Light mediators in anomaly free U (1)X models. Part II. Constraints on dark gauge bosons

We consider experimental constraints in the MeV region in order to determine the parameter space for the U(1)X extension of the Standard Model, presented in the first part of our work. In particular, we focus on the model UV-completed by cold WIMPs. We conclude that the electron anomalous magnetic moment and the neutrino trident production provide the most stringent bounds to gX2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {g}_X^2 $$\end{document}∼ 10−6 in the mass interval below the di-muon threshold. By allowing the axial-vector coupling of the dark gauge boson Z′, the interference effect with the SM gauge bosons may reduce the bounds coming from the neutrino trident production. At the same time, such coupling allows a region of the parameter space already favored both by the relic abundance and by the discrepancy between experimental result and theoretical prediction for the muon anomalous magnetic moment. We emphasize that light-Z′ interactions, non-universal for the two first lepton families, necessarily create a difference in the proton charge radius measured in the Lamb shift of the e-hydrogen and μ-hydrogen. Finally, we determine the effects of the new gauge boson on the forward-backward asymmetry in e+e−→f¯f\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \overline{f}f $$\end{document} = μ, τ, and on the leptonic decays M → jνjl+l−, where M = π, K, D, Ds, B and j, l = e, μ.


Introduction
Phenomenology of U(1) gauge bosons X µ (see e.g. [1][2][3][4][5][6][7][8][9]) is, in general, very dependent on the particle content and the X-hypercharge assignment of the fundamental theory. The canonical requirements for the formulation of an ultraviolet (UV) model, such as to be anomaly free and to recover the Standard Model (SM) fermion mass matrices, indicates the presence of new scalars and stable fermions even in minimal extensions like the Two Higgs Doublet Model (2HDM). Furthermore, in order to cancel the triangle anomalies per generation, it is common to introduce new right-handed fermions. Motivated by the existence of discrepancies related to the muonic interactions, by charging the second generation under U(1) X , we have found an appropriate theoretical framework to discuss either dark photons or Z gauge bosons. The model must contain at least two Higgs doublets and one scalar singlet [10], as a condition to recover a consistent fermion mass spectrum. Apart -1 -
Guided by our theoretical analysis of SM⊗U(1) X theories, in this paper we describe the constraints from the existing experimental data on the dark gauge boson phenomenology at the low energies (MeV regime) [11]. We first consider leptonic interactions and provide the detailed expressions for the computation of the most stringent processes. In Part I of our work we present how the set of X µ gauge bosons can be separated into Aincluding dark photons -and the Z subsets, depending whether they have axial couplings with fermions or not [12]. In the minimal dark photon phenomenology [13], only three new parameters are present, namely, the kinetic-mixing coupling, the mass of dark photon A and its branching fraction into invisibles. In the more general case when SM fields are charged under X, the parameter space is increased by at least two parameters -the gauge coupling, and the new breaking scales. The presence of dark photon leads to small corrections of Quantum Electrodynamic (QED) quantities and is commonly favored in current experimental searches, for falsifiability reasons and safety from parity-violating effects. Notwithstanding, we aim to explicitly obtain the dependence of axial couplings on the model parameters and why their effects must be suppressed. We also check what is the effect of axial couplings in the decay width of Z into dark matter candidates. In the chiral U(1) X model, Lepton Flavor Violation (LFV) is directly suppressed through the presence of a flavor matrix F. For instance, by X-charging the second generation of the right-handed fermions and selecting the element F 22 = 1, we immediately achieve that no LFV emerges at tree-level.
The analysis of section 3 is performed under the hypothesis that there is a light X µ boson below the di-muon threshold (∼ 10 − 200 MeV), whose effects can be detected for couplings of the order g X ∼ 10 −4 − 10 −1 . If this hypothesis holds, the gauge boson existence must be associated to one specific point of the most favored (less excluded) parameter space, in the plane defined by m X and g 2 X . We will see that the search for the most favored parameter planes will imply, for instance, that the fermionic dark matter candidate χ is lighter than the new vector, once the decaying channel X →χχ is open. In addition, the strong bounds from the electron anomalous magnetic moment [7] can be reduced in the vicinity of the poles, which emerge in the calculations of the (g − 2) e , provided by the presence of axial couplings. Under standard considerations for the relic density computation [14], this feature would severely restrict dark photon models coupled to dark matter fermionic candidates. A similar conclusion is reached from the neutrino trident production [3].
We denote by X µ the new and generic gauge boson, while Z and A are assigned according to the presence or absence of axial-couplings, respectively, whenever the distinction is necessary in our analysis. In section 4.1 we calculate the forward-backward asymmetry forēe →f f , which is relevant in the studies of light Z physics. In section 4.2 we make predictions for leptonic meson decay widths for M → jν jl l, with M = π, K, D, D s B and j, l = e, µ, in the framework of SM ⊗ U(1) X . Finally, section 5 contains a summary of our results. The present work considers a number of the most stringent constraints coming from the flavor physics in the MeV regime, applied for a particular U(1) X SM extension which exclusively charges the second generation of all right-handed (RH) fermion species. Here we summarize the important vertexes, necessary for the computation presented in the next section. The fermion-gauge interactions are described by: where α = 1, 2, 3, L, Q are lepton and quark isospin doublets, while l, u, d are the lepton, dtype and u-type fermion singlets, respectively. The covariant derivative D µ is introduced as The new SU(2) singlet and stable fermion, χ R , is required for the treatment of quantum anomalies. The couplings in the mass basis are dependent on two independent angles, i.e.
where g, g Y , g X are the weak couplings and τ 3 is the SU(2) generator. We use c φ ≡ cosφ and s φ ≡ sinφ. The quantum numbers Y and X are related to the U(1) Y and U(1) X parts of the SM ⊗ U(1) X gauge group. The parameter κ = g Y is the coupling resulting from the L m = 2 B Y µν B Xµν kinetic mixing term, while g SM Z = g c φ (τ 3 − s 2 φ Q). There are two Higgs doublets and a singlet in the model, denoted by: with the following hypercharges The vacuum expectation values v 0 , v X are the weak breaking scales assumed to be related to the SM v 2 = v 2 0 + v 2 X . The full reproduction of QED and the introduction of a new breaking scale, larger than the SM weak scale, allow the angles to be parametrized as: and The parameter s θ regulates the NP effects in the neutral currents and it must be a small parameter. The g X X term of eq. (2.3) operates for RH fields and generates a -3 -

