Spectral action and the electroweak $\theta$-terms for the Standard Model without fermion doubling

We compute the leading terms of the spectral action for a noncommutative geometry model that has no fermion doubling. The spectral triple describing it, which is chiral and allows for CP-symmetry breaking, has the Dirac operator that is not of the product type. Using Wick rotation we derive explicitly the Lagrangian of the model from the spectral action for a flat metric, demonstrating the appearance of the topological $\theta$-terms for the electroweak gauge fields.


Introduction
The Standard Model of Particle Physics is a powerful theory that gives a precise and effective description of all fundamental forces apart from gravity. Its predictive power and agreement with experimental results guarantee that it needs to remain the backbone of any fundamental theory of particle interactions. Yet, in contrast to General Relativity, which is deeply rooted in geometry of the space-time, the Standard Model only partially can be explained in a similar manner. The structure of gauge theory and the Yang-Mills action signifies that indeed the geometry plays there a significant role. However, the appearance of a Higgs field and the symmetry-breaking quartic potential are not directly implied by the classical geometry.
The hint that the Standard Model has more subtle structure came from noncommutative geometry and the theory of spectral triples. Founded by Alain Connes to solve significant mathematical problems related to the index, the theory is a well-structured non-trivial generalization of classical differential geometry that allows to study not only differentiable manifolds but also discrete spaces, fractals and quantum deformations of spaces from a novel point of view. Interestingly, the tools of noncommutative geometry allowed to construct models that explain the geometry of the Standard Model [1][2][3] (see also [4] and [5] for detailed discussion) and its extensions [6][7][8][9][10]. Their structure is similar to Kaluza-Klein models yet with a finite noncommutative algebra instead of the additional dimension of space-time. The geometry of the entire enhanced space-time is determined by a Dirac operator that depends on the metric and the gauge connections, and also includes the Higgs field, which plays a role of a connection over the finite noncommutative component. The spectral action then gives the full gravity and Yang-Mills action with the quartic Higgs potential and minimal couplings between the Higgs and the gauge fields [11].
The story of the noncommutative model-building is, however, not yet complete as the mostly accepted model is in the Euclidean signature and requires additional assumptions to remove the possibility of the SU (3) symmetry breaking [12,13] as well as additional projection onto the physical space of fermions (due to the fermion quadrupling in the model) [14][15][16]. In the analysis of the Lorentzian case with a slight modifications of the spectral triple rules we proved that there exists a model without the fermion doubling and with exact colour SU (3) symmetry [17]. Moreover, the non-product Dirac operator satisfied a slightly modified first-order condition which is tantamount to spin c condition under certain conditions on mass spectra of fermions. The CP-symmetry breaking in the Standard Model was then geometrically explained as the lack of reality symmetry of the finite component of the Dirac operator as witnessed by nonvanishing of the Wolfenstein parameter and the CP-phase in the neutrino sector.
In the paper we compute the spectral action for the model we presented in [17]. It needs to be stressed that this model is not of the product-type geometry and therefore the computations and results are not automatically identical to these performed in the series of papers computing the spectral action [11,18]. In addition, as we start with the Lorentzian model we need to perform a Wick rotation to be able to use heat trace techniques [19] or restrict the model to spatial and time-independent (static) components of the fields. To check the consistency of the computations we perform both operations. The new element of the spectral action, apart from slight differences in the relative coefficients, is the appearance of topological theta terms for the gauge fields in the electroweak sector. This is a characteristic new feature of this model, which is inherently chiral, especially that such terms cannot appear in the spectral action of the product geometries.

