New Limits on Light Hidden Sectors from Fixed-Target Experiments

New physics can be light if it is hidden, coupling very weakly to the Standard Model. In this work we investigate the discovery prospects of Abelian hidden sectors in lower-energy fixed-target and high-precision experiments. We focus on a minimal supersymmetric realization consisting of an Abelian vector multiplet, coupled to hypercharge by kinetic mixing, and a pair of chiral Higgs multiplets. This simple theory can give rise to a broad range of experimental signals, including both commonly-studied patterns of hidden vector decay as well as new and distinctive hidden sector cascades. We find limits from the production of hidden states other than the vector itself. In particular, we show that if the hidden Abelian symmetry is higgsed, and the corresponding hidden Higgs boson has visible decays, it severely restricts the ability of the hidden sector to explain the anomalous muon magnetic moment.


Introduction
New physics beyond the Standard Model (SM) could take many forms. To be consistent with existing experiments, any new particles must either be heavier than the electroweak scale or they must interact very weakly with us [1]. Lower-energy experiments with a very high precision or intensity are particularly well-suited to discovering the second possibility when the characteristic mass of the new physics lies below the electroweak scale [2]. The results of such experiments may also provide guidance for future searches at high energies.
An interesting variety of new physics below the electroweak scale is an exotic Abelian gauge force U (1) x that couples to the SM exclusively through a small kinetic mixing with hypercharge [3,4] L ⊃ − where X µν = (∂ µ X ν −∂ ν X µ ) and X µ is the U (1) x gauge boson. If the only new particle is the massless X µ , it can be rotated with the photon such that one linear combination decouples from the theory [3]. However, if there are new matter fields charged under U (1) x they will develop millicharges under this rotation [3]. Since light millicharged matter is very strongly constrained by direct searches and astrophysical observations [5], we will focus on a theory where U (1) x is spontaneously broken.
The simplest way to break U (1) x is with a hidden Higgs field. In this work we study a minimal supersymmetric theory in which the new gauge force is broken well below the weak scale by a pair of hidden Higgs bosons. With supersymmetry, the relatively low breaking scale can be natural if the hidden sector feels superymmetry breaking less strongly than the visible SM sector and is sufficiently small [6][7][8][9][10]. The hidden sector will also contain a dark matter candidate in the presence of R-parity. While naturalness and dark matter are attractive features, our primary motivation for investigating this particular model is that it provides a simple but non-minimal theory of a light hidden sector.
Light hidden vectors interacting with the SM through the kinetic mixing interaction of Eq. (1) have been studied extensively in recent years. The two scenarios that have received the most attention are minimal models where the new massive vector Z x decays primarily to the SM [11][12][13][14] or to a pair of dark matter particles [15][16][17][18][19]. These decay channels can also occur in the non-minimal theory we study here, but other decay channels can be dominant as well. In particular, we investigate phases of the theory where the leading vector decays are Z x → h x 1 A x , where h x 1 and A x are hidden Higgs bosons, and Z x → χ x 1 χ x 2 , where χ x 1 and χ x 2 are hidden neutralinos. The experimental signals of these additional channels have not received as much attention as the purely visible or invisible channels (although see Ref. [20]). We find that the h x 1 , A x , and χ x 2 decay products can be metastable, and that fixed-target experiments are particularly well-suited to finding them.
In this paper, we study the sensitivity of low-energy and high-precision experiments to a minimal supersymmetric Abelian hidden sector, with a focus on fixed-target experiments. We begin in Section 2 by defining the theory and characterizing the mass eigenstates and their decay modes. We also formulate four distinct sets of benchmark parameters to be studied in the sections to come. In Section 3, we investigate the existing limits on the theory from non-fixed-target experiments, including precision tests, meson factories, and cosmology. Next, in Sections 4 and 5 we study the sensitivities of current and future electron and hadron fixed target experiments to the theory in its various phases. The implications of our results for the LHC are discussed in Section 6, and we conclude in Section 7. Some technical details about the theory and our calculation of vector production and detection in fixed target experiments are collected in the Appendices A, B and C.

A Minimal Supersymmetric Hidden Sector
We consider the minimal supersymmetric hidden sector formulated in Refs. [9,21] consisting of a U (1) x vector multiplet X together with a pair of chiral multiplets H and H with charges x H,H = ±1 that develop vacuum expectation values (VEVs). In addition to a minimal Kähler potential for H and H , the interactions in the hidden sector are taken to be: with the vector terms the superpotential terms and the soft terms Note that the kinetic mixing has been written in terms of ≡ Y cos θ W , where θ W is the Weinberg angle, to agree with the most common convention. This is a different normalisation to that made in Ref. [21] (which took ≡ Y ). We assume 1 and take the dimensionful µ and soft terms to be at or below a GeV. This choice is consistent with supersymmetric naturalness for sufficiently small [9,21], and predicts a hidden vector mass of the same order. The visible sector is assumed to be the minimal supersymmetric standard model (MSSM).

Hidden Masses
For appropriate values of the model parameters, the scalar components of H and H develop VEVs, H = η sin ζ, H = η cos ζ , with ζ ∈ [0, π/2]. As a result, the hidden vector X µ obtains a mass equal to We will denote this state Z x .
Hidden symmetry breaking also gives rise to mass mixing among the scalar and fermion sectors of the theory. The remaining bosonic mass eigenstates are a pair of real scalars h x 1 and h x 2 with masses m h x 1 ≤ m h x 2 , and a pseudoscalar A x of mass m A x . 1 The fermionic mass eigenstates are mixtures of the U (1) x gaugino and the hidden higgsinos: χ x 1 , χ x 2 , χ x 3 with masses |m χ x 1 | < |m χ x 2 | < |m χ x 3 |. A more detailed exposition of hidden-sector mass mixing can be found in Ref. [21] and Appendix A.
When symmetry breaking occurs in the hidden sector, a convenient basis of parameters for the theory is {g x , µ , m x , m A x , M x , tan ζ, }. Note as well that the structure of the hidden Higgs sector mirrors that of the neutral Higgs of the MSSM [22]. As such, we obtain m h x 1 < m x , m A x < m h x 2 at tree-level, and we expect this ordering to persist at loop level provided g x is small and µ , M x ∼ m x . We will also assume that R-parity is conserved and that the gravitino is moderately heavy, so that χ x 1 is a stable lightest superpartner (LSP).

Hidden Couplings and Decays
Particles in the hidden sector interact with each other through the gauge coupling g x and with the visible (MSSM) sector via the kinetic mixing parameter . For e g x , as we assume here, hidden states nearly always decay to other hidden states when kinematically allowed. However, when the hidden channels have been exhausted, decays to SM can dominate giving rise to new signals in low-and high-energy experiments.
The main connector to the SM is the hidden vector Z x , and it can decay in various ways depending on the mass spectrum of the hidden sector. The experimental signatures of the theory can be classified according to the most probable decay mode of the vector. We will consider the four most important possibilities: Below, we will define four benchmark parameter slopes that realize each of these four cases.
An interesting feature of this theory is that the lightest scalar h x 1 is always lighter than the vector and all the other scalars (at weak coupling). The only possible hidden channels are When all these channels are forbidden, m h x 1 < 2m χ x 1 , the h x 1 state will decay to the SM. The two main contributions to the decay come from mass mixing with the MSSM Higgs bosons and from a loop of hidden vectors connecting to a pair of SM fermions. The effective Higgs mixing angle goes like (m X /m h 0 ), where m h 0 is the mass of the lighter MSSM Higgs boson, while the vector loop is proportional to 2 . In computing the h x 1 decay width, we find that the 1 We neglect CP violation in the present work.
Higgs mixing contribution tends to dominate over the loop contribution above the dimuon threshold. However, both contributions have significant suppressions, and the h x 1 state is nearly always long-lived on collider time scales. More details can be found in Appendix A.
Among the heavier scalars, there are a number of possible decay channels: where ( * ) denotes that the decay product is potentially off-shell. These decays tend to be prompt when they are two-body, but can become slow when they are forced to be three-body with a SM final state (e.g. where f is a charged SM fermion). With our assumption of R-parity conservation, the lightest hidden neutralino is the stable LSP (in the absence of a light gravitino). The heavier hidden neutralinos decay according to Again, the branching fractions of these decays depend on the spectrum and mixing of hidden states. Note as well that the lightest MSSM superpartner can decay to the hidden sector through gauge kinetic mixing (e.g. χ 0

Benchmark Slopes
This minimal supersymmetric hidden sector can give rise to a broad range of experimental signatures. To illustrate this range, we investigate the signals of four sample slopes corresponding to four different primary decay modes of the hidden vector Z x . For each slope, we set tan ζ = 3 and α x = α, and we take the remaining model parameters to be fixed ratios of the vector mass m x . These ratios and the resulting mass spectra (in units of m x ) are listed in Table 1.
Each of the four slopes corresponds to a different dominant decay channel for the hidden vector boson. They are: The vector has no two-body decays within the hidden sector, and as a result decays almost entirely through gauge kinetic mixing with the photon. This scenario occurs readily for m x < M x , µ , m A x , and has been studied in great detail [1].
This channel is now the only two-body hidden mode, and it dominates over the SM + SM channel (for e g x ). It is likely to occur for M x < m x < m A x , µ . The final state is invisible, and similar scenarios have been investigated [1].
The initial vector decay is prompt, but the h x 1 and A x products decay further to the  Table 1: Sample parameters and particle masses for the four benchmark slopes. All quantities are listed in units of the vector mass m X , and we also fix α x = α and tan ζ = 3.
SM and are typically delayed. Decays of the lighter scalar h x 1 are addressed above and in Appendix A. The pseudoscalar decays via A x → h x 1 + Z x * with Z x * → SM +SM , and it can be slow as well. An ordering m A x < m x < µ , M x is likely to produce this decay channel. This case has only been studied in Ref. [20] within a simplified model of the hidden sector.
In this case, the vector has a pair of two-body hidden decay channels to neutralinos with branching fractions BR(χ x 1 χ x 2 ) 0.94 and BR(χ x 1 χ x 1 ) 0.06. The presence of two lighter hidden neutralinos occurs readily when µ < m x < M x , m A x . The heavier of the two states tends to have a larger gaugino fraction, and this favours the The χ x 2 produced in the vector decay goes to χ x 2 → χ x 1 + Z x * with Z x * → SM+SM , and can be long-lived. We do not know of any previous studies of this decay channel.
In the sections to follow, we will investigate the current experimental bounds on these four scenarios.

Non-Fixed Target Limits
To begin, we collect limits on the four scenarios from precision tests, meson factories, and cosmology.

