The Higgs Mass and the Emergence of New Physics

We investigate the physical implications of formulating the electroweak (EW) part of the Standard Model (SM) in terms of a superconnection involving the supergroup SU(2/1). In particular, we relate the observed Higgs mass to new physics at around 4 TeV. The ultraviolet incompleteness of the superconnection approach points to its emergent nature. The new physics beyond the SM is associated with the emergent supergroup SU(2/2), which is natural from the point of view of the Pati-Salam model. Given that the Pati-Salam group is robust in certain constructions of string vacua, these results suggest a deeper connection between low energy (4 TeV) and high energy (Planck scale) physics via the violation of decoupling in the Higgs sector.


Introduction and Overview
The Standard Model (SM) of particle physics is a phenomenally successful theory whose last building block has recently been detected [1,2]. In light of the apparent discovery of the Higgs boson, we address the connection between its mass and the structure of the electroweak (EW) sector of the SM, and argue that it points to some very exciting new physics at a rather low energy scale of 4 TeV.
A long time ago, Ne'eman [3] and Fairlie [4] independently discovered the relevance of a unique SU (2/1) supergroup structure to SM physics. In this formalism, the even (bosonic) part of the SU (2/1) algebra defines the SU (2) × U (1) gauge sectors of the SM, while the Higgs sector is identified as the odd (fermionic) part of the algebra. Although the model gives the correct quantum numbers of the SM, and it represents a more unified-hence more aesthetic-version of the SM, it suffers from the violation of the spin-statistic theorem, a common problem seen in the models using supergroups. 1 In this work we adopt the superconnection approach of Ne'eman and Sternberg [5] who observed that the SU L (2) × U Y (1) gauge and Higgs bosons of the SM could be embedded into a unique SU (2/1) superconnection, and the quarks and leptons into SU (2/1) representations [6,7]. SU (2/1) in this formalism is not imposed as a symmetry; it is rather only the structure group of the superconnection. Therefore, the SU (2/1) structure can be interpreted as an emergent geometric pattern that involves the EW part of the SM, which avoids the problems with the ghosts. 1 For example, there are anticommuting Lorentz scalars (the Higgs fields) which represent ghost-like degrees of freedom in the model.
The formalism fixes the ratio of the SU L (2) × U Y (1) gauge couplings, and thus the value of sin 2 θ W , and the quartic coupling of the Higgs. The value of sin 2 θ W selects the scale Λ ∼ 4 TeV at which the superconnection relations can be imposed 2 , and renormalization group (RG) running leads to a prediction of the Higgs mass. However, the claim of Refs. [6,7] that the predicted Higgs mass is around 130 GeV turns out to be incorrect.
In this Letter, we point out that the SU (2/1) superconnection approach predicts the mass of the Higgs to be 170 GeV, which obviously disagrees with observation. Given the well-known issue with the ultraviolet incompleteness of the SU (2/1) approach [6], which implies the emergent nature of this description, we should have no qualms in introducing new physics to fix the Higgs mass.
Here, we note a connection with the Spectral SM of Connes and collaborators [8,9] in which spacetime is extended to a product of a continuous four dimensional manifold by a finite discrete space with non-commutative geometry. The SM particle content and gauge structure are described by a unique geometry, where the Higgs appears as the connection in the extra discrete dimension [10]. Curiously, the original Higgs mass prediction of the Spectral SM was also 170 GeV [11], despite the fact that the boundary conditions imposed on the RG equations were quite different: in the Spectral SM, the usual SO(10) relations among the gauge couplings are imposed at the GUT scale. In a recent paper [12] Chamseddine and Connes isolate a unique scalar degree of freedom that is responsible for the neutrino Majorana mass in their approach, which, when correctly coupled to the Higgs field, can reduce the mass of the Higgs boson to the observed value, 125 ∼ 126 GeV. 3 We argue that a similar 'fix' works for the superconnection formalism: one needs to introduce extra scalar degrees of freedom which modify the RG equations. We further point out that this can be accomplished by the embedding of SU (2/1) into SU (2/2), and thus, in effect, a left-right (LR) symmetric extension of the EW sector [13], which is also natural from the point of view of the Pati-Salam model [14]. The SU (2/2) formalism, as in the SU (2/1) case, selects the scale Λ ∼ 4 TeV via the value of sin 2 θ W . Therefore, 4 TeV in this formalism is the prediction for the energy scale of new physics, which is the LR symmetric model in this case.
We also note the peculiarity of the Higgs sector, which due to the relation between the coupling and the mass, violates decoupling [15]. When interpreted from either the emergent superconnection or the non-commutative geometry viewpoint, this violation of decoupling offers an exciting connection between the SM and short distance physics, such as string theory, via the non-decoupling of the 4 TeV and the Planck scales.
In particular, the embedding of SU (2/1) into SU (2/2) would be interesting from the point of view of string vacua, where it has been observed that the Pati-Salam group appears rather ubiquitously in a large number of vacua [16]. Though we lack a fundamental understanding of this phenomenon, it is quite intriguing in our context as it would point to a new relationship between low energy (SM-like) and high energy physics (quantumgravity-like) which is not seen in the standard effective field theory approach to particle physics.