The starting point: fermions and the algebra of the Standard Model
We begin by briefly reviewing the model as described in details in [17,20]. The particle content in the one-generation Standard Model can be conveniently parametrized in the following form: Every entry of the above matrix is a Weyl spinor (from H W ) over the Minkowski space M 1,3 . The algebra A is taken to consist of (smooth) C ⊕ H ⊕ M 3 (C)-valued functions over M 1,3 . We choose its left and right real representations: where λ, q and m are complex, quaternion and M 3 (C)-valued functions, respectively. Since left and right multiplications commute, the zeroth-order condition is satisfied, i.e.
[π L (a), π R (b)] = 0 for all a, b ∈ A. It is convenient to encode local linear operator acting on the particle content of the model, at every point of M 1,3 , as an element of where the first and the last matrices act by multiplication from the left and from the right, respectively, while the middle M 2 (C) matrix acts on the components of the Weyl spinor. Using this notation, the full Lorentzian Dirac operator of the Standard Model can be written of the form, where σ 0 = 1 2 = σ 0 and σ i = −σ i , the latter being standard Pauli matrices. D F is a finite endomorphism of the Hilbert space M 4 (H W ).
In [17] the Krein-shifted full Dirac operator of the Standard Model, D SM = βD SM , where was discussed in details. The Krein-shifted manifold component of the Lorentzian Dirac operator D in the local Cartesian coordinates over R 4 , with a flat metric, is whereas the Krein-shifted discrete part of the Dirac operator is, where M l , M q ∈ M 2 (C).
The Krein-shifted Dirac operator and the algebra were proven to satisfy the modified order-one condition: for all a, b ∈ A, The Lorentzian spectral triple for the signature (1, 3) has the standard chirality Z 2grading γ and the charge conjugation operator, J , where cc denotes the usual complex conjugation of spinors. The construction can be easily generalized for the three families of leptons and quarks by enlarging the Hilbert space diagonally, i.e. by taking M 4 (H W ) ⊗ C 3 with the diagonal representation of the algebra. The matrices M l and M q in (2.5) are no longer in M 2 (C) but in M 2 (C) ⊗ M 3 (C). Its standard presentation for the physical Standard Model is where Υ e and Υ u are chosen diagonal with the masses of electron, muon, and tau and the up, charm, and top quarks, respectively, and Υ ν and Υ d can be diagonalised by the Pontecorvo-Maki-Nakagawa-Sakata mixing matrix (PMNS matrix) U and the Cabibbo-Kobayashi-Maskawa mixing matrix (CKM matrix) V , respectively: The matrices Υ ν , Υ d provide (Dirac) masses of all neutrinos and down, strange, and bottom quarks.
As it was demonstrated in [17] the model has interesting algebraic properties, like the Morita duality (which means that the generalized Clifford algebra is a full commutant of the algebra A) provided that both pairs of matrices (Υ ν , Υ e ) and (Υ u , Υ d ) have pairwise different eigenvalues. Furthermore, without referring to additional symmetries or assumptions the model preserves the SU (3) strong symmetry and allows for the natural breaking of the CP-symmetry, which is linked to the non-reality of the mixing matrices. This is, on the level of algebra of the model, equivalent to the failure of the finite part of the Krein-shifted Dirac operator to be J -real (see [17] for details).

The gauge transformations and the Higgs
In this section we extend the model by intruding the fluctuations of the Dirac operator, that is a family of operators obtained from D SM by perturbing them with one forms, that originate from commutators with the Dirac operator. Here, both left and right representations will contribute to the gauge transformations and to the fluctuations of the Dirac operator.
A gauge transformation of physical fields is a unitary transformation of the form, for a unitary element U of the algebra A, so that the gauge-transformed Dirac operator becomes: which, after using the order-zero and order-one conditions, can be rewritten as The unitary group of the finite algebra is U (1) × SU (2) × U (3), while the unitaries of the form U LR are elements of the group being a quotient of this group by its diagonal normal subgroup To parametrize the fluctuations we first start with computing left and right ones separately: , and the representations π L and π R are of the form: where λ a ∈ C ∞ (R 4 ), q a ∈ C ∞ (R 4 , H) and m a ∈ C ∞ (R 4 , M 3 (C)).
We first notice that [ D F , π R (b)] = 0 from the very definition of the representation and the structure of this Dirac operator. Therefore, the only contribution from the right fluctuations can be parametrized as Now, we compute the left fluctuations. Starting with the ones following from the D part of the Dirac operator we immediately get Imposing the selfadjointness condition we immediately get A µ = −A µ and infer that W µ is indeed an element of isu(2) (as it is enforced to be a real linear combination of Pauli matrices). Similarly, we deduce that G µ is a U (3) gauge field.
It remains to take into account the contribution from D F . By a straightforward computation we get Since both matrices M l and M q are diagonal, they commute with diag(λ, λ). Denoting by we can parametrize those fluctuations, under the assumption of selfadjointness, as: (2.18) As a result, the fluctuations coming from all the terms can be parametrize as

