Heavy neutral leptons from kaons in effective field theory

: In the framework of the low-energy effective theory containing in addition to the Standard Model fields heavy neutral leptons (HNLs), we compute the decay rates of neutral and charged kaons into HNLs. We consider both lepton-number-conserving and lepton-number-violating four-fermion operators, taking into account also the contribution of active-heavy neutrino mixing. Assuming that the produced HNLs are long-lived, we perform simulations and calculate the sensitivities of future long-lived-particle (LLP) detectors at the high-luminosity LHC to the considered scenario. When applicable, we also recast the existing bounds on the minimal mixing case obtained by NA62, T2K, and PS191. Our findings show that for some of the effective operators considered the future LLP detectors, in particular, MATHUSLA

LLPs are present in many models that can account for unresolved problems in particle physics and cosmology, such as the mechanism of neutrino mass generation and the nature of dark matter.The main focus of phenomenological studies so far has been on various renormalizable models, including the Higgs portal, the neutrino portal, the dark photon portal, as well as on non-renormalizable models featuring long-lived axion-like particles (ALPs), see e.g.Refs.[3,19] for reviews and references.However, it is plausible that in addition to the renormalizable couplings, LLPs may interact with the Standard Model (SM) via effective, non-renormalizable interactions.In this case, the SM viewed as an effective field theory (EFT) has to be extended to include LLPs and their effective interactions.One such example is the N R SMEFT [20][21][22][23], which assumes the existence of heavy neutral leptons (HNLs), N , with masses below or around the electroweak scale, v. Recently, a dictionary of tree-level UV completions of the dimensionsix and dimension-seven operators with N R has been provided in Ref. [24].
There are two basic ways of HNL production at the LHC: (i) directly from partonic collisions, and (ii) for m N ≤ (few) GeV, in decays of mesons, which in turn are copiously produced in pp collisions.The latter way is also the dominant production channel of the HNLs at the DUNE experiment.Long-lived HNLs produced in the first way via the four-fermion operators with two quarks and either two or one N R have been studied in Ref. [25] and Ref. [26], respectively.These two sets of effective operators lead to distinct phenomenology.The pair-N R operators may enhance the production cross section while not contributing to the decay of the lightest HNL.This makes the far LLP detectors introduced above an ideal place to probe such effective interactions [25].On the contrary, single-N R operators, if large enough to enhance the HNL production, will also induce HNL decays.Still, displaced-vertex searches at the local LHC detectors (ATLAS and CMS) are very efficient to probe this set of effective operators [26].The reach of the LLP detectors to the neutrino dipole operator involving an active neutrino and N R has been recently estimated in Ref. [27].Instead, Ref. [28] has revisited the LEP limits on the dimension-five sterile neutrino dipole operator (existing for at least two generations of HNLs, N 1 and N 2 ), taking into account active-heavy neutrino mixing.
Meson decays into long-lived HNLs triggered by non-renormalizable interactions have been considered in Refs.[29][30][31][32][33].The EFT appropriate for the description of meson decays is N R LEFT [34][35][36][37], the low-energy theory where the top quark, the Higgs boson and the heavy SU(2) L gauge bosons are not present as dynamical degrees of freedom.In Ref. [29], the authors have investigated the reach of the proposed LHC far detectors to HNLs produced in the decays of D-and B-mesons via single-N R operators with a charged lepton, demonstrating that the new physics scale, Λ, as high as 100 TeV could be probed by these detectors.The same set of single-N R operators, but with the τ lepton, would lead to HNL production in τ decays at Belle II [38] (see also Ref. [39]).Pair-N R operators triggering D-and B-meson decays have been thoroughly examined in Ref. [30].It has been shown that for certain operators Λ as large as 300 TeV and active-heavy neutrino mixing squared, |U eN | 2 , as small as 10 −15 could be tested by MATHUSLA, with the reach of ANUBIS being only a factor of a few smaller.In Ref. [31], the sensitivities of FASER2, FACET, ANUBIS, CODEX-b, and MAPP, to the dimension-five sterile neutrino dipole operator have been estimated.This operator leads to two-body decays of vector mesons into N 1 and N 2 mediated by a photon, γ, and a subsequent decay of the heavier HNL N 2 → N 1 γ.Similarly, the sensitivity reach of FASER2 and FACET to the dipole operator coupling an HNL to the photon has recently been investigated in Ref. [32] for the HNLs produced from meson decays.Furthermore, Ref. [33] studied the sensitivity reach of the DUNE-ND (as well as the LHC far detectors) to the HNLs produced from decays of mesons including kaons, both in the minimal scenario and in the EFT, emphasizing on the feasibility of using the N R SMEFT to describe not only the HNLs but also the lightest neutralinos in the R-parity-violating supersymmetry.
In the present work, we perform state-of-the-art numerical simulations and derive the sensitivities of the future LLP detectors at the LHC as well as the DUNE-ND1 to HNLs produced from neutral and charged kaon decays in the N R LEFT. 2 First, we study the scenario in which the HNL production is induced by pair-N R operators, while the HNL decay proceeds via active-heavy neutrino mixing U eN .Second, we investigate the case in which both HNL production and decay are induced by the same single-N R operator structure, but with different quark flavor indices.Finally, we consider the situation where the production proceeds via a single-N R operator as well as via U eN , while the N decays via mixing only.We consider both lepton-number-conserving (LNC) and lepton-numberviolating (LNV) operators and discuss the differences.
HNLs can also mediate meson decays.In particular, LNV decays K ∓ → π ± ℓ ∓ ℓ ∓ mediated by light sterile neutrinos have been studied in Ref. [45] adopting the EFT approach.We also mention Ref. [46].Here, the authors have derived limits on EFT operators from HNL searches in kaon decays (among others).
The remainder of the paper is organized as follows.In Sec. 2, we briefly recap neutral kaon mixing, summarize the N R LEFT operators of interest, and then discuss the HNL production from kaon decays.In Sec. 3, we describe various experiments we consider and provide the details of numerical simulations.Sec. 4 contains the derived sensitivities and the relevant discussion.Finally, we provide summary and conclusions of our work in Sec. 5. Additionally, we add two appendices at the end of the paper which explain in detail the computation of the decay widths of the kaons into the HNLs and those of the HNLs into a charged lepton and a pion, in the EFT framework.
2 HNLs from kaons in effective field theory

