryons one would have 3 color flux tubes and and 3 em flux tubes. For large mass excitations one would have in linear approximation for M2 harmonic oscillator spectrum! Could linearization of mass squared formula replace linearization of effective potential function leading to harmonic oscillator model? The dimension D=3 for the nuclear harmonic oscillators would correspond to the fact that nucleons consist of 3 quarks. The free nucleon approximation would have simple justification: in good approximation one can treat the nucleons of nuclear strings as independent particles!
- Could the nuclear binding energy per nucleon correspond to a reduction of the value of n for the IR Regge trajectory of free nucleon? The mass squared formula for IR trajectory would be M2= M02(N)+ nT(π). This mechanism requires that the one has M0≤ m(N) so that one has n>0 for nucleons. For Δ n=-1 one has Δ M =Δ nT(π)/2m(N) ≃ 7.8 MeV.
- Binding energy per nucleon is below 3 MeV for nuclei lighter than 4He and has tendency to increase up to Fe. For the most abundant stable isotope of Fe with (Z,A)=(26,56) it is 8.78 MeV. For heavier nuclei neutron number N increases and binding energy per nucleon starts to decrease.
- For D one must have Δ n=0 and p-n pairing would be somehow responsible for the binding. For T the total binding energy is 8.478 MeV and could involve Δ n =-1 for one nucleon. 3He has total binding energy 7.715 MeV and also now one nucleon could have Δ n=-1. 4He has binding energy per nucleon equal to 7.07420 eV. This suggests that p-n pairing causes the reduction Δ n=-1 for all nucleons in 4He units proposed to be building bricks of nuclei.
For nuclei with odd Z and nuclei there are would be also deuteron sub-unit present and also |A-Z| unpaired neutrons. This would reduce the binding energy. The prediction is that for nuclei with N=Z with even Z the binding energy exceeds that for 4He. For heavier nuclei this can happen also for odd Z and also for N different from Z.
- The pairing of to D subunits should be rise to binding energy 2.223 MeV per deuteron unit. Why the value is so small? Could deuteron unit correspond to a smaller string tension: perhaps corresponding to k=9 as the ratio of 4He and D binding energies per nucleon would suggest. The ratio of the maximal binding energy 8.7892 MeV per nucleon to deuteron binding energy is rather precisely 8, which supports the interpretation.
- What causes the increase of the binding energy per nucleon eB up to Fe? Attractive potential energy is not an attractive interpretation in TGD framework. Some repulsive interaction should reduce the binding energy per nucleon for lighter nuclei than Fe from the value 8.8 MeV. The increase from 4He to Fe is about 1 MeV. Why does this repulsive contribution decrease up to Fe? Does it start to increase after that or is the presence of surplus neutrons the reason for the reduction? Or are both mechanisms involved?
The IR Regge trajectories considered are not the only ones as already the findings of Tatischeff and Tomasi-Gustafsson suggest and there might be trajectories with smaller string tension. The value of k=9 with string tension T(π)/8 assignable to D, which corresponds to a eB of about 1 MeV and this is roughly the total variation of the eB from 4He to Fe. Could both k=6 and k=9 flux tubes be present for given nucleon. Could the reduction of n for k=9 flux tubes take place also for 4He units as nuclei become heavier. What happens in nuclei heavier than Fe? Could the increase of neutron surplus reduce eB?
For a summary of earlier postings see Latest progress in TGD.