Light hidden mesons through the Z portal

Confining hidden sectors are an attractive possibility for physics beyond the Standard Model (SM). They are especially motivated by neutral naturalness theories, which reconcile the lightness of the Higgs with the strong constraints on colored top partners. We study hidden QCD with one light quark flavor, coupled to the SM via effective operators suppressed by the mass M of new electroweak-charged particles. This effective field theory is inspired by a new tripled top model of supersymmetric neutral naturalness. The hidden sector is accessed primarily via the Z and Higgs portals, which also mediate the decays of the hidden mesons back to SM particles. We find that exotic Z decays at the LHC and future Z factories provide the strongest sensitivity to this scenario, and we outline a wide array of searches. For a larger hidden confinement scale Λ ∼ O (10) GeV, the exotic Z decays dominantly produce final states with two hidden mesons. ATLAS and CMS can probe their prompt decays up to M ∼ 3 TeV at the high luminosity phase, while a TeraZ factory would extend the reach up to M ∼ 20 TeV through a combination of searches for prompt and displaced signals. For smaller Λ ∼ O (1) GeV, the Z decays to the hidden sector produce jets of hidden mesons, which are long-lived. LHCb will be a powerful probe of these emerging jets. Furthermore, the light hidden vector meson could be detected by proposed dark photon searches.


Introduction
A confining hidden sector, or "hidden valley," that interacts weakly with the visible sector is an intriguing possibility for physics beyond the Standard Model (BSM) [1]. There is, however, a wide range of options for such theories, and without some guiding principle it is unclear what particles and interactions should be expected to populate the hidden valley. One way to make progress is to connect the hidden sector to the resolution of one or more of the open questions of the SM, such as the naturalness and dark matter problems. The naturalness problem of the weak scale remains one of the most important motivations to explore BSM physics. However, the apparent lack of new physics (NP) signals at the Large Hadron Collider (LHC) has put significant strain on "traditional" models of supersymmetry (SUSY) and of composite Higgs. In these models light colored top partners cancel the quadratic ultraviolet (UV) sensitivity of the Higgs mass parameter. Except for some special cases, the LHC bounds on new colored particles have reached beyond 1 TeV. A possible way to evade the strong experimental constraints is that the new light particles

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We minimize model dependence by phrasing our discussion within an effective field theory (EFT), where the low-energy effects of the heavy EW-charged particles are captured by higher-dimensional operators built out of the SM and light hidden fields. The prototypical hidden sector we consider contains one Dirac fermion ψ with mass m ψ , transforming in the fundamental representation of a hidden SU(N d ) color group that confines at scale Λ m ψ , where we take N d = 3 as motivated by NN. The fermion is a complete singlet under the SM gauge symmetries, but interacts with the visible sector according to the Lagrangian L = L SM + ψ(i / D − m ψ )ψ −Ĝ a µνĜ a µν /4 + L 6 , where the covariant derivative acting on ψ is D µ = ∂ µ − ig dĜ a µ t a . For simplicity we assume CP conservation in the hidden sector, and therefore take m ψ to be real and neglect the θ-term of hidden QCD. The non-renormalizable part of the Lagrangian reads where the m 2 t /v 2 = y 2 t /2 factor manifests its origin from a solution to the naturalness problem. M is the mass of heavy EW-charged fermions, whereas c g is a dimensionless parameter (4πα d ≡ g 2 d ). The EFT description encoded by L is valid at energies Λ E M . The chiral structure of L 6 is inspired by the TT model, and will be assumed in the rest of the paper. 1 In unitary gauge, the second operator in eq. (1.1) yields a small coupling of ψ to the Z boson, g Z 2 where g Z ≡ g 2 + g 2 . The first operator in eq. (1.1) can be rewritten, by using the leading-order equation of motion for ψ, as m 2 t |H| 2 m ψ ψψ/(M 2 v 2 ). Hence, the couplings of the Higgs to hidden particles read a µνĜ a µν . (1. 3) The interactions in eqs. (1.2) and (1.3) mediate decays of the Z and h to the hidden sector. In addition, they control the decays of the lowest-lying hidden hadrons, which are light mesons [24]. We focus on a 1-flavor hidden QCD theory because this case arises most naturally in the TT model. Since the anomaly removes all chiral symmetries and therefore no light pseudo-Nambu-Goldstone bosons (pNGBs) are expected [25], several among the lightest mesons play important roles in the phenomenology. This is in contrast to the multi-flavor scenario, where the hidden pions are expected to dominate, and whose phenomenology will be the subject of a separate publication [26]. Incidentally, we note that the lightest baryon of the 1-flavor theory, (ψψψ) with spin 3/2, could be cosmologically stable due to hidden baryon number conservation and provide an interesting candidate for asymmetric dark matter, along the lines followed in ref. [27] for the FTH model with light twin bottom. 2 1 The effective operators in eq. (1.1) can be contrasted, for example, with those obtained in a Fraternal Twin Higgs (FTH) model [12] with light twin b, L FTH 6 = |H| 2 mbbb − α dĜ a µνĜ a µν /(12π) /f 2 with mb = ybf / √ 2. 2 See also ref. [28] for asymmetric dark matter in the mirror TH model, and ref. [29] for the FTH withb much heavier than the confinement scale.

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An indirect constraint on the EFT in eq. (1.1) comes from 1-loop corrections to the T parameter of electroweak precision tests (EWPT). Diagrams with two insertions of the second operator give a quadratically divergent contribution that we expect to be cut off at M , resulting in T = κN d y 2 t m 2 t /(16π 2 M 2 ), with κ a UV-dependent O(1) coefficient. For example, the fermionic sector of the TT model gives κ = 4/3, as shown in appendix A. The current constraint T 10 −3 then bounds M 0.87 TeV, while future e + e − colliders will be able to improve the sensitivity to T 10 −4 [30], corresponding to M 2.7 TeV. 3 As usual, though, the EWPT constraints can be importantly affected by additional unknown corrections. In this paper we focus on direct probes of the hidden sector, which, as we show, extend the reach to larger M in many regions of parameter space.
Identifying the most promising signatures requires a detailed understanding of the spectrum and decay patterns of the lightest hidden mesons. Depending on their masses and on the mass scale of the heavy EW-charged particles, the expected signals range from prompt two-body decays, to hidden parton showers followed by displaced decays. We analyze many of these possibilities in detail, finding that Z decays, especially, will have an impressive NP reach both at the LHC and at future e + e − colliders. While rare and exotic Higgs decays have been extensively studied at the LHC (see e.g. refs. [31,32]) and also at future Higgs factories [33,34], the sensitivity of Z decays to hidden sectors has been much less explored. Reference [35] studied the LHC reach on both prompt and long-lived decays for a hidden Abelian Higgs model, whereas ref. [36] focused on prompt decays at future Z factories in scenarios where the hidden sector contains the dark matter particle.
This paper is organized as follows. In section 2 we introduce a new version of the TT framework [10], which realizes the scenario outlined above. While the model provides important motivation, our discussion is structured so that readers whose primary interest is phenomenology may omit section 2. In section 3 we discuss the essential ingredients for our phenomenological study: the production of the light hidden mesons through Z and Higgs decays, as well as the expected pattern of hidden meson lifetimes and branching ratios. Section 4 presents the analysis of the collider phenomenology, and contains our main results. We summarize and conclude in section 5. Finally, three appendices complete the paper.