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so-called flavor matrix F, encompassing the flavor changing (FCNC) and non-universality effects in the theory. F emerges after the field redefinition The couplings x f V,A can be extracted by replacing in eq. (2.2) and eq. (2.1) the following hypercharge assignment: • Y hypercharges: • X hypercharges (the second generation receives the charges under U(1) X ): where the remaining RH fields uncharged.
For completeness, we present the quarks and leptons vector and axial-vector couplings -4 -JHEP10(2019)279 We emphasize that the effects of additional new scalar fields are considered to be negligible, in contrast with those coming from the χ fermion coupling to X µ .
X µ interactions with charged hadrons The interaction of a new dark gauge boson X µ with charged hadrons can be obtained by using the gauge principle of the QED Lagrangian, and by rotating the Abelian gauge field like which brings the kinetic mixing, at first order in , into a diagonal form. In other words, after the above transformation, the QED covariant derivative D µ = ∂ µ −ieqA µ is extended to where κ = g Y . We neglect second order terms in the small parameters. The shift will allow us to compute the inner X-bremsstrahlung from a charged hadron. We present in section 4.2 its importance in the case of the M → µνee meson decays, for M = π, K, D, D s and B.
3 Low-energy constraints

ρ parameter
The dependence of the W -boson mass on the coupling and vacuum expectation values (v.e.v) reproduces the Standard Model value at tree-level, i.e.
In the limit of vanishing new couplings, the Z-boson mass is SM-like, while the mass of X µ gauge boson becomes M 2 . We use notation cosθ w ≡ c w . In the SM these parameters are connected, such that at tree-level ρ = 1. Within SM ⊗ U(1) X theories, if the couplings g and g Y are assumed to take the SM values, or equivalently c φ ≡ c w , the Z mass parameter approaches (m 2 Z ) SM from the right, i.e. (m 2 Z ) X > (m 2 Z ) SM , which leads to a suppression in ρ. In order to find how the ρ parameter differs from unity, we can write m Z as The last step recalls eq. (2.6) and the light mass condition a 1 ḡ 2 . By relying on the above result and using the eq. (3.1) in the definition of ρ , it follows that ρ tree which cannot reach the central value of the experimental measurement [15] ρ ∈ 1.00040 (24). Nevertheless, at two sigma level one can demand 0.99992 < c 2 θ ≤ 1, i.e.
Hence, it follows the genuine smallness of the θ angle at tree-level.

