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The masses of hadrons, weak bosons, and Higgs in p-adic mass calculations

The TGD based model for the additional features of spin puzzle of proton led to a model of baryon in which flux the quark and antiquark at the ends of flux tubes connecting valence quarks to a triangular structure replace sea quarks, which represent a rather ugly feature of perturbative QCD. This led also to a model allowing to understand the successful predictions of baryon masses and magnetic moments in the old Gell-Mann quark model and also how constituent quark masses and current quark masses relate. It also turned out to possible to understand the masses baryons and mesons in p-adic framework.

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 mp/3 about 310 MeV much larger than the current quark masses of u and d quark masses of order 10 MeV.




  1. 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.



  2. 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 L0.

    m2p= ∑ m2p,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.



  3. 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 m2(tube)p mapped by canonical identification to mR(tubes)= (m2R(tubes))1/2 and analogous valence quark contributions to mass add up.

    mB= mR(tube)+∑q mR(valence,q).

    The map m2p→ m2R is by canonical identification defined as

    xp= ∑nxnpn→ xR= ∑ xnp-n

    mapping p-adic numbers in continuous manner to reals.



  4. 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.




A detailed estimate for nucleon mass using p-adic mass calculations shows the power of p-adic arithmetics even in the case that one cannot perform a complete calculation.


  1. Flux tube contribution can be assumed to be independent of flux tube in the first approximation. Its scale is determined by the Mersenne prime Mk= 2k-1, k=107, characterizing hadronic space-time sheets (flux tubes). Electron corresponds to Mersenne prime M127 and the mass scales are therefore related by factor 2(127-107)/2≈ 210: scaling of electron mass me,127= .5 MeV gives mass me,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

    me≈ [1/(ke+X)1/2] 2-127/2m(CP2) .

    ke=5 corresponds to the lowest order contribution. X<1 corresponds to the higher order contributions.



  2. By additivity of mass squared for flux tubes one has m2(tubes)= 3m2(tube,p) and mR(tubes)=31/2m(tube,R): one has factor 31/2 rather than 3. Irrespective whether mR(tubes) can be calculated from p-adic thermodynamics or not, it has general form m2(tube,p)= kp in the lowest order - higher orders are very small contribute to m2R 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

    mR= (3kp/M107)1/2× m(CP2) =(3kp/5)1/2× m(e,107).

    For kp= 6 (for electron one has ke=5) one has mR(tubes)= 949 MeV to be compared with proton mass mp= 938 MeV. The prediction is too large by 1 per cent.





  3. 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 ke1/2=51/2 with (5+X)1/2, X<1, in the formula for mR. This implies the replacement me,107→ (5/(5+X)1/2me,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(CP2) and CP2 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(p2).

Just for fun one can ask how strong conclusions p-adic arithmetics allows to draw about W and Z masses mW=80.4 GeV and mZ= 91.2 GeV. The mass ratio mW/mZ allows group theoretical interpretation. The standard model mass formulas in terms of vacuum expectation v=246.22 GeV of Higgs read as mZ= (g2+(g')2)1/2v/2 and mW= gv/2= cos(θw)mZ, cos(θW)= g/(g2+(g')2)1/2.


  1. 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 ke1/2 m127, ke=5, in the p-adic scale M89 assigned to weak bosons: this would give v= 219× me≈ 262.1 GeV: the error is 6 per cent. For ke=4 one would obtain v= 219× (4/5)1/2me≈ 234.5 GeV: the error is now 5 per cent.

    For ke=1 the mass scale would correspond to the lower bound mmin=117.1 GeV considerably higher than Z mass. Higgs mass is consistent with this bound. kh=1 is the only possible identification and the second order contribution to mass squared in mh2∝ kh+Xh must explain the discrepancy. This gives Xh= (mh/mmin)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?





  2. Could one understand cos(θw) = mW/mZ≈ .8923 as a ratio (kW/kZ)1/2 obtained using first orderp-adic mass formulas for mW and mZ characterizing the masses in the lowest order by integer k? For kW=4 and kZ=5 one would obtain cos(θW)= (kW/kZ1/2= .8944..: the error is .1 per cent. For kZ=89 one would however have mZ=v=me,89, which is quite too high. k=86 would give mZ= 92.7 GeV: the error is 1.6 per cent. For me∝ (5+Xe)1/2, Xe≈ .2 deduced from proton mass, the mass is scaled down by (5/(5+Xe)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 XZ=.05 give a correct mass.

    k=86 is however not consistent with the octave rule so that one must kZ=kW=85 with (kW,kZ)=(8,10). This strongly suggests that p-adic mass squared is sum of two identical contributions labelled by kW=4 and kZ=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 M107 and pion to k(π)=k(p)-4=111.



  3. 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 mW and mZ however replace the expression of Weinberg angle with cos(θW)= (kW+XW/(kZ+XZ)1/2 and allow to obtain correct Weinberg angle. Note that canonical identification allows this if the second order correction is of form rp2/s, s small integer.


To sum up, it is fair to say that p-adic mass calculations allow now to understand both elementary particle masses and hadron masses. One cannot calculate everything but p-adic arithmetics with mild empirical constraints fix the masses with 1 per cent accuracy.

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.

Articles and other material related to TGD.

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