Reinterpretation of searches for long-lived particles from meson decays

With a lack of concrete discovery of heavy new fields inspired mainly by supersymmetry Nilles:1983ge ; Martin:1997ns at the LHC, increasingly more attention has been shifted in recent years towards light and feebly interacting new physics (NP). Such exotic form of NP often involves long-lived particles (LLPs)¹¹1See Refs. Alimena:2019zri ; Lee:2018pag ; Curtin:2018mvb ; Knapen:2022afb for some recent summary reviews. that perhaps have so far escaped LHC searches. LLPs are predicted in a wide range of models beyond the Standard Model (BSM), and usually motivated for solving issues in the Standard Model (SM) of particle physics, such as the non-vanishing neutrino mass and the nature of dark matter (DM). For instance, a class of the so-called “portal-physics” models predict heavy neutral leptons (HNLs), dark photons, dark Higgs, and axion-like particles (ALPs), that can be either promptly decaying or long-lived, and may connect our visible SM sector to a hidden sector of the DM.

LLPs can be produced via different channels at high-energy colliders, $B$-factories, beam-dump experiments, and so on. For instance, they can be directly produced via beam collisions, e.g. electroweakino production at proton-proton colliders CMS:2020atg ; ATLAS:2022rme . LLPs can also originate from (rare) decays of mesons, electroweak gauge bosons, Higgs bosons, or perhaps heavy new particles. Testing all the different channels can usually allow for probing complementary parts of the model parameter space. In this study, we will focus on LLPs produced from mesons.

In fact, there exist a wide range of models predicting LLPs that can be produced from meson decays, such as HNLs, ALPs, and the lightest neutralino in R-parity-violating supersymmetry Dedes:2001zia ; Dreiner:2002xg ; Dreiner:2009er ; deVries:2015mfw . When an experiment publishes its results for a search, usually a certain model, say the HNLs in a minimal “3+1” scenario, is chosen to be interpreted for the experimental results. If the experimental search can, in principle, also constrain another model, traditionally a proper reinterpretation would require knowledge of the detailed signal-event chain topologies, of the main beam collisions, of the detector geometries and efficiencies, and of the cuts taken by the search. For instance, the recasting tool CheckMATE Drees:2013wra ; Kim:2015wza ; Dercks:2016npn can call programs such as MadGraph5 Alwall:2014hca or Pythia8 Sjostrand:2014zea to perform Monte-Carlo (MC) simulation including hadronization and showering, and Delphes3 deFavereau:2013fsa for fast detector simulation, followed by applying closely the experimental event-selection cuts, to determine if a parameter point of a model is excluded or allowed by the search. Or the tool SModelS Kraml:2013mwa ; Khosa:2020zar studies simplified model spectrum topologies and confronts them with the related experimental bounds, with no necessity to perform MC simulation.

For LLP searches, there are additional challenges for reinterpretation, as there are no standard definitions for LLP objects nor enough experimental information publicly available for LLP reconstruction in most cases LHCReinterpretationForum:2020xtr ; Alimena:2019zri ; Proceedings:2018het . Current efforts for addressing LLP reconstruction within standard tools include CheckMATE Desai:2021jsa , MadAnalysis Araz:2021akd , and SModelS Alguero:2021dig . Other recasting LLP simulation efforts include public repositories with specific long-lived particle searches as displaced vertices or disappearing charged tracks LLPrecastingRepo , as well as a dedicated new Delphes module to reconstruct displaced showers delphes_pr .

Here, we wonder, whether it is possible to conduct reinterpretation of LLPs from meson decays without running simulation, which is time-consuming, and also with no need to check and compare the details of the event topologies, nor the knowledge of the relevant collision and detector information; can we accomplish the job with only a computation of the LLPs’ production and decay rates? We find this is achievable under certain conditions, i.e. if the LLPs in different models have similar kinematics, e.g. all are produced from a set of mesons with similar mass and lifetime or the same meson (at the identical experiment), and have a lab-frame decay length much larger than the distance between the interaction point (IP) and the detector of the experiment.