A new tripled top model
This section presents a NN model with a confining hidden sector of light mesons, whose constituents are SM-singlet fermions. The interactions between the hidden and visible sectors are described by the EFT in eq. (1.1). The construction is an alternative realization of the TT framework proposed in ref. [10].
Tripled top models are supersymmetric extensions of the SM which include two copies of a hidden top sector, each charged under its own SU(3) color gauge group. The hidden sectors consist of vector-like SU(2) L -doublet and -singlet hidden top supermultiplets, and JHEP11(2019)031 the stabilization of the Higgs mass is achieved by means of an accidental supersymmetry in their spectrum [9,10]. In ref. [10] the scalar top partners were chosen to be complete SM singlets. However, from the point of view of naturalness there is no particular preference for EW-singlet top partners. It is straightforward to write down an alternative model where the roles of doublets and singlets are switched. 4 The superpotential of the three top sectors is then where H = H u and the subscript A labels the SM fields, while B and C denote the two hidden sectors. A Z 3 symmetry is assumed to relate the top Yukawa couplings and the SU(3) gauge couplings of the three sectors. It is softly broken to a Z 2 that exchanges the B and C sectors by the supersymmetric mass terms M and ω. The scale M is taken to be multi-TeV, while the size of ω will be discussed momentarily. We have neglected the additional superpotential terms ∼ȳ(Q c B H d u B + {B → C}), as they constitute a hard breaking of the Z 3 . The SM fields have the usual charges under the EW SU(2) L × U(1) Y , which also defines the component fields. The charges of the B and C fields are chosen to be In the above expression the "u" fields are SU(2) L singlets, while "t" states are the electrically-neutral components of doublets. The hypercharges are chosen such that "u" fields in the B, C sectors are complete SM singlets. In addition, the following form is assumed for the leading soft SUSY-breaking masses, The soft mass m is assumed to be close to M , so that the colored A stops are raised to the multi-TeV scale. On the other hand, the cancelation between M 2 and m 2 makes the hidden sector scalars Q B,C light, with masses in the few hundred GeV range. The Higgs potential in this new model is identical to the one presented in ref. [10]. However, here the light EW-doublet scalars Q B,C play the role of the top partners, cutting off the quadratic contribution to the Higgs potential from the top quark loop. For this reason, we call them "top siblings." In addition, the supermultiplets u B,C , u c B,C , which are denoted as "top cousins," are complete SM singlets, hence ω can be JHEP11(2019)031 taken very small without violating any experimental constraint. The scalar components u B,C ,ũ c B,C are still expected to receive sizable soft SUSY-breaking masses and become heavy. Conversely, the fermions remain light and, if ω is smaller than the confinement scale Λ QCD B,C ≡ Λ of SU(3) B,C , they efficiently break the hidden QCD strings and form light hadrons. This is the region of parameters we are interested in: a TT model with light singlet cousin fermions, which for brevity we simply call TT in this work.
As described, this setup successfully stabilizes the Higgs mass against the multi-TeV scale M . Yet, for it to be a complete natural theory in the UV, the peculiar pattern of opposite-sign, equal-magnitude soft mass terms in eq. (2.4) must be explained, as well as the proximity of the soft-breaking and SUSY masses in eq. (2.5). A possible origin of the special structure of soft masses was presented in ref. [10], whereas ∆ M requires a ∼ ∆ 2 /M 2 fine-tuning in the absence of a theoretical mechanism that relates the soft and SUSY masses. We do not discuss these issues any further here, since our purpose is to use the model as an example for phenomenological studies. We also note that eq. (2.4) only represents the leading soft SUSY-breaking terms in the top sector. The A sector gluino and light generation squarks must also have multi-TeV SUSY-breaking masses to satisfy LHC bounds. All other fields can receive subleading SUSY-breaking masses of a few hundred GeV which split the fermions and bosons in the supermultiplets, without spoiling naturalness.
For each of the two hidden sectors, by integrating out the heavy fields with masses ∼ M and ∼ ∆ we obtain a 1-flavor QCD with couplings to the SM dictated by eq. (1.1). However, to explicitly demonstrate how the results arise from a UV-complete model, we keep the heavy states "integrated in" in the following discussion. Since the two hidden sectors are identical, we only discuss the B sector. For simplicity, we assume the Higgs sector is in the decoupling limit at large tan β, so in unitary gauge GeV, and h denotes the physical Higgs boson. The mass matrix for the fermions is , where the rotations are given by (we use capital letters for the mass eigenstate fields) with mixing angles (2.8) The first equality in eq. (2.8) is exact whereas the others have been expanded for small ω. As a result,

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where we have expanded for small ω. The electrically-charged Ψ − B ≡ (b B , b c † B ) form a Dirac fermion with Q = −1 and mass M . 5 The mixing parameterized by R(θ R ) couples ψ B to the Z boson. In four-component spinor notation the coupling reads (g Z /2) sin 2 θ R ψ BR / Zψ BR , which to leading order in a large M expansion gives eq. (1.2). Similarly, the R(θ L ) mixing matrix leads to a ψ BL / Zψ BL coupling, but this is suppressed by an extra factor of (ω/M ) 2 , so we neglect it. After rotating to the fermion mass eigenstate basis, the top Yukawa interactions couple the light eigenstate to the Higgs boson as (y t / √ 2) sin θ L cos θ R h ψ B ψ B . Expanding in large M gives the first term in eq. (1.3).
We calculate the Higgs coupling to hidden gluons by recalling that, given a set of Dirac fermions f and complex scalars φ which transform in the fundamental of SU(N d ) and with couplings −L = f g hf f hf f + φ g hφφ hφ * φ, the 1-loop Higgs coupling to gluons reads, allowing for off-shell Higgs with four-momentum p µ h , In the fermionic term the relevant couplings are g hψ B ψ B = −y t sin θ L cos θ R / √ 2 and g hΨ 0 , for small ω the contribution of ψ B can be neglected, yielding (c g ) fermions 1 in eq. (1.3).
Finally, to calculate the second term in eq. (2.10) we must discuss the scalar sector. The mass matrices are where the terms proportional to δm 2 include in a crude way the effects of subleading SUSYbreaking masses. We expect δm 2 ∼ (100 GeV) 2 ∆ 2 , M 2 . Diagonalization is achieved where m =ȳv cos β/ √ 2, mt = ytv sin β/ √ 2 and we have kept tan β arbitrary. If ω m mt, M we would find sin θL M m/(M 2 + m 2 t ) and m ψ mtm/ M 2 + m 2 t , whereas θR and M Ψ 0 remain as in eqs. (2.8) and (2.9). In this work we setȳ = 0.

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(2.14) and mixing angles The physical masses are In addition we haveb B andb c B with charges −1 and +1, respectively, and mass ∆. From the D-term potential we obtain the couplings to the Higgs, (2.17) As in the fermion case, at small ω the contribution of the light scalars to eq. (2.10) can be neglected. The leading term originates from the T B , yielding (c g ) scalars M 2 /(4∆ 2 ) in eq. (1.3). Since ∆ M requires a ∼ ∆ 2 /M 2 accidental cancellation, a moderate tuning of O(10)% corresponds to (c g ) scalars ∼ 2 -4. In passing, we note that the current constraint on the T parameter only requires ∆ 400 GeV, as shown in appendix A, making a study of the collider phenomenology of the EW-doublet scalar top partners an interesting direction for future work. In this paper, however, we concentrate on the less model-dependent production of hidden hadrons from decays of SM particles.

Production and decays of the light hidden mesons
This section sets the stage for our study of the hidden sector phenomenology. We first discuss production of the light hidden mesons through rare Z and Higgs decays, and then analyze the expected pattern of hidden meson lifetimes and branching ratios.

Production
The coupling in eq. (1.2) gives the Z a width for decay to one hidden sector, Taking the hidden color factor N d = 3 , the corresponding branching ratio into both sectors is From eq. (1.3) we calculate the widths for Higgs decay to the light hidden fermions and gluons, The corresponding branching ratios show that Higgs decays to hidden fermions are negligible. Including 2-loop running with one flavor, the coupling is α d = α d (m h /2; Λ) 0.18 for Λ = 5 GeV, 6 and we have chosen as reference c g = 4, motivated by the TT discussion in section 2. Equations (3.2) and (3.4) assume the existence of two identical hidden sectors (labeled B and C), as in the TT model; unless otherwise noted, we retain this assumption throughout the paper. The results for a scenario with a single hidden sector are trivially obtained by adjusting appropriate factors of 2. At the LHC, the inclusive cross section for Z production is where a f = T 3 Lf and v f = a f − 2s 2 w Q f , while K Z = 1.3 is an approximate K-factor that accounts for QCD corrections [37]. The parton luminosity is L qq (τ ) = 1 τ dx x f q (x)fq(τ /x) + f q (τ /x)fq(x) and in numerical evaluations we use MSTW2008NLO PDFs [38] with factorization scale set to m Z /2. The Higgs cross section is a thousand times smaller, σ h ≈ 48.6 (54.7) pb at 13 (14) TeV for the dominant gluon fusion channel [39]. Combining these with eqs. (3.2) and (3.4) we find that the expected number of Z decays to the hidden sectors is ∼ 120 (0.18/α d ) 2 (4/c g ) 2 times larger than the analogous number for the Higgs. Turning to future electron-positron colliders, the total cross section for production at the Z pole is where K QED 0.73 accounts for QED photon radiation [40]. A Z-factory will be able to produce 10 9 to 10 12 Z bosons, corresponding to an integrated luminosity of 22.9 fb −1 (ab −1 ) for the GigaZ (TeraZ) option. On the other hand, a Higgs factory running at √ s ∼ 240 -250 GeV will yield a lower number of Higgses, ranging from 10 6 to 10 7 depending on the collider configuration. In light of these considerations, in what follows we focus on Z decays. Nonetheless, we discuss Higgs decays when they provide a useful term of comparison.

