# Constraints on asymmetric production of long-lived scalars at the Large Hadron Collider

Thomas Chehab, Louie Dartmoor Corpe, Andreas Goudelis, Abdelhamid Haddad, Louise Millot

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Thomas Chehab, Louie Dartmoor Corpe, Andreas Goudelis, Abdelhamid Haddad, Louise Millot. Constraints on asymmetric production of long-lived scalars at the Large Hadron Collider. European Physical Journal C: Particles and Fields, 2025, 85 (8), pp.824. 10.1140/epjc/s10052-025-14519-w. hal-04980503

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# Constraints on asymmetric production of long-lived scalars at the Large Hadron Collider

Thomas Chehab<fnref n="1" />, Louie Dartmoor Corpe<fnref n="2" />, Andreas Goudelis<fnref n="2" />,<fnref n="3" />, Abdelhamid Haddad<fnref n="2" />, Louise Millot<fnref n="3" />

Received: 3 April 2025 / Accepted: 14 July 2025

© The Author(s) 2025

**Abstract** Searches for pair-produced long-lived particles (LLPs) at the LHC commonly operate under the assumption that the two LLPs are identical. In this paper we entertain the possibility that the targeted final states are, instead, induced by a LLP pair with different masses and/or lifetimes. We propose a simple and intuitively-parametrised toy model in order to study such asymmetric production of LLPs. Using the recasting material of a recent search for displaced jets by the ATLAS collaboration, we demonstrate that we can set constraints on the production cross-section times branching fraction into jets for a variety of asymmetric LLP and mediator mass combinations.

## Contents

1 Introduction

2 Reference search for displaced jets with the ATLAS detector

2.1 Description of reference analysis

2.2 Underlying assumptions: the HAHM in theory and in practice

3 A simple approach for asymmetric production of LLPs

4 Numerical results

4.1 Validation: symmetric production of LLPs

4.2 Going further: asymmetric production of LLPs

5 Conclusions

References

## 1 Introduction

During the past decade, searches for new long-lived particles (LLPs), i.e. particles with a macroscopic lifetime, have evolved from a relatively “exotic” topic into an important pillar of the LHC New Physics search programme – and for good reasons. Indeed, LLPs not only represent an exciting experimental challenge, among other reasons because they touch upon the very definition of what constitutes a “particle” from an experimental standpoint,<fnref n="1" /> but are also theoretically well-motivated. Examples of classes of new physics scenarios potentially predicting the existence of exotic LLPs include Hidden Valley models [1–3], models of feebly interacting dark matter [4,5], neutrino mass models [6,7] or supersymmetric [8–13] and/or “neutral naturalness” scenarios [14–17].

Given that so far no significant excesses have been observed, and although variations depending on the specific final state being considered may exist, typical LLP searches present their results in the form of constraints on the LLP production cross-section times branching fraction ($\sigma \times \text{BF}$) as a function of the LLP lifetime and mass. To do so, analyses often use simple benchmark models which, essentially, are used as kinematic templates to guide the optimisation of the analyses – i.e. to define a set of kinematic requirements for signal selection and background rejection. It follows that the limits (published in terms of physical quantities such as cross-sections, masses, and lifetimes) only hold within the

<footnote n="1">¹ Artemis, Université Côte d'Azur, CNRS, Observatoire Côte d'Azur, BP4229, 06304 Nice Cedex 4, France</footnote>

<footnote n="2">² Laboratoire de Physique de Clermont Auvergne (UMR 6533), CNRS/IN2P3, Univ. Clermont Auvergne, 4 Av. Blaise Pascal, 63178 Aubière Cedex, France</footnote>

<footnote n="3">³ Laboratoire de Physique Subatomique et de Cosmologie (LPSC), CNRS/IN2P3, Université Grenoble-Alpes, 53 Avenue des Martyrs, 38026 Grenoble, France</footnote>

<footnote n="1">¹ As an example, and grossly speaking, experimentally a promptly produced electron is a track accompanied by energy deposits in the electromagnetic calorimeter. An electron stemming from a displaced vertex, on the other hand, may be a trackless object. From a theoretical perspective, of course, both objects are identical.</footnote>

<footnote>ª e-mail: andreas.goudelis@clermont.in2p3.fr (corresponding author)</footnote>
framework of the underlying template model, or within very
contrived embeddings in larger frameworks that contain the
template model in essentially intact/factorisable form.

