About the construction of mesons and elementary bosons in TGD Universe

It looks somewhat strange to talk about the construction of mesons and elementary bosons in the same sentence. In TGD framework the construction recipes are however structurally identical so that it is perhaps sensible to proceed from mesons to elementary bosons. Therefore I will first consider the construction of meson like states relevant for the TGD based model of hadrons, in particular for the model of the pion of M₈₉ hadron physics possibly explaining the 125 GeV state for which LHC finds evidence. The more standard interpretation is as elementary spin 0 boson, which need not however have anything to do with Higgs.

One can construct also spin 0 states and their super-partners in TGD framework and at this stage it is not clear whether some internal consistency argument could exclude them from the spectrum. Amusingly, the construction for meson like states and elementary spin 0 states obey very similar mathematics. At this moment it is not possible to select between M₈₉ and elementary spin 0 boson as an explanation for 125 GeV state. What is however clear that this analogy of Higgs has also charged companions and that the couplings to fermions are not those predicted by standard model since particle massivation is not caused by Higgs vacuum expectation value but described by p-adic thermodynamics.

As an unexpected by-product construction gives insight to the difference between (π,η) and (K,Kbar) assignable to 3+1 and 2+2bar multiplets of strong isospin symmetry group SU(2). In QCD framework both the relation of this symmetry to color group and the organization of meson states to these 3+1 and 2+2bar are poorly understood. The construction reveals that these representations in TGD framework correspond directly to the u(2) subalgebra of su(3) and its complement and that this u(2) subalgebra can be assigned with strong isospin. Therefore strong isospin group is in well defined sense subgroup of color group but acting in weak isospin spin degrees of freedom! This example shows once again how good theory relates seemingly completely unrelated things to each other.

Construction of meson like states in TGD framework

The challenge is how translate attributes like scalar and pseudo-scalar making sense at M⁴ level to statements making sense at the level of M⁴× CP₂.

In QCD the view about construction of pseudo-scalar mesons is roughly that one has string like object having quark and antiquark at its ends, call them A and B. The parallel translation of the antiquark spinor from A to B is needed in order to construct gauge invariant object of type ‾ΨOΨ, where O characterizes the meson. The parallel translation implies stringy non-locality. In lattice QCD this string correspond to the edge of lattice cell. For a general meson O is "charge matrix" obtained as a combination of gamma matrices (γ₅ matrix for pseudo-scalar), polarization vectors, and isospin matrices.

This procedure must be generalized to TGD context. In fact a similar procedure applies also in the construction of gauge bosons possible Higgs like states since also in this case one must have general coordinate invariance and gauge invariance. Consider as an example pseudo-scalars.

1. Pseudo-scalars in M⁴ are replaced with axial vectors in M⁴× CP₂ with components in CP₂ direction. One can say that these pseudo-scalars have CP₂ polarization representing the charge of the pseudo-scalar meson. One replaces γ₅ with γ₅× O_a where O_a=O_a^kγ_k is the analog of ε^kγ_k for gauge boson. Now however the gamma matrices are CP₂ gamma matrices and O_a^k is some vector field in CP₂. The index a labels the isospin components of the meson.

2. What can one assume about O_a at the partonic 2-surfaces? In the case of pseudo-scalars pion and η (or vector mesons ρ and ω with nearly the same masses) one should have four such fields forming isospin triplet and singlet with large mass splitting. In the case of kaon would should have also 4 such fields but with almost degenerate masses. Why such a large difference between kaon and (π,η) system? A plausible explanation is in terms of mixing of neutral pseudo-scalar mesons with vanishing weak isospin mesons raising the mass of η but one might dream of alternative explanations too.
   1. Obviously O_a:s should form strong isospin triplets and singlets in case of (π,η) system. In the case of kaon system they should form strong isospin doublets. The group in question should be identifiable as strong isospin group. One can formally identify the subgroup U(2)⊂ SU(3) as a counterpart of strong isospin group. The group SO(3)⊂ SU(3) defines second candidate of this kind. These subgroups correspond to two different geodesic spheres of S². The first gives rise to vacuum extremals of Kähler action and second one to non-vacuum extremals carrying magnetic charge at the partonic 2-surface. Cosmic strings as vacuum extremals and cosmic strings as magnetically charged objects are basic examples of what one obtains. The fact that partonic 2-surfaces carry Kähler magnetic charge strongly suggests that U(2) option is the only sensible one but one must avoid too strong conclusions.

