Light Mediators in Anomaly Free $U(1)_X$ Models I - Theoretical Framework

We examine theoretical features of $U(1)_X$ extensions of the Standard Model whose quantum anomalies are canceled per generation. Similarly to other versions, the theory consists of a Two-Higgs-Doublet Model plus a scalar singlet embedded into the $SM \otimes U(1)_X$ gauge group, and introduces small modifications to the $Z$-boson interactions. These changes can be minimized by exclusively charging right-handed fermions under the new Abelian symmetry, and are compensated by the neutral $X$-boson exchange. Non-universality of fermion couplings can also be achieved by requiring one single $X$-charged family. In general, $X$ gauge bosons can be separated into $A'$ (dark photons) and $Z'$ subsets, distinguished by the presence of axial-vector currents. $A'$ physics is commonly simpler to constrain and therefore favored by experimental tests. Finally, the model can be UV completed both by stable $\chi$ fermions or by right-handed neutrinos. The prior case may provide cold WIMPs in the theory.


Introduction
The existence of dark matter (DM) has been supported by numerous astrophysical and cosmological arguments (see e.g. [1]). On the other hand, experimental and theoretical efforts for direct detection have provided strong constraints on the couplings of DM candidates to the Standard Model (SM) particles. The simplest DM model can be obtained by extending the SM with an additional U (1) X gauge symmetry. This type of extension commonly requires, for the UV completion, at least one new fermion χ which can be made stable by some ad hoc dark symmetry. The symmetry forbids the appearance of tree-level couplings between χ and SM fields and a DM portal is generated exclusively through the neutral gauge bosons [2]. In the case where the boson associated with the new Abelian symmetry contains only vector couplings it is generally referred to as a dark photon.
Very often dark photons searches are performed by assuming that the new vector is on-shell and then decays into µ + µ − pairs [3]. New Physics effects in the few MeV region generally require searches for e + e − channels facing a vast set of background events. The possible discrepancies [4,5] in the region m X < 2m µ motivate proposals of the simplest U (1) X SM extension containing one weakly coupled gauge boson. As summarized by the authors of Ref. [6] this task must follow in accordance with some strong requirements, which goes from the absence of new electrically charged fields at low energies to the consistency of neutrino interactions with electrons and nucleons. In particular, the last argument will motivate the proposal of right-handed specific models.
The present work contains a detailed study of the basic properties of U (1) X models. We are guided by general principles as (i) introduction of a minimal set of free parameters and physical degrees of freedom (falsifiability); (ii) the fermions are accommodated in the SM representations and quantum anomalies are canceled per generation (as in the SM); (iii) the degree of fermion non-universality should explain observed discrepancies between theory predictions and experimental measurements, such as the flavor anomalies in B meson decays [7,8] and the muon anomalous magnetic moment puzzle [9][10][11][12][13]. In a subsequent work we consider manifestation of New Physics in the low energy regime (∼ MeV) by allowing small couplings of the new vector X to matter fields of the order g X ∼ 10 −4 to 10 −2 . In Ref. [14] one can notice, for instance, that the region 10 MeV -1 GeV still allows the presence of X with a coupling g X ≈ 10 −4 in the U (1) B−L case, whose strength might imply measurable effects in heavy meson physics. The same is not true in the case of U (1) B or the protophobic model. This distinction illustrates the impact of different X-hypercharge assignments albeit under the same gauge structure.
A minimal SM extension is readily achieved even when non-universal X-hypercharges for leptons and quarks are considered. The first consequence of arbitrary charges is the appearance of a flavor matrix F in the fermion-gauge sector. Its properties are specially interesting in the case when only one generation is charged under X. Moreover, the attempt to reduce the number of free parameters may favor the choice of a chiral U (1) X . Once more, if we aim to control the modification of neutrino interactions, the right-handed (RH) currents might be the appropriate option within this framework. In the scenario where only RH fermions are charged under U (1) X , the constraints from quantum anomalies arise exclusively from U (1) 3 X , U (1) 2 X U (1) Y , U (1) X U (1) 2 Y and SU (3) 2 U (1) X currents, once SU (2) does not play any role. This minimal version 1 shows that is not possible to charge the quarks universally in order to avoid flavor changing neutral currents (FCNC) and to preserve, at the same time, the feature of anomaly cancellations per generation. Besides, the call for fermion non-universality along with the criterion to recover a consistent CKM matrix will demand the introduction of a new scalar doublet.
In summary, here we introduce a Two-Higgs-Doublet Model (2HDM) embedded in the SM ⊗U (1) X gauge group where only the second generation of fermions is charged under X. Our main goal is to determine the allowed region of the parameter space in comparison with similar U (1) versions already presented, for instance, in [4] and [15], whose phenomenology were focused in the MeV-GeV regime. Motivated by the absence of left-handed (LH) singlets under the SM, we construct the UV completion by including in the setup a new right-handed χ R fermion charged exclusively under U (1) X . We focus on a stable fermion whose mass is generated after a symmetry breaking performed by a scalar singlet. The alternative version where a leptonic Yukawa Lagrangian generates neutrino masses is possible too and is discussed in the Appendix A.1. Notwithstanding, we choose to investigate in detail a DM model due to the following reason -the tiny couplings governing the χχ ↔ SM portal can turn a Weakly Interacting Massive Particle (WIMP) overabundant in the present universe and therefore forbidden. Hence, the relic abundance constraints are expected to cover the parameter space in the opposite direction if compared with those from direct detection, which provide us with a mechanism to maximize its excluded area and eventually rule out the particular model.
The work is divided into three sections. Section 2 introduces the general features of U (1) X theories, and motivates the variant with X-charged right-handed fermions. Section 3 discusses the experimental status for dark photons, A , and Z searches. The last section is devoted to our conclusions.

