Exploring Hidden Sectors with Two-Particle Angular Correlations at Future e⁺e⁻ Colliders

Abstract

Future $e^{+}{e}^{-}$ colliders are expected to play a fundamental role in measuring Standard Model (SM) parameters with unprecedented precision and in probing physics beyond the SM (BSM). This study investigates two-particle angular correlation distributions involving final-state SM charged hadrons. Unexpected correlation structures in these distributions is considered to be a hint for new physics perturbing the QCD partonic cascade and thereby modifying azimuthal and (pseudo)rapidity correlations. Using Pythia8 Monte Carlo generator and fast simulation, including selection cuts and detector effects, we study potential structures in the two-particle angular correlation function. We adopt the QCD-like Hidden Valley (HV) scenario as implemented in Pythia8 generator, with relatively light HV v-quarks (below about 100 GeV), to illustrate the potential of this method.

1. Introduction

Correlations play a fundamental role in the study of hadronic dynamics since the beginning of cosmic ray and accelerator physics [1,2]. Recently, the study of angular correlations has revealed a new phenomenon in heavy-ion collisions at both the BNL RHIC [3,4,5,6] and the CERN LHC [7,8,9,10,11], later also found in smaller system (proton–proton, proton–nucleus) collisions [12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28].

In particular, long-range near-side (so-called) ridges appear in two-particle angular correlations, while different theoretical explanations have been put forward to understand this initially unexpected phenomenon [29,30,31,32,33,34,35]. Almost all of those theories require the existence of an unconventional state of matter at the primary interaction of the collision (like, for example, quark–gluon plasma), ultimately yielding collective effects among final-state SM particles. On the other hand, no explicit ridge-like signal was found in the $e^{+}{e}^{-}$ data analysed by ALEPH [36] and Belle [37,38] experiments, except for recent claims in Ref. [39] at the highest available energy and high multiplicity of the former.

Motivated by such unexpected correlation structures found in high-energy collisions, we investigate possible anomalies (not limited to ridge effects) in both azimuthal and (pseudo) rapidity correlations to clarify on insights into unknown stages of matter on top of the QCD parton shower. We focus on two-particle angular correlations as a way of searching for BSM at future high-energy $e^{+}{e}^{-}$ colliders, which offer a significantly cleaner environment than hadron colliders in both theoretical and experimental respects. For instance, QCD effects on matter under extreme temperature and density conditions (for example, leading, to the gluon–gluon condensate [40]) cannot be reached in $e^{+}{e}^{-}$ collisions due to the absence of initial-state gluonic fields and beam remnants. Let us also emphasize that the scope of the this study is limited to the search for new phenomena in leptonic collisions, and does not trivially extend to hadron colliders such as the LHC.

The theoretical framework of new physics (NP) explored in this paper is based on the so-called Hidden Valley (HV) scenario, which actually encompasses a broad class of models with one or more hidden sectors beyond the SM. Thereby, new (valley) particles with masses below about 100 GeV can arise, surviving current experimental constraints (see discussion in Section 3), and contribute to the partonic cascade, ultimately yielding final-state SM particles.

2. Two-Particle Angular Correlations

The clean enough environment of $e^{+}{e}^{-}$ collisions, in contrast to hadronic collisions, is particularly well-suited for defining a reference frame whose z-axis aligns with the direction of the back-to-back jets in most events. The thrust reference frame is adopted in this study, so the rapidity y of a particle is always defined with respect to the thrust or z-axis. The azimuthal angle $\varphi$ is defined, as most ofetn considered, on the transverse plane to the thrust axis, on an event-by-event basis. Note that only rapidity and azimuthal-angle (azimuthal) differences of two final-state SM charged particles 1 and 2, $\mathsf{\Delta }y\equiv y_1-y_2$ and $\mathsf{\Delta }\varphi \equiv {\varphi }_1-{\varphi }_2$, respectively, are considered throughout the study.