Proton charge radius in the U(1) X model
The proton radius can be extracted from the comparison of the theoretical prediction and the measured value for the Lamb shift in muoninc and atomic hydrogen. The result can be expressed as a sum of independent physical contributions, i.e.
∆E| l th = δE l a + δE l b + · · · + λ l r 2 p | l , (3.5) where l = µ, e accounts for the two types of hydrogen and the last term corresponds the correction due to the finite-size of charge distribution in the proton. At leading order λ l is given by On the theoretical side, the proton charge distribution is considered to affect the effective potential defining the µ-hydrogen states, whose Lamb shift is estimated to be (see [16]) Here the first term summarizes the vacuum polarization contributions and recoil effects, while the second includes a two-photon exchange contribution. The proton puzzle denotes the difference in the solutions of eq. (3.7) for both eH and µH systems, which provides 2 r 2 p | µ = 0.84087(39) fm [16] (3.10a) r 2 p | e = 0.8758(77) fm CODATA-2010 [18] (3.10b)

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Note that the level of precision in the µH is one order of magnitude higher than in eH. The above results are obtained when no New Physics effects are included. In the present work we accommodate this discrepancy through the NP contribution in eq. (3.7), whose l.h.s. may be rewritten as where δE l 0 sums up the errors of the results in eq. (3.10), and now Moreover, from eq. (3.11) the difference between the "X" and QED frameworks can be expressed as a small deviation δ X l : In summary, a constraint on the proton radius is imposed by eq. (3.12), i.e.
In general, the correction δ X l is derived as the deviation from the Coulomb potential due to the exchange of a massive vector boson X µ , or ( [19]) with a correspondent shift in 2P − 2S given by [20] Above, |R 2i | 2 are the radial wave-functions corresponding to the states P and S, while g l , g p are the lepton and proton couplings with X µ , respectively and f (x) = x 4 (1+x) 4 [19]. The parameter a l = (αm lp ) −1 denotes the Bohr radius, with m lp denoting the reduced mass of the l − p system, and implies a µ ∼ 1.44, i.e. ∼ mµ me smaller than a e . Therefore, for m X > 10 MeV one can approximate f (x) ∼ 1. From eq. (3.14) it follows that and a proton curve is defined by which, in principle, can be solved by an attractive force (i.e. sgng p = −sgng l ) strongly coupled with muons. In the SM ⊗ U(1) X framework, and in the limit f (x) ∼ 1, the sgn g p -7 -

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must be opposite only to the non-universal part of the X µ coupling. The couplings g p and g l , extracted from eq. (2.14) and eq. (2.12e) respectively, are given by: For simplicity F τ τ may be set to 0, such that F µµ + F ee = 1, what reduces the proton curve to In figure 3 we present two examples for the 2σ favored region allowed by the current experimental results for the proton radius, 3 with the fixed parameters (c β , κ, F µµ ) = (0.8, −4g X , 1) and (c β , κ, F µµ ) = (0.8, −g X , 1). We emphasize that, even in the absence of the experimental discrepancy in eq. (3.10), any theory of lepton non-universality in the first two generations will imply, from eq. (3.11), a non-zero contribution to the Lamb shift in the l-hydrogen system and, therefore, it will impose strong constraints on these couplings in the MeV range.

M + → µ + invisibles
In this section we compare the experimentally measured decay width for K → µY (Y denotes invisible states), and the theoretical prediction for the decay width of K → µ + ν µ χχ. The narrow-width approximation (NWA) is assumed to be valid in the region where m X > 2m χ , i.e when e + e − , νν and χχ are the only directly accessible decay products of X µ . Hence the differential decay width may be determined using The dominant contributions to the decay amplitude are presented in figure 1(a,b). The Feynman's rules for the vertexes related to figure 1(a,b) are given in appendix A and they lead to the amplitudes where q = p 1 + p 3 and q 23 = p 2 + p 3 , f M are meson decay constants and V UD stands for the CKM matrix element present in the particular decay mode. As a cross-check one can prove that the gauge invariance holds by replacing the dark gauge boson X µ by the photon field through Figure 1. The Feynman diagrams (a,b) contributing to M lY , Y denoting invisibles, in the U(1) X model and the structure dependent diagram (c) in QED.
The details of the decay width calculation are given in appendix B. Assuming that the vector X µ decays into the invisible pairsχχ,νν, its contribution to the process K → µ + invisibles can be constrained by the existing experimental bound [21] with the missing energy in the interval 227.6 < m Y (M eV ) < 302.2. The vertex X µχ χ is extracted from the Lagrangian component with g R given by eq. (2.3c). Since χ R is a singlet under the SM gauge group and X χ = −1, it follows that the Feynman's rule for the X µχ χ is i(g X /2)γ µ (1+γ 5 ), where we set c θ ∼ 1. If the mass of X µ is in the MeV region, the dark boson may still decay into a electron-positron pair, whose vertex is written in eq. (A.2). In the NWA, where the factor of three accounts for the neutrino flavors. The decay amplitude coming from the diagrams in figure 2 can be written most generally as Again, one can estimate the branching ratio where, from eq. (B.1), . (3.30) The sum over l takes all l = χ, e, ν e , ν µ , ν τ . Finally, from the amplitude of eq. (3.27), one can write the general formula where the average over final state spins is made implicit. Since i.e. the decay width of X → χχ is suppressed by the difference between X and χ masses. Provided by the experimental constraint [21], we present some examples of the allowed region for m X × g 2 X in figure 3 and figure 4.