It is impractical to consider all the possible models in one paper, and therefore we choose to restrict ourselves to the HNLs and ALPs only, for illustrative purposes. We will now briefly discuss the models we consider here. Firstly, HNLs are perhaps the most hunted candidate of LLPs, which are highly motivated for various reasons. They are fermionic SM singlets that mix with one, two, or all three species of the SM active neutrinos. They can explain the non-vanishing active neutrino masses in an elegant way by the so-called seesaw mechanism and its variations Minkowski:1977sc ; Yanagida:1979as ; Gell-Mann:1979vob ; Mohapatra:1979ia ; Schechter:1980gr ; Wyler:1982dd ; Mohapatra:1986bd ; Bernabeu:1987gr ; Akhmedov:1995ip ; Akhmedov:1995vm ; Malinsky:2005bi . In the minimal “3+1” scenario of the HNLs, there are only two types of parameters in the model, i.e. the HNL mass and mixing angles, controlling both production and decay of the HNLs. Oftentimes for the purpose of performing phenomenological studies, the HNLs are assumed to mix with only one generation of the active neutrinos, and hence constraints are obtained in a plane spanned by two independent parameters, the mixing angle and the mass. Even though this choice is unrealistic as neutrino oscillation data (see Refs. Capozzi:2021fjo ; Esteban:2020cvm ; deSalas:2020pgw for recent global analyses) requires at least two massive HNLs, it is a useful minimal choice to characterize a possible experimental signal. At beam-dump experiments such as CHARM CHARM:1985nku , HNLs can arise from weak decays of various mesons that are copiously produced there. $B$-factories including Belle Belle:2013ytx and BaBar BaBar:2022cqj have also obtained bounds for HNLs from $B$-meson decays. Moreover, in recent years, a series of far-detector programs have been proposed (with some already approved) to be operated in the vicinity of different IPs at the LHC, aiming mainly to detect displaced-vertex signatures of LLPs. These include FASER Feng:2017uoz ; FASER:2018eoc , MATHUSLA Curtin:2018mvb ; Chou:2016lxi ; MATHUSLA:2020uve , and MoEDAL-MAPP Pinfold:2019nqj ; Pinfold:2019zwp , among others. Since the production rates of mesons at the LHC are huge, these experiments are estimated to be highly sensitive to LLPs from meson decays. Phenomenological studies on the exclusion limits of these far detectors on the HNLs in the minimal case can be found e.g. in Refs. Helo:2018qej ; Kling:2018wct ; Hirsch:2020klk ; Curtin:2018mvb ; Ovchynnikov:2022its ; Aielli:2019ivi ; DeVries:2020jbs .

Beyond the minimal scenario, HNLs also appear in other BSM models that often extend the SM with new gauge groups. Examples include models with an extra $U(1)$ gauge group predicting a new gauge boson $Z^{\prime}$ Chiang:2019ajm , and leptoquark models predicting heavy leptoquark particles Dorsner:2016wpm . In this class of models, the production and decay of the HNLs are usually mediated by two independent parameters, respectively, besides the HNL mass. If HNLs have masses around the weak scale and other new particles are much heavier, the effects of NP at low energies can be described in terms of the Standard Model effective field theory extended with HNLs, known as $N_{R}$SMEFT delAguila:2008ir ; Aparici:2009fh ; Liao:2016qyd . In the $N_{R}$SMEFT, one writes down non-renormalizable operators including one or more HNLs in the Lagrangian and the operator coefficients are associated with the NP scale, usually labeled as $\Lambda$. For GeV-scale HNLs produced in meson decays, the appropriate description is provided by the low-energy effective field theory extended with HNLs, $N_{R}$LEFT Bischer:2019ttk ; Chala:2020vqp ; Li:2020lba ; Li:2020wxi , in which heavy SM degrees of freedom (namely, the top quark, the Higgs and $Z$ and $W^{\pm}$ gauge bosons) are not present. In this paper, we will adopt this framework and will show how the bounds on the HNLs in the minimal scenario can be translated to those on the Wilson coefficients of the $N_{R}$LEFT. For simplicity, for the HNLs in either the minimal scenario or effective-field-theory (EFT) scenarios, we will study their association with a charged lepton or active neutrino of the electron flavor only.