(2.19)
We denote the fluctuated Dirac operator by D SM (2.20) The gauge transformation can be therefore implemented by As a result, the fields A µ , W µ , G µ , Φ transform accordingly: (2.21) We remark that in the above derivation the crucial role was played by the fact that the representation of U (1) part of the gauge group commutes with the mass and mixing matrices.
It is known that the gauge group of the Standard Model should contain SU (3) rather than U (3). This can be achieved with a further condition, the unimodularity of the representation, which, however, can be imposed in different ways. In particular, let us observe that the left action of the group is unimodular from the beginning. For the right representation one could require either the condition that it is unimodular on each fundamental component (chiral lepton and quark) or in the full representation, including all chiral fermions and families. In the first case, the unimodularity condition is equivalent to u 1 det u 3 = 1, whereas in the second case it becomes (u 1 det u 3 ) 12 = 1. In the first case, the resulting group is exactly the group of the Standard Model, whereas in the latter case it is the one described in [5,Prop. 11.4], which differs from the gauge group of the Standard Model by a finite factor. Independently, their Lie algebras agree and the finite difference does not affect the structure of the gauge fields. The unimodularity condition on the algebraic level of perturbation means that the trace of a perturbation has to vanish. This condition results in Tr(G µ ) = A µ . We therefore introduce the traceless gauge field G µ = G µ − 1 3 A µ 1 3 and then in the perturbations we can replace G µ by G µ + 1 3 A µ 1 3 , where now G µ is assumed to be a SU (3) gauge field.

The spectral action
In the considerations so far (see e.g. [5] and references therein), the spectral action for the Standard Model was computed for the Euclidean model with fermion doubling. Moreover, the assumed Dirac operator was of the product type and therefore its square was simply the sum of the squares of the Dirac operators on manifold component and on the discrete component. While this strategy can be justified by the arguments of covariance and geometric character of the action, the relative coefficients and the proportions between them may in general differ, when one considers the Lorentzian and explicitly chiral Dirac operator.
Of course, the best strategy would be to apply genuine Lorentzian approach (see [21]), however, this appear to be at the moment restricted only to scalar operators and not Diractype operators. Therefore we propose two simple, computable methods to obtain an insight into the action of the model, which is motivated by spectral methods.
The first one assumes that we restrict ourselves to the static and spatial case, computing the terms of the spectral action for the Krein-shifted Dirac operator that is restricted to the spatial part and with the gauge fields that arise exclusively through static (time independent) gauge transformations. Such restricted Dirac operator is already a hermitian elliptic operator and one can easily compute the heat trace coefficients of its square. This shall recover the action of the model for the time-independent fields, which is invariant under static gauge transformations. However, one cannot expect that all terms of the action will be present, and their coefficients correct.
The second method takes as the input the Wick-rotated Lorentzian Dirac operator D w . Such operator is elliptic, as its continuous part is just the usual Wick-rotated Dirac operator (with gauge fields) over the flat space-time. However, the discrete part of the operator (which is not Krein-shifted) is alone not hermitian but only normal. Nevertheless one can still compute the heat trace coefficients of D † w D w and then, using the Wick rotation back to the Lorentzian case recover the action functional.
In what follows we assume that we work on a flat compact manifold (torus) so all curvature terms vanish from the spectral action, and we are using a physical parametrisation of fields, described next.