Model-Independent Limits
Precision electroweak observables and lepton magnetic moments provide important limits on the mass and kinetic mixing of the hidden vector that are independent of how it decays. We review here the model-independent constraints that apply to all four cases.
Kinetic mixing ties the hidden vector to the hypercharge vector, and this induces a mixing between Z x and the Z 0 after electroweak and hidden symmetry breaking [23]. A global fit to precision data in the presence of this mixing was performed in Ref. [24], and the limit for m x 10 GeV was found to be The leading effects of a kinetically-mixed hidden vector on the anomalous magnetic moments a = (g −2) of the electron and muon were computed in Ref. [11]. The current experimental status for the muon is [25] ∆a µ = a obs µ − a SM µ = (287 ± 80) × 10 −11 .
We demand that 0 < ∆a µ < 447 × 10 −11 , so as to lie within 2σ of the observed value while also allowing for consistency with the SM. The additional contribution due to the hidden vector tends to increase the predicted value of ∆a µ , and an interesting possibility is that a hidden vector is the source of the apparent (3.6σ) discrepancy with the SM [11].
For the anomalous magnetic moment of the electron, we use [26] ∆a e = (−10.
and we impose the 2σ limit −26.7 < ∆a e · 10 13 < 5.7. This range uses the updated direct measurement of a e [27] compared with the SM prediction [28] computed with the value of of α derived from atomic measurements in rubidium [29]. The combined precision electroweak and magnetic moment bounds are shown in Figs. 1 and 2.

Meson Factories
New light states have been searched for in a diverse range of meson factories, including BaBar, Belle, KLOE, and in rare Kaon decays at Brookhaven. We examine the bounds they imply for the theory.
Case A The direct signals of the vector in this case have been studied previously. For 2m µ < m x < 10.335 GeV, there are strong limits from the BaBar search for an exotic pseudoscalar a 0 via Υ(3s, 2s) → γ a 0 with a 0 → µ + µ − [30,31] when applied to the continuum process e + e − → γ Z x with Z x → µ + µ − [14,32]. Below m x < 400 MeV, the strongest bounds come from the KLOE search for φ → η Z x with Z x → e + e − [33,34]. In both cases, the bound is approximately 3 × 10 −3 .
Our supersymmetric realization of this scenario will also induce new decay channels for mesons through the mixing of h x 1 and h x 2 with the MSSM Higgs bosons. Limits on this mixing were studied in Refs. [15,35,36], where the strongest bounds were found to come The effective Higgs mixing angle in this theory is proportional to (m x /m h 0 ), where m h 0 is the SM-like Higgs boson mass (see Appendix A). We find that the model-independent and direct limits on and m x from the effects of the vector are stronger than those from decays through Higgs mixing for m X < 10 GeV. In the same way, no useful limits are found from leptonic heavy meson decays [37].
Case B Direct bounds on an invisibly decaying hidden vector were investigated in Refs. [17,19]. The BaBar search for an invisible pseudoscalar in Υ(3s) → γ a 0 with a 0 → invisible [38] Figure 1: Non-fixed target limits on Case C. The black, green and blue regions show the model-independent exclusions of Section 3.1; the red regions the BaBar constraints discussed in Section 3.2; and the grey regions the cosmological limits from Section 3.3.
Case C The vector decay Z x → h x 1 A x has not been studied in as much detail as the other cases [20]. Both products can have long lifetimes, and the limits found for Case B can often be applied to this case as well. The BaBar Υ(3s) → γ + invisible search imposed a veto on additional activity in the electromagnetic calorimeter and on hits in the muon chamber opposite to the photon direction [38]. To approximate the exclusion implied by this search, we apply the bounds found in Refs. [17,19] with the addtional condition that γv T > 100 cm for both h x 1 and A x , where γv T is the mean boosted transverse velocity in the lab frame. The value of 100 cm is the approximate transverse radius of the BaBar electromagnetic calorimeter and muon chamber [41]. We estimate γv T =p T /m using a simple Monte Carlo simulation that takes into account the angular distribution of the production cross section [42] and the angular acceptance of photons in the search [38]. The exclusions derived in this way are shown in Fig. 1 together with the model-independent bounds. We also find that the exclusions from K + → π + + inv apply here for 10 −2 , although we do not show them in the figures.
In contrast to searches for invisible decay modes, the existing meson-factory searches for the visible decays of a hidden vector presented in Refs. [30,31,33,34,43,44] do not appear to give any relevant limits on this scenario. These searches tend to focus on leptonic decays, they typically look for one or more resonances, and they often demand that the energies of the visible products reconstruct the mass of the decaying meson or the beam energy. Vector decays in case C will typically have two visible products and missing energy (for h x 1 longlived), or six visible products (for h x 1 short-lived). When there is missing energy, the energies of the visible products will not reconstruct the total beam or decaying meson energy, as required in Refs. [30,31,33,34,43]. Missing energy will be absent when h x 1 decays promptly. However, this only occurs when m h x 1 GeV where the hidden Higgs has only a very small branching fraction to leptons. This implies that the inclusive leptonic searches of Refs. [44] are insensitive. Finally, note that the pseudoscalar A x decay involves an off-shell vector, and its decay products will not reconstruct a sharp resonance.
Case D The vector now decays primarily to Z x → χ x 1 χ x 2 with BR 12 0.94, but also to Z x → χ x 1 χ x 1 with BR 11 0.06. In the former channel, the subsequent decay of χ x 2 to χ x 1 and visible matter via an off-shell Z x can be delayed. As in Case C, we find that meson factory searches for visible decays do not give any useful bounds. However, the BaBar Υ(3s) → γ + invisible search of Ref. [38] is applicable here. When the χ x 2 decay occurs relatively promptly, γv T τ < 100 cm (with the mean value of γv T =p T /m χ x 2 estimated with a simple Monte Carlo simulation), only the Z x → χ x 1 χ x 1 channel contributes significantly. In this situation we rescale the limits on obtained Refs. [17,19]

Cosmology and Astrophysics
This theory can also be constrained by cosmological observations given a specific assumption for the evolution of the Universe. An obvious requirement for all cases is that the relic density of the lightest hidden neutralino not be too large. The relic density calculations of Refs. [45] are applicable here. In general, the thermal relic density tends to be well below critical for m χ x 1 > m X unless α x is also very small, but can become unacceptably large when m χ x 1 < m X . Additional bounds from cosmology can arise if some of the states in the theory are longlived. Lifetimes greater than τ 0.05 s will lead to decays after the onset of big-bang nucleosynthesis (BBN) and can potentially alter the abundances of light elements [46]. Late decays can also lead to spectral distortions in the cosmic microwave background radiation [47] or an excess in gamma rays [48].
In cases A-C, the longest-lived state is nearly always the h x 1 scalar. It decays mainly through mixing with the MSSM Higgs bosons to pairs of SM particles. When m h x 1 > 2m π , a significant fraction of these decays will be hadronic and will ultimately create light pions. When τ > 0.05 s, these pions can scatter with protons to create more neutrons than would otherwise be present [46,49]. Below the pion threshold, all decays will be purely electromagnetic, producing muons, electrons, or photons. These particles thermalize efficiently with the plasma, and for τ 10 4 s, they do so before potentially destroying light nuclei [46,49]. Thus, longer lifetimes are safe below the pion threshold. In Fig. 1 we show the regions where h x 1 decay could be dangerous for nucleosynthesis (or the CMB) based on these considerations. These regions are not necessarily ruled out if the yield Y = n/s of h x 1 is sufficiently small at the time of decay. However, in a thermal scenario where h x 1 begins with an equilibrium density and freezes out, the freeze-out process relies on the three-body annihilation mode h x 1 h x 1 → Z x Z x * and tends to produce a relatively large yield. The long-lived state in case D is the next-to-lightest neutralino χ x 2 , which decays through an off-shell vector. For ∆m χ = m χ x 2 − m χ x 1 > 2m π , a significant fraction of these decays will be hadronic. For smaller mass differences the decays will be electromagntic. In Fig. 2 we show the regions where this decay could potentiallly affect nucleosynthesis. Note, however, that the transfer reaction χ x 2 χ x 1 → χ x 1 χ x 1 is generally efficient, and the thermal yield of χ x 2 is typically very small.
Further constraints on all four cases can be derived from astrophysics. For masses m x 1 GeV, the strongest limits come from the observed cooling rate of supernova (SN) 1987A [14,50]. The hidden vector can be created by thermal scattering within a SN and generate new energy loss mechanisms. In case A, this vector can be long-lived and escape the SN before it decays to the SM, and small values of 10 −7 are excluded for m x 100 MeV [14]. In cases B-D, the vector will decay promptly to long-lived hidden states such as h x 1 or χ x 1 which can carry energy away if they leave the SN before decaying or scattering. For these cases, we expect limits similar to those found in Ref. [50], which studied the bounds on hidden dark matter in the form of a Dirac fermion or a complex scalar charged under U (1) x . A potential important difference in the present case is that the metastable states interact differently with the vector boson, and may be less likely to scatter before exiting the SN. We defer an analysis of this effect to a future work. The electron beam strikes the target, producing a metastable hidden sector state (dashed line); this decays to visible sector particles within the decay volume after passing through the shield. A signal requires that at least one of the visible decay products strike the detector.

Electron Fixed Target Experiments
Fixed target experiments using electron beams have played an important role in studies of hidden vectors. Significant limits have been set when the gauge bosons decay directly to the SM [14,[51][52][53][54]. Additionally, a number of experiments have been proposed to extend the search reach based on both visible [55][56][57][58] and invisible [17] decays. It is therefore natural to examine how they constrain the more general hidden sectors we consider here.
Most searches require hidden states decaying to the SM, so we begin by considering that case. The experimental geometry is sketched in Fig. 3. A signal requires the production of a metastable state at the interaction point, which decays to visible sector particles (typically required to be leptons) within the decay volume. In previous work, the long-lived state was the hidden vector itself; the lifetime requirement leads to limits being set at small kinetic mixing and/or low mass.
However, as already discussed in Section 2, our model displays a broader range of phenomenology. Of our four benchmarks, only in Case A do the standard limits apply. In all other Cases, the vector Z x decays promptly to the hidden sector. Instead, limits may be derived from other metastable particles: the lightest scalar h x 1 (Cases A-C), pseudoscalar A x (Case C) and next-to-lightest fermion χ x 2 (Case D). The subsequent limits at electron beam dumps have not been previously studied.
Another class of searches look for the production of invisible states at fixed target experiments. These have been proposed as a way to search for light dark matter [17]. The experimental geometry is sketched in Fig. 4. Production of hidden sector states proceeds as before, producing either stable or sufficiently long-lived hidden particles. The experiment would seek to observe the quasi-elastic scattering of these states with nuclei within the detector, similar to searches for the direct detection of dark matter. Such a search is highly relevant in Cases B and D, when the Z x can decay to the stable lightest fermion χ x 1 . Figure 4: Sketch of the experimental configuration for fixed target searches for invisible final states. The electron beam strikes the target, producing a stable or long-lived hidden sector state (dashed line). This then scatters inside the detector at the starred point. After some general comments, we discuss the production of the hidden sector through an on-shell (off-shell) Z x in Section 4.1 (4.2). We relate the cross sections to the expected signals in Section 4.3, and discuss relevant experiments in Section 4.4. The theoretical framework for the production of a hidden vector at electron fixed target experiments is well established [14,51,59,60] (see also Ref. [17]), and we outline the main points here to be self-contained and extend the formalism in some points for the analysis at hand. Additional technical details are presented in Appendix B. The reader who is interested primarily in results should therefore skip to Section 4.5, where we present the combined limits from electron beam dumps for our four benchmark scenarios.

