Rare Kaon Decay to Missing Energy: Implications of the NA62 Result for a $Z^\prime$ Model

Meson decays offer a good opportunity to probe new physics. The rare kaon decay $K^+ \rightarrow \pi^+ \nu\bar{\nu}$ is one of the cleanest of them and, for this reason, is rather sensitive to new physics, in particular, vector mediators. NA62 collaboration, running a fixed-target experiment at CERN, recently reported an unprecedented sensitivity to this decay, namely a branching fraction of $BR(K^+ \rightarrow \pi^+ \nu\bar{\nu}) = (11^{+4.0}_{-3.5})\times 10^{-11}$ at 68\% C.L. Vector mediators that couple to neutrinos may yield a sizeable contribution to this decay. Motivated by the new measurement, we interpret this result in the context of a concrete $Z^\prime$ model, and put our findings into perspective with the correlated $K_L \rightarrow \pi^0 \nu \bar{\nu}$ decay measured by KOTO collaboration, current, and future colliders, namely the High-Luminosity and High-Energy LHC.


I. INTRODUCTION
Mesons have played a key role in the understanding of properties of elementary particles. The introduction of strangeness along with isospin lead us to the eight-fold way, based on the SU(3) flavor symmetry. These theoretical insights have contributed to the discovery of quantum chromodynamics as we know today. Another good example is the famous θ − τ puzzle. Two different decays were found for charged strange mesons known at the time as θ + and τ + . The decay modes had different parities, but the particles were shown to have the same mass and lifetime. It was indeed a puzzling observation. Later, it was realized that weak interactions violate parity, and these two particles were actually the same K + -meson. Additionally, the Glashow-Iliopoulos-Maiani (GIM) mechanism and quark charm surfaced as an explanation of the absence of weak flavor changing neutral currents in the processes such as K + → π + νν. The discovery of CP violation in the K 0 −K 0 system further proved the importance of meson physics for our understanding of nature. Furthermore, meson systems are able to access possible new sources of CP violation that are of paramount importance for explaining the observed baryon-antibaryon asymmetry in our universe [1]. Lastly, the K + rare decay into neutrinos can efficiently probe the presence of heavy vector mediators, beyond the Standard Model (SM), [2][3][4][5][6][7][8][9] via the Feynman diagrams displayed in Fig.1. Figure 1. Feynman diagrams that illustrate how vector mediators can contribute to the K + → π + νν decay. The first diagram requires Z coupling to neutrinos, whereas the second further requires couplings to the top quark.
The meson decay K + → π + νν is a flavor changing neutral current process that occurs in the SM via the box and penguin diagrams, with the latter being dominated by the top quark contribution. Due to the GIM and loop suppression, the SM contribution to this decay is very small, reading BR(K + → π + νν) = (8.4 ± 1.0) × 10 −11 [10], while the NA62 currently imposes BR(K + → π + νν) = 11.0 +4.0 −3.5 × 10 −11 [11,12] (results collected in 2016, 2017 and 2018). Therefore, one can clearly notice that the current sensitivity achieved by NA62 is not far from the SM prediction. Having said that, the NA62 collaboration has been continuously searching for the rare kaon decay [12]. KOTO collaboration has also conducted some searches as well, but offering weaker constraints [13] 1 . To concretely show the relevance of the recent NA62 result, we will put it into perspective other existing limits in a model that features a heavy vector mediator. The model is based on the SU (3) C × SU (3) L × U (1) Y gauge group, known as 3-3-1 model. It is well motivated by the ability to naturally explain the observed replication of the generations, 1 KOTO collaboration has also recently reported the observation of three anomalous events in the K L → π 0 νν. This anomaly requires the branching ratio for this decay mode to be about two orders of magnitude larger than the SM one [4], indicating the presence of a new light and long-lived particle with mass of the order of 100 MeV. There is no such light particle in our model. Hence the KOTO anomaly will be regarded as an statistical fluke. and nicely hosting dark matter candidates [14][15][16][17][18][19][20][21], addressing neutrino masses [22][23][24][25], among other interesting phenomena [3,[26][27][28][29][30][31][32][33]. As a result of the enlarged gauge group, the 3-3-1 model have several new gauge bosons, such as W and Z bosons, which are subject to restrictive constraints rising from collider physics [34][35][36][37], muon anomalous magnetic moment [38][39][40][41], and lepton flavor violation [42,43]. We will investigate the role of the Z gauge boson in the rate K + decay and then use this to draw bounds on the Z mass using the K L → π 0ν ν, K + → π +ν ν decays.
Our work is structured as follows: In Section II, we review the 3-3-1 model and compute the Z couplings necessary to our analyses; in Section III, we discuss the computation of the branching fractions of the kaons in our model; in Section IV, we discuss our results and, finally in Section VI, we draw our conclusions.

