- The 3.7 sigma deviation from standard model predictions in the differential distribution of the B→ K
^{*}μ^{+}μ^{-}decay products.

- The 2.6 sigma violation of lepton flavor universality in B
^{+}→ K^{+}l^{+}l^{-}decays.

^{+}→ K

^{+}e

^{+}e

^{-})/B(B

^{+}→ K

^{+}μ

^{+}μ

^{-})≈ .75 holds true. Standard model expectation is very near to unity.

Scalar lepto-quark has been proposed as an explanation of the anomaly. The lowest order diagram for lepton pair production in standard model is penguin diagram obtained from the self energy diagram for b quark involving tW^{-} intermediate in which W emits γ/Z decaying to lepton pair. Lepton universality is obvious. The penguin diagram involves 4 vertices and 4 propagators and the product of CKM matrix elements V_{tb}V^{*}_{st}. The diagram involving leptoquark is obtained from this diagram by a modification.

The diagram would induce an effective four-fermion coupling bbar_{L}γ ^{μ}s_{L} μ^{+}_{L}γ_{μ} μ^{-}_{L} representing neutral current breaking universality. Authors propose a heavy scalar boson exchanges with quantum numbers of lepto-quark and mass of order 10 TeV to explain why no anomalous weak interactions between leptons and quarks by lepto-quark exchange have not been observed. Scalar nature would suggest Higgs type coupling proportional to mass of the lepton and this could explain why the effect of exchange is smaller in the case of electron pair. The effective left-handed couplings would however suggest vector lepto-quarks with couplings analogous to W boson coupling. Note that the effect should reduce the rate: the measured rate for B_{s} → μ^{-}μ^{+} is .79+/- .20: reduction would be due to destructive interference of amplitudes.

**General ideas**

Some general ideas about TGD are needed in the model and are listed in order to avoid the impression that the model is just ad hoc construct.

- In TGD all elementary particle can be regarded as pairs of wormhole contacts through which monopole magnetic flux flows: two wormhole contacts are necessary to get closed magnetic field lines. Monopole flux in turn guarantees the stability of the wormhole contact. In the case of weak bosons second wormhole contact carries fermion and antifermion at opposite throats giving rise to the net charges of the boson. The neutrino pair at the second wormhole contact neutralize the weak charges and guarantees short range of weak interactions.

- The TGD inspired explanation of family replication phenomenon is in terms of the genus of the partonic 2-surfaces (wormhole throat) at the end of causal diamond. There is topological mixing of partonic topologies which depend on weak quantum numbers of the wormhole throat leading to CKM mixing. Lepton and quark families obvious correspond to each other: L(g)↔ q(g) and this is important in the model to be considered.

The genera of the opposite wormhole throats are assumed to be identical for bosonic wormhole contacts. This can be assumed also for fermionic wormhole contacts for which only second throat carries fermion number. The universality of standard model couplings inspires the hypothesis that bosons are superpositions of the three lowest genera forming singlets with respect effective symmetry group SU(3)

_{g}associated with the 3 lowest genera. Gauge bosons involve also superpositions of various fermion pairs with coefficients determined by the charge matrix.

- p-Adic length scale hierarchy is one of the key predictions of TGD. p-Adic length scale hypothesis (to be used in the sequel) stating that p-adic primes are near powers of of 2: p≈ 2
^{k}, k integer, relies on the success of p-adic mass calculations. p-Adic length scale hypothesis poses strong constraints on particle mass scales and one can readily estimate the mass of possible p-adically scaled up variants of masses of known elementary particles.

One of the basic predictions is the possibility of p-adically scaled up variants of ordinary hadron physics and also of weak interaction physics. One such prediction is M

_{89}~ hadron physics, which is scaled up variant of the ordinary M_{107}hadron physics with mass scale which is by a factor 512 higher and corresponds to the energy scale relevant at LHC. Hence LHC might eventually demonstrate the feasibility of TGD.