General properties of U (1) X and inclusion of right-handed fermions
In this section we determine the functional dependence of the vector and axial-vector currents on the hypercharges and fundamental constants, like v.e.v's and couplings. We see that general Abelian extensions must produce Z-like, instead of A-like, vertexes, i.e.
where x V , x A are non-universal and flavor violating matrices and L χ describes the possible interactions of new fermions in the theory. The Eq.(2.1) illustrates the fact that the dark photons phenomenology comprises a subset of the X boson theory, where the axial couplings are set to zero. By keeping the complete vertex in our study, we can determine the impact of x A on the allowed parameter space. In addition, non-universality effects accommodated within the SM ⊗ U (1) X model usually spread into both quark and leptonic sectors. Apart from that, the long-lived boson is commonly assumed to decay exclusively into a darksector, i.e. Br(X → χχ) = 1, which motivates experimental searches in processes with invisible final states. By allowing the gauge boson to couple to an electron-positron pair, the parameter space for X (long-lived enough to decay out of the detector) has the impact of loosening important bounds, such as those from K → µ + invisibles [16]. Apart from the Yukawa Lagrangian and the X-hypercharge assignments, the following description and results are general in the sort of 2HDM plus scalar singlet of SM ⊗ U (1) X : • Three independent coupling constants g, g Y , g X and a kinetic mixing constant ; • Three generations of weak isospin doublets: with i = 1, 2, 3; • right-handed SU (2) L singlets: χ R , l iR , u iR , d iR ; • Y hypercharges: • X hypercharges: with the remaining RH fields uncharged.
• Higgs doublets φ 0 , φ X and singlet s: Notice that the term φ † 0 φ X s is allowed in the scalar potential.
• Electroweak Lagrangian is Anomalies A basic prerequisite of any ultraviolet complete gauge theory is that it is free of triangle anomalies [17]. The following equations summarize how this criterion can be achieved for arbitrary X charges within SM ⊗U (1) X models. Again, Q and L denote quark and lepton doublets, respectively, while the rest denoted by u R , d R , e R refer to the charges of right-handed fields: In the Standard Model the above equations are solved per generation. This property can be taken as part of the SM structure and implies that no information about a new fermion family could be found particularly through these diagrams. Here we will follow this principle. The solutions define a subset of U (1) extensions and are given by 2 [18] (2.8)

Two Higgs Doublets Requirement
The theory is designed to be (a) non-universal under X charges and (b) to be minimal in its particle content. Thus, let us first assume it is possible to generate all fermion masses through only one Higgs doublet. FCNC should not appear in the scalar sector under this property. In order to construct the Yukawa Lagrangian, one has to consider solutions to the equations which should also satisfy Eq.(2.8). In addition, the condition is necessary for generating masses for all charged fermions 3 , where the indexes denote the two non-universal X-hypercharges. Notwithstanding, the mass matrix corresponding to the above criterion can be put in a block-diagonal form, and would be inconsistent with the expected form of the mixing matrices. We must additionally fill at least two more entries by requesting the condition The solutions are obtained by solving to XL in the quadratic expressions, with Yχ = 0, and then for the gravitational and cubic case. 3 Alternatively, one could create the entries XL i − X l j = XL j − X l i along the diagonal ones XL i − X l i = X0, which would lead to the same conclusion.