II. THE MODEL
The 3-3-1 models are extensions of the standard model and are based on the following local symmetry group: where C corresponds to the color charge, L denotes the left-handed fermions and N is the quantum number of the U(1) group. The general expression for the electric charge operator in these models is written as, where λ 3 = diag(1, −1, 0), λ 8 = diag(1, 1, −2)/ √ 3, and I are the diagonal Gell-Mann matrices and the identity matrix, respectively. We took β = − 1 √ 3 because in our work we choose the model known as 3-3-1 with right-handed neutrinos (3-3-1 r.h.n) [44,45]. However, we highlight that our conclusions are also applicable to the 3-3-1 model with neutral fermions proposed in [14]. The hypercharge in this model is given as, which is identical to the standard model one. The left (L) and right (R)-handed fermionic fields of this model are represented as follows, where a = 1, 2, 3 and i = 1, 2 are the generation indexes, f a L and Q iL , Q 3L represent the left-handed leptonic and quark triplets, respectively. These fields encompass the SM spectrum like neutrinos (ν a = ν e , ν µ , ν τ ), charged leptons a = e, µ, τ , and quarks  1) and (2). Furthermore, the 3-3-1 r.h.n model contains three scalar fields χ, η and ρ in the following representations These scalar triplets in Eq. (6) are responsible for the spontaneous symmetry breaking (SSB) in the model, with the following vacuum (vev) structure, .
We will assume that v χ v η , v ρ , leading to the two-step SSB, The fermion masses rise from the Yukawa Lagrangian that reads, The quark and charged lepton masses are proportional to v = 246 GeV, where v 2 = v 2 ρ + v 2 η similarly to the SM. The fourth term leads to a 3 × 3 antisymmetric neutrino mass matrix [45], which means that the model has one massless and two degenerate neutrino mass eigenstates. Moreover, the gauge bosons W and Z acquire mass term identical to the SM as well. In addition to the SM fields, there are new massive gauge bosons predicted by the model as result of the enlarged gauge symmetry, denoted as Z , V ± and U 0 , U 0 † . The masses of these fields are, The charged (CC) and neutral (NC) currents are found to be, The second and third term in Eq. (9) violate leptonic number and weak isospin [45]. Z 1 and Z 2 are neutral physical gauge bosons, which rise from the Z and Z gauge boson mixtures. a 1R (f ), a 1L (f ), a 2R (f ) and a 2L (f ) are couplings of fermions with the Z 1 and Z 2 bosons. The mixing angle of these bosons is commonly denoted as φ and when φ = 0, the couplings of Z 1 with the leptons and quarks are the same as the boson Z in the SM. Likewise, the couplings of Z 2 in this limiting case should be the same as Z [45]. These couplings for the vertices Table I.
In order to study meson physics in our model, and investigate its connection to the Z boson, we need to extract flavor Table I. Z couplings to neutrinos and the left-handed down-type quarks, considering φ = 0. changing neutral current. To do so, we start by reviewing how the flavor changing terms arise. The quark fields the following standard rotation are, Note that only the left-chiral terms of the Lagrangian (10) lead to the quark flavor violation. The rightchiral quark couplings to Z in Eq. (10) are independent of flavor and therefore are flavor-diagonal in the mass eigenstate basis. We can write these terms in the form, with D L = (d , s , b ) T , and Changing the basis we get, Using the unitarity of the V L matrix, we finally find the coupling between the quarks d and s, which is given by, Analogously for the neutrino-Z coupling we have, In principle, we can vary the entries of the matrix V d L freely since the CKM matrix does not constrain V d L , but the product V d L V u L . So, we choose the following parametrization for the wheres ij = sinθ ij ,c ij = cosθ ij and δ i are the phases, with i, j = 1, 2, 3. For our purposes, the following entries will be important, then, the product which appears in the ∆ sd L coupling is where we leave the product |V d * L32 V d L31 | and the phase δ as free parameters.