In this paper we focus on the case of ATLAS and CMS
searches for neutral, hadronically-decaying scalar LLPs (see
Refs. [18–21] for recent examples). These searches are
designed using the Hidden Abelian Higgs Model (HAHM)
[22] as a template, assuming that a pair of identical scalar
LLPs are produced through the decay of a scalar mediator and
subsequently decay, through Yukawa-like interactions, pre-
dominantly into heavy quarks (i.e. jets) and/or $\tau^+\tau^-$ pairs.
The results are then presented in terms of constraints on the
mediator production cross-section times branching ratio into
LLP pairs decaying into jets.

There are at least three instances, or combinations thereof,
in which it would be natural to expect a substantial modifi-
cation of the published constraints:

* Multiple states (e.g. Higgs bosons) could mediate the pro-
  duction of LLPs, which, if their contributions are compa-
  rable, could alter the kinematic distributions of the decay
  products;

* Gluon fusion might not be the predominant mediator pro-
  duction mode, and jets might not be the LLP decay mode
  driving experimental sensitivity (e.g. in models involving
  vector LLPs, or, in the case of scalars, Yukawa coupling
  structures substantially departing from a minimal flavour
  violation-like hypothesis);

* The produced LLPs could, simply, *not* be identical. From
  a phenomenological standpoint, this would mean that
  their mass and/or mean proper lifetime could be differ-
  ent which would then lead to modifications in the decay
  products' kinematic and/or spatial distributions. Such a
  situation could arise, for example, in models involving
  extended scalar sectors with multiple singlets coupled to
  the Standard Model (SM) Higgs boson bilinear $H^\dagger H$.

In this paper we focus on the latter possibility, which we will
henceforth refer to as “asymmetric” LLP production. Our
analysis will be mainly driven by phenomenological consid-
erations: we aim neither at considering the “best-motivated”
SM extension nor at studying a model which can encapsu-
late all New Physics scenarios predicting the existence of
multiple LLPs. Instead, we will study how the constraints
presented in Ref. [23] translate in terms of a simple and, hope-
fully, intuitively parametrised toy model of “asymmetrically
produced” LLPs, depending on their masses and lifetimes.

The paper is structured as follows. In Sect. 2 we describe
an ATLAS search for LLPs which we will use as a reference
throughout the paper, as well as the HAHM which it uses
as a benchmark model. In Sect. 3 we propose an alternative
lifetime-parametrised model, which also has the flexibility
to allow asymmetric LLP production. This is followed by

Sect. 4 where we use the recasting material for the reference
search to derive, for the first time, limits on asymmetric LLP
production. Finally, we draw our conclusions in Sect. 5.

## 2 Reference search for displaced jets with the ATLAS detector

The analysis presented in Ref. [23] targets pairs of identi-
cal neutral long-lived particles decaying hadronically in the
ATLAS hadronic calorimeter system. It is part of a family
of searches which look for the same signature with decays
occurring in various parts of the ATLAS detector [19–21].
Each of those searches, and their various iterations, have used
the HAHM [22] (which will be briefly discussed in Sect. 2.2)
as a kinematic template since the start of LHC Run 2. In
this paper, we use the most recent calorimeter-based ATLAS
search as a reference because it has the advantage of provid-
ing some of the most-cutting edge re-interpretation material
available [24]: a machine-learning based “surrogate model”
to obtain the probability that an event would be selected
based only on truth-level kinematics, i.e. circumventing the
need for detector simulation. An independent validation of
the surrogate models for that analysis is available in Ref.
[25]. The availability of this recasting material is particu-
larly relevant because many efforts to recast LLP analyses
cannot reliably make use of tools like DELPHES [26] to sim-
ulate the effects of the detector since most LLP searches rely
heavily on the detailed response of each later of the detector
(low-level information). For this reason, it is often difficult
to estimate the sensitivity of such a search to a new model,
even when it is relatively close to the benchmark models
employed in the original analysis. The presence of the surro-
gate model in the re-interpretation material is a boon for this
type of exercise.

