### Criticality of Higgs: is Planck length dogmatics physically feasible?

While studying the materials related to Convergence conference running during this week at Perimeter institute I ended up with a problem related to the fact that the mass M

_{h}= 125.5+/- .24 GeV implies that Higgs as described by standard model (now new physics at higher energies) is at the border of metastability and stability - one might say near criticality (see this and this), and I decided to look from TGD perspective what is really involved.

Absolute stability would mean that the Higgs potential becomes zero at Planck length scale assumed to be the scale at which QFT description fails: this would require M_{h}>129.4 GeV somewhat larger that the experimentally determined Higgs mass in standard model framework. Metastability means that a new deep minimum is developed at large energies and the standard model Higgs vacuum does not anymore correspond to a minimum energy configuration and is near to a phase transition to the vacuum with lower vacuum energy. Strangely enough, Higgs is indeed in the metastable region in absence of any new physics.

Since the vacuum expectation of Higgs is large at high energies the potential is in a reasonable approximation of form V= λ h^{4}, where h is the vacuum expectation in the high energy scale considered and λ is dimensionless running coupling parameter. Absolute stability would mean λ=0 at Planck scale. This condition cannot however hold true as follows from the input provided by top quark mass and Higgs mass to which λ at LHC energies is highly sensitive. Rather, the value of λ at Planck scale is small and negative: λ(M_{Pl})=-0.0129 is the estimate to be compared with λ(M_{t})=0.12577 at top quark mass. This implies that the potential defining energy density associated with the vacuum expectation value of Higgs becomes negative at high enough energies.The energy at which λ becomes negative is in the range 10^{10}-10^{12} GeV, which is considerably lower than Planck mass about 10^{19} GeV. This estimate of course assumes that there is no new physics involved.

The plane defined by top and Higgs masses can be decomposed to regions (see figure 5 of this), where perturbative approach fails (λ too large), there is only single minimum of Higgs potential (stability), there is no minimum of Higgs potential (λ<0, instability) and new minima with smaller energy is present (metastability). This metastability can lead to a transition to a lower energy state and could be relevant in early cosmology and also in future cosmology.

The value of λ turns out to be rather small at Planck mass. λ however vanishes and changes sign in a much lower energy range 10^{10}-10^{12} GeV. Is this a signal that something interesting takes place considerably below Planck scale? Could Planck length dogmatics is wrong? Is criticality only an artefact of standard model physics and as such a signal for a new physics?

How could this relate to TGD? Planck length is one of the unchallenged notions of modern physics but in TGD p-adic mass calculations force to challenge this dogma. Planck length is replaced with CP_{2} length scale which is roughly 10^{4} longer than Planck length and determined by the condition that electron corresponds to the largest Mersenne prime (M_{127}), which does not define completely super-astrophysical p-adic length scale, and by the condition that electron mass comes out correctly. Also many other elementary particles correspond to Mersenne primes. In biological relevant scales there are several (4) Gaussian Mersennes.

In CP_{2} length scale the QFT approximation to quantum TGD must fail since the the replacement of the many-sheeted space-time with GRT space-time with Minkowskian signature of the metric fails, and space-time regions with Euclidian signature of the induced metric defining the lines of generalized Feynman diagrams cannot be anymore approximated as lines of ordinary Feynman diagrams or twistor diagrams. From electron mass formula and electron mass of .5 MeV one deduces that CP_{2} mass scale is 2.53× 10^{15} GeV - roughly three orders of magnitudes above 10^{12} GeV obtained if there is no new physics emerges above TeV scale.

TGD "almost-predicts" several copies of hadron physics corresponding to Mersenne primes M_{n}, n=89, 61, 31,.. and these copies of hadron physics are expected to affect the evolution of λ and maybe raise the energy 10^{12} GeV to about 10^{15} GeV. For M_{31} the electronic p-adic mass scale happens to be 2.2× 10^{10} GeV. The decoupling of Higgs by the vanishing of λ could be natural at CP_{2} scale since the very notion of Higgs vacuum expectation makes sense only at QFT limit becoming non-sensical in CP_{2} scale. In fact, the description of physics in terms of elementary particles belonging to three generations might fail above this scale. Standard model quantum numbers make still sense but the notion of family replication becomes questionable since in TGD framework the families correspond to different boundary topologies of wormhole throats and the relevant physics is above this mass scale inside the wormhole contacts: there would be only single fermion generation below CP_{2} scale.

This raises questions. Could one interpret the strange criticality of the Higgs as a signal about the fact that CP_{2} mass scale is the fundamental mass scale and Newton's constant might be only a macroscopic parameter. This would add one more nail to the coffin of superstring theory and of all theories relying on Planck length scale dogmatics. One can also wonder whether the criticality might somehow relate to the quantum criticality of TGD Universe. My highly non-educated guess is that it is only an artefact of standard model description. Note however that below CP_{2} scale the transition from the phase dominated by cosmic strings to a phase in which space-time sheets emerge and leading to the radiation dominated cosmology would take place: this period would be the TGD counterpart for the inflationary period and also involve a rapid expansion.

For a summary of earlier postings see Links to the latest progress in TGD.

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