The two-particle correlation function is defined as

$$C^{\left(2\right)}(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )={\displaystyle \frac{S(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )}{B(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )}},$$

(1)

where $S(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )$ stands for the density of particle pairs within the same event:

$$S(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )={\displaystyle \frac{1}{N_{\mathrm{pairs}}}}{\displaystyle \frac{d^2{N}_{\mathrm{same}}}{d\mathsf{\Delta }yd\mathsf{\Delta }\varphi }},$$

(2)

while $B(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )$ represents the density of mixed particle pairs from distinct events:

$$B(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )={\displaystyle \frac{1}{N_{\mathrm{pairs},}\mathrm{mix}}}{\displaystyle \frac{d^2{N}_{\mathrm{mix}}}{d\mathsf{\Delta }yd\mathsf{\Delta }\varphi }}.$$

(3)

Here, $N_{\mathrm{same}}$ denotes the number of events where the pairs of particle of the total number $N_{\mathrm{pairs}}$ of all events were selected per same event, and $N_{\mathrm{mix}}$ denotes the number of events from where the pairs of a total number $N_{\mathrm{pairs},\mathrm{mix}}$ of pairs with each particle per a pair from different events for all such events were selected.

Then, the azimuthal yield, $Y\left(\mathsf{\Delta }\varphi \right)$, is of particular interest, being defined by integration over a given $\mathsf{\Delta }y$ range as

$$Y\left(\mathsf{\Delta }\varphi \right)={\displaystyle \frac{{\int }_{y_{\inf }\le \left|\mathsf{\Delta }y\right|\le y_{\sup }}S(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )dy}{{\int }_{y_{\inf }\le \left|\mathsf{\Delta }y\right|\le y_{\sup }}B(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )dy}},$$

(4)

where $y_{\inf \left(\sup \right)}$ defines the lower (upper) integration limit for different rapidity intervals depending on the kinematic region of interest. The $S(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )$ (2) used for the yield calculation are defined as

$$S^{\mathrm{SM}}(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )={\displaystyle \frac{1}{N_{\mathrm{pairs}}^{\mathrm{SM}}}}{\displaystyle \frac{d^2{N}_{\mathrm{same}}^{\mathrm{SM}}}{d\mathsf{\Delta }yd\mathsf{\Delta }\varphi }}$$

(5)

or

$$S^{\mathrm{SM}+\mathrm{HV}}(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )={\displaystyle \frac{1}{N_{\mathrm{pairs}}^{\mathrm{SM}}+N_{\mathrm{pairs}}^{\mathrm{HV}}}}\left({\displaystyle \frac{d^2{N}_{\mathrm{same}}^{SM}}{d\mathsf{\Delta }yd\mathsf{\Delta }\varphi }}+{\displaystyle \frac{d^2{N}_{\mathrm{same}}^{\mathrm{HV}}}{d\mathsf{\Delta }yd\mathsf{\Delta }\varphi }}\right)$$

(6)

for the cases when there only the SM assumption is put forward or also the assumption of the HV existence, respectively, with $B(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )$ (3) being common for both as it is estimated from an enriched SM back-to-back sample, as defined in Section 4 below.

3. Hidden Valley Phenomenology

In most HV models [41], the SM gauge group sector $G_{\mathrm{SM}}$ is extended by (at least) a new gauge group $G_{\mathrm{V}}$ under which all SM particles are neutral. Hence, a new category of HV v-particles emerges charged under $G_{\mathrm{V}}$, but neutral under $G_{\mathrm{SM}}$. Moreover, ‘communicators’, charged under both $G_{\mathrm{SM}}$ and $G_{\mathrm{V}}$, are introduced to the theory allowing interactions between SM and HV particles. Communicators can be either intermediate (considerably massive) vector $Z_v$ bosons, or hidden partners of the SM quarks and leptons, generically denoted as $F_v$, assumed to be pair-produced. In this study, $F_v$ are considered fermions (of the spin 1/2) while HV v-quarks ($q_v$) considered scalars.

Different mechanisms can be considered to connect the hidden and SM sectors through communicators, for exmaple, via the tree-level channel: $e^{+}{e}^{-}\to Z_v\to q_v{\overline{q}}_v\to$ hadrons. Alternatively, communicators can be pair produced via SM ${\gamma }^{*}/Z$ coupling to a $F_v{\overline{F}}_v$ pair, yielding both visible and invisible cascades in the same event. We have checked that the latter channel significantly dominates over the former within the range of energies under study.