Spectral action -physical parametrization
Let us now write explicitly the full spectral action in terms of Yukawa parameters and Higgs field in the case of one generation of fermions. Since Φ is a quaternion it can be parametrize as where φ 1 , φ 2 are two complex fields. Then where we introduced the Higgs doublet:

The spectral action for the full static SM
We consider here the Krein-shifted operator for the static simplified Higgs model, computing the coefficients of the spectral action for its spatial part only, which is an elliptic operator. For consistency we keep the dimension-related constants in the Gilkey-Seelay-DeWitt coefficients the same as for four dimensional case. The fluctuated Krein-shifted static Dirac operator for the Standard Model splits into the lepton and the quark sector, after introducing new fields, with the lepton part where we have used the identification M 4 (C) ⊗ M 2 (C) ⊗ C ∼ = M 4 (C) ⊗ M 2 (C) and therefore omitted the third component in the expression above. This reproduces the correct hypercharges in the leptonic sector: 0, −2, −1, −1. We remark that for the left particles, the hypercharges are defined as coefficients in terms containing σ j instead of σ j .
For the quark sector we have: Again, it gives correct hypercharges for quarks: 4 3 , − 2 3 , 1 3 , 1 3 . Contributions to the action can be computed separately for the leptonic and the quark sector. The detailed computations are in the Appendix A, here we present the final result in the physical parametrization.

The full spectral action
The asymptotic expansion of the spectral action for models on a four dimensional space with a fluctuated Dirac operator D ω is given by and reduces simply to where a k are the so-called Gilkey-Seeley-DeWitt coefficients and can be computed explicitly -see e.g. [5,19] for the detailed discussion. Here f is a sufficiently regular function (see e.g. [22,Chapter 2]) with f k being its kth moment, and Λ is the cut-off parameter.
We start with the model containing only one generation of particles. In this case we get and κ is the normalization of the trace.
In case of three generations we have to change the above coefficients into , and we have to multiply the terms with field curvatures by a factor of 3. As a result Taking κ = 4 and ignoring the term related to the gravitational constant (i.e. the one ∼ Λ 4 ) we end up with a model with an effective Lagrangian L = L Higgs + L gauge , where The above result is in agreement with the one in [5,Prop. 11.9], for c = d = e = 0 in the notation used therein. Furthermore, notice also that this is consistent (up to an irrelevant global factor) with taking the static part of the Lorentzian Lagrangian for the Standard Model. Indeed, we have In particular any prediction related to the mass of the Higgs field remains unchanged.

Wick rotated model
Let us start with the full Krein-shifted Dirac operator in the leptonic sector, Its Lorentzian counterpart is of the form In what follows we perform a Wick rotation on the level of the algebra of Pauli matrices: σ j → iσ j and consequently σ j → −iσ j , while the σ 0 remains unchanged. The Wick-rotated Dirac operator in the leptonic sector is then of the form (3.5) For the quark sector we have and after performing the Wick rotation we get Again, all further details of the computations are in the appendix B, and in what follows we present only the final expressions for the Wick-rotated model.

