Problems with Holography = Holomorphy Principle

The holography = holomorphy principle (H-H) has become a key principle of TGD. For the volume action H-H yields minimal surfaces as extremals and this encouraged the conjecture that H-H holds also for Kähler action.

Even stronger conjecture would be that even curvature scalar allows minimal surfaces as extremals. The strongest conjecture would be that the H-H principle holds for any general coordinate invariant action constructible in terms of the induced geometry and spinor structure.

Marko Manninen tested the conjectured universality by using NeuroPaja, a research setup combining AI systems with computer algebra, and found that it fails in two tested cases, namely for Kähler action and for the curvature scalar.

The curvature scalar does not appear in the basic action so that this is not a problem. The failure for Kähler action is however a problem. This is due to the failure of the Kähler current to be light-like. There is a very simple remedy to the problem. The condition that the Kähler function of CP₂ restricted to the space-time surface satisfies 2-D Laplace equation in transversal degrees of freedom implies that the transversal part of the Kähler current vanishes outside lower-dimensional singular surfaces, which could correspond to wormhole throats. This implies the Poincare - and general coordinate invariant condition that the Kähler current is light-like.

Hamilton-Jacobi (H-J) structure defines a generalization of Kähler structure and complex structure as a fusion of complex and hypercomplex structures. H-J structure can be identified as a slicing defined by 2-D Minkowskian string world sheets and Euclidean surfaces orthogonal to each other. The assumption that both these surfaces are minimal surfaces is not necessary but is very attractive. It is not always true.

See the article Problems with Holography = Holomorphy Principle 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.

4.8 sigma evidence for the B meson anomaly: what could be its TGD based explanation?

Sabine Hossenfelder (see this) told that 4.8 sigma evidence for B meson anomaly manifesting itself in the decays of B meson to K^* meson plus muon pair has been reported (see this).

1. What is known?

Below is the abstract of the article telling about the finding.

An analysis of the decay amplitudes of B⁰; K^*0→ K⁺π ^-μ ⁺μ ^- is presented, using proton-proton collision data recorded by the LHCb experiment at centre-of-mass energies of 7, 8, and 13 TeV, corresponding to an integrated luminosity of 8.4 fb^-1. The amplitudes are constructed from Legendre polynomials in the μ⁺μ⁺ invariant mass squared region 1.1 < q²< 8.0 GeV²/c⁴. CP-averaged observables are obtained from the amplitudes.

Some of these observables present deviations with respect to the Standard Model, which can be interpreted as shifts in the effective Wilson coefficients. This model-independent approach enables tests of theoretical predictions that can help disentangle hadronic effects from potential contributions from physics beyond the Standard Model. This allows flexibility in the choice of q² binning for global analyses. Depending on the binning scheme, the deviation of the Wilson coefficient C₉ from its Standard Model expectation varies from 4.3σ to 4.8σ.

A large variety of models explaining the B meson anomaly have been proposed. Perhaps the simplest model for the anomaly is in terms of a heavier partner Z^' of Z⁰ boson. The anomaly reduces to that associated with the vertex involving a transformation of the b quark to s quark by CKM mixing by the emission of Z^' boson.

Since a very short distance phenomenon is in question this can be approximately described in effective QFT approach by a purely local interaction terms as product of the neutral weak current involving b and s quark and muon pair. This description is completely analogous to the description of weak interactions in terms of the four-fermion vertex in the original Fermi model.

As in the Fermi model, the interaction term is space-time integral for the product of weak currents involving Z^' propagator proportional to the inverse mass squared of Z^' having spatial dimension 2. The spatial dimension of the product of weak fermion currents coefficient of is determined by d=-6. In the parameterization the mass squared gives a proportionality constant proportional to cutoff mass squared 1/μ ². The index of C₉ is a convention for labelling various non-local interaction terms of the effective action of the standard model. There is no perturbative expansion since effective action is in question and the interaction term gives a simplified description for the loops responsible for the appearance of heavier particles in the interaction vertex making possible the anomaly.

Google Gemini informs that there are two kinds of models possibly explaining the anomaly in terms of Z^'. If the Z^' gauge coupling is comparable to the Standard Model electroweak couplings g_Z^' ∼ 0.1 - 1.0, global fits to the b → l⁺l^- data typically restrict its mass to the range (1-7) TeV. Why this range?

1. The lower bound is ∼ 1-2 TeV: direct searches at ATLAS and CMS rule out lighter Z^' bosons. If Z^' were lighter than this and coupled directly to muons, it would have already appeared as a striking resonant "peak" in dimuon μ⁺μ^- invariant mass distributions at the LHC.
2. The Upper Bound ∼ 5-7 TeV: If the mass is any higher, its effect on the low-energy Wilson coefficients C₉ would become too weak to explain the observed deviations in B-meson decays, unless the couplings are artificially dialed up to mathematically unstable levels.

If the couplings are allowed to be much larger (approaching the perturbative limit where quantum field theory calculations remain reliable), the Z^' can hide at much higher energy scales: M_Z^' ∼ 10-30 TeV In this scenario, a heavy Z^' is too massive to be directly produced at the LHC, but its virtual "ghost" signature would still be heavy enough to generate the required modifications to the Wilson coefficients. Testing a Z^' in this range requires future facilities like the Future Circular Collider (FCC) or a high-energy muon collider.

2. What could TGD say about the B anomaly?

TGD predicts a hierarchy of scale up copies of the standard model physics. p-Adic length scale hypothesis involves the conjecture is that these copies are labelled by 3 kinds of Mersenne primes: ordinary Mersenne primes M_k =2^k-1, Gaussian Mersennes M_G,k =(1+i)^k-1 and Eisensteinian Mersenne primes (see this). Ordinary hadron physics correspond to M₁₀₇ =2¹⁰⁷-1, there is evidence for M₈₉ hadron physics, which would be the scaled up version of ordinary hadron physics with 512 times higher mass scale and also for the M_G,79 hadron physics with mass scale higher by a factor 2¹⁴. The next candidate would be M₆₁ hadron physics with mass scale higher by factor 2²³. Therefore M_G,79 remains the only option.

