Light gauge boson interpretation for $(g-2)_\mu$ and the $K_L \rightarrow \pi^0 + \text{(invisible)}$ anomaly at the J-PARC KOTO experiment

We discuss a list of possible light gauge boson interpretations for the long-standing experimental anomaly in $(g-2)_\mu$ and also recent anomalous excess in $K_L \rightarrow \pi^0 + \text{(invisible)}$ events at the J-PARC KOTO experiment. We consider two models: $i$) $L_\mu - L_\tau$ gauge boson with heavy vector-like quarks and $ii$) $(L_\mu - L_\tau) + \epsilon (B_3 - L_\tau)$ gauge boson in the presence of right-handed neutrinos. When the light gauge boson has mass close to the neutral pion in order to satisfy the Grossman-Nir bound, the models successfully explain the anomalies simultaneously while satisfying all known experimental constraints. We extensively provide the future prospect of suggested models.


Introduction
The KOTO experiment at the Japan Proton Accelerator Research Complex (J-PARC) recently released their result on K L → π 0 ν ν searches [1][2][3]: four candidate events were observed in the signal region over the background estimation 0.05 ± 0.02.One of those candidate events is still suspected as a background from overlapped pulse, but other three events are distinctive in their properties from the known backgrounds.
In this paper, we check simultaneous explanations of (g − 2) µ and KOTO events with a light gauge boson X.We found that only through the mixing with photon, it cannot generates sufficient Br(K L → π 0 X) for KOTO events excess, meanwhile satisfies other experimental constraint, especially from BaBar [40] and NA64 [41] (Section 2.2).Alternatively, we investigate the two kinds of plausible interactions between a new light gauge boson X coupled to muons and quarks through the followings: • L µ − L τ gauge boson X with heavy vector-like quarks (VLQs) (Section 3): Introducing heavy VLQs at TeV scale couple to both L µ − L τ gauge boson and SM quark sector [42] is a promising way to enhance Br(K L → π 0 X).The flavour changing neutral current (FCNC) is generated at tree level due to the VLQs' nontrivial contributions to the off diagonal elements of the quark mass matrix.Then we check the consistency with existing constraints such as Br(K + → π + X), K 0 − K0 mixing, Br(K L → µ + µ − ), Cabibbo-Kobayashi-Maskawa (CKM) unitarity.
2 Decay of K and B mesons with FCNCs

Decay widths and experimental limits
We focus on the effective FCNC couplings of X boson where first term is relevant to the KOTO process, and both terms are correlated to each other under these two model frameworks that will be discussed in this work.These FCNC couplings lead to the branching ratios of rare K and B meson decays as follows [10,43]: Br(B + → K + X) m 3 where λ 1 (x, y, z) ≡ x2 + y 2 + z 2 − 2xy − 2xz − 2yz, and for K + and K L mesons the corresponding form factors ) are close to the unity.The branching ratios of K + and K L are correlated to each other, since the K L corresponds to only the imaginary part of g eff dsX , meanwhile K + is proportional |g eff dsX | 2 .The total widths for these mesons Γ K + = 5.315 × 10 −17 GeV, Γ K L = 1.286 × 10 −17 GeV, Γ B + = 4.017 × 10 −13 GeV are used to obtain the branching ratios.[44].
For m X below the muon threshold and no coupling with electron current, only neutrino pair decay mode is kinematic allowed.Furthermore the X boson can also decay invisibly into pair of hidden sector light particles.And thus in the rest of this paper, we assume that the invisible decay mode dominates the light X boson decay.The required effective coupling strength to explain KOTO events excess from above estimation is We set m X m 0 π to evade the stringent constraint from Br(K + → π + + invisible) decay, which is suffered from overwhelming K + → π + π 0 background.Therefore, it can satisfy other upper bounds from current observations of rare K and B meson FCNC decays.Taking K + for example, because of the huge K + → π + π 0 background, when the square of missing energy around pion mass q 2 m 2 π , weaker bound [45]. 2 And thus weaker GN bound, i.e.Br( π 0 , can be translated into the limit as which is still an order of magnitude larger than the prefer coupling for KOTO events.The bound for b to s coupling is from the Br(B + → K + + invisible) ≤ 1.3 × 10 −5 of Belle [46] and BaBar [47,48] requires It may provides additional constraint, if the couplings g eff sbX and g eff dsX are correlated.Explaining the KOTO event excess through K L → π 0 + X with light gauge boson m X m 0 π is still in accordance with other present experimental constraints.Further more, if this X boson carries the muonic force with coupling strength of O(10 −3 ), it can also explain the (g − 2) µ anomaly [36].One simple and vastly discussed model in the literature is the U (1) X gauge boson X kinematic mixing with SM photon or Z boson through the mixing parameters γX and ZX , respectively.However, we would like to show that this single model cannot explain the KOTO event excess under the constraint from Br(K + → π + + invisible) of E949.
The X boson couples to the SM quark current through the mixing, and then the FCNC are generated from one-loop W boson and top quark penguin diagram.The down-type FCNC transitions b → sX and s → dX are given by The q is outgoing momentum carried by X gauge boson, therefore the above vertices are suppressed by ratio m 2 X /m 2 W , where m 2 W comes from Fermi constant G F .The vertex function H 0 (x) consisting of photon component function D 0 (x i ) and Z component function D0 (x i ), are characterized by γX and ZX , giving [49][50][51] H 0 (x) = γX D 0 (x) + ZX D0 (x), (2.10) 12) The loop function D 0 (x) is determined by the sum of amplitudes iM kin.a,b,c,d .The diagrams for each amplitudes are shown in Fig. 2. Due to the unbroken U (1) EM gauge symmetry and the cancellation between the amplitudes, the resulting FCNC operator from kinetic mixing γX is proportional to the transverse part of the outgoing momentum, g µν q 2 − q µ q ν .
The diagrams which contribute to the loop-induced dsX FCNC process only with the kinetic mixing γX between X boson and SM photon.In t Hooft-Feynman gauge, the charged goldstone boson φ − contributions also should be included.
As a result, the upper bounds on the kinetic mixing from Br(K + → π + X) and the preferred value to explain KOTO excess are 16) It is clear to see that the mixing should be as large as γX ∼ O(1), to match the required effective coupling of Eq. (2.5) to explain KOTO result.Note the hierarchy between the real and imaginary components of FCNC coupling g eff dsX from the charm quark contribution which is proportional to V cs V * cd D 0 (x c ). Avoiding constraints from the upper limit of Br(K + → π + X), which corresponds to γX ∼ < 10 −2 , is not possible in the presence of charm quark contribution.Furthermore, invisible dark photon searches from BaBar [40] and NA64 [41] exclude large kinetic mixing down to γX ∼ < 10 −3 .For short summary, the KOTO event excess cannot be explained by a (invisibly decaying) light gauge boson, kinematically mixed with the SM photon.Therefore, in the next two subsections, we i) introduce heavy VLQs to enhance the coupling between L µ − L τ gauge boson and SM quarks, especially for tree-level FCNC, or ii) consider a (L µ −L τ )+ (B 3 −L τ ) gauge boson which dominantly contribute to the down-type FCNC at the loop-level.