Production Cross Section: On-Shell Vector
We first consider the production of an on-shell hidden vector. This has been the usual focus of attention, with a long-lived vector decaying to leptons; that is relevant in Case A. Our interest is on a vector that decays promptly to hidden sector scalars (Case C) or fermions (Cases B and D).
There are two Feynman diagrams at leading order, shown in Fig. 5. The improved Weizsacker-Williams (WW) approximation [59] lets us factor out the dependence on the structure of the target into an integral over form factors. For an electron of energy E e , this approximation is valid when [51] m e m x E e and E x θ 2 where E x , θ x are respectively the energy and scattering angle of the hidden vector. The production differential cross section for the full process is then related to that for electronphoton scattering: The two-to-two cross section is evaluated at the special kinematics q = q * , where q * is parallel to p e − k x . The effective flux χ contains all details of the target structure; it is also a function of the electron energy E e and vector mass m x . We plot χ/Z 2 in Fig. 6 for the E137 experiment [61] (see Section 4.4), where the target atomic number Z = 13. It is O(1-100) for m x < 1 GeV, and drops sharply above that point. More details about this function are provided in Appendix B.1.
Introducing x e ≡ E x /E e , the total differential cross section is [14] dσ where for later convenience we have defined U is the virtuality of the intermediate electron in the second diagram of Fig. 5, and is a function of both x e and θ x : Figure 7: The two leading Feynman diagrams for production of off-shell massive vector in electron-fixed target scattering. The dotted lines represent the on-shell hidden states; we integrate over the phase space of these particles in deriving Eq. (21).
This cross section is sharply peaked at θ x ≈ 0 and x e ≈ 1. Indeed, we see from Eqs. (17) through (19) that there is a singularity in that limit that is only regulated by the electron mass. The WW approximation can be understood as focusing on these singular terms.
We integrate Eq. (17) over the angular variable θ x . This gives us the differential cross section with respect to the electron energy, The total cross section for production of on-shell hidden vectors at the E137 and proposed JLab experiments are shown in Fig. 8.

Production Cross Section: Off-Shell Vector
We next consider the situation where we enter the hidden sector through an off-shell vector. This is the natural generalisation of the previous scenario. It is particularly relevant to Case B, where it is the only way to get a visible signal from the hidden sector; and Case A, where it is the only way to produce hidden states other than the hidden vector.
We focus on the production of two on-shell hidden states through the diagrams shown in Fig. 7. The dotted lines represent either h . Of these, the first is usually most important on kinematic grounds. Before fully generalising the WW approximation, we note that by integrating over the hidden state phase space and using the QED Ward identity, we can obtain a simple extension of Eq. (16) (see Ref. [17] and Appendix B.2): This expression is valid under the the conditions of Eq. (15) with the replacement m x → k 2 x . The partial width Γ HS is for a vector of mass m 2 x = k 2 x to decay to the relevant final state; explicit expressions are given in Appendix B.3. Note that if we replace the numerator with a Breit-Wigner propagator and allow the vector to go on-shell, then in the narrowwidth approximation the bracketed term in Eq. Eq. (16). We show the production cross sections for off-shell hidden vectors in Fig. 8 for two cases that are extremal among those we consider.
The full generalisation of the WW approximation is most conveniently expressed in terms of integrated matrix elements, due to non-trivial phase space factors. We denote the electrontarget and electron-photon amplitudes by M eN and M eγ respectively, where the latter is evaluated at the special kinematics q = q * . Then the WW approximation is Here, M i is the mass of the target, N i its internal degrees of freedom, the vector magnitude is roughly the minimum momentum transfer between the electron and the target. There are n hidden sector particles with momenta k i , i = 1 . . . n, and dΠ l represents the l-dimensional phase space integral including the delta function that enforces overall conservation of momentum. More details are presented in Appendix B.2.
For the case n = 2 of interest, the hidden sector kinematics may be expressed in terms of the momentum of the off-shell vector k x , and k ∆ ≡ k 1 − k 2 . Requiring that the final states be on-shell imposes The first constraint can be used to fix | k ∆ |, and the second to fix the angle between k x and k ∆ . k ∆ is then defined by k 0 ∆ and φ ∆ , with the latter being the azimuthal angle for the vector k ∆ relative to k x . The lab-frame cross sections we desire are, in the Higgsstrahlung channel: for the production of a pair of scalars: and for a pair of fermions The various N s are the combinations of mixing matrices that appear in the three-point vertices respectively. In practice we need only consider a = 1 and i = 1, j ∈ {1, 2}. They are explicitly given by R is the matrix that relates the real hidden scalar mass and gauge bases, as defined in Eq. (59). P is the equivalent matrix for the hidden fermions [21]. We have used x e and U defined as in the on-shell case with m 2 x → k 2 x . The k ∆ -dependence comes through the function In the limit that the WW approximation is valid, (p e − q) · k ∆ ≈ p e · k ∆ ; while with θ ∆ the angle between k x and k ∆ .

Expected Number of Signal Events
In the previous two sections we gave the differential cross sections for the production of hidden sector states at beam dump experiments. Here we relate those expressions to the actual experimental observables, specifically the number of events seen in the detectors. We first relate the cross sections to the number of hidden sector particles actually produced, and then discuss the acceptance factors that give the number of signals per hidden sector event.
For a fixed target experiment, we must account for interactions throughout the full depth of the target material. This must also include energy loss from the electron beam as it travels through the target. In particular, the electron energy in Eq. (17) and similar expressions satisfies E e ≤ E 0 , the incident beam energy. This effect is handled by convolving the production cross section with an attenuation function. We have the following expression for the distribution of the number of hidden vectors produced: Here, N e is the total number of electrons delivered to the target; N 0 is Avogadro's number; A is the atomic number and X 0 the unit radiation length (M/L 3 ) of the target material; s is the depth in radiation lengths within the target, of total thickness T ; and The attenuation function of Eq. (30) gives the energy distribution of an initiallly monochromatic electron beam after travelling s radiation lengths through the target material. We follow Refs. [51,62] in taking 2 with Γ the Gamma function and b = 4 3 . Since σ ∝ E 2 e (see Eq. (17)), production is dominated from electrons with the initial beam energy; and from I, this mostly occurs in the first few radiation lengths of the target.
We define an average acceptance number that is the ratio of the number of signal events to the total number of hidden sector events computed above. There are two steps in deriving these factors. First, we need to infer the kinematic distributions for the stable or metastable states from the differential cross sections of Sections 4.1 and 4.2. Next we must determine either what fraction of metastable states' decay products will give visible signals; or what fraction of hidden sector particles will scatter in the detector, depending on the nature of the experiment.
We compute these using a simple Monte Carlo. First we generate events drawn from the appropriate one of the distributions in Eqs. (17), (24), (25) and (26). We then decay through the hidden sector till we reach the stable and metastable states. In doing so we use the narrow width approximation, treating all decays as prompt and isotropic in the parent rest frame. We also track all branching ratios as necessary, though in most cases these are trivial. This effectively gives us the (meta-)stable particle kinematics.
The subsequent steps then depend on the nature of the experiment. For searches based on visible sector particles, there is another branching ratio; the experiments we will discuss triggered only on electrons, suppressing sensitivity for hidden sector masses above the muon threshold. The acceptance factor is the product of all relevant branching ratios, the probability that the metastable state decays within the decay volume, and that its visible daughter particles hit the detector with sufficient energy to pass the threshold. We compute this on an event-by-event basis. The decay probability is with L sh and L dec as shown in Fig. 3, v z and p z are the parent's velocity and momentum along the beam direction, m is its mass and τ its lifetime. The angular acceptance probability is found by extending our Monte Carlo, decaying the metastable states within the decay volume and counting what fraction of daughter particles pass the experimental cuts. To this end, we model the experiments as cylinders of radius r Acc with hard energy threshold cuts at E thr . The final number of signal events is where EV is the total number of events we generate, and HIT the set of metastable states with daughter particles observed in the detector.
For the proposed search of Ref. [17] based on invisible particles, the remaining factors are much simpler. The angular acceptance is directly found from the kinematic distribution of metastable states, asking what fraction strike the detector (including, where necessary, a survival probability). This is multiplied by the event-by-event probability that the hidden sector state scatter within the detector: with ρ the target density, m N the nucleon mass, L d = 1 m the detector depth as in Fig. 4 and σ χN the total hidden sector-nucleon scattering cross section summed over final states. This assumes that multiple scattering and attenuation of the dark sector beam is negligible (P scat 1). Scattering to different hidden sector states often dominates due to mixing matrix suppression of the elastic process. For this reason we neglect scattering from electrons, where the inelastic channels are usually kinematically forbidden.
Full details on the differential scattering cross sections are given in Appendix C. When the incident particle is relativistic we may approximate [17] σ χN ∼ 4 π 2 α α x N 2 Q 2 is the momentum transfer in the scattering process, with Q min ∼ 140 MeV and Q max ∼ 1 GeV (for the JLab experiment). N I is the mixing matrix factors for the three-point coupling of the hidden sector state to X µ , summed over possible final states. In the case when all hidden sector states are accessible, we have We take the signal yield as with EV the number of simulated particles, HIT the set of hidden particles with trajectories intersecting the detector, and P surv the probability of that state not decaying before passing through it.