The neutral kaon system
The neutral kaons K 0 (ds) and K 0 (s d) are flavor eigenstates that can be produced in strong interactions.Weak interactions cause the mixing between these two neutral states.If CP were a symmetry of the total hamiltonian H (including strong, electromagnetic, and weak interactions), CP eigenstates would also be eigenstates of H.With the convention (see e.g.Ref. [47]) Table 1.Four-fermion operators in N R LEFT, involving two quarks and two N R 's, assuming one generation of N R .LNV operator structures require "+h.c.".In the third column, we provide the number of independent real parameters, N pars , associated with each operator structure.
we can define the CP eigenstates as: where the former has CP = +1 and the latter CP = −1.However, CP is mildly violated by weak interactions and therefore the mass eigenstates |K S ⟩ and |K L ⟩, characterized by definite lifetimes, are different from |K 1 ⟩ and |K 2 ⟩: where ε is the parameter accounting for indirect CP violation in neutral kaon decays.K S (K L ) denotes the neutral kaon with the shorter (longer) lifetime.Experimentally, |ε| ≈ 2.23 × 10 −3 [48], and for the purposes of this work we can safely neglect it.Thus, in what follows we assume |K S ⟩ ≈ |K 1 ⟩ and |K L ⟩ ≈ |K 2 ⟩.

Effective field theory with right-handed neutrinos
We will work in the framework of the low-energy effective field theory extended with righthanded neutrinos, N R , dubbed as N R LEFT, see e.g.[34][35][36][37].We assume N R to be a Majorana particle and allow for both lepton-number-conserving (LNC) and lepton-numberviolating (LNV) operators.Charm and bottom meson decays triggered by four-fermion effective operators with N R have been studied in detail in Refs.[29,30].Here, we are interested in the decays of kaons induced by four-fermion interactions.These interactions can be grouped into pair-N R operators given in table 1 and single-N R operators provided in table 2. 3 Since the top quark is not in the spectrum of the N R LEFT, we have n u = 2 and n d = n e = n ν = 3, with n f denoting the number of generations of a fermion f .In addition, we assume one generation of N R .
Name Structure Generically, N R mixes with the active neutrinos at the renormalizable level.Integrating out the W boson leads to the following contribution to the effective Lagrangian: where G F is the Fermi constant, V is the CKM quark mixing matrix, and U ℓN is activeheavy neutrino mixing.For simplicity, we will assume that the HNL mixes with the electron neutrino only, and consider the single-N R operators with the first-generation leptons only. 4or charged kaon decays, the relevant CKM matrix element is V us .In what follows, we will separate this contribution from the corresponding operator in table 2 by denoting its Wilson coefficient (WC) as For a more detailed discussion of N R LEFT, including the running of the considered operators and their matching to the N R SMEFT operators, see Ref. [30].