For some values of the HV parameter space, communicators may promptly decay into a particle f of the visible sector (its SM partner) and a $q_v$ of the hidden sector according to the splitting: $F_v\to f{q}_v$. Note that the $q_v$ mass may quite strongly influence the kinematics of the visible cascade (leading to SM particles) in the same event, thereby yielding observational consequences. This mass remains currently unconstrained ranging from zero to close to the $F_v$ mass [42], so different values of $F_v$ are assumed throughout this study.

Indeed, we are especially interested in the influence of the invisible HV sector on the partonic cascade into final-state SM particles, leading to potentially observable effects. This can be understood, for example, as both visible and hidden cascades have to share the event’s total energy, thereby modifying their respective available phase spaces. For the sake of concreteness, we restrict this study to a QCD-like HV scenario with $q_v$-quarks, $g_v$-gluons, and an equivalent ‘strong’ coupling constant ${\alpha }_v$. For simplicity, ${\alpha }_v$ is assumed not running with energy but fixed to ${\alpha }_v=0.1$, as no significant differences are found in this study by varying that ${\alpha }_v$ value.

Present limits on masses and couplings in hidden sectors have been set by experimental searches at hadron colliders, particularly the LHC [43]. However, these bounds are primarily derived from searches targeting long-lived particles, events with relatively large missing transverse momentum, or relying on mass peak reconstruction. As such, bounds focus on specific signatures that probe only certain realizations of the HV scenario. For example, current limits on BSM $Z^{\prime }$ vector bosons exclude masses below 2–4 TeV. However, as noted just above, these limits do not apply here.

Acually, a broad class of models falls under the ‘umbrella’ of the HV framework, ranging from unparticles [44] to soft bomb–like phenomena [45], dark sector showers [46] and dark matter candidates [47]. Moreover, since the methodology proposed in this paper notably differs from conventional search strategies, much of the parameter space remains largely unconstrained.

In summary, the values chosen throughout the present study for the masses of the v-quarks, communicators, and the coupling constant ${\alpha }_v$ can be regarded as benchmark scenarios that illustrate the expected behavior of angular correlations and the feasibility of detection.

4. Analysis at Detector Level

To assess the feasibility of HV signal detection in $e^{+}{e}^{-}$ colliders, we rely throughout on the Pythia8 Monte Carlo event generator [48] because of its established reliability and the feature that the HV production channel is already built in.

Fast detector simulations were performed using the SGV tool [49] and the geometry and acceptance according to the extended version of the model of the ILD concept for the International Linear Collider (ILC), as described in Ref. [50]. The simulated events were provided in the same event model by the ILD concept group, and the tools available in the ILCSoft package [51] were used for the event reconstruction and analysis. The ILD reconstruction is based on the particle flow approach [52], which is considered a tool to reconstruct all individual particles produced in the final state via pattern recognition algorithms. Then, the reconstructed candidates of single particles are called particle flow objects (PFOs).

In this study, which includes detector effects, we set the centre-of-mass energy (c.m.) $\sqrt{s}$ of $e^{+}{e}^{-}$ collisions equal to $250\mathrm{GeV}$ corresponding to the planned first commissioning of the collider as a Higgs boson factory. As a further outlook, we extend our prospective study (this time only at particle level) up to $\sqrt{s}=500$ GeV and 1 TeV. The case of longitudinally polarised beams is planned to be addressed in future investigations as soon as this option becomes available.

As already notified in Section 3, the HV signal proceeds via the process $e^{+}{e}^{-}\to D_v{\overline{D}}_v\to \mathrm{hadrons}$ (where $D_v$ denotes the lightest communicator, being the hidden partner of the d-quark) as depicted in Figure 1a. In the benchmark scenarios considered here we set ${\alpha }_v=0.1$, $m_{D_v}=125\mathrm{Ge}\mathrm{V}$, with four different v-quark mass values: $m_{q_v}=0.1,10,50$ and 100 GeV. We also consider $m_{D_v}=80$ and $100\mathrm{Ge}\mathrm{V}$ with $m_{q_v}=40$ and 50 GeV respectively. At higher c.m. energies, other scenarios are also contemplated (see Section 6).