Ordinary weak bosons correspond to M₈₉ and this would mean that for M_G,79 weak physics, the mass scales of weak bosons are by a factor 32 higher than for the standard model weak interactions. For the scaled up version of the ordinary Z⁰ boson with mass .09 TeV this would predict a mass M= 2.88 TeV. This mass belongs to the region allowed by coupling strength which are of same scale as for the standard model.

2.1 What guesses can one make about the new bosons possibly explaining the B anomaly?

1. Google Gemini informs that if the couplings are in the range .1-1 for weak couplings needed for perturbation theory the mass of the new boson is in the range 1-5 TeV. There is an estimate of 2.9 TeV for the mass of the new boson.
2. TGD predicts a hierarchy of standard model physics. Weak interactions correspond to the p-adic scale associated with the Mersenne prime M₈₉. The next p-adic length scale corresponds to M_G,79. The scaling Z⁰ mass from M₈₉ to M_G,79 by factor 32= 2^(89-79)/2 gives mass of 2.88 TeV, which is rather near to the empirical estimate. M_G,79 is the only possibility since the next candidate M₆₁ would give a huge mass.
3. For this option the universality of quark and lepton couplings is the first guess. There are however indications for the violation of fermion universality and TGD allows us to consider a model explaining these violations (see this and this).

2.2. What could happen in the transformation of b quark to s quark?

Consider first what might happen the creation of a muon pair as b quark transforms to s quark.

1. The decay of B meson to muon pair or to K^* and muon pair induced by the transformation of b quark to s quark. The direct coupling b → s vanishes. Since neutral-current couplings are diagonal. CKM mixing for charged-current couplings involving the emission of a virtual W boson makes possible the b-s coupling via radiative corrections. Therefore a radiative correction involving emission of virtual W bosons must be involved.
2. Upon the emission of a W boson, the b quark temporarily transforms into a c, u, or t quark. This W boson is then absorbed and i by CKM mixing gives rise to say s quark. The virtual W boson emits an M_G,79 photon (γ₇₉) or Z⁰ boson Z₇₉, which subsequently decays into, say, a muon pair. This gives a small correction to the production rate predicted by the standard model.

With this in mind, we can re-examine the situation from the TGD perspective.

1. The standard view remains valid, but in the context of M₈₉ hadron physics, the "official" t quark would naturally correspond to the M₈₉ u quark. The t quark is replaced by the "Aleph" quark, which has a mass of 56 GeV (see this, this and this). If the coupling to the official t quark is small, Aleph quark dominates, and the mass squared scaling reduces the amplitude by a factor of (56/175)² ∼ 1/9. This constitutes a fundamental prediction that is easily ruled out. The propagator of Z_G,79 would give a reduction factor 1/M²_ZG,79. The re-identification of t quark changes the CKM matrix and this affects the coupling.
2. What about evidence for the violation of universality - a phenomenon for which some evidence exists and about which I have previously written (see this and this )? Could the Z and W bosons be states in which different genera carry different weights. In the case of γ_G,79 genus-dependent weights look problematic since the em charges of different fermion generation would be apparently different. This is because in the TGD based model the mesons are regarded as superpositions of fermion pairs associated with different generations.

2.3 Could the M_G,79 weak physics emerge in a phase transition M₈₉→ M_G,79?

This would alter the coupling of the Z boson within the W loop and also the coupling of W. Could all electroweak bosons be dark M_G,79 bosons this would be visible in the mass dependence of the loops.

What happens in the transition between different standard model copies is an interesting problem. I suspect that a phase transition rather than mixing is in question. One can take the phase transition from M₁₀₇ to M₈₉ as a role model. The transition would take place quantum criticality generating a dark M₈₉ phase with h_eff scaled up by 512. One can imagine at least two options but the following first one looks the most plausible.

1. Quantum criticality would be lost in p-adic cooling \cite{allb/tgdnewphys1} (see this and this). The most recent application is to the TGD counterpart of energy production in the Sun that would occur at the surface of the Sun (see this). The p-adic mass scale would be reduced step by step, most naturally by octaves or half-octaves and decay to "virtual hadrons" characterized by this scale would take place. The value of h_eff would also reduce gradually so that the Compton lengths would not be affected. The process would be analogous to overheating followed by cooling without a dramatic compression.
2. Second option that one can imagine is that first quantum criticality would be lost completely by a reduction of M₁₀₇ p-adic length scale to M₈₉ p-adic length scale, meaning a contraction by factor 1/512. After that explosive increase of the p-adic length scale by stepwise manner down to M₁₀₇ would take place.

Similar picture should apply to the transition for M₈₉ electroweak physics to its M_G,79 variant.

1. The emission of a Z₇₉ or γ₇₉ by virtual W boson with a mass of 2.88 TeV would result in the production of muon pairs. The decay to muon pairs correspond to final-state interactions; for the purpose of predictions, only the process leading to the emission of Z₇₉ or γ₇₉ is relevant.
2. The estimate for the mass of the emitted particle comes from the propagator of the new particle. The mass of Z would be too small. Z₇₉ propagator with much larger mass would add to the additional small contribution of the ordinary Z. The additional contribution to the amplitude is proportional to m²(t)/m²(Z₇₉). If t quark is the "official" t quark, the M²_t/M²_>ZG,79 = (0.175/2.88)² = 3.7 × 10^-3 provides the scale factor. If the official top quark is replaced by an "Aleph" quark (see this), an additional factor of 1/9 approximately equal to the mass squared ratio of Aleph quark and top quark emerges and gives a factor 4× 10^-4.
3. What about the propagator of γ₇₉ if it has a very small mass (this is if p-adic mass calculations contribute to its mass only in the second order). If Z_G,79 is nearly on mass- shell, its contribution is larger than that of γ_G,79.