Decays
For ω < Λ, the color group in each hidden sector has one light quark flavor. This theory does not predict a light pNGB: using standard J P C notation, the lightest hadrons are expected to be the s-wave 0 −+ , 1 −− mesons, and the p-wave 0 ++ meson, which we denote asP ,V , andŜ, respectively. Lattice calculations have not yet provided precise information about the mass spectrum [24], in particular no attempt to evaluate mV has been made. We make the reasonable assumption that mP mV < mŜ, and take as reference values mP , mV = 2Λ, 7 and ∆m ≡ mŜ−mV = Λ. The latter is motivated by the preliminary lattice result mŜ/mP ≈ 1.5 [24]. However, we provide results in general form, and depart from the above benchmarks whenever this has important consequences. As it will be discussed momentarily, this is especially relevant for ∆m, on which the lifetime ofŜ depends very sensitively. The mesons decay back to SM particles through the annihilation of their constituents, which proceeds via the small couplings to the Z and h bosons in eqs. (1.2) and (1.3). The resulting pattern of lifetimes and branching ratios is a crucial input to study the collider phenomenology, so we analyze it in detail. By contrast, we neglect the baryons as they are significantly heavier than the lowest-lying mesons: lattice calculations place the mass of the lightest baryon at (2.5 -3)mP [24].
TheV (1 −− ) meson decays democratically to SM fermions, through the coupling of ψ to the transverse Z boson in eq. (1.2). The width forV → ff decay is (3.8) We have summed over all SM fermions except the top quark, and taken as reference mV = 2Λ. For smaller masses the decays can be displaced: taking mV = 2 GeV we find cτV ∼ 10 cm 2 GeV mV (3.9) 7 This choice is motivated by the spectrum of SM QCD, where ΛQCD A ≈ 370 MeV according to a 2-loop RG analysis [10] and the mass of the ω meson is mω ≈ 780 MeV ∼ 2 ΛQCD A . 8 Recall that ψ(0) = R(0)/ √ 4π where R is the radial wavefunction. Our simple estimate |R(0)| = Λ 3/2 has been checked by comparing with the decays of light SM vector mesons to e + e − . Neglecting Z exchange,

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TheP (0 −+ ) decays dominantly to the heaviest SM fermion that is kinematically available, through exchange of the longitudinal mode of the Z. The corresponding width is where we have replaced m ψ with the constituent mass µ ψ = O(1) × Λ, since the chiral symmetry breaking from the condensate dominates for ω Λ. Estimating again 9 |ψ(0)| 2 = Λ 3 /(4π), for mP > 2 m b we obtain a decay length cτP ∼ 0.3 mm mP 10 GeV where we took mP = 2Λ as reference and included subleading decays to c and τ . IfP is too light to decay to bb, it can be long-lived. For example, taking mP = 2 GeV we have (3.12) In this estimate we have included decays to µ + µ − as well as to ss. For the latter, sincê P → KK is forbidden by CP invariance the leading decay is the three-bodyP → KKπ.
To approximately account for this [42,43] we multiply the perturbative width Γ(P → ss) in eq. (3.10) by where ρ denotes the phase space for isotropic 3-body decays [43], and take m s * = 450 MeV as motivated by a perturbative spectator model [44]. The resulting width for decay to strange hadrons is ∼ 0.15 of that to muons.
For the p-wave scalarŜ (0 ++ ), two competing decay channels exist. The first isŜ → ff through Higgs exchange, whose width is [16] where λ hf f = y f / √ 2. The effective hψψ coupling can be estimated by observing that for ω Λ it originates dominantly from the interaction of the Higgs to the hidden gluons in eq. (1.3). We relate the corresponding matrix element to the ψ constituent mass via the QCD trace anomaly [45], obtaining λ hψψ = 2c g µ ψ m 2 t /(3bvM 2 ) where b = 11 − 2N l /3 is the leading-order coefficient of the beta function. In our case we have N l = 1 light flavor. The second channel is the electric dipole-type transitionŜ →V (Z * → ff ). We estimate its width by consideringŜ →V γ (which actually vanishes in our setup, since Q ψ = 0) and making an appropriate replacement of couplings. We begin with [46] TheP → ff decay width can also be calculated by defining theP decay constant via 0|ψγ µ γ5ψ|P (q) = ifP q µ . This leads to the identification fP = 4 √ N d µ ψ |ψ(0)|/m 3/2 P , which for our typical assumptions yields

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where ε if accounts for the overlap of the radial wavefunctions of the initial and final mesons. One finds ε if ∼ a where a is the size of the bound states, hence for ω Λ we estimate ε if ∼ Λ −1 . The replacement of photon radiation with Z * → ff radiation is approximately captured by the substitution 15) where N f counts the SM fermions with 2m f < ∆m. Thus we obtain For ω Λ the dipole decay dominates: assumingŜ → bb is kinematically open and taking (3.17) where N f = 18 includes all SM fermions except the top and bottom. For lighterŜ the ratio in eq. (3.17) is even smaller, since we need to replace y b with the Yukawas of the light fermions. Thus we expect that decays to ff can be neglected, unless ∆m Λ. When the dipole decay dominates, theŜ decay length is (3.18)

Collider phenomenology of the light hidden mesons
The main production mechanism for the hidden mesons at colliders are the decays of the Z and Higgs bosons. Production via decays of heavy particles in the hidden sector is also present in general, at least from the EW-charged states with mass ∼ M that give rise to the effective interactions in eq. (1.1). However, other channels may also exist, for example, in the TT model discussed in section 2 the scalar top partners with mass ∼ ∆ can also play an important role. As these additional channels are more model-dependent, being very sensitive to the detailed spectrum and decay modes, we focus on the Z and h decays. These probe the most interesting range of hidden meson masses, going from ∼ m Z,h /2 down to the SM QCD scale. Very few parameters determine the phenomenology. The heavy mass M controls the strength of the interactions with the SM, 10 while the hidden confinement scale Λ sets, 10 As discussed in section 2, in the TT model M also roughly sets the mass scale of SM-colored degrees of freedom, because m ∼ M . In this case direct LHC searches for the colored stops bound M 1-1.5 TeV. However, this does not need to be the case in general, and a priori M could be significantly smaller if only EW-charged states appear at this scale. modulo O(1) coefficients that must calculated on the lattice, all hadronic quantities. These include the masses of the mesons (recall that we assume ω Λ, so our typical benchmark is mP ∼ mV ∼ 2Λ) as well as the wavefunction overlaps and the constituent mass of ψ. Another relevant parameter is the mass splitting ∆m, on which the lifetime of the scalar meson depends very sensitively (as illustrated by figure 1), although it is in principle also determined by Λ. In this parameter space we have identified several different regions, each leading to distinct phenomenological predictions.
For large confinement scale, Λ 10 GeV, the hidden mesons are not much lighter than m Z,h /2, hence phase space forces the Z and h decays to be dominantly two-body. At these larger Λ values the mesons are likely to decay promptly back to the SM for M ∼ O(TeV). This phenomenology is discussed in subsection 4.1.
If Λ 10 GeV but M is very large, roughly M 5 TeV, then some of the mesons become long-lived. The relatively large meson masses imply that ATLAS and CMS (as well as future Z factories) can be sensitive to their displaced decays, opening up an alternative strategy to detect the hidden sector. We discuss searches for heavy long-lived hidden mesons in subsection 4.2.
For smaller confinement scale, Λ 2 -3 GeV, the Z and h decays to the hidden sector result in the production of two (or more) hadronic jets, dominantly composed of light hidden mesons. In this region of parameters some or all of the hidden mesons are expected to be long-lived, so the phenomenology bears similarities with the emerging jet scenario [47]. However, the current ATLAS and CMS emerging jet searches do not apply to our signals, due to the rather soft nature of the latter. Instead, we find that the best sensitivity is obtained at LHCb. This is discussed in subsection 4.3.
If the hidden mesons are sufficiently light, then other production mechanisms become relevant, such as, for example, decays of the SM B and K mesons, as well as JHEP11(2019)031 brehmsstrahlung and Drell-Yan processes. In this case the vector mesonV , which couples to the SM by mixing with the Z boson, gives the largest signals. In subsection 4.4 we reinterpret current bounds and future searches for dark photons in our parameter space.
Finally, for very small Λ the hidden mesons are very long lived, and typically escape the detectors. The signals to look for are invisible Z and h decays. We translate the corresponding bounds to our setup in subsection 4.5.

