Today I took a more precise look at the baryonic masses. It the case of scalar mesons quarks give the dominating contribution to the meson mass. This is not true for spin 1/2 baryons and the dominating contribution must have some other origin. The identification of this contribution has remained a challenge for years.

A realization of a simple numerical co-incidence related to the p-adic mass squared unit led to an identification of this contribution in terms of states created by purely bosonic generators of super-canonical algebra and having as a space-time correlate CP_{2} type vacuum extremals topologically condensed at k=107 space-time sheet (or having this space-time sheet as field body). Proton and neutron masses are predicted with .5 per cent accuracy and Δ-N mass splitting with .2 per cent accuracy. A further outcome is a possible solution to the spin puzzle of proton.

** 1. Does k=107 hadronic space-time sheet give the large contribution to baryon mass?**

In the sigma model for baryons the dominating contribution to the mass of baryon results as a vacuum expectation value of scalar field and mesons are analogous to Goldstone bosons whose masses are basically due to the masses of light quarks.

This would suggest that k=107 gluonic/hadronic space-time sheet gives a large contribution to the mass squared of baryon. p-Adic thermodynamics allows to expect that the contribution to the mass squared is in good approximation of form

Δm^{2}= nm^{2}(107),

where m^{2}(107) is the minimum possible p-adic mass mass squared and n a positive integer. One has m(107)=2^{10}m(127)= 2^{10}m_{e}5^{1/2}=233.55 MeV for Y_{e}=0 favored by the top quark mass.

- n=11 predicts (m(n),m(p))=(944.5, 939.3) MeV: the actual masses are (m(n),m(p)=(939.6,938.3) MeV. Coulombic repulsion between u quarks could reduce the p-n difference to a realistic value.
- λ-n mass splitting would be 184.7 MeV for k(s)=111 to be compared with the real difference which is 176.0 MeV. Note however that color magnetic spin-spin splitting requires that the ground state mass squared is larger than 11m
_{0}^{2}(107).

**2. What is responsible for the large ground state mass of the baryon?**

The observations made above do not leave much room for alternative models. The basic problem is the identification of the large contribution to the mass squared coming from the hadronic space-time sheet with k=107. This contribution could have the energy of color fields as a space-time correlate.

- The assignment of the energy to the vacuum expectation value of sigma boson does not look very promising since the very of existence sigma boson is questionable and it does not relate naturally to classical color gauge fields. More generally, since no gauge symmetry breaking is involved, the counterpart of Higgs mechanism as a development of a coherent state of scalar bosons does not look like a plausible idea.
- One can however consider the possibility of Bose-Einstein condensate or of a more general many-particle state of massive bosons possibly carrying color quantum numbers. A many-boson state of exotic bosons at k=107 space-time sheet having net mass squared
m

^{2}=nm_{0}^{2}(107), n=∑_{i}n_{i}could explain the baryonic ground state mass. Note that the possible values of n

_{i}are predicted by p-adic thermodynamics with T_{p}=1.

**3. Glueballs cannot be in question**

Glueballs (see this and this) define the first candidate for the exotic boson in question. There are however several objections against this idea.

- QCD predicts that lightest glue-balls consisting of two gluons have J
^{PC}= 0^{++}and 2^{++}and have mass about 1650 MeV. If one takes QCD seriously, one must exclude this option. One can also argue that light glue balls should have been observed long ago and wonder why their Bose-Einstein condensate is not associated with mesons. - There are also theoretical objections in TGD framework.
- Can one really apply p-adic thermodynamics to the bound states of gluons? Even if this is possible, can one assume the p-adic temperature T
_{p}=1 for them if it is quite generally T_{p}=1/26 for gauge bosons consisting of fermion-antifermion pairs (see this). - Baryons are fermions and one can argue that they must correspond to single space-time sheet rather than a pair of positive and negative energy space-time sheets required by the glueball Bose-Einstein condensate realized as wormhole contacts connecting these space-time sheets.

- Can one really apply p-adic thermodynamics to the bound states of gluons? Even if this is possible, can one assume the p-adic temperature T

** 4. Do exotic colored bosons give rise to the ground state mass of baryon?**

The objections listed above lead to an identification of bosons responsible for the ground state mass, which looks much more promising.