Acknowledegements: I want to thank Martin Ebner and Marko Manninen for spontaneous AI assisted help in the analysis of the B anomaly.

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.

About the Relationship of Electroweak and Strong Interactions in the TGD Universe: Part II

The following represents the abstract of the 2^nd part of a 3-part article summarizing the view about the relationship of electroweak and strong interactions in the TGD Universe. This view has been developed together with Martin Ebner during the last almost four months.

The purpose of the second part of the article is to develop a detailed model for the relationship between electroweak and strong interactions at the level of the embedding space H developed in the first part of the article. The initiative came from Martin Ebner who wanted to find clearcut predictions related to the standard model, in particular Weinberg angle. This led to a project involving Martin and two AI agents, who did extremely valuable work identifying the numerous errors in my paper and pencil calculations and also in the earlier view about Weinberg angle estimated in my thesis dating back to 1982. The project led to a rather dramatic increase in the understanding of the relationship between electroweak and strong interactions in the TGD framework.

TGD explains the coupling pattern of electroweak interactions but gluon couplings do not appear in the embedding space and space-time variants of the Dirac action. The fact that the holonomy group of CP₂ is identifiable as a subgroup of the color group acting as isometries of CP₂ motivates the conjecture that color interactions and electroweak interactions are parts of the same pattern. The pleasant surprise was that the induced gauge couplings included not only electroweak gauge potentials but also a U(1) gauge potential assignable to strong interactions and coupling to strong isospin. The 3-D generalization of the Weinberg mixing and conformal invariance for the photon emission vertex relate the Weinberg angle and the strong interaction coupling strength α_s to the choice of the monopole harmonic and to a second parameter of the generalized Weinberg mixing.

A generalization of the Weinberg mixing is proposed. The assumption that the emission of photons having wrong helicity with respect to  the direction of fermion momentum -- lightlike  in 8-D sense -- vanishes, relates the Weinberg angle and the strong interaction coupling strength α_s to the choice of the monopole harmonic and to a second parameter of the generalized Weinberg mixing. Electric-magnetic duality states that U(1) interaction at X⁴ level corresponds to color interaction of QCD such that the two coupling strengths are inverses of each other.  A model for  WCW wave functions    as monopole harmonics at partonic 2-surfaces leads to a 3-D generalization of Weinberg rotation and an estimate of the Weinberg angle, identification of strong and weak interactions as parts of color interaction and estimate of the strong coupling strength. What is required is charge fractionation  and SL(2,C) acting as global conformal symmetries at partonic 2-surface allow representations with fractional spin.  

This view would give at the space-time level a concrete realization for the idea that strong resp. weak interactions represent orbital angular momentum-like resp. spin-like aspects of color interactions and that strong interactions relate deeply to electroweak interactions as the old-fashioned hadron physics assumed. I have speculated for decades about a kind of duality between QCD description and low energy hadronic description in terms of strong isospin and it seems that this finding makes possible a precise quantitative realization of this vision.

TGD also predicts the M⁴ counterpart of Kähler force as a new U(1) interaction possibly related to gravitation. If this interaction is strong as compared to the gravitational force, it could relate to the hadronic strings with transversal M⁴ degrees of freedom. Perhaps the more plausible option is that M⁴ Kähler gauge potential generalizes the Newtonian gravitational potential.

See the article About the Relationship of Electroweak and Strong Interactions in the TGD Universe: Part II 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.

New physics going beyond standard model

Sabine Hossenfelder told in YouTube talk "Large Hadron Collider Sees Extra Dimension, Physicists Claim" (see this) that there are indications for new particles not predicted by the standard model. Since TGD predicts a scale hierarchy of standard model physics, these findings are extremely interesting from the TGD point of view.

1. Observations

What has been observed are 4-jets in p-p collisions at few TeV energies at CMS. There are observations in two energy ranges.

1.1 T700 resonances

There are indications at LHC for BSM (Beyond Standard Model) particles with spin 0 or spin 2 appearing as resonances in 650-750 GeV range christened as T700.

Le Yaouanc and Richard have considered (see this) an explanation of these anomalies in terms of Kaluza-Klein (KK) graviton. The proposal relies on the Randall-Sundrum model with 1-D internal space. For the Randall-Sundrum option the internal space is a circle and by Uncertainty Principle the mass scale would determine the radius of the circle and would be very large as compared to the original view that the radius is of order Planck length. In TGD this scale corresponds to the length scale of string-like objects.

Spin 2 or spin 0 are possible for the states involved. For the spin 2 option, vector boson dominance as fusion of weak bosons producing a top pair decaying to ordinary quarks producing 4-jets is the natural production mechanism. For spin=0 option, gluon fusion producing heavy quark pairs, in particular top quark pairs, would be the production mechanism.

1.2 The anomalies around 3.6 TeV and 8.6 TeV

Also indications for additional have been found. Le Yaouanc and Richard (the same authors as above) propose as an explanation Kaluza-Klein gluons (see this) with masses m₁=3.6 TeV and m₂=8.6 TeV as an explanation. Standard physics does not favour 3.6 TeV gluons but the authors suggest that if these particles can decay also to other heavy quark pairs besides top pair, this is not a problem. KK quarks are proposed to have masses 1 TeV 2.1 TeV: the ratio is .58 to be compared with m_u/m_d= .55. The mass ratio of the KK gluon masses is m₁/m₂ is .42.

2. Possible TGD based explanations of the anomalies

TGD predicts a hierarchy of scaled variants of standard model physics so that these anomalies a very interesting from the TGD point of view. Mersenne primes coming as 3 varieties, ordinary, Gaussian and Eisensteinian (see this), are in a special role concerning p-adic mass scales.