Two-body prompt decays
For large values of the confinement scale, Λ ∼ 10 GeV, we expect all mesons to decay promptly if M ∼ O(TeV), as shown in figure 1. Furthermore, the hidden mesons are sufficiently heavy that the Z boson dominantly decays to two-body final states. We ex-pectPŜ to be the leading mode, because using an EFT for the mesons we can write the unsuppressed couplingĝ Z Z µ (Ŝ ∂ µP −P ∂ µŜ ). 11 In our analysis we focus primarily on Z → (P → bb)(Ŝ →V ff ) followed byV → ( = e, µ), which leads to bb + X. We study this final state both at the LHC and at future Z-factories. At the Z-factories we also considerV → νν, leading to bb + missing momentum + X. This last analysis is presented in appendix B.
Subleading modes include decays to a (pseudo-)scalar and vector, which require one flip of the ψ chirality. From the effective couplings cVŜ (P )ĝ Z µ ψ Z µV µŜ (P ), with where the decay to longitudinally-polarizedV is assumed to dominate. TheVP mode produces final states similar to those of the dominantPŜ mode (albeit with slightly different kinematics), so we do not discuss it further. We do analyzeVŜ, focusing on Z → (V → )(Ŝ →V ff ) followed byV → . This yields a very clean 4 -lepton+X final state, which plays an important role at the LHC.
The last remaining two-body decay isVV , mediated by an interaction of the form cVVĝ Z (m 2 V /m 2 Z ) µνρσ Z µVν ∂ ρVσ where cVV is a dimensionless coefficient. This coupling can, in general, arise from anomalous Wess-Zumino terms (see ref. [48] for a recent discussion of light anomalous vectors). We have extracted a factor of m 2 V /m 2 Z , which is expected in our setup, to make manifest the smooth decoupling for mV m Z of the decay width Γ(Z →VV ) = c 2 ). We define f XY ≤ 1 as the fraction of Z decays to the hidden sectors that yield the XY final state. Rather than attempt to accurately estimate f XY , based on the discussion above we simply take as reasonable benchmarks fPŜ ∼ 1 for the leading mode and fVŜ, fVV ∼ 0.1 for subleading modes.
Z → bbµµ + X at the LHC. While the LHC collaborations have not yet performed a dedicated search for this final state, we can glean some information from the searches for 11 Matching the Z →PŜ decay width to the one for Z → ψψ in eq. (3.1), we can identify parametricallŷ

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h → aa → bbµµ, where a is a light pseudoscalar [49,50]. The CMS analysis [49] imposes softer cuts on the muons and b-jets, namely and is therefore better suited to retain sensitivity to our signal than the ATLAS analysis [50].
To estimate the total acceptance times efficiency (A ) tot for our signal to pass the basic selection of eq. (4.1), we implement our model in FeynRules [51] and simulate the process pp → Z → (P → bb)(Ŝ →V ff → µµff ). TheŜ →V ff decay is described by an effective coupling structureŜV µ f γ µ P L f and we take f = u to capture the dominant decay to quarks. The signal is generated using MadGraph5 v2.6.6 [52], at leading order in QCD including up to one additional parton. Events are showered using Pythia8 [53] and detector response is modeled with Delphes3 [54]. For the latter we use the CMS card, but lower the p T threshold to 5 GeV for the muons and 10 GeV for the jets; in addition, we apply a flat total b-tagging efficiency bb = 3 × (0.4) 2 ≈ 0.5, which was estimated from the requirements described in ref. [49]. With these settings we reproduce within 20% the total acceptance times efficiency for the h → aa → bbµµ signal to pass the basic selection, which is (A ) tot ∼ 5% for m a = 20, 40 GeV.
We consider two benchmark mass spectra, where σ(pp → Z) was given in eq. (3.5) and the branching ratios are BR(P → bb) ≈ 0.88 and BR(V → µµ) ≈ 0.034 in this meson mass range. Note that we have taken BR(Ŝ → V ff ) 1, as expected from eq. (3.17). Assuming the integrated luminosity used in the CMS analysis, L = 35.9 fb −1 , we find N I, II S = 3.4, 3.9 fPŜ (2 TeV/M ) 4 . Key distributions for the signal are shown in figure 2.
To estimate the reach we exploit the fact that the CMS paper provides (see their figure 3, middle-right panel) the expected background yields after the basic selection, down to m bbµµ = 75 GeV. We find that the best sensitivity is achieved by selecting m bbµµ ∈ We stress that these bounds are obtained without exploiting the characteristic feature m bb ≈ m µµ ≈ mP ,V of our signal, which would permit a further suppression of the background. They are, therefore, an extremely conservative illustration of the reach. We do not attempt a dedicated analysis here, but encourage the experimental collaborations to undertake it. Keeping the transverse momentum cuts as low as possible will play an important role: we have checked that softening slightly the CMS cuts to p µ1,2 T > 17, 8 GeV (corresponding to the thresholds for the dimuon trigger [49]) and p b1,2 T > 15, 15 GeV increases the signal efficiency by a factor 1.9 (1.8) for benchmark I (II). Conversely, the efficiency for our signal to pass the moderately harder cuts employed in the ATLAS selection [50] is 5-10 times smaller than for the CMS selection.
Z → 4µ + X at the LHC. For the Z → (V → )(Ŝ →V ff → ff ) signal we use the results of the CMS search for a light Z in Z → 4µ events [55]. The basic event selection requires 4 µ with p µ T > 5 GeV, |η µ | < 2.4, of which ≥ 2 with p µ T > 10 GeV and ≥ 1 with p µ T > 20 GeV, zero total charge, m µ + µ − ∈ [4, 120] GeV for all combinations. (4.5) In addition, the total invariant mass must lie within m 4µ ∈ [80, 100] GeV. The µ + µ − pair with invariant mass closest to m Z is defined as Z 1 , and the other pair as Z 2 . Depending on its mass, the Z is then typically reconstructed as Z 1 or Z 2 . These requirements are not well suited to our scenario, for two reasons. First, the m 4µ > 80 GeV cut removes the bulk of our signal. The ff pair carries a significant amount of energy, pushing m 4µ well below m Z . Second, the presence of two dimuon resonances causes the Z 1,2 reconstruction to either completely fail to produce a peak, as in benchmark I, or be inefficient, producing peaks at m(Z 1,2 ) = mV but also important tails that reduce the sensitivity, as in benchmark II. Therefore, we retain the basic selection of eq. (4.5), but propose dedicated cuts to target our signal. The m 4µ distribution for the signal and SM background after the basic selection are shown in the left panel of figure 3. The simulation parameters are identical to the bb + X analysis. For the background we generate qq → 4µ in the SM with up to one additional parton, including in the normalization an approximate K-factor of 1.3 and a further rescaling factor of 1.15 to match the expected number of events in the m 4µ ∈ [80, 100] GeV window quoted by CMS [55]. This simplified prescription allows us to obtain agreement at the level of ∼ 20% with the shape of the background quoted by CMS, which we consider sufficient for our scope. We then require, in addition to eq. (4.5), that (4.6) The distribution of m µ + µ − , averaged between the two values that are within 1 GeV of each other for each event that passes this additional selection, is shown for signal and background in the right panel of figure 3. Finally, we require that the average m µ + µ − ∈ [mV − 0.5 GeV, mV + 0. To conclude, we note that the Z →VŜ signal discussed here shares some features with Z → A h D studied in ref. [35], where A and h D are a dark photon and dark Higgs, respectively. However, in ref. [35]   inclusive in the decays of the A . Here we have followed a different strategy, in particular we did not attemp to reconstruct theŜ invariant mass peak.
The Z → (V → )(V → ) decay gives a very similar signature, except the total invariant mass of the four leptons peaks at m Z . We apply the same event selection described in eqs. Z → bb + X at Z factories. Turning to the prospects at future Z factories, we analyze first the ee → Z → (P → bb)(Ŝ →V ff → ff ) final state, where includes both electrons and muons. The main SM background is ee → bb , with amplitude at O(g 4 w ). We generate both signal and background using MadGraph5, interfaced with Pythia8 for parton showering. Detector simulation is performed with Delphes3, using the CEPC card (with, in particular, a b-tagging efficiency b = 0.8) but lowering the jet p T threshold to 5 GeV and applying the same jet energy scale that we used for the LHC. 12 After the basic selection

