The model for hadron masses revisited

The blog of Tommaso Dorigo contains two postings which served as a partial stimulus to reconsider the model of hadron masses. The first posting is The top quark mass measured from its production rate and tells about new high precision determination of top quark mass reducing its value to the most probale value 169.1 GeV in allowed interval 164.7-175.5 GeV. Second posting Rumsfeld hadrons tells about "crackpottish" finding that the mass of B_c meson is in an excellent approximation average of the mass of Ψ and Υ mesons. TGD based model for hadron masses allows to understand this finding.

1. Motivations

There were several motivations for looking again the p-adic mass calculations for quarks and hadrons.

1. If one takes seriously the prediction that p-adic temperature is T_p=1 for fermions and T_p=1/26 for gauge bosons as suggested by the considerations of blog posting (see also this), and accepts the picture about fermions as topologically condensed CP₂ type vacuum extremals with single light-like wormhole throat and gauge bosons and Higgs boson as wormhole contacts with two light-like wormhole throats and connecting space-time sheets with opposite time orientation and energy, one is led to the conclusion that although fermions can couple to Higgs, Higgs vacuum expectation value must vanish for fermions. One must check whether it is indeed possible to understand the fermion masses from p-adic thermodynamics without Higgs contribution. This turns out to be the case. This also means that the coupling of fermions to Higgs can be arbitrarily small, which could explain why Higgs has not been detected.

2. There has been some problems in understanding top quark mass in TGD framework. Depending on the selection of p-adic prime p≈ 2^k characterizing top quark the mass is too high or too low by about 15-20 per cent. This problem had a trivial resolution: it was due to a calculational error due to inclusion of only the topological contribution depending on the genus of partonic 2-surface. The positive surprise was that the maximal value for CP₂ mass corresponding to the vanishing of second order correction to electron mass and maximal value of the second order contribution to top mass predicts exactly the recent best value 169.1 GeV of top mass. This in turn allows to clean up uncertainties in the model of hadron masses.

2. The model for hadron masses

The basic assumptions in the model of hadron masses are following.

1. Quarks are characterized by two kinds of masses: current quark masses assignable to free quarks and constituent quark masses assignable to bound state quarks (see this). This can be understood if the integer k_q characterizing the p-adic length scale of quark is different for free quarks and bound quarks so that bound state quarks are much heavier than free quarks. A further generalization is that the value of k can depend on hadron. This leads to an elegant model explaining meson and baryon masses within few percent. The model becomes more precise from the fixing of the CP₂ mass scale from top mass (note that top quark is always free since toponium does not exist). This predicts several copies of various quarks and there is evidence for three copies of top corresponding to the values k_t=95,94,93. Also current quarks u and d can correspond to several values of k.

2. The lowest mass mesonic formula is extremely simple. If the quarks characterized by same p-adic prime, their conformal weights and thus mass squared are additive:

   m²_B = m²_q1+ m²_q2.

   If the p-adic primes labelling quarks are different masses are additive m_B = m_q1+ m_q2.

   This formula generalizes in an obvious manner to the case of baryons.

   Thus apart from effects like color magnetic spin-spin splitting describable p-adically for diagonal mesons and in terms of color magnetic interaction energy in case of nondiagonal mesons, basic effect of binding is modification of the integer k labelling the quark.

3. The formula produces the masses of mesons and also baryons with few per cent accuracy. There are however some exceptions.
   1. The mass of η' meson becomes slightly too large. In case of η' negative color binding conformal weight can reduce the mass. Also mixing with two gluon gluonium can save the situation.

   2. Some light non-diagonal mesons such as K mesons have also slightly too large mass. In this case negative color binding energy can save the situation.

2. An example about how mesonic mass formulas work.

The mass formulas allow to understand why the "crackpottish" mass formula for B_c holds true.

The mass of the B_c meson (bound state of b and c quark and antiquark) has been measured with a precision by CDF (see the blog posting by Tommaso Dorigo) and is found to be M(B_c)=6276.5+/- 4.8 MeV. Dorigo notices that there is a strange "crackpottian" co-incidence involved. Take the masses of the fundamental mesons made of c anti-c (Ψ) and b anti-b (Υ), add them, and divide by two. The value of mass turns out to be 6278.6 MeV, less than one part per mille away from the B_c mass!

The general p-adic mass formulas and the dependence of k_qon hadron explain the co-incidence. The mass of B_c is given as

m(B_c)= m(c,k_c(B_c))+ m(b,k_b(B_c)),

whereas the masses of Ψ and Υ are given by

m( Ψ)= 2^1/2m(c,k_Ψ)

and

m(Υ)= 2^1/2m(b,k_Υ).

Assuming k_c(B_c)= k_c(Ψ) and k_b(B_c)= k_b(Υ) would give m(B_c)= 2^-1/2[m( Ψ)+m( Υ)] which is by a factor 2^1/2 higher than the prediction of the "crackpot" formula. k_c(B_c)= k_c( Ψ)+1 and k_b(B_c)= k_b( Υ)+1 however gives the correct result.

As such the formula makes sense but the one part per mille accuracy must be an accident in TGD framework.

1. The predictions for Ψ and Υ masses are too small by 2 resp. 5 per cent in the model assuming no effective scaling down of CP₂ mass.

2. The formula makes sense if the quarks are effectively free inside hadrons and the only effect of the binding is the change of the mass scale of the quark. This makes sense if the contribution of the color interactions, in particular color magnetic spin-spin splitting, to the heavy meson masses are small enough. Ψ and η_c have spins 1 and 0 and their masses differ by 3.7 per cent (m(η_c)=2980 MeV and m(Ψ)= 3096.9 MeV) so that color magnetic spin-spin splitting is measured using per cent as natural unit.

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.