Both M₈₉ hand physics, for which mass scales of hadrons are scaled up by factor 2⁹=512 and M_G,79 hadron physics for which the mass scales of hadrons are 2^{14 times those of the ordinary hadron physics are interesting.}

M_G,79 is special in the sense that k=79 corresponds also to Eisensteinian Mersenne prime M_E,79: the corresponding mass scale is near to that for the electron (see this). This is due to the fact that for Eisensteinian Mersenne primes p-adic mass scales come as powers of 3 (3^1/2) rather than octaves (half-octaves).

Note that for M_G,73 resp. M₆₁ the hadron mass scales are by factor 2¹⁷ resp. 2²³ higher than for ordinary hadron physics.

2.1 T700 anomaly

Consider first the T700 anomaly for which interpretation as KK graviton is proposed (see this and this). The production mechanism would be vector boson fusion (VBF).

In TGD, the natural candidate for the BSF particle would be as a meson of M₈₉ hadron physics. TGD suggest that the top quark actually corresponds to the u or d quark of M₈₉ hadron physics and that real top to the Aleph anomaly (see this and this). This predicts an entire spectrum of mesons and the scaling argument allows us to estimate their masses. One expects that the top pair gives rise to a pion-like state.

The masses of M₈₉ would be obtained from the masses of ordinary mesons scaled by a factor 512. The meson a₂(1320) is the lightest meason with spin J=2 and would be scale up to a state with mass of 675.8 GeV, which could explain T700.

2.2 The states with masses 3.6 TeV and 8.6 TeV

Consider next the states with masses 3.6 TeV and 8.6 TeV, suggested to correspond to KK gluons (see this). The large mass scales make M_G,79 hadron physics look like a natural option. The scaling of the mass of the top quark identified as a quark of M₈₉ hadron physics would predict a mass of 5.6 GeV for a M_G,79 u quark.

Also electroweak physics should appear as a scaled variant. The scaling of the ordinary Higgs mass .125 TeV having k=89 by 2^{Δ k}, Δk=(89-79)/2=5 would give m(H₇₉)= 4 TeV, not very far from the lower mass 3.6 TeV.

The scaling of M₈₉ pion mass, if identified as m(π(89))= 2m_t=.350 TeV (assuming additivity of quark masses), would give m(π(79))= 11.3 TeV, which is too large. If mass squared as conformal weight is additive instead of mass for bound states, as assumed in the model for the weak bosons as superpositions of quark pairs (see this), this estimate is scaled down by 2^-1/2 and would give m(π(79))=7.9 TeV, not far from 8.6 TeV.

By the same logic, M₈₉ pion would have mass m(π(89))= 2^1/2m_t= .248 TeV. There is no evidence for this kind of state.

See the articles About the relationship of electroweak and strong interactions in the TGD Universe: part III and Official top quark and toponium as particles of M₈₉ hadron physics? or the chapters Comparing the S-matrix descriptions of fundamental interactions provided by standard model and TGD and About the relationship of electroweak and strong interactions in the TGD Universe: part III.

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.

About the Relationship of Electroweak and Strong Interactions in the TGD Universe: Part III

The following represents the abstract of the 3^rd part of a 3-part article summarizing the view about the relationship of electroweak and strong interactions in the TGD Universe. This view has been developed together with Martin Ebner during the last almost four months.

Recall that in the second part of the article a generalization of the notion of Weinberg rotation to a 3-dimensional rotation involving besides Weinberg angle a second angle parameter is developed. In the TGD framework, strong and weak interactions are very strongly related and the model relates the Weinberg angle and the strong interaction coupling strength to the choice of the monopole harmonic and to a second parameter of the generalized Weinberg mixing.

In the third part of the article a model estimating the masses of weak bosons and their mass ratio is developed. The model relies on the basic ideas of TGD, in particular p-adic thermodynamics and the fact that weak bosons are not fundamental fields in TGD and can be seen as analogs of bound states of fermion pairs coding the decay amplitudes to fermion pairs. Estimates for masses and mass ratio emerge if one accepts the TGD proposal that the official top quark corresponds to the u quark of M₈₉ hadron physics, with a d partner of the same mass scale, and the real top quark of M₁₀₇ hadron physics corresponds to a new quark explaining the Aleph anomaly.

In the Appendix various notions appearing in all three parts of the article are discussed. The topics include conformal involution highly relevant to the TGD based topological view about family replication and the connection of TGD view of particle reactions at parton level with quantum homology of Gromov--Witten. Besides holomorphic 4-surfaces predicted by holography = holomorphy vision, Lagrangian minimal surfaces are allowed if the classical action is identified as the sum of volume term and Kähler action. Therefore basic facts about these surfaces are summarized. There is also a summary of exceptional unitary representations of SL(2,C) serving beside scalar monopole harmonics as a possible candidate for the wave functions for the position of the fermion line at 3-D light-like partonic orbits.

See the article About the Relationship of Electroweak and Strong Interactions in the TGD Universe: Part III 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.

About the Relationship of Electroweak and Strong Interactions in the TGD Universe: Part I

The following represents the abstract of the 1^st part of a 3-part article summarizing the view about the relationship of electroweak and strong interactions in the TGD Universe. This view has been developed together with Martin Ebner during the last almost four months. The model has evolved by trial and error process from a general view about weak and strong interactions as different aspects of color interactions coded by CP₂ geometry. Without Martin Ebner and his two AI agents the work would not have been possible.

The purpose of this three-part article is to develop a more detailed view of the relationship between electroweak and strong interactions at the level of the embedding space H. In the first part of the article the general vision is described. In the second part a concrete model for the Weinberg angle is developed, leading to a vision about the relationship of weak and strong interactions in the TGD framework. In the third part the masses of W and Z⁰ and their ratio are estimated, and questions related to the relationship of weak and strong interactions are discussed.

The TGD view of electroweak and strong interactions based on CP₂ geometry allows to interprete see strong and weak interactions as different aspects of color interaction During last almost four months we have worked together with Martin Ebner a view about how this view relates to the standard model.