### 2.1 Description of reference analysis

We will now briefly describe the reference analysis [23],
which is an extension of a similar analysis presented in Ref.
[20]. In both cases, the full LHC Run 2 dataset is used to
search for pairs of displaced jets originating from hadronic
decays of neutral LLPs in the hadronic part of the calorime-
ter. The displaced jets are expected to be largely trackless
(since the LLP is neutral) and with a high proportion of
energy in the hadronic part of the calorimeter with respect to
the upstream electromagnetic calorimeter (since the search
focuses on decays in the hadronic part). These selections are
helpful to reduce QCD multijets, which only rarely produce
such signatures, but which still make up the dominant back-
ground on account of their overwhelming production cross-
section in proton collisions. The data were collected with
dedicated triggers which also exploit this calorimeter energy
ratio topology. Both analyses make use of a neural network (NN), whose inputs include deposits in each layer of the calorimeter as well as tracker and muon system information. It classifies jets as likely to have originated from QCD processes, signal, or beam-induced background (such as beam halo muons which travel parallel to the LHC beam and strike the detector without passing through the interaction point).

The search presented in Ref. [20] was optimized for cases where the decay products of each LLP (for instance two b-quarks) are very collimated, leading to a single “merged” jet in each case. This is optimal for cases where the mediator is heavy, but breaks down when the mediator is relatively light, because a large fraction of LLP decays can lead to two resolvable jets. Hence, the analysis was re-optimised in Ref. [23], to allow one of the jets to be “resolved”. The analysis also contained channels where the mediator was produced in association with a W or Z boson, but we do not consider those channels further in this work: we focus, instead, exclusively on the “CR+2J” channel (defined in Ref. [23]), optimised for cases where the mediator is produced via gluon fusion. This channel makes use of event-level machine-learning to exploit the NN scores for the jets most likely to originate from a signal process, along with additional event-level information, to classify events as signal-like or not. Some additional cleaning requirements (on signal-like jet timing, momentum and position, etc...) are used to remove any remaining events likely to have originated from cosmic ray muons or beam-induced background. The resulting event-level score is then used with an orthogonal variable measuring the proximity of tracks to the jets in the events, to form an ABCD plane with which a data-driven background estimate can be made. The signal is mostly concentrated in region A of the plane, while the background (which at this stage is composed purely of QCD multijets) is distributed in the other regions in an uncorrelated way. A simultaneous fit of the signal and background yields (which should obey the $A = B \times C/D$ relation) is performed for the final statistical analysis. The signal efficiencies in region A are extrapolated to arbitrary lifetimes using an exponential reweighting method. We note that this method, adopted for reasons of computational efficiency (i.e. to minimize Monte Carlo event generation and detector simulation) clearly assumes that the mean proper lifetime is uncorrelated with the rest of the model parameters. This, of course, is not true and will be further commented upon in what follows. In the absence of an excess, the $CL_s$ [27] method was used to set constraints on $\sigma \times \text{BF}$ as a function of the LLP mass, for a set of benchmark mediator masses. As mentioned above, the surrogate models provided by the analysis and validated in Ref. [25], provide an easy-to-use proxy to the analysis selection efficiency for scenarios with different kinematics than the original template model, such as the one we study in this paper.

## 2.2 Underlying assumptions: the HAHM in theory and in practice

As we already mentioned, the microscopic model that was chosen as a kinematic template in Refs. [20,23] was the HAHM [22]. The HAHM extends the SM with an additional $U(1)_X$ gauge group under which all SM particles are neutral and which is spontaneously broken by the vacuum expectation value of a $SU(3)_c \times SU(2)_L \times U(1)_Y$-singlet scalar field. The model predicts the existence of an additional gauge boson ($Z'$) along with a new scalar particle ($S$) which mixes with the SM Higgs boson. This mixing, in turn, induces couplings of the new state to the SM fermions.