Quite generally, one can argue that one should speak about M

_{89}physics in which exotic variants of weak bosons and scaled up variants of hadrons appear. There would be no deep distinction between weak bosons and M_{89}hadrons and elementary particles in general: all of them would correspond to string like objects involving both magnetic flux tubes carrying monopole flux between two wormhole throats and string world sheets connecting the light-like orbits of wormhole throats at which the signature of the induced metric changes.

- TGD predicts dark matter hierarchy based on phases with non-standard value h
_{eff}=n× h of Planck constant. The basic applications are to living matter but I have considered also particle physics applications.

- Dark matter in TGD sense provides a possible explanation for the experimental absence of super partners of ordinary particles: sparticles would be dark and would be characterized by the same p-adic mass scales as sparticles

- TGD predicts also colored leptons and there is evidence for meson like bound states of colored leptons. Light colored leptons are however excluded by the decay widths of weak bosons but also now darkness could save the situation.

- I have also proposed that RHIC anomaly observed in heavy ion collisions and its variant for proton heavy ion collisions at LHC suggesting string like structures can be interpreted in terms of low energy M
_{89}hadron physics but with large value of h_{eff}meaning that the M_{89}p-adic length scale increases to M_{107}p-adic length scale (ordinary hadronic length scale).

- Dark matter in TGD sense provides a possible explanation for the experimental absence of super partners of ordinary particles: sparticles would be dark and would be characterized by the same p-adic mass scales as sparticles
- TGD view about gauge bosons allows to consider also lepto-quark type states. These bosons would have quark and lepton at opposite wormhole throats. One can consider bosons which are SU(3)
_{g}singlets defined by superpositions of L(g)q(g) or L(g)qbar(g). These states can be either M^{4}vectors or scalars (all bosons are vectors in 8-D sense in TGD by 8-D chiral symmetry guaranteeing separate conservation of B and L). Left handed couplings to quarks and leptons analogous to those of W bosons are suggested by the model for the anomalies. Vector lepto-quarks can be consistent with what is known about weak interactions only if they are dark in TGD sense. Scalar lepto-quarks could have ordinary value of Planck constant.

**A TGD based model for the B anomaly in terms of lepto-quarks**

It is natural to approach also the anomaly under discussion by assuming the basic framework just described. The anomaly in the decay amplitude of B→ Kμ^{-}μ^{+} could be due to an additional contribution based on a simple modification for the standard model amplitude.

- In TGD framework, and very probably also in the model studied in the article, the starting point is the penguin diagram for lepton pair production in B→ Kμ
^{-}μ^{+}decay involving only the decay b→ sl^{+}l^{-}by virtual tW state emitting virtual γ/Z decaying to lepton pair and combining with t to form s.

- The diagram for lepton pair production involving virtual lepto-quark is obtained from the tW
^{-}self-energy loop for b. One can go around the W^{-}branch of the loop to see what must happen. The loop starts with b→ tW^{-}followed by W^{-}→ l^{-}(g_{1}) νbar(g_{1}) producing on mass shell lepton l^{-}(g_{1}). This is followed by νbar(g_{1})→ s X(Dbarνbar) producing on mass shell s. The genus of the virtual neutrino must ge g=1 unless

leptonic CKM mixing is allow in the W decay vertex.

After this one has X=∑Dbar(g)νbar(g) → Dbar(g

_{2})νbar(g_{2}). Any value of g_{2}is possible. Finally, one has tDbar→ W^{+}and W^{+}νbar(g_{2})) → l^{+}(g_{2}). There are two loops involved and four lines contain a heavy particle (two W bosons, t, and X). The diagram contains 6 electroweak vertices whereas the standard model diagram has 4 vertices.

- All possible lepton pairs can be produced. The amplitude is proportional to the product V
_{tb}V^{*}_{tD(g2)}implying breaking of lepton universality. The amplitude for production of e^{+}μ^{-}pair is considerably smaller than that for μ^{+}μ_{-}and τ^{+}μ_{-}as the experimental findings suggest. If neutrino CKM mixing is taken into account, there is also a proportional to the matrix element V^{L}_{l(g1)νg=1}.