The full spectral action
We consider now the full model with three generations of particles. In this case, using the above results, we get and where the parameters a and b are as before: Notice that by construction these parameters are non-negative. Taking κ = 4 and considering the first terms of the asymptotic expansion (and neglecting the gravitational terms) we end up with the following Lagrangians for gauge fields and the field H: Since the Wick rotation was performed in three spatial directions on the level of Pauli algebra, when going back from the Minkowski signature (1, 3) into the Euclidean one we have to change spatial derivatives and coordinates according to ∂ j → −i∂ j and A j → −iA j , respectively, and in order to preserve the spin structure we have to change the Minkowskian structure constants ε jkl M into their Euclidean counterparts: so that for these terms we have in a complete agreement with our result. The global minus sign (together with the additional −i factor from the measure) is related to the definition of an Euclidean action: iS M = −S E . Next, let us consider the remaining term: Taking into account the additional global sign from the identification of iS M with −S E , we finally see that the Lorentzian counterpart of 6ε jkl F jk F 0l − 3ε jkl Tr (W jk W 0l ) is Therefore, the spectral action for this model contains terms which can be interpreted as the so-called θ-terms in the electroweak sector [23][24][25]. We remark that from the above derivation of the spectral action not only the presence of such terms is deduced but also the numerical value of the electroweak vacuum angle is fixed by the model. The presence of such terms was linked with the CP-violation [24], especially for the discussion of the baryogenesis process. In contrast to the usual considerations in the physical formulation of the Standard Model no CP-violating θ-term in the QCD sector is present here. It will be interesting to see what are the physical limitations, e.g. on the energy scales on which such model is valid, from the perspective of the presence of the electroweak θ-terms. The CP-violation was present in this model also on the level of algebra by the lack of the J -symmetry [17]. We remark that the explicit form of the potential V (H) differs from the one in the standard derivation [5], where the coefficient in the quadratic term |H| 2 contained −2f 2 instead of 6f 2 , which we have in the present model. In the usual formulation f is assumed to be, besides the others, a non-negative on the positive real half-line, so then f 2 is also non-negative therein. If we would not allowed for any relaxation of this principle, our model will not predict the Higgs mechanism, or in other words the model could be valid only in a sector with the Higgs potential of the form the Higgs potential will not possess a non-trivial minimum. On the other hand, having the possibility of using function f which has negative second moment gives rise to an effective action for the Standard Model with the Higgs mechanism implemented in a completely similar manner as in the usual product-like almost-commutative geometry [5]. Since all the derivations were made on the algebraic level we could, by linearity, extend the applicability of the usual methods into the case with f being a difference of two positive functions. However, the discussion of the analytical aspects is required in order to establish the range of validity of these computational methods -see [22] for some further discussion of these aspects which are beyond the scope of this paper.
Allowing for the negative value of f 2 there is no further difference in the numerical value of the Higgs mass, which can be computed from the derived Lagrangian using the standard tools based on the renormalization group equation [5,11]. This is because the difference in the numerical value of f 2 in the coefficient for the |H| 2 term does not affect any relation between the mass of the W boson, the Higgs vacuum expectation value v and the coupling constant g 2 for the W boson field.

Conclusions and outlook
The presented noncommutative geometric model describing the particle interaction appears to be the closest to the observed Standard Model. The derived bosonic spectral action gives all correct terms and, in addition, new, topological θ-terms. While the latter has no significance for the dynamics of the model, it may play a role for the quantum effects for the electroweak sector. These terms are, in principle, not excluded and have been discussed in literature [23][24][25]. The result signifies also that computing the spectral action for the Wickrotated Lorentzian Dirac operator is important. Possibly, the next step shall be to compute the genuine Lorentzian spectral action using the tools that are at present available for the Laplace-type operators [21]. Furthermore, possible relations of non-product geometries with bundle-like structures over noncommutative manifolds [26] as well to the inclusion of gravity for this non-product geometry (see [27] for a link between nonproduct geometries and gravity) shall also be explored and examined. Finally, it shall be interesting to see possible extensions of the model, both in the direction of scalar conformal modifications that can help to fix the Higgs mass as well as extensions of the Pati-Salam type [28,29].

A.1 Leptonic sector
In the leptonic sector we have where, As a result, following the notation of [19], we have ω j = 1 2 a j , so that Next we compute, We get then and furthermore, where κ is the normalization of the trace (i.e. everything within the bracket is computed for the unnormalized trace) and W 2 = W jk W jk . Moreover, As a result, in the leptonic sector we have Using the parametrization from the section 3.1 in this sector we then have and as a result Next, we decompose the W field in terms of Pauli matrices,

so that
Tr Next, we compute Let us now verify whether these terms can be written using the covariant derivative of the Higgs doublet, We check, The only terms that are potentially different that the ones in the coefficient a 4 are but simple calculation shows that and in a complete agreement with a 4 . Therefore,

A.2 Quark sector
In this sector we have Therefore, As a result, where we use short notation G 2 = G ij G ij , and the full second contribution reads, As a result, in the quark sector we have In a completely similar manner as for the leptonic sector we derive: and Its trace is therefore Furthermore, we have where now F 2 = F µν F µν = F jk F jk + 2F 0j F 0j and similarly for W 2 .
Next, we have where the summation is performed over Euclidean indices.
As a result, in the leptonic sector we have Using the parametrization from section 3.1 we can further write and (B.11)

B.2 Quark sector
For the quark sector, starting from (3.8), we get (B.12) In this case we therefore have (B.13)

Moreover
, and