   2. Could one identify O_a as Killing vector fields for u(2)⊂ su(3) or for its complement and in this manner obtain two kinds of meson states directly from the basic Lie algebra structure of color algebra? For u(2) one would obtain 3+1 vector fields forming a representation of u(2) decomposing to a direct sum of representations 3 and 1 of U(2) having interpretation in terms of π and η the symmetry breaking is expected to be small between these representations. For the complement of u(2) one would obtain doublet and its conjugate corresponding to kaon like states. Mesons states are constructed from the four states U_iDbar_j, barU_iD_j, U_ibarU_j, D_ibarD_j. For i=j one would have u(2) and for i≠ j its complement.

   3. One would obtain a connection between color group and strong isospin group at the level of meson states and one could say that mesons states are not color invariants in the strict sense of the world since color would act on electroweak spin degrees of freedom non-trivially. This could relate naturally to the possibility to characterize hadrons at the low energy limit of theory in terms of electroweak quantum numbers. Strong force at low energies could be described as color force but acting only on the electroweak spin degrees of freedom. This is certainly something new not predicted by the standard model.

3. Covariant constancy of O_a at the entire partonic 2-surface is perhaps too strong a constraint. One can however assume this condition only at the the braid ends.
   1. The holonomy algebra of the partonic 2-surface is Abelian and reduces to a direct sum of left and right handed parts. For both left- and right-handed parts it reduces to a direct sum of two algebras. Covariant constancy requires that the induced spinor curvature defining classical electroweak gauge field commutes with O_a. The physical interpretation is that electrowak symmetries commute with strong symmetries defined by O_a. There would be at least two conditions depending only on the CP₂ projection of the partonic 2-surface.

   2. The conditions have the form

      F^ABj^a_B=0 ,

      where a is color index for the sub-algebra in question and A,B are electroweak indices. The conditions are quadratic in the gradients of CP₂ coordinates. One can interpret F^AB as components of gauge field in CP₂ with Abelian holonomy and j^a as electroweak current. The condition would say that the electroweak Lorentz force acting on j^a vanishes at the partonic 2-surface projected to CP₂. This interpretation looks natural classically. The conditions are trivially satisfied at points, where one has j^a_B=0, that is at the fixed points of the one-parameter subgroups of isometries in question. O_a would however vanish identically in this case.

   3. The condition F^ABj^a_B=0 at all points of the partonic 2-surface looks un-necessary strong and might fail to have solution. The reason is that quantum classical correspondence strongly suggests that the color partial waves of fermions and planewaves associated with 4-momentum are constant along the partonic surface. The additional condition F^ABj^a_B=0 allows only a discrete set of solutions.

      A weaker form of these conditions would hold true for the braid ends only and could be used to identify them. This conforms with the notion of finite measurement resolution and looks rather natural from the point of view of quantum classical correspondence. Both forms of the conditions allows SUSY in the sense that one can add to the fermionic state at partonic 2-surface a covariantly constant right-handed neutrino spinor with opposite fermionic helicity.

   4. These conditions would be satisfied only for the operators O_a characterizing the meson state and this would give rise to symmetry breaking relating to the mass splittings. Physical intuition suggests that the constraint on the partonic 2-surface should select or at least pose constraints on the maximum of Kähler function. This would give the desired quantum classical correlation between the quantum numbers of meson and space-time surface.

4. The parallel translation between the ends connecting the partonic 2-surfaces at which quark and antiquark reside at braid ends is along braid strand defining the state of string like object at the boundary of CD. These stringy world sheets are fundamental structures in quantum TGD and a possible interpretation is as singularity of the effective covering of the imbedding space associated with the hierarchy of Planck constans and due to the vacuum degeneracy of Kähler action implying that canonical momentum densities correspond to several values for the gradients of imbedding space coordinates. The parallel translation is therefore unique once the partonic 2-surface is fixed. This is of outmost importance for the well-definedness of quantum states. Obviously this state of affairs gives an additional "must" for braids.

The construction recipe generalizes trivially to scalars. There is however a delicate issue associated with the construction of spin 1 partners of the pseudo-scalar mesons. One must assign to a spin 1 meson polarization vector using ε^kγ_k as an additional factor in the "charge matrix" slashed between fermion and antifermion. If the charge matrix is taken to be Q_a=ε^kγ_k j^a_kΓ^k, it has matrix elements only between quark and lepton spinors. The solution of the problem is simple. The triplet of charge matrices defined as Q_a=ε^kγ_k D_kj^a_lΣ^kl transforms in the same manner as the original triplet under U(2) rotations and can be used in the construction of spin 1 vector mesons.

Generalization to the construction of gauge bosons and spin 0 bosons

The above developed argument generalizes with trivial modifications to the construction of the gauge bosons and possible Higgs like states as well as their super-partners.