In absence of leptonic CKM mixing (mixing explains the recently reported production of μ

^{+}e^{-}pairs in the decays of Higgs) only μ^{-}l^{+}(g) pairs are produced. The possibility to have g≠ 1 is also a characteristic of lepton non-universality, which is however induced by the hadronic CKM mixing: lepto-quark couplings are universal.

Note that the flavour universality of the gauge couplings means in the case of lepto-quarks that Lq pairs superpose to single SU(3)

_{g}singlet as for ordinary gauge bosons. If L(g)q(g) would appear as separate particles, only μ^{+}μ^{-}pairs would be produced in absence of leptonic CKM mixing.

- The diagram for lepton pair production involving virtual lepto-quark is obtained from the tW
- A rough estimate for the ratio r of lepto-quark amplitude A(b→ sl
^{-}(g_{1})l^{+}(g_{2}) to the amplitude A(b→ sl^{-}(g)l^{+}(g) involving virtual photon decaying to l^{+}l^{-}pair is

z =[X

_{1}/X_{2}]× [F_{1}(x_{X},x_{t})/F_{2}(x_{t})],

X

_{1}=V^{L}_{l1ν(g=1)}[∑_{g}V^{L}_{l(g2)ν(g)}

V^{*}_{D(g)t}]V_{tD(g2)}g_{ X}^{2}g_{W}^{2},

X

_{2}=V^{*}_{dt}e^{2},

x

_{X}= m^{2}(X)/m^{2}(W) , x_{t}= m^{2}(t)/m^{2}(W) .

The functions F

_{i}correspond come from the loop integral and depend on mass ratios appearing as the argument. The factors X_{i}collect various coupling parameters together.

- The objection is that the model predicts a contribution to the scattering of leptons and quarks of the same family (L(g)-q(g) scattering) by the exchange of lepto-quark, which is of the same order of magnitude as for ordinary weak interactions. This should have been observed in high precision experiments testing standard model if the mass of the lepto-quark is of the same magnitude as weak boson mass. 10 TeV mass scale for lepto-quarks should guarantee that this is not the case and is probably the basic motivation for the estimate of the article. This requires that the ratio of the loop integrals appearing in z is of the order of unity. For a processional it should be easy to check this. Since the loop integral in the case of scalar lepto-quark studied in the article has the desired property and should not depend on the spin of the particles in the loops, one has good reasons to expect that the same holds true also for vector lepto-quarks.

Without precise numerical calculation one cannot be sure that the loop integral ratio is not too large. In this case one could reduce the gauge coupling to lepto-quarks (expected to be rather near to weak coupling constant strength) but this looks like ad hoc trick. A more adventurous manner to overcome the problem would be to assume that lepto-quarks represent dark matter in TGD sense having effective Planck constant h

_{eff}=n× h. Therefore they would not be visible in the experiments, which do not produce dark matter in elementary particle length scales.

- The proposal of the article is that lepto-quark is scalar so that its coupling strength to leptons and quarks increases with mass scale. If I have understood correctly, the motivation for this assumption is that only in this manner the effect on the rate for e
^{+}e^{-}production is smaller than in the case of μ^{+}μ^{-}pair. As found, the presence of CKM matrix elements in lepto-quark emission vertices at which quark charge changes, guarantees that both anomalous contributions to the amplitude are for electron pair considerably smaller than for muon pair.

- Consider first a mass estimate for
*dark vector lepto-quark*assumed to have weak boson mass scale. Even the estimate m(X)∼ m(W) is much higher than the very naive estimate as a sum of μ^{-}and s masses would suggest. Quite generally, if weak bosons, lepto-quarks, and M_{89}hadrons are all basic entities of same M_{89}physics, the mass scale is expected to be that of M_{89}hadron physics and of the order of weak mass scale. A very naive scaling estimate for the mass would be by factor 512 and give an estimate around 50 GeV. If μ^{-}mass is scaled by the same factor 512, one obtains mass of order 100 GeV consistent with the estimate for the magnitude of the anomaly.

