Search for muon-philic new light gauge boson at Belle II

Motivated by the long-lasting $3.5\sigma$ discrepancy in the anomalous magnetic moment of muon, we consider a new muon-specific force mediated by a light gauge boson, $X$, with mass $m_X<2m_\mu$ and the coupling constant $g_X \sim (10^{-4}, 10^{-3})$. We show that the Belle II experiment has a robust chance to probe such a light boson in $e^+ e^- \to \mu^+ \mu^- + X$ channel and cover the most interesting parameter space explaining the discrepancy with the planned target luminosity, $\int dt \ {\cal L}=50~{\rm ab^{-1}}$. The clean signal of muon-pair plus missing energy at Belle II can be a smoking gun for the new gauge boson. We expect that the (invisibly decaying) muon-philic light ($m_X<2 m_\mu$) gauge boson can be probed down to $g_X \geq 5 \times 10^{-5} \ (1.5 \times 10^{-4}, \ 4 \times 10^{-4})$ for 50 (10, 1) ab${}^{-1}$ search.


Introduction
After the Higgs discovery in 2012, we are now entering the new era of particle physics. The main goal now is to uncover physics beyond the standard model (SM) even though there are still more rooms to improve the precision of the measurements especially in the Higgs quartic and cubic couplings as well as the top quark (pole) mass, which are crucial to determine the stability of our universe [1,2]. 1 Even without any theoretical prejudice, we are actually facing the observational problems, which enforce us to modify or enlarge the standard model. In particular, the significant discrepancy in the anomalous magnetic moment of the muon remains one of the largest anomalies in particle physics [5][6][7][8]: a exp µ − a SM µ = (268 ± 63 exp ± 43 the ) × 10 −11 , (1.1) where the errors are from experiment and theory prediction, respectively. Many well motivated theoretical solutions to fit the data have been proposed [9][10][11][12][13] but no one has been experimentally confirmed so far [14]. It is well-known that light, weakly coupled particles can bring theoretical predictions into agreement with observations [10]. With a simplified interaction with muon, L = −g X X µμ γ µ µ, the light (m X ∼ < 2m µ ) gauge boson (X µ ) contribution to the anomalous magnetic moment of muon at one-loop level is The integration is easily done numerically and found to be positive and close to unity when m X /m µ ∼ < 1 so that ∆a X µ ∼ g 2 X /8π 2 ∼ 3 × 10 −9 . Hence g X ∼ 5 × 10 −4 is desired. This sets up the ball-park range of parameters for our study. (see Fig. 1 When we target to the new light gauge boson, X µ , we don't really need a huge centerof-mass frame energy of the LHC or other future experiments but rather a precise measurement at a relatively low energy experiment. In this letter, we would focus on the Belle II experiment [16], which has been just started and will get scientific data in coming years [17]. Indeed, as we will show in detail, the Belle II experiment would be an ideal place for our purpose. Most dark photon searches at low-energy colliders have considered the mono-photon process e − e + → γA which depends on the kinetic mixing γA between the Standard Model photon and the dark photon A [18,19]. For the muonic force such as gauged L µ − L τ , the similar mono-photon channel has been considered for 'minimal' gauged L µ − L τ whose kinetic mixing is induced by only SM µ and τ loops [20,21]. To be specific for muonic force, we have considered the X-bremsstrahlung process e − e + → µ − µ + X, X → (invisible) in the muon pair production, which is independent on the kinetic mixing γX . This paper is organized as follows. In the next section (Sec. 2) we first set up our theoretical model, a minimal model of muon-philic gauge boson, where the necessary interactions and the most relevant parameters are introduced. We are taking the anomaly-free condition into account for consistency while requiring the model to remain minimal. In Sec. 3, we study the signature at Belle II experiment in e + e − → µ + µ − X channel then optimize the signal/background taking the spectral shape and the missing transverse energy / E T and the missing mass m 2 miss cuts of muons into account. We show the potential coverage of the Belle II experiment in comparison with other relevant experiments. We finally conclude in Sec. 4.
ruled out regardless of whether it decays visibly or invisibly [15].

