There are more things in theory and phenomenology, my friend, than are covered in more popular blogs.

More things in theory

• Is there a theory that properly combines quantum field theory with gravity?  

• Why do we observe precisely the SM-particles, and why not more? Or less?  

• What accounts for the dark matter that astrophysicists observe? Why do all fermions appear in three copies that are identical,save for their masses?  

• What keeps the Higgs boson mass stable when considering loop corrections?  

• Why is the mass of the neutrinos so small compared to that of—say— the top quark? 

In the past decades the academic community has witnessed the birth of a plethora of theories that address one or more of the above questions. Some of them entail only minor modifications to the SM, others require us to radically reconsider the origin of the laws of nature. The hope is that there will scientific progress in the upcoming years via the falsification of many such theories by the results of the Large Hadron Collider. We live in fascinating times indeed! 

In this thesis we focus on an alternative way to obtain particle theories such as the SM on the one hand, and on a particular extension of the SM on the other. In fact, it is the combination of both that we are after. The first of these comes from the field of noncommutative geometry (NCG). Historically, this is a branch of mathematics, but it has applications in physics. From the latter point of view it can be considered as a generalization of Einstein’s theory of General Relativity in the sense that it admits spaces to exhibit some notion of noncommutativity. A particular class of noncommutative geometries —called almost-commutative geometries (ACGs)— does a marvelous job at describing gauge theories, of which the SM is an example. These ACGs are constructed by combining a commutative geometry, consisting of a curved space(time) on which there ‘live’ fermions, with a so-called finite noncommutative geometry... The constraints that are imposed on these ACGs by the axioms of NCG translate to properties of particle theories that are actually observed in experiments. NCG thus provides us new ways, of geometrical nature, to understand theories such as the SM. The latter in fact comes out as very natural in this context. 

The aforementioned extension of the Standard Model encompasses supersymmetry. This line of thought was once devised to ‘get the most’ out of quantum field theories, by using all its possible symmetries. Applying it to the SM in particular requires extending it with a set of new particles, one for each particle that we have currently observed. This leads to a theory that provides an answer to some of the fundamental open questions, raised above. The theory is called the Minimal Supersymmetric Standard Model (MSSM). At the LHC, experimenters vigorously look for signals that hint at its validity. To date, however, these have not been observed. Despite this lack of experimental success, the MSSM remains one of the prime candidates for a ‘beyond the Standard Model’ theory. 

A natural question to ask is then if noncommutative geometry and supersymmetry go well together,i.e. if the framework of NCG admits models that exhibit supersymmetry. This question has already been around for some time, but despite several previous attempts by others, its answer was still inconclusive.  

This PhD thesis is devoted mainly to address this subject and combining NCG and supersymmetry. We have restricted ourselves to the class of almost-commutative geometries,in combination with the spectral action principle, a combination that was of immense value in obtaining the SM from NCG. In addition, we have restricted our analysis  to finite KO-dimension 6, that allows us to solve the fermion doubling problem in 4 space-time dimensions.  

We have first turned to general extensions of the SM in NCG, supersymmetric or not. We have translated several physical demands (anomaly-freedom, correct hypercharges) and properties (the existence of a GUT-point) into constraints on the multiplicities of particles. The SM only satisfies these constraints when three right-handed neutrinos are added to the particle content. Although the MSSM particle content is anomaly free, it does not yield a GUT-point nor does it give the correct hypercharges. This last problem can be solved by introducing the notion of R-parity —one that is characteristic for supersymmetry— in the context of almost-commutative geometries and modifying some expressions accordingly.  

In order to answer the question of whether a certain ACG exhibits supersymmetry or not, a distinction must be made between the almost-commutative geometry itself and its associated action functional. Necessary for supersymmetry is the equality of fermionic and bosonic degrees of freedom. At the level of the ACGs, this leads us to the identification of supersymmetric building blocks and a diagrammatic approach to manage calculations. These are additions to the ACG (consisting of components of the finite Hilbert space and Dirac operator) that yield degrees of freedom eligible for supersymmetry. Since this demand must hold both on shell and off shell, we are forced to introduce (non-physical) auxiliary fields by hand, since the spectral action is interpreted as the on shell action. The requirement for the total action to actually be supersymmetric (i.e. its variation under the supersymmetry transformations vanishes) then depends on the value of the components of the finite Dirac operator. In total,we have identified five such building blocks. For each of them, the action that results corresponds in form to a term in the superfield method, in which supersymmetry is most often phrased... In the process, all formal properties of and demands on almost-commutative geometries are respected.  

Characteristic for this approach is that each new addition to an ACG provides extra contributions to the prefactors of terms that were previously already present in the action. This requires reassessing all interactions with each newly added building block. To manage this, we have set up a list with all possible terms that occur in the action and all possible contributions to them from each building block. The action is then supersymmetric if for a particular set of building blocks all the pre-factors of the terms that occur in the action can be equated to the value required for supersymmetry. At least for the most straightforward situations (a single building block of the second type, a single building block of the third type) this is not the case; the set up turns out to be over-constrained. An interesting phenomenon occurring is that in some cases the demand of a supersymmetric action puts constraints on the number of particle generations.  