HNL production in kaon decays
The pair-N R operators O dN from table 1 involving d and s quarks trigger the following two-and three-body decays: In appendix A, we provide the formulae for the twobody decay widths and the three-body decay amplitudes.The three-body decay widths are then computed according to the procedure explained in Refs.[29,30].
π 0 e + ν e , K S/L → π ± e ∓ ν e .The branching ratios of the decays into final states with at least one charged particle have been measured, whereas for K L → π 0 νν a stringent upper limit on the branching ratio has been obtained.We summarize the current experimental results in table 3. We will take them into account when deriving the limits on the WCs of the pair-N R and single-N R operators.In the case of a measured branching ratio, we will require the new contribution not to exceed twice the experimental error, whereas for K L → π 0 N N , we will demand that its branching ratio is smaller than 3.0 × 10 −9 in accordance with the current upper bound on the branching ratio of K L → π 0 ν ν [48].
In figure 1, we display the branching ratios of kaon decays triggered by the LNC operator O V,RR dN,21 and by the LNV operator O S,RR dN,21 .In each case, we assume that the corresponding WC is either real (left panel) or purely imaginary (right panel).For the LNC operator, a real (purely imaginary) WC does not allow for K S → N N and K L → π 0 N N (K L → N N and K S → π 0 N N ), as can be inferred from eqs. (A.15) and (A.22) (eqs.(A.16) and (A.21)). 5For small m N , the allowed two-body decay is suppressed, since the corresponding decay width scales with m 2 N , see eqs.(A.15) and (A.16).In this figure, we fix the absolute value of the operator coefficient to 10 −6 v −2 in order to comply with the measurements reported in the left panel of table 3, in particular, with that of ) is such that it is below the existing constraints.For HNLs lighter than the kaons, the leading constraints on the mixing parameter come from NA62 [49], PS191 [50], and T2K [51], which set upper limits at the level of |U eN | 2 ∼ 5 × 10 −10 .For even lighter HNLs, below the pion mass, PIENU [52] has ruled out parameter space corresponding to |U eN | 2 > 10 −7 ∼ 10 −8 .Moreover, with the chosen value for the mixing, its contribution is comparable to the contributions of most operators for c O ∼ 10 −5 v −2 , cf. eq.(2.5).For the scalar operator O S,RR udeN,12 , the constraints from NA62 set limits on the operator to c O ∼ 10 −6 v −2 .
In figure 3, we show the branching ratios for the same processes as in figure 2, but for the LNV operators O V,RL udeN,12 , O S,RL udeN,12 , and O T,LL udeN,12 .In the absence of mixing, the results are identical to those for the corresponding LNC operators (switched on one at a time).In the presence of mixing, there is an interference between the effective operator generated by new physics and the four-fermion interaction (see eq. (2.4)) arising from integrating out the W boson.Its effect is more pronounced for the vector-type operators, and it is stronger for the LNV operator, as can be understood from eq. (A.17).

