Reconstruction of the standard model with classical conformal invariance in noncommutative geometry

In this paper, we derive the standard model with classical conformal invariance from the Yang--Mills--Higgs model in noncommutative geometry (NCG). In the ordinary context of the NCG, the {\it distance matrix} $M_{nm}$ which corresponds to the vacuum expectation value of Higgs fields is taken to be finite. However, since $M_{nm}$ is arbitrary in this formulation, we can take all $M_{nm}$ to be zero. In the original composite scheme, the Higgs field itself vanishes with the condition $M_{nm} = 0$. Then, we adopt the elemental scheme, in which the gauge and the Higgs bosons are regarded as elemental fields. By these assumptions, all scalars do not have vevs at tree level. The symmetry breaking mechanism will be implemented by the Coleman--Weinberg mechanism. As a result, we show a possibility to solve the hierarchy problem in the context of NCG. Unfortunately, the Coleman--Weinberg mechanism does not work in the SM Higgs sector, because the Coleman--Weinberg effective potential becomes unbounded from below for $m_{t}>m_{Z}$. However, viable models can be possible by proper extensions.

1 Introduction 2 Generalized gauge theory with ordinary exterior derivative in M 4 × Z N Here, we define several definitions and main formalization in the generalized gauge theory with ordinary exterior derivative. The following construction of theory is basically based on the formalization by Morita and Okumura [34].
In the N-sheeted Minkowski space M 4 × Z N , the coordinates are denoted by (x µ , n = 1 − N). In this setup, the exterior derivative is enlarged as d = d + d 5 as follows: where f is an arbitrary matrices f = f nm . The differential form dy is dependent to n: dy = Diag(dy 1 , dy 2 , · · · , dy N ). Basically, "zero (one)-form" of dy is represented by diagonal (off-diagonal) matrix f = f n δ nm (f = f nm , f nn = 0). M † nm = M mn with M nn = 0 is the matrix corresponds to the typical scales of the discrete spaces and determines the pattern of the symmetry breakings. Since M nm are arbitrary parameters, the formulation still works when M nm = 0. In this case, the Higgs boson has no vevs at tree level, as shown later. From now on, M = 0 and d = d is assumed. Then the nilpotency of d is obvious.
As the wedge products of differential dx µ and dy n [34], dx µ ∧ dx ν = −dx ν ∧ dx µ , dx µ ∧ dy n = −dy n ∧ dx µ , dy m ∧ dy n = dy n ∧ dy m = 0, (2) are assumed. This dy n does not have the noncommutative property, dy n f nm = f nm dy n , because the noncommutative differential algebra is undertaken by the matrix algebra, such as in many papers [5,34].
In the original paper by Connes and Lott [4], and its successors [5,6,8] treat the gauge and Higgs bosons as some kind of composite fields. In this picture, a gauge (Higgs) field consists of "constituent fields" a i n (x), b i n (x): Although this formulation is conceptually interesting, the detailed dynamics of the binding mechanism in Eqs. (3) and (4) is not specified. Indeed, this kind of composite theory has been explored in the induced gauge theory [35][36][37], preon models [38][39][40], composite Higgs models [17][18][19][20], and so on. However, it is highly nontrivial whether the system of the gauge and Higgs bosons (3) and (4) is phenomenologically viable. Additionally, as we can be seen from Eq. (4), the scalar field Φ(x) vanishes Φ(x) → 0, in the limit of M nm → 0. Then, we can not write down a theory of the scalar fields without vevs in this composite scheme.
For these reasons, it seems to be more natural to regard the gauge and Higgs bosons as elemental fields, like several papers [41][42][43][44]. The elemental generalized connection A(x) is introduced as Henceforth, we often omit the argument x if there is no confusion. In the matrix forms, or in components, Here, A n is the k n × k n matrix-valued gauge bosons of some gauge groups G 1 n × · · · × G r n . Then, the scalar filed H nm is k n × k m matrix.
The covariant exterior derivative is defined by and requiring D ′ G = GD with G nm = G n (x)δ nm , the gauge transformation of A is found to be In components, Then, A n µ (H nm ) transforms as a usual gauge (bi-fundamental scalar) field. The gauge transformation of the fifth component becomes an incomplete form because d 5 = 0 1 . Note that the gauge invariance does not forbid the mass term of H nm . We impose the following Hermite condition to the connection, for the sake of the Hermiticity of the Lagrangian, The field-strength two-form F is defined as a usual form: or, in components, It should be emphasized that we can not interpret finite M nm as vevs of the Higgs scalars, in the elemental scheme with the matrix formalization. The reason is as follows. When we calculate the extended curvature F with finite M nm , Higgs interaction terms proportional to dy l ∧ dy m are found to be Since this curvature is not written in only (M + H) nm , we can not identify (M + H) nm ≡ Φ nm as the Higgs fields with vevs, like in the composite scheme.
In order to build the gauge-invariant Lagrangian, the inner products of two-forms are calculated to be 2 [34] while other products between the basis two-forms to be vanish. Summarizing these results, the bosonic Lagrangian is described as , g n and E n is the gauge coupling and the unit matrix of the nth gauge fields A n . Tr denotes the trace over both external linear space with n, m = 1 − N and internal gauge space. In components, where F n µν = (∂ µ A n ν − ∂ ν A n µ + [A n µ , A n ν ]) , and tr denotes the trace over internal gauge spaces. By rescaling the boson fields we obtain the final Lagrangian with canonical kinetic terms and the following self interaction term: Then, in this scheme, the self couplings of the Higgs scalars are comparable to square of the gauge couplings.
The fermionic Lagrangian is also constructed from the generalized connection. The Dirac operator in this space is produced by replacing the basis forms dx M in the covariant derivative (8) to the gamma matrices Γ M : 3 where Γ M = (γ µ , iγ 5 ) satisfies the Clifford algebra {Γ M , Γ N } = 2g M N . The fermion fields, assigned in each n = 1 − N, are represented by row and column vector and the fermionic Lagrangian is defined by a bi-linear form: which satisfies L † F = L F with the rescaling (23).

