No SUSY dark matter

LUX group has reported that one leading dark matter candidate has disappeared (see this). Lubos tells more about this. The candidate is light fermion - so called neutralino predicted by SUSY models as a candidate for dark matter. What makes it a candidate is that it stable against decays if R-parity is conserved: this implies that neutralino can disappear only via pair annihilation. This is also a further blow against N=1 SUSY paradigm in its standard form implying among other things the non-conservation of baryon and lepton number or both.

The result of course does not mean that there would be no dark matter. It only says that the main stream of particle physics community has been at completely wrong track concerning the nature of dark matter. As I have patiently explained year after year in this blog, dark matter is not some exotic particle this or that. Dark matter is something much deeper and its understanding requires a generalization of quantum theory to include hierarchy of Planck constants. This requires also a profound generalization of the notion of space-time time.

In particular, all standard particles can be in dark phase characterized by the value of Planck constant, and the main applications are TGD inspired quantum biology and consciousness theory since dark matter with large value of Planck constant can form macroscopic quantum phases. Also dark energy in TGD sense is something very different from the standard dark energy. Dark energy in TGD Universe corresponds to Kähler magnetic energy assignable to magnetic flux tubes carrying monopole flux. These magnetic fields need no currents to generate them, which explains why cosmos can full of magnetic fields. Superconductors at the verge of breakdown of superconductivity and even ordinary ferromagnets might carry these Kähler monopole fluxes although monopoles themselves do not exist.

The result of LUX was expected from TGD point of view and does not exclude particles dark in TGD sense. Even dark particles at the mass scale of tau lepton and even at mass scale of 7-8 GeV can be considered and the CDF anomaly reported few years ago could have explanation in terms of dark variant of tau-pion identifiable as pion like bound state of colored tau leptons: also for other leptons analogous states have been reported (see this). The experimental signatures of this kind of particles are however very different from the dark particles that LUX was searching for and could explain some reports about evidence for dark matter in ordinary sense.

The lesson to learn is that one can find only what one is searching for in recent day particle physics. Particle phenomenologists should return to the roots. Challenging the cherished beliefs - even the beliefs about what QCD color is - is painful but is the only way to make progress.

A general model for metabolism

The general strategy in attempts to understand metabolism is based on the assumption that a very large class of anomalous phenomena rely on same basic mechanism. This includes life as a phenomenon, water memory and homeopathy, free energy phenomena involving over-unity phenomena related to the dissociation of water, lightning and ball lightning, anomalous effects associated with rotating magnetic systems, phenomena related to UFOs (light balls), even remote mental interactions. One must have a unified explanation for all these phenomena based on a real theory.

Plasmoid as primitive life form would the underlying connecting thread between these phenomena so that all the listed phenomena would involve life and prebiotic (or possibly postbiotic!) life. This gives very strong constraints on the model. Plasmoid should consists of the analogs of linear biomolecules, it should metabolize and communicate, in TGD Universe it should have magnetic body, and even genetic code might be realized. In particular, the simplified analog of biological metabolism would be at work. In living matter photosynthesis relies on the splitting of water whereas cell respiration relies on the reversal of this process producing carbon di-oxide and water. Something very similar should happen in free energy systems involving electrolysis, and the fact that water splitting occurs also in several free energy phenomena suggests that these processes are analogous to photosynthesis and store energy to "molecules" analogous to various linear biomolecules, in particular sugars. Even the counterpart of ADP-ATP process might be realized.

TGD suggests a very general model for the metabolism of pre-biotic systems (or post-biotic ones:the identification depends on what general vision about evolution is adopted) identifed as plasmoids consisting of cyclic linear structures formed by exotic water molecules. For a dark water molecule one proton would be dark and dark protons of the neighboring exotic water molecules would bind to form a linear structure identifiable as dark nucleus: this picture is a direct generalization of nuclear string model (see this, this, and this). These linear structures would define the analogs of linear biomolecules. This metabolism would be more fundamental than ordinary biochemical metabolism and form a yet unkown part of the latter. One cannot exclude the possibility that also other than water molecules contain dark protons: the signature would be the presence of apparently unallowed covalent bonds due to the fact the dark proton is not visible. In the following I will discuss the basic principles involved.

1. Three possible models for liberation of metabolic energy

One can imagine three different models for the liberation of metabolic energy.