Now, both conditions combined imply
hence breaking the (a) criterion. Naturally, in the U (1) X extensions one must introduce a larger scalar sector, in comparison to the SM, due to the creation of the longitudinal polarization for the massive X µ boson. In summary, the Eq.(2.12) will state that any version containing only one Higgs doublet is necessarily universal in the fermion families.

Kinetic mixing
Once is assumed to be a small parameter, it is convenient to translate its dependence directly into the coupling constants, thus leaving the kinetic Lagrangian in a diagonal form.
The task can be achieved through the field redefinition i.e., by rewriting Eq.(2.6a) i.e. the crossed terms vanishes and the mixing effect is converted into a new term in the covariant derivative: where, up to first order, one may write g Y ≡ κ.

Couplings and masses of gauge bosons
In the previous section we showed that the non-universal model must contain at least two Higgs doublets, here denoted as φ 0 and φ X , as a necessary condition to recover the correct mass spectrum of the fermions. In addition, a singlet s is required to couple to the fermion χ R (or to generate the mixing between the second and the remaining generations of RH neutrinos, see A.1) as well as to break a residual U (1) in the potential which could leave the theory with a massless pseudo-Goldstone boson at tree-level [15]. The gauge boson masses are extracted from the kinetic piece of the scalar Lagrangian once the scalars acquire a vacuum expectation value. In terms of ladder operators the covariant derivatives can be written as 4 Once the relation Q = T 3 + Y to the electric charge matrix is preserved, it follows that The charged currents are untouched and result for the W mass The neutral fields must mix and their mass matrix is extracted from the symmetric The above real symmetric matrix eigenvectors define an orthonormal basis and compose the orthogonal matrix V which rotates the fields from the gauge to the mass basis. Although the choice of parametrization for V is not physical, there are options which can make the analysis simpler. Consider, for example, the choice made in terms of the three Euler angles in the usual zxz rotations by the angles (φ, θ, ψ) (using notation sinα ≡ s α and cosα ≡ c α ) The angle θ introduces the mixing of B X µ with the remaining gauge fields, i.e. θ is the angle between the B X µ and the z-plane where the W 3 µ − B Y µ mixing occurs. All three angles can be written using the couplings, vev's and scalar charges.
One can notice from Eq.(2.20) that the block (W 3 µ , B Y µ ) has a null determinant. This substructure of M 0 implies a zero entry in one eigenvector, which fixes one of the angles. Therefore, by taking c ψ = 0; s ψ = −1 and applying a phase redefinition (2.23) The minimal coupling in the covariant derivative can be presented by where the vectors are defined as and The parametrization of Eq.(2.23) corresponds to By taking g, g Y to be the same as in the SM, the angle φ must correspond to the weak mixing angle, related to the electric charge by On the other hand, the Z-couplings will be replaced by , thus making explicit how θ (i.e the small parameter s θ ) tunes the change in the Z interactions due to the presence of a new neutral gauge boson. Finally, the interactions with the new X µ is governed by Once the parametrization of Eq.(2.23) is established, the above results are general for 2HDM-like models. The fermion interactions with the new X µ gauge boson can be determined using Eq.(2.30). It contains a term proportional to the SM Z-coupling, weighted by s θ , and its last piece will regulate the amount of flavor violation in the gauge sector, along with the gauge symmetry-mass basis rotation matrices. One can notice that, after electroweak symmetry breaking (EWSB), the chiral version with RH fields still generates the gauge boson vertexes with LH currents via the kinetic coupling κ. The gauge interactions with X-neutral fields are further suppressed by s θ .
Using the parametrization Eq.(2.23) and the choice of Eq.(2.5), s θ can be written as (2.32) From these definitions the neutral gauge boson masses, in terms of couplings and v.e.v's, are given by Now one can verify that, if g X , κ ḡ In the same limit the angle may be written in a simplified expression: As mentioned before, the above description is general in 2HDM embedded in a U (1) X extension of the SM. The main feature of this study is: when the charges under X are nonuniversal, there must appear flavor changing neutral currents both in the interactions with Z µ and X µ , doubly suppressed in the first case by the factor s θ g X . Moreover, regarding the lepton flavor violation (LFV), the first effect of charging LH fields under X is that a number of free parameters appear, due to the matrix elements of V l L which rotates these fields to their mass eigenstates. Therefore, the choice for charging only one chiral fermion provides a minimal description of flavor violating processes, although it certainly does not resolve the entire combination of operators. For instance, in the case of R K ( * ) anomalies in B physics, , which can be generated from a particular choice for LH and RH hypercharges (see section 2.3).
Once more, we choose to charge RH fields in order to preserve SM-like (for s θ = 0) the Z boson interactions with LH fields. From the phenomenological point of view, we also opted for charging the second generation only. Apart from the anomalous magnetic moment, both for electrons and muons, we can include the effect in the proton charge radius measured from the Lamb shift in e-hydrogen and µ-hydrogen among the most stringent constraints to the model [5].
In this framework, LFV will be mediated by the currents When e R represents the mass eigenstates, such that e R = V l R l R , the term is converted to (2.38) Therefore, the matrices introducing flavor violation and non-universality are given by We see that only a coherent explanation about the possible alignment between flavor and mass eigenstates could confirm the assumption of small FCNC processes in the model. Some hints in different sectors of the Lagrangian could be selected, for instance, through the CKM matrix, defined as with implicit summation on k. Thus, if the mass and flavor eigenstates were approximately aligned, simultaneously for U-and D-type LH quarks, all the non-diagonal (ND) elements of V CKM would be suppressed compared to the diagonal ones. The inverse, however, is not true -the presence of phases could suppress ND elements of the CKM but with large terms in the above summation facing a negative interference. Phenomenologically, since the Wolffenstein parameter λ 0.22, ND CKM terms are in fact smaller. Once we do not have any particular information on the matrices components, nothing can be affirmed about interferences or suppression. The same holds, for example, for the PMNS matrix.
In the following we consider U (1) X models, at low-energies (Mev-GeV), originally proposed to explain the muon anomalous magnetic moment discrepancy and the proton charge radius discrepancy (see e.g. [19,20]). These anomalies are often treated as a signal of lepton flavor non-universality. Thus, in the framework described above, we focus on the g X component of the g R coupling.

