**1. Basic facts about proton spin crisis**

In the EMC experiment the contributions of u,d, and s quarks to the proton spin were deduced from the deep inelastic scattering of muons from polarized proton target (see this). What was measured, were spin asymmetries for cross sections and the conclusions about parton distribution functions (see this) were deduced from the experimental data from the muon scattering cross sections using Bjorken sum rule testing QCD and Ellis-Yaffe sum rule assuming vanishing strange quark contribution and testing the spin structure of the proton. Bjorken sum rule was found to be satisfied reasonably well. Ellis-Yaffe sum rules related to the spin structure of the proton were violated.

It was found that the contributions of u quarks were positive and those of s quarks (assuming that they are present) and d quarks negative and the sum almost vanished when the presence of s was assumed. The Gell-Mann quark model predicts that u-quarks contribute spin 2/3 and d-duarks -1/6 units (ℏ) to the proton spin. For the fit allowing besides u, d contributions, also s contributions, the contributions were found to be 0.373, -0.254 and -0.113. The sum was 0.006 and nearly zero. For protons the contribution is roughly one half of Gell-Mann prediction. For d quark the magnitude of the contribution is considerably larger than the Gell-Mann prediction -1/6≈-.16.

The Wikipedia article creates the impression that the proton spin crisis has been solved: the orbital angular momentum would significantly contribute to the spin of the proton. Also sea partons, in particular gluon helicity polarization would contribute to the proton spin. This might well be the case.

**2. Dark sea partons and proton spin crisis**

I have considered possible TGD inspired solutions of the proton spin crisis already earlier. One can however also consider a new version involving dark sea quarks.

- The possibility that sea partons are dark in the TGD sense, forces us to ask what was really measured in the EMC experiment leading to the discovery of the proton spin crisis. If sea partons are dark, only the quark distribution functions corresponding to quarks with ordinary value of h
_{eff}appearing in the coupling to muon would contribute? This should be the case in all experiments in which incoming particles are leptons.Assuming that also valence quarks can be part of time strange, the results of the EMC experiment assume that most of the proton spin could reside at the polarized dark sea. Note however that also orbital angular momentum can explain the finding and in the TGD framework color magnetic flux tubes could carry "stringy" angular momentum.

- For this option one could identify the measured cross section in terms of scattering from quarks with h
_{eff}=h. It has been proposed that valence quarks are large scale structures (low energy limit) and sea quarks are small scale structures (high energies) inside valence quarks.In the TGD framework, this suggests that valence quarks correspond to a larger p-adic prime than sea quarks. This does not imply that valence quarks have large h

_{eff}. Large h_{eff}for the sea partons would increase their size so that, contrary to the expectations from the Uncertainty Principle, they could contribute to hadron-hadron scattering with large momentum transfer in long length scales.

**2.1. How to represent ordinary quarks at the level of color MB?**

One should understand how the color interactions for which the perturbation series does not converge at the level of ordinary matter are transferred to the dark magnetic body at which the perturbation series converges. The color of the ordinary quarks should be neutralized and transferred to the color of dark quarks at color MB.

- The valence quarks have an ordinary value of h
_{eff}and the perturbation series does not converge. One should have a concrete realization for the transfer of color interactions at the level of valence quark to the level of the sea quarks with large h_{eff}. If only dark gluons exist, the color interactions take place at the level of the color MB and one the perturbation theoretic coupling would be α_{s}= β_{0}/4π.The physical mechanism in question should map valence quarks to dark valence quarks at the MB.

Also color confinement could take place at the level of the color MB and induce it at the valence quark level. The ordinary electroweak interactions should take place between valence quarks q

_{o}("o" for "ordinary") but also a dark variant of ew interactions between dark quarks is possible and indeed assumed in TGD inspired quantum biology. Could the mechanism be as follows? - Consider a free hadron. The color MB contains dark sea quark q
_{d}("d" for "dark") and antiquark q_{d}^{*}with opposite charges and spins such that q_{d}^{*}combines with q_{o}to form an entangled color singlet meson-like state.q

_{d}would carry the same quantum numbers as q_{o}. Quark quantum numbers would be transferred by entanglement to the color MB! Color confinement would take place at the level of MB and induce color confinement at the level of valence quarks.A stronger assumption would be that this state is spin singlet: this would imply automatically a vanishing average spin for the valence quarks but would not be consistent with the EMC determination of Δ S

_{i}. This suggests that only color singlet entanglement between q_{d}and antiquark q_{d}^{*}makes sense. This option might be consistent with the QCD picture about the spin crisis of the proton.