Experiments and simulation
A whole list of far detectors have been proposed for operation in the vicinity of various interaction points (IPs) of the LHC, either during Run 3 or the HL-LHC phase; some of them have even been approved and are running, including FASER and MoEDAL-MAPP1.Since they are all sensitive to signatures of tracks stemming from displaced decays of LLPs taking place inside their fiducial volumes, we will perform the numerical analysis taking into account these proposals comprehensively.
We classify these detectors according to their associated IP at the LHC.For the ATLAS IP, ANUBIS [8], FASER [4,5], and FASER2 [3] are relevant.ANUBIS6 is a detector proposed to be installed inside one of the service shaft above the ATLAS IP.It has a cylindrical shape of 56 m height and 18 m diameter.Being close to the ATLAS IP, it is expected to suffer from certain background sources such as neutral kaons.Nevertheless, exclusion bounds at 95% confidence level (C.L.) are usually shown with 3 signal events, assuming zero background, and we will follow the practice in this paper.FASER is currently collecting data at the LHC; first results can be found in Refs.[55,56].It is a small cylindrical detector installed in the very forward position along the beam direction, with a distance of 480 m from the IP.Further, an upgraded program of FASER, FASER2, has been suggested to be built at the site of the proposed Forward Physics Facility [3].It is larger than FASER and has hence better acceptances to LLPs.ANUBIS and FASER2 should collect in total 3 ab −1 integrated luminosity data, while FASER will have order of 150 fb −1 integrated luminosity.
Near the CMS IP, MATHUSLA [1,11,12] and FACET [10] have been proposed.MATHUSLA would be an experiment on the ground, with about 100 m distance from the CMS IP.It would have a huge effective volume of 100 m × 100 m × 25 m.FACET is suggested to be a sub-system of the CMS experiment; with a cylindrical shape, it has a distance of 101 m from the CMS IP, enclosing the beam pipe.Moreover, with a radius of 0.5 m and a length of 18 m, FACET is relatively large compared to FASER.Both MATHUSLA and FACET would be running during the HL-LHC and will hence have an integrated luminosity of 3 ab −1 .
Finally, for the LHCb IP, we have CODEX-b [9], and MoEDAL-MAPP1 and MoEDAL-MAPP2 [6,7].CODEX-b has been proposed as a cubic box with a dimension of 10 m × 10 m × 10 m covering the pseudorapidity range [0.2, 0.6], roughly 10 m away from IP8. MoEDAL-MAPP1 is a small detector of about 140 m 3 in a gallery of negative pseudorapidity with respect to the LHCb IP, under operation during Run 3, and MoEDAL-MAPP2 is an enlarged version of MAPP1, to be running during the HL-LHC period.The integrated luminosity associated with the LHCb IP is lower compared to that of ATLAS or CMS.CODEX-b and MoEDAL-MAPP2 will have an integrated luminosity of 300 fb −1 while MoEDAL-MAPP1 has only 30 fb −1 .
For a more detailed summary of these proposals, we refer the reader to e.g.Refs.[2,29,57], as well as to the respective references proposing the detectors.
In addition, at DUNE, a proton beam of energy 120 GeV hits a fixed target with 1.1 × 10 21 POTs (protons on target) expected per year.The produced mesons decay either promptly or with a macroscopic distance into the LLPs (which are the HNLs in this study) inside a decay pipe which is 26 m downstream from the IP and has a length of 194 m and a radius of 2 m.The produced LLPs should then travel a long distance before decaying inside the DUNE-ND which is further downstream with a distance of 574 m from the fixed target, and has a length of 6.4 m and a width of 3.5 m.We take an operation duration of 10 years as benchmark in this study, expecting thus in total 1.1 × 10 22 POTs.
In order to obtain the kinematics of the HNLs produced from kaons, we make use of the tool Pythia8 with the module SoftQCD:all.The kaons are generated in pp collisions with a center-of-mass-energy 14 TeV, and are set to decay exclusively in the signal-event channels.Pythia8 provides the boost factor and the polar angle of each simulated HNL.We note that since kaons are themselves long-lived, we let Pythia8 decide the decay positions of the kaons and thus take into account the production position of the HNL (= the decay position of the kaon) as well as the polar angle and boost factor of the HNL, in the computation of the HNLs' average decay probability inside the far detectors.The total signal-event rates at each detector can be computed with the following formula where the summation goes over different kaons, N K is the total number of kaons, n = 1, 2 is the number of HNLs produced in the considered kaon decays, BR(N → visible) is the visible decay branching ratio of the HNL, and ϵ is the average decay probability in a far detector.For the LHC experiments, we estimate the number of kaons N K , with the help of the tool EPOS LHC [58] provided in the CRMC simulation package [59], to be N K ± = 2.38 × 10 18 , N K S = 1.31 × 10 18 , and N K L = 1.30 × 10 18 , over the whole 4π solid angle.For the DUNE experiment, we follow Refs.[33,60] and conclude that the total numbers of kaons produced at DUNE for 10 years are N K ± = N K S = N K L = 5.76 × 10 21 over the whole 4π solid-angle range.The computation procedure of ϵ is based on the exact formulae given in Refs.[29,30,[61][62][63] and the further development presented in Ref. [33] for LLPs from kaons.As the kaons, especially K ± and K L , travel macroscopic distances, the infrastructure surrounding the IPs may affect the kinematics of the kaons.
To simplify the analysis, we neglect the influence of any magnetic fields present at the LHC IPs or the magnetic horns at DUNE.Additionally, we introduce cut-offs for the production positions of the HNL at the LHC that are included in the signal-event rates N S , resulting in conservative estimates.Here, we provide a brief summary of the cut-offs we employ in our simulation; for more detail, see Ref. [33].For example, a lead shield covering the total fiducial volume in order to veto neutral background events is placed approximately 5 m in front of CODEX-b.Hence, we require the kaons to decay before reaching the shield.
For detectors in the far forward region (FASER, FASER2, and FACET), we employ the beamline geometry of the ATLAS and CMS IP, which involve absorbers for charged and neutral particles in order to protect beamline infrastructure.The hadron calorimetry of ATLAS and CMS restricts the decay region of the kaons for ANUBIS and MATHUSLA, respectively.Lastly, for MoEDAL-MAPP1 and MAPP2, the natural rock between the IP and the detectors is the limiting factor.Hence, the kaons are required to decay within the 3.8 m wide beam cavern.We note that such cut-offs are not needed for the DUNE-ND, as a decay pipe for the long-lived mesons to decay in should be instrumented.Finally, we discuss the procedure we apply for recasting the bounds on the HNLs in the minimal scenario, obtained in some past searches, into those on the HNLs in the EFT scenarios considered here.In general, we follow the approaches laid out in Refs.[46,64,65] (see also Ref. [66]).We consider three searches at NA62 [49], PS191 [50], and T2K [51].Since these searches all require a prompt charged lepton, they are applicable only to the single-N R scenarios.We first consider the NA62 search, which looked for HNL production in K + decays to positrons and missing energy, assuming the proper lifetime of N is larger than 50 ns.The search obtained bounds on BR(K + → e + N ) and hence those on the active-sterile neutrino mixing, as functions of the sterile neutrino mass.We simply convert BR(K + → e + N ) to the production Wilson coefficient of the single-N R scenario in question, for each mass value, and obtain the corresponding recast bounds.Both the PS191 and T2K searches are for both a prompt charged lepton and a displaced vertex at the detached detector.For PS191, we extract the sensitivity curve presented in the plane |U eN | 2 vs. m N for the signal process K + → e + N , N → e − π + and its charge-conjugate channel, and for the T2K near detector ND280, the signal process K ± → e ± N , N → e ± π ∓ is considered.By rescaling the production and visible decay rates, we obtain the recast bounds in the EFT parameter space.

