A new view of hadrons inspired by the nuclear physics model

The proposed revision of the view of the construction of nuclei makes it possible to get rid of the non-perturbative nuclear nightmare and is extremely simple. The recent QCD based view of hadron physics leads to a similar nightmare. Could the proposed lego brick picture of nuclear physics extend to the level of hadrons?

1. Could the original Gell-Mann model baryons consisting of heavy constituent quarks find a justification. Could the difference between constituent quarks and current quarks be due to different p-adic mass scales?
2. Could the masses of hadrons be just sums of quark masses and masses of bonds which are not far from pion masses. p-Adic mass calculations (see this) lead to a formula for the mass squared of leptons and quarks. For a given p-adic mass scale mass squared is in the lowest order approximation integer: m²= Am_p², where m_p is the p-adic mass scale. Apart from the effects due to CKM mixing, the values of these integers for leptons (e,ν_e) and the lightest quarks (u,d,s) are

   (A(e),A(ν),A(u),A(d),A(c),A(s))= (5,4,5,8,14,17) .

   These values represent lower bounds and perturbative p-adic second order contribution to A can be at most one unit. What is remarkable is that the masses of electron and u quark for the same p-adic length scale are identical in the lowest order and electron and neutrino masses are nearly identical. The large electron-neutrino mass difference would be due to different p-adic mass scales.

   If the p-adic length scales of u,d, and s quarks are same, their mass ratios in the lowest order are m(u)/m(d)=(5/8)^1/2 and m(s)/m(d)= (17/8)^1/2.

3. Can one predict the masses of pions and lightest baryons p, n,Λ from this input by assuming that the masses of quarks and pion-like bonds are additive and selecting the p-adic mass scale to be that associated with Gaussian Mersenne prime corresponding to k=113, to k=109 or to k=107 assigned with light baryons?

Consider first the masses of pions.

1. If quark masses are additive, one obtains identical masses m(π^-/+)/m_p= m(π⁰)=(8^1/2 +5^1/2)× m(k)= ((8/5)^1/2 +1)m_e , m_e= .5 MeV in lowest for neutral and charge pions. m(k is the p-adic mass scale p≈ 2^k, most naturally k=113. Note that in the case of neutral pions averaging for pairs uu^* and dd^* is involved.
2. Since one has A(u)= A(e)=5, for k=113, which has been identified a the nuclear p-adic length scales, the mass of u quark is obtained by scaling the mass of electron by the factor 2^(127-113)/2= 2⁷. This gives m(u(113))≈ 64 MeV. The mass of the d quark would be m(d)= (8/5)^1/2m(u)≈ 80.9 MeV. For k= 113, pion mass would be m(π)= m(u)+m(d)= 144.9 MeV, which is quite near to 140 MeV for neutral pions. Note that there would be no Cabibbo mixing for pions.

Consider next the masses of the proton, neutron and Λ.

1. The flux tube contribution to the mass would be the sum of the masses identifiable as pions and would be given by m(tubes)= 3× 140=420 MeV, that is m_p/2, exactly one half of proton mass. The simplex model assumes that the quark contribution is the same as the meson contribution. This suggests mass equipartion or a kind of dynamical supersymmetry relating pion and quark contributions to the mass of the nucleon. The masses of proton and neutron m(p)=2m(u)+ m(d)= (2+ (17/5)^1/2)m_e and m(n)= m(u)+2m(d)=(1+25^1/2) m(u) so that neutron proton mass splitting would be quite too large.
2. The first candidate for the solution of the problem is provided by the same mechanism as used to minimized energy in the construction of nuclei: n-π_L⁰ with a larger mass were replaced by p-π_L^- pairs, where π_L^- has the mass of electropion (quarks correspond to k=127 characterizing also electron). One can replace d quarks with u-π^- pairs so that the masses uud and udd are identical. The contribution of quarks to the total mass of the nucleon would be 3m(u)/2= 193 MeV for k=113. For k=111 the contribution is 384 MeV and by Δ m=36 MeV smaller than the nucleon mass ≈ 940 MeV.

   Intriguingly, if the mass equals to the average mass m(u,k=111)+m(d,k=111))/2= m(π(k=113)) of u and d quarks, k=111 gives the same contribution as pions and one obtains proton mass correctly. The masses of nucleons would come out correctly apart from differences relating to pion charge which is 4 MeV. The masses of n resp. p is 939.57 MeV resp. 938.27 and the mass difference is 1.3 MeV.