The SU(2/1) formalism and the Higgs mass
Here we summarize the superconnection approach to the SM based on the supergroup SU (2/1) [6,7]. Obviously, this supergroup has as its bosonic subgroup the EW gauge group SU (2) L × U (1) Y . What is highly nontrivial is that the embedding of SU (2) L × U (1) Y into SU (2/1) also gives the correct quantum numbers for all the physical degrees of freedom. Furthermore, the Higgs sector comes out naturally as a counterpart of the gauge sector. These have natural analogs in the Spectral SM as well [8,9,12], as already emphasized in the conclusion to the review Ref. [6]. We concentrate on the superconnection formalism which should be understood as an emergent framework, because of the fundamental ultraviolet incompleteness of gauged supergroup theories.
We start by defining the supercurvature as F = d J + J · J where J is the superconnection, which is of the form Since we would like to embed SU L (2) × U Y (1) and the Higgs into SU (2/1), M and N are respectively 2 × 2 and 1 × 1 g-even submatrices valued over one-forms, while φ and φ are respectively 2 × 1 and 1 × 2 g-odd submatrices valued over zero-forms. The superconnection J is written as J = iλ a s J a , a = 1, 2, · · · , 8. The generators λ a s are matrices with supertrace zero. Therefore, they are the usual SU (3) λ-matrices except for λ 8 s which is To obtain the superconnection we need to make the identifications J i = W i (i = 1, 2, 3) and J 8 = B, where W i and B are one-form fields corresponding to the SU L (2) and U Y (1) gauge bosons. The zero-form fields are identified as Then, the superconnection is Here, W = W i τ i (where τ i are the Pauli matrices) and I is a 2 × 2 unit matrix, and Φ = φ + φ 0 T . To obtain the supercurvature F , we recall the rule for supermatrix multiplication [5,7] A where |A| denotes the Z 2 grading of the differential form A. Then, the supercurvature (after introducing the dimensionless coupling g, J → gJ ) reads as The Action reads as follows where the ⋆ on F ⋆ denotes taking the Hermitian conjugate of the supermatrices and the Hodge dual (denoted as * ) of the differential forms, and λ ≡ 2g 2 . Note that we need to break SU (2/1) explicitly in order to introduce the Higgs mass. In 4 dimensions we have the following explicit form of the Lagrangian (given the metric g µν = diag(1, −1, −1, −1)): Note that the explicit forms of the curvature strengths and the covariant derivatives have the standard forms: To switch to the common SM convention we rescale g and g ′ as g, g → g/2, g ′ /2 (which is the missing part in [7]) which also changes our constraint at the symmetry breaking energy to λ = g 2 /2. 5 Now we address the prediction for the Higgs mass. In what follows we use the relation M 2 H = 8M 2 W (λ/g 2 ) and the RG equations for λ and top Yukawa coupling g t which are where g ′ , g, and g s are the U (1) Y , SU (2) L and SU (3) c coupling constants, respectively, h t = √ 2M t /v, and M t = 173.4 GeV is the mass of the top quark. We will follow Ref. [3] to find the boundary condition on λ. To find the scale of emergence of SU (2/1) (Λ s ), we find the scale where the group theoretical value for θ W , g = √ 3g ′ (sin 2 θ W = 0.25), holds. We use where the respective constants b i read as: If we do not make these rescalings at this point then we need to make appropriate ones in Eq. (8).
Setting the number of fermion families to n f = 3, and the number of Higgs doublets to n H = 1, we find Λ s ≃ 4 TeV (note that g 1 = g ′ , g 2 = g, g 3 = g s ). Using Eq. (8) with the boundary conditions λ = g 2 2 /2 at 4 TeV and h t = √ 2M t /v at M Z , we find that λ(M Z ) ≃ 0.24 and thus M H ≃ 170 GeV. The numerical values (MS) we use in this calculation [17] The SU(2/2) embedding Given the incorrect mass of the Higgs and the fact that the superconnection approach suffers from ultraviolet incompleteness, and thus it has to be considered only as an emergent description, we now introduce new emergent physics to correct the Higgs mass. In this section, we use SU (2/2) instead of SU (2/1) to do the embedding. (From the Spectral SM viewpoint SU (2/2) would correspond to a symmetric non-commutative geometry.) In this case, the embedded gauge group is We follow the same route as in the previous section and find the energy scale of the new physics predicted by this structure. We also make the simplifying assumption that this energy scale is also the energy scale at which SU (2) L × SU (2) R × U (1) B−L breaks to the SM. First, we find the superconnection we need. Given the generators of SU (2/2), J can be expressed as J = iλ a s J a , a = 1, 2, · · · , 15. We make the following identifications: J 1,2,3 = W 1,2,3 Here W i L and W i R are 1-forms and the others are 0-form fields corresponding to the left-and right-handed gauge bosons and the bidoublet Higgs field. As a result, we obtain the superconnection, a 4 × 4 supermatrix, in the following form where This leads to the following expression for F (after rescaling J as gJ ) where where λ ≡ g 2 . In 4 dimensions, with the metric g µν = diag(1, −1, −1, −1), the Lagrangian (again with the rescaling g → g/2) becomes where now λ = g 2 /4. To relate λ to the SM λ, we look at the potential term at the symmetry breaking scale where Φ acquires vacuum expectation values (VEVs) Φ = κ 0 0 κ ′ , so that V ( Φ ) = 2 λ |κ| 4 + |κ ′ | 4 . We equate [18]. Assuming |κ| ≫ |κ ′ |, 6 we find λ ∼ = 2 λ, and the constraint becomes λ = g 2 /2, which is the same as that for the SU (2/1) case. Similarly, the prediction for sin 2 θ W can be shown to be the same as in the SU (2/1) case [19].
The SU (2/2) structure has to be broken explicitly in order to introduce the Higgs mass, which is similar to the SU (2/1) case. Additionally, we need to introduce extra scalars in the triplet representation of SU (2) L,R which are necessary in LR symmetric models in order to break the SU (2) R × U (1) B−L to U (1) Y by appropriate VEVs. 7 These triplets may be remnants of a larger geometrical structure, e.g. SU (N/M ).