(g − 2) e
The contribution to the electron anomalous moment a e , coming from the new dark gauge boson X µ is equivalent to a shift in the fine-structure constant, as already discussed in ref. [7] The r.h.s. is the relative correction to the measurement of α −1 which should not exceed 0.5 ppb [1]. The contribution of the X µ gauge boson to the electron magnetic moment in the dipole function can be written as where (3.35) Since the limit m X m e is valid in our analysis, we can set the bounding curve  Figure 3. The allowed region for the proton radius explanation, using the bound in eq. (3.24). Under the narrow-width approximation the vector X µ decays into the missingχχ,νν pairs. Here m X = 3m χ and F τ τ = 0.
Parameter space. As discussed in Part I, we have to find out how to fix a plane in a five-dimensional parameter space assuming that the model can explain the selected experimental discrepancies. If we insist to explain the proton charge radius puzzle, one has to require sgn g X = −sgn κ.
In the examples depicted in figure 3, it is evident how stringent are the bounds from (g−2) e . However, due to the interplay of the contributions coming from the vector and axial-vector couplings, the curve can be minimized through the root equation with a fixed F, i.e. around the roots there is almost no effect from the dark boson X to the fine structure constant. If F ee = 0 the solutions to eq. (3.38) are for κ = ng X . Due to the condition of eq. (3.37) only n = −7/5c 2 β may be a solution for the proton puzzle, as presented in figure 4. We denote κ 0 = −7/5c 2 β g X . Note that for κ 0 value, the (g − 2) e bound reduces the discrepancy of the proton puzzle from 5σ to 2σ. In the case where κ is not inside the range of eq. (3.39), then the electron anomalous magnetic moment gives the most stringent bound on the parameter space.
In the following calculations we have written the electron vector and axial-vector couplings to X µ in the form In addition, the decay of the vector boson X µ into neutrino pairs is allowed, with couplings

(g − 2) µ
In figure 5 we present the most general four contributions to the muon anomalous magnetic moment. Since the new vector boson is neutral, the diagram (b) is the only one not contributing to our case. The diagram (c) is present for charged scalars and (d) for neutral Higgs. In this section we want to find the necessary conditions for the correct sign of the (g − 2) µ discrepancy. In the following, h + , h 0 generically denote the charged and neutral scalars present in the theory.
Following the work of [22], a general fermion-X µ vertex can be written as For simplicity, we do not include the suppressed flavor violating processes, i.e. F = µ. For m F = m µ , the integral linked to the first diagram is given by . (3.44) As mentioned before, we work under the assumption of very large Higgs masses, where a µ is dominated by [a µ ] a . On the other hand, the above integral leads to a wrong negative sign