_{g}bosons in the case of hadrons. As will be found, the simplest option in which they are not present allows one to understand CKM mixing in terms of SU(3)

_{g}gluon exchanges.

**2.2 How to understand the standard QCD view about the proton spin crisis in the TGD framework?**

If spin-isospin quantum entanglement gives a spin singlet, valence quark spin does not contribute to proton spin at all. This view is in conflict with the QCD view about the values of Δ s and their summation to a small value. Could one understand the QCD values in the TGD framework by giving up the assumption of spin singlet property of entanglement? There would be only color entanglement between q_{o} and q_{d}, and spins would be opposite but the state would belong to a direct sum of vector and singlet representation of SU(2).

Could one modify the entanglement between quarks q_{o} such that one can explain the EMC findings?

- Gell-Mann model cannot be correct at the level of details but would predict correctly that baryons correspond to irreps of spin and isospin. In particular, protons would be spin- and isospin doublets. The entanglement between spin degrees of freedom and between isospin degrees of freedom of quarks should be more general than that in the Gell-Mann model. Is this possible?
- Consider the nucleon as a tensor product of 3 quarks as tensor products of 3 spin and isospin doublets giving rise to a spin and isospin doublet. The sums of individual isospin and spin components correspond to those of baryon: for the proton uud, udu, and duu can serve as building bricks of the state. The needed antisymmetrization is in color degrees of freedom.
In the case of a nucleon, the spin S

_{z}and isospins I_{3}must sum up to +/- 1/2. This leaves 3× 3=9 complex coefficients in case of proton/neutron (uud/udd). The state is defined only modulo anoverall complex coefficient: this leaves 7 complex coefficients.The values of Casimir operators S(S+1) and I(I+1) are fixed: these conditions can be written as eigenvalue conditions for ∑

_{i}(S_{i}(S_{i}+1) + 2∑_{i≠ j}s_{i}• s_{j}= S(S+1) and ∑ I_{i}(I_{i}+1)+ 2∑_{i≠ j}I_{i}• I_{j}= I(I+1). These 2 conditions leave 5 complex parameters. - A more straightforward approach is group theoretic. The tensor product 2\otimes 2 \otimes 2 decomposes as 4 ⊕ 2
_{1}⊕ 2_{2}. 4 is totally symmetric and the doublets have mixed symmetries. At least formally, one can construct from 2_{1}⊕ 2_{2}a proton state for which the conditions for Δ s from the EMC experiment hold true?The superposition of these representations can be parametrized as cos(θ)exp(iφ)2

_{1}⊕ sin(θ)exp(iφ) 2_{2}. Same applies in the isospin degrees of freedom so that one would have 4 parameters. In Nature, only single nucleon doublet appears and there might be some trivial reason for this. Could the superposition of these two representations be selected by some principle or could also the other representation and therefore also superposition be realized in Nature. - The conditions on the values of Δ s
_{i}coming from the EMC experiment give 2 constraints leaving a 3-D complex space of solutions.

**2.3 A model for the representation of a general particle at its magnetic body**

The challenge is to generalize the model for baryons so that it would also apply to bosons and leptons.