The observed Higgs boson mass from SU(2/2)
Let us now discuss how the observed Higgs mass comes about. We have seen that both SU (2/1) and SU (2/2) embeddings predict the scale of new physics as ∼ 4 TeV, provided in the latter case that the SU (2/2) emerges at (1 − λ 2 HS /λ λ S ) ≥ 0, while λ remains always small for the correct Higgs mass.

Fermions
The leptons can be incorporated in the SU (2/1) or SU (2/2) construction by taking advantage of the vector space isomorphism between the Clifford algebra and the exterior algebra. Defining the Dirac operator as / D = / ∂ · I+ g 2 / J , 8 where / J is J with the one-forms in it contracted [10], L f = ψi / Dψ gives the necessary terms (including the Yukawa terms) for both constructions. Not surprisingly, we have a relation for the Yukawa couplings (Y = g/ √ 2) from the embedding, just like the one we have for sin 2 θ W . This prediction of the Yukawa coupling universality is a common problem in the literature and it should be lifted via some suitable mechanism. For example, there might exist some mixing with new degrees of freedom at around 4 TeV which may change the running of the couplings and still satisfy the constraint at this scale.

Conclusion
In this Letter we have discussed an emergent superconnection formulation of the EW sector of the SM and its minimal extension which accommodates the observed Higgs mass based on the supergroups SU (2/1) and SU (2/2), respectively. The SU (2/1) formalism unifies the Higgs and the gauge sectors (of the EW part) of the SM. It gives a geometric meaning to the low energy