1. The simplest TGD based model is as a phase transition increasing the value of p-adic prime p assignable to the space-time sheet at which particle is topologically condensed:
   
   
   1. Particle "drops" to a larger space-time sheet with larger p-adic prime p₁ with p₁/p≈ 2^k. The problem is that different particles need not drop simultaneously so that coherent liberation of energy is not automatic consequence of the assumption.
      
      
   2. The space-time sheet itself suffers a phase transition increasing its p-adic length scale. In absence of interactions (particles in box) the energies are scaled down by factor 2^-k and the difference is liberated as usable energy. Coherent liberation of energy is achieved automatically. If the particle insider the space-time sheet is free in good approximation a model as particle in box applies, and if the expansion of the space-time sheet takes place adiabatically, the quantum numbers characterizing the state of the particle do not change in the transition. As a consequence, the energy E = k∑_i n_i²hbar²/2mL_p² is reduced as L_p ∝ p^1/2 increases to L_p1, where p₁/p≈ 2^k holds true. The difference of vacuum energies is liberated as usable energy in coherent manner: this is of special significance in living systems. This has led to the identification of p-adic length scales that would correspond to fundamental metabolic quantum with value about .5 eV. Entire hierarchy of metabolic quanta is predicted.
      
      
2. The space-time sheet could also carry magnetic energy and particles are expected to be in cyclotron states and perhaps form a cyclotron Bose-Einstein condensate. In this case the phase transition reduces the value of B but preservers the magnetic flux so that B→ B/2^k, p₁/p≈ 2^k, takes place. This scales down the energies of cyclotron states by the same scaling factor 2^-k as in the case of free particle. The liberated energy is in good approximation just the cyclotron energy for large enough values of k. Coherence is achieved automatically. The value of the fundamental metabolic energy quantum and the value of endogenous magnetic field of about B_end=.2 Gauss deduced from the experiments of Blackman and others fix the value of h_eff. It would be proportional to particle mass number A.
   
   
3. The earlier model for the liberation of cyclotron energy was based on the assumption that the value of B is not changed but that the value of magnetic quantum number n changed. If n is reduced one achieves liberation of energy. Coherence of the transition might produce problems now. Both models can explain the observations of Blackman and others concerning the effects of ELF radiation on vertebrate brain since the spectrum of photons energies inducing effects correspond to cyclotron energies for the latter option and in excellent approximation to it for the previous model. The mechanism is however quite different.
   
   

This phase transition for the larger space-time sheet can take place in two steps.

1. First a phase transition increasing h_eff of the background space-time sheet by n=2^k occurs. This leaves ZPKE invariant but scales up the size of the space-time sheet by 2^k/2. The interpretation would be as "electric expansion" of Brown's gas. No energy transfer takes place since both kinetic and magnetic energies are invariant under the scaling of hbar. Note however than in the original situation the magnetic field can be very strong so that zooming up from microscopic scales can happen.
   
   
2. After this a phase transition reducing Planck constant back to h but increasing p-adic length scale by 2^k occurs. The size scale of the background space-time sheet is not affected but the zero point kinetic energy is reduced by factor 2^-k and liberated as usable energy. This phase transition would take place for the dark component of Brown's gas in the melting of the metal and other similar phenomena. Also the liberation of metabolic energy in living matter could correspond to this phase transition.
   

This model for electric expansion, implosion, and energy liberation assumes nothing about the particles involved since dark particle means ordinary particle topologically condensed on dark space-time sheet and having wave function de-localized in the n-sheeted structure. For instance, water can be dark in this sense. One could indeed consider the possibility that the vapour phase identified as charged water cluster is just water containing positive ions H³₊ or protons and electrons and that phase transition to large hbar phase expands the space-time sheet at which water is topologically condensed at evaporates the water. Ordinary liquid to gas transition could proceed in the same manner and involve liberation of ZPKE at the second step of the process. In the general case the binding energy involved with the formation of the denser phase could compensate for the energy gain in the increase of the p-adic prime so that the melting would require energy feed.

2. Model for the building bricks of plasmoids

I have already earlier discussed a model for dark proton sequences as primitive life forms. The observation discussed by Moray B. King inspired a more detailed formulation of the model of plasmoids identified as primitive life forms in TGD framework.