M 4 × Z 2 toy model
As a typical example, we present M 4 × Z 2 toy model. For the following connection A, the gauge transformation is defined by The field-strength is calculated as where F n = dA n + A n ∧ A n and D µ H nm = ∂ µ H nm + A n H nm − H nm A m .
With the rescaling (23), the Lagrangian with canonical kinetic terms is found to be where we rename H 12 → H. The self-coupling of the Higgs boson is λ = g 2 1 g 2 2 /(g 2 1 + g 2 2 ).

Reconstruction of the standard model without mass scale
In this section we proceed to the reconstruction of the SM without mass scale in the noncommutative geometry. For simplicity, we consider only one generation and omit internal flavor space. The introduction of the Yukawa interaction is found in many literatures, such as in Ref. [45]. The internal gauge space is assumed to be eight dimension. The identity matrix is represented where 1 n is the identity matrix of the n dimensional space. Each subspaces are correspond to those of SU(2) L,R and SU(3) c × U(1) B−L respectively. In this space, the elemental extended connection is defined in 2 × 8 = 16 dimensional space (we rename the index of the discrete space (1, 2) → (L, R)): Here, where σ a are the Pauli matrices with a = 1 − 3 and λ ′α are the generator of SU(3) with α = 1 − 8, which is embedded in the 4 × 4 representation space as follows: Here, λ α is the Gell-Mann matrices which satisfies tr[λ α λ β ] = 2δ αβ . The hypercharges Y L,R are The gauge transformation matrix G = diag(G L , G R ) is represented by where γ α , α a , β, are gauge transformation function of the SM gauge groups SU(3) c × SU(2) L × U(1) Y respectively. From Eqs. (41) and (42), the gauge transformation property of H is determined Then, generally H transforms as (1 + 8, 2, ±1/2) representation under the gauge groups of the SM. This Higgs model inspired NCG (including the original composite scheme) allow a color-octet Higgs scalar. Phenomenologically it is interesting possibility and several author discussing on such scalar bosons [46,47], we exclude this alternative. Imposing the following constraint, with a 2 × 2 matrix h, the gauge transformation of the Higgs boson becomes Then we assign Therefore, it leads to some restricted class of the two Higgs doublet models (2HDM) [48]. If we impose an additional constraint H u =H d , whereH = iσ 2 H * , it result in the SM which has only one Higgs doublet.
When the connection A n is a direct sum of multiple gauge bosons like (36), (37), there are several methods to assign the different gauge couplings. Here, we assume the gauge coupling matrix S is dependent to the internal space as follows: From the Lagrangian (21) with rescaling of the connections, the bosonic Lagrangian is calculated as where Here, the gauge couplings are found to be 1 and the Higgs self coupling is These couplings are not independent and the following formula holds, 4 9 1 In principle, we can break this relation taking the gauge coupling matrix S R = (x R , y R ) ⊗ ( 1 3 a R 1 3 , b R ). Indeed this is the most general form of S R which commutes with A R . In any case, the size of the Higgs self coupling is roughly equal to square of the gauge couplings, λ ∼ g 2 .
The Higgs potential is found to be This is rather restricted form of the general potential of the 2HDM.

The fermionic sector
The construction of the Dirac Lagrangian is achieved by the discussion along Eqs. (25)- (28). In this space, the SM fermions are assigned as where the color index is omitted. The covariant derivative is written as By rescaling the connections A L,R → igA L,R , H → i √ λH, the Dirac Lagrangian (28) is found to be where D / L,R are the covariant derivatives Then, we found that the Yukawa coupling is related to the self coupling of the Higgs y = √ λ ∼ g in this toy model.

Conclusions
In this paper, we derived the standard model with classical conformal invariance from the Yang-Mills-Higgs model in NCG. In the ordinary context of the NCG, the distance matrix M nm which corresponds to the vacuum expectation value (vev) of Higgs fields is taken to be finite. However, since M nm is arbitrary in this formulation, we can take all M nm to be zero. In the original composite scheme, the Higgs field itself vanishes with the condition M nm = 0. Then, we adopt the elemental scheme, in which the gauge and the Higgs bosons are regarded as elemental fields. It should be emphasized that we can not interpret finite M nm as vevs of the Higgs scalars, in the elemental scheme with the matrix formalization. Since the extended curvature F nm (17) is not written in only (M + H) nm , we can not identify (M + H) nm ≡ Φ nm as the Higgs fields with vevs, like in the composite scheme. By these assumptions, all scalars do not have vevs at tree level. The symmetry breaking mechanism will be implemented by the Coleman-Weinberg mechanism [30].
As a result, we show a possibility to solve the hierarchy problem in this context of NCG. Unfortunately, the Coleman-Weinberg mechanism does not work in the SM Higgs sector, because the Coleman-Weinberg effective potential becomes unbounded from below for m t > m Z [31]. However, viable models can be possible by proper extensions such as in Refs. [21][22][23][24][25][26][27][28]. We leave it for our future work. Recent observation shows that the self coupling of the Higgs boson is very close to the critical value, λ(M Pl ) ≃ 0 [32,33]. Although the self coupling (56) is finite in this model, the zero self coupling λ(M Pl ) = 0 is achieved by imposing the condition dy n ∧ dy m = 0. It might suggest that the Higgs boson is a remnant of some noncommutative theory at the Planck scale.