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for a wide range of the c β parameter, and have to be compensated by additional contributions. Therefore, we assume that scalar masses are relatively large and compute the scalars contributions to the moment function in the region where the asymptotic approximation to the integrals is fairly valid, i.e. m h > 20 m µ . The bounds on the Higgses couplings to Z are coming from the LHC analyses, as already considered by the authors of ref. [23]. The Yukawa Lagrangian can be parametrized as Both diagrams (c) and (d) can contribute to the muon anomalous magnetic moment. For the diagram(c) we have to specify F = ν or m F = 0. The coupling C P is in fact present in our model for both neutral and charged scalars and, in the neutral case, it is purely imaginary. For m F = m ν = 0 it follows that and for neutral scalars in the diagram (d) F = µ, one gets If we consider m h + , m h 0 m µ the integrals converges to a simplified form: (3.51) Therefore, the charged scalars cannot provide the correct sign and their interactions have to be suppressed, either by their large masses or by their negligible couplings to muons. In summary, in order to explain the (g − 2) µ discrepancy only diagram (a) and (d) contribute. Again, there is a range for c β in which the (a) integral already gives the correct sign for the a µ discrepancy, allowing all the scalars to live in the decoupling limit. For instance, if we do not take into account the constraint coming from the proton radius, then we can use all solutions derived from eq. (3.39). Thus, for κ = 3 2 c 2 β g X , and c β < 0.9, light neutral scalars with masses in the range m h 0 ∈ (10 − 100)m µ are required to restore (g − 2) µ , for |C 0 | S ∼ |C 0 | P ∼ 10g X and different values for c β . Charged scalars are still disfavored. Since c 2 β = v 2 X /v 2 , a minimal model at low-energies (MeV) is well supported by a small scale v X . For the completeness of our study, let us mention that the pairs κ = − 7 5 c 2 β g X and c β > 0.7, as well as κ = 3c 2 β g X and c β > 0.99, both can solve, through (a) diagram only, the discrepancy of (g − 2) µ .  Figure 6. The trident production in the equivalent photon approximation (EPA). In addition, there are the reciprocal diagrams where the real photon is attached to µ − .

K µνe + e −
The decay width of K µνe + e − , denoted as Γ Kµ2ee , is obtained from the distribution dΓ Kµ2ee dmee , integrated over the electron-positron invariant mass m ee > 145MeV [15]. We use the narrow-width approximation in a way that, for a fixed m X = m ee , the contribution from X → ee should not exceed the uncertainty of the total Γ Kµ2ee . By demanding that the decay rate has to be smaller than the experimental uncertainty, it actually implies that no enhancement should be seen in the region m ee > 145 MeV. The analysis is similar to the calculation of Γ KµY , now with for 145 MeV < m X < 2m µ . In figure 3 and figure 4 the excluded region is marked by the yellow color.

Neutrino trident production
We determine the cross-section for the neutrino trident production in the Equivalent Photon Approximation (EPA) [3,[24][25][26], i.e. by connecting it with the scattering of a real photon and the neutrino beam. We then include the bound from the CHARM-II experiment [27]. The total amplitude for the scattering of a real photon and a neutrino beam in γν → νµ + µ − includes the six diagrams of figure 6 for the exchange of W, Z and X bosons. The neutrinos spins are summed, while one takes the average over the photon polarization, i.e.
In the case of the neutral currents, both Z and X, the neutrino vector and axial-couplings are related by where z = Z, X. The amplitude can be simplified and presented as with The constants C V,A summarize the contributions coming from all diagrams These expressions are exact in the sense that the Z mass parameter depends on the new g X , s θ . Once the vertices are defined as in eq. (2.9), by taking c θ ≈ 1 and neglecting second order terms on the small parameters, the SM contribution to neutrino trident production is given by The phase-space for the scattering of a real photon and a neutrino can be defined through the invariants [28] (see figure 6) (3.59) In the EPA, the total cross-section can be written as [29] σ , (3.60) where q ≡ |q|, Z is the atomic number of the target and F (q 2 ) is the electromagnetic form-factor introduced in [3,29]. The details of the integration over phase space are given in appendix B. In eq. (3.60) the integrals over q (with q ≡ q 2 ) and s = (p a + p b ) 2 are derived from the probability of creating a virtual photon with momenta q, defined by [3] P (q 2 , s) = Z 2 e 2 4π 2 ds s dq 2 q 2 F 2 (q 2 ) (3.61) and the electromagnetic form factor F (q 2 ) = 1 + 1.21 q 2 is given in ref. [24], where m p is the proton mass. In the CHARM-II experiment, a neutrino beam with the mean energy E ν ∼ 20 GeV is scattered by a glass target (Z = 10). We require that the contribution coming from the interference of the SM and the SM ⊗ U(1) X to the total cross-section should be inside the one standard deviation region, i.e.