Current and Prospective Experiments
Let us first briefly review the experiments that have been used or proposed to set limits on low energy hidden sectors.
Previous Beam Dumps Several previous beam dump experiments have been used to set limits on hidden vectors decaying directly to leptons [14,51]. These include E137 [63] and E141 [61] at SLAC, E774 [64] at Fermilab, KEK [65], and Orsay [66]. We use the experimental parameters as given in Table 2. The energy thresholds are taken from the relevant experimental papers; they were usually ignored in previous work, as they are trivial for searches based on Z x → e + e − . For our more complex decay chains they must be included. Note that all these experiments were only sensititve to e ± , except for E141 which was only sensitive to positrons. We demand that the number of expected events be less than N 95% .
Present and Future Visible Searches A number of experiments are currently running or have recently been proposed to search for low energy hidden sectors. This includes MAMI [52], APEX [53,55], HPS [57,58], a proposal at the CERN SPS [67,68], and Dark-Light [56]. Regretably, these experiments do not set any limits on our model beyond those that apply in Case A. For APEX, MAMI and HPS, a crucial tool used to discriminate signal from background is the requirement that the observed electron-positron pair have combined energy equal to the beam energy. This is based on the sharp peak in the cross sections at  Table 2: Parameters of the electron beam dump experiments we consider. The parameters E 0 and E thr are in GeV, while L sh , L dec and r Acc are in metres.
x e ≈ 1 in Eq. (17). However, if the hidden vector decays to hidden sector particles-or is not produced on-shell-then the initial beam energy will be distributed among a larger number of particles. Hidden sector events in our model will then not pass the signal cuts.
The proposed CERN SPS search uses different discriminators, but with a similar philosophy. Calorimeters will be positioned before and after the decay volume, and the total energy in them must sum to the incident beam energy. For the small scale of the decay volume (∼ 5 m) the hidden Higgs is effectively stable, so all non-trivial hidden sector events will involve irreducible missing energy from either h x 1 or χ x 1 . We thus expect that no events will pass the experimental cuts. It is also planned to search for invisible decays [68]; this likely would have some sensitivity to cases B-D, but we defer a full investigation of this for a future paper.
For DarkLight, the situation is slightly different. DarkLight uses a peak in the electronpositron invariant mass spectrum to discriminate the signal. In our model, the pseudoscalar A x and fermion χ x 2 have three-body decays, so will not pass this cut. The scalar h x 1 will pass the cut; however, DarkLight requires a relatively prompt decay. In the mass range to which DarkLight is senstive (less than 100 MeV), the scalar has a decay length of 10 m for 0.1. It follows that there will be no sensitivity below this kinetic mixing.

JLab Inivisible Search
The proposed search at JLab [17] would have the experimental parameters listed in Table 2. Note that for this experiment, the proposed detector would have a square cross section so r Acc should be interpreted as the side of that square. We follow the original proposal in taking three different choices for sensitivity, corresponding to different possibilities for background subtraction. In order from least to most optimistic, we consider sensitivity to 20000, 1000 and 40 events respectively.

Limits
We now present current and prospective limits for our four benchmark scenarios. For clarity, we show only limits from electron beam dump experiments, as well as from the electron and muon anomalous magnetic moments (to aid in comparisons with other results). For composite plots showing all limits we find, see Section 7. We show limits in the m x -plane for benchmarks A and B in Fig. 9, and benchmarks C and D in Fig. 10. For benchmarks A through C, we also show the mass of the lightest scalar h x 1 on the upper horizontal axis; for benchmark D, we show the mass of the lightest fermion χ x 1 . The general features of all exclusion plots from visible searches are straightforward. At large kinetic mixing, the exclusion contours are set when the lifetime of the metastable particle becomes too short. They are approximately contours of 2 m x , except near particle thresholds. At small kinetic mixing, the boundary is mainly set when the production cross section becomes too small, which is only weakly dependent on mass. Finally, there is a lower limit on all these searches when the long-lived particle can no longer decay to electrons. While decays to photons might fake electrons in the experimental detectors, the lifetimes are highly suppressed with the result that no limits from these searches can be set.
Case A This benchmark corresponds to the previously-studied case, where the hidden vector may only decay to the visible sector. We therefore include past limits for electron beam dumps taken from Ref. [51] in Fig. 9. We find no new limits beyond those. Additional constraints could in principle arise from the production of hidden sector scalars or fermions through an off-shell vector; but a combination of kinematic and coupling suppressions result in this not being the case. The strongest such limits we find come from long-lived h x 1 decays at the E137 experiment. These are shown as the red shaded region in Fig. 9, but are weaker than the current constraints from a e . 3 Figure 10: Exclusion plots from electron beam dumps for benchmark slopes C (left) and D (right). Also shown are limits from the electron and muon anomalous magnetic moments; these are shown using the same notation as in Fig. 9, but are omitted from the keys for space. See the text for more details.
Case B Similarly to Case A, there are no new limits from previous experiments. The limits from long-lived scalars that would exist are comparable to those in Case A, and inferior to those from a e . However, we now have promising prospective limits from the JLab search. Even in the most pessimistic background scenario, we can extend the exclusion contours to exclude much of the region preferred by a µ . This is unsurprising; the vector here decays invisibly to the stable χ x 1 χ x 1 final state, precisely the motivating case for that search. The limits here are not directly comparable to the plots of Refs. [17,19]: that work fixed the mass of the stable hidden particle and varied the vector mass, while we vary both with a fixed mass ratio.
Case C This benchmark offers a number of interesting new and prospective limits, that go beyond the constraints from a e and a µ . The improved reach is because we can produce unstable hidden sector states through an on-shell hidden vector. The phenomenology is richer because the vector decays to two particles, h x 1 and A x , that are both potentially long-lived.
We first address the limits from searches for visible decays. In Fig. 10 we only show the limits from E137; the limits from the other experiments listed in Table 2 are strictly inferior. We show separately the limits from the scalar (red) and pseudoscalar (blue) decays. The scalar limits are sensitive to the Higgs mixing induced from D-terms as discussed in Section A.3. The limits with (without) this mixing are shown in darker red with solid boundary (lighter red with dashed boundary), and labelled "SUSY" ("SM") in the key. The two cases are equivalent below the muon threshold, as the mass mixing is unimportant there (see Fig. 25). Above the muon threshold, the limits with the mixing are weaker due both to the hidden scalar decaying too quickly and the branching ratio to electrons is being more strongly suppressed.
The limits from pseudoscalar decays extend to lower kinetic mixings than those from the scalar decays. This is because the A x width is less suppressed than the h x 1 , so smaller values of are needed to have macroscopic decay lengths. There are two regions where the scalar limits are superior. At low mass, the threshold for the decay A x → h x 1 e + e − is crossed before that for h x 1 → e + e − ; while at high mass and moderate , the pseudoscalar is too short-lived to produce an observable signal. Additionally, the limits from the pseudoscalar are less generic for two reasons. First, it is possible that the decay A x → χ x 1 χ x 1 may be allowed and the corresponding scalar decay forbidden. Second, if CP is violated in the hidden sector and we choose a value of tan ζ slightly closer to 1, the decay A x → h x 1 h x 1 may become relevant. Finally, we have the prospective limits from the JLab invisible search. These limits are dominated from the scattering of the scalar h x 1 in the detector. As such, while these limits are generically weaker than the pseudoscalar limits from E137, they might still be relevant if the latter are evaded as discussed above.
Case D This benchmark has non-trivial limits from both visible and invisible searches. Both come from production of an on-shell vector that decays to χ x 2 χ x 1 . Visible searches look for the decay χ x 2 → χ x 1 e + e − , suppressed for the same reasons (phase space and 2 ) as the pseudoscalar in Case C. This slope is the only one of our four where visible searches other than E137 set limits, though that experiment still gives the strongest constraints. Of the five experiments in Table 2, the limits from the Orsay experiment are strictly inferior to those from the KEK experiment, so we do not show them in Fig. 10.
Invisible searches for this benchmark are most sensitive to scattering of the χ x 1 in the detector. Of the two vector decay modes to the lightest fermion, the χ x 2 χ x 1 channel dominates due to its larger branching ratio. While there remains a region of parameter space where a µ can be explained that is not excluded by visible searches, it can be excluded by the JLab search.

Hadronic Fixed Target Experiments
Hadronic fixed target experiments can also be used to constrain low-energy hidden sectors [15,16,20,[70][71][72][73][74]. This is particularly relevant for hidden sectors with GeV-scale masses, where most constraints from electron experiments are limited by kinematics. As with electron beam dumps, most previous studies assumed the hidden vector either decays directly to the SM or to stable invisble particles. When long-lived scalars were considered [15,20], they were required to decay only through hidden vector loops. The possible limits from hidden sector pseudoscalars and fermions have not been studied.
The searches for visible particles discussed in this section have the general configuration shown in Fig. 11. This is similar to Fig. 3, but the decay volume and detector are combined. Searches for the scattering of invisible particles have the same configuration of Fig. 4, with an incident beam of protons rather than electrons, and some of the experiments we consider placed the detector off the beam axis.
We discuss production of hidden sectors at these experiments in Section 5.1 and the calculation of acceptances in Section 5.2. This includes review of material covered elsewhere, both for completeness, and to extend some expressions for our more general hidden sector. In Section 5.3 we discuss relevant experiments. The reader who is mainly interested in results should skip to Section 5.4, where we present and discuss the exclusions we find for our four benchmark slopes.