Gauge interactions with fermions
The gauge interactions with fermions are described by: In terms of mass eigenstates the covariant derivative can be written as where the coupling constant components proportional to the identity -i.e. those depending only on the charges assignment under the SM gauge group -have been separated and describe flavor universal vertexes: The charged currents occur entirely like in the SM, weighted by the CKM matrix. The new physics (NP) effects are limited to the neutral currents. The parameter s θ introduces the size of NP contributions in comparison with the SM ones. Therefore, it can be a small parameter. Since we choose to charge RH fields only, the amount of flavor violating processes in both Z and X interactions is related to the g X X µ term and is therefore exclusive to RH sector, taking place in the second generation. Defining the vector of fermion fields f = (f 1 , f 2 , f 3 ) and rotating the system to the mass basis, the general currents depending on the X charges can be fully separated via: The matrices or summarize the amount of flavor violation and fermion non-universality in the model. Once more, in the scenario where flavor is aligned to the mass eigenstates, i.e. when the absolute value of diagonal elements of V f R are larger than the non-diagonal ones, the flavor violating processes must favor the second generation in the final state, since it would include at least one factor of (V f R ) 22 . Moreover, the diagonal elements also fix the amount of LFV by or, one can define, Due to unitarity of V f R , the trace of F f is equal to X f : Naturally, the closer one of the diagonal entries is to X f , smaller the LFV is predicted by the model. Notice that, unlike the CKM matrix, F does not enclose all the physical processes involving RH fields and the matrices V R must be independently present in the scalar interactions.
The interaction with Z follows a similar pattern but it is doubled suppressed by s θ g X ≈ g 2 X , i.e. flavor changing and non-universality are dominated by X µ interactions. All the non-diagonal vertexes are summarized in the second line of Eq.(2.44), represented by the matrix F, and it is useful to separate the diagonal currents in a simplified form. Here these terms will be written like such that 51) By replacing the electric charges and hypercharges: Clearly, the root for x l A = 0 results in the purely vectorial leptonic vertexes. In addition, by charging LH currents one may generate LH FCNC bi-linears in Eq.(2.44) and enable the effective operators favored by the R K ( * ) flavor anomalies [8].
In our model we emphasize the interactions mediated by a light X µ (m X ∼ 10 2 MeV). If compared to the dark vector exchange, the effects from the remaining new fields, like H, H s , χ 0 r , φ + presented in section 2.4.2, are negligible due to their presence in the decoupling limit [21]. In the second part of this work, the free parameters coming from Yukawas are constrained by experimental bounds and not fixed along the analysis.
X µ interactions with charged hadrons In order to calculate the contribution coming from the inner X-bremsstrahlung from a charged hadron, one must first perform the transformation which converts the QED covariant derivative  where κ = g Y and c θ ≈ 1. The remaining terms include a Z interaction suppressed at second order in the small parameters.

