Following the Higgs and top quark trails through the "big desert", chasing a complex scalar singlet and right-handed neutrinos...

 ... riding two loop renormalisation group flow, guided by the spectral standard model

The Lagrangian of the spectral standard model comes from the most general form of the Dirac operator consistent with axioms of noncommutative geometry plus an additional constrain called the first order condition. This Lagrangian possesses three important features distinguishing it from the minimal standard model. First, the couplings of the model are not totally arbitrary and there are relations between them at the unification scale. These relations are consistent with grand unified theories such as SU(5) unified theory. Second, there is a singlet scalar field present in the spectral action. It is shown that this field can help the situation with low Higgs mass which is not otherwise consistent with the unification of spectral action in high energies [6]. We will see in this letter that the results improve if the scalar field is taken to be complex. It is also seen that such an extra scalar field can be responsible for dark matter particle [1, 14]. Finally, right-handed neutrino appears into the picture automatically as well as its Yukawa interaction. These terms are needed to give a small mass to the left-handed neutrino by see-saw mechanism and usually are added to the standard model by hand...

In [6], the singlet scalar field was assumed to be real... In fact, the reality condition on the singlet field is not necessary and we assume the singlet to be a complex field in this work. Our consideration shows the model in its most general form is consistent with the current experimental values of the Higgs and top quark masses. Furthermore, we use 2-loop renormalisation group (RG) equations to compare the following cases: when the added singlet is a complex field, when it is real, and the pure standard model with neutrino mixing. We show that while running RG equations from unification scale toward current experimental energies, the model with added complex singlet behaves slightly better than the other two cases. Yet, like the standard model itself, one can only attain the experimentally observed gauge couplings at low energies within some percent of accuracy. This agrees with the separations of the standard model gauge at the unification scale when we start from experimental values and run them upward...

We ... show ... that... a few extra terms are added to the RG equations due to the complex singlet field, ... their effect on the negativity of the Higgs self-coupling at high energies can be substantial. The reason we cannot predict what exactly happens for the coupling is that the experimentally unknown right-handed neutrino Yukawa coupling contributes in the RG equations as well. This coupling also plays a role in determining the Higgs and top quark masses at low energies. What we can do is to follow its effect by following RG equations down and looking at the particle masses. The proper value of right-handed neutrino Yukawa coupling - turns out to be between 0.411 and 0.455 at unification scale ... The resulting value for this coupling at Z boson mass region is also between 0.517 and 0.530, while Yukawa coupling of the top quark is about 0.995. Besides, the values of scalar sector couplings are derivable in this scale from RG equations. We argue that in this acceptable range of the couplings, although vacuum instability is not cured, but the situation is improved by the presence of the complex scalar field. We use two-loop equations and near to the leading order three-loop equations to assess the loop correction effects in presence of a complex or real singlet field.

Each line shows the suitable values of unification scale U (logarithmic scale normalized by Z boson mass) and the squared ratio between the Yukawa couplings of tau-neutrino and top quark labelled n, in order to retrieve the experimental values of Higgs (dotted lines) or top quark (solid lines) masses at low energies from two-loop renormalisation group running down. Each set of three lines is for a specific unification gauge coupling value g and is illustrated with a particular thickness. It can be inferred from this figure that within a reasonable range of g, the lines associated with top quark and Higgs always have a collision point. Therefore suitable n and U can always be found to fit the low energy values for the Higgs and top quark masses. This is true for both real and complex cases which are distinguished by blue and green lines.

Comparison of the Higgs self-coupling running from unification scale down to the electroweak scale for the standard model with neutrino mixing and for the spectral standard model containing a real or complex singlet scalar.  The figure shows that the addition of new fields could improve the stability problem of the Higgs vacuum. (Warning : the unified gauge coupling value used here to show the effect of the scalar singlet on the eletroweak vacuum stability does not lead to the correct gauge couplings at low energy. This is a generic feature of spectral standard model which can be overcome abandoning the "big desert" hypothesis with a Pati Salam type scalar extension [10] not studied here)

We stress that {these} results are not merely derivable from the standard model plus a complex singlet. The reason is that in our considerations, we use the initial conditions predicted by the spectral action approach [8]. Moreover here a neutrino coupling is present in RG equations and contributes to the values of particle masses. The form of potential is also restricted and is different from extended standard model cases with complex singlet described in the literature. In our case, the results for stability are slightly better (e.g. compare with [11, 14, 17, 20]).

{slightly adapted from} Implications of the complex Singlet field for Noncommutative Geometry model

Hosein Karimi Khozani (Submitted on 3 May 2017)

Next time at the Pati-Salam caravanserai

... in the simple version that we considered here ... one could not fully {retrieve} the gauge couplings at low energies. Yet, the...  better results {presented here} with this minimal change in the settings of the standard model might urge us to investigate {extended} models derived from noncommutative geometry principles {without the order one condition for example)... Further investigations showed in 2014 that imposing a generalized version of Heisenberg uncertainty relations leads to Pati-Salam models as the most general possible outcome of this approach [10]. The model we considered here is the simplest special case of that general theory. The Pati-Salam model has a rich content of beyond SM fields that might help the situation and will be the subject of our further investigations.

 Id.

A personal comment (updated on May 9)

The article quoted here was written in all likelihood by a Ph D. student of Ali Chamseddine, one of the father of the spectral standard model. It does a nice job in adding a piece to the fascinating picture puzzle of unification of fundamental forces and also to the spectral unification model I was dreaming about in the last post.

I regret nevertheless there is no mention of the doctoral work of Shane Farnsworth who has proposed an original motivation for considering as complex the scalar singlet derived from the spectral action principle by Chamseddine and Connes in their 2010 article (see here for a vivid description of their collaboration). 

In light of the promissing evidence already uncovered, it will be a tremendous achievement to compute the renormalisation group equations from the specific Pati-Salam extension of the spectral standard model and a fantastic piece of information to find out proper initial conditions able to recover all known low energy data after running down from the spectral unification scale.  

Even more exciting would be exploring the quite constrained scalar spectrum in spectral Pati-Salam models and testing in parallel the vanilla leptogenesis scenario. It has been already tested successfully in nonsupersymmetric SO(10) models based on a hierarchical or a compact right-handed neutrino spectrum with normal ordered masses. This kind of test was shown - under the last hypothesis at least - to be able to fix all the parameters of the model thanks to neutrino mixing data and the amount of baryon asymmetry in the visible universe!