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The total acceptance times efficiency for the signal is (A ) I, II tot = 12%, 11%, whereas the expected background yield at TeraZ is of 65 and 49 events, respectively. The resulting 95% CL bounds (calculated using Poisson statistics) are, in terms of the branching ratio BR(Z →PŜ → bb + X), 2.6 × 10 −8 (1.2 × 10 −10 ) at GigaZ (TeraZ) for benchmark I, and 2.9 × 10 −8 (1.3 × 10 −10 ) for benchmark II. We stress that these constraints are derived assuming that the hidden mesons decay promptly. This is satisfied in most of the parameter space that can be probed at GigaZ, where the bounds translate to since in this Λ range the pseudoscalar has a lifetime mm for M 5 TeV. For larger M theP becomes long-lived, requiring a change in experimental strategy at TeraZ (discussed in subsection 4.2). Nonetheless, the above bounds on the Z branching ratio can be relevant to other scenarios, where the decays remain prompt even for very small rates. Finally, a comment is in order about the potential sensitivity of LEP1 data to this final state. The 4 LEP experiments recorded a combined total of 1.7 × 10 7 events at the Z pole [40], hence after accounting for BR(Z → νν) = 0.20 we estimate that ∼ 2.2 × 10 7 Z bosons were produced. This yields the number of Z →PŜ → bb + X events, N S ≈ 28 fPŜ (2 TeV/M ) 4 (A ) tot . Given the relatively low b-tagging efficiency at LEP1, b ∼ 0.3 [40], we expect < 3 events and therefore no constraint for M = 2 TeV.
Higgs decays. We expect the dominant Higgs two-body decays to be h →PP ,VV ,ŜŜ, since h →VP ,VŜ violate C and h →PŜ violates CP . Here we concentrate on h →VV and h →PP , which appear most promising. We discuss, drawing from the existing experimental and theoretical literature, a few searches for these decays that provide useful points of comparison with our results for Z decays presented above. Defining f h XY ≤ 1 as the fraction of Higgs decays to the hidden sectors that yield the XY final state, we take f ĥ VV , f ĥ PP ∼ 0.3 as benchmarks. We begin with h →VV → bbµµ, for which we apply the CMS search of ref. [49] that we used to estimate the reach in Z →PŜ. The CMS result is a bound BR(h → aa → bbµµ) 2 × 10 −4 , weakly dependent on m a in the range [20, 62.5] GeV. Neglecting the difference in acceptance between pseudoscalar and vector, this translates into M 0.55 TeV (f ĥ VV /0.3) 1/4 (α d /0.24) 1/2 (c g /4) 1/2 , where we use BR(V → bb, µµ) ≈ 0.15, 0.034 and take as reference the coupling strength corresponding to Λ = 10 GeV. This current constraint on M is weaker than the one we obtained with the extremely conservative analysis of Z →PŜ, see eq. (4.4).
A second channel we consider is h →VV → . Here we directly apply the results of the ATLAS analysis in ref. [56], which targeted (among others) the h → Z d Z d signal [57], where Z d is a dark photon kinetically mixed with hypercharge. Using 36.1 fb −1 of data, the search set a constraint BR(h → Z d Z d ) 10 −4 for m Z d in the range [15,50]  this current bound on M is significantly weaker than those from Z → 4µ (+X). Due to the democratic decays ofV to SM fermions, h →VV yields a variety of other final states, many of which were discussed in the extensive survey of ref. [32]. Looking ahead to future Higgs factories, a particularly appealing prospect is the possibility to probe h →PP → 4b, which has a relatively large branching ratio due to BR(P → bb) ∼ 0.9, and for which the sensitivity at FCC-ee will reach down to BR(h → (bb)(bb)) = 3 × 10 −4 for mP in the range [20,60] GeV [33]. This corresponds to the bound M 1.4 TeV (f ĥ PP /0.3) 1/4 (α d /0.24) 1/2 (c g /4) 1/2 , much weaker than what can be achieved from Z →PŜ even at GigaZ, see eq. (4.11). These results illustrate in a quantitative manner the superior sensitivity of Z over h decays in probing this region of parameters.

Two-body displaced decays
For Λ ∼ 10 GeV but larger M , at least some of the hidden mesons become long-lived particles (LLPs) for collider purposes, requiring different experimental strategies. In this subsection we discuss searches for the long-lived mesons at ATLAS and CMS, as well as at future Z factories. As M is increased, theP andŜ become long-lived first (the latter especially if ∆m is even moderately smaller than Λ, see figure 1), whereas theV remains prompt. We exploit this feature to propose HL-LHC searches where a mostly-hadronically decaying LLP (P orŜ) is produced in association with aV , whose prompt decay to µ + µ − allows for efficient triggering and suppresses the SM background to a negligible level. A similar search was proposed in ref. [35], taking as benchmark the process Z → A h D , where the dark photon A decays promptly to leptons and the dark Higgs h D is the LLP. Furthermore, ATLAS has recently published [58] a search for h → ZZ d , where Z → and the LLP Z d decays in the hadronic calorimeter (HCAL).
In addition, we extend our analysis to TeraZ, where very large scales M > 10 TeV can be probed. In this region of parameters theV , too, can decay at a macroscopic distance from the interaction point. However, the trigger does not pose a problem and we expect that the combination of a (possibly) displaced dilepton pair fromV → with a displaced vertex (DV) fromP orŜ will remove any SM backgrounds. As a consequence, when going from HL-LHC to TeraZ we do not make dramatic changes to our method for deriving projections.
A general difficulty is that the meson cτ and BR(Z, h → ψψ) have inverse scaling with M , so that macroscopic decay lengths necessarily correspond to very small branching ratios to the hidden sectors. Nevertheless, the very large statistics that will be collected at the HL-LHC and TeraZ and the absence of backgrounds result in promising sensitivity.