Results
For presenting the numerical results, we choose several benchmarks characterized by the different couplings responsible for the HNL production and decay.We summarize them in table 4. In total, we consider four scenarios with pair-N R operators (benchmarks B1 and B2, each having two sub-cases) and six scenarios with single-N R operators (benchmarks B3-B8).
For benchmark B1 (B2), HNL production is governed by the LNC (LNV) pair-N R operator O V,RR dN,21 (O S,RR dN,21 ), while HNL decay proceeds via active-heavy neutrino mixing U eN .For each of these two operators, we consider two cases: (i) real WC and (ii) purely imaginary WC, cf.figure 1 and the related discussion.First, in figure 4, we fix the absolute value of the WC to 10 −6 v −2 (to respect the constraint coming from K + → π + νν) and display the projected exclusion limits in the plane |U eN | 2 vs. m N for three signal events.The curves correspond to 95% C.L. limits (under the assumption of zero background).For the LNC operator, the sensitivities are approximately two orders of magnitude weaker in the case of purely imaginary WC (B1.2, top-right plot) than in the case of real WC of the same size (B1.1, top-left plot).This can be understood from figure 1, which shows that for the same size of the WC, BR(K S → N N ) is about two orders of magnitude smaller than BR(K L → N N ), whereas the production numbers of K S and K L are nearly the same.The difference in the sensitivities is much milder for the LNV operator, for which the branching ratios of the relevant kaon decays differ only slightly between the cases of real and purely imaginary WC.
Among the considered LLP detectors, the best limit comes from MATHUSLA, which can probe |U eN | 2 down to 4 × 10 −8 (1.4 × 10 −9 ) at m N ≈ 0.23 GeV for B1.1 (B2.1).It is followed by ANUBIS, which has a factor of a few weaker sensitivity.MAPP2, FACET and CODEX-b provide very similar exclusion limits, which are approximately two orders of magnitude weaker than the expected limits from MATHUSLA.Finally, FASER2 and  MAPP1 have the weakest depicted sensitivities. 7However, all the LHC far detectors are incomparable to the DUNE-ND which can probe |U eN | 2 down to the levels stronger than MATHUSLA by up to two orders of magnitude in these scenarios.This superior performance is mainly due to the much larger production rates of the kaons at DUNE.We also show the existing limits on active-heavy neutrino mixing obtained by the NA62 [49], T2K [51], and PS191 [50] experiments.Though derived under the hypothesis of the minimal mixing case, these limits apply to the considered EFT scenarios as well.As can be seen, NA62 outperforms all the LHC far detectors and already touches the (naive) type-I seesaw band, where the values of m N and U eN yield the light neutrino mass m ν = 0.05-0.12eV.We find that the DUNE-ND can still be sensitive to parameter space beyond the current bounds, especially in the scenarios of the benchmark B2, where it covers the type-I seesaw band.
In figure 5, we fix |U eN | 2 = 10 −10 and show the exclusion limits in the plane WC vs. m N .For benchmark B1.1 (B1.2),MATHUSLA will be able to probe the WC as small as 2 × 10 −5 v −2 (2 × 10 −4 v −2 ) for m N in the range 0.22-0.24GeV.These numbers translate to the new physics scale Λ of 55 (17) TeV.We also show the constraint originating from the measurement of the branching ratio of K + → π + νν.It is complementary to the projected limits for m N ≲ 0.16-0.17GeV.We recall that the NA62, PS191, and T2K exclusion limits on active-heavy mixing cannot be reinterpreted into the limits on the WCs of the pair-N R operators, since these interactions do not lead to a prompt charged lepton.Next, we consider the single-N R operator benchmarks summarized in the right part of table 4. We start with benchmarks B3 and B4, for which we assume that both HNL production and decay proceed via the same effective operator, but carrying different quark flavor indices.The indices 12 lead to the HNL production in kaon decays, while the indices 11 realize the decay N → e ∓ π ± .For these benchmarks, we also assume that there is no active-heavy neutrino mixing.We consider two single-N R operators O V,RR udeN and O S,RR udeN , both conserving the lepton number.