Scalar potential
The scalar potential has the same features as the the given in Ref. [21] (see Eq. (2.26)), e.g. the absence of pseudo-Goldstone bosons ensured by the µ-dependent cubic coupling below: It is convenient to consider a gauge-fixing Lagrangian before we analyze the physical spectra, since it can provide useful tools for the diagonalization of the potential.

R ξ gauges
The longitudinal components of the W boson arise from the contributions of both Higgs doublets. The charged scalars will mix due to the Lagrangian The gauge-fixing Lagrangian can be chosen as This Lagrangian produces, for instance, the term such that, after an integration by parts, it must cancel the equivalent piece in L S . The remaining terms are The determinant of the mixing matrix above is obviously zero. The zero eigenvalue is linked to a physical charged scalar and the non-zero one to a Goldstone boson with mass The scalars mass matrix (M 2 W ) ξ is given in the basis (ϕ + 0 , ϕ + X ) and can be diagonalized by matrix The Goldstone theorem implies that the matrix R W ξ must be orthogonal to the mixing matrix derived from the potential. Hence, they can be simultaneously diagonalized and, as discussed in the following section, the matrices R ξ may be sufficient to diagonalize the entire system.
The construction of a gauge fixing Lagrangian for the neutral Z and X bosons is not as straightforward as in the previous example, although it is based on the same procedure. The difference comes from the the mixing matrix which contains two dependent parts on the gauge fixing parameters. In other words, there must be a set of bi-linears in the Z and X component of the following Lagrangian The masses can likewise be written as m 2 Z = k z 2 k and m 2 X = k x 2 k . After introducing the gaugefixing part, the scalars shall mix analogously to the charged case, but here with both Z and X components containing independent ξ parameters. The mass matrices of both gauge bosons are not commuting and the total mass matrix must be diagonalized at once, such that the two non-zero eigenvalues embody the gauge-fixing parameters. Explicitly, we see that the matrix is given by (2.64) In the absence of ξ X , for instance, the expression results in just one non-zero eigenvalue as L g.f. ⊃ ξ Z 2 m 2 Z χ 2 Z , and vice-versa. Therefore, we can write this part of the gauge fixing Lagrangian using two scalars χ p and χ a such that The remaining scalar is physical, whose mass is determined from the potential and will be denoted as χ r . The matrices coming from Eq.(2.64) are orthogonal to the mixing matrix coming from the potential. Finally, the gauge fixing Lagrangian can be chosen as (2.67)

Scalar spectra
As elaborated in the previous section, the theory contains four Goldstones φ + g , φ − g , χ p , χ a , three physical (pseudo) scalars (φ + , φ − , χ r ) and three Higgses (H, H, H s ), corresponding to the ten original degrees of freedom. The vacuum stability equations are extracted from the linear terms of the real neutral scalars (H χ , H 0 , H s ) and lead to the conditions: • H χ : • H s : Mass matrix -charged scalars In the basis (φ + 0 , φ + χ ), using the vacuum stability equations (2.68), (2.69) and (2.70), the squared mass matrix of charged scalars can be written as and such that Therefore, there is a charged scalar φ + such that Neutral scalars First, we write the mass matrix generated by the gauge-fixing part of the Lagrangian in the following form On the other hand, from the potential and the vacuum stabiltiy conditions, in the basis (χ 0 , χ X , χ s ), it follows that A cross-check can be performed by using the orthogonality of the above matrices. The mass of the physical scalar is given by i.e. in the absence of a term breaking the residual U (1) symmetry of the potential, a massless pseudo-Goldstone boson would survive at tree-level in the model. Once the matrices in Eq.(2.74) and Eq.(2.75) commute, a particular matrix R ξ , rotating the gauge-fixing Lagrangian, can be further applied to M χ and will result in a block-diagonal matrix, then diagonalized via a second R V . In other words, the total mixing matrix can be written like  This assertion is proved by considering two real and symmetric commuting matrices A, B.
and finally, Therefore B is block-diagonal whose dimension is given by the eigenvalues degree of degeneracy.
The Higgs For the real scalars, the vacuum stability equations lead to the following matrix  23 , where λ 0s and λ Xs are both positive numbers, a condition to leave the potential bounded from below. Therefore, the fields denoted by H, H and H s have their masses given by Here the choice for the indexes is motivated by the region where v 0 ∼ v X , such that (2.83)

