A spectral geometry for the Standard Model without the fermion doubling

We propose a simple model of noncommutative geometry to describe the structure of the Standard Model, which satisfies spin${}_c$ condition, has no fermion doubling, does not lead to the possibility of color symmetry breaking and explains the CP-violation as the failure of the reality condition for the Dirac operator.


I. INTRODUCTION
The Standard Model of particle interactions is certainly one of the most successful and best tested theory about the fundamental constituents of matter and the forces between them.
Even though we still have no satisfactory description of the strong interactions in low-energy regime and there are some puzzles concerning masses and character of neutrinos as well as there are some experimental signs that could point out to new physics, the Standard Model appears to be robust and verified. Yet neither the content of fermion sector, the mixing between the families and the fundamentally different character of the Higgs boson from other gauge bosons appear to have a satisfactory geometrical explanation.
One of the few theories that aimed to provide a sound geometrical basis for the structure of the Standard Model, explaining the appearance of the Higgs and symmetry-breaking potential, was noncommutative geometry (see [1][2][3]). Constructed with the core idea that spaces with points can be replaced with algebras provided a plausible explanation of the gauge group of the Standard Model and the particles in its representation as linked to the unitary group of a finite-dimensional algebra. Merged with the Kaluza-Klein idea that the physical spacetime has extra dimensions, the geometry of the finite-dimensional algebra (in the noncommutative sense) gave rise to the Higgs field understood as a connection, and the Higgs symmetry-breaking potential appeared as the usual Yang-Mills term in the action.
The original model, which is based on the construction of a product geometry, with the resulting geometry being the tensor product of a usual "commutative" space with the finitedimensional noncommutative geometry suffers from two problems. Firstly, in the original formulation it is Euclidean. Secondly, the product structure leads to the quadrupling of the degrees of freedom in the classical Lagrangian [4,5]. Moreover, the conditions put on the Dirac operator for the finite geometry are not sufficient to restrict the class of possible operators to the physical one, leaving the possibility for the non-physical SU(3)-breaking geometries [6][7][8][9]. Though the latter problems appear to have at least a partial solution [8] we believe that they can be completely avoided if the noncommutative geometry behind the Standard Model is assumed to be spin c only.
In what follows we present a spin c description of the geometry for the Standard Model, which does not require fermion doubling, satisfies the spin c duality for spinors provided that the mass matrices and mixing matrices are non-degenerate. The crucial role is then played then not by the Lorentzian Dirac operator but rather by its Krein-shift D, the product of the Krein space fundamental symmetry β and the Dirac operator D. This operator can be understood as the selfadjoint component of the Krein decomposition of the Lorentzian Dirac operator, D = β D. Moreover, we link the breaking of the J-condition between the real structure and the Dirac operator to the appearance of the CP-symmetry breaking in the Standard Model.

II. DIRAC OPERATOR FOR THE STANDARD MODEL
The Dirac operator for the four-dimensional Minkowski space is of the form D = iγ µ ∂ µ , with the gamma matrices satisfying the relation γ µ γ ν + γ ν γ µ = 2η µν , where η µν is the standard Minkowski metric of signature (+, −, −, −). We use the conventions of [8], so that γ 0 is selfadjoint and the remaining gamma matrices are antiselfadjojnt.
The Lorentz-invariant fermionic action, which leads to the Dirac equation, is where ψ = ψ † γ 0 and D = γ 0 D. The operator, D is a symmetric operator, which we call Krein-shift of the Dirac operator. This follows from the properties of the Lorentzian Dirac operator D, which is Krein-selfadjoint [10], D † = γ 0 Dγ 0 , where γ 0 is the fundamental symmetry of the Krein space. Written explicitly in the chiral representation it becomes where σ µ andσ µ are the standard and associated Pauli matrices, σ 0 = σ 0 , σ k = −σ k .
The Lorentzian Dirac operator and the related Lorentzian spectral triple have the standard Z 2 -grading γ and the charge conjugation operator given, where cc denotes the usual complex conjugation of spinors. The operators D, γ, J satisfy the usual commutation relations for the geometry of the signature (1, 3): whereas for the Krein-shifted operator we have The so-far accepted and tested experimentally action for the Standard Model of fundamental interactions can be viewed as the extension of the action for a single bispinor to a family of particles, with the additional terms in the action arising from a slight modification of the Dirac operator by an endomorphism of the finite-dimensional space of fermions.
Before we discuss this extension and the conditions it satisfies we recall the notion of Riemannian spectral triples and spin c -spectral triples, which are a bigger class than these arising from generalisation of the spin geometry only.

