Extended Standard Model in multi-spinor field formalism: Visible and dark sectors

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . To generalize the Standard Model so as to include dark matter, we formulate a theory of multispinor fields on the basis of an algebra that consists of triple-tensor products of elements of the Dirac algebra. Chiral combinations of multi-spinor fields form reducible representations of the Lorentz group possessing component fields with spin 1/2, which we interpret as expressing three ordinary families and an additional fourth family of quarks and leptons. Apart from the gauge and Higgs fields of the Standard Model interacting with the fermions of the three ordinary families, we assume the existence of additional gauge and Higgs fields interacting exclusively with the fermions of the fourth family. While the fields of the Standard Model organize the “visible sector” of our universe, the fields related with the fourth family are presumed to generate a “dark sector” that can contain dark matter. The two sectors possess a channel of communication through the bi-quadratic interaction between visible and dark Higgs fields. After experiencing a common inflationary phase, the two sectors follow a reheating period and weak-coupling paths of thermal histories. We propose scenarios for dark matter that have a tendency to take relatively broad interstellar distributions and examine methods for the detection of the main candidate particles of dark matter. The exchange of superposed fields of the visible and dark Higgs bosons induces weak reaction processes between the fields of the visible and dark sectors, which enables us to have a glimpse of the dark sector. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

To generalize the Standard Model so as to include dark matter, we formulate a theory of multispinor fields on the basis of an algebra that consists of triple-tensor products of elements of the Dirac algebra. Chiral combinations of multi-spinor fields form reducible representations of the Lorentz group possessing component fields with spin 1/2, which we interpret as expressing three ordinary families and an additional fourth family of quarks and leptons. Apart from the gauge and Higgs fields of the Standard Model interacting with the fermions of the three ordinary families, we assume the existence of additional gauge and Higgs fields interacting exclusively with the fermions of the fourth family. While the fields of the Standard Model organize the "visible sector" of our universe, the fields related with the fourth family are presumed to generate a "dark sector" that can contain dark matter. The two sectors possess a channel of communication through the bi-quadratic interaction between visible and dark Higgs fields. After experiencing a common inflationary phase, the two sectors follow a reheating period and weak-coupling paths of thermal histories. We propose scenarios for dark matter that have a tendency to take relatively broad interstellar distributions and examine methods for the detection of the main candidate particles of dark matter. The exchange of superposed fields of the visible and dark Higgs bosons induces weak reaction processes between the fields of the visible and dark sectors, which enables us to have a glimpse of the dark sector.