In the current study at $\sqrt{s}=250\mathrm{Ge}\mathrm{V}$, the $e^{+}{e}^{-}\to q\overline{q}$ background arises from the inclusive production of all SM quark species except for the top quark, whose production is kinematically forbidden at this energy (see Figure 1).

In a preceding investigation [53], the HV signal and background were examined at particle level. However, no initial state radiation (ISR) was included, although it plays a crucial role in the study as shown below. Four-fermion production (dominated by $e^{+}{e}^{-}\to WW$) was neither considered, although this process contribution was found to be negligible. In the present study, we extend the investigation to detector level, now taking into account the effect of ISR. This inclusion enables the production of the large enough cross-section process $e^{+}{e}^{-}\to \gamma Z$, where the Z boson is produced on-shell and subsequently decays. The cross-sections for both the HV and SM processes were estimated using Pythia8 generator, as shown in

Table 1. Inclusive cross-sections for $e^{+}{e}^{-}\to D_v{\overline{D}}_v$, $e^{+}{e}^{-}\to q\overline{q}$ and $WW\to 4q$ processes at $\sqrt{s}=250\mathrm{Ge}\mathrm{V}$, with different assignments for the $m_{D_v}$ and $m_{q_v}$ masses. Efficiencies of the selection criteria, and event-averaged charged-track multiplicities along with $\mathrm{RMS}$ of the multiplicities are also shown. See text for details.

Process	${\mathit{m}}_{{\mathit{D}}_{\mathit{v}}}$ (GeV)	${\mathit{m}}_{{\mathit{q}}_{\mathit{v}}}$ (GeV)	${\mathit{\sigma }}_{\mathrm{PYTHIA} 8}$ (pb)	Efficiency (%)	$\langle {\mathit{N}}_{\mathbf{ch}}\rangle$
$e^{+}{e}^{-}\to D_v{\overline{D}}_v$	125	0.1	$0.13$	36	$12.4\pm 3.7$
	125	10	$0.12$	36	$12.4\pm 3.7$
	125	50	$0.12$	42	$11.4\pm 3.5$
	125	100	$0.12$	42	$6.5\pm 2.1$
	100	50	1.29	42	$11.1\pm 3.4$
	80	40	1.54	36	$18.0\pm 4.9$
$e^{+}{e}^{-}\to q\overline{q}$ with ISR			48	≲0.01	$9.9\pm 3.4$
$WW\to 4q$			$7.4$	≲0.001	−

.

In view of quite the small cross-sections for HV production as shown in Table 1, specific cuts need to be applied to maximise background suppression while retaining as much of the signal as possible. Displaced vertices resulting from the production and decay of long-lived particles are not considered in this study, as the focus is on prompt decays of v-particles and their impact on the partonic cascade leading to final-state SM particles. Following a similar strategy to that in Ref. [54], a set of selection cuts has been defined. The final selection-cut efficiency, reported in Table 1, demonstrates a quite a drastic reduction of the SM background, while the HV signal remains largely preserved.

The following selection cuts are applied in this study: consider.

• $\mathrm{S}1$: constraints are set on the number of displaced vertices per jet (to equal to 0) and the number of neutral and charged PFOs (below 22 and below 15, respectively);
• $\mathrm{S}2$: cuts are applied on the reconstructed ISR photon candidates angle ($|cos{\theta }_{{\gamma }_{\mathrm{ISR}}}|<0.5$) and energy ($E_{{\gamma }_{\mathrm{ISR}}}<$ 40 GeV);
• $\mathrm{S}3$: constraints on the di-jet invariant mass ($m_{jj}<130$ GeV) and on the energy of the most energetic jet ($E<80$ GeV);
• $\mathrm{B}1$: a thrust value $T>0.95$ is imposed.