Production
Hidden sector production in these experiments can roughly be divided into two domains [15], according to the hidden sector mass scale. If the hidden states have masses below Λ QCD , then production will mainly occur through the decay of mesons and baryons produced in the initial collision; we term this the "low mass" region. In the complementary "high mass" region, we can resolve the quark content of the proton and produce an s-channel hidden vector through qq fusion.
The low mass regime was previously considered in Ref. [15]. We extend some of their results to include new decay modes, in particular Eq. (41). Production of hidden sector states occurs primarily through the secondary decay of hadrons produced at the interaction point. These decays can proceed either through kinetic mixing with the hadron decay product or with the hadron itself. In the former class, for any hadron M with an electromagnetic decay of the form M → γ + . . ., there will be a companion decay M → Z x( * ) + . . .. In the second, a vector meson can mix with the Z x e.g. through a mutual mixing with the photon, allowing it to decay into the hidden sector. These possibilities are illustrated in Fig. 12. The most important example of the first category is the diphoton decay of a neutral pion. This has maximal branching ratio, and pions are produced in abundance in any hadron experiment. To extend the reach above the pion mass, we also consider the diphoton decays of the heavier pseudoscalars η, η ; and the baryonic decay ∆ → N γ. When the hidden vector is on-shell, its production has branching ratio This assumes that the additional decay does not change the total width, a good approximation for 0.1. We also have the possibility of decays involving an off-shell hidden vector; these have branching ratio where Γ HS is the partial width from Appendix B.3 appropriate to the hidden final states; m HS is the sum of those masses; and Br 2 refers to the on-shell expression from Eq. (39) with m 2 x replaced by q 2 . The second class of hadronic decays into the hidden sector involve the mixing of a vector meson with the hidden vector. This is exemplified by the ρ meson. As is well-known, the ρ mixes the the photon; successive insertions of the ρ-γ and γ-X µ mixings then lead to a ρhidden vector mixing, which allows ρ to decay to hidden states. This leads to a ρ branching ratio The propagator is regulated by the total hidden vector width Γ x , which is relevant as m x → m ρ . Analogous results hold for other neutral vector mesons.
The total number of hidden particles produced is proportional to the number of different hadron species produced. When possible, we use measurements or estimates of this from the experimental collaborations themselves. Otherwise, we model the production as dominantly Figure 13: Feynman diagrams for the three parton-level processes that produce two hidden sector particles at hadronic beam dumps in the high mass regime. All proceed through an s-channel hidden vector to produce hidden scalars (a), fermions (b) or a scalar-vector pair (c).
in the first interaction length at full beam energy. This lets us relate the hadron production cross sections and yields using a simplified version of Eq. (30): where N p is the total number of protons on target, X nuc is the nuclear interaction length, and σ N is the per-nucleus cross section. We then have N X = Br(M → X) × N M , summed over hadron flavour M .
In the high mass regime, we can produce two-body hidden sector final states through either an on-or off-shell vector, through the processes shown in Fig. 13. We have the scalar, fermion and (for the intermediate vector off-shell) scalar-vector final states. If the vector is on-shell, the tree-level partonic cross section iŝ with Br x the hidden vector branching ratio to the final state of interest. If the vector is off-shell, the equivalent expression may be written in terms of the vector partial widths from Appendix B.3:σ withŝ the partonic center of mass energy. We find the per-nucleon cross sections by convolving Eqs. (43) and (44) with the CTEQ 10 parton density functions (pdfs) [75] in the usual way, We convert to a per-nucleus cross section by assuming a simple incoherent scaling, σ N ≈ Z σ p + (A − Z)σ n .
Though we have named this the high-mass regime, we are concerned with particle masses of one to a few GeV. This sets the energy scale for the partonic cross sections, and it is still close to Λ QCD . One might then be concerned about possible higher-order corrections in α S . We computed these terms, and show show the tree and one-loop cross sections for on-and off-shell processes in Fig. 14. These plots assume a factorisation scale µ F = m V (on-shell) or √ŝ (off-shell); varying this represents an additional theoretical uncertainty. This can be as large as an O(1) effect on the overall production rate [71]. We use the NLO cross sections in computing all limits, and note that our limits are self-consistently smooth between the lowand high-mass regimes.
As with hadron production, we obtain the total number of hidden states produced by assuming all interactions take place in the first interaction length with full beam energy. The equivalent of Eqs. (30) and (42) is

Acceptances
As for electron beam dump experiments, the experimental acceptance is a product of three terms: an angular acceptance factor that the (meta-)stable particle hits the detector; a survival probability; and the probability of either decaying or scattering within the detector, as appropriate. We operate with samples of hidden sector events rather than the kinematic distributions directly.
In the low mass region, we first construct a set of hadron events, then decay them to through the hidden sector spectrum using the narrow width approximation. Our samples are built on uniform grids in the parent hadron phase space, then weighted by the hadronic kinematic distributions. Our choice of weighting function differs by experiment. For LSND and MiniBooNE, we use parameterisations defined in the lab frame as functions of the meson momentum p lab and θ lab . For LSND we use the Burman and Smith fit f BS [76], appropriately weighted to account for the use of different target materials over the experiment's lifetime. For MiniBooNE, we use the Sanford-Wang distributionf SW [77]. For all other experiments, we use the modified Bourquin-Gaillard parameterisation f BG [78]. This is defined in terms of the CoM frame rapidity y com and transverse momentum p T,com , so after constructing our initial grid we must boost to the lab frame. Lastly, we decay each parent hadron to the hidden sector multiple times, so as to avoid a rare decay at a high-weight point unduly affecting our results.
In the high mass region, we use an unweighted sample of events construced with standard tools. We implemented our model in FeynRules 2.0 [79], and generate events using MadGraph 5.1.5.13 [80] for the processes of Fig. 13. Since we can produce any hidden sector particle directly in a 2 → 2 process, we do not consider secondary production through the decay of heavier hidden sector particles. This should be a small correction due both to suppressed production of heavier states, and the secondaries having larger production angles relative to the beam axis.
Once we have generated a sample of hidden sector events, the remaining analysis is analogous to Section 4.3. For visible searches, we need modify the decay probability P dec of Eq. (32) by replacing L dec with L d . The angular acceptance is also simplified, as we only need the metastable state to cross the detector. In the high mass region, the numer of signal events is given by with EV the number of generated events and HIT the subset of hidden sector particles that hit the detector, with daughter particles that pass the experimental cuts. In the low mass region, the expression is slightly modified: with w i the weighting associated with the event containing the particle i and w tot the total weighting summed over all events. The weights in turn are given by with p i lab etc. the parent hadron kinematics, and δp lab etc. the sampling interval in phase space.
For searches for dark matter scattering in the detector, the analysis here is almost identical to to the electron beam dump case. There only modification of note refers to the LSND experiment, which searched for electron scattering with the final state electron in a given energy range. This demands that we use the full scattering cross sections of Appendix C, with Eq. (35) a poor approximation. It is also useful to rewrite the scattering probability as P scat = n e,N L d σ χe,N .
With this correction, the total signal yield in the high mass region is given by Eq. (38). In the low-mass region, we modify it to include the weighting,  Table 3: Experimental parameters descriibing the beam and detector position and size. Energies E p are in GeV, lengths L sh and L d are in m, and the area A acc is in m 2 . N p is the number of protons on target recorded or expected.  Table 4: Hadron cross sections we take for low mass hidden sectors. Except as mentioned in the text, we take σ π ≈ 25-30 σ η from Ref. [89], and the multiplicities of other hadrons from Pythia 8.180 [90].
with w i as given in Eq. (49).

Experiments
We list general properties of the experiments we find set non-trivial limits in Table 3, and hadron production cross sections in Table 4. Properties relevant to visible searches are in Table 5, and to invisible scattering searches in Table 6. Because there is a relatively large variation in the configuration and searches for each experiment, we discuss them individually below.
CHARM The CHARM experiment featured a 400 GeV proton beam impacting a copper target. Searches where performed for an axion (pseudoscalar) decaying to γγ, e + e − or µ + µ − in Ref. [81]; and for heavy neutrinos decaying as ν h → νl + l − in Ref. [82]. These cover the relevant topologies of two-and three-body decays with and without additional unobserved states. No events where observed in either search.
The CHARM detector was positioned 5 m (10 mrad) off the beam axis and had a square transverse cross section. For low mass hidden resonances, we normalise our results to N det π = 2.9 × 10 17 from the experimental collaboration [81], where N det π is the number of pions produced on a trajectory that would intersect the detector. For other pseudoscalar mesons, we use the cross sections quoted in Ref. [70]; for vectors, the measurements of Ref. [91]; and for fermions, we estimate N ∆ ≈ 0.03N π from the relative multiplicities in Pythia 8.180 [90].
Limits from these CHARM searches have previously been interpreted in terms of vector    Table 3 or the text for references.
kinetic mixing. Ref. [70] only considered the decay Z x → l + l − , with the hidden vector produced in psuedoscalar meson decay. Ref. [20] did consider production and decay of hidden scalars, but in a non-supersymmetric context and at high masses. Our results are then different due to the mass mixing, as discussed in Appendix A.
MINOS The MINOS experiment [83] uses the 120 GeV NuMI beam on a graphite target. We consider possible signals in the near detector, positioned 1.04 km away. We model the detector as a cylinder of radius 1 m and take a recent value [84] for the total number of protons on target. For hadron production we take the cross section for pC → π + . . . from Ref. [92]; using Eq. (42) gives N π ≈ 0.99N p .
We consider the possibility of signals at MINOS from both hidden sector scattering and decay. For decays, we follow Ref. [12] and consider a sensitivity to O(10) muon pairs with a detection efficiency comparable to that for scattering [93]. Searches for invisible particle scattering must compete against a large neutrino background. We follow Ref. [71] and show prospective limits for 10 3 and 10 4 events in our plots, corresponding to different estimates for background rejection.
ν-Cal I The ν Calorimeter I experiment took data from a beam dump at the U70 accelerator, where a 70 GeV proton beam was delivered to an iron target. Searches for axions and light Higgs bosons [85] were reinterpreted in terms of hidden vectors decaying to leptons in Refs. [72,73]. We find the pion production cross section by rescaling the measurement of Ref. [94] as discussed in [73]: From Eq. (42) we find N π ≈ 0.96N p . Also, our choice for the η production cross section is consistent with its non-observation in Ref. [94].
The combination of a relatively small distance between the target and the detector and a moderately large energy are useful in setting limits when the mass mixing of Appendix A dominates the h x 1 decay. However, we find that the limits from meson decays are almost always inferior to the corresponding limits from CHARM.
INGRID INGRID [86] is the near on-axis detector in the T2K experiment [95]. We use an estimate for the prospective total lifetime N p , and approximate the detector efficiencies as similar to MINOS. The detector has a non-trivial cross shape transverse to the beam direction. As with MINOS, a large neutrino background would demand a dedicated study to reject backgrounds, which we approximate by considering limits for 10 3 and 10 4 events.
LSND The LSND experiment used a 800 MeV proton beam on two different targets over its lifetime, water and a high-Z metal [87]. When computing the acceptance efficiencies, we use a weighted mean of the two distribution functions f BS for the different target materials. We model the detector as a cylinder of diameter 5.7 m. Because of the low beam energy, we only consider limits from hidden states produced in pion decays; it is assumed that the production of heavier hadrons is highly kinematically suppressed. We use an estimate N π ≈ 10 22 of the number of pions produced from [16].
A search for neutral current scattering on the full LSND data set observed O(300) events above background, of which O(200) where expected from neutrinos [88]. We conservatively place limits for 1000 hidden sector events as unambiguously ruled out [15,16]. Along with MINOS, we consider the possibility of both hidden sector decay (to electrons) and scattering, as shown in Tables 5 and 6, though we find that the limits from visible decays are only as good as those from CHARM.
Other Experiments Searches for heavy neutrinos decaying to leptons were carried out at NOMAD [96] and PS-191 [97], and reinterpreted as searches for hidden vectors in Ref. [74]. We find that the limits from other hidden decays then Z x → l + l − are inferior to those from CHARM. Refs. [15,16,71] proposed looking for hidden particles scattering at MiniBooNE [98] and in the ND280 detector at T2K [99]. For our benchmarks, the former is always inferior to MINOS and the latter to INGRID. Finally, we note that future experiments such as Project X [100] or AFTER@LHC [101] would likely place further exclusions on our paramter space. The nature of those limits is beyond the scope of this work.