My own AI skills are meager and at this age computation is far from being a pleasure. Martin together with two AI agents has made possible to develop by trial an error method a model free of any obvious problems. The model leads to a generalization of Weinberg rotation. Excellent estimates for the basic coupling parameters of the standard model and for the W/Z mass ratio emerge.

The following is the abstract of the firest part of the article having 3 parts.

The purpose of this three-part article is to develop a more detailed view of the relationship between electroweak and strong interactions at the level of the embedding space H. In the first part of the article the general vision is described. In the second part a concrete model for the Weinberg angle is developed, leading to a vision about the relationship of weak and strong interactions in the TGD framework. In the third part the masses of W and Z⁰ and their ratio are estimated, and questions related to the relationship of weak and strong interactions are discussed.

General view. Quantum TGD involves three levels corresponding to embedding space H=M⁴× CP₂, space-time X⁴ and "world of classical worlds" (WCW). Holography principle at X⁴ level implies the absence of path integral and related divergence problems. Holography = holomorphy (H-H) principle allows an exact solution of field equations. Also Lagrangian minimal surfaces are allowed as candidates for preferred extremals.

Bosons are not primary fields and correspond at H and WCW levels to bound states of fermions. At H and X⁴ level bosons correspond to induced spinor connections and apparently only electroweak gauge bosons are present. These couplings however involve implicitly vector boson couplings identified as U(1) counterparts of gluons by electric-magnetic duality. The challenge is to identify the gluon propagators at the level of X⁴.

Fermions at the H level are identified as modes of H spinor fields forming a hierarchy of color multiplets. This predicts a hierarchy of standard model physics. At the X⁴ level fermions correspond to induced spinor fields.

Construction of scattering amplitudes. Interactions are contact interactions with contacts defined by the 2-D string world sheets as intersections of space-time surfaces, including self intersections. Fermion lines correspond to intersections of string world sheets and partonic orbits.

Zero energy ontology (ZEO) and the associated mild classical non-determinism are fundamental. The loci of classical non-determinism as intersections of 3-D edges of space-time with 3-D partonic orbits define partonic 2-surfaces as geometric vertices. Feynman graphs are replaced with analogs of Brownian motion for 3-surfaces and associated fermion lines. Fermion propagators are free propagators in H. Bosonic propagators correspond to WCW correlation functions.

Q CD serves as a role model. The notions of fragmentation and hadronization are generalized. H level corresponds to observed hadrons and leptons. At X⁴ level quark gluon plasma is replaced by induced spinor fields for quarks and leptons. ZEO gives rise to statistical aspects appearing also in QCD.

Standard model and TGD. The Higgs mechanism is a key element of the standard model. p-Adic thermodynamics replaces it at the level of WCW. At the X⁴ level massivation is described in terms of warping distinguishing between GRT and TGD.

Kähler-Chern-Simons action (KCS) and corresponding modified Dirac action define the dynamics for the partonic orbits. M⁴ possesses an analog of Kähler structure and contributes to this action an additive part.

Color confinement involves two steps. First color triplets/singlets are formed from quark/leptonic color multiplets and then color triplets form hadrons. The color information is coded by WCW wave functions for the position of the fermion line and its edge at the partonic 2-surface. The 3 members of the color triplet are associated with the 3 coordinate patches of CP₂.

The condition stating that the emission amplitude for conformal photons vanishes is postulated and states conformal invariance. Electric-magnetic duality states that U(1) interaction at X⁴ level corresponds to color interaction of QCD such that the two coupling strengths are inverses of each other. A model for WCW wave functions as monopole harmonics at partonic 2-surfaces leads to a 3-D generalization of Weinberg rotation and an estimate of the Weinberg angle, identification of strong and weak interactions as parts of color interaction and estimate of the strong coupling strength.

See the article About the Relationship of Electroweak and Strong Interactions in the TGD Universe: Part I 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.

Considerable progress in understanding of neutrino masses

Juno collaboration working in China has made very interesting progress in determination of neutrino masses (see this). Google Gemini informs that the mass squared differences are unexpectedly small. The mass squared differences are reported to be Δm₂₁²=(7.50+/- 0.12)×10^-5 eV². Δm₃₁²= 2.509 ×10^-3 eV². Assuming .5 eV mass for electron neutrino, the mass differences would be about Δm₂₁=1.61× 10^-5 eV and Δm₃₁ =6.13×10^-4 eV.

In the TGD framework, p-adic thermodynamics predicts lepton masses (see this) apart from p-adic mass scale, which corresponds to a power of 2. In the TGD context the finding would mean that the p-adic length scales are the same for all 3 neutrinos. For charged fermion families (e,μ,τ) masses are predicted to differ also because they correspond to different genera g=(0,1,2). This also implies that the CKM mixing is very nearly maximal.

The first guess for p-adic mass scale would be k=167, which is the largest of the Gaussian Mersenne primes M_G,k =(1+i)^k-1, k = 151, 157, 163, 167 appearing in the length scale range 10 nm-2.5 μm assignable to the DNA. The mass of k=167 electronic neutrino is .49 eV in absence of CKM mixing. The mass for maximal mixing would be 1.285 eV, which is perhaps too large. For k=163 the mass would be scaled down to .255 eV. k=163 is a very special prime in that it also corresponds to an Eisensteinian Mersenne prime. TGD predicts an entire scale hierarchy of standard model physics, and one can ask whether the p-adic mass scales for neutrinos could correspond to Eisensteinian Mersenne primes. There is empirical evidence for the appearance of this hierarchy as time scales of biological evolution (see this).

See for instance the article A refined view of the phenomenology of hadron physics and p-adic mass calculations and the chapter Particle Massivation in TGD Universe.

The flatness of the rotation curve of the Milky Way is challenged

Sabine Hossenfelder (see this) told about the recent findings challenging the view that the rotation curve of the Milky Way is flat. The findings are discussed in the article "The rotation Curve of the Milky Way: State of the Art, the Keplerian Decline Debate, and Implications for Dark Matter" by Alexandro Melchiorri and Ruchika (see this).