Besides the HNLs, there are other possible LLPs that may be produced from meson decays, such as ALPs. ALPs are the pseudo-Nambu-Goldstone bosons associated with an approximate global symmetry that is spontaneously broken at a high scale $\Lambda$. They are mainly motivated as DM candidates Preskill:1982cy ; Abbott:1982af ; Dine:1982ah ; Panci:2022wlc , and their mass and couplings are independent and can range across orders of magnitude Kim:2015yna ; DeMartino:2017qsa ; Rubakov:1997vp . At low energies, the ALP interactions are described by the corresponding EFT Georgi:1986df ; Choi:1986zw . This theory has gained significant attention in recent years, see e.g. Refs. Brivio:2017ije ; Chala:2020wvs ; Bauer:2020jbp ; Bauer:2021mvw ; MartinCamalich:2020dfe ; Calibbi:2020jvd , in part because of the lack of observation of NP at the LHC. The (pseudo-)Goldstone nature of the ALP manifests itself in its derivative couplings to SM fermion currents. In particular, the flavor off-diagonal couplings to up (down) quarks can trigger the decays of $D$($B$)-mesons to a lighter meson and the ALP. The latter can be long-lived, decaying e.g. via its couplings to charged leptons, cf. Ref. Dreyer:2021aqd . In this work, we will show how the searches for HNLs in the minimal scenario can constrain the parameter space of the ALP EFT.

Since all these LLPs are produced from meson decays, they should have similar kinematics, as long as they come from a set of similar mesons such as $D$-mesons ($D^{0}$, $D^{\pm}$ and $D_{s}$) or only $B$-mesons ($B^{0}$, $B^{\pm}$ and $B_{s}$), or even the same meson, at the same experiment. This is legitimate because the LLPs stemming from mesons of similar masses and lifetimes are expected to have similar boost factor and polar angle distributions and hence similar decay probabilities in the detector, given the same LLP proper lifetime and mass.²²2The spin of the LLP, the number of LLPs in each signal meson decay, as well as whether the meson decay is two-body or three-body, should have an impact on the distribution. However, as we will see when we discuss numerical results, this impact is small. Therefore, we do not discuss it in detail here. Moreover, from the experimental exclusion limit point of view, the small coupling regime (or heavy NP scale) is almost always harder to probe than the large coupling one (or light NP scale), and in the small coupling regime, the LLPs are usually very long-lived given their small masses as required by kinematic constraints (production from meson decays). Therefore, our requirements for the simple reinterpretation as mentioned above are fulfilled most of the time, for the LLPs originating from mesons. In this work, as a proof-of-principle example, we will show how to reinterpret bounds on the minimal HNL scenario (in the plane mixing angle vs. mass) produced from charm and bottom meson decays, into the model-parameter planes of (i) the $N_{R}$LEFT and (ii) the ALP EFT. For the $N_{R}$LEFT, we will consider operators that include either one or two HNLs, i.e. single-$N_{R}$ or pair-$N_{R}$ operators.

In the next section, we elaborate on the reinterpretation method we propose for LLPs produced from meson decays. Sec. 3 contains a brief introduction to both the past and future experiments we consider, as well as relevant experimental bounds for the theoretical scenarios we are studying in this work. Then in Sec. 4, we show the results of reinterpreting the sensitivity limits on HNLs in the minimal scenario into HNLs in the EFT and into ALPs. Finally, we discuss the advantages and limitations of the proposed reinterpretation method and provide an outlook, in Sec. 5.

To explain the reinterpretation procedure, we start with a general discussion on searches for the HNLs in the minimal scenario. First, once an experimental search for the minimal HNL scenario is finished, say, without a discovery, the $95\%$ (or sometimes $90\%$) confidence level (C.L.) exclusion limits corresponding to a certain number of signal events, $N_{S}$, are determined according to the numbers of observed events and expected background events. These bounds can be converted to model parameters with the following formula,

where $N_{N}$ is the total number of HNLs produced from certain meson decays, $\epsilon$ is the product of the detector acceptance and efficiencies, and $\text{BR}(N\to\text{vis.})$ is the decay branching ratio of the HNLs into certain final states visible/detectable/observable in the detector chamber. $N_{N}$ and BR$(N\to\text{vis.})$ can just be computed from the model parameters, mixing angle squared $|V_{eN}|^{2}$ and mass $m_{N}$, and $\epsilon$ should be determined by a proper simulation together with the computation of the proper lifetime of the HNL, $c\tau_{N}$, from $|V_{eN}|^{2}$ and $m_{N}$. Thus, the experimental search can obtain bounds in the plane $|V_{eN}|^{2}$ vs. $m_{N}$. Therefore, from the published results of the experimental search, we obtain a three-dimensional (3D) array of data in $(m_{N},|V_{eN}|^{2},N_{S})$, with $N_{S}$ fixed at a certain single value (e.g. $N_{S}=3$ for $95\%$ C.L. exclusion limits with zero observed event) and $m_{N}$ and $|V_{eN}|^{2}$ varying. Alternatively, if we could perform the simulation including the experimental event-selection cuts by ourselves, we could also obtain another set of 3D data for the same set of variables, but with $|V_{eN}|^{2}$ fixed at a certain value, and $m_{N}$ and $N_{S}$ varied.

