A symmetry breaking mechanism by parity assignment in the noncommutative Higgs model

We apply the orbifold grand unified theory (GUT) mechanism to the noncommutative Higgs model. An assignment of $Z_{2}$ parity to the"constituent fields"induces parity assignments of both the gauge and Higgs bosons, because these bosons are treated as some kind of composite field in this formalism. As a result, some of the gauge bosons and the colored triplet Higgs boson receive heavy mass comparable to the GUT scale, and the gauge symmetry is broken. No particles appear other than the SM ones in the massless states.


Introduction
The discovery of the Higgs boson [1,2], a great triumph of science, has completed the particle contents of the Standard Model (SM). However, the theoretical origin of the Higgs boson is still unclear. Innumerable theories and models have been considered so far, in order to explain the existence of the boson and clarify the underlying physics beyond the SM.
Among these attempts, one interesting possibility is the Yang-Mills-Higgs model inspired noncommutative geometry, founded by A. Connes [3,4]. As a shortened name, we call this model noncommutative Higgs model (NHM). In this picture, the underlying spacetime is assumed to be multi-sheet of ordinary Minkowski space M 4 × Z n . The Higgs boson is interpreted as a gauge boson along a discrete extra dimension, that has noncommutative differential algebra. For example, in the simplest model M 4 ×Z 2 , the coordinates are represented as (x µ , y = ±1). Thus y 2 = 1 holds and it leads to the anti-commutative relation y dy = −dy y. This noncommutative differential algebra generates nonzero Higgs mass and the spontaneously symmetry breaking (SSB) mechanism.
As a typical example, in this paper, we implement the orbifold GUT mechanism by Y. Kawamura [19] to the SU(5) NHM. In the previous standard studies of NHM, the symmetry breaking pattern is determined by selecting the distance matrices M nm . By contrast, we imposed an assignment of Z 2 parity to the "pre-gauge fields". It induces both of parity assignments of gauge and Higgs bosons, because these bosons are treated as some kind of composite fields in this formalism. This is the main difference from the original orbifold GUT model that treats the assignments of bosons as independent conditions. As a result, a part of the gauge bosons and colored triplet Higgs boson receive heavy mass comparable to GUT scale, and the SU(5) gauge symmetry is broken. No particle appear other than the SM ones in the massless states. This paper is organized as follows. In the next section, we review the generalized gauge theory in NHM. In Section 3, we present the symmetry breaking mechanism by parity assignment, and discussions on a new field and the proton decay. Section 4 is devoted to conclusion.
2 Generalized gauge theory on M 4 × Z n Here, we define several definitions and main frame of formulation in the generalized gauge theory on M 4 × Z n . Since the formulation is same as those in [15], detail description is omitted. The essential difference of formulations between the original paper [13,14] and [15] is the flavor dependence of the distance matrices M nm , and differential algebraic structure is almost equivalent.

Differential calculus
On the n-sheeted Minkowski space M 4 × Z n , the coordinates are represented by (x µ , y = 1 − N), and a function on n-th sheet is expressed as f (x, n) ≡ f n . In this space, the exterior derivative is generalized to d = d + d χ introducing extra exterior derivatives d χ and the differential forms χ m . They are defined as follows; where the matrix M † nm = M mn (n = m) represents the distance between two sheets. In order to keep the nilpotency of the generalized exterior derivative d 2 = 0, the differential forms dx µ and χ m should be anti-commutative dx µ ∧ χ m = −χ n ∧ dx µ , and the d χ should also be nilpotent d 2 χ = 0. To proof d 2 χ = 0, and to keep the differential algebra consistent, there are several "index shifting rules" between f n , M nm and χ m . These shifting rules are source of the noncommutativity that corresponds to the relation ydy = −dyy in other formulations [5][6][7][8]. Precise shifting rules and a proof of d 2 χ = 0 is presented in [15].