III. RIEMANNIAN AND PSEUDORIEMANNIAN SPECTRAL TRIPLES
A Riemannian finite spectral triple [11] built over a finite-dimensional algebra A is a collection of data (A, D, H, π L , π R ), where π L is the representation of A on H, π R is the representation of A op (the opposite algebra to A) on H such that: for all a ∈ A and b ∈ A op .
We say that the spectral triple is of spin c (see [12] and compare with the classical result or of Hodge type if By the generalized Clifford algebra Cl D (π L (A)) (and similarly Cl D (π R (A))) we understand the algebra generated by π L (a) and [D, Of course, genuine Riemannian geometries require further assumption that the operator D has a compact resolvent. In the case of Lorentzian or, more generally, pseudoriemannian geometries, we might follow the path of [10] extending the definition of Lorentzian real spectral triples to Lorentzian spin c geometries.

IV. FERMIONS AND THE ALGEBRA OF THE STANDARD MODEL
Let us recall a convenient parametrization of the particle content in the one-generation Standard Model [12]: where each of the entries is the Weyl spinor over the Minkowski space with a fixed chirality.
As the algebra A we take the algebra of functions over the Minkowski space, valued in C ⊕ H ⊕ M 3 (C) and chose the following two representations of the algebra: where λ, q, m are complex, quaternion and M 3 (C)-valued functions, respectively. The representation π L acts by multiplying Ψ from the left whereas π R acts by multiplying Ψ from the right. This is the reason that we transpose m so that π R is indeed a representation. Observe that since left and right multiplication commute then [π L (a), π R (b)] = 0 for all a, b ∈ A, i.e.
the zero-order condition is satisfied. Due to the simplicity of the notation at every point of the Minkowski space we can encode any linear operator on the space of particles as an operator where the the first and the last matrix act by multiplication from the left and from the right and the middle M 2 (C) matrix acts on the components of the Weyl spinor.
The full Lorentzian Dirac operator of the Standard Model is, in this notation, of the form where D F is a finite endomorphism of the Hilbert space M 4 (H W ). Suppose then that for all a, b ∈ A. As any element in π L (A) commutes with π R (A) it suffices to find all D F that are selfadjoint, commute with the elements from π R (A) and anticommute with Γ. It is easy to see that such operators are restricted to where M l , M q ∈ M 2 (C).

A. The spin c condition
The Krein-shifted Dirac operator satisfies first order condition, yet it still may not provide the spin c spectral geometry. We shall look for necessary and sufficient conditions that the commutant of the (complexified) Clifford algebra, Cl D (π L (A)), generated by π L (A) and Therefore the common commutant of both parts will be the commutant of the full Clifford algebra.
From the decomposition it is easy to see that the commutant of second part are the functions in id ⊗ M 2 (C) ⊗ (C ⊕ M 3 (C)) and therefore, the common part are functions valued in id ⊗ id ⊗ (C ⊕ M 3 (C)), which indeed is the algebra π R (A).

B. Three generations
Let us consider three families of leptons and quarks, that is the Hilbert space M 4 (H W )⊗C 3 with the diagonal representation of the algebra. The only difference from the previous section is that the matrices M l and M q are no longer in M 2 (C) but in M 2 (C)⊗M 3 (C). As the algebra acts diagonally on the Hilbert space (with respect to the generations) we can again repeat the arguments of [14] and argue that the spin c condition will hold if algebras generated by π L (A) and D l , D q , respectively, will be full matrix algebras, that is ( independently for lepton and for quarks. Since the arguments we have used here are analogous to ones used in the discussion of full conditions (4.2.2 in [14]), we infer the same condition for the Hodge property to be satisfied.
Both M l and M q can be diagonalized, yet because of the doublet structure of the left leptons and quarks the components (up/down) cannot be diagonalized simultaneously. The standard presentation of the mass matrices for the Physical Standard Model is then where Υ e and Υ u are chosen diagonal with the masses of electron, muon, tau and the up, charm, top quarks, respectively and with diagonal matrices Υ ν , Υ d providing (Dirac) masses of all neutrinos and down, strange, bottom quarks, where U is the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix (PMNS matrix) and V is the Cabibbo-Kobayashi-Maskawa mixing matrix (CKM matrix).
As was indicated also in [14] the sufficient condition to fulfill the Hodge property is that Assume now that D F is such that the spin c condition holds. We shall describe now the procedure of the doubling of the triple, so that the resulting real spectral triple satisfies the Hodge duality and is the finite spectral triple of the Standard Model studied so far as the finite component of the product geometry.
Consider H 2 SM = H SM ⊕ H SM with the representation π L ⊕ π R . We define the real structure J as the composition of the hermitian conjugation with the Z 2 action exchanging the two copies of H SM , so that J(M 1 ⊕ M 2 ) = M * 2 ⊕ M * 1 . It is clear that the conjugation by J maps the representation of the algebra A to its commutant. We extend Γ so that the relation JΓ = ΓJ holds and extend the Dirac operator D F in the following way, Clearly D ′ anticommutes with Γ and commutes with J. The Clifford algebra, that is the algebra generated by π L ⊕ π R and the commutators with D ′ is Cl D F (π L (A)) ⊕ π R (A). Due to the fact that before the doubling we had the spin c -condition it is clear that the commutant of the Clifford algebra contains π R (A) ⊕ Cl D F (π L (A)). It is therefore sufficient to verify that there are no other operators T that map H SM to H SM , which would satisfy that they commute with the representation of Cl D F (π L (A)) ⊕ π R (A). Identifying the Hilbert space as C 16 ⊕ C 16 we see that the first component of Clifford algebra is M 4 (C) ⊕ M 4 (C) (3) (action diagonally on C 16 , the notation B (n) means that we take n-copies of the algebra B) and the second is C (4) ⊕ M 3 (C) (4) . Since all these algebras are independent of each other there exists no operator intertwining their actions, hence the commutant is exactly the one indicated above.