(The results for the corresponding LNV operators are the same under the assumption of zero active-heavy neutrino mixing.)For graphical presentation, we choose to set the production and decay couplings equal, i.e. c V,RR udeN,12 = c V,RR udeN,11 for B3, and c S,RR udeN,12 = c S,RR udeN,11 for B4, and show the derived exclusion limits in the plane WC vs. m N , see figure 6.In addition to the sensitivities of the proposed LLP detectors at the LHC and the DUNE-ND, we show the recast bounds from NA62 [49], T2K [51], and PS191 [50], obtained according to the procedure explained at the end of Sec. 3. Except for FASER, of which the sensitivity is comparable to that of NA62, all the far detectors at the LHC and the DUNE-ND will have better reach to these scenarios than NA62, which excludes WC values larger than approximately (1-2)×10 −5 v −2 ((9-20)×10 −7 v −2 ) for benchmark B3 (B4).For the vector-type operator and m N ≈ 0.35-0.40GeV, DUNE-ND and MATHUSLA will probe the effective couplings as small as 4 × 10 −8 v −2 and 9.5 × 10 −8 v −2 , respectively, and FASER down to 1.4 × 10 −5 v −2 , with the sensitivities of the other experiments lying between these extremes.Translating these numbers to the new-physics scale Λ, we find that DUNE (MATHUSLA, FASER) will be sensitive to Λ as high as 1230 (798, 65) TeV.For the scalar-type operator, the reach in the N R LEFT WC is around one order of magnitude better, owing to the larger branching ratio of K ± → e ± N in this case, see figure 2. The new-physics scales, which could be probed by DUNE (MATHUSLA, FASER) for m N ≈ 0.25-0.35GeV, are in excess of 3000 (2000,200) TeV.The projected limits from DUNE and MATHUSLA are approximately more than ten times more stringent than the limits derived on the effective interactions in Ref. [45] from the LNV decays K ∓ → π ± ℓ ∓ ℓ ∓ mediated by N .
Finally, we turn to benchmarks B5-B8, where both an effective operator and activeheavy neutrino mixing contribute to HNL production, while the HNL decay proceeds via mixing only, see table 4. Here, we consider both LNC vector (B5) and scalar (B6) operators, and LNV vector (B7) and scalar (B8) operators, since the interference between the effective interaction and the mixing term is slightly different in the LNC and LNV cases, cf.figures 2 and 3.In figure 7, we fix the corresponding WC to either 10 −5 v −2 for vector operators (to satisfy the constraint coming from K + → e + ν e ), or 10 We also display the recast bounds from NA62, T2K, and PS191.The NA62 limit is more stringent than the expected limits from the future LLP detectors for m N ≲ 0.35 (0.41) GeV in B5 and B8 (B6 and B7).For larger m N , MATHUSLA takes over, excluding new parts of the parameter space.In particular, for B7, the projected MATH-USLA exclusion limits can probe the seesaw target region for 0.25 GeV ≲ m N ≲ 0.45 GeV.DUNE can, however, probe |U eN | 2 values smaller than the current bounds by up to about 2 orders of magnitude, in B5, B7, and B8 benchmarks.For B6, (only) DUNE can exclude a small part of the parameter space at m N ≳ 0.4 GeV.
In figure 8, we fix |U eN | 2 = 10 −10 and show the exclusion limits in the plane WC vs. m N .We again depict recast bounds from NA62, T2K, and PS191.The recast NA62 bound covers the ranges which will be accessible to the future LLP detectors for m N ≲ 0.45 GeV.For larger HNL masses, MATHUSLA will probe an unexplored region of the parameter space, ruling out the WC ≳ 2 × 10 −5 v −2 (≳ 3 × 10 −6 v −2 ) for the vector-type (scalar-  type) operators.On the other hand, the DUNE-ND can exclude new parameter space for m N ≳ 0.25 (0.35) GeV in benchmarks B5, B7, and B8 (B6), showing again the much better constraining power than the future LLP far detectors.In the plots for the benchmarks B5, B7, and B8, we observe a unique funnel feature in the DUNE sensitivity reach for m N roughly between 0.36 GeV and 0.4 GeV.This arises from the fact that in this mass range with |U eN | 2 = 10 −10 , even in the absence of the effective operators considered, more than three signal events are predicted at DUNE (see figure 6 of Ref. [33]), 9 as long as the interference between the minimal mixing and the EFT operators for the HNL production is constructive.Indeed, the funnel feature of the DUNE sensitivities do not appear in B6 (the upper right plot), exactly as a result of the destructive interference in this case.this effect better, we choose to show, in all these four plots, an additional sensitivity curve for DUNE corresponding to |U eN | 2 = 8 × 10 −11 which is slightly smaller than the default 10 −10 value we have chosen.We now observe that the funnel feature in B5, B7, and B8 has disappeared, and in the plot for B6, we find the two DUNE curves for |U eN | 2 = 10 −10 and 8 × 10 −11 almost completely overlap as a result of the two values' closeness.We further see, that the NA62 constraints for B6 and B8 exclude WC ≳ 10 −6 v −2 , justifying the WC choice in figure 7. Overall, for benchmarks B5-B8, the future far detectors will access new parameter space for larger HNL masses, while DUNE shows much more promising sensitivities even for m N as low as 0.25 GeV.