Focusing on the scalar sector, the concrete process through which displaced jets can in principle arise in the HAHM (albeit with an extremely small cross-section) is gluon fusion-induced (potentially virtual) SM Higgs boson ($h$) production, followed by the production of two exotic scalar states $S$, which in this case plays the role of the LLP. In the analyses presented in Refs. [20,23], the implementation of the HAHM in MadGraph [28] presented in Ref. [29] was employed, where gluon fusion was approximated by an effective interaction of the SM Higgs boson with a pair of gluons, based on the prediction obtained in the heavy top quark loop limit, as illustrated in Fig. 1. The LLPs can, in turn, decay into SM particles with a macroscopic lifetime, provided the mixing with the SM Higgs boson is small enough. It should be noted that, once all of the HAHM free parameters are fixed, the LLP mean proper lifetime (along with all the cross-sections, partial widths and branching ratios, for that matter) is a predicted quantity.

Using this HAHM implementation, the event generation in Refs. [20,23] proceeds as follows:

* Events are generated for chosen combinations of $m_{h,S}$, the masses of the two scalars, which are treated as free parameters. In other words, the SM Higgs Boson is co-opted into the role of a generic, gluon fusion-induced scalar mediator. In such cases, we denote the particle instead by $\Phi$,<fnref n="2" /> whereas the role of LLPs is undertaken by the exotic scalars.

* For each chosen set of $m_{h,S}$ values, a single event sample is actually generated, with a mean proper lifetime chosen to maximise the number of observable decays in the calorimeters, regardless of the values of the underlying parameters. To be even more explicit: with the typical mass and coupling values actually chosen for the initial event generation in the ATLAS searches, it turns

<footnote n="2">² It follows that, when events are generated with a mediator mass assumption other than 125 GeV, there is actually no SM-like Higgs boson in the model; in this instance, the HAHM is just meant to be used as a template for kinematics.</footnote>
<figure><img src="image_2.png" /><figcaption>Fig. 1 Gluon fusion-induced mediator production in the HAHM in the dynamical (left) and infinite mass (right) top quark limit</figcaption></figure>

out that the HAHM actually predicts promptly decaying exotic neutral scalars. The macroscopic lifetime is imposed post-hoc, by replacing the decay position in the Les Houches Events file by hand.

These comments motivate the following considerations. The HAHM is a well-established, fully consistent (e.g. gauge-invariant and renormalizable) model of New Physics. However (i) its usage in practice introduces a set of phenomenological inconsistencies described in the bullets above and (ii) these inconsistencies are opaque both to the analysis team and to the community at large. Could an alternative simplified model be considered which, in a sense, actually “accepts its nature” as serving as a mere kinematic template and which manages to address at least some of these concerns? This would perhaps come at the expense of theoretical consistency, but at least would provide some more insight into the event generation procedure, in the sense of being parametrised in terms of more physically intuitive quantities such as the LLP lifetime itself.

Clearly, the answer to this question is neither easy, nor universal. In what follows we will propose such a framework for our case-study analysis, being nonetheless fully conscious of the fact that there might be cases in which it may not be possible to employ “toy models” which are as simple.<fnref n="3" />

### 3 A simple approach for asymmetric production of LLPs

Let us now turn to our main question, i.e. how the constraints obtained in Refs. [20,23] translate in the case in which a gluon-fusion-produced scalar mediator decays into a pair of different scalar LLPs. In order to describe the process, we consider the Lagrangian

$$ \mathcal{L} = \mathcal{L}_f + \mathcal{L}_{QCD} - \frac{m_{\Phi}^2}{2} \Phi^2 - \frac{m_{S_1}^2}{2} S_1^2 - \frac{m_{S_2}^2}{2} S_2^2 \\ + \frac{1}{\Lambda} \Phi G_{\mu\nu} G^{\mu\nu} + \kappa v \Phi S_1 S_2 \\ + \sum_f \left( y_1 \frac{\sqrt{2} m_f}{v} S_1 f \bar{f} + y_2 \frac{\sqrt{2} m_f}{v} S_2 f \bar{f} \right) \quad (3.1) $$