1. The key observation was that the model for dark nuclei (see this and this), in particular dark proton, predicts counterparts of DNA, RNA, tRNA, and amino-acids and also vertebrate genetic code follows naturally. This together with nuclear string model led to the vision that life appears already at the level of dark variants of nuclei. The observed anomalous H_1.5O stoichiometry of water in atto-second scale supports the view that dark protons appear in ordinary water.
   
   
2. This model was first introduced to explain water memory and homeopathy. The basic idea was that the process creating homeopathic remedy induces the analog of molecular evolution for the dark proton sequences, which in turn provide representations for the molecules appearing in environment. These representations would be fundamental also for the functioning of immune system of living matter. The dark life could provide R&D laboratory for living matter allowing to test say various gene candidates and transcribe them to ordinary biological DNAs if they are successful in the virtual model world. Evolution would not be random but directed just as evolution of technologies.
   
   
3. The latest step in the process (see this) was the proposal that cell membranes involve dark proton sequences providing a representation of dark DNA and connected by magnetic flux tubes to the units of DNA in genome. These two DNA representations would be identical. Quite generally, dark and ordinary biomolecules might be connected by magnetic flux tubes.
   
   

This picture does not yet provide model for the metabolism of the building bricks of plasmoids. Something very much analogous to the splitting of sugars to carbon di-oxide and water is however expected. Since carbon is not present now, this leaves only the option that the linear dark structures are nothing but exotic form of water for which the proton of one hydrogen atom of each water molecule is dark. These dark protons would combine by strong interactions to a nuclear string and O-H groups would be attached to them. The cyclic analog of DNA, RNA, or amino-acid realizing genetic code would be the outcome. The stoichiometry H_1.5O observed in atto-second time scale would be achieved in average sense if the portions of exotic and dark water are same. The prediction is that dark water is heavier than ordinary water: the molecular weight would correspond to average length of the dark water cycle. This is consistent with the observations about Brown's gas.

Plasmoid should also possess a magnetic body. This requires a currents rotating along the cyclic structures. The obvious identification of the current is as dark supra currents assignable to dark protons so that the building bricks of plasmoid would be analogous to super-conducting rings.

3. Model for the metabolism of plasmoids

The proposed dark analogs of basic biomolecules would be created through the analog of photosynthesis involving the splitting of water to H + OH followed by H→ H_dark and by recombination to a sequence of dark water molecules. The process would be analogous to translation of mRNA to amino-acids and could proceed by an analogous mechanism. The process would be spontaneous since the energy of cyclotron states would not change in h→ h_eff= 2^k× h.

Metabolic energy would be liberated in the decay of the exotic water back to water with h_eff=h and p-adic prime scaled by about 2^k. This process is completely analogous to the splitting of various linear biomolecules in metabolism in order to obtain metabolic energy. This process would explain the ability of cool Brown's gas to melt metal for instance. When fossil fuels are used, the outcome is carbon di-oxide and water. Now only water is obtained so that this form of free energy might not contribute to the warming of environment.

The process differs from ZPE in that it does not provide any endless source of energy. Since water is in practice an unlimited natural resource, this shold not be a problem. A closed cycle at the level of visible matter is obtained only if the reverse phase transition transforming the water with h_eff=h and p-adic prime p₁≈ 2^k/2p to that with h_eff=2^k× h and p-adic prime p takes place spontaneously.

The irradiation with carrier frequency f_h and modulation frequency f_l such that one has f_l/f_h= 2^-k is one possibility which I have proposed. Dark solar radiation at magnetic flux tubes with magnetic field B_end=.2 Gauss (guess from the experiments of Blackman; also many other values can be considered) could provide automatically the needed pulsed radiation inducing the phase transition. The most optimistic option is that this transition occurs even in the case of closed system in which water circulates.

Before attempting to identify reasonable candidates for f_l and f_h it is useful to consider estimates for h_eff/h=2^k. Note that this assumption might be too strong: the vision about evolution as emergence of number theoretical complexity suggests that so called Fermat integers defining polygons, which are constructible using ruler and compass, define favored values of h_eff/h=n (see this). These integers are expressible as products of different Fermat primes F_n= 2^2k+1 and power of 2. The known Fermat primes correspond to k= 0,1,2,3,4 and are 3,5,17,257,65537. Only the two lowest ones differ significantly from power of two.This raises the question whether also the scale hierarchies 3^1/2L(k), 5^1/2L(k), and 15^1/2L(k) are important besides p-adic length scale hierarchy L(k)= 2^k/2R_CP2. They could be associated with the algebraic extensions of p-adic numbers involving 3^1/2 and5^1/2.