Model I: gauged L µ − L τ with heavy VLQs
We focus on the extension of SM gauge group by a new abelian and anomaly free U (1) Lµ−Lτ with the associated X vector gauge boson [29][30][31][32].As the original gauge symmetry is leptonic so that it does not allow the direct coupling to hadrons, the X boson still can couple to the SM quark sector through the dimension-6 operators with cutoff Λ at TeV scale.When a scalar Φ carrying +1 charge under U (1) Lµ−Lτ [42] is introduced, the relevant dimension-6 operators are explicitly given as where q L = (u L , d L ), d R , and u R are the SU (2) L doublet and singlet quarks with flavour index i, j.In general, the coupling λ q,d,u i,j are 3 × 3 complex matrices, which potentially violate flavour and CP symmetries.After Φ gets a VEV Φ = v Φ / √ 2, the U (1) Lµ−Lτ is spontaneously broken, and then the hadronic current violating flavour symmetry is generated are given in Eq. (3.10).At the same time, the X boson obtain a mass m X = g X v Φ , where g X is the U (1) Lµ−Lτ gauge coupling.After all, the effective action is given as where where g X 5×10 −4 and m X ≤ 2m µ is still allowed region for the (g −2) µ anomaly [32,52], and 2L = (ν µ µ) T L and 3L = (ν τ τ ) T L are the second and third generation lepton doublets in the SM, respectively.To explain the KOTO events excess, additional X boson couplings to J µ(had) X will be generated by introducing VLQs at TeV scale, which mix with SM quarks.The general expression for the hadronic current induced from VLQs, which carry U (1) Lµ−Lτ charges and couple to SM quark sector through new scalar Φ.The model has been previously suggested in Ref. [42] to explain the lepton universality violation (LUV) in rare B meson decay B → K * l − l + (l = e, µ) and has been applied to K 0 L → π 0 + (invisible) with enhanced coupling to top quark [10].We follow Ref. [42] and introduce VLQs with the gauge charges (SU (3) c , SU (2) L ) (Y,Q ) of the interaction eigenstates are assigned as where Y and Q are SM hypercharge and U (1) Lµ−Lτ charge, respectively.Then the Yukawa interactions between VLQs and SM quarks are written as that will induce the mixing between VLQs and SM quarks.In order to maintain the electroweak invariant, they shall satisfy the relation After Φ gets VEV, these Yukawa interactions contribute to the off diagonal elements of the up-type and down-type quark mass matrices where the masses for VLQs come from They can be diagonalized by performing the bi-unitary transformation: and induce the FCNC interactions for Eq./,(3.1) which are illustrated in Fig. 3.Here we only keep the leading terms of order Furthermore, the CKM unitarity within 3 × 3 SM quark block will be violated due to the extension of quark sector.In fact, the CKM matrix is extending to 5 × 5 and relates to SM CKM matrix as where the zero diagonal element is from SU (2) L singlet VLQs ŨL and DL .The unitarity condition still hold in the 5 × 5 CKM matrix, but it would be violated in the 3 × 3 block of CKM, that can be tested by current precision measurements of CKM elements, for example the deviation from unity should be less than O(10 −3 ) under current measurements on 1st row of CKM, i.e.
3.1 Explanation of KOTO events We have added six VLQs: Q L , QR , ŨL , U R , DL , and D R .Only QR gives most relevant contribution to K → πX, which involves the left-handed down quarks mixing among 1st and 2nd generations.From Eq. (3.7), we can see QR generates non-zero offdiagonal elements on the upper-right corner, meanwhile, ŨL and DL contribute on the lower-left corner.According to these patterns, as it was expressed in Eq. (3.10), that QR induce the larger left-handed quarks mixing, while ŨL and DL induce the larger righthanded mixing.As a consequence, introducing only QR is the efficient way to enhance the K → πX.They give the tree-level FCNC effective interactions as However, QR also induces non-trivial FCNC for the up-type quarks due to the relation of Eq. (3.6), but, the FCNC constraints among up-quark sector are not as stringent as the down-quark sector.To explain the KOTO event excess, the effective coupling g eff dsX | VLQ shall satisfy Eq. (2.5) and gives 3.2 constraints The CP violation in Kaon mixing process might put strong bound on the FCNC between the 1st and 2nd generations in down-quark sector.In terms of six-dimensional operator the upper bound of the FCNC coupling (g eff dsX ) 2 can be translated into the lower bound on the scale Λ ds .The lower bound on Λ ds comes from the experimental constraints on the mass difference ∆m K and the mixing coefficient K .We quote limits from [60] |Re as the constraint from Kaon mixing in this work.Nevertheless, the FCNC coupling induced by heavy VLQ can contribute to the FCNC operator as and it gives the upper bounds at m X = m π 0 .The KOTO desired (and allowed by K + → π + X branching ratio measurement) region satisfies Kaon mixing constraints, with large difference of the order of magnitude.If we assume g X 5 × 10 −4 and v Φ 260 GeV, they give ) and still comfortably survives.
For chirality-flipping operator, the new physics bound becomes slightly stronger, but the KOTO desired values are not excluded.Even in the presence of both QR and DL (and mixing to the SM s and d quarks), we find that the Kaon mixing constraint is not sensitive to our bulk part parameters.The Kaon mixing constraint can be translated into the bound on flavour-changing couplings to both left-handed and right-handed quarks through the effective operator (s which still satisfies the Kaon mixing constraints from Eq. (3.20) and Eq.(3.22).
Since we impose the mixings between the heavy vector-like quark QR and the lefthanded SM quarks s L and d L , it naturally provides up-type quark interactions including flavour violating components as due to the SU (2) L gauge invariance.If we assume O(10 −3 ) of real and imaginar components of yukawa couplings Y Qd and Y Qs , then we get very tiny couplings for g eff ucX ∼ O(10 −17 ).It is obviously safe from current upper bounds obtained by D meson mixing constraints.