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where the dependence on M cancels out. For reference, the HL-LHC (running at √ s = 14 TeV) will produce approximately 1.8 × 10 11 Z and 1.6 × 10 8 Higgs bosons, hence we concentrate on Z decays and specifically on Z →PŜ, which yields aP → ff DV (where about 90% of the time f = b, and otherwise f = τ, c), together withŜ →V ff . The decays of bothŜ andV are assumed to be prompt. We selectV → µµ, performing a parton-level MadGraph5 simulation. We require the muon pair to pass the dimuon trigger requirements: p µ T > 17, 8 GeV and |η µ | < 2.5 [49]. 13 Furthermore, we impose |ηP | < 2.5 and for each surviving event we integrate theP decay probability distribution, determined by the four-momentum ofP and its proper lifetime in eq. (3.11), over the volume of the detector where the DV can be reconstructed. The latter is taken to be an annulus with radii r ∈ [1,30] cm, approximately corresponding to the capability of the ATLAS inner tracking detector (ID) [60], with efficiency for hadronic DV reconstruction equal to a constant DV . We take DV = 20 (10)% as an optimistic (conservative) benchmark, and by requiring 3 signal events -corresponding to a 95% CL exclusion if the background is negligible -we derive the solid (dashed) red curve in the left panel of figure 5. Note that for M ∼ 11 TeV, the largest scale accessible at the HL-LHC, the chosen benchmark spectrum gives cτŜ ∼ 1 mm and cτV ∼ 0.6 mm, hence the dimuon pair can still be considered prompt.
The TeraZ analysis proceeds along similar lines, but we includeV decays to both electrons and muons, requiring E > 10 GeV and |η | < 2.3, as well as |ηP | < 2.3. For the DV coverage we consider two options, one identical to the LHC to facilitate the comparison, and one where efficient hadronic DV reconstruction is extended down to r = 1 mm, which is expected to be easily achievable in the absence of pileup (see e.g. ref. [61] for a recent discussion). In the left panel of figure 5, these two scenarios are shown in solid blue and dashed blue, respectively. Both assume a constant efficiency DV = 20%, and negligible background.
Displaced scalar decays. Due to the ∼ (∆m) 7 dependence of eq. (3.16), the lifetime of the scalar mesonŜ is very sensitive to the mass splitting with theV , and even a mild hierarchy ∆m < Λ results in macroscopic decay lengths for M few TeV. To probe this LLP at the HL-LHC we choose the Z →VŜ decay, selectingV → µµ which is assumed to be prompt. The mostly-hadronic DV 14 fromŜ →V ff (orŜ → ff ) is reconstructed either in the ID or in the HCAL + muon spectrometer volume, taken to be r ∈ [2, 7.5] m [60]. The inclusion of the outer detector is important due to the long lifetime for small ∆m: for example, for ∆m = Λ/2 = 5 GeV we find cτŜ ∼ 60 cm at M = 10 TeV. The selection requirements on the muons are the same as in the analysis of displacedP , and we assume a constant DV = 20% across the whole detector volume. At TeraZ we include bothV → ee, µµ and assume a similar DV coverage, except for the already-mentioned 13 We have checked that because of the higher transverse momentum thresholds for the dielectron trigger, p e T > 23, 12 GeV [59], the contribution ofV → ee would be relatively suppressed by about one order of magnitude, so we neglect it.
14 When calculating the expected signal rate we subtract the fraction of events whereŜ → (V → νν)ff , in which case the tracks resulting from ff alone are likely too soft for DV reconstruction. corresponding to ∼ 300 Z events and ∼ 4 h events after 3 ab −1 at 14 TeV for the reference parameters. Clearly, observing displacedV decays at the HL-LHC will be extremely challenging. The prospects may be better at TeraZ, thanks in particular to the likely improved DV reconstruction efficiency at small displacements.

Decays to hidden jets
For sufficiently small confinement scale, roughly Λ 2-3 GeV, Z and Higgs decays to the hidden sector result in parton showers, producing jets of hidden mesons. Unless M is very low, the mesons are long-lived (see figure 1) and therefore the hidden jets include a significant fraction of displaced vertices, realizing emerging jet-like phenomenology [47]. 15 As the vector mesonV decays democratically to SM fermions, LHCb is especially well JHEP11(2019)031 suited to probe our type of emerging jets, by resolving the decay of a singleV → µµ inside the jet cone with the Vertex Locator (VELO). The sensitivity is maximal for lifetimes of O(cm). On the other hand, tracker-based searches for emerging jets at CMS [63] and ATLAS must rely on hard cuts that suppress our signals to a negligible level. Therefore we need to require the production of an associated object, for example pp → ZV with V an EW gauge boson that decays leptonically, or pp → Zj with a hard initial-state radiation jet. This ensures efficient triggering, but significantly reduces the signal rate while the backgrounds remain appreciable, and we find that these searches cannot compete with the LHCb sensitivity. 16 Reach at LHCb. In Z → 2 hidden jets events, the VELO can detect a singleV → µµ DV within one of the jets [64]. This requires the vector meson to have transverse decay length between 6 and 22 mm [65] and pseudorapidity within the LHCb coverage, η ∈ [2, 5]. We simulate Z decay events in Pythia8, choosing the benchmark mP = mV = 1 GeV and Λ = µ ψ = 0.5 GeV. We impose that the average numbers of mesons in each jet satisfy NP / NV = 1/3 , as expected from a counting of the spin degrees of freedom, and for simplicity we neglect heavier hadrons in the shower, including theŜ and other mesons, as well as the baryons. We expect the baryons to make up a small, invisible component of the jets, thus only slightly reducing the signal rate. We find that the simulation produces an average of ∼ 7 hidden mesons in a jet. EachV → µµ DV with pV T > 1 GeV, transverse decay length between 6 -22 mm and η ∈ [2, 5] is assumed to be reconstructed with constant efficiency µµ = 0.5 . The efficiency degradation due to the overlap with other nearby tracks is not explicitly mentioned in preceding studies [64,65], but it is likely to reduce µµ significantly. Therefore, we reject non-isolated events where the µµ vertex is accompanied by one or moreV visible decays with transverse decay length shorter than 22 mm and ∆R < 0.4. Because NP is small and theP decay length is much longer than that ofV , we neglect the effects ofP decays.
The SM background to such displaced µµ vertices is expected to be about 25 events for 15 fb −1 of integrated luminosity [65]. To estimate the ultimate sensitivity achievable at LHCb, we also compute the constraints by assuming negligible SM background. The results are shown in figure 6, where the solid (dashed) curves correspond to standard (negligible) background. Due to the moderate typical boost factor, the sensitivity is optimal for cτV 1 cm, where it reaches Z branching ratios down to O(10 −7 ) at the HL-LHC. In our setup, the bounds translate to M 1.6 (2.0) TeV for L = 15 (300) fb −1 assuming the standard background count, while in the background-free case we find M 1.8 (2.7) TeV.
At future Z factories, searches for emerging jet signals will greatly benefit from the straightforward triggering. The corresponding analysis is described in appendix C, where we find that at TeraZ the sensitivity will reach BR(Z → ψ B,C ψ B,C ) ∼ O(10 −8 ) for cτV ∼ O(1-10) cm. As figure 6 shows, in our setup this will mostly probe parameter space already accessible at LHCb. In more elaborate scenarios, however, the Z branching ratio to hidden fermions and the vector meson lifetime may be decoupled, for example by extending the model to allow additional decay channels forV . In this case, a future Z factory could provide the crucial test.

Vector meson as dark photon
The vector mesonV couples to the SM through the Z boson, with a coupling structure similar to that of a dark photon γ D . For a broad class of dark photons, the interactions with the SM can be written as where A D is the dark photon field and J EM and J NC are the electromagnetic and weakneutral currents, respectively. 17 If a single source of kinetic mixing generates both of the operators in eq. (4.14), then the coupling to J EM is the more sensitive probe of the interaction. This is the case that is most frequently studied. However, in our scenario the constituent fermions do not couple to the photon at tree level and only the second operator in eq. (4.14) is relevant, which can be seen as originating from an effective Z-V mass mixing term ε Z m 2 Z Z µV µ [66]. This coupling can lead to a variety of signals in low-energy experiments, such as parity violation tests and flavor-changing neutral current (FCNC) meson decays, in particular B → Kγ D and K → πγ D [66,67]. By matching to eq. (3.7) we identify