Yukawa Lagrangian
The choice of non-universal charges requires the inclusion of at least one additional SMlike Higgs, a necessary condition to obtain the correct fermion mass spectra. In addition, one cannot preserve the quarks neutral under X. Actually, we avoid adding a third scalar doublet by assigning the same hypercharge to two fermion generations. Following the proposal of Ref. [15], we charge one generation whilst the remaining ones remain neutral. This approach reduces the flavor matrix F to its minimal version. Therefore, here the new Higgs doublet, charged both under X and Y , fills the second Yukawa matrix column of a U (1) X specific to the second fermion family. The Yukawa Lagrangian in the quark sector, from the notation of Ref. [21], can be given by with i ∈ [1, 2, 3] and j ∈ [1,3]. Note that X U + X D = 2X Q is valid per generation. In general, U (1) X models preserve total equivalence in their gauge structure and are distinguished by the functional form of the equations (2.53) under the kinetic and gauge couplings. In other words, the gauge sector can generically be represented by Eq.(2.50) such that a particular X-hypercharge assignment will then be converted into the independent functions defining the fermion couplings of (2.53). The version we have selected, for instance, generates at tree-level the same set of vertexes as that of [15]. As in their case, we request a new singlet scalar s living at the high scale and it might generate Majorana mass terms for neutrinos. In addition, it explicitly breaks a residual global U (1) in the potential. Although we aim to favor "muon-specific" processes, our X-hypercharge choice X q i = X l i = 0 for i = 1, 3, X s = −X c = 1 and X µ = −X νµ = 1, will still disperse its effects into different flavors. Notwithstanding, under the same constraints, the two models are expected to produce completely independent parameter spaces. In the next part of this work we show the comparison between dark photons and Z physics (see section 3) facing the same and most stringent bounds in the MeV regime, coming from the electron and muon anomalous magnetic moment [19] and the neutrino trident production [22].
The mass Lagrangian (2.84) can be written in terms of flavor vectors as Again, Y 0 has filled the first and third columns, while Y X has non-zero elements in the second one. By rotating the quark vectors as with real, non-negative and diagonal matrices defining the quark masses. For completeness, the interactions with neutral scalars can be put in the form Since there are no interactions among quarks and the singlet s, the vertices can be written as Thus, in the mass basis, The matrix R h rotates M 2 h according to Section 2.4. Finally, (2.90) The same follows for down-quarks and charged leptons. For instance, The lepton vectors are rotated as l R → V lR l R , l L → V lL l L , such that V † lL (Y l 0 c β + Y l X s β )V lR defines their mass matrix.

Parameter space
The multi-dimensional free parameter space in models beyond the Standard Model is commonly larger than the simple g X × m X planes. In order to avoid a redundant criterion for fixing these planes, it is important to consider all relations emerging in the gauge sector and connecting the remaining variables at tree level. The procedure is equivalent to the reduction of the dimension of a multi-variable set through some associated set of independent equations. In fact, a natural relation encompasses coupling constants and energy scales which, in general, may be directly fitted by the observable connected to it. In addition to that, one can also permute some of the variables. Such a replacement does not reduce the dimension of parameter space, but it might lead to a more convenient use of the model. Let us consider the SM example, which is initially described by the P set (2.92) After the W 3 -B mixing, the angle parameterizing the eigenvectors can be used in place of g Y , i.e.
In all vertexes, g Y must be written as g Y (g, s w ) (in fact g Y does not depend on v). Now, from the Z pole mass one can perform a fit of the parameters which eliminates, for instance, any dependence on v (i.e. v = v(g, s w )). Thus, (2.94) Next, once the charged currents are coupled only through the g coupling, it can be related to the Fermi constant G F at the low energy limit, i.e. P → [g, Finally, from the requirement that the theory must reproduce the electromagnetic interactions one last independent equation is given by (2.95) Therefore, the gauge sector of the Standard Model is fully determined. We must note that the W pole mass was not used in any of the steps presented above and it emerges as a prediction of the model.
As mentioned before, the P set is defined by the variables entering in the New Physics effective couplings and includes the mixing matrices. In our model, P is necessarily larger than in the SM but it still allows a significant reduction. Initially, it follows that Similarly to the previous example, the constants g, g Y are solved in terms of the remaining elements. Since in the asymptotic limit m Z depends only on v it might be convenient to preserve c β in the analysis. Finally, the v s breaking scale can be replaced by m X . We end up with a five-dimensional parameter space, namely The kinetic mixing constant is independent and may be replaced by the new mixing angle θ. Accordingly, there must be a region for κ where the Z interactions are exactly described as in the SM, i.e. where s θ = 0.