where $\mathcal{L}_f$ and $\mathcal{L}_{QCD}$ denote the SM fermion gauge interaction and QCD Lagrangians, respectively, $\Phi$ is the mediator, $G$ is the gluon field strength tensor, $\Lambda$ is a scale governing gluon fusion-induced mediator production, $S_{1,2}$ are the LLPs, $\kappa$ is a dimensionless coefficient describing the interaction strength between the LLPs and the mediator, $y_{1,2}$ are the LLP Yukawa-like couplings to the SM fermions $f$ and $v$ is the SM Higgs boson vacuum expectation value. Note that we have chosen the normalisation factors such that the coupling of $S_{1,2}$ to the Standard Model fermions be proportional to the masses of the latter.

Now, in this framework the mediator and LLP partial decay widths acquire a simple form

$$ \Gamma_{\Phi}^{jj} = \frac{2m_{\Phi}^3}{\pi \Lambda^2} \quad (3.2) $$

$$ \Gamma_{\Phi}^{\text{LLP}} = \frac{\kappa^2 v^2 \sqrt{\lambda(m_{\Phi}^2, m_{S_1}^2, m_{S_2}^2)}}{16 \pi m_{\Phi}^3} \quad (3.3) $$

and

$$ \Gamma_{S_i}^{ff} = \sum_f \left( \frac{C_f y_i^2 m_f^2 (m_{S_i}^2 - 4m_f^2)^{3/2}}{\pi v^2 m_{S_i}^2} \right) \quad (3.4) $$

where $C_f = 1(3)$ for leptons (quarks). The latter of these expressions (in which the sum runs over all SM fermions) can be easily inverted in order to obtain the overall couplings $y_i$ as a function of the LLP lifetime which, in turn, can be promoted to an input parameter of the model. The “symmetric” scenario can, then, be recovered simply by choosing the two masses and mean proper lifetimes to be the same.

<footnote n="3">³ One such example are, e.g. searches which are performed having in mind processes involving multiple gauge bosons, in which gauge invariance can become important for Monte Carlo event generation.</footnote>
All in all, the model described by the Lagrangian (3.1) is governed by seven free parameters which we choose to be: the mediator and LLP masses, the two LLP lifetimes and the parameters $\Lambda$ and $\kappa$ which describe the mediator interactions with gluons (for $\Lambda$) and $S_{1,2}$ (for $\kappa$), respectively.<fnref n="4" />

It is clear that from a theoretical standpoint, the Lagrangian (3.1) as such is not fully consistent. Concretely, it is not gauge-invariant. However, it can arise fairly straightforwardly as part of the total Lagrangian within different UV-complete and/or well-motivated extensions of the Standard Model, such as in models involving multiple exotic $U(1)$'s broken by $SU(3)_c \times SU(2)_L \times U(1)_Y$-singlet scalars that mix with the SM Higgs boson through Higgs portal-like interactions (the original HAHM is, in fact, such an example, albeit corresponding to the symmetric limit of our setup) or in models aiming to explain the hierarchy of fermion masses (for a recent proposal cf. e.g. Ref. [30] and references therein). Moreover, we believe that it has the following practical advantages:

* It focuses on the particle content which is, in practice, relevant for the experimental search at hand. For example, the $Z'$ part of the HAHM has been discarded: searches for long-lived gauge bosons are, of course, extremely interesting but would most likely be optimally performed through final states involving leptons. Moreover, even fully hadronically decaying gauge bosons would typically be produced through $q\bar{q}$ annihilation which would, in turn, imply very different kinematics compared with gluon fusion-produced objects.

* It is intuitive in the sense that, at least as far as the LLP properties are concerned, it can be easily parametrised in terms of physically measurable quantities (masses, lifetimes). We consider this to be an important point, since it can allow experimental groups to better control the event generation process and, e.g., generate initial event samples which are as “phenomenologically consistent” as possible.