Model
To incorporate the muonic new force for muon-philic new gauge boson, we extend the SM by including a new U (1) X gauge symmetry. The Lagrangian now contains the kinetic term, mass term and the gauge interaction term for the gauge boson, X µ , of the new gauge symmetry: where X µν = ∂ µ X ν − ∂ ν X µ denotes the field strength tensor of the new gauge interaction and g X is the gauge coupling constant. The U (1) X current is given by the charge assignment of the SM fields (and extra fields too, in principle). The kinetic mixing between U (1) X and SM U (1) Y gauge bosons induce the small electromagnetic current contribution ∼ γX J µ EM but we do not focus on it since they are much more suppressed by both γX and g X . As a simple but consistent example, we may take the leptonic symmetry, X = (L µ −L τ ), which is anomaly free. In this case, the new gauge boson couples with the muonic and tauonic currents with their corresponding (left-chiral) neutrinos [22]: It is important to notice that as long as the new boson is light below the muonic threshold, m X ∼ < 2m µ ≈ 2 × 105.7 MeV, the X µ boson would decay mainly to neutrinos (i.e. ν µνµ , ν τντ ) because all other channels are kinematically forbidden.
It may be worth considering other potentially interesting options free from anomaly. The first, seemingly minimal, option is the solely muonic symmetry U (1) Lµ , which couples to only muon and muon-neutrino at low-energies. This option looks indeed good enough for phenomenological studies of muonic force. However, as pointed out in [23,24], regardless of the UV structure (content of anomaly-cancelling fermion), they would be strongly constrained by Wess-Zumino counterterm contributions to exotic Z → γX decays [25] from the 4-dimension operator g X g 2 µνρσ X µ B ν ∂ ρ B σ and FCNC processes such as B → KX, K → πX [26,27] from the other operator g X g 2 µνρσ X µ (W a ν ∂ ρ W a σ + 1 3 g abc W a ν W b ρ W c σ ). Another potentially interesting option for UV completion free from anomaly is U (1) Lµ−B i=1,2,3 , which would open not only leptonic but also hadronic interactions. This case is also highly constrained by e.g. proton beam-dump experiment [28]. 3 Thus, to avoid unnecessary complication in our analysis, we will focus on the U (1) Lµ−Lτ case below.
In addition, one can naturally extend the list of interactions mediated by muon-philic X gauge boson, including dark sector particles. It provides possible scenarios of light dark matter at sub-GeV scale [32]. If one considers additional interactions between X and the dark sector particles, N χ (vector-like) fermions χ i for example, as the width of X boson can be enhanced as Γ X,total = (1 + δ NM ) · Γ Minimal where and Γ Minimal = m X g 2 X /12π is the total width of minimal gauged L µ − L τ case. N χ is the number of fermion species in the dark sector.
Before studying the future perspectives of finding the muon-philic new gauge boson at Belle II experiment, we first consider the existing constraints in the kinematic range of our interest from various experiments as follows: • Z-pole precision measurement. The X boson can contribute to the Zµ + µ − vertex correction at one-loop level thus modifying the muonic decay width of Z boson by where the loop-function is with the polylogarithmic function of order 2 being Li 2 (x) = − x 0 dt t ln(1 − t) [33]. We set the bound for this correction taking the precision measurement at Z-pole as where we used the values [6] Br(Z → e − e + ) = 3.3632 ± 0.0042%, (2.8) The bound is depicted in Fig. 2 on the top left as a slowly growing line (in magenta). Even after removing phase space suppression due to the lepton masses, Γ(Z → τ − τ + ) still has some tension from the averaged value of leptonic decay width. If we specify our case as U (1) Lµ−Lτ , it gives slightly stronger bound. However, in any case, the bounds from virtual corrections are much weaker than ν-trident production bound.
• Neutrino trident production. (νN → νN µ + µ − ). The neutrino-nucleon scattering experiments effectively provide the stringent constraint to the light gauge boson parameters which couple to the muon and the neutrino(s). The total cross section of ν-trident production νN → νN µ + µ − with X boson, in the light X boson limit (m X < m µ √ s), is given by [34] (2.10) The CCFR experiment using a ν-beam with E ν 160 GeV has obtained the result σ CCFR /σ SM = 0.82 ± 0.28 [35]. The bound is depicted in Fig. 2 by the purple line slightly above the ±2σ band of (g − 2) µ .