Now, assuming the HNLs are in the large decay length limit, such that its boosted decay length, $\lambda_{\text{decay}}=\beta\gamma c\tau=\beta\gamma c\hbar/\Gamma_{\text{tot.}}$, with $\Gamma_{\text{tot.}}$ being the HNL total decay width, is much larger than the distance from the detector to the IP, $\epsilon$ would be simply linearly proportional to the total decay width of the HNL, $\Gamma_{\text{tot.}}$ in general. This can be understood as follows. $\epsilon$ is proportional to the detector acceptance to the HNL, $P[\text{decay}]$, which can be essentially expressed with the following formula:

where $L$ is the distance from the IP to the detector, $\Delta L$ is the length inside the detector that the HNL traverses if it does not decay inside the detector, and the second-last equality holds for $\lambda_{\text{decay}}\gg L$. One sees that in the large decay length limit, the exponential decay distribution can be approximated as linearly proportional to the HNL total decay width. As a result, we can derive from Eq. (1) the following relation:

where the last equality stands because $\text{BR}(N\to\text{vis.})=\Gamma_{\text{vis.}}/\Gamma_{\text{tot.}}$ where $\Gamma_{\text{vis.}}$ denotes the sum of partial widths of the HNLs into visible final states. Now considering the HNLs in the $N_{R}$LEFT, we can write a similar relation,

where the primed variables are the EFT-counterparts of those in Eq. (3). We assume for now that the production and decay Wilson coefficients are different in the $N_{R}$LEFT. Note that the same relation can also be written for other types of LLPs than the HNLs, such as the ALPs, but we will stick to the HNLs in the EFT in the rest of this section, for convenience of discussion.

Here, we should comment that as long as we work in the large decay length limit, and the HNLs in both cases are produced from the same type of mesons ($D$-mesons only or $B$-mesons only, for instance), the kinematics of the HNLs in the two cases are sufficiently similar so that we can ignore the differences for our purpose and assume that the proportionality is shared between Eq. (3) and Eq. (4). This allows us to take the ratio of Eq. (3) to Eq. (4) and to reach the following relation:

or equivalently,

We further note that in a commonly seen case where $N_{S}=N^{\prime}_{S}$, Eq. (6) can be simplified to be:

As discussed above, we already have a list of 3D data in $(m_{N},|V_{eN}|^{2},N_{S})$. For each $m_{N}$, the corresponding value of $|V_{eN}|^{2}$ allows to compute $N_{N}$ and $\Gamma_{\text{vis.}}$. Further, $N_{S}$ is known, and $N^{\prime}_{S}$ is usually either equal to $N_{S}$ or can be determined from the experimental search depending on the final states of the LLP decays. As a result, for the $N_{R}$LEFT case, once we fix the production Wilson coefficient determining $N^{\prime}_{N}$, we can derive $\Gamma^{\prime}_{\text{vis.}}$ with the usage of Eq. (6) or Eq. (7). It is then a straightforward exercise to convert these bounds on $\Gamma^{\prime}_{\text{vis.}}$ into the corresponding bounds on the decay Wilson coefficient in the $N_{R}$LEFT.

We stress again this approximation holds only if in both the minimal scenario and the $N_{R}$LEFT scenario:

1. 1.

   the HNLs are produced from the same type of mesons, ensuring the same kinematics, and

2. 2.

   the HNLs are long-lived relative to the detector distance from the IP, ensuring working in the linear regime for the exponential decay distributions.