Second p-adic mass scale estimate assumes vector or scalar lepto-quark with mass scale not far from 10 TeV. Ordinary μ

^{-}corresponds to Gaussian Mersenne M_{G,k}, k=113. If p-adically scaled up variant of lepton physics is involved, the electron of the p-adically scaled up lepton physics could correspond to M_{89}. If muons correspond to Gaussian primes then the scaled up muon would correspond to the smallest Gaussian Mersenne prime below M_{89}, which is M_{G,79}. The mass of the scaled up muon would be obtained from muon mass by scaling by a factor 2^{(113-79)/2}=2^{17}=1.28× 10^{5}giving mass of order 10 TeV, which happens to be consistent with the conservative estimate of the article.

See the chapter New Particle Physics Predicted by TGD of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy" or the article Have lepto-quarks been observed in the decays of B mesons?.

## 8 comments:

I would propose a different view on LeptoQuarks, by a Geometrical proposal for the sub-quantum structure of 6 Leptons and all Quarks interpreted as compound Photon/Electron Positron propeller combinations.

Muon- and Tau particles seem to show up as family members of the Down respectivily Charm Quarks.

https://www.flickr.com/photos/93308747@N05/14604202116/in/photostream

The extreme stability of the proton could be related to the complexity or binding interlocking of the d- quark ( three particles) which I did also recognize as the Muon.

A second reason could be the assumed symmetric geometry of the proton with the sensitive u-quark ( with only 2 interlocking particles) in the middle and sturdy d-quark at the outside.

See perhaps also: http://vixra.org/pdf/1103.0002

Hi Leo,

I wonder how you get electroweak quantum numbers and quark color in this framework. Same question about flavour universality.

Matti,

I am an architect focussing of Form AND Function of complex knot particles, in relation to known decay examples.

For the W and Z, I would suggest to look at the W an Z knots presented at figure 10, of the article I already mentioned:

http://vixra.org/pdf/1103.0002

For the quark colour differences: see figure 4 at the same article.

For the leptons , see: figure 3.

Robert Finkelstein wrote an article about such knotty ideas called:

THE ELEMENTARY PARTICLES AS QUANTUM KNOTS IN ELECTROWEAK THEORY Robert J. Finkelstein

http://arxiv.org/pdf/0705.3656v1.pdf

I think weneed lots of ontological sound speculations about the vacuum and black holes.

AND I think that Stephen Hawking did not calculate with the possibility of a chiral oscillating Higgs field vacuum lattice combined with propeller shaped Fermions. Then, Electrons and positrons could both pushed away from the BH horizon at different distances, forming two charged separated spheres. With quark ( plasma) formation in between. So Black Holes could be charge splitters violating the 2e law af thermodynamics, combined with a continuous microscopic big bang plasma creation process!

The result I try to describe :

1: Black holes are the same as Dark Matter, they all consume photons, even gravitons and the Higgs field, but REPEL Fermions due to their propeller shape. They produce electric charged plasma.

2: Dark Energy is the oscillating ( Casimir) energy of the Higgs Field equipped with a tetrahedron lattice structure with variable Planck length..

3: Quantum Gravity = Dual Push gravity= Attraction (Higgs-Casimir opposing Graviton push).

4: The Big Bang is a Splitting dark matter Big Bang Black Hole (BBBH), splitting into smaller primordial BBBH Splinters forming the Fractalic Lyman Alpha forest and evaporating partly into a zero mass energetic oscillating Higgs particle based Higgs field.

5: Dual PBBSs hotspots, produce central plasma concentration in electric Herbig Haro systems as a base for star formation in open star clusters as a start for Spiral Galaxies.

6: Spiral Galaxies will keep both Primordial Dark Matter Black Holes as Galaxy Anchor Black Holes (GABHs) at long distance.

7: After Galaxy Merging, these GABHs are the origin of

Galaxy- and Magnetic field complexity and distant dwarf galaxies .

8: Black Holes produce Plasma direct out of the Higgs field because two Higgs particles are convertible into symmetric electron and positron (or even dual quark-) propellers (by BH horizon fluctuations).

9: The chirality of the (spiralling) vacuum lattice is the origin our material universe. (propeller shaped positrons merge preferentially first with gluons to form (u) Quarks to form Hydrogen.