D. The reality and the CP-violation
Let us take the real structure J acting on finite part just by the complex conjugation, that is, the real structure implemented on M 4 (H W ) simply as id ⊗ J ⊗ id. Of course, it does not implement the usual zero-order condition, however, we still have a milder version of the zero-order condition in the following form: We have already observed what are the commutation relations between D and J (and hence J). Next let us see whether similar commutation relations can be imposed on D F . As both J 2 as well as the anticommutation with Γ are fixed, we see that by imposing the same KO-dimension (6) for the Euclidean finite spectral triple as for the Lorentzian spatial part we shall have J D F = D F J. This condition is very mild and means that the mass matrices M l and M q have to be real. In case of one generation of particles it implies that masses of fermions have to be real, which is hardly very restrictive.
Yet the situation changes when we pass to three generations as already discussed above when considering the spin c -condition. Since J acts by complex conjugation then the requirement D F J = J D F still is equivalent to the matrices M l , M q having only real entries. Using the standard parametrization described above, this leads to the reality of the physical masses.
However, since in the case of three generations the matrices Υ ν , Υ d are not diagonal we must ensure that both U and V mixing matrices as real.
If this is the case, then all phases in the standard parametrization of these matrices should vanish, which physically will have the interpretation of the CP symmetry preservation. However, in case of the CKM matrix it implies that the Wolfenstein parameterη has to vanish, but experimentally it is known thatη = 0.355 +0.012 −0.011 [15]. The CP-violating phase δ ν CP in the neutrino sector, originated from the PMNS matrix, was determined to be δ ν CP /π = 1.38 +0.52 −0.38 [15,16], that strongly confirms the CP symmetry breaking. Therefore, the existence of CPviolation may be interpreted as a shadow of J-symmetry violation in the non-doubled spectral triple.

E. Twisted (pseudoriemannian) spectral triple
We have verified that the Krein-shifted Dirac operator satisfies the order-one condition (7).
It appears that this is equivalent to the Lorentzian Dirac operator D ST = β D ST satisfying a twisted version of the order-one condition, that is, where [x, y] β = xy − βyβ −1 x and β = id ⊗ γ 0 ⊗ id. This follows directly from a simple computation, which uses β 2 = id:

V. CONCLUSIONS
Let us stress that the geometry of the Standard Model, as discussed above, is not a product of spectral triples. Nevertheless, it has interesting features, which we summarize here with an outlook for the future research directions.
When restricted to the commutative algebra of real-valued functions (and its complexification) we obtain the even Lorentzian spectral triple with a real structure, so that the Dirac operator satisfies order-one condition and which is of KO-dimension 6 (compatible with the signature (1, 3)).
On the other hand, the restriction of the spectral triple to the constant functions over the Minkowski space gives a Euclidean even spectral triple, which, fails to be real. The failure of the real structure to satisfy the commutation relation with the (Krein-shifted) finite part of the Dirac operator is tantamount to the appearance of the violation of CP symmetry in the Standard Model.
Neither of the restrictions does satisfy the spin c -condition, as in both cases we still consider the full Hilbert space. Yet the full spectral triple satisfies the spin c -condition in the following sense: the Clifford algebra generated by the commutators of the Krein-shifted Dirac operator with the representation π L of the algebra has, as the commutant, the right representation of the algebra π R .
There are several possible ramifications of the above observations. First is the disappearance of the product structure: yet even if the triple is not a full product, then possibly it can have some structure of a quotient "spectral geometry". It will be interesting to classify all possible covers and all Dirac operators for them. In the presented spectral triple the family of allowed Dirac operators that satisfy the spin c condition is much closer to physical reality as it does not include any color-symmetry breaking operator unlike [7] and, moreover, the conditions are exactly the same as for the Hodge duality. The failure of the finite spectral triple to be real is then a geometric interpretation of the CP-symmetry breaking in the Standard Model. Finally, the disappearance of the product structure may have deep consequences for the spectral action. We postpone the discussion of possible effects on the physical parameters of the model for the forthcoming work.
Acknowledgements: The authors thank L.Dąbrowski for helpful comments.