1. The condition that cold nuclear fusion is possible via the TGD based mechanism requires dark variant of weak interactions corresponds to scaled up p-adic length scale of order atomic size. The condition that weak bosons are effectively massless in atomic length scale gives one estimate for h_eff/h. The condition that weak scale characterized by M₈₉ is increased to that characterized by M₁₂₇ gives h_eff/h=2⁴⁸≈ 2.8× 10¹⁴.
   

   
   

2. Second estimate for h_eff/h follows from the condition that cyclotron energy for given charged particle is of the order of metabolic energy quantum. For proton B_end=.2 Gauss gives f_c=300 Hz. The energy is about .5 eV for h_eff/h= 1.37×10¹⁴ rather near to h_eff/h= 2⁴⁷ which is by a factor of 1/2 smaller than the previous estimate. It is however clear that the estimates are internally consistent: skeptic would see this as a pure accident and some-one taking anthropic principle seriously as an outcome of evolution in very general sense. Note that for electron the metabolic energy quantum would be about 938 eV suggesting that keV energy scale assignable to the dark weak interactions has its own metabolic energy quantum.
   

   For ion of mass number A and ionization z the value producing the same value of metabolic quantum is A/z× 1.37× 10¹⁴. An alternative assumption is a hierarchy of metabolic quanta coming as z/A multiples of the fundamental metabolic energy quantum for a fixed value of h_eff/h. The condition that the metabolic energy quantum is above thermal energy of photon at physiological temperature for which peak wavelength for blackbody radiation corresponds to energy of .13 eV. This gives A/z≤ .5/.13=3.84. The estimate is too stringent since Ca⁺⁺ with A/z=20 should allow metabolic energy quantum above the thermal energy. This suggests that h_eff/h characterizes given ion and that its multiples coming as power of two are allowed.
   
   

3. For h_eff/h=n=2^k_dark with k_dark ∈ {47,48} dark electron would have p-adic length scale L(k), k=127+ d_dark∈ [174,175]. This corresponds to a Compton length l_c∈ [28,4] μm. That this corresponds to the size scale of cell gives additional support for the vision. Note also that for electron the size scale of CD identified as secondary p-adic time scale associated with M₁₂₇=2¹²⁷-1 corresponds to .1 seconds, which defines a fundamental biorhythm. Proton Compton length would be scaled to the range [15,21] nm (10 nm defines the thickness of the cell membrane) and light current quarks with energy of 5-20 MeV to the size scale of cell nucleus.
   
   

A reasonable guess is that the candidates for f_h and f_l should satisfy the condition f_h/f_l=2^k, k=47 or k=48. f_h can be deduced from the estimate for h_eff.

1. Schumann frequency 7.8 Hz is the first candidate for the modulating frequency. This would give UV frequency f_h≈ 1.1 × 10¹⁵ Hz corresponding to energy of 9.7 eV for k=47, which corresponds to the energy scale for covalent bonds. The energy scale of hydrogen atom is 13.6 eV.
   
   
2. For the cyclotron frequency of DNA (which depends only weakly on the length of the DNA sequence due to the constant charge density per unit length) of about 1 Hz (the frequency of heart beat) one would obtain f_h=1.4× 10¹⁴ Hz for k=47, which corresponds to energy of 1.4 eV and is just below the visible range starting around 1.65 eV. The scaling of this energy by 3^1/2/2, 5^1/2/4, and15^1/2/4. For k=48 the energy would be to 3.3 eV, which is quite near to the UV end 3.36 eV of visible portion of spectrum. Again one can ask whether just accidents are in question.
   

   Allowing the generalization of the p-adic length scale hypothesis one obtains 7 photon energies in the visible range corresponding to the scalings of 1.4 eV by [3^1/2/2, 5^1/2/4, 5^1/2/2, , 15^1/2/4, 3^1/2, 2, 3^1/2/8,5^1/2] giving E/eV= [1.71, 1.57, 2.21, 1.91, 2.42, 2.71, 2.80, 3.13]. Note that 2 eV corresponds to red light and metabolic energy quantum of .50 eV to k=51. An interesting question is whether these special frequencies relate to the peak wave lengths for color vision.
   