10: The first Supernovas produce medium sized Black Holes as the base for secondary Herbig Haro systems and open star clusters.

11: ALL Dark Matter Black Holes are supposed to be CHARGE SEPARATORS with internal positive charge and an external globular shell of negative charged Quark electron plasma.

12: The lightspeed is related to gravity fields like the earth with long extinction distances to adapt with the solar gravity field.

See also: vixra.org/author/leo_vuyk

In addition:

Based on such a simple object ( propeller and process) oriented ontology, Black Holes could be imagined as charge splitters violating the 2e law af thermodynamics, combined with a continuous microscopic big bang plasma creation process!

You mention knots. I believe that knottiness makes sense at elementary particle level but that effects are extremely delicate. According to an estimate that I made years ago, they are totally out of reach of recent day technology since effects on mass

values are so incredible small.

The scattering amplitudes in the generalisation of twistor approach to TGD based on 8-D twistors can be seen as analogs of braid invariants. The fermion lines identified as light-like boundaries of string world sheets at the orbits of partonic 2-surfaces are braided.

Braiding would however have nothing to do with elementary particle quantum numbers and would not be visible at QFT limit. Since the only fundamental fermionic vertex is 2-vertex there are no infinities and OZI rule holds true: hence braids.

TGD can be seen as a generalisation of braid theory in this approach: almost topological QFT. The construction of scattering amplitudes would be analogous to that for knot invariants: in particular, the problems caused by non-planar diagrams would be solved by recursion transforming non-planar diagrams to superpositions of planar diagrams. Nima et al indeed suggest reduction to planar ampitudes for MHV amplitudes.

The idea about association of function to objects of size of elementary particle looks weird. Certainly the notion of function goes outside the realm of existing physics since it requires the notions of intention, goal, and behavior, and consciousness.

In Zero Energy Ontology however systems are 4-D and even elementary particles with non-standard value of Planck constant (h_eff/h=n) could have self lasting for few quantum jumps (repeated state function reductions at same boundary of CD). This would give rise to self-organisation and "behaviour" in very primitive sense. Even elementary particle or at least its dark variant with non-standard Planck constant could have behavioral patterns in very primitive sense. I however find difficult to speak about purpose of life of electron;-).

Hi Matti,

Thank you for your interesting reaction, about the purpose of life of the electron I deformed some of the ideas of Bohm and Vigier described in::

http://vixra.org/pdf/1111.0061v1.pdf

called:

Black hole horizon curvature dependent balance between plasma creation and e-e+ annihilation in Quantum FFF Theory.

At Page: 26, I describe the parallel of my propeller fermions, pushed around by the oscillating Higgs field with Bohm Vigier ideas::

“David Peat described in his book "Infinite Potential" the gist of the problem, as follows: "Bohm argued that his "quantum potential" guides the electron in a non-mechanical way. But how exactly it operates was less clear.

Vigier favored explaining the process in terms of some sort of underlying mechanism- a subquantum

fluid, perhaps. In his opinion the electron exchanges energy and momentum with

this fluid and in this sense the electron is "pushed along" (by the fluid).

Bohm went along with Vigier for a time, but eventually he felt that his theory was being

forced too far back in the direction of Classical Physics. The two men simply drifted apart in

their philosophies."

...For Vigier, however, the crisis of the theory lay in the problem of nonlocality. The question

of nonlocality is basic to the original debate between Bohr and Einstein on "Independent

elements of reality".

Einstein argued that if the quantum objects are far enough apart, they must be independent;

since quantum theory could not account for this fact, then the theory had to be incomplete.

Bohr countered that quantum systems are an "Un-analysable Whole". This wholeness, present

in Bohm's theory, implied non-locality."

However, in this paper, it is assumed that Dual Anti-Mirror Symmetry and BigBangEntanglement, (BBE) is the base for consciousness on all levels of existence. At the same time, we must be able in the future to

measure the "Un-analysable Whole" of multiple Copy Universes, by more precise human

intention/reaction experiments as Benjamin Libet did before.

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