• Rare kaon decay at Beam-dump experiments. Rare kaon decay at NA62 beam-dump experiment provides upper bound for muon-philic light bosons by rare kaon decay with a significant feature of some kinematic variables. Current bound comes from the 10 8 charged kaons and it gives the upper bound as g X ∼ < 10 −2 [36] in the parameter range of our interests (also shown in Fig. 2 by yellow line), although it is above the bound from neutrino trident experiment. 4 • BaBar 4µ channel search. The BaBar experiment have explored [37] the muon-philic gauge boson by using the 4µ channel (e − e + → µ − µ + X, X → µ − µ + ), although the result is valid for the case m X > 2m µ . This is depicted in Fig 2 by the green colored (wiggly) region above 2m µ .
• Constraints from Big Bang Nucleosynthesis (BBN). A light X boson coupled to neutrinos can directly enhance the number of relativistic degree of freedom in the BBN era for m X ∼ < O(1) MeV. Even in the heavier case m X ∼ O(1−10) MeV, the presence of muon-philic X boson can affect the effective number of the light neutrino species N eff by providing additional energies to ν µ ,ν µ (and also ν τ ,ν τ in L µ − L τ case) from the decay process X → ν µ(τ )νµ(τ ) after all SM neutrinos are decoupled from SM thermal bath at T ν,dec 1.5 MeV [38,39]. The deviation of the effective neutrino number ∆N eff comes from the difference between the tempreature T of the thermal bath of (ν µ ,ν µ , ν τ ,ν τ , X) and the temperature T of the thermal bath of (ν e ,ν e , γ). This process is an analogy to the photon heating by the e − e + → γγ annihilation.
Requiring ∆N eff < 0.7 (0.1), it disfavors the case m X ∼ < 5.3 (10) MeV [38] 5 . The lower bound for m X corresponding to ∆N eff < 0.7 is shown in Fig. 2 as the orange dotted line. Because X gauge boson can be in thermal equilibrium with other SM particles at early times as long as the coupling between X and ν µ,τ are g X ∼ > 4 × 10 −9 [39], this lower bounds on m X is valid in the range of our interest.
At low mass region (m X < 2m µ ), e − e + → γX, X → (invisible) is the main channel of the minimal dark photon search [19]. The discovery potentials in the same channel e − e + → γX at Belle II experiment also have been explored [20,21].
However, the kinetic mixing between X boson and SM U (1) Y is not determined, unless we assume that µ-and τ -lepton loops only contribute to the kinetic mixing γX which is the minimal mixing case. For instance, other heavy fermion loops (with the fermions with mass splitting M ψ − M ψ ) also can contribute to the mixing as γX ∼ eg X 16π 2 ln(M ψ /M ψ ) and the total kinetic mixing depends on UV structure. If one does not impose the kinetic mixing values between U (1) X and SM hypercharge gauge boson as X ∼ O(10 −4 ), the bound for purely light muon-philic force is not completely determined by low-energy e − e + -collider experiments up to now.
Similarly, other indirect bounds of muonic force for m X < 2m µ , which comes from the electron-neutrino scattering process [40] such as Borexino experiment [20] and the white dwarf cooling [41], also depend on the kinetic mixing between X gauge boson and SM U (1) Y gauge boson, since these bounds assume ν l -e − scattering via t-channel with the mixing γX . Another advantage of considering the parameter region m X < 2m µ is to avoid stringent constraint from cosmic microwave background (CMB), once the X boson becomes a portal to the fermionic dark matter (DM). In general, the process where DM annihilate into a pair of X in s-wave with X decays to charged leptons, can easily contaminate the CMB observations. The upper bound for the annihilation cross section from CMB can be estimated by < σv > /m DM 4.1 × 10 −28 [cm 3 /s/GeV] [42], which rules out the thermally produced DM lighter than 100 GeV. However, the CMB stringent constraint can be avoided, if X decays only into invisible channels. In addition, since the kinetic mixing γX ∼ O(10 −5 ) from µ− and τ −loops is small enough, the process DM + DM → X + X → X + γ could not give significant modification to CMB spectrum. Eventually, the bound from CMB can be satisfied in the parameter region m X < 2m µ .