Summary
In this work, we have studied the potential of present and future far-detector experiments at the LHC and the beam-dump-type experiment DUNE, for probing long-lived heavy neutral leptons (HNLs) of Majorana nature produced from rare kaon decays, in the theoretical framework of low-energy effective field theory extended with sterile neutrinos (N R LEFT).We have focused on dimension-6 effective operators consisting of a pair of quarks together with a charged lepton and an HNL, or a pair of HNLs.Besides the effective operators, we also take into account the minimal mixing parameter between the HNLs and the standardmodel active neutrinos.For simplicity, we assume in this work that there is only one kinematically relevant HNL, that the HNL mixes with the electron neutrino only, and that for the effective operators also only the first-generation leptons are considered.
There are both lepton-number-conserving and lepton-number-violating operators; we have investigated both of them and elaborated on their differences.We have computed the kaons' decay branching ratios into the HNLs with the considered effective operators, as a function of the HNL mass, the effective couplings, and the mixing parameter U eN .In addition, the decay rates of the HNLs in the N R LEFT are calculated with care, including the interference between the EFT operators and the minimal mixing contributions.We have further performed detailed Monte-Carlo simulations, in order to determine the acceptance of the LHC far detectors and the DUNE-ND for the long-lived HNLs.The LHC experiments include ANUBIS, CODEX-b, FASER and FASER2, FACET, MATHUSLA, and MoeDAL-MAPP1 and MAPP2.In particular, because of the long-lifetime nature of the kaons (K ± , K S , and K L ), we cannot assume they decay essentially at the IPs; we have thus taken into account their decay positions in the simulation when we compute the decay probability of the HNLs in the detector fiducial volumes.Moreover, for the various experiments, we have properly placed a cut-off position for each LHC far detector beyond which the kaons are vetoed.
For a series of benchmark scenarios classified by the number of HNLs in the operators as well as the Lorentz structure of the operators, we have obtained numerical results.Besides the projection for the considered experiments, we have recast existing bounds on the HNLs in the minimal mixing scenario into those on the HNLs in the considered EFT benchmarks.We find that for the pair-N R scenarios, the existing bounds from NA62 are already so strong that it has excluded all the parameter space that could be probed by future LHC far-detector experiments, but DUNE can still be sensitive to new parameter space in the case of scalar-type operators (benchmarks B2.1 and B2.2).On the other hand, for the single-N R benchmarks, the studied future experiments are sensitive to regions of parameter space currently unexcluded.Particularly, for benchmarks B3 and B4, the projected limits on the effective couplings can be orders of magnitude stronger than the existing bounds.In all these benchmarks, we find the projected sensitivities for DUNE-ND are stronger than those for the LHC far detectors by various degrees in different benchmarks, mainly in virtue of the much larger production rates of the kaons at DUNE.To summarize, our findings in this work show that for long-lived HNLs in the EFT framework produced from kaons, the DUNE-ND is expected to have sensitivities much more promising than the present and future LHC far detectors.Nevertheless, the LHC far detectors can probe unexcluded parameter space in some scenarios, motivating their construction and operation during the HL phase of the LHC.