Dark photons vs. Z gauge bosons
The full set of dark gauge bosons X µ can be divided into two subsets, namely the one composed out of dark photons, here denoted by A , coupled exclusively to vector currents. The second subset comprises general Z bosons whose couplings include axial-vector components. In the following paragraphs we briefly summarize the current theoretical status as well as the results of experimental searches for the effects of these fields [4].
Dark Boson Searches and Future Experiments From our study in the Section 2.3, a general property of the vector and axial-vector couplings is that both contain universal and non-universal parts. The LEP searches [23] can primarily test possible electron couplings to dark fields by looking for recoil energy in a nucleus and therefore can be used to place bounds on the universal part. On the other hand, experiments such as M u3ee [24], devoted to test LFV via the decay channel µ → ee + e − , can place bounds on the flavor matrix 6 of the particular model and will cover the range 10 MeV< m A < 80 MeV. The BaBar collaboration has also performed A searches [3] and their results highly constrain dark photons with mass above the di-muon threshold.
In the minimal dark photon case, i.e. where the fields are neutral under X and the couplings defined exclusively by the kinetic mixing constant, one can mention the results of the KLOE experiment [25], in which the searches were performed in φ → ηX, η → π + π − π 0 , X → e + e − , with the dark photon mass in the range 50 MeV < m A < 420 MeV. The NA48/2 collaboration [26] has covered a 9 MeV < m A < 120 MeV range in kaon decays K → ππ 0 and K → µνπ 0 ; The Run 3 of LHCb plans to search for dark photons in the charm meson decays D * → D 0 A (A → e + e − ) and is scheduled for 2021-2023 [27]. The DarkLight experiment [28] will be sensitive to 10 MeV< m A < 100 MeV by e-H scattering producing on-shell dark photons. It is scheduled for 2018-2020. Similarly, the Heavy Photon Search [29] will scatter an electron beam on a Tungsten target, and is scheduled for 2020. Finally, the NA62 experiment [30] can place bounds on the light Z couplings by measuring the rate of the rare decays K → πνν.
Constraints The long-standing discrepancy between the measured and theoretically predicted muon anomalous magnetic moment [9][10][11][12][13] is at the level of ∼ 3.5 − 4 σ. Many approaches of physics beyond the SM were used to resolve this discrepancy by assuming only one new mediating particle [31][32][33][34]. In the second part of this work we will present the application of different versions of the present model under the most stringent bounds in the MeV regime. Among these processes we can mention the electron anomalous magnetic moment [19], νe scattering [35], parity non-conserving observables in Z phenomenology, neutrino trident production [22] and the missing energy searches in K → µY [16]. In Fig.  3 the differential decay width dΓ M µY /Γ µν for M = K, D s is presented, motivated by the work [16], with the fixed values (c β , κ, F µµ , m χ ) = (0.8, −4g X , 1, 3m X ). In (b) the differential decay width for D s → τν τ (τ → µν τνµ ) must overshadow the dΓ DsµY normalized by Γ Ds→µν .
In order to maximize the parameter space covered in our analysis, our strategy includes DM considerations applied to a stable χ fermion, in principle lighter than Z . In addition, we notice that lepton non-universality in the first and second families will necessarily imply a discrepancy in the proton charge radius estimated from the Lamb shift in the e-hydrogen and µ-hydrogen system [20], such that a precise measurement of such processes must correspond to one of the most severe bounds for non-universal dark boson theories.
Finally, effects of dark fields in purely leptonic processes support DM searches at future lepton colliders. The same physics may still give some effects in the leptonic meson decays such as M → µνee for M = K, D, D s , B. In the subsequent part of this work, we compute the SM branching ratios for these channels and compare them to the results from the Z -boson exchange.