Expected sensitivity at Belle II
The muon-philic gauge bosons are exclusively produced in muon-associated channels thus is less constrained compared with the model with universal couplings to fermions. In the Belle II experiment, muons are pair-produced and X boson can be radiated away from muon as in Fig. 3. Finally, X → νν and do not leave a detectable signal so that we regard it as an invisible particle (INV) and exploit appropriate kinematical variables such as missing transverse energy ( / E T ) and missing-mass-squared (m 2 miss ). Since the cross section of e + e − → µ + µ − X is proportional to g 2 X = 4πα X ∼ 8π 2 ∆a X µ so that we can almost directly check whether the X boson would be responsible for the anomalous magnetic moment of muon from the measurement at Belle II experiment. We provide some details about the expected sensitivity of muon-philic X boson search in the µ − µ + +INV channel at the Belle II experiment. e e γ * /Z * µ − µ + ν ν X Figure 3: The X gauge boson production in e + e − → µ + µ − X channel.

Signal
The signal process is a muon-pair production with the real emission of a light X boson as a final state radiation. Thus, most of X bosons are very soft and collinear (along with muons' momenta). The signal cross section is [33] where σ   Including X and muon masses, we utilize the splitting function of the X emission in the process µ ± → µ ± + X for massive partons (m µ , m X = 0) [43][44][45] as follows: in the small mass limit m X √ s (See Fig. 4). Note that factor of 2 comes from X boson emission by both µ − and µ + . Here, z ≡ E X /( √ s/2) is the energy fraction carried by the emitted X boson, within kinematically allowed range and the total cross section is consistently given by integrating the spectral splitting function D µ ± →X (s, z) as In principle, the signal (µ − µ + + INV) has a peak in the missing-mass-squared around m 2 miss m 2 X . The decay width of X boson is given by and this width is very small (Γ X→νν ∼ g 2 X m X /12π m X ) in the region of our interests (g X ∼ < 10 −3 ) and the narrow width approximation (NWA) is valid in our event analysis. In this case, we are sure that the produced X bosons are on-shell, and spectral shape of m 2 miss will be very clear. 6 However, once the detector resolution is involved, the peak of missing mass becomes much broad with Gaussian smearing [46]. The tracking resolution of muon momenta in the central drift chamber (CDC) detector is given as at Belle II experiment, where p µ ± is momentum of the muon track [47]. We use σ p µ ± /p µ ± = 0.005 in our event analysis at the detector level. For typical momentum of muons p µ ± 3−5 GeV, the momentum resolution is about σ p µ ± 15 − 25 MeV. Thus, at the low X boson mass region (m X ∼ < 50 MeV), it is hard to expect that the signal peak is distinguished from the backgrounds without additional kinematic cuts to remove relatively huge SM backgrounds.

SM backgrounds and kinematic cuts
The main µ − µ + + / E T backgrounds are as follows: • e − e + → µ − µ + (γ ISR,FSR ) 6 If the coupling of X to dark sector is large as gD ∼ O(1) and a number of species of light (2mχ i ∼ < mX ) dark sector particles are coupled to X boson (Nχ 1), then the width for additional Dirac fermions χi in the dark sector coupled to X gauge boson, for example. Thus, the finite width effect becomes significant in this case. However, for relatively small value of width Γ X,total ∼ < mX , the production cross section σ(e − e + → µ − µ + X, X → χχ) is almost constant (even after the / E T and m 2 miss cuts) because the narrow width approximation (NWA) is valid. Thus, our conclusion about the sensitivity of gX is indeed independent to the detail of the dark sector in most cases.