Introduction
Quarks and leptons exist in threefold family modes with color and electroweak symmetries. Recent observations by WMAP [1] and Planck [2] have established that our universe consists of more dark matter of an unknown nature than visible matter composed of ordinary quarks and leptons. To investigate the origin of such a rich spectrum of visible fermions and the real identity of dark matter, we generalize the Dirac theory of spinor fields and develop a theory of multi-spinor fields on the basis of an algebra, A T , which consists of all the triple-tensor products of elements of the Dirac algebra A γ . We call the algebra A T triplet algebra and the multi-spinor field a triplet field. The triplet algebra A T can be decomposed into three mutually commutative subalgebras, i.e., an external algebra defining the external properties of fermions and two internal algebras that have the respective roles of prescribing family and color degrees of freedom. We choose the external algebra so that it is isomorphic to the Dirac algebra A γ and all of its elements are separately invariant under the action of the permutation group S 3 , which works to exchange the order of A γ elements in the tensor product. The internal algebras for family and color degrees of freedom form the Lie algebras PTEP 2013, 123B02 I. S. Sogami su(4), which have very fine substructures with "su(3) plus su(1)" conformations that are no longer reducible under the group S 3 .
Reflecting the structure of the triplet algebra A T , the triplet field makes up a reducible representation of the Lorentz group, including sixteen component fields with spin 1 2 , which has degrees of freedom of four families and four colors. The family mode and the color symmetry of the triplet field have substructures with "three plus one" formations. Namely, the triplet field possesses the modes of three families and an additional fourth family of tricolor and colorless fermions. Hereafter, we call the three-family mode triple mode and the fourth family mode single mode. The existence of the single mode is a unique characteristic of the current theory of multi-spinor fields.
The electroweak symmetry of the Standard Model (SM) is incorporated by introducing two types of compound fields, called the L-field and R-field, which consist of left-handed triplet fields and right-handed triplet fields, respectively. We demand that the triple mode of the L-field (R-field) is composed of left-handed doublets (right-handed singlets) of the electroweak symmetry SU L (2) and that the electroweak hypercharges Y of the gauge group U Y (1) are assigned so as to cancel chiral anomalies in each family. It is necessary, however, to go beyond the SM in order to determine the physical interpretation of the single mode of the L-and R-fields.
There is no experimental evidence for the existence of fermions other than three families of ordinary quarks and leptons. This means that, if the additional fermions belonging to the single mode exist in the range of energy that is presently attainable by experiment, they are sterile with respect to the interactions mediated by the gauge and Higgs fields related to the SM symmetry (1). Accordingly, we hypothesize that the single mode of the R-field (L-field) contains right-handed doublets (left-handed singlets) of an L-R twisted symmetry SU R (2) and that hypercharges Y 1 of a new gauge group U Y (1) are assigned so that chiral anomalies are canceled in the family.
To qualify the interactions of quarks in the triple and single modes, we have to take the observed characteristics of hadron spectra into account. If the ordinary mechanism of confinement based on the color SU c (3) symmetry were applied to both family modes, there might emerge exotic hadrons bearing hybrid quantum numbers of gauge symmetries G EW = SU L (2) × U Y (1) and G EW = SU R (2) × U Y (1). So far no such hadrons have been found. Therefore, the quarks in the single mode are required to interact exclusively with confining gauge fields of another color symmetry, expressed hereafter as SU c (3).
These suppositions lead us to the viewpoint that, while the fermion fields of the triple mode and the gauge and Higgs fields of the SM symmetry G give birth to our visible sector including baryonic matter, the fermion fields of the single mode and the gauge and Higgs fields of the symmetry G = SU c (3) × SU R (2) × U Y (1) work to create a dark sector that can comprise dark matter. To develop the renormalizable gauge field theory describing the structure of the two sectors and their mutual relations, we postulate that no basic field can share both attributes characterizing each sector. For example, the field with "charges" of both gauge symmetries G and G is predicted not to exist, since observation of the effects of such a field entails a denial of the darkness of the dark sector.
For the present formalism to give a realistic theory for a unified description of the universe, it should have effective ways and means, apart from gravity, to observe dark phenomena from the visible sector. The present theory possesses a natural channel for weak communication between the PTEP 2013, 123B02 I. S. Sogami two sectors, which is opened by means of the bi-quadratic interaction of the visible and dark Higgs fields related, respectively, with the G EW and G EW gauge symmetries.
The two sectors are presumed to experience a common inflationary phase in the primordial universe and then follow weak-coupling paths of thermal histories after a reheating period. The similarity of the gauge groups G and G enables us to assume that the symmetry G is also broken after the Weinberg-Salam (WS) mechanism. Breakdowns of the symmetries G EW and G EW are specified, respectively, by the scales and ( < ). The symmetry G is broken down at to the low energy symmetry SU c (3) × U Q (1), leaving the same number of bosonic fields as in the SM. The U Q (1) gauge field induces phenomena similar to electromagnetism. This suggests that the dark sector consists of dark radiations and dark materials analogous to ordinary atoms and molecules in the visible sector.
We examine two scenarios for the emergence of dark matter, which tends to have relatively extensive interstellar distributions. The channel for communication between the two sectors is opened through exchanges of superpositions of the fields of visible and Higgs bosons. We inquire into possible ways to observe effects that can prove the existence of the dark sector.
In Sects. 2 and 3, the triplet algebra and its subalgebras are described in detail. We introduce the triplet field in Sect. 4. The G EW and G EW gauge symmetries are incorporated in terms of the chiral sets of the triplet fields and breakdowns of these symmetries are examined in Sect. 5. We investigate the emergence of the dark matter and its detection in Sect. 6 and discuss future problems in Sect. 7.

