Probing the muon g_\mu-2 anomaly, L_{\mu} - L_{\tau} gauge boson and Dark Matter in dark photon experiments

In the L_{\mu} - L_{\tau} model the 3.6$\sigma$ discrepancy between the predicted and measured values of the anomalous magnetic moment of positive muons can be explained by the existence of a new dark boson Z' with a mass in the sub-GeV range, which is coupled at tree level predominantly to the second and third lepton generations. However, at the one-loop level, the Z' coupling to electrons or quarks can be induced via the \gamma -Z' kinetic mixing, which is generated through the loop involving the muon and tau lepton. This loophole has important experimental consequences since it opens up new possibilities, in particular for the complementary searches of the Z' in the ongoing NA64 and incoming dark photon experiments with high-energy electrons. An extension of the model able to explain relic Dark Matter density is also discussed.

At present there are several signals that new physics beyond the standard model (SM) exists. The most striking is the observation of Dark Matter (DM). Among the many models of DM, for a review, see e.g. [1] - [4], those that motivate the existence of light vector(scalar) bosons with a mass m d ≤ O(1) GeV are rather popular now [5,6]. The main idea is that in addition to gravity a new interaction between visible and dark sector exists which is mediated by this gauge boson [6].
Anther possible hint in favour of new physics is the muon g µ − 2 anomaly, which is the 3.6 σ discrepancy between the experimental values [7,8] and the SM predictions [9,10,11,12] for the anomalous magnetic moment of the muon.
Among several extensions of the SM explaining the anomaly, the models predicting the existence of a weak leptonic force mediated by a sub-GeV gauge boson Z that couples predominantly to the difference between the muon and tau lepton numbers, L µ − L τ , are of general interest. The abelian symmetry L µ − L τ is an anomaly-free global symmetry within the SM [13,14,15]. Breaking L µ − L τ is crucial for the appearance of a new relatively light, with a mass m Z ≤ 1 GeV , vector boson (Z ) wich couples very weakly to muon and tau-lepton with the coupling constant α µ ∼ O(10 −8 ) [16]- [29]. The Z interaction with L µ − L τ vector current given by leads to additional contribution to the muon anomalous magnetic moment [30] where and α µ = e 2 µ 4π . The use of the formulae (2,3) allows to determine the coupling constant α µ which explains the value of the g µ − 2 anomaly [16] - [29] and it does not contradict to existing experimental bounds for m Z ≤ 2m µ [29]. Namely, for m Z m µ [30] α µ = (1.8 ± 0.5) × 10 −8 .
For another limiting case m Z m µ the α µ is In addition to the case of the g µ − 2 anomaly, there are also other implications of Z [16]- [29]. For example, in neutrino sector, the L µ − L τ model can provide a natural explanation of a zeroth-order approximation for neutrino mixing with a quasi-degenerate mass spectrum predicting a maximal atmospheric and vanishing reactor neutrino mixing angle [31,32,33], small masses of neutrinos and its oscillations by extending the model with the left-right gauge symmetry [34], the R K puzzle in LHCb data and the g µ − 2 anomaly can be simultaneously explained with the 10 MeV Z which also induces the nonstandard matter interactions (NSI) of neutrinos [35]. The later could also provide LMA-Dark solution to solar anomaly, which also requires NSI [36]. Recently, it has been pointed out that specific features of cosmic neutrino spectrum reported by the IceCube Collaboration can be explained by a mass of the MeV scale [37,38], which can be of interest for the search at Belle II [39]. Moreover, below we show that the L µ − L τ model with a 10 MeV Z boson interacting with a light DM can also explain the observed relic DM density. All these solutions employ a SM extension with a gauge L µ − L τ model.
It is generally assumed that searches for the L µ − L τ gauge boson are difficult as it couples only to the muon and tau lepton family. The relevant bounds can be extracted form the measurements of the neutrino trident production ν µ N → ν µ µ + µ − N [20,21], from the search for a muonic dark force at BABAR [40], and from the data of the Borexino experiment [23]. Currently, the allowed Z mass region for the explanation of the g µ − 2 anomaly is constrained to m Z 400 MeV from by the neutrino trident production [41,42] The direct search for such Z by using the reaction µZ → µZZ ; Z → invisible of the Z production in high-energy muon scattering off heavy nuclei at the CERN SPS was proposed in Ref. [49]. The experiment is expected to improve the sensitivity to the coupling α µ by a few orders of magnitude and fully cover the parameter region referred with Eqs. (4) and (5).
Let us now discuss the mixing between the Z and ordinary photons. An account of one-loop diagrams, which is in our case propagator diagrams with virtual µ-and τ -leptons in the loop, leads to nonzero γ − Z kinetic mixing 2 F µν Z µν where is the finite mixing strength given by [19] = 8 3 Here e is the electron charge and m µ , m τ are the muon and tau-lepton masses respectively.
It should be stressed that we assume that possible tree level mixing tree 2 F µν Z µν is absent or much smaller than one-loop mixing 2 F µν Z µν . To be precise, we assume that there is no essential cancellation between tree level and one loop mixing terms | tree + | ≥ | | .
For m Z m µ the value e µ = (4.75 ± 0.8) · 10 −4 from Eq. (4) leads to the prediction of the corresponding mixing value Thus, one can see that the Z interaction with the L µ − L τ current induces at one-loop level the γ − Z mixing of Z with ordinary photon which allows to probe Z not only in muon or tau induced reactions but also with intense electron beams. In particular, this loophole opens up the possibility of searching the new weak leptonic force mediated by the Z in experiments looking for dark photons (A ).