The combined S1, S2 and S3 cuts are applied to the evaluation of $S(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )$ (2), and for the evaluation of $B(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )$ (3), a combination of S1, S2 along with $\mathrm{B}1$ are applied. The selection efficiencies for different combinations of these cuts applied to different processes are given in

Table 2. Breakdown of accumulated efficiencies of the selection criteria. For simplicity, the efficiencies only for one HV model is given, since results are quite similar for others ($e^{+}{e}^{-}\to D_v{\overline{D}}_v$ with $m_{D_v}=125\mathrm{Ge}\mathrm{V}$ and $m_{q_v}=100\mathrm{Ge}\mathrm{V}$). The results for the two SM processes (see Table 1) are also given. Here, we distinguish between the $q\overline{q}$-RR (dominated by radiative return events) and the $q\overline{q}$-cont. in the continuum (i.e., with $m_{q\overline{q}}^{\mathrm{inv}}\approx \sqrt{s}$ at partonic level and before parton shower. See text for further details.

	Efficiency (%)
Cuts	HV	$q\overline{q}$-RR	4q	$q\overline{q}$-cont.
S1, i.e., multiplicity cut	89	30	64	40
S1 and S2, i.e., S1 and ISR photon cuts	42	3.8	12	8.8
S1, S2 and S3, i.e., S1, S2, energy and invariant mass cuts	42	≲0.01	≲0.001	−
S1, S2 and B1, i.e., S1, S2 and thrust cuts	−	−	−	5.6

.

Figure 2 shows three-dimensional plots of the two-particle correlation function $C^{\left(2\right)}(\mathsf{\Delta }y,\mathsf{\Delta }\varphi )$ (1) for the SM (Figure 2a) and HV (Figure 2b) scenarios, before and after cuts. For a reference sample in the HV scenario, we set $m_{q_v}=100\mathrm{GeV}$, $m_{D_v}=125\mathrm{Ge}\mathrm{V}$ and ${\alpha }_v=0.1$ alongside the decay $D_v\to d+q_v$ initiating the partonic (both visible and invisible) shower. As expected, a near-side peak shows up at ($\mathsf{\Delta }y\simeq 0$, $\mathsf{\Delta }\varphi \simeq 0$), receiving contributions mainly from track pairs within the same jet. On the other hand, an away-side correlation ridge around $\mathsf{\Delta }\varphi \simeq \pi$, and extending over a relatively large rapidity range, results from back-to-back momentum balance, actually, not related to NP effects. After cuts, a near-side ridge (with two pronounced and symmetric bumps) shows itself for $1.6<\left|\mathsf{\Delta }y\right|<3$ at $\mathsf{\Delta }\varphi \simeq 0$ for the SM scenario, similar to that in the pure HV scenario. This structure arises from the ISR effect, since the effective c.m. energy approaches the Z mass, and thus the resonant production highly enhances the production cross-section. It becomes of paramount importance to take this effect into account in this study mimicking possible HV signatures in angular correlations.

In order to examine in more detail the possibility of discriminating the HV signal from the pure SM background, we depict in Figure 3 the yield $Y\left(\mathsf{\Delta }\varphi \right)$ (defined in Equation (4)) for both $0<\left|\mathsf{\Delta }y\right|\le 1.6$ and $1.6<\left|\mathsf{\Delta }y\right|<5$ ranges: before (Figure 3a) and after (Figure 3b) cuts. Note that the HV signal for various masses $m_{D_v}$,$m_{q_v}$ is considered together with the SM background, while a standalone findings of the SM background are also presented. A visible difference becomes apparent for $0<\left|\mathsf{\Delta }y\right|\le 1.6$: a sizeable peak at $\mathsf{\Delta }\varphi \approx \pi$ characterises the HV scenario, unlike the pure SM case. This remarkable discrepancy of shapes may potentially serve as a valuable signature of a hidden sector, complementary to more conventional BSM searches, as claimed in this study.

5. Prospects on the Experimental Sensitivity

Let us stress that a search for BSM physics in high-energy collisions relying on angular correlations (as proposed in this paper) offers several advantages with respect to more conventional methods, for example based on an excess of events in cross sections or invariant mass peaks. In particular, yield distributions (4) benefit from an almost total cancellation of reconstruction efficiencies and detector acceptances, luminosity and cross-section dependence, and so on. However, modelling uncertainties could be a limiting factor for these observables. In order to better understand this issue, we performed a study assuming different scenarios for the foreseen statistic and systematic uncertainties.