In order to better understand the second comment, let us give a concrete example: LLP decay widths are, necessarily, extremely narrow. The $S_1$ or $S_2$ width can, in principle, have an impact on the decay products' kinematic distributions and, hence, on the ensuing constraints. Even if, for reasons of computational efficiency, only a single event sample per $m_{\Phi, S_1, S_2}$ combination is generated, through the parametrisation that we propose it is completely straightforward to choose macroscopic $S_1$ or $S_2$ lifetimes and the associated decay widths will automatically and consistently be adjusted to be narrow. Let us also be clear on two points: first, as long as a single event sample is generated for each $m_{\Phi, S_1, S_2}$ combination, some amount of inconsistency is bound to persist; however, and at least, all constraints will be obtained under the assumption of a very narrow LLP width. Secondly, in practice we have found that, at least in this instance, the LLP width can vary over orders of magnitude without affecting the resulting constraints. In some sense this is expected, as long as $\Gamma_{S_1}^{ff}$ remains below the experimental energy resolution. However, the validity of this statement depends on one hand on the underlying model and, on the other hand, on the parameter values employed for the initial event generation.

This, in our view, is one of the major advantages of the framework that we propose: the user can choose from the start the LLP lifetime as an input parameter and the LLP width is modified accordingly and consistently, at least for the original event generation sample.

# 4 Numerical results

Let us now turn to our numerical analysis. In order to generate events, we have implemented the model described by the Lagrangian (3.1) in MadGraph5_aMC@NLO (MG) [28] by means of the FeynRules package [31].<fnref n="5" /> Throughout the subsequent analysis we will be employing the NNPDF2.3LO set of Parton Distribution Functions whereas showering and hadronization are simulated with Pythia 8 [32] with all settings chosen at their default values.

## 4.1 Validation: symmetric production of LLPs

The first step is to validate the theoretical and computational framework that we propose. Concretely, in this context the validation involves two distinct tasks:

* First, we must ensure that our template model yields the same results as the HAHM in the “symmetric” limit, i.e. when the two LLPs are assigned identical masses and lifetimes.

* Second, for reasons of computational efficiency in what follows, we will be adopting the approach followed by the ATLAS collaboration: generating one event sample per ($m_\Phi, m_{S_1}, m_{S_2}$) combination and sampling the LLP proper lifetime from an exponential distribution, regardless of the predicted LLP (and mediator) width. Thus, we need to establish that, at least within the scope of our analysis, both approaches yield quantitatively similar results.

<footnote n="4">⁴ Note that in principle the latter two parameters can also be traded for the mediator partial widths into gluon and LLP pairs, respectively. Each choice offers different advantages. For example, promoting the partial widths to input parameters would make it easier to ensure that Footnote 4 continued the mediator width remains smaller than its mass. Retaining the original Lagrangian parameters, on the other hand, allows for an easier assessment concerning the EFT scale $\Lambda$ value that is being probed.</footnote>

<footnote n="5">⁵ The corresponding model files are publicly available in https://github.com/FeynRules/Models/tree/main/AsymmetricLLPs.</footnote>
<figure><img src="image_3.png" /><figcaption>**Fig. 2** Validation plots comparing the selection efficiency obtained from the HAHM model kinematics, and the asymmetric model while treating the lifetime as a dependent or independent parameter. The peak efficiency tends to increase with mediator mass, since the LLPs result-</figcaption></figure>

<figure><img src="image_4.png" /></figure>

ing from heavier mediators are more likely to pass the trigger energy thresholds of the reference search selection, and are easier to distinguish from low-energy QCD backgrounds

In order to perform these validation exercises we first choose four benchmarks sets of masses (with $S_1$ and $S_2$ masses and lifetimes set to be equal), and we use the surrogate models described in Sect. 2 to obtain efficiencies for the events to be selected in region A of the reference analysis. In Fig. 2 we present the obtained efficiencies as a function of lifetime, focusing on the relative changes between (i) the raw HAHM (labeled “HAHM (independent $c\tau$)”), as used in the original ATLAS analysis, (ii) the symmetric limit of the model presented in Sect. 3 with a consistent treatment of the LLP lifetime, i.e. maintaining all relations between $c\tau$ and the other parameters of the model (labeled “asymmetric model (dependent $c\tau$)”) and (iii) the same model, but instead treating the lifetime as an independent parameter by sampling from an exponential distribution (labeled “asymmetric model (independent $c\tau$)”).

