SU(5) orbifold GUT in noncommutative geometry

In this paper, we appliy the orbifold GUT mechanism to the SU(5) model in noncommutative geometry, including the fermonic sector. Imposing proper parity assignments for"constituent fields"of bosons and fermions, the couplings between fermions and the heavy bosons $X_{\mu}, Y_{\mu}$, and $ H^{c}$ are prohibited by the parity symmetry. As a result, the derived fermionic Lagrangian is just that of the standard model, and proton decay is forbidden at tree level. If quantum fluctuation respects the parity symmetry, the process will be naturally suppressed or even forbidden completely.


Introduction 2 Generalized gauge theory on M 4 × Z N
In this section, we present a basic formulation of generalized gauge theory on M 4 × Z N . The original papers utilize the Dirac operator and the Clifford algebra [10,16,17]. However, we follow the formulation using the inner product of the differential forms, developed in [18,19]. The formulation presented here is quoted from Ref. [18,19], and is basically the same.

Differential calculus and generalized gauge field
The background spacetime M 4 ×Z n is the direct product of the ordinary four-dimensional Minkowski space M 4 and discrete space Z n , with the coordinates (x µ , n = 1 − N). In this space, the generalized exterior derivative d is defined as follows; where χ † k = −χ k and M † nm = M mn (n = m) are assumed. These matrices determine the distance between two Minkowski spacetimes and the pattern of the spontaneous symmetry-breaking.
This is the source of the noncommutativity that corresponds to the relation y dy = −dy y in other formulations [12][13][14][15]. For the "one form" M nm χ m f (x, m), the Leibniz rule is modified to be where d χ l M nm = M nm M ml χ l .
Equation (3) corresponds to the usual graded Leibniz rule of the differential form, where ∂ξ is the order of the differential form ξ.
Utilizing this calculus, we can prove the nilpotency of d that is indispensable to constructing the gauge theory. By the definition of d = d + d χ , the nilpotency condition is rewritten as Enforcing the ordinary anticommutative relation dx µ ∧ χ m = −χ n ∧ dx µ , the condition is reduced to Here, we impose χ m ∧ χ k = +χ k ∧ χ m due to the noncommutative property of the background spacetime. Nevertheless, In the last line, another index shifting rule, F kn χ n M ml = F kn M nl χ n [18], is applied. As a result, The proof of the nilpotency in the general case is also presented in [19].
Next, we consider the generalized gauge field A(x, n) in this space: Here, a i (x, n) is square-matrix-valued function and the summation over i is assumed to be a finite sum. In the components, the gauge and Higgs fields are represented as According to [16][17][18], we impose the normalization condition i a † i (x, n)a i (x, n) = 1, which leads to the following Hermitian condition: The gauge transformation property of the a i (x, n) is assigned to a fundamental representation under the nth gauge transformation, where g(x, n) = g −1 (x, n) † is an arbitrary unitary matrix associated with the gauge group on the nth M 4 space. From Eq. (17), the gauge transformation of A(x, n) is derived as the standard form with M ′ nm = M nm . In particular, the following back-shifted Higgs field, transforms as a bifundamental representation, and is identified as a physical Higgs boson with a vacuum expectation value. Regarding this connection A as a building block, we can construct the field-strength two-form F , and the gauge-invariant Lagrangian Here, g n are independent coupling constants introduced on each nth space. The Lagrangian (22) is subdivided into four terms: The first term is the pure Yang-Mills term with independent coupling constants, the second is the Higgs kinetic energy term, the third represents the self-coupling of Higgs H nm , and the fourth term describes interactions among different Higgs, H nm and H ml . The derivation and explicit formula of Eq. (22) is in Ref. [18,19].

Fermionic Lagrangian
Next, we proceed to the fermion sector to construct the full Lagrangian. At first, we introduce the generalized spinor one form Dψ, and the covariant derivative D acting on the spinor field ψ(x, n) by Here, A f (x, n) is the differential representation for the fermions ψ(x, n) such that Note We also define the extra derivative of the fermion as which leads to Here where g f (x, n) is the gauge transformation function corresponding to the representation of ψ(x, n). Due to this, A f µ (x, n) and H f nm should transform as From Eqs. (27), (28), and (29), we can easily verify that Dψ(x, n) is gauge covariant: In order to obtain the Dirac Lagrangian by the inner products of differential forms, the original paper introduce the following associated spinor one-form [19]: Here, c Y is a real, dimensionless constant which relates to the Yukawa coupling constant. It is obvious thatDψ(x, n) is also gauge covariant, Finally, we introduce the inner products for spinor one-forms, Here,Ā n = A † n γ 0 denotes the usual Lorentz conjugate of the spinors, while other inner products vanish.
Summarizing these considerations, the Lorentz and gauge-invariant Dirac lagrangian is constructed by taking the inner product and the summation over n = 1 − N: In particular, the last term of Eq. (36) provides the Yukawa couplings constant y = c Y α 2 between Higgs and fermions.

