This led to the question about weak boson and Higgs masses, which had remained poorly understood p-adically: success story continued. The secret of the success is that p-adic arithmetics combined with some very mild physical assumptions is extremely powerful constraint and leads to predictions having 1 per cent accuracy. These calculations are contained by the previous blog post but due to their importance I decided to post them separately.

** Nucleon mass**

This model could also allow to understand how the old-fashioned Gell-Mann quark model with constituent quarks having masses of order m_{p}/3 about 310 MeV much larger than the current quark masses of u and d quark masses of order 10 MeV.

- I have proposed that the current quark + color flux tube would correspond to constituent quark with the mass of color flux tube giving the dominating contribution in the case of u and quarks. If the sea quarks at the ends of the flux tubes are light as perturbative QCD suggests, the color magnetic energy of the flux tube would give the dominating contribution.

- One can indeed understand why the Gell-Mann quark model predicts the masses of baryons so well using p-adic mass calculations. What is special in p-adic calculations it is mass squared, which is additive as essentially the eigenvalue of scaling generator of super-conformal algebra denoted by L
_{0}.

m

^{2}_{p}= ∑ m^{2}_{p,n}

This due to the fact that energy is replaced by mass squared, which is Lorentz invariant quantity and conformal charge. Mass squared contributions with different p-adic primes cannot be added and must be mapped to their real counterparts first. On the real side is masses rather than mass squared, which are additive.

- Baryon mass receives contributions from valence quarks and from flux tubes. Flux tubes have same p-adic prime characterizing hadron but quarks have different p-adic prime so that the total flux tube contribution m
^{2}(tube)_{p}mapped by canonical identification to m_{R}(tubes)= (m^{2}_{R}(tubes))^{1/2}and analogous valence quark contributions to mass add up.

m

_{B}= m_{R}(tube)+∑_{q}m_{R}(valence,q).

The map m

^{2}_{p}→ m^{2}_{R}is by canonical identification defined as

x

_{p}= ∑_{n}x_{n}p^{n}→ x_{R}= ∑ x_{n}p^{-n}

mapping p-adic numbers in continuous manner to reals.

- Valence quark contribution is very small for baryons containing only u and d quarks but for baryons containing strange quarks it is roughly 100 MeV per strange quark. If the dominating constant contribution from flux tubes adds with the contribution of valence quarks one obtains Gell-Mann formula.

- Flux tube contribution can be assumed to be independent of flux tube in the first approximation. Its scale is determined by the Mersenne prime M
_{k}= 2^{k}-1, k=107, characterizing hadronic space-time sheets (flux tubes). Electron corresponds to Mersenne prime M_{127}and the mass scales are therefore related by factor 2^{(127-107)/2}≈ 2^{10}: scaling of electron mass m_{e,127}= .5 MeV gives mass m_{e,107}≈ .5 GeV, the mass electron had if it would correspond to hadronic p-adic length scale.

p-Adic mass calculations give for the electron mass the expression

m

_{e}≈ [1/(k_{e}+X)^{1/2}] 2^{-127/2}m(CP_{2}) .

k

_{e}=5 corresponds to the lowest order contribution. X<1 corresponds to the higher order contributions.

- By additivity of mass squared for flux tubes one has m
^{2}(tubes)= 3m^{2}(tube,p) and m_{R}(tubes)=3^{1/2}m(tube,R): one has factor 3^{1/2}rather than 3. Irrespective whether m_{R}(tubes) can be calculated from p-adic thermodynamics or not, it has general form m^{2}(tube,p)= kp in the lowest order - higher orders are very small contribute to m^{2}_{R}at most 1/p. k is a small integer so that even one cannot calculate the its precise value one has only few integers from which to choose.

The real mass from flux tubes is given by

m

_{R}= (3k_{p}/M_{107})^{1/2}× m(CP_{2}) =(3k_{p}/5)^{1/2}× m(e,107).

For k

_{p}= 6 (for electron one has k_{e}=5) one has m_{R}(tubes)= 949 MeV to be compared with proton mass m_{p}= 938 MeV. The prediction is too large by 1 per cent.

- Besides being by 1 per cent too large the mass would leave no room for valence quark contributions, which are about 1 per cent too (see this). There error would be naturally due to the fact that the formula for electron mass is approximate since higher order contributions have been neglected. Taking tis into account means replacing k
_{e}^{1/2}=5^{1/2}with (5+X)^{1/2}, X<1, in the formula for m_{R}. This implies the replacement m_{e,107}→ (5/(5+X)^{1/2}m_{e,107}. The correct mass consistent with valence quark contribution is obtained for X=.2. The model would therefore fix also the precise value of m(CP_{2}) and CP_{2}radius.