   

A macroscopic variant of photosynthesis using the possibly existing dark photons at the flux tubes of B_end=.2 Gauss can be imagined. The flux tubes of B_end could correspond to those of B_E with nominal value .5 Gauss if a weakening of the field value takes place inside living matter. Note that in case of h_eff/h∼ 10¹⁴ this field value would correspond to about 10¹⁰ Tesla for the ordinary value of hbar (a field strengths assignable to supernovas!) and assignable to electron Compton scale.

The sequences of these two phase transitions involved with dark metabolism would be very much analogous to ..-ATP-ADP-ATP-... "Karma's cycle". There is also a strong analogy with breathing and even sleep-wake-up cycle and longer bio-rhythms. p-Adic fractality forces to ask whether all these rhythms involve the same dark metabolic cycle but in different scales. Increase of h_eff indeed corresponds to an increase of "IQ" in TGD inspired theory of consciousness and its reduction to its lowering. This could quite concretely correspond the experience of becoming tired. There is also a close analogy with the state function reduction sequence in ZEO. State function reductions occur alternatively at the opposite boundaries of causal diamond (CD) of given scale and I have proposed an interpretation in terms of generalized sleep-awake cycles.

4. Does dark biology represent pre- or post-biotic evolution?

The discovery of dark proton realization of genetic codons (see this and this) was an accident and I am still puzzled about whether the vertebrate genetic code can really emerge from dark nuclear physics or is it only a curiosity or self deception. The first interpretation for the dark code is as a code associated with prebiotic evolution (see this). This is suggested by the enormous simplicity for the analogs of counterparts of linear biomolecules, and the fact that the utilization of metabolic energy means that these "molecules" decay to ordinary water. In this view life would have migrated from dark space-time sheets to visible space-time sheets. This higher level life would be gradually migrating to lower levels in the hierarchy and taking visible matter to its control and that biological evolution represents a step in this process.

There are however some objections against this view. The dark code corresponds to vertebrate code, which can be seen as an outcome of along genetic evolution. There are also other codes, which are less perfect (see the chapter of "Genes and Memes" representing a number theoretic approach to genetic code). For instance, the meaning of the codeword is context dependent for some codons and Peter Gariaev has proposed that this context dependence is a more general phenomenon. One would expect that prebiotic code is much simpler than genetic code and I have considered a model for how genetic code might have emerged from more primitive codes with 4 and 16 code words as a "product code" (see this).

These objections inspire the question whether life could migrate from lower to higher scales. The dark genetic code would in this framework correspond to the emergence of a new level in evolution - perhaps identifiable as cultural evolution. This would explain why dark variant of the genetic code corresponds to vertebrate code. One could also solve Fermi paradox (see this) due to the fact that no signs of intelligent life have been observed in cosmos and probabilistic estimate suggests that cosmos is full of life. The answer could be very simple: in some stage the civilization transforms to dark matter invisible to us! The civilizations are there but living on magnetic flux quanta and probably communicate with us telepathically. The higher evolutionary level would also conform with the fact that the spatial and temporal scales of consciousness are much longer than for the consciousness assignable to visible manner. This could allow also to understand the mystery of crop circles. To my opinion many of them are genuine, and the interpretation as some kind of cognitive representations analogous to those realized in brain is highly suggestive. Certainly these representations would represent mental images of conscious entities, which are at higher evolutionary level than us.

Many great leaps in evolution have occurred via crisis periods involving extinction. Could it be that gradual transition to dark matter based life could be begin as a response to the recent crises of human kind? The gradual transition of life to the dark matter level would indeed solve the energy problem by coupling us to the energy sources assignable to the dark matter hierarchy at various magnetic bodies. It would also solve the problem caused by the climate warming if it is indeed is due to the liberation of CO₂ as fossil fuels are used. The dark matter "molecules" as analogs of biomolecules and hydrocarbons would produce only water when used.

See the new chapter Macroscopic quantum coherence and quantum metabolism as different sides of the same coin: Part II of "Bio-Systems as Conscious Holograms" for details.

The TGD variant of the model of Widom and Larsen for cold fusion

Widom and Larsen (for articles see the Widom Larsen LENR Theory Portal) ) have proposed a theory of cold fusion (LENR), which claims to predict correctly the various isotope ratios observed in cold fusion and accompanying nuclear transmutations. The ability to predict correctly the isotope ratios suggests that the model is on the right track. A further finding is that the predicted isotope ratios correspond to those appearing in Nature which suggests that LENR is perhaps more important than hot fusion in solar interior as far as nuclear abundances are considered. TGD leads to the same proposal and Lithium anomaly could be understood as one implication of LENR (see this). The basic step of the reaction would rely on weak interactions: the proton of hydrogen atom would transform to neutron by capturing the electron and therefore would overcome the Coulomb barrier.