Limits
We now present current and prospective limits for our four benchmarks from hadronic beam dumps. Composite plots showing all the limits we find are shown in Section 7. We give limits for Cases A, B and D in Figs. 15, 16 and 19. For Case C, we show limits separately for constraints from scalars (Fig. 17) and pseudoscalars (Fig. 18) for clarity. In Figs. 15, 16 and 17, we split the constraints for m x > 0.1 GeV with (left) and without (right) the Figure 15: Case A, with (left) and without (right) the Higgs mass mixing induced by the SUSY D-terms. See the text for more details. Note that the limits from ν-Cal I extend to lower masses than shown, but are inferior to CHARM in that region.
inclusion of D-term Higgs portal mass mixing. As discussed in Appendix A, the presence of this mass mixing heavily alters the h x 1 lifetime above the muon threshold. Hence all experimental searches based on a long-lived scalar will differ between these two possibilities.
We note some general features of our results before discussing each case in detail. Limits from the hidden Higgs decays are alomst always upper bounds on . This is because for kinetic mixings not ruled out by e.g. EWPTs, the scalar decay length is large. Signals are then limited by production (∝ 2 ) and, when relevant, scaterring probability (∝ 2 ). The exception comes only for large masses m h x 1 > 2m µ , when the lifetime can become short enough for the scalar to decay within the shield.
At low masses, limits from LSND tend to dominate due to its large luminosity and the absence of kinematic suppression. At intermediate masses, limits from CHARM are stronger; and at high masses above the muon threshold, the ν-cal I search can also be relevant. We show current limits derived here in shaded regions, and use unshaded contours to denote prior exclusions or prospective limits derived here. For MINOS, we label possible limits from visible searches as "MINOS(VIS)". For both MINOS and INGRID, we use "10 3 " and "10 4 " for contours of that many expected scattering events. We omit any results whose exclusions would be entirely within those of another experiment; this means that not all experiments appear in all plots. Case A In this benchmark the hidden vector will decay to the SM, so we include limits for this from NOMAD and PS-191 [70], ν-calorimeter I [72,73], and CHARM [74]. These extend to lower values of the kinetic mixing than the channels we consider here, but the exclusions cannot reach large values of as the Z x lifetime becomes too short.
The limits from LSND come from scattering of the h x 1 ; at such low masses it is stable over LSND length scales. We have two new regions for CHARM, both from hidden Higgs decays. The darker region with dashed boundary shows the limits when the hidden sector is produced through the decay of a pseudoscalar meson, or through qq fusion at high masses. The light region with dotted boundary shows limits from the decay ρ → Z x h x 1 . The resonance at m x ≈ m ρ is clearly visible. These limits are stronger as the arise from a two-body decay, which compensates for the smaller ρ multiplicity.
The limits from INGRID, and from MINOS due to dark particle scattering, also come from the ρ. While the decay ρ → χ 1 χ 1 is possible, the large fermion mass means our limits are set by h x 1 , which is effectively stable in this mass range. At high masses, we can clearly see the effect of the Higgs mass mixing by comparing the left (with) and right (without) sides of Fig. 15. With the mass mixing, the limits from qq fusion at ν-Cal I are strictly superior to those from CHARM, and we do not show the latter. We also have potential limits from decays to muons at MINOS. The unusual shape of these limits is due to a resonance in the Higgs decay near the K 0 mass. Without the mass mixing, we can exclude higher masses and the higher centre of mass energy of CHARM becomes important. Figure 17: Case C, with (left) and without (right) the Higgs mass mixing. We show here only limits from hidden sector scattering and from h x 1 decays.

Case B For visible searches, this benchmark point differs from Case A in two regards.
The main difference is that there are no longer any limits from Z x → l + l − , as previously discussed. For limits from other hidden particle decays, apart from small differences in the ratio m h x 1 /m x and the mixing matrix R, the two sectors are effectively the same. It follows that the associated limits are nearly identical.
However, the possibility for the hidden vector to decay to two hidden fermions, Z x → χ x 1 χ x 1 , substantially enhances the limits from dark sector scattering. We can easily see this by comparing the LSND, MINOS and INGRID exclusions and prospects between Figs. 15 and 16. At low masses, hidden states at MINOS and INGRID limits are dominantly produced from pions, as this can now produce hidden sectors through successive two-body decays. Above the pion mass the contributions from the ρ and η are comparable, and at high masses quark fusion is the dominant channel. Finally, we note that the limits here are not directly comparable to those shown in e.g. Refs. [1,19]. Those limits assumed a fixed χ x 1 mass and varied the vector mass, while we fix the mass ratio and vary them together.
Case C The ability to produce the scalars through an on-shell vector substantially enhances the limits we derive. Additionally, the A x can be long-lived and is also produced in the on-shell decay of a hidden vector, so we also find limits from its decays. To avoid our plots becoming too cluttered to read, and because they are independent of the Higgs mass mixing, we separate the limits from A x decays into Fig. 18.
In Fig. 17 we show all other limits in this benchmark. In particular, the exclusions from Figure 18: Case C, limits from pseudoscalar decays only. The limits from ν-Cal I at low masses and mixings are worse than the CHARM limits, but beyond that are only approximate. h x 1 decays at both CHARM and ν-Cal I extend to higher masses and lower mixings than in cases A and B. We note that the limits we find at CHARM without the mass mixing are comparable to those derived in Ref. [20]. The limits at low masses at CHARM come from pseudoscalar decays, which can now produce hidden states through two-body decays.
The limits from scattering experiments come from both h x 1 and A x , which give roughly equal contributions. This is to be expected from Eqs. (35) and (37). Hidden states are again dominantly produced by pions when allowed, and from η and ρ decays when not.
The limits from the pseudoscalar are interesting in that the exclusion contours have relevant upper bounds. The A x decay is less suppressed than the hidden Higgs, and so it can be sufficiently short-lived to decay before reaching the detector. This will happen at large mass and kinetic mixing. The ν-Cal I experiment sets better limits in this domain for the simple reason that the detector was closer to the interaction point. At low masses, the limits from CHARM dominate, with the ρ resonance visible.
Case D Unlike the three previous cases, there are no limits in this benchmark from metastable scalars. All limits come from the production of hidden sector fermions. We have significant limits both from χ x 2 decays to leptons, and from χ x 1 scattering (χ x 2 scattering is a subleading effect). Finally, because we have no limits from h x 1 we have no sensitivity to the Higgs mass mixing, and hence show only a single plot in Fig. 19. Figure 19: Limits in Case D. Limits from both hidden particle decays and scattering come from production through on-shell vector decays.
The constraints from visible searches are dominated by CHARM and low mass and ν-Cal I at high mass. Hidden fermion production at CHARM is dominated by pion decay when kinematically accessible, and by ρ decay when not. The ρ wins out over the η mostly due to higher angular acceptances. Hidden fermion production from pseudoscalars requires two two-body decays, and so they tend to have a larger angle relative to the beam axis. Like the pseudoscalar in Case C, the fermion decays are sufficiently less suppressed than the h x 1 for our exclusion contours to have an upper bound. This occurs when the χ x 2 lifetime is so short, it decays before reaching the detector. The superiority of the ν-Cal I experiment at high masses is due to its location closer to the detector.
Limits from scattering experiments are straightforward. Of particular note is their sensitivity at high mass, where the χ x 2 is decays too promptly. In this region the MINOS and INGRID limits can be very important. Otherwise, limits from visible searches win so long as we are above the electron threshold.
6 Implications at the LHC Signatures of this theory at the LHC were investigated in Ref. [21]. When the MSSM superpartners are near the TeV scale, hidden states can be produced readily in supersymmetric cascades Refs. [7,[102][103][104]. In general, the creation of a pair of MSSM superpartners will initiate a pair of visible cascades down to the lightest SM superpartner (LSMP). Each LSMP will subsequently decay to the hidden sector, thereby initiating a hidden cascade. Depending on the decay properties of the hidden states, this may produce additional visible activity in the event and reduced missing energy (/ E T ). The extent to which this occurs is related to the signatures of the hidden sector in low-energy experiments.
MSSM superpartners connect to the hidden sector through the supersymmetric kinetic mixing of Eq. (3) between the Bino and the U (1) x gaugino. When the LSMP is a neutralino with a significant Bino component, the decays , A x , are likely to be prompt [103]. As in Ref. [21], we will focus on this case here. Other species of LSMPs tend to have delayed decays, possibly leading to charged tracks [103]. The decay of an LSMP neutralino populates the four bosonic states equally, while the relative fractions of the hidden neutralino species depends on their hidden Higgsino contents [21].
Decays of the χ 0 1 LSMP will contribute to missing energy, but they can also produce visible signals. The latter occurs when a vector, on-or off-shell, is created in the cascade, and decays to SM states. In contrast, direct decays of h x 1 to the SM are usually too slow to occur within the LHC detector. Since we assume m χ 0 1 m x , the boson and the fermion produced by a χ 0 1 decay will be highly boosted by an amount γ ∼ m χ 0 1 /2m x . Any visible products created along the subsequent boson and fermion decay chains will therefore be highly boosted and collimated. Following Ref. [21], we will call these hidden-valley (HV) jets. An HV jet may contain leptons or hadrons, and it can be prompt or displaced. Each χ 0 1 decay can produce zero, one, or two HV jets. The average number of HV jets per χ 0 1 decay depends on the spectrum of the hidden sector, and is related to the signals expected in low-energy experiments. In case A, the HV jets come mostly from on-shell vectors and can be prompt or delayed, with γ cτ ∼ 500 µm Hidden vectors are produced directly from the Bino decay, or in a two-step decay χ 0 1 → A x → Z x . The heavier scalar preferentially decays h x 2 → h x 1 h x 1 . At least one HV jet will be present in 3/4 of supersymmetric events.
In case B, nearly all LSMP decays are completely invisible. With the exception of h x 1 , the dominant decay channel of all the bosons in the theory is S x → χ x 1 χ x 1 . While the h x 1 state does decay to the SM, it is typically too slow to be seen in the LHC detectors.
Case C can yield HV jets when A x pseudoscalars are created in cascades. These decay through A x → h x 1 +Z x * with Z x * producing an HV jet. However, these HV jets are frequently displaced or delayed when the boost from the χ 0 1 decay is included. Case D is similar to case C, but now the HV jets come mainly from the vector in χ x 2 → χ + 1 Z x * decays. Again, these are likely to be delayed. This is especially true given that our limits in these benchmarks tend to force the kinetic mixing to be small, 10 −4 -10 −3 . Figure 20: Composite plot of all limits we find for Case A. The exclusions from electron beam dumps, and from proton beam dumps at low m x and , come from the decay Z x → l + l − . These are as cited in Sections 4 and 5, plus the exclusions from the MAMI [52] and APEX [53] testruns. We do not include the new electron limits from Fig. 9, as they are inferior to those from the anomalous magnetic moment. Prospective limits from electron beam dumps are the union of the APEX [55], HPS [57], DarkLight [56], MAMI and MESA [105] projections. The limits from BaBar are from Refs. [1,13,14,31]. Not shown are limits from KLOE [33,34] and WASA-at-COSY [106], which are slightly weaker than the CHARM limits.