K
The CP-conserving Kaon rare decay K L → µ + µ − has similar short-distance part contribution to K L → πν ν, from Z-penguins and box diagrams.However, important long-distance contributions from two-photon intermediate state are difficult to precisely be calculated and separated from short-distance part.Therefore, here we just require the additional contributions from our model of Br(K L → µ + µ − ) do not exceed the current experimental observation [44] Br(K L → µ + µ − ) EXP = (6.84± 0.11) × 10 −9 . (3.25) For the short-distance part, the effective Hamiltonian from SM and VLQ are [61] where α EM ≡ e 2 4π and the loop functions , and η Y = 1.026 ± 0.006.Then the branching ratio is where the first two terms in the square bracket are from SM short-distance part, and the third one comes from VLQs contribution.Here we defined where P 0 (Y ) = Y NL /λ 4 0.138 and λ ≡ V us = 0.22453 ± 0.00044.One obtains By insert these values, we obtained SM short-distance contribution Combining the VLQ and SM contribution and using KOTO preferred region from Eq. (3.13) it gives Br(K L → µ + µ − ) = 9.931 × 10 −10 , which does not modify much.Under the preferred parameter values for KOTO event excess, VLQs contribution to Br(K L → µ + µ − ) is less than O(10 −12 ), which is two orders of magnitude below the current experimental sensitivity.