Challenges of the model

The model has to meet several challenges.

1. The electron capture reaction p+e→ n+ν is not possible for ordinary atom since the mass difference of neutron is 1.3 MeV and larger than electron mass .5 MeV (electron has too small kinetic energy). The proposal is that strong electric fields at the catalyst surface imply renormalization effects for the plasmon phase at the surface of the catalyst increasing electron mass so that it has width of few MeVs (see this. Physically this would mean that strong em radiation helps to overcome the kinematical threshold for the reaction. This assumption can be criticized: the claim is that the mass renormalization is much smaller than claimed by Widom and Larsen.
   
   
2. Second problem is that weak interactions are indeed very weak. The rate is proportional to 1/m_W⁴, m_W∼ 100 GeV whereas for the exchange of photon with energy E it would be proportional to 1/E⁴. For E∼ 1 keV the ratio of the rates would be of the order of 10^-48!
   

   This problem could be circumvented if the transition from proton to neutron occurs coherently for large enough surface patch. This would give rate proportional to N², where N is the number electrons involved. Another mechanism hoped to help to get high enough reaction rate is based on the assumption that the neutron created by the capture process has ultra-low momentum. This is the case if the mass renormalization of electron is such that the energies of the neutrons produced in the reaction are just above the kinematical threshold. Note however that this reduces the electon capture cross section. The argument is that the absorption rate for neutron by target nucleus is by very general arguments proportional to 1/v_n, v_n the velocity of neutron. Together these two mechanisms are hoped to give high enough rate for cold fusion.
   
   

3. The model must also explain why gamma radiation is not observed and why neutrons are produced much less than expected. Concerning gamma rays one must assume that the heavy electrons of the plasmon phase assigned to the surface of the catalyst absorb the gamma rays and re-emit them as infrared light emitted to environment as heat. Ordinary electrons cannot absorb gamma rays but heavy electrons can (see this), and the claim is that they do transform gamma rays to infrared photons. If the neutrons created in LENR have ultra-low energies their capture cross sections are enormous and the claim is that they do not get out of the system.
   
   

The assumption that electron mass is renormalized so that the capture reaction can occur but occurs only very near threshold so that the resulting neutrons are ultraslow has been criticized (see this).

TGD variant of the model

TGD allows to consider two basic approaches to the LENR.

1. Option I involves only dark nucleons and dark quarks. In this case, one can imagine that the large Compton length of dark proton - at least of order atomic scale - implies that it overlaps target nucleus, which can see the negatively charged d quark of the proton so that instead of Coulomb wall one has Coulomb well.
   
   
2. Option II involves involves both dark weak bosons and possibly also dark nucleons and dark electrons. The TGD inspired model for living matter - in particular, the model for cell membrane involving also Z⁰ membrane potential in the case of sensory receptor neurons (see this) - favors the model involving both dark weak bosons, nucleons, and even electrons. Chiral selection for biomolecules is extremely difficult to understand in standard model but could be understood in terms of weak length scale of order atomic length scale at least: below this scale dark weak bosons would be effectively massless and weak interactions would be as strong as em interactions. The model for electrolysis based on plasmoids identified as primitive life forms supports also this option. The presence of dark electrons is suggested by Tesla's cold currents and by the model of cell membrane.
   

   This option is fixed quantitatively by the condition that the Compton length of dark weak bosons is of the order of atomic size scale at least. The ratio of the corresponding p-adic size scales is of order 10⁷ and therefore one has h_eff∼ 10¹⁴. The condition that h_eff/h=2^k guarantees that the phase transion reducing h_eff to h and increasing p-adic prime p by about 2^k and p-adic length scale by 2^k/2 does not change the size scale of the space-time sheet and liberates cyclotron magnetic energy E_n(1-2^-k) ≈ E_n.
   
   

Consider next Option II by requiring that the Coulomb wall is overcome via the transformation of proton to neutron. This would guarantee correct isotope ratios for nuclear transmutations. There are two options to consider depending on whether a) the W boson is exchanged between proton nucleus (this option is not possible in standard model) or b) between electron and proton (the model of Widom and Larsen relying on the critical massivation of electron).