χ relic abundance
In this section we compute the relic abundance for a thermally produced χ-fermion from a set of fixed points in the parameter space. Before we proceed to the details, it is important to emphasize that our computation will follow the standard scenario presented in [14]. Since Cold WIMPs freeze-out in the pre-Big Bang Nucleosynthesis (BBN) epoch, the results rely on a number of assumptions concerning the early history of the universe during an interval from which no information can be collected. Therefore, they may substantially differ between independent cosmological models. These assumptions are commonly centered, for instance, on the expansion rate of the universe at the freeze-out, the kinetic and chemical equilibrium state before the decoupling, and on the dominant interactions for the WIMP production in the plasma, if thermally produced [30]. In the following, we set up the framework which determines a bounding curve to the χ parameters, in accordance with the current measurement of Ω CDM . We start with the exact solution of the Boltzmann equation, aiming to examine the sensitivity of the allowed band on small variations of the couplings. We then consider the approximate formula for the weakly interacting massive particles (WIMP) relic density [30] with the thermal average computed at the freeze-out temperature T f.o. mχ 20 . In the general case, the attempt to create a direct bounding curve (i.e. g X < f (m 2 X ) for the analytical function f ) is difficult due to the presence of the X boson as a resonance. In other words, the coupling g X cannot be easily factorized, once it enters the decay width in the Breit-Wigner propagator. We perform the integration of by describing the evolution of the comoving abundance Y, whose value at chemical equilibrium is given by [14] Y eq = 45g 4π 2 Above, the M P = 1.22 × 10 19 GeV is the Planck mass. The variable x ≡ mχ T , where T is the photon temperature. It is commonly taken from x = 1, which defines the boundary for the condition Y = Y eq , to the present value. According to ref. [14], we choose for the χ mass range, 10 MeV < m χ < 80 MeV, such that the effective degrees of freedom g 1/2 * has a small deviation from g 1/2 * ≈ 7/2 due to QCD quark-hadron phase transition, which we neglect in our calculation. In eq. (3.65), the constant g accounts the degrees of freedom for the particles present at the equilibrium. We consider the dominant channels including ν, e, µ, χ. On the other hand, the h eff (T ) encloses the effective degrees of freedom for entropy and, in principle, it should sum over all species present in the plasma. In practice, however, the -16 -

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species with large energies are suppressed by their distribution function such that we may sum over photons in addition to the particles at the equilibrium. 4 The thermally averaged annihilation cross-section is regulated by the diagram of figure 10 with χχ in the initial states and f summed over the particles at chemical equilibrium. Co-annihilations are discarded. The expression for σv can be written as [14] σv = x 8m 5 where, as in eq. (3.65), the k i are the order i modified Bessel functions. The cross-section σ is a sum over f such that is the differential cross section for each fermion f in the final state. Again, the X µ resonance is dominant over the remaining mediators and is implemented as a Breit-Wigner vector whose χ couplings are x χ V = x χ A = g X . The total amplitude squared can be integrated over the polar angle and results in the expression .
(3.69) The solution of eq. (3.64) then has to be translated in the present time. The abundance Y 0 is related to the WIMP relic density through [30] Ω χ h 2 = 2.755 × 10 5 Y 0 m χ MeV (3.70) and must be consistent with the current measurement Ω CDM h 2 = 0.1131(34) [31] of the cold dark matter density. In figure 7(a) we present the integrand of the thermal average in eq. (3.67) for different temperatures, under a fixed point in the parameter space, dominated by the resonance. 5 In figure 7(b) we present the solution of eq. (3.64) for the WIMP mass m χ = 30 MeV, g X = 4 × 10 −3 and a set of couplings x. It depicts the point where the low temperatures hinder the abundance to follow the evolution of its equilibrium value, and the particle decouples. The horizontal black band is the 3σ limit for the current relic abundance. The well-known pattern of the figure 7(b) reveals that the relic density gets 4 The relevant species are considered to be at thermal equilibrium with the plasma and h eff (T ) = i hi(T ) where gi accounts the number of spin degrees of freedom of i, while Ei, mi and ηi = 1(−1) denote its energy, mass and Fermi-Dirac (Bose-Einstein) statistics, respectively. 5 The numerical integration can be optimized by hiding the pole via the approximation for a Breit-Wigner function whose pole is xp .