Generalized connection
In several class of models [5][6][7][8], the gauge and Higgs bosons are regarded as elemental fields. By contrast, Connes's original paper and its successor [3,13,14] treats these bosons as some kind of composite fields. This formulation is effective in particular to construct realistic GUT models. Here we adopt this composite formulation, that is also quoted from [15]. In this picture, a gauge field consists of "constituent fields" or "pre-gauge fields", defined on the same sheet, besides a Higgs field does on the different sheets. The quotation mark " " means that the detailed dynamics of the binding mechanism is not specified. The above picture is described by utilizing generalized connection one-form A, defined as Here, the "pre-gauge field" a i (x, n) is square-matrix-valued continuous function defined on the n-th sheet, and the summation over i is assumed to be a finite sum. The last line in Eq. (1) defines the gauge and Higgs fields as follows, Afterwards, we use a notation A(x, n) = A n , A(x, n) = A n , and a i (x, n) = a i n as shortened forms. As in [13][14][15], we impose the following Hermitian condition From Eq. (4) we assume Φ † nm = Φ mn and χ † m = −χ m . For the later convenience, the back-shifted Higgs field is also introduced. The field strength two-form is defined as follows where because of d 2 = 0. After some calculation, the explicit form of the F n contains the following three pieces: where It contains the gauge boson, Higgs boson, and the new field X ′ nml (x). The treatment of X ′ nml is decided by whether X ′ nml is dependent function of Higgs fields H nm or not. If X ′ nml can be written as a some function of the Higgs fields X ′ nml = f (H nm ), X ′ nml is treated as a Higgs interaction terms. If not, X ′ nml is regarded as a auxiliary field that does not have a kinetic term, and then it will be eliminated from Lagrangian by the equation of motion ∂L/∂X ′ nml = 0.
In order to determine the gauge transformation of "composite fields", at first the "pre-gauge fields" a i n is assigned to a fundamental representation under the n-th gauge transformation; where g n = g(x, n) = (g(x, n) † ) −1 is an arbitrary unitary matrix associated with the gauge group on the sheet n. From Eq. (12), the generalized connection and field strength will transform as the standard form with M ′ nm = M nm . In addition, Eq. (13) implies the back-shifted Higgs field Eq. (5) transform as a bi-fundamental representation;

The Lagrangian
In consequence, the gauge-invariant Lagrangian is formulated by where independent coupling constants g n are introduced for gauge fields on each n-th sheet.
In order to calculate Eq. (16), the metric of the space M 4 × Z n is specified as dx µ , dx ν = g µν , χ n , dx µ = 0, χ n , χ m = −δ nm α 2 n , where g µν = diag(+, −, −, −). Then the inner products of two-forms are found to be while other inner products among the basis two-forms to be zero. Finally we split X nml (x) in Eq. (11) into two terms according to n = l and n = l for convenience; where n = m = l = n. From Eq. (16), we obtain the final expression of the (bosonic sector of) Lagrangian [15] The Lagrangian Eq. (22) is subdivided to four terms corresponding to the decomposition Eq. (8); The first term is the pure Yang-Mills term with independent coupling constants, the second is the Higgs kinetic energy term, the third represents self-coupling of Higgs H nm , and the fourth term describes interactions among different Higgs H nm and H ml .

SU (5) GUT breaking by parity assignment
In this section, we review a SU(5) GUT in the NHM shortly, and implement the orbifold GUT mechanism in this toy model. In the formalism presented the previous section, SU(5) GUT model requires N ≧ 3. This is because that N ≧ 3 realizes more than two independent M nm 's, that corresponds to two energy scales of SU (5)  At first, "pre-gauge fields" a i 1 , a i 2 are assumed to be complex 5×5 matrices and a i 3 does a real-valued continuous function, that satisfy Eq. (4) i a i † n a i n = 1. Additionally, we consider the parity symmetry between n = 1 and n = 2 sheets of spacetime, and impose the following parity condition for the "pre-gauge fields"; where P = diag (−1, −1, −1, +1, +1). In order to break the gauge symmetry, this parity assignment found to be unique under proper assumptions, that is discussed later.
Accordingly, the SU(5) gauge fields at each three copies of M 4 are calculated as follows where T a (a = 1, · · · , 24) are the generators of SU (5).
In order to determine Higgs fields, we set the matrix M nm as where M(µ) is the energy scale at the stage of GUT (SM) symmetry breaking SU(5) → SM (SM → SU(3) c × U(1) em ). Thus the back-shifted Higgs fields are found to be From Eqs. (29, 30) it is found that H 13 is 5 × 1 matrix-valued field transforming like 5 representation under SU (5), and H 12 is 5 × 5 matrix-valued field that is linear combination of the field like 1 and 24 representation. The discussion on the field Σ(x) is presented later. Note that both of the parity assignments of the gauge boson Eq. (25) and the Higgs boson Eq. (30) are determined by the condition of the "pre-gauge fields" Eq. (23). They are independent conditions in the original orbifold GUT model.
In order to calculate the Lagrangian Eq. (22), it is necessary to consider which X ′ nml (x) are independent of Higgs field and which X ′ nml (x) are not. Here we refer to only dependent X ′ nml (x) that should be kept in the Lagrangian; Since other X ′ nml (x) are auxiliary fields independent of Higgs, corresponding P nm (x) and Q nml (x) are entirely eliminated in Eq. (22) by ∂L/∂X ′ nml = 0. Substituting these results into Eq. (22), we found the final form of the Lagrangian , and we have assumed the 1 ↔ 2 symmetry g 2 1 = g 2 2 = g 2 , α 2 1 = α 2 2 = α 2 , and set g 2 3 = g ′ 2 , α 2 3 = β 2 . In particular, the mass term of the 5 representation Higgs is computed as Similarly the gauge boson masses are whereâ runs the broken generators except those of SU(3) c × SU(2) L × U(1) Y . Eqs. (36), (37) shows that the parity assignment condition Eq. (23) a i 2 = P a i 1 P invokes SU(5) symmetry breaking, and provides the colored triplet Higgs and broken gauge bosons with heavy mass of order M. Therefore, it is reasonable to consider that the symmetry breaking by the condition Eq. (23) corresponds to the orbifold GUT mechanism [19] in the NHM. This is our main result. Of course, this result holds only in the tree-level. However, if the parity symmetry imposed between first and second sheet does not broken by the quantum correction, the doublet -triplet splitting holds at quantum level and the SM Higgs doublet remains light.