- The vision about MB as a receiver of sensory information from the biological body and control of it has been applied in biology and the fractality of the TGD Universe suggests that this picture applies in all scales. Hence the idea that MB of the particle carrying dark matter serves a universal representation of the ordinary particle is attractive.
- Color entanglement can bind the q
_{o}and q_{d}^{*}in a stable way. What about leptons which are color singlets? The TGD view of color comes to rescue here. In TGD, color is not a spin-like quantum number but at the level of H corresponds to color partial waves for H spinor fields. There are two alternative proposals for what leptons could be.- For the first option, leptons correspond to second H-chirality for H spinors. The color partial waves correlate with the electroweak quantum numbers in a wrong way for both quarks and lepton chiralities. The physical states assignable to partonic 2-surfaces involve super symplectic generators carrying color in such a way that physical leptons are color singlets and quarks are color triplets.
Lepton states involve an action of super symplectic generator O on the lepton spinor OL

_{o}^{c}such that the O transforms as the conjugate of the color representation associated with color partial wave L_{o}^{c}. L_{o}would be essentially the inner product of O and color partial wave L_{o}^{c}and therefore a color singlet. In the case of quark q, q_{o}would be obtained by projection color triplet from q_{o}= P_{3}(Oq).The inner product of L

_{o}^{c}and L_{d}^{*c}defines a color entangled color singlet. - The second option is that fundamental leptons correspond to color singlets formed from 3 antiquarks. The 3 leptonic antiquarks do not reside at separate wormhole contacts having two wormhole throats identified as partonic 2-surfaces but reside at a single partonic wormhole. The mechanism proposed for hadrons can be applied to quarks. This option can explain matter antimatter asymmetry: antimatter as antiquarks could bind to leptons. A small CP breaking predicted by TGD in principle allows this.

- For the first option, leptons correspond to second H-chirality for H spinors. The color partial waves correlate with the electroweak quantum numbers in a wrong way for both quarks and lepton chiralities. The physical states assignable to partonic 2-surfaces involve super symplectic generators carrying color in such a way that physical leptons are color singlets and quarks are color triplets.
- This approach works also for bosons since all bosons can be realized as a quantum superposition of fermion-antifermion pairs in the TGD framework (note that graviton involves two pairs). Electroweak bosons involve pairs q
_{o}q^{*}_{o}: the contraction with respect to color gives entanglement. Also lepton pairs are involved: now the contractions are of the form L_{o}^{c}L_{o}^{*c}.The construction of B

_{o}^{c}B_{d}^{*c}reduces to the formation of color entangled pairs q_{o}q_{d}^{*}and L_{o}^{c}L_{d}^{*c}. Gluons, with SU(3)_{g}gluons included, can be formed as a color octet pairing of quarks and antiquarks and G_{o}^{c}G_{d}^{*c}pairing can be formed as in the case of baryons.

The basic physical idea deserves restating: the simple and perturbatively convergent dynamics at the level of MBs for the dark images X_{d} of the particles induces the dynamic of particles X_{o} by stable color quantum entanglement. The MB of the dark particle would be the boss and the dynamics of the ordinary particle would be shadow dynamics in accordance with the general vision about induction as the basic dynamical principle of TGD.

One open question is whether the ordinary matter follows the dynamics of dark particles instantaneously or whether the time scales of the dynamics of dark matter and ordinary matter can be different in which case only the asymptotic states would realize the proposed correspondence between X_{d} and X_{o}.

**3. Could SU(3) _{g} gluons induce CKM mixing of quarks and leptons?**

The above simple model did not say anything about the possible presence of SU(3)_{g} gluons at the color magnetic MB. Even if they are not present, the exchange of SU(3)_{g} g>0-bosons between entangled q_{o} and q_{d}^{*} could increase the genus of both q_{o} and q_{d}^{*} (note the genus is counted as negative for antiquarks).

At the level of the ordinary matter this could give rise to what looks like CKM mixing whereas no mixing would take place for q_{d}. This process generalizes to the case of leptons since L_{o}^{c} and L_{d}^{*c} are colored states for both identifications of leptons.

The g>0 gluon exchange involves a transformation of the dark g>0 gluon to an ordinary g>0 gluon. This process is assumed to occur for dark photons in the TGD inspired model for quantum biology: bio-photons would be an outcome of this process for dark photons.

Some CKM mixing angles are rather large. If the CKM mixing is solely due to this process, the masses of the g>0-gluons must be considerably smaller than weak boson masses so that mass scale could be around 100 MeV, say.

See the article What it means if a Higgs-like particle decaying to eμ pairs exists? or the chapter with the same title.

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

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