CKM unitarity
Before considering the SM quarks mixing with VLQs, we assume that the 3 by 3 block of quark mass matrix corresponding to SM is diagonalized, as shown in Eq. (3.7).Hence the 3 by 3 block of SM CKM matrix satisfies unitarity.
After SM quarks mixing with VLQs, the couplings with W boson are modified as where SU (2) singlet DL and ŨL do not couple to W boson.And then the CKM is modified accordingly The Therefore, the VLQs modifications of the SM corresponding CKM matrix is of order O(10 −4 ), it is still compatible with the present observational precision of |V ud |.
4 Model II: gauged In the presence of (at least) two species of heavy right-handed neutrinos N 2,3 , we can consider a possible anomaly-free extension of the gauge group where X1 and X2 are the gauge bosons which belong to gauged U (1) Lµ−Lτ and U (1) B 3 −Lτ in the gauge eigenbasis, respectively, and are the conserved currents of U (1) Lµ−Lτ and U (1) B 3 −Lτ , respectively. 3Here, q 3L = (t b) T L is the third generation left-handed quark doublet in the SM.Similar to single X gauge boson cases, one can impose the general kinetic mixing between SM gauge bosons (γ and Z) and new gauge bosons X1,2 with dimension-four operators γX i Bµν Xµν ) although sizeable values of γX i are constrained by dark photon searches [40,41].See Appendix A for the detailed formulation in the presence of generic kinetic and mass mixing.To obtain the physical spectrum and interactions, we diagonalize them from gauge eigenstates to mass eigenstates.As a result, we obtain a simple pair of light gauge bosons as where Here, the ratios i and each couplings g X,i are determined by the model parameters M 2 i , δ M 2 12 , ĝX i and 12 for gauge eigenstates Xi .In this work, we focus on the phenomenological setup of an effectively light gauge boson in the ranges of 100 MeV ∼ < m X ∼ < 165 MeV with a gauge coupling g X to (L µ − L τ ) + (B 3 − L τ ) current where is a small ratio between muon and top quark couplings, rather than the complete two gauge boson construction starting from the gauge eigenstates.

Explanation of KOTO events
In the presence of X coupled to SM top quark, it significantly enhances FCNC at one-loop level.This is contrary to the SM photon case, because there is no cancellation among the diagrams from the symmetry.In the presence of B 3 coupling, effective FCNC couplings are where is the loop function of X gauge boson induced penguin diagram in the limit m d,s,b,X m W,t with x t = m 2 t /m 2 W (See Appendix B).We show the diagrams that contribute to down-type s → dX FCNC transition in Fig. 4a and Fig.   K from the measurement of rare B meson decays B 0 → K * 0 l − l + (l = e, µ) with TeV scale X gauge boson and heavy vector-like fermions [62].
up to O(λ 5 ) in the expansion of the Wolfenstein parameters.The upper bounds on g X from Br(K + → π + X), Br(B + → K + X) and the required value for KOTO are (From Br(B + → K + X) upper limit for q 2 = m 2 π 0 ) (4.12) Considering a (L µ − L τ ) + (B 3 − L τ ) gauge boson with 5 × 10 −4 ∼ < g X ∼ < 10 −3 , 0.01 − 0.03 and m X 100 − 165 MeV, we have a simple interpretation for (g − 2) µ and KOTO events.We show this value of top quark coupling is consistent with other current constraints from other FCNC decays such as K L → µ + µ + , B s → µ + µ + and neutral K, B, and D meson mixings.
One can analogously consider B 2 (the baryon number of second generation) gauge coupling to make FCNC via charm quark contribution as ), and obtain FCNC couplings up to O(λ 5 ) in the expansion of the Wolfenstein parameters again.However, it cannot provide a desired Br(K 0 L → π 0 X) value, avoiding Br(K + → π + X) constraint at the same time because the imaginary part is three order of magnitude smaller than the real part in g eff dsX | B 2 .Similar to the minimal kinetic mixing case, shown in Section 2.2, charm quark contribution spoils loop-level FCNC explanation of KOTO excess without changing the mixing structure in the quark sector.

Constraints
In this section, we consider possible constraints and summarize them in Fig. 5.In Fig. 5, we show the preferred region of parameters (m X , g X and ) and the current experimental constraints.In model II, we have two allowed regions (120 MeV ∼ < m X ∼ < 160 MeV and 250 MeV ∼ < m X ∼ < 350 MeV) for the KOTO events, although higher mass region cannot explain (g − 2) µ simultaneously, due to the experimental constraints from 4µ search from BaBar [64] and the search of the muonic force coupled to b → sX FCNC vertex at LHCb [65].
G / a h / Y 5 j + a 0 x c w x + A P t + w e S V a E I < / l a t e x i t > x X q z P q 3 Z I r p m p T M n 4 A + s 7 x + w R p 7 i < / l a t e x i t > mixing < l a t e x i t s h a 1 _ b a s e 6 4 = " v o 9 u Y z t J 0 a e 5 U F b 6 q s G / a h / Y 5 j + a 0 x c w x + A P t + w e S V a E I < / l a t e x i t > x X q z P q 3 Z I r p m p T M n 4 A + s 7 x + w R p 7 i < / l a t e x i t > mixing < l a t e x i t s h a 1 _ b a s e 6 4 = " v o 9 u Y z t J 0 a e 5 U F b 6 q s Blue shaded band is the region for KOTO desired Br(K 0 L → π 0 X) value.Red shaded band is the required value for (g − 2) µ .The constraints from Br(B + → K + X) (purple), Br(K + → π + X) (orange), GN bound (magenta), D 0 − D0 mixing (green) and muonic force search in 4µ channel (gray dashed) are also shown.We show two different cases of the ratio between the muonic and the hadronic couplings as = 0.012 (Left panel) and = 0.035 (Right panel).See the main text for details.
In model II, we have loop-induced down-type FCNC couplings as Eq.(4.7) and Eq.(4.8), contributing to mixings of neutral mesons.Nevertheless, there are upper bounds from B − B0 and K 0 − K0 mixings, which are converted into the Wilson coefficients of sixdimensional operators (s L γ µ b L ) 2 and (s L γ µ d L ) 2 respectively.The experimental upper bounds are [60] |Re ) ) d mixings as well as Eq.(3.15-3.16)for K 0 − K0 mixing.Due to the loop and CKM suppressions, they give very weak upper bounds, g X 3 ∼ < 1.11 from B meson mixing and g X 3 ∼ < 0.14 from Kaon mixing, at m X m π 0 .
In addition to down-type FCNC couplings (g eff , there are also tree-level up-type (left-handed) FCNC couplings due to SU (2) L gauge invariance as where U L D † L = V SM CKM and we assume U L = V SM CKM , D L = 1 in our model.It generates sizeable tree-level up-type FCNC interactions The coupling g eff ucX | B 3 ,tree-level can be constrained by D 0 − D0 mixing as [60] Re which are the real and imaginary part of the Wilson coefficient for the operator (ū L γ µ c L ) 2 .At m X m π 0 , the constraints are translated into g X ∼ < 1.55 × 10 −2 , which is not enough to constrain the KOTO required value.
The couplings g eff utX | B 3 ,tree-level , g eff ctX | B 3 ,tree-level makes a FCNC decay of top quark.However, the branching ratio is much smaller than current experimental sensitivities from LHC searches.

