Thinking one more (six? ;-) time(s) about the Higgs mass

A farewell to low energy susy
We think that there is one clue, perhaps, provided by the latest result from the Large Hadron Collider:[5,6] the mass, 125 GeV, found for the Higgs particle, is a very special value. The Higgs mass has been the one unknown parameter in the Standard Model, but, since the vacuum value of the Higgs field is precisely known from the W and Z mass and Fermi’s interaction constant for the weak force, having the Higgs mass now also yields the Higgs self-interaction parameter, λH. The value of λH, following from the given value of the Higgs mass, appears to be very special. One can compute how λH changes as we look at different energy scales. It is a running coupling parameter. If mH ≈ 125 GeV, the Higgs self-coupling parameter runs almost to zero at very high energies [8]
This means that the renormalization group β function rapidly runs to zero at high energies, a property it will have in common with the other β functions that describe how the gauge field forces run to zero. Apparently, our field theory of the subatomic particles, at sufficiently high energies, becomes scale-invariant 
Why is this? What do we have to infer from that?  
If there is scale invariance at higher energies, the masses of heavy particles would be at odds with this, and this could explain why we have not seen them, and also the massive superpartners of the SM particles, expected by many investigators, would be at odds with this symmetry. Is this why we see no heavy particles at all? Then, what about the preferred explanation of ‘dark matter’? It was always assumed to consist of WIMPs, ‘Weakly Interacting Massive Particles’. WIMPs would also have to be forbidden. So, perhaps LHC gave us a hint, but the hint is difficult for us to understand. It is more likely that the role played by scaling transformations will be a more subtle one, [9] and that it will not forbid the occurrence of heavier masses in the system, if the particles associated with these heavy masses, interact sufficiently weakly, as, indeed, is likely to be the case with the dark matter particles.
It is also the case with gravity. In units where = c = 1, Newton’s constant GN has dimension length-squared, or inverse mass-squared. The associated length, called Planck length, is very small, and accordingly, the associated mass, the Planck mass, is very large... This length unit is very tiny, even at the scale of subatomic particles, and the mass is very large compared to subatomic particles. Nevertheless, one can elegantly restore exact scale invariance in perturbative quantum gravity. This, we do by observing that only the truly constant quantities in a theory, such as interaction constants, must be dimensionless if we want scale invariance. In contrast, the metric tensor gµν, which determines the distance scales between neighboring points in space–time in terms of metres, is actually a dynamical variable. Like many other dynamical variables of a theory, it may have nontrivial dimensions even if the theory itself is scale invariant. 

Writing 

gµν (x, t) ≡  ω2(x, t)gµν(x, t) , (7)
we can take the field ω2(x, t) to have dimension of a length — just like all other fields in the Standard Model — while all components of the tensor gµν(x, t) are kept dimensionless. 
In perturbative quantum gravity, we now keep gµν close to the identity matrix ηµν ≡diag(−1,1,1,1). In that case, however, we must postulate that ω stays close to one in the geometrically flat vacuum, while in general it may fluctuate. Or, 
we say:
<∅|ω(x, t)|∅> = 1 . (8)
This means that scale invariance is a gauge symmetry that is spontaneously broken, a situation that one often encounters in quantum field systems. 
Actually, in gravity, we also have general coordinate transformations, so we can also say that the theory has local conformal invariance, which is spontaneously broken. In that case, Eq. (8) expresses the fact that we have a BEH mechanism here. We can choose the gauge such that ω2(x, t)=1, or we can fix the gauge constraint in some other way. In either case, we have a new local gauge symmetry, and this is a very important observation shedding a different light on quantum gravity. 
One can even argue that gravity herewith becomes a renormalizable theory [15], but before arriving at such a conclusion, one would have to add kinetic terms for the gµν field [10,11,12]. Such terms can be written down (the Weyl action), but that ruins positivity: the Weyl action adds a negative metric massive spin-2 particle to the system, something that will be difficult to accept, as this is believed to make our theory internally inconsistent. 
The Weyl action, however, seems to be such a fundamental interaction, that some of us suspect it can be used anyway, inviting us to think again about stability of theories and the exact role of indefinite metric particles...  in a local conformally invariant theory.
Gerard ’t Hooft
published 9 June 2016

Sur un air de géometrie spectrale
Thanks to David Broadhurst for stressing the following point during my lecture at a summer school in Les Houches. 
Whilst introducing gauge fields from noncommutative spin manifolds (aka spectral triples) I first explained how the Dirac operator can be seen as a metric on a (possibly noncommutative) space described via Connes’ distance formula. Then the action of a unitary in the algebra of coordinates was given as a gauge transformation on the Dirac operator, generating a pure gauge field. 
What David noticed was that this is in compelling agreement with Weyl’s old idea of gauge invariance. Indeed, the term Eichinvarianz was preceded by Maβstabinvarianzin the original work (see Yang’s review ...). This is precisely the notion captured by noncommutative geometry: a gauge transformation actually acts on the metric (the Dirac operator) but leaves the distance function invariant.
Perturbations of the metric and Weyl’s Eichinvarianz
posted in Blog on June 19, 2014 by Walter van Suijlekom


A long Hello again to Weyl invariance
In this note we will show the intimate relationships between Weyl anomalies, the dilaton and the Higgs field in the framework of spectral physics. The framework is the expression of a field theory in terms of the spectral properties of a (generalized) Dirac operator. In this respect this work can be seen in the framework of the noncommutative geometry approach to the standard model of Connes and collaborators..., as well as of Sakharov induced gravity [5] (for a modern review see [6]).  
We start with a generic action for a chiral theory of fermions coupled to gauge fields and gravity. The considerations here apply to the standard model, but we will not need the details of the particular theory under consideration. It is known, and this is the essence of the noncommutative geometry approach to the standard model, that the theory is described by a fermionic action and a bosonic action, both of which can be expressed in terms of the spectrum of the Dirac operator. In [7] two of us have shown that if one starts from the classic fermionic action and proceeds to quantize the theory with a regularization based on the spectrum, an anomaly appears. it is possible that the full quantum theory is still invariant by correcting the path integral measure. This is tantamount to the addition of a term to the action, which renders the bosonic background interacting to the dilaton field. The main result of that paper is that this term is a modification of the bosonic spectral action [3]. In this case the theory is still invariant. 
In this paper we have a shift of the point of view. We still consider the theory to be regularized in the presence of a cutoff scale, but we consider this scale to have a physical meaning, that of the breaking of Weyl invariance. We then consider the flow of the theory at a renormalization scale, which is not necessarily the scale which breaks the invariance. The theory has a dilaton, and the Higgs field. 
The dilaton may involve a collective scalar mode of all fermions accumulated in a Weyl-non-invariant dilaton action. Accordingly the spectral action arises as a part of the fermion effective action divided into the Weyl non-invariant and Weyl invariant parts. 
We calculate the dilaton effective potential and we discuss how it relates to the transition from the radiation phase with zero vacuum expectation value of Higgs fields and massless particles to the electroweak broken phase via condensation of Higgs fields. The collective field of dilaton can provide the above mentioned phase transition with EW symmetry breaking during the evolution of the universe.
(Submitted on 16 Jun 2011)

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