SU(5) grand unified theory
In this section, we review an SU(5) GUT in the NCG, and implement the orbifold GUT mechanism for the fermonic sector. Since the SU(5) GUT model has two symmetrybreaking scales, the model requires N ≧ 3, which realizes more than two independent M nm s. Then we choose N = 3 to construct the SU(5) GUT [16,18]. The indices n, m, l run the values 1, 2, 3 only. At the beginning, a i (x, (1,2)) are assumed to be complex 5×5 matrices and a i (x, 3) is a real-valued continuous function that satisfies Eq. (16), i a † i (x, n)a i (x, n) = 1. Moreover, a parity symmetry between n = 1, 2 is imposed with the following parity condition for the fields: where P = diag (−1, −1, −1, +1, +1). In order to break the gauge symmetry, this parity assignment is found to be unique under proper assumptions [28]. The SU(5) gauge fields at each discrete point are calculated from the a i (x, n)s as where T a (a = 1, · · · , 24) are the generators of SU (5). In order to eliminate the redundant U(1) generator, the following traceless condition is imposed: TrA(x, 1) = TrA(x, 2) = 0.
The matrix M nm are fixed on as where M(µ) corresponds to the energy scale of GUT (SM) symmetry-breaking. These M nm s determine the following back-shifted Higgs fields: Here, the field H 13 (x)(H 12 (x)) is a 5 × 1 (5 × 5) matrix transforming like the 5 (1 plus 24) representation under SU (5). Substituting these results into Eq. (22), it is found that the Lagrangian contains the following mass term of the 5 representation Higgs [28], and the gauge boson masses Here, a =â +ǎ,â runs the generators of SU(3) c × SU(2) L × U(1) Y , andǎ runs the broken generator except for those of SU(3) c × SU(2) L × U(1) Y . Equations (46) and (47) show that the parity assignment condition Eq. (37) a i 2 = P a i 1 P invokes SU(5) symmetrybreaking, and provides the colored triplet Higgs and broken gauge bosons with heavy mass of order M. Therefore, it is adequate to regard that this symmetry-breaking by the condition (37) corresponds to the orbifold GUT mechanism [27] of the GUT in NCG.