Conclusions
We have seen that the minimal anomaly free SM ⊗ U (1) X models may provide solutions to explain existing anomalies in the leptonic sector at low energies. In the UV complete version presented here, the quantum anomalies are canceled per generation. The chiral Ds → μνX(X → χχ, 3νν) (b) Figure 1. The Differential decay width dΓ M µY normalized by Γ µν for M = K, D s . The curves are plotted under the fixed values (c β , κ, F µµ , m χ ) = (0.8, −4g X , 1, 3m X ). In Fig.(b) the channel D s → τν τ (τ → µν τνµ ) hides the distribution generated by X →χχ, 3νν.
X-hypercharges for a single family require a second Higgs doublet and a scalar singlet in order to provide to the model a consistent fermion mass spectra. Right-handed fermions are incorporated, being constrained by the neutrino interactions, while by charging the second generation under U (1) X , we find a convenient framework to explain the discrepancies involving muons. Finally, we considered X µ bosons either as dark photons A , or Z gauge bosons, according to the role played by the vector-axial currents, aiming to test the common assertion that Z physics might be disfavored by parity non-conserving effects. This work is a theoretical introduction which will be accompanied by a more complete phenomenological analysis, developed under the dark matter considerations applied to a light (M eV ) and stable dark χ R fermion. It establishes both the notation and the relations between the new parameters which might be useful for future studies.

A.1 Possibilities for the new fermion χ R and neutrino masses
In the version with Majorana fermions the model must include a new sterile generation such that the mass Lagrangian can be written as: The brackets in the Yukawa Eq.(A.1b) lead to a symmetric Majorana mass matrix. The conjugated field is defined by in terms of a general realization of the charge conjugation operator C, such that The Lagrangian can be rewritten like In principle the mass matrix elements are complex numbers. However, in the minimal 2 × 2 version a redefinition of the χ Rα fields can absorb their phases, leaving the final matrix real and symmetric: The mass matrix arises after EWSB, given by with V χ R being unitary, a criterion to give M χ with real and positive eigenvalues. The fields are rotated as χ R → V χ R χ R ≡ V χ R χ R , where in the r.h.s. the same notation is used for the mass states. In conclusion, L χ ⊃ m k 2 χ Rk C † χ Rk + h.c. . have well defined mass. Note that these fields are protected to decay due to a global U (1) χ . The portal into SM is created by both Z and X interactions. above, the index "c" denotes charge conjugated and is used to recover that Majorana mass terms may appear as long as the charges satisfy X α + X β = 0. In the above variant this piece is composed by the sterile neutrinos of first and third generations. Since ν 2R is charged under X, the mixing with the additional fermions is inserted via the singlet s and under the condition X α + X 2 + X s = 0. In this section we briefly present some general aspects of the see-saw mechanism which concludes the construction of the model. By including three generations of RH neutrinos, after the EWSB the Dirac and Majorana mass terms can be included in the Lagrangian (see [36])

Neutrino masses
which can be in this form due to ψ c ψ c = −ψ C −1 Cψ = ψψ, where in the last step the anti-commuting property of fermion fields has been considered 7 (see 3.52 [36]). Here the matrix M L = 0 once there is no Majorana mass terms for LH neutrinos. The mass matrix can be rotated by the following transformations 15) In the framework where M D M R , the matrices V L and V R may be taken approximately unitary, with V RL , V LR suppressed by ≈ M D M R . In summary, the light neutrinos must mix LH flavor states only, while the heavy states mix the remaining RH fields. The matrix M R is extracted from the part which can be presented in the basis The mass eigenstates are denoted as χ Rα = (V R ) αβ ν Rβ . Once the rotation occurs among ν R 's, the fields χ are the degrees of freedom present in the interactions. As for the LH neutrinos, the χ spinors are given by The case M D M R is valid both from the presence of small Yukawas as from the difference between the electroweak and the Majorana scales. In the first scenario, small Yukawas, which control the decay of the new mass eigenstates into SM particles (apart from the suppressed V LR ), will dictate how reliable it is to assume at least one generation of χ fields as a dark matter candidate.
The last part concerns neutrino interactions and a block diagonal U is assumed to rotate the system to the mass basis, such that the heavy neutrinos mixes the RH fields only, and the block V νR is approximately unitary (see [36]). The universal elements are weighted by g I R (ν R ) = g I Z (ν R ) = 0 , (A. 20) since ν R are singlets under the SM gauge group. Thus,  Figure 2. The flavor changing matrix element |V d | 21 . The region above the lines is constrained according to [37].