Γ D + and D
The coupling g eff ucX | B 3 ,tree-level also promotes FCNC decay of the charged D meson.The branching ratio of the decay D + → π + X is given by where is the form factor obtained from chiral perturbation theory of heavy hadrons [66].We use f D = 200 MeV, f π = 130 MeV and g D + Dπ = 0.59 in our calculation, following the analysis given in Ref. [63].We set our upper bound by requiring Γ(D + → π + X) < Γ D + ,total − Γ D + ,K 0 using the inclusive value of the branching ratios, to avoid a significant modification of the total width of D + meson. 4At m X m π 0 , it gives only a weak upper bound g X ∼ < 1.31 × 10 −2 , and thus not sensitive to KOTO and (g − 2) µ preferred region.
Before we go through following detailed analysis, we provide brief results of this subsection here.For rare meson decays K L /B s /B d → µ + µ − , the upper bound on FCNC couplings are about g 2 X O(10 −5 ) and thus are insensitive to our bulk part parameter region, g 2 X ∼ O(10 −8 ).Since the dominating uncertainties come from theoretical calculations, the upper bounds are determined by the condition where the X boson contribution does not exceed the SM contribution for each decay channel.
For K L → µ + µ − , as in Section 3.2.2,we write down the short-distance part of the effective hamiltonian as and the upper bound is given by demanding that the new physics contribution is smaller than the SM prediction value, as follows: and it gives g 2 X ∼ < 1.72 × 10 −5 for m X m π 0 .The preferred values of 0.026 and g X 5 × 10 −4 for KOTO and (g − 2) µ gives g 2 X O(10 −8 ), therefore the K L → µ + µ − decay branching ratio is not sensitive to our model parameters.
For B s → µ + µ − , we have and with the same criterion.The upper bound is g 2 X ∼ < 1.92 × 10 −3 for m X m π 0 , which is even weaker than Kaon constraints.For B d meson, the branching ratio is given by where F B d F Bs ≈ 210 MeV.Thus, B d meson decay gives a similar upper limit value of the coupling g X .

Expected sensitivities in future experiments
The most promising way to probe the KOTO preferred parameter region in model II is the rare decay of the charged B meson (B + → K + X(→ inv.)) search at Belle II.The strongest upper bound on Br(B + → K + X) comes from Belle [46] and BaBar [47,48], which corresponds to Br(B + → K + X) ∼ < (1.3 − 1.6) × 10 −5 with the data of 492 ab −1 and 418 ab −1 , respectively.By a simple rescaling for the upper limit as g upper.X ∝ ( dtL) −1/4 , we show the expected limit at Belle II in Fig. 6.We also include the expected limits on the muonic force from Belle II using 4µ channel [64] and µ − µ + X(→ inv.) channel [36,67], and neutrino-trident production at DUNE [38] for a (L µ − L τ ) + (B 3 − L τ ) gauge boson X, Kaon decays (K + → µνX) at NA62 [37], M 3 (Muon Missing Momentum) based at Fermilab [39], and ATLAS detector as muon fixed-target experiment [68] for comparison.We show the expected sensitivities in Fig. 6.
For both muonic (L µ ) and hadronic (B 3 ) coupling, most of (g − 2) µ and KOTO desired region can be probed by Belle II through Br(B + → K + X) and 4µ channel searches, with the data of 50 ab −1 integrated luminosity.Note that we assume similar systematic uncertainties in Br(B + → K + X) and 4µ channel search of muonic force.Thus, the actual limit could be different from our estimation, depending on experimental environment at future experiments.