III. KAON DECAYS
The rare Kaon decay modes K + → π + νν and K L → π 0 νν are considered golden modes in flavor physics, as they are very well understood theoretically and are sensitive to new physics contributions. In the SM both decays occur only at loop level and are dominated by Z penguin and box diagrams. The corresponding branching ratios have been calculated at a high precision, including NNLO QCD, electroweak corrections and also non-perturbative and isospin breaking effects [47][48][49][50][51].
The decay K + → π + νν is CP conserving, whereas K L → π 0 νν is CP violating. In the 331 model, the new sources of flavor and CP violation which contribute to these decays come from the Z interactions with ordinary quarks and leptons, as discussed above. Although these couplings induce the transitions at tree level, they are suppressed by the large Z mass.
Following the notation of Ref. [52], we can write the branching ratios for the Kaon decay modes K → πνν as, and, In these expressions λ denotes the Cabibbo angle, κ + and κ L are given by, κ + = (5.21 ± 0.025) · 10 −11 λ 0.2252 X eff describes the contribution of short distance physics, where, and, The first term in Eq. (22) represents the SM contribution, which is dominated by Z-penguin and box diagrams, and includes QCD and electroweak corrections. The factor η X is close to unity, η X = 0.994, and The 331 contribution is enclosed in the second term of Eq. (22), with ∆ sd L and ∆ νν L given in the Eqs. (14) and (15), respectively. If Z is absent we have ∆ νν L = ∆ sd L = 0 and the SM result is recovered.
The decays K + → π + νν and K L → π 0 νν are related to each other in the SM, via isospin symmetry, since both transitions are ruled by the same short distance operator. This interdependence leads to the Grossman-Nir limit [53], For a fixed value of BR(K + → π + νν), this theoretical bound provides an upper limit for BR(K L → π 0 νν), which is typically still stronger than the current experimental limits. The bound remains valid in SM extensions in which the new physics is much heavier than the Kaon mass. In particular, the 331 model obeys the bound, as we will show in the next section.
Having presented the formulas that summarizes the predictions of the 331 model regarding meson FCNC processes, we can now discuss the implications of the experimental searches for the flavor violating Kaon decays on the parameter space of the model. δ Figure 2. Branching ratio versus Z mass for kaon long (left panel), BR(KL → π 0 +νν), and for the K + , BR(K + → π + +νν) (right panel). The gray band provides the excluded region by KOTO experiment, and the blue region provides the allowed by the NA62 experiment. The colored bar represents the variation of the δ phase, for details see the text.