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where we have assumed mV m Z and included an extra factor of √ 2 to account for the two hidden sectors B, C. 18 Very recently, ref. [67] derived strong FCNC decay bounds on ε Z for a generic light vector X µ . In our setup, though, the NP contribution to a given SM final state should be smaller than the corresponding perturbative rate, BR(B → Kff ) NP BR(B → Kψψ), where f is a SM fermion and we have taken B decays as example. This allows us to conservatively estimate where we assumed dominance of the Z penguin over the box amplitudes and used eq. (1.2). An analogous expression applies to K decays. For any f , the r.h.s. of eq. (4.16) is below a percent if M > 1 TeV, showing that the precision needed to probe the NP is well beyond the current one [48]. 19 We do not expect non-perturbative corrections to change this conclusion. Looking ahead, it is interesting to ask whether future dark photon searches [68,69] will be able to probe theV . In general, dark photon production is simply accomplished by taking a visible photon production channel and replacing one visible photon with a dark photon. For instance, a fixed-target setup with target nucleus N and beam particle e produces bremsstrahlung, eN → eN γ. Consequently, dark photons γ D can be produced through eN → eN γ D . Similarly, lepton colliders produce a visible photon recoiling off a dark photon through e + e − → γγ D , and hadron colliders exploit Drell-Yan production qq → γ D . In addition, as discussed above, the dark photon can be produced in meson decays if kinematically allowed. Comprehensive reviews can be found, for example, in refs. [70,71]. From eq. (4.14) we read that in order to gain a first, crude impression of the future reach, we can treat theV as a kinetically mixed dark photon with "effective coupling" where we have neglected the differences between the electric and weak charges of the SM fermions. However, theV has an appreciable decay width to the SM neutrinos, since it mixes with the Z rather than the photon. This reduces theV rates to visible particles, especially for smaller mV , where decays to light quarks are cut off by the meson masses.
To account for this we rescale ε eff by BR(V → visible) 1/2 as a function of mV . In figure 7 we show whereV falls in the standard dark photon parameter space. The solid lines trace out the relationship between the vector mass and ε eff for three M benchmarks, assuming mV = 2Λ. The gray-shaded region indicates the existing bounds on dark photons, as collected in ref. [69]. The dashed lines denote projected limits from a 18 A small coupling to the electromagnetic current is induced by the magnetic dipole operator ∼ αW m 2 t m ψ ψσ µν ψ eFµν /(πM 4 ), generated at one loop through the W boson and the electrically-charged heavy fermion Ψ − . Neglecting O(1) factors, we estimate ε/εZ ∼ αW e m 2 Z /(πgZ M 2 ) ≈ 10 −5 (2 TeV/M ) 2 . 19 Reference [67] exploited the emission of the longitudinal mode of Xµ, which can lead to ∼ (mEW/mX ) 2 enhancement. However, our vector mesonV is a composite particle, which "dissolves" into its constituents at energies not far above its mass, so such enhancement does not apply.

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Existing Bounds collection of future experiments: we expect that FASER2 [72], SeaQuest [73] and NA62 [69], as well as ultimately SHiP [74], will be sensitive to hidden vector mesons with masses below a few GeV. These preliminary results provide solid motivation for a more detailed analysis of this physics, which we leave as an interesting direction for future work.

Invisible decays
If the hidden mesons are very long-lived, as is expected at small Λ, then invisible Z and Higgs decays constitute the main search strategy. Currently, the strongest constraint comes from the LEP1 measurement of the Z invisible width, requiring ∆Γ inv Z < 2 MeV at 95% CL [40]. Using eq. (3.1), this translates to M > 0.8 TeV for m ψ = 0 . From eq. (3.4) we obtain then the current upper bound on the invisible Higgs branching ratio, BR(h → g B,CĝB,C ) 1.4 × 10 −2 (α d /0.24) 2 (c g /4) 2 , which is likely out of reach at the LHC even in the high-luminosity phase [75].
Turning to future colliders, an FCC-ee run in TeraZ mode will be able to improve the precision on the number of light neutrino species by a factor 7 [76]. This roughly corresponds to a reduction by the same factor of the uncertainty on Γ inv Z , leading to a 95% CL projected constraint ∆Γ inv Z 0.4 MeV, 20 or M 1.2 TeV. A more careful analysis including correlations would affect this estimate only mildly. This method is severely limited by the systematic uncertainty affecting the measurement of the integrated luminosity [76], and in fact the ultimate precision can already be achieved at GigaZ. Runs at higher center-of-mass energy will be able to constrain ∆Γ inv Z through radiative return e + e − → Zγ events, although current estimates suggest an improvement of only 20% on the uncertainty with respect to the measurement at the Z pole [76].
Finally, we should mention that FCC-hh running at 100 TeV will be able to achieve an impressive limit on the Higgs branching ratio to new invisible particles, BR(h → 20 We have assumed the future measurement will agree with the SM prediction. Recall that at LEP1 Γ inv Z was measured 2 σ below the SM [40], so the current bound on ∆Γ inv Z is stronger than the expectation.

Conclusions
Motivated by a possible realization of the tripled top model for neutral naturalness, we have studied a confining hidden sector where one fermion ψ is light compared to the confinement scale Λ. The latter is taken in the range 0.1 GeV Λ 15 GeV, as generally motivated by 2-loop naturalness considerations. The couplings of the hidden sector fields to the SM are mediated by EW-charged particles with TeV-scale mass M , and are described at low energies by a handful of dimension-6 operators. These determine both the production and the decays of the hidden sector mesons through the Z and Higgs portals. Since the theory does not possess light pNGBs, several of the lightest mesons are important for phenomenology, resulting in a complex pattern of signatures. We performed a survey of these, identifying several regions of the parameter space -which is characterized primarily by Λ and M -with distinctive phenomenological properties. We have found that the Z portal, which was not considered in the previous hidden valley literature, has dramatic implications for the prospects to detect the hidden sector. In particular, the enormous numbers of Z events that will be collected at the HL-LHC and at a future Z factory, allow for unprecedented sensitivity. For large Λ ∼ O(10) GeV, the Z decays to two-body hidden meson final states, followed by prompt decays back to the SM. We showed that adaptations or extensions of current LHC searches can probe M up to ∼ 3 TeV in the high-luminosity phase, whereas a future GigaZ run at the Z pole will be able to reach M ∼ 5 TeV. For even larger M the hidden mesons become long-lived, and we have found that a TeraZ run will probe scales up to M ∼ 20 TeV for negligible SM background. These results demonstrate in a quantitative way the power of a future Z factory to directly probe the confining hidden sector. For smaller Λ ∼ O(1) GeV the Z decays produce jets of hidden mesons, which are typically long-lived. At the LHC the best sensitivity to this scenario is obtained at LHCb, by resolving single displacedV → µµ decays within the jets, whereV is the hidden vector meson. In addition, since theV couples to the SM through mixing with the Z, it can be probed by an array of planned experiments that will search for dark photons at the intensity frontier.
Our work can be extended in a number of ways. First of all, we stress that our phenomenological study is intended to be only an initial survey of the many possibilities available, and several areas deserve to be analyzed in greater detail. At the LHC, searches for Z decays to mesons with masses in the 10-40 GeV range strongly benefit from keeping the selection cuts as soft as possible, and we encourage ATLAS and CMS to extend their analyses in that direction. Another topic that warrants further attention is the future experimental sensitivity to light hidden vectors mixed with the Z boson (such as ourV ), which we have only sketched briefly, but should be analyzed in a systematic way. In addition, in this paper we have focused our attention on the Z portal to the hidden sector,

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which constitutes a main novelty of our setup, while Higgs decays have typically played a marginal role. This picture may change at the FCC-hh, however, thanks to the large sample of Higgs bosons that will be collected (approximately 3 × 10 10 for 30 ab −1 at 100 TeV [77]); the associated new physics reach certainly deserves to be investigated. Finally, our results reinforce the relevance for phenomenology of 1-flavor (hidden) QCD, whose properties can only be reliably studied using non-perturbative methods. We believe this theory merits attention from the part of the lattice QCD community with interest in BSM physics, and further results beyond the partial ones of ref. [24] would be most welcome.

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The pieces of the D-term scalar potential that contain the Higgs field are Focusing on the fields without the "c" superscript, rotating to the mass eigenbasis and taking ω → 0, we find that the relevant coupling is −y t m t T * B h +b B +h.c., which renormalizes δZ + and thus , while M 2 T was defined in eq. (2.16) and M 2 b = ∆ 2 . This is the dominant correction to T in this model. Numerically, requiring T S, B+C 10 −3 gives ∆ 400 GeV, which can be seen as a rough current lower bound from EWPT on the masses of the EW-doublet scalar top partners.