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overabundant for small couplings, a feature that will help us to maximize the excluded region of our parameter space. Although an underabundant sector is disfavored by Ω CDM , it actually informs us about the necessity of completing the theory with additional dark matter candidates, and is not entirely ruled out. One can also notice from the figure how sensitive Y 0 is to small variations of g X . We consider that the use of eq. (3.63) is appropriate for our precision level, and it may simplify our analysis. For large temperatures (x < 20) and small couplings (g X 10 −2 , c β ∈ [0.4, 0.95]) the integral is dominated by the region around the resonance and is sufficiently narrow to use a Dirac delta approximation. 6 In order to illustrate the importance of the relation between the X µ and χ fermion masses, in figure 8(a) we present the favored band for the current relic abundance by taking the relation m X = ym χ for different values of y, at the freeze out temperature x f.o. = 20. In figure 8(b) a similar set of lines is computed for particular choices of c β .

Outlook
The results of the previous section, for two fixed points, were summarized in figure 9. In plot (a), c β = 0.6 cannot resolve the (g − 2) µ discrepancy. Notwithstanding, we can verify the impact of the axial-vector couplings interference in the cross-section of the neutrino trident production (dark-blue region) in comparison with the dark photon case (lightblue). Moreover, the choice of the parameter κ = 3 2 c 2 β g X renders irrelevant the bound coming from (g − 2) e . In plot (b) the favored region for the explanation of the (g − 2) µ discrepancy (in gray) is presented, for c β = 0.95, while the circled area highlight the overlap between the solution for the muon anomaly and the relic abundance Ω CDM favored bands. The parameter space that we covered hitherto should be further tested. In the following subsection 4.1, we consider the forward-backward asymmetry in electron-positron collision to fermion anti-fermion, e + e − →f f , in order to determine how the observable is sensitive on the SM ⊗ U(1) X parameters. Finally, in subsection 4.2 we illustrate the impact of one set of parameters, allowed by figure 9(b), to the leptonic decays M → jν j i + i − , where M = π, K, D, D s , B and i, j = e, µ.

Parity non-conserving observables
Light Z physics requires the use of the LEP data, where the effects of Z interactions are suppressed by its large mass. Here, we have to constrain the region for the light Z and its small couplings. As we mentioned in the previous section, in scattering processes, the dark photon effects can be considered as a correction to the fine-structure constant. When the axial-vector coupling of the dark gauge boson is present it can be tested in the angular asymmetries of the differential cross-section. The forward-backward asymmetry is defined as .  Figure 7. (a) The differential thermal average dominated by a narrow resonance. In the example, the X boson mass is m X = 90 MeV, while the dark fermion mass is m χ = 30 MeV. In (b), the horizontal black band presents the 3σ region allowed by the current measurement of cold dark matter density. The abundance Y 0 is sensitive to small variations of the coupling constant g X , such that Ω CDM provides a strong bound for SM ⊗ U(1) X theories. The remaining parameters are fixed as (c β , κ, F µµ ) = (0.6, 1.5c 2 β g X , 1).
Here we will focus on the energy region far from both Z and X peaks, i.e. 2m µ √ s m Z and we must compute the generic diagram in figure 10 for V = γ, Z, X. For simplicity, the Feynman rule for the vertexf f V µ can be written as The amplitude can be expressed as with The dominant contribution to A(θ) originates from the interference of contributions coming from γZ and γX in the numerator, while the cross section coming from the photon mediator (γ) gives the dominant contribution in the denominator, i.e.
In the CM reference frame it results in  Here c θ is the scattering angle, and k the space-momentum of the particles in the final state. In the region √ s m µ the contribution coming from X exchange can be written In our case, by assuming a Z ∼ g and a X ∼ g X , the region around √ s ∼ m Z 10 would severely constrain g X ∼ 10 −1 g. In summary, the precise measurement of forward-backward asymmetry, in the region described above, might provide an additional limit to the coupling g X , as well as a possible test of lepton-flavor universality.