Inseparable from supersymmetry is its breaking, required to give (realistic) masses to the particles appearing in the theory. We observe that soft supersymmetry breaking interactions appear automatically in the spectral action. Hence, NCG provides a new soft supersymmetry breaking mechanism. There are in fact only two supersymmetry breaking sources: the trace of the finite Dirac operators squared, which yields mass-like terms for the scalars, and gaugino masses. The second is the most prominent one. Interestingly, the gaugino masses provide a cascade of other soft breaking interactions, each of them associated to one of the five building blocks. In particular, they also give contributions to the scalar mass terms. These are of opposite sign with respect to those from the trace of the finite Dirac operator squared. This is required for the scalar mass terms to have the right sign needed to prevent them from maximally breaking the gauge group.  

This sets the stage for answering the central question, concerning a noncommutative version of the MSSM. There exists a set of building blocks whose particle content corresponds to that of the MSSM and whose fermionic interactions coincide with those of the MSSM. However, the relevant constraints on the four-scalar interactions that were mentioned above can only be satisfied for a non-integer number of particle generations. Thus, the almost-commutative geometry whose particle content is equal to that of the MSSM has a spectral action that is not supersymmetric. 

Properties of this theory hint at possible extensions of the MSSM that do satisfy all constraints, but to find it (or any other positive example of a supersymmetric NCG for that matter) requires a more constructive, and possibly automated, approach. If such a search would yield one or more positive results, these will —due to the stringency of this approach— at least enjoy a very special status.

Supersymmetry and the Spectral Action

On a geometrical interpretation of the MSSM

Thijs van den Broek

Thesis defended on 5 September 2014 (p 121)

More things in phenomenology

Although the Standard Model takes a prominent place [14,16,7] within the possible models of almost-commutative geometries one can to go further and construct models beyond the Standard Model. The techniques from the classification scheme developed in [14] were used to enlarge the Standard Model [24,25,23,26], but most of these models [24,25,23] suffer from a similar shortcoming as the Standard Model: The mass of the SMS is in general too high compared to the experimental value. Here the model in 26 will be of central interest, since it predicted approximately the correct SMS mass. In the case of finite spectral triples of KO-dimension six [2,5] a different classification leads to more general versions of the Standard Model algebra [7], under some extra assumptions. Considering the first order axiom as being dynamically imposed on the spectral triple one finds a Pati-Salam type model [9] . From the same geometrical basis one can promote the Majorana mass of the neutrinos to a scalar field [8,11] which allows to lower the SMS mass to its experimental value.

The model we are investigating extends the Standard Model [6] by N generations of chiral X₁-and X₂-particles and vectorlike V_c/w-particles. It is a variation of the model in [26] and the model in [27] . For details of the following calculations as well as the construction of the spectral triple we refer the reader to [27]. In particular its Krajewski diagram is depicted in figure 4 ? The necessary computational adaptations to the model in this publication are straightforward. The gauge group of the Standard Model is enlarged by an extra U(1)_X subgroup, so the total group is G=U(1)_Y×SU(2)_W×SU(3)c×U(1)_X. The Standard Model particles are neutral with respect to the U(1)_X  subgroup while the X-particles are neutral with respect to the Standard model subgroup G_SM=U(1)_Y×SU(2)_W×SU(3)_c. Furthermore the model contains two scalar fields: a scalar field in the SMS representation and a new scalar field carrying only a U(1)_X charge. They induce a symmetry breaking mechanism G→U(1)_em×SU(3)_c... 

From the experimental point of view the question is of course the detectability of the new particles (at the LHC?) and whether the dark sector contains suitable dark matter candidates. One would expect more kinetic mixing of U(1)_X  to the hypercharge group U(1)_Y compared to the model [27] due to the fact that here the V_c -particles may have a different masses and therefore mix the abelian groups. This poses the question how Z₀-like the Z_X-boson are and whether they are already excluded (at least for the mass range explored in this publication). The model certainly has a rich phenomenology and is one of the few known models to be consistent with (most) experiments, with the axioms of noncommutative geometry and with the boundary conditions imposed by the Spectral Action.

 Example for the mass eigenvalues m_H (bottom) and m_ϕ (top) of the scalar fields H and ϕ with respect to v₂. Here the top quark mass is taken to be m_t(m_Z ) = 173.5 GeV and the experimental SMS mass mexp. = 125.5 GeV. We obtain a heavy scalar with m_ϕ = 320 GeV and a U(1)_X-scalar boson mass m_Zx = 172 GeV.

Noncommutative Geometry in the LHC-Era

Christoph A. Stephan

(Submitted on 14 May 2013)