Here is part of the abstract of the article:

For four decades, the roughly flat rotation curve (RC) of the Milky Way (MW) stood as local evidence for an extended dark matter (DM) halo. Gaia DR3 has changed the picture: several analyses now find a velocity decline beyond R ≈ 15 kpc, and the most radical interpretation claims a nearly Keplerian fall-off (v ∝ R^-1/2) that excludes a flat RC at 3σ and implies a total dynamical mass of only ≈ 2 × 10¹¹M_sun-three to five times below pre-Gaia estimates. Taken at face value, this would make the MW exceptional among comparable spirals and challenge both ΛCDM and MOND. It would also alter predictions for direct-detection experiments, although the local DM density is not fixed by virial mass alone.

The findings challenge not only the ΛCDM model and MOND but also the TGD view of the galaxies, galactic dark matter, and galaxy formation (this).

The TGD based model (see this) for the formation of galaxies and of also smaller structures relies on the notions of cosmic string and flux tube.

1. Galactic dark matter is identified as the volume energy and Kähler magnetic energy assignable to long cosmic strings orthogonal to the galactic plane. Also cosmic strings parallel to the plane can be considered and collisions of nearly orthogonal cosmic strings could have given rise to spiral galaxies.

   Galaxies would correspond to tangles along the cosmic strings. Thickening of the cosmic string would have liberated energy and produced ordinary matter somewhat like in the decay of the inflaton field in cosmic inflation. The cylindrical geometry of the model predicts flat velocity spectrum but also implies testable deviations from the halo model.

2. The radius of the visible part of the Milky way is about R=100,000 ly. The critical distance at which the rotation curve ceases to be flat is 15 kpc, that is 48,923 light-years, roughly R/2. The average distance between galaxies is L= 1-few million ly and suggests that the average distance of galaxies along cosmic strings is roughly L.
3. In the modification of Λ CDM, the dark mass halo should be concentrated near the centre. This implies that the dark matter of the galaxy is about 3/4 of the recent estimate and smaller than the cosmological average. For MOND the challenge is much more difficult to overcome. In TGD the galactic dark matter is along the long cosmic string so that the problem disappears.

How to explain the finding in the TGD framework?

The TGD based model for the formation of galaxies and of also smaller structures relies on the notions of cosmic string and flux tube.

1. Galactic dark matter is identified as volume energy and K\"ahler magnetic energy assignable to long cosmic strings transversal (not necessarily orthogonal) to the galactic plane. Also cosmic strings parallel to the plane can be considered and collisions of transversal cosmic strings could have given rise to spiral galaxies \cite{btart/3pieces}.

   Galaxies would correspond to tangles along the cosmic strings. Thickening of the cosmic string would have liberated energy and produced ordinary matter somewhat like in the decay of the inflaton field in cosmic inflation. The cylindrical geometry of the model predicts flat velocity spectrum but also implies testable deviations from the halo model.

2. The radius of the visible part of the Milky way is about R=100,000 ly. The critical distance at which the rotation curve ceases to be flat is 15 kpc, that is 48,923 light-years, roughly R/2. The average distance between galaxies is L= 1-few million ly and suggests that the average distance of galaxies along cosmic strings is roughly L.
3. In the modification of ΛCDM, the dark mass halo should be concentrated near the centre. This implies that the dark matter of the galaxy is about 3/4 of the recent estimate and smaller than the cosmological average. For MOND the challenge is much more difficult to overcome. In TGD the galactic dark matter is along the long cosmic string.

One can consider several modifications of this model as an explanation of the findings.

1. The monopole flux tubes and cosmic strings as extremely thin flux tubes are necessarily closed. If the return flux is along a single parallel monopole flux tube, there would be a violation of the planar asymmetry in the rotation curve caused by the weakening of gravitational force at one side and strengthening at the other side. The effect would increase at large distances and cause an approximate Keplerization. The rotation velocity would have a strong dependence on the direction in the galactic plane.
2. Could one think that the return flux is branched to less heavier flux tubes returning along a wall or could be also along thin cylindrical flux sheets? In this case one can however argue that the effect of a constant cylindrical mass distribution would however vanish in the entire interior so that this option fails.
3. The most plausible option is that a lattice of parallel cosmic strings transversal to a plane defined by the galactic plane causes Keplerization. Monopole flux tubes bring in mind ferromagnets and antiferromagnets. The cancellation of the gravitational fields of neighboring cosmic strings would cause approximate Keplerization of the velocity curve. One cannot exclude the possibility that there is a deformed 3-D network of galaxies and there are orthogonal cosmic strings along all 3 directions and connecting galaxies as vertices of a cubic lattice. Galaxies could be associated with the intersections of cosmic strings and TGD view suggests. The fluctuation spectrum of CMB should reflect the symmetries of the cubic lattices. The symmetries of the cube suggest 3-possible Axes of Evil related by symmetries of the cube. This would be a much weaker prediction than a single preferred plane in cosmological scales.

What kind of lattice?

One can consider several kinds of 2-D lattices of cosmic strings.

1. Triangular and hexagonal lattices having the same symmetries, rectangular lattices with or without center points, and oblique lattices, can be considered. The lattices spanned by a parallelogram appear as moduli spaces of toric conformal structures and of elliptic Riemannian surfaces (see this).
2. The square lattice would conform with the approximate 4-fold rotation planar symmetry assignable to the quadrupole harmonics (l=2). The quadrupole in the distribution of temperature fluctuations can be visualized as a collection of 4 charges at the vertices of parallelogram in the plane orthogonal to the Axis of Evil.