Triplet algebra and external subalgebra
Let us call the triple-tensor products of the bases 2 1, γ μ , σ μν = i 2 (γ μ γ ν − γ ν γ μ ), γ 5 γ μ , and γ 5 = iγ 0 γ 1 γ 2 γ 3 = γ 5 of the Dirac algebra A γ = γ μ primitive triplets [3], and define the triplet algebra A T by all of the linear combinations of primitive triplets. In other words, the triplet algebra is generated in terms of the 12 primitive triplets γ μ ⊗ 1 ⊗ 1, 1 ⊗ γ μ ⊗ 1, and 1 ⊗ 1 ⊗ γ μ as follows: The transpose (Hermite conjugate) of the primitive triplet is defined by the triple-tensor product of its transposed (Hermite conjugate) elements of A γ , and the trace of the primitive triplet is given by the product of the traces of its A γ elements as

External algebra
We introduce a set of four primitive triplets defined by [4,5] which satisfy the anti-commutation relations From these triplets, the external algebra A is constructed as follows: and The algebra A is, evidently, isomorphic to the original Dirac algebra A γ and all its elements are severally invariant under the action of the group S 3 . The Hermite conjugate of μ is defined by †

Centralizer of the external algebra
To explore the structure of the triplet algebra, it is relevant to introduce the centralizer of the external algebra A as follows: The primitive triplets of the centralizer are the triple-tensor products of even numbers of the elements γ μ for arbitrary μ [3]. Namely, the centralizer is given by With the mutually commutative subalgebras A and C , we have the following decomposition of the triplet algebra as Apparently, the centralizer C is commutative with arbitrary generators M μν of the Lorentz transformation. Note that the internal properties of fundamental fermions must be fixed independently of the inertial frame of reference in which observations are made. This means that the elements of the centralizer C satisfy a necessary condition for the generators specifying internal attributes of fundamental fermions. possesses two sets of mutually commutative su(2) subalgebras as follows:

Internal algebras for family and color degrees of freedom
and By taking the triple-tensor products of elements of the respective subalgebras A σ and A ρ so as to be included in the centralizer C , we are able to construct two types of su(4) algebras that are commutative and isomorphic with each other. From those algebras, we select the appropriate subalgebras to describe the family and color degrees of freedom of fundamental fermions.

Subalgebra for family degrees of freedom
Taking the sums and differences of primitive triplets made out of the subalgebra A σ , we can construct 15 elements belonging to the centralizer C as follows [3,4]: and These elements are proved to satisfy the commutation relations and the anti-commutation relations of the Lie algebra su(4), where f (4) jkl and d (4) jkl are the antisymmetric and symmetric structure constants characterizing the algebra. The elements π j are self-adjoint and have the traces By inspecting the explicit forms of the elements in Eqs. (13) and (14), we can confirm that the algebra is closed under the action of the S 3 group. To examine the substructure of A (4) π in detail, it is relevant to introduce the projection operators and which satisfy the relations (t) + (s) = I and for a, b = t, s. Although it is laborious, we can prove by direct calculations the following equations for the operators (a) and π j ( j = 1, 2, . . . , 8). These identities imply that π j are simultaneous eigen-operators of (t) and (s) with the respective eigenvalues 1 and 0.
It is now possible to constitute two subalgebras as and The elements of A (t) are confirmed to satisfy the commutation relations and the anti-commutation relations where f (3) jkl and d (3) jkl are the structure constants of the Lie algebra su(3). Accordingly, the sets A (t) and A (s) form, respectively, the Lie algebras su(3) and su(1).
By using the equations for (a) and A (a) , we can prove the relations and for a, b = t, s. Direct inspection of all elements of Eq. (13) ascertains that the algebra A (t) is irreducible under the action of the permutation group S 3 . We interpret that the algebras A (t) and A (s) have functions to classify, respectively, the triple and single modes of family degrees of freedom. Note that the projection operator (t) on to the triple mode can be subdivided as follows: where These operators i (i = 1, 2, 3) work to sort each family out of the triple mode in the so-called interaction states, to which the triple modes of mass eigenstates are related by the action of the algebra A (t) . By renaming the operator (s) as we obtain the four set of operators i (i = 1, . . . , 4) satisfying the relations