The fact that the γ−Z mixing of Eq.(7) is at an experimentally interesting level is very exciting. We point out further that a new intriguing possibilities for the complementary searches of the Z in the currently ongoing experiment NA64 [45,46] exists. Indeed, the NA64 aimed at the direct search for invisible decay of sub-GeV dark photons in the reaction e − + Z → e − + Z + A ; A → invisible of high energy electron scattering off heavy nuclei [47,48]. The experimental signature of the invisible decay of Z produced in the reaction e − + Z → e − + Z + Z ; Z → invisible due to mixing of Eq. (6) is the sameit is an event with a large missing energy carried away by the Z . Thus, by using Eq. (6) and bounds on the γ − A mixing the NA64 can also set constraints on coupling e µ .
Another possible way to search for the Z is based on production and detection of its visible decay mode, Z → e + e − , which can also occur at the one-loop level. The flux of Z s would be generated in a high intensity beam dump experiment through the mixing with photon produced either directly in the dump [51] or, e.g., in the π 0 , η, η decays [52]. The Z s would then penetrate the dump without significant interaction and decay in flight into e + e − pairs which can be observed in a far detector. For a given flux dΦ(m Z , E Z , N P OT )/dE Z of Z 's from the dump the expected number of Z → e + e − decays occurring within the fiducial length L of a far detector located at a distance L from the beam dump is given by where E Z , P Z , and τ Z are the Z energy, momentum and the lifetime at rest, respectively, where the decay rate of the Z into neutrino, Γ(Z → νν) ( ν = ν µ , ν τ ) and e + e − pairs, and respectively. Using Eqs. Thus, the advantage of searching for Z in a missing-energy type experiment, e.g. such as NA64, is that its sensitivity is roughly proportional to the mixing squared, 2 associated with the Z production in the primary reaction and its subsequent invisible decay, while for the visible case it is proportional to 2 × Br(Z → e + e − ). The factor 2 is coming from the Z production process and another suppression factor Br(Z → e + e − ) = O(10 −4 ) from the Z → e + e − decay in the detector. Similar arguments are also valid for the experiments that searched for the A in particle decays, because their exclusion area is 10 −4 − 10 −3 for the mass range 1 m A 200 MeV [8]. As a consequence, taking into account the previous discussions, in any beam dump or decay experiment using electrons or quarks as a source of Z s through the mixing of (6), the number of visible Z → e + e − signal events would be highly suppressed resulting in a weak bound on α µ .
Similar considerations results in rather modest constraints on invisible decays of Z which one can extract from the present results of dark-photon experiments searching for the invisible A decays [8]. For example, the bound on the coupling α µ from the K + → π + + missing energy decay is at the level α µ ≤ O(10 −3 ), which is several orders of magnitude below the value from Eq.(4).
Finally, note that in order to cover the range 10 −5 for the Z → e + e − decays the trick would be to try to run a corresponding experiment in a very short-length beam dump mode. A good example of such approach is the AWAKE experiment, which plan to search for dark photon decays A → e + e − with a 50 GeV electron beam by using short W-dump and a detector located at a distance L a few meter [55]. This experiment would be very complementary to the Z searches in invisible decay mode provided the accumulation of 10 16 EOT is feasible. Another experiment, which potentially might be sensitive to the values around of those from of Eqs.(6),(5) for the masses m Z 100 MeV, is the HPS [56], which currently aims at reaching the sensitivity 10 −5 for the A → e + e − decays.
Let us show now that an extension of the L µ − L τ model is able to explain today DM density in the Universe. Consider the simplest SM extension with an additional complex scalar field φ d 1 . The charged dark matter field φ d interaction with the Z field is The annihilation cross section φ dφd → ν µνµ , ν τντ for s ≈ 4m 2 DM has the form 2 We use standard assumption that in the hot early Universe DM is in equilibrium with The dark matter relic density can be numerically estimated as [58] Ω DM h 2 = 0.1 (n + 1)x n+1 where < σv rel >= σ o x −n f , x f = m DM T dec and 1 The annihilation cross-section for scalar DM has p-wave suppression that allows to escape CMB bound [57]. 2 Here we consider the case m Z > 2m DM . c = ln 0.038(n + 1) For the case where dark matter consists of dark matter particles and dark matter antiparticles the DMDM → SM particles annihilation cross sestion σ = σan 2 . Numerically we find that Here the coefficient k(m DM ) depends logarithmically on the dark matter mass m DM and k DM ≈ 0.5(0.9) for m DM = 1(100) M eV . For instance, for m A = 2.2 m DM we have As a consequence of (14) we find that for m Z m µ the values 2 = (2.5 ± 0.7) · 10 −6 and explain both the g µ −2 muon anomaly and today DM density. We can rewrite the equation Moreover an extension of the L µ − L τ model allows to explain relic Dark Matter density for m Z O(10) MeV, which strengthen motivation for the experimental search of the L µ − L τ mediator of the DM production in invisible decay mode. Finally, we note that if the Z couples to light DM, then an additional contribution from the invisible decay mode Z → dark matter increases the Z → invisible decay rate as a consequence for m Z ' > 2m µ visible decay Z ' → µ + µ − is suppressed.
This work grew in part from our participation in the 2nd Annual Physics Beyond Colliders workshop. We wish to thank organizers of this conference for their warm hospitality at CERN. We thank members of the PBC BSM working group, in particular G. Lanfranchi, J. Jaeckel, and A. Rozanov, for discussions and valuable comments. We are indebted to Prof. V.A. Matveev and our colleagues from the NA64 and AWAKE Collaborations for many for useful suggestions.