Parameter
Central value 0.9706 0.02422 0.1198 0.0398 0.68 1.10 relevant matrix elements read .11)where P = p K + p π is the sum of the kaon and pion 4-momenta, and q = p K − p π is their difference.The three form factors (f + , f 0 , B T ) can be parameterized in terms of q 2 as follows (see Ref. [69] for more detail) f + (q 2 ) = f + (0) + Λ + q 2 m 2 π , (A.12) B T (q 2 ) = B T (0) 1 − s Kπ T q 2 , (A. 14) and the numerical values of the parameters entering these expressions are reported in table 5.

A.2 Two-body decays
The partial decay widths of K S/L → N N mediated by the pair-N R operators O dN given in table 1 read:  We do not provide the full expression for the modulus square of amplitude (A.25) summed over spins, since it is rather cumbersome.However, it is straightforward to obtain it, especially, assuming only one operator (either V, S, or T) at a time.In the limit of zero mixing (c mix = 0), as in the case of two-body decay in eq.(A.17), the LNV single-N R operators (switched on one at a time) lead to the same results as their LNC counterparts.Finally, the three-body decay widths are computed following the procedure explained in Refs.[29,30].

B HNL decays
Effective operators in N R LEFT not only enhance the HNL production but also trigger their decay.Assuming only one generation of N R , the pair-N R operators in table 1 cannot make the HNL decay, whereas the single-N R operators in table 2 do contribute to it.
In this appendix we provide the partial decay width of N → ℓ − π + , which is the only kinematically allowed channel if HNLs are produced in kaon decays and if m N > m π + m ℓ .
In the computation we take into account the contribution from (i) the O udeN operators in table 2, (ii) eq.(2.4), i.e. the standard mixing to active neutrinos, and (iii) the interference terms between (i) and (ii).The final expression reads The previous result can be derived from the amplitude leading to eq. (A.17) by substituting K with π and perfomring the interchange p π ↔ p N along the computation.The decay constants are given by f π = 130.2 MeV [68] and f S π = m 2 π mu+m d f π .We have also used the notation introduced in appendix A for the WC, and c mix is the coefficient defined in eq.(2.5).
In the scenarios outlined in table 4, the decay of the HNL is governed by either the neutrino mixing parameter (scenarios B1-B2 and B5-B8) or one of the WC of the effective operators (scenarios B3 and B4).In cases where the former applies, eq.(B.1) reduces to the one in the minimal scenario [70].Conversely, in the latter cases, eq.(B.It is worth mentioning that in scenarios B1-B2 and B5-B8, there are additional decay channels for N , such as the purely leptonic ννν and νℓℓ channels, which enhance the HNL total decay width.The possible open channels depend on the HNL mass [70].Meanwhile, in scenarios B3-B4, the total decay width becomes twice the result in eq.(B.2), since the charge conjugated channel is also open for Majorana N .
21 and O S,LR dN,21 (one at a time) would lead to the same results as for O V,RR dN,21 and O S,RR dN,21 , respectively.In figure 2, we plot the branching ratios of kaon decays induced by the LNC operators O V,RR udeN,12 , O S,RR udeN,12 , and O T,RR udeN,12 , with respective WC c O , as well as by active-heavy neutrino mixing U eN .We show three representative cases: (i) U eN = 10 −5 and c O = 0, (i) U eN = 0 and c O = 10 −5 v −2 , and (iii) U eN = 10 −5 and c O = 10 −5 v −2 .The value of the operator coefficient ensures that the branching ratios are compatible with the errors in the measurements presented in the right panel of table 3, most importantly, with that of K + → e + ν e .The value U eN = 10 −5 (|U eN | 2 = 10 −10

Figure 1 .
Figure 1.Branching ratios of kaon decays triggered by the LNC (top) and LNV (bottom) pair-N R operators.In the left (right) panel, the corresponding WC is purely real (imaginary).

Figure 2 .
Figure 2. Branching ratios of kaon decays triggered by the LNC single-N R operators with electron, as well as by active-heavy mixing U eN .

Figure 3 .
Figure 3. Branching ratios of kaon decays triggered by the LNV single-N R operators with electron, as well as by active-heavy mixing U eN .

Figure 4 .
Figure 4. Projected exclusion limits in the plane |U eN | 2 vs. m N for pair-N R operator benchmarks B1.1 and B1.2 (top), and B2.1 and B2.2 (bottom).The absolute value of the corresponding WC has been fixed to 10 −6 v −2 .The current bounds from NA62, T2K, and PS191, as well as the type-I seesaw target region where m ν = 0.05-0.12eV, are also shown.

Figure 6 .
Figure 6.Exclusion limits in the plane WC vs. m N for single-N R operator benchmarks B3 (left) and B4 (right).The production and decay couplings have been assumed to be equal.The recast bounds from NA62, T2K, and PS191, as well as the constraint originating from the measured branching ratio of K + → e + ν e , are also shown.

Figure 7 .
Figure 7. Projected exclusion limits in the plane |U eN | 2 vs. m N for single-N R operator benchmarks B5 and B6 (top), and B7 and B8 (bottom).The absolute value of the corresponding WC has been fixed to 10 −5 v −2 and 10 −6 v −2 for vector and scalar operators, respectively.The recast bounds from NA62, T2K, and PS191 are also shown.

Figure 8 .
Figure 8. Exclusion limits in the plane WC vs. m N for single-N R operator benchmarks B5 and B6 (top), and B7 and B8 (bottom).The active-heavy neutrino mixing parameter has been fixed as |U eN | 2 = 10 −10 .The recast bounds from NA62, T2K, and PS191, as well as the constraint originating from the measured branching ratio of K + → e + ν e , are also shown.For the predictions on the sensitivities of DUNE (brown), we show an extra curve in dotted line style corresponding to |U eN | 2 = 8 × 10 −11 .

2 K 12 ,
+ 4f K f S K m N Im a V 21 Im a S 21 − a S

2 K 12 .
+ 4f K f S K m N Re a V 21 Re a S 21 + a S (A.16)

Table 2 .
Four-fermion operators in N R LEFT, involving two quarks, one charged lepton, and one N R .Both LNC and LNV operator structures require "+h.c.".For one generation of N R , there are 36 independent real parameters associated with each operator structure.

Table 4 .
Benchmarks for the scenarios with pair-N R (left) and single-N R (right) operators.

8
For DUNE-ND, the sensitivity reach