Conclusions
We have considered limits on a supersymmetric hidden sector from low energy precision experiments. In particular, while our model includes the commonly-considered scenarios of a hidden vector decaying to the SM (Case A) and invisibly (Case B), it also includes the posibility of Z x decaying to hidden states that themselves decay visibly. We divided this latter case into two further scenarios, according to whether the vector decays to hidden scalars (Case C) or fermions (Case D). Many of the standard limits either do not apply or must be modified in these cases. We have substantially extended (Case C) or performed the first study of (Case D) the exclusions that apply in these cases.
Additionally, we have studied how the presence of a non-trivial hidden sector can affect the limits even in Cases A and B. Producing hidden sector states through off-shell vectors, or through the decay of a vector meson, grant access to new detection channels. Of particular importance is the production and decay of the hidden sector Higgs, which sets new limits in phenomenologically interesting regions of parameter space. We have also studied how limits from hidden scalar decays vary depending on whether D-term mixing with the visible Higgs Figure 21: Composite plot of all limits we find for Case B. The limits from BaBar and prospective exclusions from Belle II are estimated from Refs. [17,19,38] and [19] respectively. Limits from invisible kaon decays [11,40] are comparable to limits from CHARM for m x ∼ 80-100 MeV, but otherwise weaker. Prospective limits from ILC are beyond the scope of this work, but are likely slightly better than those from JLab [17,107].
(as expected in our supersymmetric realization) is included or not.
We show summary plots containing all our limits in Figs. 20-23, together with some additional limits taken from the literature. We do not include the limits from Section 3.3 from cosmologically late decays since they depend on the detailed cosmolgical history. To simplify our plots, we combine all exclusions from electron fixed target experiments into one region, and from proton beam dumps into another. Similarly, we show a single line for prospective limits from current and future experiments, amounting to the most optimistic scenarios we considered: specifically, observation of 40 events at JLab, 1000 events at INGRID and 10 muon pairs at MINOS. For Cases A-C, we show separately the high mass (m x > 0.1 GeV) region both with (left) and without (right) the Higgs mass mixing.
In Case A, the most important new feature in Fig. 20 is the region excluded by CHARM from decays ρ → h x 1 Z x . These are the strongest limits for 30 MeV m x 150 MeV and 250 MeV m x 400 MeV. In particular, they exclude almost all of the region of parameter space where the hidden vector can explain the apparent discrepancy in the muon anomalous magnetic moment. We also note that the prospective limits from INGRID are stronger than those from DarkLight/MESA for m x 40 MeV.
In Case B (Fig. 21), the exclusions from CHARM are the strongest limits between 60 MeV m x 400 MeV. They again exclude most of the a µ -preferred region, with the rest Figure 22: Composite plot of all non-cosmological limits we find for Case C. We divide the limits from electron and hadron beam dumps into two regions; the upper regions can be excluded based on h x 1 decays alone, while the lower only on A x decays. Limits from invisible kaon decays are much weaker than the limits shown. If the limits from late decays discussed in Section 3.3 are included, they rule out all masses m x 1 GeV except for a narrow region just above the h x 1 → µ + µ − threshold.
excluded by LSND. Unlike in Case A, there is some parameter space sensitive to the Higgs mass mixing that is not excluded by BaBar: a region around m x ≈ 1 GeV can be excluded by MINOS with the mass mixing, but not without it.
The exclusion of the a µ -preferred region can be avoided in generic hidden sectors in two ways: giving the hidden vector a Stückelberg mass, or making the hidden Higgs decay to the hidden sector. If the hidden vector has a Stückelberg mass, then there is no hidden Higgs and the limits obviously do not apply. Higgs decays of the form h x 1 → χ x χ x also trivially sidestep these constraints. Note, though, that in Case A this is only possible if m h x 1 > m x , which is not possible in our minimal supersymmetric model. Finally, a more complex hidden sector might force the hidden states produced in the ρ decay to return to the visible sector only after a complex hidden cascade; such a scenario might suppress the angular acceptance enough that no limits can be set.
In Case C (Fig. 22), we have limits that substantially extend the reach and type of those of Ref. [20] in the supersymmetric hidden sector we consider . We have divided the excluded regions from fixed target experiments in two. The upper regions can be excluded based solely on the hidden Higgs, while the lower are only excluded by the pseudoscalar decays. As discussed in Section 4.5, the latter are more model-dependent and, in particular, can be avoided if CP is violated in the hidden sector. In either case, we find that both electron and Figure 23: Composite plot of all non-cosmological limits we find for Case D. Limits from invisible kaon decays are much weaker than the limits shown. Including the limits from Section 3.3 would exclude m x 3 MeV, below which the decay χ x 2 → χ x 1 e + e − is kinematically forbidden. However, their remains significant allowed parameter space at small even after this.
proton experiments comfortably exclude the region of parameter space where this benchmark can explain a µ .
Finally, in Case D (Fig. 23), we find the first limits on hidden sectors where the vector decays to hidden fermions with visible decays. We have a diverse mix of limits from searches for visible and invisible final states, and reach to low values of thanks to the relatively short χ x 2 lifetime. Most interestingly, uniquely among our benchmarks this model has substantial parameter space still allowed where the a µ discrepancy can be explained; this is due to the hidden Higgs decaying within the hidden sector. This region of parameter space will be probed by JLab and can be definitively excluded by INGRID.
The model presented here strikes a convenient balance between simplicity and flexibility. From a supersymmetric standpoint, it is minimal. Despite this, it not only encompasses the most common hidden sectors previously considered, but as repeatedly observed it extends those to situations with new and ineteresting phenomenology. For this reason, it serves as a convenient framework for studying general hidden sectors, including non-supersymmetric ones. We hope that the new hidden sector phenomenology observed here might encourage our experimental colleagues to perform dedicated searches for them.

A Masses and Mixings in the Hidden Sector
In this appendix we describe in more detail the masses and mixings within the minimal supersymmetric hidden sector formulated in Refs. [9,21] and investigated in this work. We also show how the hidden states connect to the visible MSSM sector, and we describe the calculation of the decay of h x 1 to SM states.
A.1 Masses for → 0 In the absence of kinetic mixing, → 0, the hidden and visible sectors decouple and it is straightforward to obtain the mass eigenstates. The scalar components of H and H are assumed to develop real VEVs near or below the GeV scale, with ζ ∈ [0, π/2]. As a result, the hidden vector X µ obtains a mass equal to The remaining physical bosons are a pair of real scalars h x 1 and h , and a pseudoscalar A x . 4 In terms of the vector and pseudoscalar masses, the scalar mass matrix is 5 This matrix is identical in structure to the MSSM scalar Higgs mass matrix, and implies the tree-level relations as well as We will also write where R is the real orthogonal matrix that diagonalizes the masses.
Spontaneous symmetry breaking also mixes the U (1) x gaugino with the fermion components of H and H to yield three hidden neutralinos χ The entire spectrum of the theory can be specified by the seven input parameters {g x , µ , m x , m A x , M x , tan ζ, }.

A.2 Mixing with the MSSM
When the kinetic mixing parameter is non-zero, the mixed term in Eq. (3) induces kinetic mixing between hidden and visible gauge bosons and gauginos, as well as further mixing among the auxiliary fields. In components, where c W = cos θ W refers to the weak mixing angle. The kinetic terms can be brought to their canonical form by making field redefinitions at the expense of generating new interactions among the visible and hidden sectors.
The gauge kinetic terms can be diagonalized to leading order in by the transformations where the fields A µ and Z µ are defined as in the SM, and t W is the tangent of the Weinberg angle. These transformations are not unique, but they have the advantage that the mass mixing induced between Z and X is very small, suppressed by both and m 2 x /m 2 Z , and has a negligible effect on the mass eigenvalues.
For the gauginos, neglecting a possible soft mass mixing term, the kinetic mixing is most cleanly removed by the redefinition 6B to leading order in . This transformation generates a coupling between the visible neutralinos and the hidden sector. The induced mass mixing between the visible and hidden sectors is again negligible, suppressed by both and the ratio of the hidden to visible neutralino masses.
Turning next to the scalars, the last term in Eq. (60) mixes the hidden and visible D terms. Let us represent the visible sector scalars by φ i with hypercharge Y i , and the hidden scalars by ϕ j = {H, H } with U (1) x charge x j . Upon integrating out the auxiliary fields, the D-term potential becomes [108] where c = 1/ 1 − 2 Y and s = Y c . The last term in this expression is precisely a Higgs portal interaction.
For η v (as we assume here) the modified D-term potential has a negligible effect on electroweak symmetry breaking, but it plays an important role in inducing symmetry breaking in the hidden sector. When the MSSM Higgs fields develop VEVs, driven almost entirely by the larger MSSM soft terms, an effective Fayet-Iliopoulos term is generated for U (1) x which pushes the H and H fields to develop VEVs of their own. 7 Expanding the visible and hidden Higgs fields about their VEVs, the Higgs portal interaction also induces a mass mixing between the visible and hidden mass eigenstates. This has only a very small effect on the mass eigenvalues, with deviations from the → 0 values suppressed by at least m 2 x /m 2 z . For this reason it is an excellent approximation to separately diagonalise the scalars in the visible and hidden sectors, and treat the mixing as a mass insertion. To be explicit, we will assume that the visible Higgs sector has the form of the MSSM [22], with mass eigenstates H ± , A 0 , and The mass mixing term is then 8 In the MSSM Higgs decoupling limit, α = β − π/2, this reduces to