Most dominant background process is µ − µ + γ, which has typically O(100) pb of the production cross section, although all of them actually can be removed using kinematic cuts. To remove µ − µ + γ ISR and µ − µ + γ FSR backgrounds, we reject all events with / E T < 1.67 GeV or with the photon energy in the center-of-mass frame E γ > 1.0 GeV where the electromagnetic calorimeter (ECL) has high efficiency [47]. This kinematic cut removes most of the µ − µ + γ ISR,FSR backgrounds. One notices that, at the center-of-mass energy √ s = 10.58 GeV, the resonant production of D and B mesons are not negligible. Indeed, J/ψ meson can be produced with a photon and decay into µ − µ + (or τ − τ + ), and its contribution to total µ − µ + γ production cross section is ∼ 0.12 % [48,49]. However, due to its small / E T , most of the J/ψ background events are removed by requiring / E T > 1.67 GeV. For muonically decaying tau-pairs τ − τ + (→ µ − µ + ν µνµ ν τντ ), the cross section is σ(e − e + → τ − τ + , τ ± → µ ± ν τ (µ)νµ(τ ) ) ≈ 27.79 pb (3.11) with the collision energy √ s = 10.58 GeV [49], and they contribute as a significant background. Although the final state (µ − µ + +INV) is the same as the signal mode, its energy spectrum is completely different. The muons come from the decay of taus, and the muon energies at the muon pair center-of-mass frame have broad continuum distributions. In the center-of-mass frame of the electron-positron collision, the differential cross section of the (muonically decaying) tau pair production [50,51] is given by 1 σ d 2 σ(x, cos θ, P e ) dx d cos θ = f (x) − P τ (cos θ, P e ) · g(x), (3.12) where x = E µ /E τ and E τ = √ s/2. The distribution is given by Here,ĝ l v andĝ l a is the vector and axial-vector couplings to the charged leptons. We use the Mitchel parameters ρ µ = 3 4 , ξ µ = 1, δ µ = 3 4 as the prediction in the Standard Model [51]. The anisotropic contribution is negligible because off-shell photon (not Z) is dominant channel for √ s m Z and initial electron and positron beams are not polarized. We use TauDecay [52] library to make FeynRules [53] model file which allows to perform τ decays with polarization. Most events in this background are in the region m 2 miss ∼ > (0.6 GeV) 2 , which is beyond the region of our interest (m X < 2m µ = 0.211 GeV). If one imposes the condition / E T > 1.67 GeV, the remaining m 2 miss values become even larger. Thus, we can safely ignore tau-pair background after m 2 miss cuts.
There are also off-shell W and Z involved process (e − e + → µ − µ + νν). The cross section is ∼ 7 × 10 −2 fb. However, it is 4-body production channel and highly off-shell, so after / E T and m 2 miss cuts, no background events remain, even at the integrated luminosity of 50 , and iv) W -and Zinvolved process (bottom).

Event Analysis
We use MadGraph5 aMC@NLO [54] for background and signal event generation. We use our own FeynRules [53] model file for X gauge boson coupled to muon (and neutrino), to generate µ − µ + X signal events. Event analyses have been performed for the following sets of Monte Carlo events (5 × 10 5 events for each set): where η * γ is the photon rapidity in the center-of-mass frame and the muon rapidity in the center-of-mass frame η * µ ± is given in the range −1.60 < η * µ ± < 1.21 for all events. All rapidity cuts are considered in the center-of-mass frame so that all muons are within both CDC (17.0 • < θ lab. The most dominant background, µ − µ + + unobserved γ (with |η * γ | < 1.94), comes from the photon detection inefficiency at ECL, which is usually 1 − γ 0.05. Main sources of this inefficiencies are the small gap between the barrel and endcap region, photons not converting in the crystal (with a probability of 3×10 −6 ), and so on. Because KLM detector also can be used to detect photon, with the photon identification by using ECL and KLM together, the inefficiency is suppressed up to 1 − γ = 10 −4 − 10 −6 [55]. In fact, it provides an improved sensitivity limit on the "single-photon" search at Belle II (e − e + → γX) up to γX ∼ 3 × 10 −4 . In addition, imposing / E T and m 2 miss cuts and muon detection efficiency for these background events, the expected µ − µ + (γ) event number is ∼ 18.56 As we mentioned in the previous section, / E T , E γ and m 2 miss cuts are used to remove all background events. Comparison for signals and backgrounds under these kinematic variables are shown in Fig. 6 for / E T and Fig. 7 for m 2 miss . Also, we show correlations between / E T and m 2 miss in Fig. 8 and Fig. 9 for backgrounds and signals, respectively.
respectively and / E cut T = 1.67 GeV, (m 2 miss ) cut = 0.4 GeV 2 /c 4 as we mentioned. We also reject all events including muons with momentum below 0.6 GeV/c in the lab frame and assume that the detection efficiency at the K L and muon (KLM) detector is (p µ ± ) = 0.9 for p ± > 0.6 GeV/c [55]. We focus on cases of integrated luminosity L dt = 1, 10, 50 ab −1 . Expected 3σ sensitivity limits at Belle II are shown in Fig. 10. We assume the photon detention inefficiency 1 − γ = 10 −6 and show other detection inefficiency cases.