1. Now one must form bi-linears from fermion and anti-fermion at the opposite throats of the wormhole contact rather than at the ends of magnetic flux tube. This requires braid strands along the wormhole contact and parallel translation of the spinors along them. Hadronic strings are replaced with the TGD counterparts of fundamental strings.

2. For electro-weak gauge bosons O corresponds to the product ε^k γ^k Q_i, where Q_i is the charge matrix associated with gauge bosons contracted between both leptonic and quark like states. For gluons the charge matrix is of form Q_A= ε_kγ^k H_A, where H_A is the Hamiltonian of the corresponding color isometry.

3. One can also consider the possibility of charge matrices of form Q_A=ε^kγ_k D_kj^A_lΣ^kl, where j^A is the Killing vector field of color isometry. These states would compose to representations of u(2)⊂ u(3) to form the analogs of (ρ,ω) and (K^*,Kbar^*) system in CP₂ scale. This is definitely something new.

4. In the case of spin zero states polarization vector is replaced with polarization in CP₂ degrees of freedom represented by one of the operators O_a already discussed. One would obtain the analogs of (π,η) and (K,kenooverlineK) systems at the level of wormhole contacts. Higgs mechanism for these does not explain fermionic masses since p-adic thermodynamics gives the dominant contributions to them. It is also difficult to imagine how gauge bosons could eat these states and what the generation of vacuum expectation value could mean mathematically. Higgs mechanism is essentially 4-D concept and now the situation is 8-dimensional.

5. At least part of spin zero states corresponds to polarizations in CP₂ directions for the electroweak gauge bosons. This would mean that one replaces ε_kγ^k with j_a^kΓ_k, where j_a is Killing vector field of color isometry in the complement of u(2)⊂ su(3). This would give four additional polarization states. One would have 4+2=6 polarization just as one for a gauge field in 8-D Minkowski space. What about the polarization directions defined by u(2) itself? For the Kähler part of electroweak gauge field this part would give just the (ρ,ω) like states already mentioned. Internal consistency might force to drop these states from consideration.

The nice aspect of p-adic mass calculations is that they are so general: only super-conformal invariance and p-adic thermodynamics and p-adic length scale hypothesis are assumed. The drawback is that this leaves a lot of room for the detailed modeling of elementary particles.

1. Lightest mesons are lowest states at Regge trajectories and also p-adic mass calculations assign Regge trajectories in CP₂ scale to both fermions and bosons.

2. It would be natural to assign the string tension with the wormhole contact in the case of bosons and identifiable in terms of the Kähler action assignable to the wormhole contact modelable as piece of CP₂ type vacuum extremal and having interpretation in terms of the action of Kähler magnetic fields.

3. Free fermion has only single wormhole throat. The action of the piece of CP₂ type vacuum extremal could give rise to the string tension also now. One would have something analogous to a string with only one end, and one can worry whether this is enough. The magnetic flux of the fermion however enters to the Minkowskian region and ends up eventually to a wormhole throat with opposite magnetic charge. This contribution to the string tension is however expected to be small being proportional to 1/S, where S is the thickness of the magnetic flux tube connecting the throats. Only if the magnetic flux tube remains narrow, does one obtain the needed string tension from the Minkowskian contribution. This is the case if the flux tube is very short. It seem that the dominant contribution to the string tension must come from the wormhole throat.

4. The explanation of family replication phenomenon (see this) based on the genus of wormhole throat works for fermions if the the genus is same for the two throats associated with the fermion. In case of bosons the possibility of different genera leads to a prediction of dynamical SU(3) group assignable to genus degree of freedom and gauge bosons should appear also in octets besides singlets corresponding to ordinary elementary particles. For the option assuming identical genera also for bosons only the singlets are possible.

5. Regge trajectories in CP₂ scale indeed absolutely essential in p-adic thermodynamics in which massless states generate thermal mass in p-adic sense. This makes sense in zero energy ontology without breaking of Poincare invariance if CD corresponds to the rest system of the massive particle. An alternative way to achieve Lorentz invariance is to assume that observed mass squared equals to the thermal expectation value of thermal weight rather than being thermal expectation for mass squared.

It must be emphasized that spin 0 states and exotic spin 1 states together with their super-partners might be excluded by some general arguments. Induced gauge fields have only two polarization states, and one might argue that that same reduction takes place at the quantum level for the number of polarization states which would mean the elimination of F_LbarF_R type states having interpretation as CP₂ type polarizations for gauge bosons. One could also argue that only gauge bosons with charge matrices corresponding to induced spinor connection and gluons are realized. The situation remains open in this respect.

For background and more details see the chapter New Particle Physics Predicted by TGD: Part I or the short article The particle spectrum predicted by TGD and TGD based SUSY.