IV. RESULTS
In this section, we present our results for the FCNC processes using Eqs. (20) and (21), for the 331 r.h.n model, but we emphasize that our findings are also applicable to the 3-3-1LHN [14], which is a 3-3-1 version that features heavy neutral leptons instead of right-handed neutrinos. Anyway, we compare our results with the last limits from KOTO [54] and, NA62 Run1 (2016 + 2017 + 2018) 2 [11,12]. In Fig. 2, we display the branching for K L (right panel) and K + (left panel) versus the Z mass for the 331 r.h.n model, namely combining the standard model contribution and the new one provided by the new Z boson, as mentioned above. We compute both branching varying the Z mass, the phase δ (color bar) and we fix the product |V d * L32 V d L31 | = 10 −3 . From the left-panel of Fig. 2 one can conclude that the KOTO experiment excludes at most Z masses around 400 GeV, which occurs for δ > 1.4. However, from the right-panel of Fig. 2, we find that Z masses below 3 × 10 3 GeV might be excluded depending on the value adopted for the phase δ. For δ → π/2, the NA62 sensitivity to heavy Z mediators severely weakens. Hence, NA62 yields complementary limits to other existing probes [35,55,56]. We can also see the presence of a dip due to destructive interference between the Z and the SM contributions, where the branching ratio lies bellow the SM predicted value. In the plot it occurs for Z masses in the 600 − 900 GeV range, but in general its depth and location depends on the combination of |V d * L32 V d L31 | and δ. Notice that there is no dip when δ = π/2, while it reaches its maximum depth for δ = 0.
We have drawn quantitative conclusions based on a particular value for V d * L32 V d L31 . To assess how our bounds change for different choices of this product we examine the range10 −4 < V d * L32 V d L31 < 10 −1 . In Fig. 3 we show exclusion plots in the plane |V d * L32 V d L31 | versus m Z , for fixed δ = 0 (leftpanel) and δ = π/2 (right-panel). Here we combined limits from NA62, KOTO, and LHC, High-Luminosity LHC (HL-LHC) and High-Energy-LHC (HE-LHC). The strongest collider bounds on the Z mass stems from the resonant production of Z decaying into dileptons [40].
There is an interesting interplay between collider and flavor physics. While collider bounds rely mostly on the interaction strength between the Z boson and fermions, which is fixed in 3-3-1 models, the flavor bounds weaken with |V d * L32 V d L31 |. The collider bounds displayed in Fig. 3 are conservative, as they take into account the presence of Z decays into right-handed neutrinos and exotic quarks, which can be light enough so that the decays are not kinematically forbidden. The original lower mass bound reads m Z > 4 TeV [55], but those exotic decays were ignored in [55]. If more decay channels become available, then the bound weakens. Here, in order to take into account this uncertainty on the Z decay modes, we assume conservatively a branching fraction of 50% into charged leptons, which leads to the grey region in Fig. 3 in agreement with [41]. We also show the prospects for the HL-LHC, with 3 ab −1 of integrated luminosity, and the HE-LHC, which corresponds to an integrated luminosity of 15 ab −1 at a center-ofmass energy of 27 TeV. These projected collider limits were obtained using the code described in [57], which can be used to forecast lower mass limits for resonance searches, which is precisely our case.
We found that the NA62 bounds from the decay K + → π + νν are rather restrictive, providing stronger limits than the ones from colliders over a significant region of the parameter space. Looking at left-panel of Fig.3  if |V d * L32 V d L31 | is of the order 10 −2 or larger. In the same vein, NA62 enforces the product |V d * L32 V d L31 | to remain below ∼ 10 −3 , when m Z ∼ 2 TeV.
On the other hand, these parameters are completely unconstrained by KOTO when δ = 0, since the contribution from Z to the decay K L → π 0 νν vanishes in this case. In the absence of new CP violating sources, the BR(K L → π 0 νν) takes the same value as in the SM, since this process will occur only through the contribution to CP violation from the SM via CKM matrix.
This can be easily understood with Eq. (21), from which we see that BR(K L → π 0 νν) depends only on the imaginary part of X ef f , in particular, on Im(V * ts V td ) when δ = 0. Still in the left-panel of Fig. 3 we notice that in the range shown, the Grossman-Nir limit does not appear, because the suppressed values of BR(K L → π 0 νν) makes this bound easily satisfied for practically any reasonable value of |V d * L32 V d L31 |. However, as δ increases from 0 to π/2, this bound becomes more relevant, likewise the KOTO exclusion region gradually grows, reflecting the enhancement in the K L → π 0 νν decay, while the NA62 exclusion region slightly decreases. Nevertheless, even with maximum enhancement at δ = π/2 (right plot), the KOTO bounds remain less constraining compared to NA62 and the Grossman-Nir limit.

V. DISCUSSION
Our conclusions relied on the presence of flavor changing interactions involving the Z boson, but as the model features a large scalar sector, that could be new sources of flavor changing interactions rising from the heavy scalar fields. These new contributions have been shown to be subdominant. Thus can be safely ignored. Moreover, the entire reasoning was based on the 3-3-1 model with right-handed neutrinos, but our results are also applicable to the 3-3-1 model where the right-handed neutrinos are replaced by heavy neutral fields. This occurs because these models have the same neutral current. In summary, our finding are relevant for two different models and solid irrespective of the presence of scalar fields.

VI. CONCLUSION
In this work we explored the FCNC processes mediated by a Z gauge boson featuring both in the 3-3-1 r.h.n and 3-3-1LHN models. We computed the K + and K L decay rates to missing energy, considering the extra contributions from the Z in addition to the SM contribution, leaving the quark mixing matrix and the Z mass as free parameters. We performed a complementary analysis using the results from NA62, KOTO, and the LHC (current and prospects) to set bounds on the 331 r.h.n parameters. We found that the last result of the NA62 experiment was able to constrain a large region of the parameter space, setting lower limits on the Z mass that can be more stringent than those from dilepton searches at the LHC. For example, we can impose m Z > 10 TeV for |V * L32 V L31 | ∼ 10 −1 , while |V * L32 V L31 | < ∼ 2 × 10 −3 for m Z = 3 TeV. These results apply for δ = 0, where the sensitivity of NA62 is maximum. In the case when the new CP violation effects are large our constraints weaken.