In the procedure above, it has been assumed that the production and decay couplings of the LLP in the new model (the ALP and HNL in the EFTs as in this work) are unrelated. However, it is also often the case that they are related. Labeling these two Wilson coefficients as $P_{\text{prod}}$ and $P_{\text{decay}}$, we discuss briefly here, without loss of generality, the case that they are equal: $P_{\text{prod}}=P_{\text{decay}}$. We first fix a value for $P_{\text{prod}}$ and derive a limit on $P_{\text{decay}}$ following the algorithm given above. In the general case, the derived value of $P_{\text{decay}}$ is unequal to $P_{\text{prod}}$. We notice the fact that if we chose a value of $P_{\text{prod}}$ larger (smaller) by a random positive number $x$, we would obtain a bound on $P_{\text{decay}}$ stronger (weaker) also by $x$, assuming the production and decay rates of the LLP are proportional to $P_{\text{prod}}^{2}$ and $P_{\text{decay}}^{2}$, respectively. This means if we multiply $P_{\text{prod}}$ by $\sqrt{P_{\text{decay}}/P_{\text{prod}}}$ to reach the geometric mean of $P_{\text{prod}}$ and $P_{\text{decay}}$, this new production coupling would lead to a new bound on $P_{\text{decay}}$ of equal value. We emphasize again that all the manipulation here works only if the boosted decay lengths of the LLPs in the relevant parts of the parameter space are sufficiently large.

In this section, we provide details of existing searches to be recast with our reinterpretation method, as well as comment on existing upper limits on the branching ratios of (semi-) invisible decays of $D$- and $B$-mesons. The latter will provide strong constraints on the couplings in our selected benchmark scenarios. Additionally, we briefly describe the future LLP far detectors, whose sensitivities to the minimal HNL scenario will be reinterpreted into the $N_{R}$LEFT and the ALP EFT in Sec. 4.

The minimal “3+1” scenario assumes the existence of one HNL, $N$, that mixes with the active neutrinos $\nu_{\ell}$, $\ell=e,\,\mu,\,\tau$. The interaction Lagrangian of interest reads

where $g$ is the $SU(2)_{L}$ gauge coupling, $\theta_{W}$ is the weak mixing angle, $V_{\ell N}$ is active-heavy mixing, whereas $U_{\ell i}$ is active-light mixing (the PMNS mixing matrix). The sum over $\ell=e,\,\mu,\,\tau$ and $i=1,\,2,\,3$ is implied. Note that we have neglected the terms quadratic in $V_{\ell N}$. Production and decays of the HNL are governed by its mass $m_{N}$ and the mixing angles $V_{\ell N}$. In phenomenological studies, it is often assumed that the HNL mixes with one active flavor at a time. In this case, the model is characterized just by two independent parameters, mass and mixing $V_{\ell N}$. For this work, we focus on the mixing with the electron neutrino only. Most of the searches for HNLs are interpreted in the framework of this simple model. For a generalization to the case of several HNLs, see e.g. Refs. Abada:2018sfh ; Abada:2022wvh . Using the method formulated in Sec. 2, we will translate the bounds on the parameter space of the minimal scenario to the constraints on other (non-minimal) scenarios. We will consider two examples featured with long-lived fermions and (pseudo-)scalars, respectively: (i) the EFT with HNLs and (ii) ALPs.

In Fig. 1, we present bounds on the HNLs in the minimal scenario which dominantly mix with the electron neutrino, shown in the plane $|V_{eN}|^{2}$ vs. $m_{N}$. The left and right plots contain results for the HNLs produced from rare decays of $D$- and $B$- mesons, respectively. In both plots, the exclusion limits of the LHC far detectors are at 95% confidence level for 3 signal events with vanishing background, obtained by MC simulation following the procedures given in Refs. DeVries:2020jbs ; Beltran:2022ast . In the right plot, we have included results from the Belle experiment Belle:2013ytx with 711 fb^-1 integrated luminosity, and in the left, the results are overlapped with the existing bounds obtained at CHARM CHARM:1985nku , since these two experiments gave the leading bounds on the minimal HNLs produced from $B$- and $D$-mesons’ decays, respectively.

For the far detectors, as we have performed MC simulations, we have three-dimensional data sets of $(m_{N},|V_{eN}|^{2},N_{S})$, across the whole kinematically allowed mass range. For this 3D dataset, we fix $|V_{eN}|^{2}$ to a small value such as $10^{-9}$, in order to ensure working in the large decay length limit. However, for the existing bounds from Belle and CHARM, we do not have this full simulation information and can hence only make use of the published exclusion limits directly. These datasets then allow us to perform reinterpretation into other models, such as the HNLs in the EFT, and ALPs, which are produced from bottom or charm meson decays.