A.3 Decays of the Light Scalar
The h x 1 scalar is the lightest R-even state in the hidden sector. When it is lighter than twice the mass of the lightest neutralino χ x 1 , it will decay exclusively to the SM. The three leading contributions to the decay width are shown in Fig. 24. The first two are discussed in detail in [12], and require a pair of insertions of in the amplitudes as well as the coupling of the hidden scalar to the hidden vector,  The third contribution to the decay comes from the Higgs portal interaction of Eq. (65) [21]. This piece has only one power of in the amplitude, but also receives suppression by m 2 x /m 2 Z and small SM Yukawa couplings. In the absence of interference with the loop-mediated twobody channel and working in the MSSM decoupling limit, the Higgs-portal-induced decay width is where Γ(h 0 SM , m h = m h x 1 ) is the width the SM Higgs would have if its mass were equal to m h x 1 . The branching fractions will also follow those of a SM Higgs with m h = m h x 1 . Decays through mixing with the heavier Higgs will generally be suppressed unless tan β is close to unity, when the mixing with h 0 vanishes.
In Fig. 25 we plot the total decay length cτ of h x 1 as a function of its mass. For this plot, we use = 10 −3 and the hidden sector benchmark C, together with the decoupling and large tan β limit of the MSSM. The results for benchmarks A and B are qualitatively similar, while for benchmark D the scalar decays promptly to the hidden sector, h x 1 → χ x 1 χ x 1 . This result includes the interference between the loop and Higgs-portal contributions to the amplitude. We also show in Fig. 25 lines corresponding to what the decay length would be with the Higgs-portal alone or with the loop-induced vector-mediated decay alone.
For 2m π < m h x 1 1 GeV there is a significant uncertainty in the Higgs width to hadronic modes. For the mass mixing contribution, we use the parametrization of Ref. [109] to compute the total rate and the branching fractions. A comparison of this and other estimates is given in Ref. [36] (see also [110]). For the loop diagram, we do not compute any contribution for hadronic decays below Λ QCD ≈ 2.5 GeV; this represents an additional theoretical uncertainty.
We would like to estimate the size of this uncertainty, and its corresponding effect on the limits we extract in Sections 4 and 5. When we have no mass mixing, our limits are surprisingly robust to the omission of any hadronic decay modes below Λ QCD . The upper bounds on we find in this region of parameter space come when cτ h x 1 ∼ 10 4 m. This is large compared to the typical size of experiments, especially once a boost of γ ∼ 10 -100 is included. The decay probability Eq. (32) is then Increasing the total width will increase the number of hidden states decaying within our detector. This is offset by the fact that the extra width is to hadrons, while all searches tagged on leptons. We thus pay a branching ratio penalty ∝ 1/Γ, cancelling the factor from P dec .
This argument holds provided that the width does not increase too much, specifically that the approximation (69) is valid. This is why our exclusions change with the inclusion of mass mixing; those limits are set for parameters where the scalar lifetime cτ h x 1 ∼ 10-100 m. Consequently, those limits are more sensitive to uncertainties in the hidden Higgs width. The sensitivity is greatest to increases in the width, which would suppress signals by having more decays within the shield. Decreasing the width would make the limits stronger, approaching the exclusions without mass mixing. We note that our choice for the Higgs hadronic width is both the most recent and among the largest of those discussed in Ref. [36], suggesting the main theoretical uncertainty makes our limits conservative.

B Details of Hidden Vector Production at Electron Beam Dumps
In Section 4, we sketched the calculation of the number of hidden sector events at an electron beam dump. Here we provide some of the details omitted there.

B.1 Effective Flux
We begin by considering the effective flux χ, introduced in Eq. (16) that contains the form factors describing the details of the target. This quantity was defined in Ref. [59] as Here, t = −q 2 ranges between t min = (m x /2E e ) 2 and t max = m 2 x ; M f is the mass of the target final state, and ranges between M i and M 2 up ≈ M i (M i + 2E e ). W 1,2 are form factors describing the target and including both elastic and inelastic behaviour, and t aux ≡ (p e · k X − 1 We use analytic approximations for the form factors taken from Refs. [59,111]. These are given in terms of the partially integrated quantities G: The form factor W 1 has a negligible contribution to hidden vector production [14]. We split G 2 into a sum of elastic and inelastic contributions, which in turn are the product of atomic and nuclear contributions. The elastic term is The parameters a and d are well approximated as a ≈ 111/(m e 3 √ Z) and d ≈ 0.164 GeV 2 / 3 √ A 2 , with Z and A the usual atomic and mass numbers. The terms in the first (second) brackets describe the effects of atomic screening (finite nuclear size).
For the inelastic term we take, based on Ref. [111], The first set of brackets again describes the atomic screening factor, with a ≈ 773/(m e 3 √ Z 2 ). The remaining terms describe the inelastic nuclear form factors, in terms of a prefactor C and nucleon functions G el 2p,n . The prefactor is with Q defined by while the nucleon functions are where t 0 = 0.71 GeV 2 and τ = t/(4m 2 p ). Note that Refs. [14,17,51] used a different form for G inel 2 , specifically the expressions given for elastic scattering from protons in Ref. [59]. The difference between the two choices is small, as the elastic contribution to G 2 is dominant. We compare the two in Fig. 26 using the scaled flux F = 4 3 introduced in Ref. [17]. The difference is only a few percent for the mass range of interest, with our parameterisation the smaller of the two.

B.2 Off-Shell Production
The general process for hidden sector production through an off-shell vector is shown in Fig. 27 (a simple generalisation of Fig. 7). We can write the matrix element as M 2→n+2 = 1 q 2 T N µ (p N , k N ) S µν e (p e , q, k e ) N x ν (k 1 , . . . k n ) , Figure 27: The two leading Feynman diagrams for production of off-shell massive vector in electron-fixed target scattering. The unlabelled lines represent the n hidden sector states ultimately produced through the hidden vector.
with T N , S e and N x respectively the target fermion line trace, the electron line trace, and the terms related to the hidden sector. Squaring this and integrating over the k N phase space lets us introduce form factors: with M i the target mass and E ef the final state electron energy.
To generalise the WW approximation, we follow Ref. [59] in rewriting the phase space integral over k e into an integral over M 2 f , t = −q 2 and an angular degree of freedom φ S . This is done by making use of a special frame S, defined as the frame where u ≡ p e + p N − k x vanishes. Specifically, φ S is the azimuthal angle of k e relative to p e − k x in this frame. We also note that the hidden sector factors N have no dependence on k e , while the form factors depend only on t and M 2 f . This brings us to the following expression for the integrated matrix element: S µρ e (S νσ e ) * (p e , k x , M 2 f , t, φ S ). (80) Note that the only difference so far with the derivation of Ref. [59] is that in that case, the integral over k 1 · · · k n is replaced by a single integral over k x , and the function N x ρ (N x σ ) * is replaced by −η ρσ .
At this point, we sketch the derivation of Eq. (21). We introduce a trivial integral over k x using a delta function. This lets us rewrite the integral over N x ρ (N x σ ) * in terms of the vector partial decay widths: The last term vanishes when contracted with S µρ e via the QED Ward identity, while with Γ x the partial width for a vector of mass m 2 x = k 2 x to decay to the relevant hidden sector final state. From this, the usual WW argument leads to Eq. (21).
The WW approximation is a Taylor expansion of the (integral of the) electron line trace about the minimal momentum transfer t min , leading to the two approximations dφ S 2π S µν e (S eµν ) * (p e , k x , M 2 f , t, φ S ) ≈ S µν e (S eµν ) * (p e , k x , M 2 i , t min ) , As noted before, q * is parallel to p e − k x . It follows that k e is also parallel to p e − k x , allowing us to omit φ S from the right-hand side of Eqs. (83) and (84). This approximation can be checked using the explicit expressions for S µν e . It follows from the presence of the collinear pole when the electron is nearly on-shell, as well as the photon propagator that results in the dominance of processes where t ≈ t min .
The equivalent approximations for the off-shell case are easy to guess: S ρµ e (S ν eρ ) * (p e , k x , M 2 f , t, φ S ) ≈ S ρµ e (S ν eρ ) * (p e , k x , M 2 i , t min ) , and − dφ S 2π S µρ e (S νσ e ) * (p e , k x , M 2 f , t, φ S ) ≈ t − t min 2t aux S ρµ e (S ν eρ ) * (p e , k x , M 2 i , t min ) . (86) Once these are put into Eq. (80), Eq. (22) is immediate. The validity of these approximations can also be checked by direct computation. We comment on a few points in the derivation.
The easiest way to analyse these expressions is to expand in a tensor basis that respects the QED Ward identity. For example, the tensor in Eq. (85) can only depend on p e , q * , k x and the metric, so we may define four functions f i through with (for example) x p e · k x p µ e p ν e + p e · k x η µν − p µ e k ν x − k µ x p ν e , L µν 2 = k 2 x q * · k x q * µ q * ν + q * · k x η µν − q * µ k ν x − k µ x q * ν , The approximation of Eq. (85) is then no more than replacing the functions f i of Eq. (87) by their values at the minimal kinematics t = t min , M 2 f = M 2 i . We know that these functions are finite and non-zero at this point, as it corresponds to the physical point for photonelectron scattering; while the factor of t −2 in Eq. (80) leads the contribution from this region to generically dominate.
The approximation of Eq. (86) is slightly more complicated. The tensor expansion is naïvely more involved, as we also have a possible dependence on p N . However, because q * is parallel to p e − k x , we can rewrite terms involving p µ N in terms of p µ e − k µ x and q * µ ; the basis L µν i is still sufficient. So we have the expansion The WW approximation involves using not the first term in the Taylor expansion of the g i , but the second. The leading term is suppressed; specifically, it is not singular when the intermediate electron goes on-shell. The term linear in t is singular in that limit, so is the first term we keep.
The final step is to relate the linear terms in Eq. (88) and the leading terms in Eq. (87). It is clear that Eq. (86) will hold if This identification only holds approximately, to leading order in small parameters δ/E where E ∈ {E e , E x , E e − E x } and δ ∈ √ t aux , The assumption that δ/E is small is also part of the WW approximation.

C Hidden Sector-Visible Sector Scattering
In this appendix we provide relevant details on hidden sector-visible sector scattering. We are interested in the 2 → 2 processes shown in Fig. 28, with the two sectors coupled through a t-channel Z x . The dotted lines represent one of a number of possible initial and final hidden states: χ x i → χ x j (with both i = j and i = j); A x → h x a ; and h x a → A x or Z x . The visible sector particles can be either nucleons or electrons.
We define Q 2 ≡ −q 2 x > 0. Additionally, m i (m f ) are the initial (final) hidden sector masses; and E i , p i the initial hidden sector energy and momentum. The differential cross sections are dσ HS The functions I HS are, for fermion scattering: for a pseudoscalar initial or final state: and for the h a x → Z x channel: In the limit E i m N m 2 i , m 2 f , m 2 N and Q 2 , all these expressions have the same approximate limit: This approximation is required to derive Eq. (35).