   Octupole 8 has charges at the vertices of a cube projecting to 3+3=6 charges in the plane dissecting the diagonal of the cube giving rise to a hexagon. Axis of Evil as a direction of cosmic string is therefore in the direction of the diagonal of the cube which is in the plane of the solar system (ecliptica). The hexagonal symmetry could be understood if the direction of the Axis of Evil is along ecliptica having a second body diagonal as a normal .The angle between ecliptic is 57.4 degrees and rather near to the angle between the galactic plane and ecliptica.

3. The fluctuation patterns for the CMB temperature (quadrupole and octopole) on the sky and divide the sky to two hemispheres relative to us. The octupole variation reflects itself as a division of the ecliptic to 8 angular sectors such that hot and cool sectors alternate.

The TGD view of dark matter as a phase with a larger h_eff suggests an interpretation of this kind of lattice.

1. Many-sheetedness with respect to CP₂ (space-time surface as multi-sheeted graph over CP₂) could mean this kind of lattice: different flux tubes would be sheets over CP₂. This option could be consistent with the cosmic estimate for the density of the dark matter.
2. 15 kpc= 48,923 ly gives an idea about the density of parallel cosmic strings orthogonal to the galactic plane. The distance of cosmic strings should be larger but not too much so. The radius for the visible part of the Milky Way radius is 100,000 ly and roughly twice this scale so that the distance between cosmic strings should be larger than the radius of the visible galaxy.
3. One could image a lattice of parallel cosmic strings with lattice spacing of 100-200 ly. Cosmic strings would have galaxies as tangles along them with average separation of 1-few million ly. The distance between parallel cosmic strings would be roughly by a factor 1/10 smaller if it corresponds to galactic radius. Cylindrical symmetry is however violated. The effect is largest at large enough distances and should reflect the symmetry of the planar lattice.
4. Could the galaxies along cosmic strings form 3-dimensional lattice like structure, at least in an approximate sense? The size of the visible part of the Milky Way is about 50-70 thousand ly. Total diameter is about 100,000-120,000 ly. The typical distance L between galaxies along the string would be about 1-3 million ly and roughly 10 times larger than the galactic radius R. The horizontal distance of cosmic strings could be also L in a reasonable approximation.

A quantitative estimate

One can get a very rough quantitative estimate for the logarithmic derivative of the rotation velocity at distance ρ= 15 pc≈50 ly taking the distance between centers of neighboring galaxies to be a ≈ 1 Mly. This gives x\equiv ρ/a= 1/20. Additional parameters are the rotation velocity v_K in absence of dark matter determined by galactic mass and the actual rotation velocity v_R. Google gives the estimate v_K/v_R= .57.

1. Approximating the orbits with circular orbits, Kepler's law gives

   v= (ρ F)^1/2 .

   The logarithmic derivative y== dlog(v)/dlog(ρ) = (ρ dv/d_ρ)/v is given by

   y= (1/2)[1+ (ρ dF/dρ)/F] .

2. Consider a star in the galactic plane at the line connecting the 3 cosmic strings of the lattice so that the forces due to neighboring strings are maximal. Assume that the star is near to the string at right so that the string at right tends to reduce the force due to the middle string and and the string at left tends to increase it. Besides the force caused by ordinary galactic matter, the star also experiences the forces F_M, F_R, F_L due to the 3 neighboring parallel strings in the middle, right and left. These forces are

   F_M= k/ρ , F_R= -k/(a-ρ) , F_L=k/(a+ρ).

   Here k denotes string tension: the rotation velocity in presence of only a single cosmic string would be v_S²= k (c=1).

   The total force is the sum of ordinary Kepler force F_K due to the visible mass of the galaxy and of string contributions

   F= F_K+ F_M+F_R+F_L = k₁/ρ² +k/ρ-k/(a-ρ)+k/(a+ρ) .

   It is convenient to introduce the dimensionless parameters t,x,u,v,W

   t= v_K/v_R ≈.57, x= ρ/a ≈1/20, u= x/1+x, v= x/1-x} , w= k₁/kρ .

   u and v are small parameters u≈1/20 and v≈1/20 is true in a good approximation. Using these parameters F can be written as

   F= (k/ρ)(w+ 1-v +u) .

3. The value of k can be estimated from Kepler's law v_R²=ρ F using the estimate v_K=tv_R, t=.57 and x ≈1/20.

   k= (1-t²)/(1-v+u) v_R² .

   From this the parameter w= k₁/(kρ) = v_K²/k can be estimated as

   w= v_K²/k = (1-v+u)/(1-t²) t² .

   This gives an estimate for the string tension parameter as k== v_S² as k= v_K²/w =v_R² (1-t²)/(1-v+u) ≈v_R²(1-t²). For t=.57 and x= 1/20 this gives k= .75v_R². For a single cosmic string without ordinary galactic mass one would have k=v_R².

Consider now the estimation of the parameter y=dlog(v_R)/dlog(ρ) = (ρ dv_R/dρ)/v_R providing an estimate for how rapidly the velocity starts to decrease with distance. Using Kepler's law v_R= (ρ F)^1/2 one can write

(ρ dv_R/dρ)/v_R= (1/2)[1+(ρ dF/dρ)/F] .

This gives for y=dlog(F)/dlog(ρ) and y=dlog(v_R)/dlog(ρ) the expressions

(ρ dF/dρ)/F = (-t²-1-u²+v²)(t+1-v+u) ,

y=(ρ dv_R/dρ)/v_R = (1/2)[ 1+ (-t²-1-u²+v²)/(t²+1-v+u)] .

The values of u and v are near x=1/20 and u-v ≈2x². In the approximation u= v=0 one obtains y=(1/2)[ 1+ (-t²-1/(t²+1)]=0 as expected. The value of y is y=.04. This should make it possible to test whether the assumptions of the simplest model are realistic.

Galactic warping and the satellite galaxies as evidence for the presence of cosmic strings?

Galactic warping and the satellite galaxies could be interpreted as evidence for the presence of cosmic strings.