3. Could this be achieved by the TGD counterpart of CKM mixing (see this, this, this, this, and this), which is certainly present. In TGD, CKM mixing is caused by the different topological mixings of the partonic 2-surfaces at which quarks reside. In a good approximation, the mixing is present only for the lowest quark genera (g=0 (sphere), which corresponds to u (d) and g=1 (torus), which corresponds to c (s). CKM mixing would be essentially the difference of the topological mixings. In the case of Cabibbo mixing, the mixing angle θ_c would be different θ_c= θ_u-θ_d of the topological mixing angles θ_u and θ_d.

   The condition is that the mixing of u quark and scaled down c quarks is such that the light mixed state has mass m(π(k=111))=2m(π). One would have

   c_u² +s_c²(14/5)^1/2m(u(k=111)) = m(π(k=113))== m(π) .

   Here one has (c_u,s_u)=(cos(θ_u), sin(θ_u) and m(u(k=111))=128 MeV and m(π)=140 MeV. This gives s_u² = m(π)-m(u(111))/m(u(111)/((14/5)^1/2-1) giving s_u=-/+ .1392. For the Cabibbo angle s_c= .2250 this gives s_d= s_u+/- s_c. For positive s_u this gives s_d= -0.0858. In (see this){padmass3} I have discussed a model for the topological mixing of quarks assuming that mass squared values are averages of different mass squared values of the topologically mixed particles with a given p-adic length scale. In the recent case, the mixing cannot occur for the mass squared values: this would lead to a negative value for s_c².

4. This proposal resembles the Gell-Mann model in which constituent quarks would give the entire mass of the nucleon. The situation is the same now if the constituent quarks are identified as quark-flux tube pairs. The QCD inspired view replaces constituent quarks with current quarks and divides them to valence quarks and sea quarks. Due to the technical problems of the non-perturbative QCD one cannot build a concrete model. Current quark masses would be in the range 5-10 MeV.

   In the TGD framework, valence quarks could correspond to the quarks with mass scale k=111 and sea quarks would have small p-adic mass scale. Nuclear physics suggests electron mass scale as a mass scale of sea quarks: in this case the current quark masses would be m(u)=m_e and m(d)= (8/5)^1/2m_e. The total sea quark mass would be measured in few MeVs: of order .1 per cent.

5. In case that the topological mixing does not completely take care of the equipartition of the pion and quark contributions to the mass, the missing Δ m≤ 36 MeV could be assigned to the light sea quarks and corresponds to 3.8 per cent of the total mass of the nucleon. The estimates for this contribution vary but are few percent of nucleon mass. It is also known that sea quarks carry only a very small longitudinal momentum fraction and valence quarks carry 1/3 of longitudinal momentum. This would conform with the interpretation of the valence quarks as q+π-structures and sea quarks as light quarks of mass of order electron mass appearing as bonds in nuclei. They could correspond to flux loops with length of order electron's p-adic length scale L(127), which is of the order of electron Compton length.

6. Can one understand the mass m(Λ)= 1116 MeV of Λ baryon containing also strange quark s? The mass difference m(Λ) -m(n)≈ 178 MeV cannot correspond to the mass difference m(s)-m(d), which in absence of topological mixing would be maximal and in this case given by ((17/5)^1/2 -(8/5)^1/2) m(u,111)≈ 81 MeV. This is too small to explain the Λ-n mass difference.

   Could energy minimization be achieved by replacing the s-π⁰ pair with p-K^- pair solve the problem? Kaon mass is 493 MeV so that m(Λ)-m(p) would be equal to m(K)-m(π)≈ 353 MeV for k=113. This is too large by a factor 2. For k=115 one would obtain mass difference 176.1 MeV to be compared with real mass difference m(Λ) -m(n)≈ 178 MeV!

If this picture is correct, the p-adic length scale hypothesis would make it possible to build mesons from quarks with masses predicted by and various mesons whose masses would be sums of quark masses in the case of charged mesons and averages of sums in the case of neutral mesons. It is essential that quarks and mesons can exist in several p-adic mass scales related by a power of 2. Different topological mixings for U and D type quarks would explain CKM mixing. In mesons topological mixing would not occur.

See the article About Platonization of Nuclear String Model and of Model of Atoms or the chapter with the same title.

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

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.