Fermionic sector
Under the Z 2 parity symmetry 1 ↔ 2, the 5 and 10 representation fermions in the SU(5) model are assigned at each point as follows: where a 1 / √ 2 is the normalization coefficient for the final expression, and only one generation is assumed for simplicity. In particular, Eq. (48) indicates that ψ 10 is assigned to even charge under the parity symmetry. In components, the fermions ψ 10 , ψ 5 are represented as The subscripts L, R denote that they are chiral fermions. These are the ad hoc chiral conditions explained in the introduction. Utilizing the orbifold mechanism, if a fivedimensional theory has a vector representation ψ L,R , there remain only chiral fermions in a low energy four-dimension theory. However, in this paper, the parity assignment is spent to break SU(5) symmetry, and then we retain the chiral condition Eq. (49). Each ψ(x, n) transforms under the gauge transformation respectively as ψ g (x, (1, 2)) = g(x, (1, 2)) ⊗ g(x(1, 2)) ψ(x, (1, 2)), where g(x, n) is the gauge transformation function belonging to SU (5). In fact, Eq. (51) seems to be an ad hoc condition. This point will be discussed in the next subsection.
With this in mind, the generalized spinor one-form Dψ(x, n) in Eq. (26) is taken to be The associated spinor-one form is written as Summarizing the above discussion, we can obtain the Dirac Lagrangian Note that the coupling between Σ and ψ 10 disappears by the chirality condition,ψ 10 ψ 10 = 0. In particular, the explicit form of the Yukawa interactions are just those of the SM: where H α SM = i(σ 2 ) αβ H * β , with H * β = (H 4,5 ) * . The Yukawa coupling constant is y ′ d = 2α 2 c d .
In Eq. (58), it is clear that up-quarks are still massless. Thus, we consider the uptype Yukawa interactions hereafter. Since the Dirac Lagrangian is written as an inner product form, the up-type Yukawa interaction requires the introduction of a completely antisymmetric fermionψ 10 , which is transformed as a 10 * representation of SU (5). In components,ψ 10 is expressed asψ where ǫ ijklm is the completely antisymmetric tensor of SU (5), and ψ c 10 is the charge conjugation of ψ 10 . We simply assign theseψ 10 and ψ 10 to each space as From Eqs. (60) and (61), the gauge transformation properties of ψ ′ (x, n) are: Then, the covariant spinor one-form in Eq.(26) is found to be Similarly, the associated spinor one-form is written as where c u is the related up-type Yukawa coupling constant. Accordingly, we obtain the Dirac Lagrangian for the second assignment: with y ′ u = 2c u α 2 . In particular, the explicit form of the Yukawa interactions are just those of the SM: whereH SM = iσ 2 H SM * . Finally, summarizing the results of Eqs. (58) and (70), we obtain the final form of the Dirac Lagrangian of the SU(5) GUT: Here, ψ SM represents the SM fermions q L , u R , d R , l L , e R , 1 and the Yukawa couplings are found to be y u = 8α 2 c u a 2 / √ 6 and y d = √ 2α 2 c d . This fermionic Lagrangian is just that of the SM. The couplings between fermions and the heavy bosons X µ , Y µ , and H c are prohibited by the parity symmetry. As a result, proton decay is forbidden at tree level. If quantum fluctuation respects the parity symmetry, the process will be naturally suppressed or even forbidden completely. Although the process is too suppressed to detect in this case, other GUT groups such as SO (10), and other parity assignments, might allow the baryon number violating interactions.
Meanwhile, the application of the orbifold GUT mechanism to the fermion sector may be meaningful for model building. In early papers on the NCG [13,[15][16][17], the ad hoc chiral condition is usually imposed on the fermions in order to produce chiral Yukawa couplings. However, if we consider the NCG model as a model with an extra dimension, especially with several branes [21], these chiral fermions will generate a brane anomaly. This is a mirror fermion problem [31]. By the orbifold GUT mechanism, we can eliminate the mirror fermion in proper situation. This enables us to implement new kind of theory in NCG: family unification in the extra dimension [9].

Discussion
In this subsection, we comment on an obscure point in the above construction of the Lagrangian. In fact, the conditions Eq. (51) and (63) are seem to be ad hoc conditions. The down-type Lagrangian (58) is also schematically described as Here, D M Γ M is the extension of the Dirac operator with M = 0, 1, 2, 3, 5. The 33 component of the matrix ([A µ + P A µ P ]γ µ ) is introduced only for fermions in this formulation. However, in the original paper on SU(5) GUT in NCG [16,17], the authors treated fermions as a matrix rather than a vector in Eq. (74): 1 We normalized the kinetic term of decuplets ψ 10 as a 2 1 + a 2 2 + a 2 where 1 √ 2 , and 1 √ 6 are the normalization coefficients like a 1,2,3 . This corresponds to the situation where the fermion ψ 5 is treated as a "link field," or an intrinsic differential oneform ψ = ψ(x)χ 1,2 . In this case, we can anticipate that the undesirable gauge interaction will cancel between two ψ 5 s, and then the orbifold GUT mechanism also works successfully without the 33 component term ([A µ + P A µ P ]γ µ ). Otherwise, we can also solve this point by adding an additional noncommutative extra dimension (then the model would be a six-dimensional theory), such as M 4 × Z 2 × Z 2 [22]. Introduction of ψ 5 in two separated points leads to proper cancellation of the gauge interaction.

Conclusions
In this paper, we applied the orbifold GUT mechanism to the SU(5) model in NCG, including the fermonic sector. Imposing proper parity assignments for "constituent fields" of bosons and fermions, the couplings between fermions and the heavy bosons X µ , Y µ , and H c are prohibited by the parity symmetry. As a result, the derived fermionic Lagrangian is just that of the SM, and proton decay is forbidden at tree level. If quantum fluctuation respects the parity symmetry, the process will be naturally suppressed or even forbidden completely.
Moreover, the application of the orbifold GUT mechanism to the fermion sector may be meaningful for model building. In early papers on the NCG [13,[15][16][17], the ad hoc chiral condition is usually imposed on the fermions in order to produce chiral Yukawa couplings. However, if we consider the NCG model as a model with an extra dimension, especially with several branes [21], these chiral fermions will generate a brane anomaly. This is a mirror fermion problem [31]. By the orbifold GUT mechanism, we can eliminate the mirror fermion in the proper situation. This enables us to implement a new kind of theory in NCG: family unification in the extra dimension [9].