Summary and Conclusion
The long-standing (g − 2) µ anomaly and recent J-PARC KOTO event excess can be explained in single framework by a light (m X < 2m µ ) gauge boson X, where its mass is near the neutral pion mass in order to avoid the stringent GN bound and Br(K + → π + + invisible) upper limit.The X boson has to couple to both lepton and quark sectors, and we investigated possibilities from two model frameworks, i) gauged L µ − L τ with heavy VLQs, ii) gauged L µ − L τ + (B 3 − L τ ) with mixing of two gauge bosons.Both frameworks provide allowed parameter regions for (g − 2) µ and KOTO, and satisfy the current experimental constraints.We would like to summarize our results in the following list.
• The simple model from U (1) X boson mixing with SM photon cannot interpret the KOTO event, meanwhile satisfying the constraint from Br(K + → π + + invsible).
• In gauged L µ − L τ with heavy VLQs, the (g − 2) µ prefers gauge coupling g X = 5 × 10 −4 , and KOTO event excess requires 2 TeV mass VLQs carrying complex < l a t e x i t s h a 1 _ b a s e 6 4 = " q j x l 0 e F S 0 Y W 9 h S a 1 9 q s p v D x M S T s u + 5 u t M j q i t n t S H 5 n 1 a P

DU NE
< l a t e x i t s h a 1 _ b a s e 6 4 = " U e 7 2 M C q V J j 5 0 e C K J p T I i e r D v r w X q 2 X j 5 H x 6 x R Z h X 9 g P X 2 A b s + n q I = < / l a t e x i t > Figure 6: The sensitivity limit expected in future experiments, for model II with a (L µ − L τ ) + (B − L τ ) gauge boson X.All solid lines belong to B 3 coupling and dashed lines to L µ coupling.Blue shaded band is the region for KOTO desired Br(K 0 L → π 0 X) value.Red shaded band is the required value for (g − 2) µ .From the existing upper limits of Belle and BaBar, we show the Belle II (with the data of 50 ab −1 integrated luminosity) expected upper limits from i) Br(B + → K + X) (magenta) for hadronic coupling and muonic force searches using ii) 4µ channel (green) and iii) µ − µ + X(→ inv.) channel (purple) [36,67].We also show the limit from for ν-trident production at DUNE (brown) [38], Kaon decays (K + → µνX) at NA62 (orange) [37], M 3 (yellow) [39] and ATLAS (black) [68] for comparison.We set = 0.012 as an example case.FCNC Yukawa couplings of Im(Y Qs Y * Qd ) 2.74 × 10 −7 , which is compatible with constraints from K 0 − K0 mixing, K L → µ + µ − , and CKM unitarity.
03 and m X 120 − 160 MeV provides simple interpretation for both (g − 2) µ and KOTO events.Meanwhile, it satisfies the GN bound, Br(K + → π + + invisible), and Br(B + → K + +invisible) upper limits.In near future, this preferred parameter region will be explored by the B + → K + + invisible search at Belle II.On the other hand, the muonic force region will be tested by the e + e − → µ + µ − X → µ + µ − + invisible channel at Belle II, ν-trident production at DUNE, Kaon decay at NA62, Muon beam dump experiment and muonic decay of W /Z at ATLAS.
The observation of K 0 L → π 0 + (invisible) decay events are based on the analysis of the 2016-2018 KOTO data, where the current sensitivity reaches a single event for K L branching ratio of ∼ O(10 −10 ).The enhanced data collected by KOTO experiment in 2019 is expected to improve the statistical uncertainty in near future [1].Furthermore, several upcoming experiments on rare Kaon decays, such as KOTO step-2 [69] and KLEVER using CERN SPS beam for the K L production during the period of LHC Run 4 [70,71], have been proposed and the projected sensitivity can reach branching ratio of ∼ O(10 −13 ) so that it will fully cover the SM prediction ∼ O(10 −11 ).Combining with the various and extensive searches on the muonic force [34][35][36][37][38][39] Because the matrix is symmetric and real, it can be diagonalized by an orthogonal matrix O as O T M 2 O = M 2 diag .In particular, the photon remains massless, the orthogonal matrix has the form: where O 3×3 is a 3 × 3 orthogonal matrix which we can construct using the eigenvectors (normalized to be a unit vector) x i of the mass matrix µ 2 , Analytically, we also can decompose the orthogonal matrix as given by in the Refs.[73,74], (A.37) The method of calculating θ i (i = 1, 2, 3) in [73,74] is also reviewed in Section A.3.Finally, the gauge eigenstates ( Â, Ẑ, X1 , X2 ) are related with the mass eigenstates (A, Z 1 , Z 2 , Z 3 ) as or inverted relation is given as The approximated form of mass matrix µ 2 and corresponding O 3×3 in the limit m 2 i v with respect to the vector (1, 0) T , which is equivalently the angle with positive x axis.We also define and θ 1 is given by The sign combination of θ 2 and θ 3 is determined to be with the smallest difference between θ 1(1) and θ 1(2) .
B One-loop FCNC induced by a light gauge boson coupled to third generation quarks In model II (Section 4), X gauge boson coupled to a combination of fermion numbers (L µ − L τ ) + (B 3 − L τ ) generally induces the monopole and the dipole FCNC terms at one-loop level as which F i (x t ) (i = 1, 2, 3, 4) are the (dimensionless) function of order 1.At low energy, q is the momentum of the produced X gauge boson and q 2 = m 2 X m 2 t , m 2 W .We focus on the F 1 (x t )( dγ µ P L s)X µ term, which is dominant in our case.
In Fig. 4, we show the diagrams which contribute to FCNC vertex and the amplitudes are given by iM B 3 a,b = X * µ (q)• d(iΓ )s.For the diagram a (Fig. 4a), one obtains one-loop amplitude as d(iΓ and the vertex correction Γ (B 3 )µ a is explicitly given by [75]  In the limit m 2 d , m 2 s , q 2 (= m 2 X ) m 2 t , m 2 W , we approximate the loop-induced vertex as is the loop function of order 1.