Subalgebra for color degrees of freedom
The isomorphism between the commutative subalgebras A σ and A ρ results in another su(4) algebra consisting of the triple-tensor products of the elements of A ρ . First, replacing σ a and π j with ρ a and λ j in Eqs. (13)- (18), we find a new set of generators forming the Lie algebra su(4) in the centralizer C as Then, similar procedures of replacement and relabeling can be applied to all equations in the previous subsection. From Eqs. (30) and (31), there follow the four elements which we interpret as projection operators for the extended color degrees of freedom including tricolor and colorless fermion states. Similarly, from Eqs. (19) and (20), we obtain the projection operators for the tricolor quark state and the colorless lepton state 3 as follows: and which are invariant under the action of the group S 3 . The operator for baryon number minus lepton number is determined by which has eigenvalues 1/3 and −1.
For the sets of elements π j , i , and (a) describing the family degrees of freedom, there exist the sets of elements λ j , i , and (a) for the color degrees of freedom, in a one-to-one correspondence, in the centralizer C . Both of them have the isomorphic structure of the Lie algebra su(4) with PTEP 2013, 123B02 I. S. Sogami the "su(3) plus su(1)" subalgebras whose elements show the same behavior under the action of the group S 3 .
Correspondingly to Eqs. (23) and (24), we obtain the subalgebras su(3) and su(1) out of A (4) λ for the quark and lepton degrees of freedom as follows: and The elements of A (q) satisfy the commutation relations and the anti-commutation relations Analogously to Eqs. (27) and (28), there hold the relations and for a, b = q, .
Here we postulate that the algebras A (q) and A ( ) carry functions to classify the quark and lepton states, in parallel with the algebras A (t) and A (s) sorting the triple and single family modes. Note, however, that this parallelism, which holds at the algebraic level, cannot be retained at the level of the Lie group, as discussed in the next section.

Triplet fields
After formation of the external and internal algebras in the preceding sections, it is now possible to introduce the triplet field (x) on the space-time point (x μ ), which behaves like the tripletensor product of the Dirac spinor field. Under the proper Lorentz transformation x μ = μ ν x ν , where λμ λ ν = η μν and det = 1, the triplet field (x) and its adjoint field (x) = † (x) 0 are transformed as follows: where the transformation matrix is given by with the generators μν in Eq. (6) and the angles ω μν in the μ-ν planes. The Lorentz invariant scalar product is formed as For discrete space-time transformations such as space inversion, time reversal, and charge conjugation, the present scheme retains exactly the same structure as the ordinary Dirac theory. The "three plus one" constructions of the internal algebras permit the triplet field (x) to possess the orthogonal decompositions as follows: , where (t) and (s) represent the triple and single modes, and (q) and ( ) express the quark and lepton states.
To embody how the algebras A (q) and A ( ) for quark and lepton states act on the triple and single modes, it is necessary to make up the algebras for each state by for a = t, s and j, k, l = 1, 2, . . . , 8. In the present formalism, the quarks in the triple and single modes are presumed to be confined separately by different interactions associated with the color gauge groups SU c (3) and SU c (3). Those symmetry groups are defined by the exponential mappings of the algebras (t) A (q) and (s) A (q) as follows: where θ j (x) and θ j (x) are arbitrary real functions of space-time. The scalar product for the triplet field in Eq. (46) is invariant under the action of these groups on the quark states in the triple and single modes, i.e., As stated in the previous section, the present theory shows parallelism between the color and family degrees of freedom at the algebraic level. If this parallelism persists up to the group level, there arises the theoretical possibility that claims to gauge also the family degrees of freedom. Namely, it is necessary to gauge the SU (3) symmetries induced through the exponential mappings of the algebras (q) A (t) and ( ) A (t) defined by which are isomorphic to Eq. (48). These family groups act on the quark and lepton states in the triple mode, i.e., It is appealing and theoretically possible for us to postulate that both the color and family gauge symmetries hold in a sufficiently high energy regime. At the present stage of the theory, however, it is markedly difficult to formulate whole processes of breakdown of the family symmetry so as to describe the various phenomena of flavor physics in low energy scales. Therefore, we follow here the ordinary scheme of the SM and refrain from gauging the family symmetry.
The left-handed and right-handed fields of the triplet field are given as follows: with the chirality operators h = L , R defined by Combining them with the projection operators of family modes, we can also introduce the operators In order to extract the internal component fields out of the triplet field, we conveniently import Dirac's bra-ket symbols for the projection operators a and ih as follows: For the triplet field and its conjugate, the projection operators act as where aih (x) = aih| (x) and¯ aih (x) = ¯ (x)|aih are chiral component fields, andh implies thatL = R andR = L. Then, the decomposition of the bilinear scalar and vector forms of the triplet fields can be achieved as follows: and¯