The results are presented in Fig. 2. They show that for the lifetime range in which the ATLAS searches are sensitive, the three approaches yield consistent results. These findings confirm two key points. First, the asymmetric model correctly reproduces the original HAHM when the two LLPs have identical masses and lifetimes. Although some deviations occur at low lifetimes, they remain beyond the sensitivity of current ATLAS searches. The second point is that consistently varying the LLP lifetimes and widths is, of course, theoretically sound, but in practice treating them as independent parameters does not significantly affect the results. This latter option is, on the other hand, computationally much more efficient, and indicates that the approach taken by the CMS and ATLAS collaborations is justified within the parameter ranges where they are sensitive.
<figure><img src="image_5.png" /><figcaption>**Fig. 3** The observed 95% CL exclusion on cross-section times branching fraction for an asymmetric model with a 200-GeV mediator, as a function of the two LLP decay lengths ($cτ$, in meters) and masses in GeV</figcaption></figure>

(shown above each cell). The purple contour delimits regions excluded at 0.1 pb, and regions inside the yellow contour are excluded at the level of 1 pb

## 4.2 Going further: asymmetric production of LLPs

Having validated our theoretical and computational framework, we can now use it to go beyond the scenarios which are possible within the original HAHM template model. Namely, we now set the masses and lifetimes of the two LLPs to be different, and we use the surrogate models of Ref. [23] to extract efficiencies. Since, moreover, we have established that dissociating the LLP widths and lifetimes does not modify the results in any noticeable manner, we simply generate one set of events per ($m_{S_1}$, $m_{S_2}$) pair (choosing a macroscopic mean proper lifetime) and subsequently scan over $cτ$ independently. The LLP mass pairs are chosen on a grid of possible values, which depend upon the mediator mass. At each point in the plane of the two LLP $cτ$ values and for each point on the grid of LLP masses, we can obtain the selection efficiency for region A of Ref. [23] using the surrogate model.

Moreover, the reference analysis published the observed and expected number of events in region A. We hence convert this efficiency value to an observed $CL_s$ exclusion value on the allowed cross-section times branching fraction, as shown in Figs. 3 and 4 for mediator masses of 200 GeV and 600 GeV respectively. This allows us to set limits on asymmetric LLP production for the first time.

For the 200-GeV case (Fig. 3) we see, as expected, that the 0.1 pb exclusion contours (violet) are broadly circular as a function of the logarithms of the two LLP lifetimes. The looser exclusion contour, at 1 pb (yellow), has a “corner pipe” shape, dropping off when either LLP has a too-large lifetime (which implies a lack of reconstructible objects in the detector<fnref n="6" />), or when both have too small lifetimes (leading

<footnote n="6">⁶ This is a feature of the reference search (the “CR+2J” channel of Ref. [23]) which requires *both* LLPs to decay before the calorimeters, to suppress QCD backgrounds.</footnote>
<figure><img src="image_6.png" /><figcaption>**Fig. 4** The observed 95% CL exclusion on cross-section times branching fraction for an asymmetric model with a 600-GeV mediator, as a function of the two LLP decay lengths ($cτ$, in meters) and masses in GeV</figcaption></figure>

(shown above each cell). The cyan contour delimits regions excluded at 0.01 pb, and regions inside the yellow contour are excluded at the level of 0.05 pb

to zero displaced trackless jets). The centroid of the exclusion contours shifts to higher lifetimes as the mass of the corresponding LLP increases, due to time dilation effects. Nevertheless, the excluded region at 0.1 pb corresponds to LLP lifetimes between 1 and 10 m for most scenarios, which of course is the range where the calorimeter-based search is expected to be most sensitive.