- TGD predicts exotic bosons, which can be regarded as super-conformal partners of fermions created by the purely bosonic part of super-canonical algebra, whose generators belong to the representations of the color group and 3-D rotation group but have vanishing electro-weak quantum numbers. Their spin is analogous to orbital angular momentum whereas the spin of ordinary gauge bosons reduces to fermionic spin. Thus an additional bonus is a possible solution to the spin puzzle of proton.
- Exotic bosons are single-sheeted structures meaning that they correspond to a single wormhole throat associated with a CP
_{2}type vacuum extremal and would thus be absent in the meson sector as required. T_{p}=1 would characterize these bosons by super-conformal symmetry. The only contribution to the mass would come from the genus and g=0 state would be massless so that these bosons cannot condense on the ground state unless they suffer topological mixing with higher genera and become massive in this manner. g=1 glueball would have mass squared 9m_{0}^{2}(k) which is smaller than 11m_{0}^{2}. For a ground state containing two g=1 exotic bosons, one would have ground state mass squared 18m_{0}^{2}corresponding to (m(n),m(p))=(1160.8,1155.6) MeV. Electromagnetic Coulomb interaction energy can reduce the p-n mass splitting to a realistic value. - Color magnetic spin-spin splitting for baryons gives a test for this hypothesis. The splitting of the conformal weight is by group theoretic arguments of the same general form as that of color magnetic energy and given by (m
^{2}(N),m^{2}(Δ))= (18m_{0}^{2}-X, 18m_{0}^{2}+X) in absence of topological mixing. n=11 for nucleon mass implies X=7 and m(Δ) =5m_{0}(107)= 1338 MeV to be compared with the actual mass m(Δ)= 1232 MeV. The prediction is too large by about 8.6 per cent. If one allows topological mixing one can have m^{2}=8m_{0}^{2}instead of 9m_{0}^{2}. This gives m(Δ)=1240 MeV so that the error is only .6 per cent. The mass of topologically mixed exotic boson would be 660.6 MeV and equals m_{0}^{2}(104). Amusingly k=104 happens to corresponds to the inverse of α_{K}for gauge bosons. - In the simplest situation a two-particle state of these exotic bosons could be responsible for the ground state mass of baryon. Also the baryonic spin puzzle caused by the fact that quarks give only a small contribution to the spin of baryons, could find a natural solution since these bosons could give to the spin of baryon an angular momentum like contribution having nothing to do with the angular momentum of quarks.
- The large value of the Kähler coupling strength α
_{K}=1/4 would characterize the hadronic space-time sheet as opposed to α_{K}=1/104 assignable to the gauge boson space-time sheets. This would make the color gauge coupling characterizing their interactions strong. This would be a precise articulation for what the generation of the hadronic space-time sheet in the phase transition to a non-perturbative phase of QCD really means. - The identification would also lead to a physical interpretation of super(-conformal) symmetries. It must be emphasized the super-canonical generators do not create ordinary fermions so that ordinary gauge bosons need not have super-conformal partners. One can of course imagine that also ordinary gauge bosons could have super-partners obtained by assuming that one wormhole throat (or both of them) is purely bosonic. If both wormhole throats are purely bosonic Higgs mechanism would leave the state essentially massless unless p-adic thermal stability allows T
_{p}=1. Color confinement could be responsible for the stability. For super-partners having fermion number Higgs mechanism would make this kind of state massive unless the quantum numbers are those of a right handed neutrino. - The importance of the result is that it becomes possible to derive general mass formulas also for the baryons of scaled up copies of QCD possibly associated with various Mersenne primes and Gaussian Mersennes. In particular, the mass formulas for "electro-baryons" and "muon-baryons" can be deduced (see this)

For more details about p-adic mass calculations of elementary particle masses see the chapter p-Adic mass calculations: elementary particle masses of the book "p-Adic Length Scale Hierarchy and Dark Matter Hierarchy". The chapter p-Adic mass calculations: hadron masses describes the model for hadronic masses. The chapter p-Adic mass calculations: New Physics explains the new view about Kähler coupling strength.