To estimate the statistical uncertainties, we assumed a collected luminosity of 100 fb⁻¹, which approximately corresponds to one year of data taking of ILC in the H20-staged scenario [55]. For the systematic uncertainties, we identified two potential main sources: the detector performance modelling and the fragmentation (pertaining to the final hadronisation of the partonic cascade) uncertainties. For both of these scenarios, only educated guesses based on current knowledge can be made. For instance, the detector performance modelling uncertainty is evaluated by a bin-by-bin comparison of the estimated yield distributions at particle and detector levels. The absolute value of the difference is taken as uncertainty. The estimated size of this uncertainty is of the order of 5–10% in each bin. This comparison should account for the kinematic resolution (including angular) on the track reconstruction, as well as acceptance effects. All detectors proposals for future Higgs factories foresee high-performing and low mass tracker systems, with near-$100\%$ tracking efficiency and transverse momentum resolution of $\sigma (1/p_T)=2\times {10}^{-5}$ GeV⁻¹$⨁1\times {10}^{-3}{p}_T^{-1}{sin}^{-1/2}\theta$ or better [50]. For those reasons, to stress is that this approach highly overestimates the experimental uncertainties therefore, it represents a worst-case scenario. For the fragmentation uncertainty, we followed the same recipe as for the detector-effects modelling, but this time we compared the predictions of the yield at the particle level calculated using two different fragmentation models implemented in Pythia8 and Herwig 7.3 generators [56,57]. In this case, the estimations lead to a subdominant contribution of the uncertainty of about a third of the estimated detector modelling uncertainties. The total uncertainty, calculated also bin by bin, is composed of the addition in quadrature of the statistical uncertainty for the expected number of events, and the systematic uncertainties of $Y_{\mathrm{SM}}$ since only Pythia8 generator includes Hidden Valley production. The simualtion results due to outcome of these assumptions are summarised in Figure 4, with ${\sigma }_Y$ representing the estimated uncertainty as explained just above.

As soon as all the points discussed just above are taken into account, it is straightforward to calculate the sensitivity of the observable by comparing the HV and SM predictions for Y-yields with the expected uncertainties. Already at 100 fb⁻¹, the sensitivity is found to be mostly limited by the systematic uncertainties as the detector performance systematics is found to dominate over all the other uncertainties. However, the discovery power remains almost intact, especially near the sizeable Y peak at $\mathsf{\Delta }\varphi \approx \pi$ with $0<\left|\mathsf{\Delta }y\right|\le 1.6$. As an exercise, we compared the estimated sensitivity with a much more optimistic scenario in which we improved the systematic uncertainties by one order of magnitude with respect to the current estimates assuming a significant improvement in detector-performance-related uncertainties. Dedicated studies on the fragmentation modelling of HV processes is to be also required for further understanding. Assuming this order of magnitude improvement, the sensitivity considered to quite drastically increase and the discovery power to reach for most of the phase space available. Let us emphasise that this is only an educated guess that to serve as yet another motivation to pursue the best possible design of future collider detectors and progress on Monte Carlo tools development to minimise modelling uncertainties.

6. Different Energies and Colliders

Up to here, we were focused on a future $e^{+}{e}^{-}$ collider operating at $\sqrt{s}=250$ GeV, but those machines later may run at higher energies. To this end, we extended the study to $\sqrt{s}=0.5$ and 1 TeV, at particle level for now.

As the HV signal $e^{+}{e}^{-}\to Q_v{\overline{Q}}_v\to \mathrm{hadrons}$, we have considered the two extreme cases: the lightest communicator $D_v$ and the heaviest one $T_v$ which decays into $q_{v}t$. The findings confirm that intermediate masses yield intermediate results. Besides the $q\overline{q}$ and $WW\to 4q$ backgrounds, we also took into account the production of $t\overline{t}$, which, actually becomes relevant at these energies.

Table 3. Inclusive cross-sections of the HV signal and SM background at $\sqrt{s}=500$ GeV and 1 TeV. Cross-section predictions show no dependence on the $m_{q_v}$ value unless it is too large to make the process kinematically inaccessible. The signal corresponds to $D_v$ and $T_v$ pair production with the $D_v$/$T_v$ masses set to equal $\sqrt{s}/2$. Whenever kinematically allowed, $t\overline{t}$ production has been included as a SM background source in addition to lighter flavours considered at $\sqrt{s}=250$ GeV.