A.2 Original tripled top model with singlet top partners
The contribution of the hidden fermions, expanding for m t ω M , is T fermions,B+C ≈ L T m 2 t /M 2 at the leading order. For the scalars, the roles of the S and S c sectors are reversed with respect to the model considered in this paper. We neglect the "S" sector in eqs. (36) and (37) of ref. [10], which is supersymmetric and whose contribution is therefore subleading to T fermions,B+C . On the other hand, the light scalars in the "S c " sector have sizable mixing and the associated correction is a priori relevant. To calculate it, the starting point is eq. (A.1) with M ↔ ω. Focusing on the fields with the "c" superscript, we rotate to the mass eigenbasis, expand h 0 = h 0 + (h − iπ 3 )/ √ 2 and neglect quartic interactions, which at one loop give rise to tadpole diagrams that do not renormalize the Goldstone wavefunctions. We arrive then at the relevant couplings y t ω i from which we obtain Expanding to leading order in m 2 t , we find Numerically, this correction is well below the current bound T 10 −3 : e.g., for ω = 500 GeV and ∆ = 300 GeV we have T S c ,B+C +1.7 × 10 −4 , computed using the full numerical expression in eq. (A.4). B Analysis of Z → bbνν + X at Z factories

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The prompt signal ee → Z → (P → bb)(Ŝ →V ff ) at a Z-factory was discussed in subsection 4.1, focusing onV → . However, given the very clean collider environment it is also interesting to considerV → νν, which benefits from the larger BR(V → νν) ≈ 0.20. The resulting final state is bb + missing momentum + X, for which the irreducible SM background is ee → bbνν with amplitude at O(g 4 w ). As basic selection we simply require 2 b -jets with E b > 10 GeV and |η b | < 2. We have defined m tot = (p 1 b + p 2 b + p miss ) 2 , where the missing four-momentum is p µ miss = (E miss , p miss ) with E miss = √ s − i E i ( √ s = m Z ) and p miss = − i p i . The sums in the definitions of E miss and p miss run over all reconstructed objects, in particular only jets with p T > 5 GeV are included. 21 By design, if the only reconstructed objects in the event are the two b's then m tot = m Z , so m Z − m tot is effectively a measure of additional activity in the event. Additionally we impose E miss > 30 GeV, p miss T > 20 (10) GeV for I (II), and veto extra b-jets with p b T > 5 GeV and |η b | < 2.3. These further requirements should suppress the reducible bb background, see figure 9. The total acceptance times efficiency for the signal is (A ) I, II tot = 9.3%, 8.8%, whereas the expected bbνν background yield at TeraZ is of 130 and 390 events, respectively. The resulting 95% CL bounds are, in terms of the branching ratio BR(Z →PŜ → bb + invisible + X), 3.3 × 10 −8 (2.2 × 10 −10 ) at GigaZ (TeraZ) for benchmark I, and 3.8 × 10 −8 (3.9 × 10 −10 ) for benchmark II. Comparing to the results obtained in ref. [36] for Z → φ d A → (bb)(χχ) with χ an invisible particle, we find that our BR bounds are only slightly weaker (by a factor ∼ 2 at TeraZ), even though 21 The resulting p miss T = p 2 miss,x + p 2 miss,y displays a slightly harder distribution compared to the / E T provided by Delphes. The latter is not used in this analysis. For the larger values of M that can be probed at TeraZ, theP becomes long-lived, which will require a different experimental approach. An important caveat is that the above constraints were obtained neglecting the reducible bb background, to illustrate the best sensitivity that can be achieved. With our simulation settings, however, we find that the selection we described in this appendix can only partially suppress bb, whose inclusive cross section is approximately 6.5 nb. After all cuts, at TeraZ we estimate ∼ 3 × 10 6 remaining events for analysis I, and ∼ 5 × 10 7 events for analysis II. These yields are 10 4-5 times larger than the irreducible ones, and if taken at face value they would lead to a severe degradation of our results. However, we expect that specifically designed cuts, a more refined modeling of the detector, and advanced analysis techniques will enable a further suppression of the bb contamination while preserving a high signal efficiency. A detailed analysis of this aspect is left for future work.

C Emerging jet search at Z factories
In this appendix we derive projected limits on BR(Z → ψ B,C ψ B,C ) from the search for emerging jets (EMJs) at Z factories. We follow closely the recent CMS analysis of EMJs at the LHC [63]. For each jet, the variables IP 2D and α 3D are defined, calculated using the information of charged tracks with p T > 1 GeV. IP 2D is defined to be the median transverse impact parameter (D xy ) of the tracks, while α 3D is the sum of the p T of the tracks that belong to the primary vertex (PV), divided by the scalar p T sum of all tracks in the jet. Only tracks satisfying is the uncertainty of D xy [82]. In order to validate our approach, we first try to reproduce the EMJ-1 benchmark results in ref. [63]. The LHC tracker resolutions of D xy and D z are taken from refs. [82,83]. The EMJ signal and (b-)jet pair backgrounds are simulated using Z decays with M Z = 250 GeV, and each jet is required to have p T > 50 GeV and |η| < 2. Applying the EMJ-1 selection we obtain a light jet misidentification rate j 2 × 10 −3 for track multiplicity ∈ [6, 10] and 4 × 10 −4 for track multiplicity ∈ [11,16], which are close to those reported in ref. [63]. Similarly, for b-jets we find b 4 × 10 −2 and 2 × 10 −3 for the lower-and higher-track multiplicity bins. However, at the LHC a search for only Z → 2 EMJs is not feasible due to trigger requirements, and we must resort to production in association with additional particles, such as ZZ → 2 EMJs + 2 or Zj → 2 EMJs + hard jet. The resulting sensitivity on the Z branching ratio to the hidden sectors is at the level of 10 −(4-5) , weaker than the LHCb reach described in section 4.3.
At a future Z factory, the search reach will depend on the detector resolution. For a conservative estimate, in our analysis we use the LHC tracker resolution described above. The EMJ signal and Z-pole di(b-)jet backgrounds are simulated using Pythia8, with the same benchmark parameters adopted in the LHCb analysis of section 4.3 (in particular, mP = mV = 1 GeV). We require p T > 5 GeV and |η| < 2 for each jet. SinceV has a ∼ 40% branching ratio to neutrinos and theP lifetime is very long, a significant fraction of the EMJ energy is carried by invisible particles, reducing the signal efficiency. On the other hand, theV has a ∼ 6% branching ratio to ee and µµ pairs, which are rare in SM jets. The lepton tracks can thus be used to better separate the EMJs from the SM jets. For a target cτV , each jet is required to satisfy IP 2D ∈ [cτV /10, 25 cm] and α 3D < 0.25. To avoid backgrounds from long-lived SM hadrons, such as K S [47], only EMJ candidates with track multiplicity N track ≥ 5 are accepted if they contain 0 or 1 lepton track. The cut on N track is loosened to ≥ 3 if an EMJ candidate has ≥ 2 lepton tracks. We plot the emerging jet efficiency and SM jet misidentification rates versus the target cτV in the left panel of figure 10, where the solid (dashed) curves correspond to the case with 0 or 1 (≥ 2) lepton track(s). Lepton tracks fromV → decays clearly help to discriminate the EMJs from the SM backgrounds, especially in the case of jets originating from light SM quarks. Due to its higher efficiency and lower misidentification rates, the lepton-track-rich channel provides the leading sensitivity; the largest background comes from SM Z → bb decays. The combined limits on the Z branching ratio are shown in the right panel of figure 10. For cτV ∼ few cm the sensitivity reaches BR(Z → ψ B,C ψ B,C ) ∼ 10 −6 (10 −8 ) at GigaZ (TeraZ). Improved detectors and the use of information from calorimeters or the muon system may further extend the reach. Emerging jet searches at the LHC may also benefit from LLP-specific triggers and advanced analysis strategies. The study of these topics is beyond the scope of this paper, and is deferred to future work.
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