1. The angle between Axis of Evil and the normal of the Milky Way middle plane is θ=30°. One expects that in the outer region of the galactic plane the stringy force becomes significant and tends to turn the normal of the orbital plane to the direction of Axis of Evil. Is there any evidence for this?

   The galactic plane is known to be warped, which means it has the shape of an S-curve. The Northern edge of the warp bends 4.5 kpc upward, and the Southern edge bends 3.0 kpc downward. The horizontal distance between Northern and Southern edges of "S" along the middle plane of the Milky Way is in the range 28-42 kpc. Therefore the angle between the line connecting the edges and the Milky Way middle plane is in the range θ_W=15.0-10.1 degrees so that θ_W∼ θ/2 is true.

2. If galaxies are formed in the collisions of 2 nearly orthonormal cosmic strings, there could be satellite galaxies associated with the second cosmic string in a plane nearly orthogonal to the Milky Way plane. If 3-D lattice is present, even 3 nearly orthogonal cosmic strings can be considered.

   According to Google Gemini, the Milky Way is surrounded by dozens of dwarf satellite galaxies, ranging from the well-known Large and Small Magellanic Clouds to ultra-faint collections of stars. A significant majority of the Milky Way's brightest satellite galaxies share a highly coordinated arrangement. Most of the "classical" (brightest) satellite galaxies - including the Magellanic Clouds, Fornax, Carina, Leo I, and Sculptor—are confined to a thin, structured plane known as the Vast Polar Structure (VPOS) or the Disk of Satellites (DoS). High-precision data from the Gaia space observatory confirms that 7 to 9 out of the 11 classical dwarfs are actually co-orbiting within this structure like a giant wheel spinning around our galaxy.

   The plane VPOS has an angle θ of roughly 75^° to 90^° (nearly perpendicular) with respect to the flat, middle plane of the Milky Way. Hence it could be associated with a cosmic string nearly orthogonal to the Axis of Evil. θ=60^° would mean that the two cosmic strings are orthogonal and VPOS is in the plane orthogonal to the second cosmic string.

Axis of Evil

Cosmic microwave background has a mysterious property, which is in conflict with the cosmological principle.

1. The fluctuation patterns (dipole, quadrupole and octupole) characterized by moments are are aligned along the same direction, the Axis of Evil (see this), which is defined as the normal vector for these planar fluctuation patterns and is directed along the ecliptica and points toward the constellations Leo and Virgo. Since CMB was created much before the solar system, this kind of correlation is impossible in the standard cosmology.
2. Galactic plane and the plane of the solar system (ecliptic) are tilted and intersect at an angle about 60 degrees. If the Axis of Evil as direction of cosmic strings is transversal to the galactic plane and parallel to the ecliptic it has an angle of 60 degree with respect to the galactic plane.
3. Is it possible to understand why this angle is 60 degrees?
   1. The ecliptica would contains the first body diagonal D1. The angles between body diagonals are 54.74 degrees. The angle \phi between the Milky Way plane and the plane of the solar system is 60.2 degrees? Could the normal of the galactic plane correspond to a second body diagonal D2 in a situation without symmetry breaking and could there be a slight breaking of cubic symmetry replacing 54.74 degrees with 60.2 degrees?
   2. One can also imagine another way to understand why \phi is near to 60 degrees. Cube has 8 face diagonals, which do not intersect a given body diagonal and have an angle of 60 degrees with it. The directions of diagonals of the opposite faces are the same so that there are 3 face diagonals define normals of 3 planes. Could one of these 3 face diagonals define the direction of the normal of the galactic plane.

What could cause the Axis of Evil?

1. Axis of Evil as direction of parallel cosmic string would be along the ecliptic and the projection to the ecliptic would consist of hot and cool lobes.
2. Axis of Evil means that temperature and mass density fluctuations break translational symmetry along the direction of cosmic strings. If this direction were normal to the galactic plane, a large mass in the galactic plane could explain the asymmetry. Could the galaxies along the cosmic strings, having a typical distance of million light years, somehow give rise to the anomaly?

   Could the distances of the nearest galaxies at opposite sides be different and cause a local anomaly? Or could the linear density of galaxies vary in the direction of the Axis of Evil causing global anomaly. Why so? Could one think that the formation of galaxies was a wave, a kind of phase transition, propagating in the direction of cosmic strings?

An interesting question relates to the length scale in which the parallel cosmic strings form a lattice, or more generally, in which length scale they could form a cubic lattice with galaxies as nodes connected by cosmic strings which are in 1, 2 or 3 orthogonal directions.

1. In the TGD framework, the cosmic strings would have existed already before the formation of galaxies. The planes assignable to galaxies, in particular the Milky Way, and stellar system would be determined to a high degree by this primordial cylindrical symmetry.
2. The plane of ecliptic as a preferred plane appears in cosmic scales, which suggests that it must have existed before the time when CMB was generated and reflects the presence of an effective 2-dimensionality of the Universe. If the density of galaxies varies along the cosmic strings, the CMB fluctuations would reflect this variation. This looks unrealistic.
3. A weaker proposal is that there is a 2-D or even 3-D network of nearly orthogonal cosmic strings and the CMB fluctuation spectrum has the symmetries of the cube as approximate symmetries. This would mean the existence of 2 or 3 nearly orthogonal planes for galaxies and stellar systems having angles of nearly 60.2 degrees with respect to corresponding galactic planes. As found, 57.4 degrees and 60 degrees are the options. Galaxies would be formed in the topologically unavoidable intersections of non-parallel cosmic strings giving rise to the nodes of a lattice. A kind of cosmic condensed matter physics would be in question. I have already earlier proposed a lattice-like structure of stars this) motivated by the observed gravitational hum and it could have also formed by a similar mechanism in a scale of about 5 ly. Also the galactic lattice could contribute to the gravitational hum.

See the article The flatness of the rotation curve of the Milky Way is challenged or the chapter About the recent TGD based view concerning cosmology and astrophysics.