Model	Process	${\mathit{\sigma }}_{\sqrt{\mathit{s}}=500\mathbf{GeV}}$ (pb)	${\mathit{\sigma }}_{\sqrt{\mathit{s}}=1\mathbf{TeV}}$ (pb)
HV	$e^{+}{e}^{-}\to D_v{\overline{D}}_v$	$2.4\times {10}^{-2}$ $(m_{D_v}=250\mathrm{Ge}\mathrm{V})$	$4.4\times {10}^{-3}$ $(m_{D_v}=500\mathrm{Ge}\mathrm{V})$
	$e^{+}{e}^{-}\to T_v{\overline{T}}_v$	$9.5\times {10}^{-2}$ $(m_{T_v}=250\mathrm{Ge}\mathrm{V})$	$1.8\times {10}^{-2}$ $(m_{T_v}=500\mathrm{Ge}\mathrm{V})$
SM	$e^{+}{e}^{-}\to q\overline{q}$ with ISR	11	2.9
	$e^{+}{e}^{-}\to t\overline{t}$	$0.59$	$0.19$
	$WW\to 4q$	3.4	1.3

presents the cross-sections obtained from Pythia8 simulation. The masses of $D_v$ and $T_v$ were set to equal $\sqrt{s}/2$, and results obtained for different $m_{q_v}$ are shown in Table 3. No significant variations are seen around this mass assignment to the communicator. According to expectations, the contribution from the SM backgrounds is found to decreasewith the c.m. energy. For the HV signal, a reduction of the cross-section by two orders of magnitude is obtained at $\sqrt{s}=1$ TeV.

This study is also relevant for the $e^{+}{e}^{-}$ option of the Future Circular Collider (FCC) operating at c.m. energy of $\sqrt{s}=240\mathrm{Ge}\mathrm{V}$, i.e., close to the one mainly considered here. Studies with a preliminary version of the ILD concept adapted to FCC have been started [58]. The limited coverage in the forward region makes ISR-photon detection less efficient; however, the pertinent optimisation of selection cuts recovers sensitivity to an HV signal [58]. Optimization of the ILD design for adaptation to FCC conditions is also to be performed. Similar prospects should be expected for the Chinese Circular Electron Positron Collider (CepC), which shares several design features with the FCC. All Higgs factories concepts (linear or circular) share quite ambitious designs for their detectors, featuring high-quality tracking and vertexing capabilities, including quite large acceptance, as discussed briefly in Section 5. Thus, the method proposed in this study is applicable across various collider designs.

7. Conclusions

The study of particle correlations in high-energy colliders provides valuable insights into matter under extreme temperature and density conditions, somehow reproducing the early-universe conditions when quarks and gluons had not yet bound to form hadrons. On the other hand, this kind of investigation may become a complementary tool to other conventional searches to uncover the existence of new phenomena including BSM physics in Refs. [59,60] and have been studied in this paper.

Motivated by the experimental observation of unexpected structures shown in angular correlations from hadronic collisions, we have explored at detector level the discovery potential for hidden sectors at future $e^{+}{e}^{-}$ colliders using two-particle angular correlations. Specifically, we have considered a QCD-like HV model containing not excessively heavy v-quarks and v-gluons, which may interact with the SM sector via communicators of mass, typically below about 1 TeV. We have focused on $D_v{\overline{D}}_v$ pair production at $\sqrt{s}=250$ GeV, of the order of the expected mass for the lightest communicator in this scenario. We briefly extended our study here at the particle level to higher energies (up to 1 TeV), which pointed at promising prospects.

To conclude, the results obtained here show that two-particle azimuthal correlations in a $e^{+}{e}^{-}$ Higgs factory could indeed become a useful tool for discovering NP as kinematically accessible. Although a specific HV model has been employed in this tudy, other types of hidden sectors believed to yield similar signatures. In addition, collective effects stemming from a source different from BSM cannot be excluded in this kind of studies. Such searches, based on rather diffuse signals that spread over a large enough number of final-state particles, to be contemplated as complementary to other more conventional methods, thereby increasing the discovery potential of Higgs factories.