Figure 3 :
Figure 3: The diagrams which contribute to dsX FCNC process with X gauge boson coupled to a heavy vector-like quark Q (model I).

. 13 )
by fixing g X = 5 × 10 −4 and v Φ = 260 GeV to give m X = g X v Φ 135 MeV close to neutral pion mass.The Yukawa coupling strengths are estimated to be Y Qs Y Qd 5.2 × 10 −4 , if we choose m Q 2 TeV, which is heavy enough to satisfy all the current mass lower bound from the VLQs direct searches at the LHC[54][55][56][57][58][59].
first-row of CKM |V ud | 2 +|V us | 2 +|V ub | 2 1 are known with highest precision and good agreement with unitarity.According to recent calculation of inner radiative correction with reduced hadronic uncertainty, the updated value of |V ud | = 0.97366(15) has been obtained [53].The preferred values of input parameter to explain KOTO events are Y Qd = Y Qs = 5.2 × 10 −4 , Y Qb = 0, v Φ = 260 GeV, and m Q,D 2 TeV.It gives the mixing angle between VLQs Q L and d quark of 4b .

3 U
(1)B 3 −Lτ , flavoured B − L for third generation fermions, has been considered to resolve the lepton universality violation in R ( * )

Figure 4 :
Figure 4: The diagrams which contribute to the loop-induced dsX FCNC process with X gauge boson coupled to B 3 (model II).The loop-induced down-type FCNC (Left and Middle panels) and tree-level up-type FCNC (Right panel) are shown.
t e x i t s h a 1 _ b a s e 6 4 = " + b k b 8 t f S y x 2 4 u y R 6 / w x S K 9 A p N 5 O / o w n o w 3 4 9 P 4 m p U m j H n P A f q F R P I b X c + u 9 Q = = < / l a t e x i t > t e x i t s h a 1 _ b a s e 6 4 = " s M K m 2 G g V 0 j e 2 Z / C M O 8 N B 6 g

Figure 5 :
Figure 5: The preferred region for KOTO result and (g −2) µ with a (L µ −L τ )+ (B 3 −L τ ) gauge boson X.All solid lines belong to B 3 coupling and dashed lines to L µ coupling.Blue shaded band is the region for KOTO desired Br(K 0L → π 0 X) value.Red shaded band is the required value for (g − 2) µ .The constraints from Br(B + → K + X) (purple), Br(K + → π + X) (orange), GN bound (magenta), D 0 − D0 mixing (green) and muonic force search in 4µ channel (gray dashed) are also shown.We show two different cases of the ratio between the muonic and the hadronic couplings as = 0.012 (Left panel) and = 0.035 (Right panel).See the main text for details.

KOTO < l a t e x i t s h a 1 _
b a s e 6 4 = " + b k b 8 t f S y x 2 4 u y R 6 / w x S K 9A p N 5 U = " > A A A C H X i c d V D L S g M x F M 3 U V 6 2 v q k s 3 w S K 4 k D I p x d a d 6 E Z w 0 Q q t C p 2 h Z N K 0 B j M P k j t i G e Z H 3 P g r b l w o 4 s K N + D d m a g d U 9 E D g c M 6 5 y c 3 x I i k 0 2 P a H V Z i Z n Z t f K C 6 W l p Z X V t f K 6 x v n O o w V 4 1 0 W y l B d e l R z K Q L e B Q G S X0 a K U 9 + T / M K 7 P s 7 8 i x u u t A i D D o w j 7 v p 0 F I i h Y B S M 1 C / X E 2 d y S U + N P D e x q 7 Z t E 0 L 2 M k I a + 7 Y h B w f N G m m m D v B b S E 5 b n V a a 9 s u