Gauge symmetries G and G in multi-spinor field formalism
In the WS theory of electroweak interaction, the left-handed chiral components of electron and neutrino fields constitute the doublet representation of the electroweak SU L (2) symmetry and the right-handed component of the electron field forms its singlet. Since the discovery of neutrino oscillation, the right-handed component of the neutrino field has also been added as another singlet. To integrate the WS scheme for the G EW and G EW symmetries into our theory, we have to introduce the new Pauli su(2) algebras as follows: which act, respectively, on the triple and single modes of the triplet fields and generate the SU L (2) and SU R (2) groups.

Multi-spinor field theory for G and G symmetries
Fundamental representations of the SU L (2) and SU R (2) groups are given by two types of compound fields, L-field L and R-field R , which are respectively composed of left-handed and right-handed triplet fields. The L-field contains the triple mode consisting of left-handed doublets of SU L (2) and the single mode consisting of left-handed singlets of SU R (2). In contrast, the R-field possesses the triple mode composed of the right-handed singlets of SU L (2) and the single mode composed of the right-handed doublet of SU R (2). The L-field L and R-field R constitute fundamental representations of the G and G symmetries in the forms in which and U (D) are used, respectively, to express the doublet and the up (down) singlet. The operation of transpose t is applied to line up the family modes in the horizontal direction. In order to name the fermions in the single mode, we assign new symbols u and d for up and down quark states, and ν and e for up and down lepton states. Then, by refraining to specify the color degrees of freedom, we can display the components of the quark and lepton parts of the L-field, (q) L and ( ) L , as follows: and ( ) Likewise, the quark and lepton parts of the R-field, (q) R and ( ) R , are expressed by the components as (66) We are now able to write down the kinetic and gauge part of the Lagrangian density of all fermions in terms of the L-and R-fields as follows: where the covariant derivatives take the forms and μ (x)) are gauge fields of the G EW (G EW ) symmetry with coupling constants g (2) and g (1) (g (2) and g (1) ). The hypercharges for the triple and single modes, Y and Y , can be expressed by in which y and y take 0, 1, and −1, respectively, for the doublet , the up singlet U , and the down singlet D.
The Lagrangian density for the gauge fields is, as usual, constructed by summing up all the Lorentz invariant bilinear forms of the separate field strengths of G and G symmetries. For the sake of brevity, we leave out its analysis here and will give brief notes on the results of the WS mechanism at the end of this section.
To break down the gauge symmetries G EW and G EW , we require two types of Higgs doublets as where φ(x) and φ (x) have the hypercharges (Y = 1, Y = 0) and (Y = 0, Y = 1), respectively. The Lagrangian density of the Yukawa interaction is given as follows: in which the covariant derivatives act as follows: and The Higgs potential is generally given by where λ, λ , and λ I are the constants of self-coupling and mutual interaction. The bi-quadratic interaction term, 2λ I (φ † φ )(φ † φ), plays the key role of relating the otherwise independent visible and dark sectors. It is crucial to recognize that there is no reason to exclude this cross-interaction term from V H . The invariance under the G EW and G EW symmetries and the condition of renormalizability allow this term to exist. We have to grasp the raison d'être of this interaction term in both the lower and higher energy regions than the scale . In the next subsection, we will see its role in breaking both symmetries G EW and G EW , leaving effective interactions between the resulting visible and dark Higgs bosons in the lower energy region. In Sect. 7, we will discuss the virtual quantum effects arising from the bi-quadratic term that cause interactions between the fermions in triple and single modes and also between the visible and dark gauge fields. In an early reheating period, those interactions lead all quanta of the fields of the universe to a state of thermal equilibrium. 12  With the dark hadron ++ , the helium-like atom is formed as which produces the molecule (He ) 2 = He He .
In the matter-dominant phase of the universe, the dark materials consisting of dark atoms, molecules, and their ions coexist with the baryonic materials and reinforce formations of astronomical compact objects as well as cosmological large-scale structure through gravitational interactions. It is crucial to recognize that the dark materials are stable as a whole and no dark nuclear reaction takes place in all those astronomical and cosmological processes in our scenario of the dark matter.