The 600-GeV case (Fig. 4) exhibits similar features, but the exclusion contours are shifted to much lower values than in the 200-GeV case: 0.01 pb and 0.05 pb (cyan and yellow contours, respectively). The reference analysis is known to be more sensitive to heavy mediator scenarios, and correspondingly much lower cross-sections can be ruled out. Another interesting feature concerns the fact that the size of
the 0.01 pb exclusion contour expands substantially as either of the LLP masses increases: this can be understood since heavier LLPs will produce more central jets, which are more efficiently selected by the analysis. If, however, the masses of both LLPs increase (which happens around the diagonal of Fig. 4), eventually neither of them is produced with enough transverse momentum to be detected, and consequently the sensitivity almost vanishes.

One additional point is worth commenting upon: as we have stressed throughout the previous discussion, the Lagrangian presented in Eq. (3.1) is intended to serve as a reasonably well-motivated and intuitively parametrised kinematic template in order to study asymmetric LLP production. Any (UV-) completion thereof is expected to be subject to additional constraints which can only be quantified once the structure of the underlying theory is fixed. However, even this minimal Lagrangian does predict a complementary probe of the mediator itself since the latter, on top of LLPs, can also decay into (gluon-induced) jet pairs. A detailed analysis of this topic falls beyond the scope of this paper, especially since the exact interplay between different constraints depends, at the end of the day, on the full set of decay modes of the mediator. However, and in order to provide at least some rough estimates concerning how they fare against each other, let us note the following:

* Assuming a fiducial value of $\kappa = 1$, and barring cases of extreme phase space suppression, the mediator decays into LLP pairs are necessarily prompt.

* For the same value $\kappa = 1$, and taking the example of a 600 GeV mediator, a 0.01 pb cross-section times branching fraction into LLPs limit corresponds to a $\Lambda$ scale of $10^6$ GeV. Decays into LLP pairs remain dominant for $\Lambda \sim 10^4$ GeV which, according to the latest ATLAS dijet analysis [33] would render LLP searches a more powerful probe down to these scales. However, below $\Lambda \sim 10^4$ GeV, it is likely that dijet searches will provide the leading constraints.

* For smaller mediator masses, an additional (ISR) jet is needed in order to boost the final state dijet system. This scenario was studied in Ref. [34] and the ensuing constraints were found to be comparable with the heavy mediator case. Hence, in this case too we expect LLP searches to provide the leading constraint down to comparably low values of $\Lambda$.

## 5 Conclusions

In this paper, we studied the constraints that can be set by searches for displaced jets at the Large Hadron Collider on the cross-section times branching fraction of “asymmetric”

LLP production, i.e., the production of a pair of non-identical long-lived particles. In passing, we commented upon a number of subtleties concerning the (unavoidable) usage of template models from the experimental collaborations and the applicability of existing constraints on different New Physics models.

We introduced a new toy model to study these asymmetric hadronic decays of long-lived scalars, addressing shortcomings of a commonly-used benchmark model. Our model, which allows the mediator to decay to two different long-lived scalars, uses LLP lifetimes as input parameters, rendering it more user-friendly. Using publicly available recasting material from a recent ATLAS search for displaced jets to calculate selection efficiencies, we validated that the lifetime-parametrised model in the symmetric limit yields consistent results with the existing benchmark models and that the lifetime can be treated as an independent parameter within the experimental search range. We then presented constraints on various combinations of LLP lifetimes and masses, for two mediator mass choices. Our findings show that existing searches can set constraints on the production cross-section times branching fraction of non-identical LLPs, at levels of 0.1 (0.01) pb for a 200 (600) GeV mediator, for LLP lifetimes ranging from 10 cm to 10 m. Smaller mass configurations are more efficiently constrained at shorter LLP lifetimes, while heavier masses are better constrained at longer lifetimes. As either LLP becomes sufficiently short- or long-lived, the constraints weaken.

**Acknowledgements** TC and LM would like to thank the Laboratoire de Physique de Clermont Auvergne for hospitality during the preparation of this paper. LC and AG are grateful to A. M. Teixeira for enlightening discussions.

**Data Availability Statement** This manuscript has no associated data. [Authors' comment: Data sharing not applicable to this article as no datasets were generated or analysed during the current study.]

**Code Availability Statement** This manuscript has no associated code/software. [Authors' comment: Code/Software sharing not applicable to this article as no code/software was generated or analysed during the current study.]

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Funded by SCOAP³.
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