(g 2 )µ ± 2 <
r 9 b N 6 5 f B o W k c R b a F t t I s I a q B D d I L a q I s Y u k M P 6 A k 9 W / f W o / V i v X 5 F C 9 Z 0 Z h P 9 g P X + C c w M n q w = < / l a t e x i t > l a t e x i t s h a 1 _ b a s e 6 4 = " w y6 2 E g R m z d k j W t / / P 8 X U 1 W N R O J 8 = " > A A A C J X i c b V D N S 8 M w H E 3 n 1 5 x f V Y 9 e i k O Y o K O t Z Z v g Y e j F 4 w T 3 A W s Z a Z Z 1 Y U l b k l Q Y Z f + M F / 8 V L x 4 c I n j y X z H d d t D N B 4 H H e + + X / P L 8 m B I h T f N L y 6 2 t b 2 x u 5 b c L O 7 t 7 + w f 6 4 V F L R A l H u I k i G v G O D w W m J M R N S S T F n Z h j y H y K 2 / 7 o L v P b T 5 g L E o W P c h x j j 8 E g J A O C o F R S T 7 9 J 3 d k l X R 7 4 X m q W r 2 s V 2 6 l c m G X T r F q 2 l R G 7 6 l w 5 k 1 J w a Z / 3 X J a 4 M b N d Q Q I G J z 2 9 m A U z G K v E W p A i W K D R 0 6 d u P 0 I J w 6 F E F A r R t c x Y e i n k k i C K J w U 3 E T i G a A Q D 3 F U 0 h A w L L 5 0 t O D H O l N I 3 B h F X J 5 T G T P 0 9 k U I m x J j 5 K s m g H I p l L x P / 8 7 q J H N S 8 l I R x I n G I 5 g 8 N E m r I y M g q M / q E Y y T p W B G I O F G 7 G m g I O U R S F V t Q J V j L X 1 4 l L b t s O W X n w S n W b x d 1 5 M E J O A U l Y I E q q I N 7 0 A B N g M A z e A X v Y K q 9 aG / a h / Y 5 j + a 0 x c w x + A P t + w e S V a E I < / l a t e x i t >e + e !µ + µ X(! µ + µ ) < l a t e x i t s h a 1 _ b a s e 6 4 = " O b C b f H Q 5 z k E N 9 B O l 0 L 4 g d T e N b 4 c = " > A A A C S 3 i c d V D L T g I x F O 2 g + M A X 6 t J N I z H R o K S j + G B n d O M S E 1 E S Z i C d c o G G z i N t R 0 M m / J 8 b N + 7 8 C T c u N M a F H W D h 8 y R t Ts 6 5 9 7 b 3 e J H g S h P y Z G W m p r M z s 3 P z u Y X F p e W V / O r a t Q p j y a D G Q h H K u k c V C B 5 A T X M t o B 5 J o L 4 n 4 M b r n 6 f + z S 1 I x c P g S g 8 i c H 3 a D X i H M 6 q N 1 M p 7 i T M a 0 p B d z 0 1 d H y 9 k = " > A A A C M n i c b V D L S g M x F M 3 4 r P V V d e k m W A R F K T N a a W d X 6 k Z x o 2 C 1 0 B l L J k 3 b 0 M y D 5 I 5 a h v k m N 3 6 J 4 E I X i r j 1 I 8 z U L n x d C J y c c 8 / N z f E i w R W Y 5 p M x M T k 1 P T O b m 8 v P L y w u L R d W V i 9 U G E v K G j Q U o W x 6 R D H B A 9 Y A D o I 1 I 8 m I 7 w l 2 6 Q 0 O M / 3 y m k n F w + A c h h F z f d I L e J d T A p p q F 4 4 T Z z S k J X u e m 5 g l u 3 p w s F / d N U u m a e / Z F Q 1 s 2 7 Y q V u o A u 4 W k L t O t + t U O d i T v 9 Y F I G d 7 g E 3 1 v b q f t Q j F z Z Y X / A m s M i m h c p + 3 C g 9 M J a e y z A K g g S r U s M w I 3 I R I 4 F S z N O 7 F i E a E D 0 m M t D Q P i M + U m o 2 1 T v K m Z D u 6 G U p 8 A 8 I j 9 7 k i I r 9 T Q 9 3 S n T 6 C v f m s Z + Z / W i q F b d R M e R D G w g H 4 9 1 I 0 F h h B n + e E O l 4 y C G G p A q O R 6 V 0 z 7 R B I K O u W 8 D s H 6 / e W / 4 G K v Z J V L 5 b N y s V Y f x 5 F D 6 2 g D b S E L V V A N H a F T 1 E A U 3a F H 9 I J e j X v j 2 X g z 3 r 9 a J 4 y x Z w 3 9 K O P j E 7 K n p j s = < / l a t e x i t > e + e !µ + µ X < l a t e x i t s h a 1 _ b a s e 6 4 = " I A 9 G R j 0 w z W j e D 4 J T D g b z

o w n o w 3 4 9
P 4 G r c m j I l n F / 0 p 4 / s H Z R e l / Q = = < / l a t e x i t > ✏ = 0.012 < l a t e x i t s h a 1 _ b a s e 6 4 = " v 2 x E 4 P 2 B H I C j p g n S h h x 8 5 x T + w P n 8 A T k g k h Y = < / l a t e x i t > R L Q b i 4 5 g F I z U K n t 5 c 7 i k o b p h k B P b 9 4 n n V b e I X S W u 6 9 Y M I d t u z X e K J v A b y I / 3 d 9 y i a J U r j k 2 G w I O I 7 + 9 6 I