Leptonic meson decays: M → l ν l ll
The experimental measurement of the branching ratio for K µνee gives the most stringent bound of figure (9). This result motivates us to extrapolate this analysis and make predictions for the bottom and charm mesons. In this section we present our results for the total and differential branching ratio of the purely leptonic decays M → l ν l l + l − , or simply M l 2ll , where M denotes the mesons M = π, K, D, D s , B and l , l = e, µ. The diagrams in figure 1 are dominant in that process and can be separated into the inner-bremsstrahlung (IB) amplitude (a,b) and a structure dependent (SD) amplitude (c). The off-shell vectors X µ , A µ mediate the interaction with the lepton pair l + l − . The IB amplitudes can be computed in a general form for all mesons by following the work of [32]. The SD diagram can be parametrized as where V UD is the CKM element linked to the particular meson's quark structure, and In general, the IB contribution is dominant in M µ2ee and suppressed in M e2µµ . We compute the differential branching ratio for D s decay, in the SM framework, including both IB + SD interference. The form-factors are given by: 7 (4.8) In figure 11 we plot the results for different mesons, keeping IB contributions only. The results are summarized in table 1. We provide cuts on the di-lepton invariant mass in order 7 We acknowledge D. Melikhov for providing us with these form-factors. to connect them to possible experimental limitations. These results for the SM branching ratio might be interesting if precision measurements are reached. In figure 11(b) we present the results for the D s meson including the SD part. In these last examples, the di-leptons are the e + e − and µ + µ − pairs and the integrated values are presented in table 2.
We also compute the branching ratios for M µ2ee (M = π, K, D, B), coming from the dominant IB amplitudes, in which the photon mixes with the X boson. The branching ratios below test our IB formulas in the SM ⊗ U(1) X framework for the point of figure 12, namely (g 2 X , m X ) = (4 × 10 −4 , 60): BR(D µ2ee ) IBX = 6.54 × 10 −8 , BR(B µ2ee ) IBX = 1.78 × 10 −10 , (4.9) Once more, the increase in the differential branching ratios is tinny and it can eventually be observed if high precision measurements are performed. In order to provide a possible test for lepton flavor universality we introduce in a kinematic region far from resonances. In the SM the ratio is close to the unity for q 2 (2m µ ) 2 and different leptons in the final state. This case might be potentially interesting, and it is kinetically allowed only for B τ 2ll . For m 2 ll far from the X µ pole, any non-universality effects are negligible if compared with the SM prediction. For instance, from the parameters selected for the analysis presented in figure 12, and (1500) 2 MeV 2 < m 2 ll < (1600) 2 MeV 2 , we find Finally, in figure 12 we present the normalized differential branching ratio for Ds µ2ee , both in the SM ⊗ U(1) X and in the SM frameworks. Around the resonance, the probability for measuring the di-lepton mass in the interval 58 MeV < m ee < 62 MeV is equal to P = 2.54%, for (g 2 X , m X ) = (4 · 10 −4 , 60), allowed by figure 9(a), in comparison with P = 0.63%, in the SM.

Conclusion
We have explored a number of processes at low energies (MeV-GeV) in order to constrain a SM ⊗ U(1) X theory, UV-completed by cold WIMPs and whose U(1) X is chiral for righthanded fermions. The most stringent bounds are obtained from the electron anomalous magnetic moment and the neutrino trident production. Dark photons, as U (1)  bosons, may be fully restricted by these observables if we assume the standard scenario for the computation of the χ relic density, described in section 3.8. On the other hand, the theories where the X µ bosons are coupled to the fermion fields through both vector and axial-vector currents (Z bosons) are not in conflict with the limits from the neutrino trident production, but can still be ruled out by (g − 2) e . However, a possible interference between the vector and axial-vector couplings might make the shift in the fine structure constant negligible. Under this condition a certain choice of parameters can be found for three tuned planes, which allow the explanation of the (g − 2) µ discrepancy for specific values of the c β angle. In particular, the pairs κ = − 7 5 for c β > 0.7, κ = 3 for c β > 0.99 and κ = 3 2 for c β > 0.95, solve the discrepancy in the muon system without the introduction of light scalars.
We propose to measure the forward-backward asymmetry in e + e − →f f , for f = µ, τ , far from the Z boson peak at low energies. We also suggest the measurement of the branching ratios to the purely leptonic meson decays M → jνll, for M = D, D s , B. The ratio BR(M → jν j µ + µ − )/BR(M → jν j e + e − ) in definite regions of the di-lepton invariant mass shows a deviation from the SM prediction and might serve as an important test of lepton flavour universality, in particular for M = B and j = τ .  Figure 11. Differential branching ratio as a function of the di-lepton invariant mass in the SM. In the plot (a), the IB diagrams are dominant. In (b), the IB and SD contributions are presented for D s decays.

A Feynman rules
The Feynman rules for the vertexes presented figure 1(a,b) may be generically written as BR(Dsμνee)X =1.09·10 -6 Figure 12. The normalized differential branching ratio corresponds to the probability P = 2.54% of measuring the di-lepton mass in the interval 58 MeV < m ee < 62 MeV at the resonance (g 2 X , m X ) = (4 × 10 −4 , 60), one allowed region from figure 9 (a). P = 0.63%, in the SM framework.

B Decay width
The general expression for the decay width can be written as dΓ = sp