Detection of the effects of the dark hadron
The visible and dark sectors possess a channel for mutual communication through exchanges of the superposed fields H and H . We have to inquire how to open the channel and contrive methods for observations of the effects of the dark sector. In the scenario specified by the condition in Eq. (87), we consider a method to detect the effect of the − hadron constituting the main component of dark matter. There are direct and indirect ways to detect the effects induced by the d quark in the − hadrons.
Let us examine first the interaction phenomena between the dark and visible matter. Figure 1 shows a scattering process between the dark quark d and the nucleon N induced by exchanges of the superposed fields H and H in Eq. (83), which consist of the visible and dark Higgs bosons h and h . When the dark hadron − encounters a heavy element, such as xenon or germanium, situated in a low noise environment on the ground and penetrates deeply into its nucleus, the dark quark d and the nucleons N interact through the exchange of H and H . The amplitude of this process is roughly estimated to be proportional to where y d is the Yukawa coupling constant for the dark quark and y eff is an effective coupling constant of the Higgs boson with the nucleon [9,10], which has the dominant contribution of the top quark loop-correction produced by gluons inside the nucleon.
In the low energy regime, where the energy of the dark quark d is less than the masses of the fields H and H , the elastic scattering process of Fig. 1   to the square of its energy. Rare phenomena induced by this sort of scattering process of the dark matter with the nucleon can be observed by the ground experiments designed for the direct detection of weakly interacting particles [11][12][13][14].
The cross section of the elastic scattering process between the dark hadrons − in the dark matter wind and the nucleon N inside the target element with mass number A is enhanced approximately by the factor 3 2 × A 2 , provided that the effective coupling constants y eff to the proton and neutron are approximately equal. Direct detection enables us to estimate the values of the coupling constants, λ I /(λλ − λ 2 I ), and the product of the VEVs v and v . For theoretical analysis of the experimental data, it is necessary to take into account the bound state effects of the nucleon wave functions in the target element.
Corresponding to the scattering process induced by virtual exchanges of the fields H and H in Fig. 1, it is theoretically possible to picture the decay processes of the H and H fields produced through Brehmsstrahlung from the accelerated dark hadron − , as shown in Fig. 2. This kind of production process can take place only when the dark hadron − has sufficiently high energy, as realized in the LHC experiment for the Higgs search. To utilize such processes for experiments of indirect detection of dark matter, it is also necessary to acquire reliable techniques from the LHC experiments to precisely identify the process from its decay products.

Discussion
By generalizing the Dirac concept of spinor fields, we have developed a unified theory of multi-spinor fields that can describe the whole spectrum of fields in the ordinary visible sector and the additional sets of fields constructing the dark sector. As shown in Sects. 3 and 4, the triplet algebra with the restriction of the S 3 irreducibility has the unique feature of the "three plus one" structure for both the color and family degrees of freedom. The triplet field possesses the component fields of the triple and single modes with the tricolor quarks and colorless leptons. With the chiral representations in Eq. (62), we have formulated a successful unified theory that can describe the flavor physics of the visible sector and astrophysical phenomena related to both the visible and dark sectors.
Here it is relevant to explain why we did not choose a simpler extension of the SM in which the additional fermion multiplet is identified with the sequential 4th family. This sequential model is realized by the following chiral representations of the triplet fields as where (a) , U (a) , and D (a) (a = t, s) are the doublet, up singlet, and down singlet of the Weinberg-Salam symmetry G EW , respectively. In this model, the family structure is described by the Lie algebra su(4) in Eq. (18).