**What about the masses of Higgs and weak bosons?**

p-Adic mass calculations give excellent predictions for the fermion masses but the situation for weak boson masses is less clear although it seems that the elementary fermion contribution to p-adic mass squared should be sum of mass squared for fermion and antifermion forming the building bricks of gauge bosons. For W the mass should be smaller as it indeed is since neutrino contribution to mass squared is expected to be smaller. Besides this there can be also flux tube contribution and a priori it is not clear which contribution dominates. Assume in the following that fermion contributions dominate over the flux tube contribution in the mass squared: this is the case if second order contributions are p-adically O(p^{2}).

Just for fun one can ask how strong conclusions p-adic arithmetics allows to draw about W and Z masses m_{W}=80.4 GeV and m_{Z}= 91.2 GeV. The mass ratio m_{W}/m_{Z} allows group theoretical interpretation. The standard model mass formulas in terms of vacuum expectation v=246.22 GeV of Higgs read as m_{Z}= (g^{2}+(g')^{2})^{1/2}v/2 and m_{W}= gv/2= cos(θ_{w})m_{Z}, cos(θ_{W})= g/(g^{2}+(g')^{2})^{1/2}.

- A natural guess is that Higgs expectation v=246.22 GeV corresponds to a fundamental mass scale. The simplest guess for v would be as electron mass k
_{e}^{1/2}m_{127}, k_{e}=5, in the p-adic scale M_{89}assigned to weak bosons: this would give v= 2^{19}× m_{e}≈ 262.1 GeV: the error is 6 per cent. For k_{e}=4 one would obtain v= 2^{19}× (4/5)^{1/2}m_{e}≈ 234.5 GeV: the error is now 5 per cent.

For k

_{e}=1 the mass scale would correspond to the lower bound m_{min}=117.1 GeV considerably higher than Z mass. Higgs mass is consistent with this bound. k_{h}=1 is the only possible identification and the second order contribution to mass squared in m_{h}^{2}∝ k_{h}+X_{h}must explain the discrepancy. This gives X_{h}= (m_{h}/m_{min})^{2}-1≈ .141,

Higgs mass can be understood but gauge boson masses are a real problem. Could the integer characterizing the p-adic prime of W and Z be smaller than k=89 just as k(π)=111=k(p)-4 is smaller than kp?

- Could one understand cos(θ
_{w}) = m_{W}/m_{Z}≈ .8923 as a ratio (k_{W}/k_{Z})^{1/2}obtained using first orderp-adic mass formulas for m_{W}and m_{Z}characterizing the masses in the lowest order by integer k? For k_{W}=4 and k_{Z}=5 one would obtain cos(θ_{W})= (k_{W}/k_{Z}^{1/2}= .8944..: the error is .1 per cent. For k_{Z}=89 one would however have m_{Z}=v=m_{e,89}, which is quite too high. k=86 would give m_{Z}= 92.7 GeV: the error is 1.6 per cent. For m_{e}∝ (5+X_{e})^{1/2}, X_{e}≈ .2 deduced from proton mass, the mass is scaled down by (5/(5+X_{e})^{1/2}giving 90.0 GeV, which is smaller than 91.2 GeV: the mass is two large by 2 per cent. Higher order corrections via X_{Z}=.05 give a correct mass.

k=86 is however not consistent with the octave rule so that one must k

_{Z}=k_{W}=85 with (k_{W},k_{Z})=(8,10). This strongly suggests that p-adic mass squared is sum of two identical contributions labelled by k_{W}=4 and k_{Z}=5: this is what one indeed expects from p-adic thermodynamics and the representation of gauge bosons as fermion-antifermion bound states. Recall that also for hadrons proton and baryonic space-time sheet correspond to M_{107}and pion to k(π)=k(p)-4=111.

- There can be also corrections characterized by different p-adic prime: electromagnetic binding energy between fermion and anti-fermion forming Z boson could be such a correction and would reduce Z mass and therefore increase Weinberg angle since W boson does not receive this correction. Higher order corrections to m
_{W}and m_{Z}however replace the expression of Weinberg angle with cos(θ_{W})= (k_{W}+X_{W}/(k_{Z}+X_{Z})^{1/2}and allow to obtain correct Weinberg angle. Note that canonical identification allows this if the second order correction is of form rp^{2}/s, s small integer.

See the article Two anomalies of hadron physics from TGD perspective or the chapter New Physics predicted by TGD: Part I.

For a summary of earlier postings see Latest progress in TGD.

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