1. Option II.1. Proton transforms to neutron by exchanging W boson with the target nucleus.
   
   
   1. In this case kinematics poses no obvious constraints on the process. There are two options depending on whether the neutron of the target nucleus or quark in the neutral color bond receives the W boson.
      
      
   2. If electron and proton are dark with h_eff/h=n=2^k in the range [10¹²,10¹⁴] the situation can change since W boson has its usual mass from the point of view of electron and proton. hbar⁴/m_W⁴ factor in differential cross section for 2-to-2 scattering by W exchange is scaled up by n⁴ (see the appendix of kenocitebthe/Iztykson so that effectively m_W would be of order 10 keV for ordinary hbar.
      
      
   3. One can argue that in the volume defined by proton Compton length λ_p≈ 2^-11 λ_e ∈ [1.2, 12] nm one has a superposition of amplitudes for the absorption of dark proton by nucleus. If there are N nuclei in this volume, the rate is proportional to N². One can expect at most N∈ [10³,10⁶] target nuclei in this volume. This would give a factor in the range 10⁹-10¹².
      
2. Option II.2: Electron capture by proton is the Widom-Larsen candidate for the reaction in question. As noticed, this process cannot occur unless one assumes that the mass of electron is renormalized to have a value in a range of few MeV. If dark electrons are heavier than ordinary, the process could be mediated by W boson exchange and if the electron and proton have their normal sizes the process occurs with same rate as em processes.
   

   If electron and proton are dark with h_eff/h=n ∈ [10¹²,10¹⁴] the situation can change since W boson has its usual mass from the point of view of electron and proton. 2-to-2 cross section is proportional to hbar⁴ and is scaled up by n⁴. One the other hand, the naive expectation is that |Ψ (0)|²∝ m_e³/h_eff³ ∝ 1/n^-3 for electron is scaled by n^-3 so that the rate is increased by a factor of order n∈ [10¹²,10¹⁴] (electron Compton length is of order cell size scale! instead of Angstrom) from its ordinary value. This is not enough.
   

   On the other hand, one can argue in the volume defined by proton Compton size one has a superposition of amplitudes for the absorption of electron. If there are N dark electrons in this volume, the rate is proportional to N². One can expect at most 10⁶ dark electrons in the volume of scale 10 nm so that this could give a factor 10¹². This would give amplification factor 10²⁶ to the weak rate so that it would be only by two orders of magnitude smaller than the rate for massless weak bosons.
   
   

There are also other strange features to be understood.

1. The absence of gamma radiation could be due to the fact that the produced gamma rays are dark. For h_eff/h ∈ [10¹²,10¹⁴] the energy frequency of 1 MeV dark gamma ray would correspond to that of photon with energy of [1,.1] μeV and thus to radiowave photon with wavelength of order 1 m and frequency of orer 3× 10⁸ Hz. In Widom-Larsen model the photons would be infrared photons. The decay of the dark gamma ray to a bunch of ordinary radiowave photons should be observed as radio noise. Note that Gariaev has observed transformation of laser light scattered from DNA to radio wave photons with frequencies down to 1 kHz at least.
   
   
2. The absence of the neutrons could be understood if they are dark. The absorption cross section is proportional to hbar³ giving a huge amplification factor in the range [10⁹,10¹²]. This implies that they are absorbed by nuclei coherently in a a volume of order 1.2-12 nm so that an additional amplification factor N²∈ [10⁹,10¹²] would be obtained. And the capture rate is amplified by a factor in the range [10¹⁸,10²⁶]. Effectively this corresponds to the assumption of Widom and Larsen stating that neutrons have ultra-low momentum.
   

The natural question is why h_eff us such that the resulting scale as photon wavelength corresponds to energy in scale 10-100 keV. The explanation could relate to the predicted exotic nuclei obtained by replacing some neutral color bonds connecting nucleons with charged ones and exchange of weak boson would affect this replacement. Could the weak physics associated with h_eff∈ [10¹²,10¹⁴] be associated with dark color bonds? The reported annual variations of the nuclear reaction rates correlating with the distance of Earth from Sun suggest that these variations are induced by solar X rays (see this).

For background see the chapter Nuclear Physics and Condensed Matter of "p-Adic Length Scale Hypothesis and Dark Matter Hierarchy".