Discussions
By imposing the condition Eq. (23), the additional Higgs field Σ(x) emerges (in the case of P = 1, H 12 (x) = i a i 1 Ma i 1 = M and then Σ(x) disappear ). To investigate whether it is possible to eliminate the Σ field or not, let us consider the most general form of the parity assignment Here we assume all of O, P, Q, R are diagonal and commutative, and O 2 = P 2 = Q 2 = R 2 = 1, that fulfills the Hermitian condition Eq. (4). OP QR = 1 is also imposed so as to Σ(x) → 0 at proper gauge transformation. In this case, gauge symmetry breaking in the Lagrangian Eq. (35) requires the conditions O = Q and P = R. Since important parts are only the differences of them, we can set O = P = 1. Then Q = R holds by OP QR = 1 and then it leads to the condition Eq. (23). On the other hand, in order to eliminate the Σ field, O = Q should be holds. Then, these two condition, gauge symmetry breaking and elimination of Σ field are incompatible. Under these proper assumptions, we conclude that it is impossible to avoid this kind of field to implement the orbifold GUT mechanism in this formalism. In order to probe the gauge transformation property of this Σ field, the parity matrix P is decomposed to the linear combination of hyperchage Y and identity matrix I as follows Then, the Higgs field found to be Since Y is generator of SU(5), the first term in Eq. (42) behaves as an adjoint Higgs field. Finally, we will mention to the nucleon stability. In this toy SU(5) model, heavy gauge boson of broken symmetry X µ , Y µ and colored triplet Higgs H c induces nucleon decay. However, this is the intrinsic problem in SU(5) and we can extend the lifetime of the nucleon by other mechanism, e.g. supersymmetrization. Whereas, the couplings between fermions and X µ , Y µ , H c bosons can be forbidden by the proper parity assignment of fermions in the normal orbifold GUT [20]. Then there is a possibility that these couplings can also be prohibited in this formalism. However, the femionic sector in NHM has subtleties and there are several definition [8,12,13]. We leave the construction of the fermionic sector as a future work.

Conclusions
In this paper, we have implemented the orbifold GUT mechanism in the noncommutative Higgs model. An assignment of Z 2 parity to the "pre-gauge fields" induces both of the parity assignments of the gauge and Higgs bosons, because these bosons are treated as some kind of composite fields in this formalism. This is the main difference from the original orbifold GUT model that treats the assignments of bosons as independent conditions. As a result, a part of the gauge bosons and colored triplet Higgs boson receive heavy mass comparable to GUT scale, and the SU(5) gauge symmetry is broken. No particle appear other than the SM ones in the massless states. Actually, the nucleon decay is a problem in this model. However, there is a possibility that couplings between fermions and X µ , Y µ , H c bosons could be forbidden by the proper parity assignment of fermions. We leave the construction of the fermionic sector as a future work.